CONSTITUTIVE LAWS OF PLASTIC DEFORMATION AND FRACTURE

Mechanical Behavior of Materials Managing Editor:

A.S. Krausz, University of Ottawa, Ontario, Canada

Advisory Board:

M.F. Ashby, F.R.S. (Cambridge, U.K.), E.W. Hart (Ithaca, New York, U.SA), J.P. Hirth (Columbus, Ohio, U.S.A.), E. Krempl (Troy, New York, U.SA), R.S. Rivlin (Bethlehem, Pennsylvania, U.SA), J.H. Wiener (Providence, Rhode Island, U.S.A.), M.L. Williams (Pittsburgh, Pennsylvania, U.S.A.)

Aims and Scope:

This monograph series contains volumes dealing with the mechanical behavior of ceramics, metals, polymers, and their composites. Individual volumes will discuss fundamental as well as applied concepts from both the continuum and microstruc­tural viewpoints. This comprehensive coverage embraces the relation between the phenomenological description of mechanical properties and materials structure.

Publications :

1. A.S. Krausz and K. Krausz: Fracture Kinetics of Crack Growth. 1988 ISBN 90-247-3594-7

2. A.S. Krausz, J. I. Dickson, J-P. A. Immarigeon and W. Wallace (eds.): Constitutive Laws of Plastic Deformation and Fracture. 1990

ISBN 0-7923-0639-2

Volume 2

Constitutive Laws of Plastic Deformation and Fracture 19th Canadian Fracture Conference, Ottawa, Ontario, 29-31 May 1989

edited by

A.S.KRAUSZ CluJirman of the Editorial Committee

J. I. DICKSON Ecole Poly technique, Montreal

J-P. A. IMMARIGEON National Research Council Canada

and

W.WALLACE National Research Council Canada

KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

Library of Congress Cataloging in Publication Data

CanadIan Fracture Conference (19th: 1989 : Ottawa, Ont.) ConstitutIve laws of plastIC deformatIon and fracture I 19th

CanadIan Fracture Conference, Ottawa, OntarIO, 19-31 May 1989 ; organIzing and editorial com.lttee. A.S. Krausz (chaIrman) ..• let. a 1. 1.

p. em. -- (MechanIcal behavIor of materIals; v. 2)

1. Deforlatlons (Mechanlcs)--Congresses. -Congresses. 3. Plastlclty--Congresses. II. TItle. Ill. serIes.

2. Fracture .echanlcs-1. Krausz, A. S.

TA417.6.C33 1989 620. I' 123--dc20 89-71616

ISBN-13: 978-94-010-7380-6 001: 10.1007/978-94-009-1968-6

e-ISBN -13: 978-94-009-1968-6

Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands.

Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.

Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A.

In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved

© 1990 by Kluwer Academic Publishers

Softcover reprint of the hardcover 1st edition 1990

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner.

OBJECTIVES

Much effort has been devoted to the development of consitutive laws for plastic deformation and fracture to predict the behaviour of materials during complex fabrication or service histories. This conference focused on the time, temperature, and environment dependent aspects of these laws and takes a unified approach in recognizing deformation, crack initiation, and crack growth as related processes. The concept is adopted that constitutive laws should be developed from the understanding of the effects of microstructure since the behaviour of materials is controlled at this level. Therefore, it is important that deductions can be made from this scale in describing the macroscopic aspects of flow and fracture behavior. The conference emphasized that models should reflect the physics of the deformation and fracture processes and that physically rigorous, rather than empirical, constitutive laws should be developed for reliable predictions through extrapolations.

ORGANIZING

AND

EDITORIAL COMMITTEE

A.S. Krausz (Chairman) University of Ottawa

J.I. Dickson (Secretary-Treasurer) Ecole Poly technique (Montreal)

J.-P.A. Immarigeon National Research council Canada

W. Wallace National Research council Canada

M. Boroczki (secretariat) National Research council Canada

SPONSORS

National Research Council Canada

Canadian Committee for Research on the strength and Fracture of Materials

Orenda Division of Hawker siddeley Canada

E. Aifantis z.s. Basinski J. Bratina C.E. Coleman J.J. Jonas E. Krempl W.R. Tyson

ADVISORY COMMITTEE

Michigan Technological University, USA McMaster University, Canada University of Toronto, Canada Atomic Energy of Canada, Canada McGill University, Canada Rensselaer Polytechnic Institute, USA CANMET-EMR, Canada

HIGH TEMPERATURE FATIGUE R.M. Pelloux

CONTENTS

MICRO-MECHANICAL INFLUENCES ON THE FATIGUE CRACK GROWTH 15 BEHAVIOR OF NICKEL BASE SUPERALLOYS Randy Bowman and Stephen D. Antolovich

MICROSTRUCTURES AND CRACK OPENING IN A NICKEL-BASE 27 SUPERALLOY DEFORMED BY IN-SITU TENSILE TESTS M. Ignat and J. Pelissier

MECHANICAL BEHAVIOR MODELING OF A NICKEL BASE SINGLE- 35 CRYSTAL SUPERALLOY Jean-Yves Guedou et Yves Honnorat

THE CYCLIC DEFORMATION OF PWA 1480 SINGLE CRYSTALS AS A 43 FUNCTION OF TEMPERATURE, STRAIN RATE AND ORIENTATION Walter W. Milligan and Stephen D. Antolovich

ANISOTROPIC MECHANICAL BEHAVIOR MODELING OF A NICKEL-BASE 49 SINGLE CRYSTAL SUPERALLOY P. Poubanne

THE MECHANICAL PROCESSES OF THERMAL FATIGUE DEGRADATION 57 IN IN-100 SUPERALLOY N.J. Marchand, W. Dorner and B. Ilschner

INCLUSION OF DSA MODELING CAPABILITY IN UNIFIED VISCO- 67 PLASTICITY THEORIES, WITH APPLICATION TO INCONEL 718 AT 1100 OF N.N. El-Hefnawy, M.S. Abdel-Kader and A.M. Eleiche

CYCLIC DEFORMATION AND LIFE PREDICTION USING DAMAGE 77 MECHANICS A. Plumtree and G. Shen

NON-LINEAR STRUCTURAL MODELING: INTERACTIONS BETWEEN 87 PHYSICAL MECHANISMS AND CONTINUUM THEORIES Norman J. Marchand

FATIGUE CHARACTERISTICS OF SiCp-METAL MATRIX COMPOSITE 101 S.B. Biner

ON CONSTITUTIVE RELATIONSHIPS FOR FATIGUE CRACK GROWTH 109 A.J. McEvily

EFFECT OF MICROSTRUCTURE ON THE SHORT CRACK GROWTH IN 117 Al-2024-UA AND Al-8090-UA D. Downham, G.W. Lorimer and R. Pilkington

x

CYCLIC PLASTIC INSTABILITY IN PURE ALUMINUM AND ALUMINUM ALLOY 7075 T6: EFFECTS OF TEMPERATURE, STRAIN RATE, AND WAVEFORM P. Li, N.J. Marchand and B. Ilschner

PROCESSING DEFECTS MTD THE FRACTURE OF CERAMICS AND DESIGNED CERAMIC/CERAMIC COMPOSITES Patrick S. Nicholson

INDENTATION CREEP IN SEMI-BRITTLE MATERIALS N.M. Everitt and S.G. Roberts

ON THE FRACTURE BAHAVIOR OF ROCK SALT U. Hunsche

AN EXAMINATION OF CONSTITUTIVE LAWS BY HIGH TEMPERATURE CREEP OF ENGINEERING MATERIALS K Maruyama, C. Tanaka and H. Oikawa

A CREEP CONSTITUTIVE EQUATION OF A SINGLE CRYSTAL NICKEL­BASED SUPERALLOY UNDER <001> UNIAXIAL LOADING M. Maldini and V. Lupine

A CREEP CONSTITUTIVE MODEL OF DISLOCATION THERMAL ACTIVATION C.D. Liu, Y.F. Han and M.G. Yan

DETERMINING A CONSTITUTIVE EQUATION FOR CREEP OF A WOOD'S METAL MODEL MATERIAL Mark Belchuk, Dan Watt and John Dryden

DISLOCATION CRACK-TIP INTERACTION INFLUENCE ON SUBCRITICAL CRACK GROWTH W.W. GERBERICH, T.J. FOECKE AND M. Lii

THE EFFECT OF QUENCHING PROCEDURES ON MICROSTRUCTURES AND TOUGlThTESS OF TEMPERED 4Cr5MoSiVI (AiSI H13) STEEL Y.L. Yang and X.Z. Feng

FRACTURE TOUGHNESS MODELING FOR MATERIALS WITH COMPLEX MICROSTRUCTURE Asher A. Rubinstein

THERMAL ACTIVATION AND BRITTLE FAILURE OF STRUCTURAL STEELS Bernard Faucher and W.R. Tyson

THE USE OF ELASTIC-PLASTIC STRESS FIELDS TO DESCRIBE MIXED MODE 1111 BRITTLE FRACTURE IN STEEL T.M. Maccagno and J.F. Knott

125

133

147

155

165

173

181

189

197

207

215

223

229

xi

MICROSTRUCTURE AND FRACTURE CHARACTERISTIC OF ALUMINIUM - 237 ZINC - TITANIUM ALLOYS A.M. ELSheikh

THREE-DIMENSIONAL ASPECTS OF THE FRACTURE PROCESS ZONE 245 AND CAUSTICS T.W. Webb, D.A. Meyn and E.C. Aifantis

ON THE BEHAVIOR AND THE MODELIZATION OF AN AUSTENITIC 253 STAINLESS STEEL 17-12 Mo-SPH AT INTERMEDIATE TEMPERATURE DESCRIPTION OF DISLOCATION-POINT DEFECT INTERACTIONS P. Delobelle

APPLICATIONS OF A THEORY OF MOBILE DISLOCATION DENSITY 263 TO THE STUDY OF RATE-SENSITIVE DEFORMATION Thomas H. Alden

THERMODYNAMICALLY CONSISTENT CONSTITUTIVE LAWS IN 273 PLASTICITY INCLUDING DAMAGE Th. Lehmann

COMMENTS ON MODELING PLASTIC DEFORMATION OF LOW 279 CARBON STEEL Jerzy T. Pindera

MODELING OF PLASTIC DEFORMATION OF METALS AT MEDIUM 285 AND HIGH STRAIN RATES WITH TWO INTERNAL STATE VARIABLES J.R. Klepaczko

APPLICATION OF CONTINUUM SLIP APPROACHES TO 295 VISCOPLASTICITY David L. McDowell and John C. Moosbrugger

CONSTITUTIVE LAWS PERTAINING TO ELECTROPLASTICITY 305 IN METALS H. Conrad, W.D. Cao and A.F. Sprecher

PLASTIC DEFORMATION AND FRACTURE OF CONTINUOUSLY CAST 5083 313 ALUMINUM ALLOY INGOT T. Takaai, A. Daitoh, Y. Nakamura and Y. Nakayama

MODELLING THE INFLUENCE OF MICROSTRUCTURAL INHOMOGENEITY 321 ON HIGH TEMPERATURE DEFORMATION AND FRACTURE D.S. Wilkinson

CONSTITUTIVE EQUATIONS FOR STRENGTH AND FAILURE AT 333 ELEVATED TEMPERATURES AND STRAIN RATES IN AUSTENITIC STAINLESS STEELS N.D. Ryan and H.J. McQueen

xii

MODELING OF FLOW BEHAVIOR OF THE NICKEL BASE SUPERALLOY 341 NKl7CDAT AT ISOTHERMAL FORGING CONDITIONS Y. Combres and Ch. Levaillant

CONSTITUTIVE LAW FOR CALCULATING PLASTIC DEFORMATIONS 349 DURING CZ SILICON CRYSTAL GROWTH C.T. Tsai, V.K. Mathews, T.S. Gross, O.W. Dillon, Jr. and R.J. De Angelis

CONSTITUTIVE RELATIONS FOR DEFORMATION AND FAILURE 357 OF FAST REACTOR CLADDING TUBES I.J. Ford and J.R. Matthews

HIGH TEMPERATURE FATIGUE

R.M. Pelloux Massachusetts Institute of Technology

Department of Materials Science and Engineering Cambridge, MA 02139 USA

1. INTRODUCTION: Demand for high performance power units has led to marked increases in metal operating temperatures which reach now .085 TM (OK). In parallel to these high steady state temperatures the fast temperature transients during the start or stop cycles have resulted in sharp temperature gradients at the surface of the components. These surface conditions are very demanding of the fatigue performance of advanced high temperature alloys. As a consequence, high temperature fatigue has been the subject of intensive research during the last 25 years. (see references 1, 2, 3) and excellent progress has been achieved in the understanding and the modelling of the phenomenon.

2 LOW TEMPERATURE FA TIGUE: High temperature fatigue research work for the most part has been an extension of the low temperature fatigue research methodology. Thus for the purpose of this review it is worthwhile first to summarize what we know about the micro­mechanisms of fatigue at Tffm ::;; 0.35. The reader is referred to text books (Ref. 4, 5) and to proceedings of fatigue and fracture conferences (6, 7). A short summary of the current knowledge of fatigue is best approached by following the fatigue damage sequence. The dislocation structures resulting from fatigue damage, as seen with the TEM, are fairly well understood for pure metals at low and high cyclic plastic strain ranges. Much work remains to be done to assess and quantify the fatigue damage of two phase alloys. The high concentration of vacancies which is created during cyclic deformation has not been investigated extensively. At elevated temperatures these vacancies may play an important role in assisting diffusion and accelerating oxidation.

The formation of persistent slip bands (PSB) has been extensively researched by Laird (7), Mughrabi (7) and Neumann (7). This stage of surface damage preceeds crack initiation along the PSB'S and stage I crack growth. There is limited knowledge about the rates of crack growth in stage I because measurements of crack depth are difficult and also because in many engineering alloys (two phase alloys) the extent of stage

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 1-13. © 1990 Kluwer Academic Publishers.

2

I is very limited. Stage II (mode I) of fatigue crack growth (FCG) has been extensively studied and has provided a large amount of data for a variety of alloys. It is well accepted that stage II growth is due to trans granular crack advance by alternating shear at the crack tip. Plastic shear bands accommodate the finite crack tip opening displacement (CTOD). Although CTOD'S are difficult to measure it is assumed that the CTOD is controlled by the crack tip stress - strain fields which in turn are unique functions of the far field stresses and crack geometry. The difference between the calculated CTOD and the effective CTOD has been accounted for by Ritchie (7) who proposes different mechanisms of crack tip displacements. These crack tip mechanisms (crack tip branching, oxide induced closure etc ... ) will play a large role at high temperatures because of the complex microstructures of advanced high temperature alloys.

FCG near the threshold ~K for long cracks and FCG in the short crack regime have received a lot of attention during the last 10 years. The main results of the work are that grain boundaries and microstructural features limit the validity of the similitude principle which is the foundation of LEFM. The short crack phenomena can also be expected to be important in high temperature fatigue white oxidation effects will be dominant.

The other aspects of low and high temperature fatigue which need more research work include overload effects (crack acceleration and crack retardation), corrosion fatigue, multi-axial fatigue and fully plastic (LCF) crack growth.

The main result of this extensive research work is that we are now able to optimize the microstructures of alloys to improve their fatigue performance. For instance, we control the volume fraction and size of inclusions, we avoid cyclic softening alloys whenever possible, we search for optimum grain sizes and crystallographic textures. The fatigue data base is reported in the form of an allowable design stress amplitude versus the initial flaw size (ai) for a given fatigue life. This plot, known as a Kitagawa diagram, gives a measure of the criticality of the initial defect size. Similar diagrams will be needed for high temperature design but they will have to include the effect of creep hold times.

3. HIGH TEMPERATURE FATIGUE: For T{fM ~ 0.35 the microstructural instabilities will play a large role in high temperature fatigue. These instabilities include strain ageing, precipitation strengthening, phase coarsening and phase embrittlement. The recovery of vacancies and of dislocations will be accelerated at high temperatures which in turn will accelerate the rates of formation of the instabilities. The migration, sliding and cavitation of grain boundaries which are strongly dependent

upon the strain rates and the cycling rates play an important role in high temperature fatigue. Finally, oxidation at the free surfaces, along the PSB's, at the crack tips, along the grain boundaries and in the matrix will contribute in a powerful way to the damage and performance of high temperature alloys.

4. HIGH TEMPERATURE LOW CYCLE FATIGUE: The extensive studies of low cycle fatigue (LCF) under controlled total and/or plastic strain ranges have been motivated by the importance of the start-stop cycles in determining the life of many high temperature components. LCF testing requires the recording of the stress-strain hysteresis loops in the low and high strain regimes at high temperatures. The wave shapes (sine, square, triangular), frequency and temperature are the main test variables. Large data bases, often limited to a few alloy systems have been used by different authors to establish phenomenological models of life prediction. For instance Coffin (8) uses the data base of A286 to derive a frequency modified equation.

This equation has 6 empirical constants.

Tests in vacuum with the same alloy A286 showed that the frequency effects in A286 at 600°C are due to intergranular oxidation effects and not to plastic strain rate effects. Manson (9) proposes the strain range partitioning (SRP) method whereby the plastic strain range can be divided into four components of strain (creep-creep, plastic-plastic, creep-plastic and plastic-creep) with four independent life-plastic strain relationships such as

(Xij N .. = a .. ~£

1J 1J

(2)

This method requires eight independent coefficients to be determined experimentally. Extensive work on SRP has shown that it applies best to very ductile materials, in the high strain regime with a minimum amount of oxidation. The test procedures are quite complex and although the SRP model is academically pleasing it does not provide a unique solution to high temperature fatigue life predictions.

The Ostergren (9) model takes into account the hysteresis energy liO't

by using the parameters Ii£p x 2 where Ii 0' t is the stress range. This

model has been shown to provide a fairly good fit to a large body of data.

3

4

Most low cycle fatigue models are essentially crack initiation models. They are most useful when a creep-fatigue damage summation is needed such as the linear summation rule which is given by

n D fatigue + D creep = Nf + tr (3)

where Nf is the fatigue life under pure fatigue and tr the creep rupture

time under pure creep.

5. FA TIGUE CRACK GROWTH: The other practical approach to high temperature fatigue is to deal only with fatigue crack growth. This approach is usually conservative but if one is able to measure the growth rates of very short cracks the propagation (FCG) life gives a good estimate of the total fatigue life. A large number of FCGR tests for nickel base superalloys shows that for the high strength alloys the da/dn data is well correlated with A K the elastic stress intensity factor over a wide range of AK's. Two parameters, test temperature and test frequency play a major role, the FCGR da/dn increasing both with increasing test temperature and with decreasing frequencies. The FCGR are usually well quantified in the range of AK from AK threshold to AK = (l-R) Kc. but the test frequency is a key factor in controlling the life. As the frequency decreases the fracture path changes from transgranular to intergranular and the fatigue process changes from time independent to time dependent. Figures 1, 2, 3, 4, 5, 6 (Ref. 13) illustrate clearly the FCGR performance of alloy X750 a high temperature alloy. At very low frequencies fatigue cracking becomes a process of creep crack growth and the data is best reported as

da da da -=- -dn x frequency dt dn At cycle

(4)

The transition from time independent to time dependent fatigue is a strong function of the test environment. At atmospheric pressure in air,

~~ is controlled by the rate of crack tip oxidation rather than by creep

cavitation ahead of the crack tip.

There is a lack of good theoretical models to account for the transition from mechanical fatigue to creep-oxidation crack growth. However a large body of high temperature FCGR data is available in the COST 50 Proceedings (10)(11). In order to take into account the

da temperature dependance of the modulus E, dn is often given as a

8K function of E(f). The near threshold data usually shows that 8Kth is

higher at high temperature in air than in vacuum due to the wedging effect of the oxide films which promote crack tip closure.

In summary the fatigue crack growth rates are best expressed by

~: - :: (fatigue) + ~: •. 8t(creep) + ~:. 8t (oxidation) (5)

One of the most difficult aspects of high temperature fatigue is to predict the rate of crack propagation for fatigue with different (long or short) hold times. During a hold time there is competition between creep damage, oxidation damage and crack tip blunting and or branching. Usually oxidation induced crack growth is the main component of the crack growth per cycle. However little is known today about the transient rates of oxidation of crack tip fresh surfaces during crack opening.

There is little or no data available for the short crack regime at high temperatures. This problem remains to be investigated experimen tally.

Most of the FCGR data reported deals mainly with isothermal fatigue, however thermo-mechanical fatigue (TMF) with fluctuating temperatures between T max and T min is more typical of real life fatigue problems. In most TMF analyses it is common to use da/dn at T max as an upper bound for da/dn. This approach may not always be conservative, thus realistic TMF tests may have to be performed.

6. APPLICATION OF HIGH TEMPERA TURE FATIGUE TO SINGLE CRYSTALS AND DIRECTIONALLY SOLIDIFIED NICKEL BASE SUPERALLOYS: During the last few years single crystals and CDS) directionally solidified alloys have been used because of their improved creep resistance (12). The fatigue performance of these advanced alloys has been surprisingly good because they are grown and stressed along the [001] direction which is a low modulus direction. For a given 8£t the anisotropy of the elastic constants leads to a low 8£p in the [001] direction as compared to the [111] and [110] directions. As a consequence for a given 8£t fatigue lives are a strong function of the primary orientations of single crystals or of a DS bar. Figures 8, 9, 10, 11, 12, 13 illustrate the fatigue performance of single crystals and DS alloys. (Ref. 12. 13, 14)

5

6

A recent study of fatigue performance of a DS alloy (15) showed that the fatigue cracks always initiate at the sites of large intermetallic and interdendritic phases where preferred oxidation sites are activated. Oxidation appears to favor cracking at intermetallic phases and thermal grooving thus creating a local notch effect. By increasing the rate of solidification of a DS alloy it is possible to improve the fatigue performance of the alloy by decreasing the size and volume fraction of the intermetallic phases and consequently to minimize the size and number of oxidation sites.

The high cycle fatigue crack growth rates in single crystals are a function of test bar orientation. At low temperature the fatigue crack is strongly crystallographic along (111) planes and the correlations between da/dn and t.K are quite difficult because of combined modes I, II, III cracking. At high temperatures, above gOO°C, the fracture path proceeds by mode I cracking independently of the secondary orientation and the rates are not affected by small variations of orientation of the primary growth direction. For the DS alloys interactions between the fatigue crack front and the grain boundaries slow down the FCGR. Extensive work is in progress with single crystals and DS super-alloys.

7. CONCLUSIONS:

• High temperature fatigue is a complex phenomena where time and temperature dependant fatigue, creep and oxidation damages play a large role in the initiation and propagation of fatigue cracks.

• Preferred oxidation of grain boundaries and of second phase particles at and near the free surfaces accelerate crack initiation.

• Fatigue crack growth rates are time in dependant at high frequencies and they approach creep crack growth rates at low frequencies. The transition from a trans granular fracture path at high frequencies to an intergranular path at low frequencies appear to be controlled by the diffusion of oxygen along the grain boundaries with a strain rate dependant contribution of grain boundary sliding.

• The empirical models of high temperature LCF require a large number of empirical constants to fit the data to the modeling equations. The models are valid for interpolations only. Their validity for long time extrapolations is quite limited.

The most promising approach to high temperature fatigue research is to measure the rates of crack growth for short and long cracks and to correlate the crack growth rates with global parameters (t.K, t.J) or

local parameters (Lio,LiEp,a) depending upon the geometry of the part and the nature of the time dependant damage.

References

1. Time Dependant Fatigue of Structural Alloys: A General Assessment ORNL 5073 (Oakridge Nat'l. Lab. publication). 2. Fatigue at High Temperature, R.P. Skelton editor (Applied Science Publisher). 3. High Temperature Fatigue, Properties and Prediction, R.P. Skelton Editor (Elsievier Applied Science). 4. Metal Fatigue in Engineering, Fuchs and Stephens, (Wiley In terscience). 5. Fatigue and Microstructure , ASM (1978. 6. International Conference on Fracture, ICF 7 (Houston 1989). 7. Proceedings of Fatigue 87, EMAS publishers (1987). 8. Proceedings ICF 2 (1969) London Chapman and Hall (LC Coffin p643). 9. W.J. Ostergren ASTM Standardization News 4(10) 1966 (p 327) 10. High Temperature Alloys for Gas Turbines, Coutsouradis ed. Applied Sciences (1978). 11. High Temperature Alloys for Gas Turbines, Brunetaud ed. Reidel (1982). 12. A. Diboine, J.M. Peltier, RM. Pelloux; Proceedings of MECAMAT (Oct. 87) Dourdan, France (lv-71). 13. F. Gabrielli, R. M. Pelloux, Met. Trans. A. Vol. 13A, June 1982, p. 1083-1090. 14. Superalloys II - Ed. Sims, Stoloff, Hagel, Wiley Interscience. 15. L. Terranova, MIT, MS Thesis, February 1989, "Fatigue Behavior of Nickel Base Directionally Solidified Superalloy at Elevated Temperature".

7

8

};:1 ~

'" ~ 1§ 10-4 >.< u <to oc V

w => <.D

~ U.

10-5

10-6

,/

10

INCONEL X-750

./

V= 10 Hz ./ ./

./

./ 'v""':,<) ./

)'l.. /' ./

./

• 25 ·C AIR

• 25 ·C VACUUM

0650·C AIR

<> 650 ·C VACUUM

(t.K)2 112 crOD = 0,186:'

E cyo

20 30 40 50 60

STRESS [NTENS lTY RANGE, ~lPa 1m

Figure I: Effect of temperature test and environment on the fatigue crack growth rate of Inconel X-750 (triangular waveshape, R = 0.05).

'" d >-u "-E E

W I-

~ I I-". 2i! <.D

"" ~ u w ~ ;:: "'" LL

10-1

10-2

./

10- 3

INCONEL X-750

T = 650 ·C

AIR

/ ./

./ <..,y·O

~~-$' <> ./'

o 10 Hz

o 1 Hz

6 0,10 Hz

<> 0,01 Hz

10 20 30 40 50 60 STRESS I NTENS lTY RANGE, MPa 1m

Figure 2: Frequency effect on the fatigue crack growth rate of Inconel X-750 at 6500

C, in air (triangular waveshape, R = 0.05).

10-2

10

I

INCONEL X-750 )

T = 650'C //0.01 Hz

/ /

/

/ /

20

/ /

/

/.---- '\

--- AIR

- VACUUM

30 40 50 60 70 80 STRESS I NTENS ITY RANGE, MP F

Figure 3: Frequency effect on FCGR of Inconel X-7S0, tested in vacuum at 6S0° C (triangular waveshape, R = O.OS).

10

t E

~ '" I "" ~ u

tl3 5 10-1

10

INCONEL X-750

T = 650 'C

,

9° 0, 0:' .6

,/0 p

0/"

0: :' ,

.... 1 IN AIR

,',6

20 30 40 50 60 70 80 STRESS I NTENS ITY, MP. J1if

Figure 4: Creep crack growth rates vs. stress intensity factor in Inconel X-7S0 tested in air and in argon at 6S0° C.

9

10

= -... e e I uJ I t-

~ I '" I j;; 10

~ I

'" I

u / "" "" u

CREEP

10 20 30 '10 50 60 STRESS INTENSITY. MPa.m

Figure 5: Comparison of crack growth rates da/dt in creep and in fatigue with triangular waveshape (R = 0.05). Ref. 13.

],0

O,g

~ 0,8

E

l;!!O.7 ;'2

~ 0.6

~

~ 0.5

0,11

0,3

0,2

0,]

A· FATIGUE IN AIR. 0,01 Hz B· CR'EP IN AIR

IIlCONEL X-ISO ~ )·650'[ ~

K • 23 ~Pa;ffi ;3 :r

g :=

w ~

~ ~ 0

~ I w ~ z ~ 0 ~

~ ~

~ ~ ~ c:; -

c· FATIGUE IN VACUUII, 0,01 ii,

O· CREEP I N ARGON

Figure 7: Comparison of FCGR 0.01 Hz creep and CCGR at 6500 C in air, vacuum, and argon at Kcr.eg. = 23 MPa {ill. Ref. 13.

INCONEL X-ISO

ilK • 30 MParm R • O,OS

/ T • 650°C

\ ~

1-1 ~~-~-~ )" 25°C -"'it----..::.,..;.-::.:.-::...---D-

REF ,[4) I

CREEP 6S0 °C IN ARGON

Figure 6: Frequency effect on FCGR at ~K = 30 MPa ~m. Ref. 13.

)fTR (AIR)

10 FREQUENCY, Hz

V1 V1 <.u

'" >-V1

(a) (b)

LOW TEMPERATURE

HIGH TEMPERATURE

(c)

Figure 8: Cyclic stress-strain hysteresis loops: (a) continuous cycling; (b) creep-fatigue cycle with compressive stress dwell; (c) out-of-phase themomechanical fatigue cycle. Ref. 2.

1 a

0.5

Strain range

(percent)

02

0.1 1000 2000

Conventionally cast

5000 10.000

Alumintde coating Tmax ·1900 o F 11038°C)

T min' 80QoF 1427°CI

20.000 50.000

Cycles to failure - Nf

Figure 9: DS superaUoys (CB and SC) have superior themal fatigue life (Nt) compared to CC superalloys. Ref. 14.

11

12

32.9 o· ~...:.r:;;:"J'T'"'...l;r---'T-+-r--+-+----l 12271

15 D 200 25° 30° 35 0 40° 45.0 10111

Elastic modulus of SC superalloy PWA 1480 as a function of orientation at room temperature

<001> <011>

E=43x106 psi <111>

E=33x106 psi <011>

<111>

Figure 10: Orientation dependence of elastic modulus of a nickel base superalloys. Ref. 14.

/

Stress Oy

" '/ /

/ / /

/ / / /

/ " Strain

/ {lIEplos /

/ (lIEplpOly

Strain limit

___ Directionally solidified

- - - - Polycrystalline

Figure 11: Hysteresis loop for polycrystalline alloy compared to a DS alloy.

Relative thermal fatigue

life

10,000 r-

1000r-

100 r .---

10r-

Poly- D.S. crystal

E(106 psi) 33 18

Figure 12: Fluidized bed testing demonstrates that directional solidification enhances thermal fatigue life.

1.25.--------------------, 1.0 09 0.8 0.7

0.6

0.5

TOTAL 0.4

STRAIN ('\) 0.3 ~;;;;;

<1'1>

N, (CYCLES)

200~------------------------------, Fig. 13: Orientation Dependence of 1800° F Strain controlled LCF

150

100

&TE {hkl)

(103 pSI)

r=0.939

o (113)

o (011)

A {112}

<> (t1l)

H, (CYCLES)

13

Figure 14: 1800° F Strain controlled LCF Life.

MICRO-MECHANICAL INFLUENCES ON THE FATIGUE CRACK GROWTH BEHAVIOR OF NICKEL BASE SUPERALLOYS

Randy Bowman and Stephen D. Antolovich

School of Materials Engineering Mechanical Properties Research Lab

Georgia Institute of Technology Atlanta, Georgia 30332-0245

ABSTRACT: The micro-mechanisms responsible for influencing fatigue crack propagation (FCP) in nickel base superalloys were investigated. Four experimental alloys were developed such that the lattice mismatch (~), antiphase boundary energy (APBE), and volume fraction (Vf) of 7' precipitates were systematically varied. Heat treatments were also employed to obtain two distinct 7' sizes. Monotonic and low cycle fatigue tests revealed the material response of each alloy. Closure was measured by several techniques to investigate the possible effects on the FCP rate of these alloys.

Constant amplitude cyclic loading revealed distinct differences in the FCP response of the four alloys. Precise load-displacement determinations indicated that crack closure was not responsible for these differences. The strength-normalized results indicate that those micro­structures which can best accommodate damage are most resistant to crack growth. This is consistent with the accumulated damage model of FCP. Alloys with low Vf' low 0, and low APBE exhibited FCP rates that were approximately 50 times lower than for other treatments. FCP rates were dramatically reduced for those compositions and heat treatments that promoted planar, reversible slip. The effects of individual microstructural features on FCP rates were also determined.

1. INTRODUCTION: Fatigue in turbine discs arises from variations in both thermal and mechanical stresses which occur during a flight. The highest stresses are experienced at the bore of the disc early in the flight cycle while the temperature is in the range of 200-300~. Stresses in the rim are lower but occur at a higher temperature, 500-600~. In the late 1960's, low cycle fatigue (LCF) was considered to be the major life limiting factor for over 75% of the major structural components. The LCF approach to useful remaining life predictions is an extremely conservative approach. Using this technique, a component is retired from service after the design life (based on a statistical

15

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 15-26. © 1990 Kluwer Academic Publishers.

16

analysis of LCF data) has been reached. The LCF based criteria does not consider the additional component life resulting from sub- critical crack propagation after initiation. Alternatively, a 'retirement for cause' (RFC) philosophy allows a component to remain in service until a fatigue crack of specific size has been detected.

Retirement for cause requires good non-destructive testing and accurate prediction of fatigue crack propaga­tion lives of components containing cracks utilizing fracture mechanics principles. The economic benefits of such an approach have been the driving force for the vast amount of research performed on nickel base alloys and for the development of turbine disc alloys with improved crack propagation resistance [1].

Fatigue crack propagation (FCP) has been chosen as the target property of this study since with the introduction and acceptance of RFC philosophy it has become, for the first time, design-critical for turbine disks. Furthermore, disks constitute up to 30% of the total engine weight; therefore any material savings are greatly magnified.

Not only is quantification of fatigue damage difficult, but defining it remains an elusive goal. The most sensible way to make progress in defining and improving FCP resistance is to work on microstructurally simple systems in which important parameters can be varied systematically. Only then can the importance of slip mode, precipitate coherency, crystal structure, etc., be established as they relate to fatigue.

2. BACKGROUND: 2.1 Deformation of Ni Base Supera1loys

Cutting of the r' by dislocations produces an antiphase boundary (APB) resulting in an overall increase of energy. Paired dislocations are very common during particle cutting where the first dislocation creates the APB and passage of the second restores the stacking sequence thus eliminating the fault. Alloys with combinations of large r' particles, low volume fractions and high lattice mismatch promote deformation by Orowan looping. In order for shearing to occur the stress necessary for particle cutting must be less than the Orowan looping stress. Both the shearing and looping stresses are a function of the r' properties and particle size thus allowing the deformation process to be controlled through manipulation of microstructure and chemistry. Particle shearing causes dislocation pairing, makes cross-slip difficult and thus promotes inhomogeneous planar deforma­tion. Cutting of fine precipitates leads to softening in the active slip bands.

The parameters described above can be controlled by composition and heat treatment modifications as described elsewhere [2].

2.2 Fatigue Crack Propagation Antolovich and co-authors (3,4] proposed modifications

to an earlier theory [5] in which FCP was viewed as an LCF process occurring out to a distance ahead of the main crack tip. In this model, FCP is caused by damage accumulation in small elements that undergo reversed yielding. The crack then advances by some distance when sufficient damage has accumulated in this "process zone". It was found that longer FCP lives were associated with larger process zones. The larger the process zone the smaller the average strain and the greater the number of cycles required to accumulate a critical amount of damage and advance the crack. Large grain sizes increase the process zone and should reduce FCP rates. This effect can be magnified by promoting slip reversibility (e.g. low mismatch and low APBE) .

Alternatively, other authors [e.g. 6-8] attribute the lower FCP rates in coarse grain materials to increased crack closure. Closure reduces the stress intensity due to crack tip shielding of the remotely applied load. These studies found that the improvements in FCP resistance with the larger grains did not exist at high R -ratios where closure does not occur. Coarse and fine grained materials had nearly the same growth rates for R = 0.8. The explana­tion was that the larger grain sizes increased the fracture surface roughness resulting in more roughness induced closure. Also, the larger grains and correspondingly rougher surfaces had longer effective crack paths. Both of these effects would supposedly contribute to a reduction in the FCP rates.

3. RESEARCH PROGRAM: 3.1 Alloy Compositions

The controlled microstructural variables included APBE (T), mismatch (0), and volume fraction of T' (Vf)' In addition, grain size (not reported in detail here) and T' size were controlled by heat-treatments. These variables were chosen since they have all been shown to influence the deformation mode and, presumably, damage accumulation. The compositions and relative target levels for the control variables are shown in Table I.

Table I. Alloy Compositions in Weight Percent and Associated Properties

17

AIJ..QY Ni Al Ti Mo Cr B r 0 Vf--

1 Bal 2.35 <.01 <.01 13.83 .0037 low low low 2 Bal 4.92 <.01 <.01 14.18 .0042 low low high 3 Bal 2.96 2.58 <.01 9.39 .0037 low high low 4 Bal 1. 24 3.71 9.91 13.21 .0060 high low high

3.2 Mechanical Testing Tensile specimens were tested to failure under strain

control at a rate of 50%/min at room temperature to provide information on yield stress and Young's modulus. In addition, TEM examination of the tested specimens provided information concerning the deformation mode for each

18

composition. Two types of LCF tests were conducted. The first was

a total strain controlled (O-tension-O) test to failure while the other was a interrupted constant plastic strain controlled test. All specimens were electrolytically polished prior to testing [9].

FOP tests were performed at room temperature using a closed loop servo-hydraulic test machine. Testing started in the near-threshold region and covered approximately three decades in growth rate. Crack lengths were monitored with a d-c potential drop system. In addition, crack closure was measured by compliance (load/displacement) techniques. Testing was performed at an R-ratio of 0.1 and 0.8 to examine the effect of closure. Full experimental details are given elsewhere [9J.

4. RESULTS AND DISCUSSION: For ease of presentation, the alloys were numbered according to their composition and gamma prime size. For example, in Table II, alloy number one with small r' size is referred to as 18 while the large r' material is 1L, etc. A summary of the alloys' properties is presented in Table II.

TABLE II. Measured Microstructural Properties

grain size r' size r 6 Vf Ur ef

Alloy (m) (m) (ergs/cm (%) (%) (Mpa (%)

18 52 0.08 56 .09 15 209 54.9 1L 55 0.50 80 .07 18 225 49.6

2S 51 0.09 87 .07 27 611 36.6 2L 52 0.62 101 .04 25 689 31.2

3S 36 0.07 96 .21 21 650 32.8 3L 42 0.54 120 .18 18 747 31.6

4S 23 0.07 420 .18 25 650 48.4 4L 34 0.68 403 .14 22 643 42.3

4.1 Fatiaue Crack Propagation In Fig. la, the FCP response of the small r' alloys

tested at an R-ratio of 0.1 at room temperature in air is presented. Correlation of FCP rates as a function of 6K describes a material's response to cyclic loading. Crack extension at a given value of 6K is a function of both the amount of damage imposed at the crack tip and the material's intrinsic ability to accommodate the damage. For a given stress level, an alloy with a low yield stress but high resistance to crack extension can exhibit the same FCP rate as an alloy with a high yield strength and low crack growth resistance. Most FCP models predict an inverse relationship between crack growth rate and yield strength. It is therefore useful to attempt to normalize the FCP

results with respect to yield strength thereby eliminating the differences in strength, Fig. lb.

Examination of the fracture surface, Fig. 2, shows that roughness of the fracture surface arises due to cracking along slip bands which formed during the test. Roughness measurements of the fracture surfaces as a function of ~K are presented in Fig. 3. The roughness parameter, RL, is a measure of the actual crack path divided by the projected crack path. Comparison of these results with Fig. 1 shows that the specimens with the roughest surfaces have the greatest resistance to crack growth while the specimens with a relatively smooth fracture surface have the highest growth rate. Initially this seems to imply that FCP is controlled by a roughness induced closure effect whereby the specimen with the largest surface roughness will have the most roughness induced crack closure and thus the lowest crack growth rate. This interpretation is discussed below.

10-3

10-1 C - ALLOY 1. lor r, low 4, low V, .Ito -AJJ.J1'f 2. low r,m d, bleb Vf • - wm 3. lair r.lU&h I. medlum Vf + - .ALLOY 4. hlp r, low I. bi&tL V,

(a)

10-11

1(/

/ 0+-.1' .o'

+~ ,,4

/ .. ' I ... :,. . .

Ii . I //

1/·/ ,/ ., "' : , 8

f Ii / ;

.' :

a-AIJ..OY 1. loW' r.knr I. low V, 6 ... wm 2, low r.loW' 6, Iugh Vf • "".Al.LOY 3, low r. hl&h I, m.echum V1' + ... ALLOY 4, hlP r. low I. high V,

10-1 10-6 t'lK'/(u,.E) (m)

(b)

Figure 1 - FCP response of small r' materials for R = 0.1. In (b) the data is normalized with respect to yield strength and elastic modulus.

19

20

Figure 2 - Optical micrograph of alloy 4S fracture profile.

rr.-"j

"' " 0: .a "" " ~ ... " " 0: ;:J

2.0 ,---------------,

16

1.6

1.4

1.2

c ALLOY 1. low r. low 6, low Vt .:. ALLOY 2.10w r, low o. hIgh Vt • ALLOY 3, low r, high d, medlUm Vr + ALLOY 4. hlgh r. low 0, lugh Vf

1.0 '---_-'-_-'-_--' __ .1--_--"-_--' 10.0 15.0 200 250 30.0 35.0 40

ilK (MPaVrri)

Figure 3 - Fracture surface roughness as a function of stress intensity level.

4.2 Closure To investigate the importance of roughness, closure

measurements were performed [9] at R=O.l and at R=0.8, Fig. 4. Load-line/displacement measurements indicated a max closure load of approximately 0.12P~x for all alloys at near threshold regions which decreased to zero in the mid­Paris regime (Table III). Since the closure loads were similar and relatively low for all alloys, no significant difference in the growth rates was noted when plotted vs. 6K eff (IlKeff = 6K IlIax - 6K cl)' The apparent correlation between surface roughness and FCP rates (i.e. large roughness associated with low growth rates) can not be explained on the basis of roughness induced closure, however appealing this explanation may be.

At R = 0.8 (no closure), the relative FCP rates were unchanged although the absolute values of da/dN at R = 0.8 were higher due the larger mean stress. It is clear that the observed differences in growth rates for these alloys are due to "intrinsic" differences in fundamental micromechanical processes and not to "extrinsic" effects such as closure as is often cited for other systems. With all "extrinsic" factors eliminated, the dominant features controlling FCP can be identified.

10-0

1(i

a .. ALLOY 1, low r, low I!, low VI 6=ALLOyz,lowr,lo1r4,higb. V, • ,.. AlLOY 3, low r, high I!, medium V, + = ALLOY 4, high r, low I!, hiah V,

~~t (MPavrn) let

'Ol' R ~ " = Z

"'-~ "0

10-' ,--------------,

10-'

10-'

0= ALLOY 1.10"" r,lo",. I!, 10'1'1" V, 6 = ALLOY 2, low r, low 6, hIgh V, • = ALLOY 3, low r, htgh 6, medlum V, ;. = ALLOY 4, hlgb. r, low d, high V,

(al (bl Figure 4 - FCP response of small r' materials. a) results

plotted versus lIKeff thus accounting for crack closure. bl material response at R = 0.8.

4.3 Low Cycle Fatigue

21

Total strain controlled tests resulted in an initial hardening response (increasing load) for alloys 1-3 followed by gradual softening over the remaining life of the tests. This behavior suggests that particle shearing is the dominant deformation mechanism for these alloys. Conversely, alloy 4 hardened to a saturation level, indica­tive of deformation by Orowan looping. The lives for alloys 1-3 were all very similar at the same strain levels (around 7500 cycles) whereas the life of alloy 4 (low strength, high ductility) was nearly 4 times as long.

TABLE III. Closure loads at various lIK levels at R=O.1.

closure load (P olP max)

Alloy lIK 20 30 40 50 60 70

lS 0.08 0.07 0.07 1L 0.08 0.05 0.05 nil nil nil

2S 0.12 0.10 0.10 2L 0.10 0.09 0.06 0.05 0.05 nil

3S 0.10 0.10 0.08 3L 0.10 0.08 0.07 0.07 0.03 nil

4S 0.10 0.07 0.07 4L 0.10 0.08 0.06 0.05 0.05 nil

22

4.4 Deformation Structures TEM examination of the interrupted, constant plastic

strain controlled tests indicated that the ability of alloys 1 and 2 to accommodate strain without the subsequent development of damage was very good {low dislocation density} whereas alloy 4 had nearly 5 times the dislocation density for the same imposed plastic strain. Alloys 1 and 2 had similar dislocation densities, any differences being hidden in the inaccuracies of the dislocation density measurement technique. The ability of a particular alloy to accommodate damage (as measured by the dislocation density) is directly related to the alloy's ability to resist crack growth. Specifically, alloy 1 had the lowest crack growth rate when normalized with respect to strength and it also had the lowest dislocation density under constant plastic strain controlled conditions. In this case many cycles must be imposed at the crack tip to reach the critical damage level necessary to advance the crack. Alloy 4 has poor resistance to damage and requires few cycles to accumulate the required damage necessary for crack advance.

Alloys 1S,2S, and 3S deformed by shearing {i.e. dislo­cation pairing, planar slip} while 4S deformed by looping. Representative micrographs are shown in Fig. 5. For shearing to occur the stress must be less than the Orowan looping stress. For alloy 4S, the APBE is sufficiently high to prevent shearing. Any proposed model must account for different deformation modes depending on the specific alloy system and test conditions. Those factors which control FCP when the precipitates are sheared may not operate when dislocation by-pass occurs complicating interpretation of microstructural influences on FCP.

4.5 FCP of Small I' alloys From the da/dN vs. ~K plots {Fig. Ib}, it is clear

that alloy 1S {low r, low 0, low Vf} is most resistant to crack advance. When normalized as described above, the FCP rate of this alloy system is at least two orders of magni­tude slower than the others. The low volume fraction of precipitates results in a larger mean free path between obstacles for the mobile dislocations. The imposed plastic strain is therefore more easily accommodated resulting in less damage accumulation and thus greater resistance to crack advance. The efficiency of the r' as obstacles to dislocation motion is reduced further in this alloy by the low values of APBE and o. Conversely, alloy 2S has the same APBE and 0 but a much higher Vf of y'. This combina­tion results in relatively higher crack growth rates. The ability of this material to accommodate strain at the crack tip is reduced by the small mean free path of the dislo­cations.

23

(a) (b)

Figure 5 - Representative deformation structures illustrat­ing a) shearing, alloy 23 and b) looping, alloy 43.

The decreased resistance to FCP due to higher APBE (alloy 43) is due to the difference in deformation mode caused by the APBE. In the other 5 series alloys, which have various combinations of APBE and Vf' particle shearing occurs, Fig. 5, whereas for alloy 45 with a high APBE particle by-pass is the dominant deformation mode and no APBE is created (i.e. no energy increase). With particle looping, the contribution of 0 toward inhibiting dislocation motion becomes more pronounced since in the looping regime the eRSS is directly related to mismatch.

For small r' precipitates, alloys 25 and 3S had differences of 3-5 times in FCP rates due only to changes in mismatch and a slight difference in Vf. In fact, from the previous argument, increased Vf is seen to increase the FCP rate when normalized with respect to u ys and E due to a lowering of the mean free path. All other factors being constant, a lower volume fraction of precipitates should reduce the FCP rate. Therefore, the differences in the FCP response of alloys 2S and 3S is even greater if the difference in mean free path is accounted for. The increase in FCP rate for the higher mismatch alloy is a result of the increased resistance to dislocation motion due to the enhanced strain field around the precipitate and/or a different deformation mode.

4.6 Modelling of the FCP Process With a large amount of data available it is usually

possible to describe the test results in mathematical form based on a purely empirical basis such as the Paris equation. The values of the various coefficients are experimentally determined and are not based on physical concepts. The primary use for such equations is to provide a data correlation technique and provide a means for

24

representing the data. These equations cannot be used as prediction tools if the mechanical properties are changed or the extrinsic parameters vary. Also, such equations do not provide insight for alloy development and thermo­mechanical treatments which would enhance the properties.

Alternatively, a mathematical model can be based on fundamental concepts and combined in such a way as to predict results outside the scope of the laboratory experi­ments. In these cases, however, the mechanisms are highly simplified and many assumptions are required to solve the problem. Such models have the advantage of predicting values based on fundamental properties which can be determined in most materials.

Correlation of the mechanical properties to the micro­structures in this study has suggested that FCP is a result of two competing mechanisms. One of these mechanisms is the applied driving force which provides the energy for crack growth. This driving force can arise from applied loads, environmental effects (e.g. corrosion), temperature, etc. In this study the primary driving force was a constant cyclic crack tip stress field described by 6K or 6K eff. The driving force may be expressed, in this instance, as a function of 6K, i.e. ~(6K).

Opposing this driving force is the material's intrinsic resistance to crack growth. This resistance term may be expressed functionally by, ¢(u ys ' E, ~, cf, n'); where u ys = yield stress, E = Young's modulus, ~ = degree of slip reversibility, cf = cyclic fracture strain, and n' = cyclic strain hardening exponent.

A high yield stress, modulus, and cyclic strain hardening exponent result in a high resistance to crack extension due to the decrease in plastiC strain present at the crack tip for equivalent stress intensities as discussed previously. The degree of slip reversibility is a major contributor to FCP resistance by allowing a greater accommodation of damage as the degree of reversibility increases. This reversibility term is difficult to quantify but certainly involves such factors as Vf, 0, r, stacking fault energy, dislocation back stresses, grain size, etc. Finally a material with a large fracture strain is best suited to accommodate large amounts of plastic deformation before failure.

It should be pointed out that the resistance terms are not totally independent variables. For example, slip reversibility will certainly affect strain hardening and as will be seen later, those factors which affect slip revers­ibility also affect yield strength.

There are certainly other factors not included in the above functional relationships which also affect FCP. Since only the microstructural variables described previously were controlled and measured inclusion of other terms would be inappropriate.

Fatigue crack growth is thus the result of a damage function, ~, which combines both the imposed driving force and the resistance function such that;

and da

g ( ~ ) dN

which implies that the resistance to crack growth is increased (low da/dN) by a low driving force or a high material resistance. The exact form of equation 2 is currently being developed utilizing the results of this study.

25

( 1 )

( 2)

SUMMARY: Constant amplitude FCP tests were performed on each of four alloy compositions with two different r' sizes to investigate the effects of microstructure and compo­sition.

The major findings and observations were:

1) Crack closure concepts do not explain differences in the FCP rates for both near threshold and Paris regime propagation in the model Ni base alloys studied here.

2) FCP rates are dramatically low for those compositions and treatments that promote planar, reversible slip.

3) In this study, alloys having high volume fraction, low APBE, and low mismatch exhibited FCP rates that were approximately fifty times lower than other alloys.

4) Internal resistance to damage ahead of a crack is achieved by low volume fraction of precipitates, low lattice mismatch, and low anti-phase boundary energy. FCP resistance is increased by a planar deformation mode. However, in a planar slip material, on a strength/modulus normalized basis, restricting dislocation motion decreases the alloy's ability to accommodate damage and increases the FCP rate.

5) At the same strength level, it has been demonstrated that the FCP rate can be reduced by at least a factor of 50.

6) The implications of this study are that FCP rates in alloys of practical interest can be significantly reduced by heat treatment and modest compositional changes.

ACKNOWLEDGEMENTS: The authors are grateful to the AFOSR for financial support of this research (Grant # AFOSR 84-0101, Dr. Alan H. Rosenstein Program Manager). They would also like to thank Dr. Hugh Gray of the NASA-Lewis Research Center for assistance in the alloy development phase of the work and Mr. W. Couts of Wyman-Gordon for his help in

26

processing the alloys. Dr. Bowman would also like to thank the Wyman-Gordon foundation for the Fellowship which was awarded to him.

REFERENCES: 1. J.E. King, Mat, Sci~~nd_Tech., Vol. 3, 1987, pp. 750-764. 2. The~u2eralloys~ C.T Sims and W.L. Hagel eds., John Wiley and Sons, Inc., New York, 1972. 3. G.R. Chanani, S.D. Antolovich, and W.W. Gerberich, Met. Trans., Vol. 3, 1972, pp. 2661-2672. "4. S.D. Antolovich, A. Saxena, and G.R. Chanani, Eng.i!:ac. Mech., Vol. 7, 1975, pp. 649-652. "S-:--F:"A. McClintock, Frac;J:ure of Solids, D.C. Drucker and J. Gilman Eds., John Wiley and Sons, N.Y., 1963, pp. 65-102. 6. G.T. Gray III, J.C. Williams, and A.W. Thompson, Met~ Trans, Vol 14, 1983, pp. 421-433. 7. Jian Ku Shang, J.-L. Tzou, and R.O. Ritchie, Met. Tra~, Vol. 18, 1987, pp. 1613-1627. 8. R.D. Carter, E.W. Lee, E.A. Starke Jr., and C.J. Beevers, Met. Trans., Vol. 15, 1984, pp. 555-563. 9. R. Bowman, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, Georgia, 1988.

MICROSTRUCTURES AND CRACK OPENING IN A NICKEL-BASE SUPERALLOY DEFORMED BY IN-SITU TENSILE TESTS.

* M. IGNAT and J. PELISSIER

L.T.P.C.M. - E.N.S.E.E.G., Domaine Universitaire, B.P. 75 ST MARTIN D'HERES

*DMG - Centre d'Etudes Nucleaires de Grenoble

ABSTRACT: This paper describes some resul t.s concerning in-situ experiments, performed on single-crystalline samples of a nickel based superalloy.

tensile N~'

The deformation developped during the in-situ tests was heterogeneous. It was principally localized in narrow bands. Direct observation of dislocation motion in these bands was difficult. The deformation bands were in the begining of microcracks. r~icrostructures in the deformation bands were dependent on sample's orientation.

Different dislocation arrangements formed in the sample, and particularly near a crack-tip were observed. Their analysis indicated evidence of the activation of non common slip systems in the super alloy •

1. INTRODUCTION: An important fact deduced from the investigations carried on single crystalline ~/k" superal10ys, concern their increasing of strength with temperature. This particular property, has lJeen I inked to the !'" particule size, as to the orientati on of the straining axis with respect to the sample. (1).

Although in-situ experiments on single crystals of the ~'phase

have been performed (2,3), as far as we know, on ~/t I superalloys this type of tests has only been attempted once before (4).

In order to contribute with some microstructural information about the dislocation mechanisms as the microcrack opening processes in

'i\ / l:' ' super alloys , we performed ill-situ tests. The samples presented different orientations in relation to their

tensile axis as to their main faces. The temperatures used for the tes ts were choosed so that one was under the critical temperature in which the alloy increases it strenght: 823 K, and the other at the maximum strenght temperature for our alloy which is about 1023 K.

2. EXPERIMENTAL METHODS: The nickel based superalloy investigated here has the CMSX-2 denominati.on. It is a two phase alloy, who contains about 70 % volume faction of ~' precipitates the Ni Al intermetallic L1 phase. These ordered preCipitates coherent in3the nickel rich CFt matrix phase have a cubic shape. Their faces are parallel to(010j

27

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 27-33. © 1990 Kluwer Academic Publishers.

28

cristallographic planes and their mean size are 0.4 10-6 Iml. This micrsotructure is obtained after an ONERA patented heat treatment (5).

From single crystals of the CMS-X2 alloy, thin ~~il microsamples were ~utted. The dimensions of the samples were: 6 10 tm] long, 2 10- em] large and 0.15 10-3 lm] thick after beeing mechanically thinned. When electrolytically thinned, in a classical jet electropolishing device, the samples presented a central hole with thin zones for electron microscopy observations.

When submitted to the in-si tu tests, these thin samples showed that , the deformation was located in the central thin zones, near the hole. As a matter of fact, following the solution for a thin plate with a central hole, submitted to a tensile test, the highest local stresses will develop tangentially to the central hole and parallel to the tensile axis (6).

Concerning the tensile stage and the heating system used for our in-situ experiments, a detailed desciption is included elsewhere (7).

The samples we used in our experiences had main faces parallel to {100} , {110} and (Ill} crystallographic planes. In this paper we ar~ describing experimental results concerning only the {110} and {1ll} samples. They were deformed with a tensile axis parallel to their longitudinal axis: [lOOJ and [1121 respectively. For both type of samples the straining experiments were performed at 823 K (550°C) and 1023 K (750°C). The accelerating voltage used in the Electron Microscope was 1 MeV. Fom the begining each test was recorded on a video tape. The onset of the deformation bands and consecutive microcrack opening processes was observed. As the dislocation movements in the deformation bands was difficult to observe on the TV screen, the deformed microstructures were analysed afterwards in a 200 kV transmission electron microscope. In the following we shall describe and comment the principal observations on the 200 kV transmission electron microscope.

3. EXPERIMENTAL RESULTS: The microstructures obtained during the in­situ experiments, related to the crystallography of the main faces of the samples are described here.

For the (Ill} type samples, their longitudinal axis was a [112] direction and they were pulled parallel to it. The microstructure of these deformed samples showed the following features:

At 823 K (550°C), the deformation bands developped when pulling the sample corresponded to {Ill} octaedral slip systems. Increasing the tensile force, conduced to crack opening and propagation. The microcracks followed the above mentionned planes, but we also noticed short "saw tooth" crack paths, corresponding to octaedral to cubic crack plane transitions.

At the intersections of these planes, dislocations were emitted in octaedral and cubic planes. Their 110lj straight directions beeing coplanar to the cubic and octaedral planes. The Burgers vectors of these dislocations has not yet been determined.

At 1023 K (750°C), the cracks opened exclusively on {Ill} planes, without deviation on cubic planes. On the contrary, numerous traces corresponding to slip or rearrangement of dislocations on cubic planes were observed (Figure 1). The direction of the dislocations in the

slip planes was in general parallel to IllO] directions. The observations performed in weak beam conditions showed that they were dissociated.

Looking at the crack tip, lying on L1IIJ plane, we noticed that the dislocations ewi tted in front of the crack could cross slip on (011] planes. As expected, the direction of the cross slipping djslocations was common to the crack plane and the slip plane.

Figure 1 : Microstructure observed on a (lll} sample deformed in­si tu at 1023 K. The dislocations are aligned on cubic planes:'6 The white arrow correspond to g: [200]; the black line to 0.3 10 fm] approximati vely. T is the tensile axis (BF., 1 Mev.).

The (0111 type samples were pulle~ along their [100J longitudinal axis. At 823 K (550 0 C), the cracks were parallel to [1111 crystallographic planes, as the observed deformation bands. Ahead of these deformation bands, the dislocations rearranged in networks as shown in Figure 2: a twist subboundary is build up by different types of a /2 11011 screw dislocations. Other deformation bands showed complex stalking faul ts, left by the shearing of the ~'precipitates byl112J partials. When the octaedral slip bands intersected one to each other, complex tangles of dislocations were formed, thus reducing their mobility (Figure 2,b).

At 1023 K (750 o C) the main crack extended parallel to a {111} plane. Short shear bands parallel to {100} planes, interconnected the main {Ill} crack paths. In the zones of the sample where the cracks stopped, slip bands corresponding to non common (101) slip system were observed. The propagation of these bands showed (as the cracks) short deviating paths on cubic planes (Figure 3). The dislocation lines in these bands had [110 J type dirt'ctions.

29

30

Figure 2 Microstructures corresponding to a [011 J type sample deformed at 823 K. The dark line in (a) and (b) represents 0.2 10-6 [m] (lMeV). (a) Weak beam conditions g,2g. The white arrow

corresponds to g: [200J. (b) Bright field. The dark arrow corresponds to

g: [200J. 'j' .is the tensile axis.

Figure 3: Dislocation microstructure of a (011) sample deformed at 1023 K. The deformation bands, corr1fpond to (110)(1101 non common slip. The dark line represents 0.1 10- [ml (1 Mev; e S !! to !011!). T is the tensile axis.

4. DISCUSSION: Some comments on the previous described microstructures are developped in the following.

If considering first the crack opening and propagation modes, the in-si tu experiments showed that the main crack plane corresponded always to the {Ill} plane belonging to the octaedral system presenting the highest Schmid factor. This with respect to the direction of the

31

32

applied stress. In spite of this, the propagation mode of the 11111 cracks showed to be dependent on the type of sample and test temperature. For instance, the saw tooth mode for the {Ill} samples as the slip bands, observed both at 823 K (550°C) can be also explained through Schmid factor (S.F.) considerations. As a matter of fact, when pulling along the rl12] direction a {Ill} sample, the SF has a maximum value for the (ill)[10l1 and (111)(011J octaedra1 systems: 0.41. It is followed then by the (010) t10lJ and (100) f011) cubic systems, with a SF of 0.35. The ratio among this systems remains favourable to the octaedral one. Meanwhile we observed short slip or crack paths on cubic planes. This slip becames possible, when the tensile axis mooves only few degrees from the [112] direction towards the (1111 direction, inversing the S.F. ratio favorably to cubic slip. The displacement of the stress axis could be accounted as an effect of local stress fields,acting for exemple close to the intersections of the octaedral systems we noted before.

When increasing the test temperature to 1023 K, intense cubic slip was observed. Meanwhile the octaedral systems was involved only in the opening of the main crack. At 1023 K the observation of the cubic slip agree with the thermally activation of cube cross-slip in 'phase. Moreover, high local stresses may enhance the entire deviation of the cross-slipped superlattice dislocations forming glissile configurations on the cubic planes, in the begining of the slip bands we observed. The other cross slip we noted: for dislocations emitted at the tip of {Ill} crack, who cross-slipped in a (011) plane, could be explained by the local high stresses constricting the partials of an emitted superdis10cation. This allowing it to cross-slip on the (011) plane.

With respect to the (110} samples: they confirmed that the crack opening process is always initiated on octaedral planes. For 1-hese [110} samples, when pulled alon& the [100) axis, the maximum S.F. of 0.5, corresponds to [101] llOl) systems; which are followed by the octaedra1 ~1111 [llOJ systems who have all a SF of 0.41. In this case, we noted again that: below 1023 K and above this temperature the main crack planes were always parallel to {Ill} planes.

Which respect to the slip bands at 823 K, as at 1023 K for the {II O} samples, no cubic sl ip was observed. This agrees with the SF considerations (null for this orientation). But the important feature for this sample orientation was the intense slip on non common [110] ~lQl slip systems. It was activated only at 1023 K. The activation of these type of slip systems has been reported frequently for CFC single phase materials. It is worth noting, that for superalloys, they were reported only once, from tensile creep microstructures (8).

5. CONCLUDING REMARKS: In this paper we described our first results obtained from in-situ tests on a superalloy. The attention was focused on the deformation bands, leading to crack opening processes; and on the dislocation microstructures deve10pped in the deformation bands.

The most relevant results were: - main cracks always propagaded on octaedral systems, favourably

oriented with respect to the applied stress. - as expected the cubic slip was active at the highest test

tempera ture (1023 K). This temperature corresponded to the critical

t.emperature for thermally activating cube cross slip. - the cube cross slip was dependent on the samples orientation :

it was activated for {Ill} samples strained along a (l12J direction, and enhanced when the temperature was increased. It was inexistant for the {110] samples strained along a (lOOJ direction.

- non common (1011 llOl} slip was clearly put in evidence for {OIl} samples, and also noticed adjacent to the crack tip of a {Ill} type sample.

Futher developpements involving other type of samples, and precise identification of Burgers vectors will be developped. This will clarify the effects of high local stresses and the propagation mode of crack in this alloy •

REFER~;NCES : 1 P. Caron, Y. Onto, Y.G. Nakagawa, T. Khan, Proceedings of the 6th International Symposium on superalloys, Seven Springs USA, (1988). 2. N. Clement, J.Micr.Spectr.Electr., 11, (1986) 195-203. 3 Ph. Lours, B. de Mauduit, A. Beneteau, N. Clement, D. Caillard, Proceedings of the 8th International Conference on the Strength of Metals and Alloys. Tampere, finland, (1988) 251-256 .• 4. M. Ignat, F. Louchet, J. Pellissier, Proceedings of the 7th ICSMA Conference, Montreal, Canada, (1985). 5. P. Caron, T. Khan, Mater.Sci.Eng. (1983) 61 173-178. 6. S. Timoshenko, J.N. Goodier, "Theory of Elasticity" Mc Graw Hill (1951). 7. J. Pelissier, J.J. Lopez, P. Debrenne, Sixth International Conference, Electron /f;icroscopy, (1980) Vol. 4, 30-33. 8. C. Carry, J.L. Strudel, Acta Met. (1977) Vol. 25 767-777.

33

MECHANICAL BEHAVIOR MODELING

OF A NICKEL BASE SINGLE-CRYSTAL SUPERALLOY

Jean-Yves GUEDOU et Yves HONNORAT

"Materials and Processes" Department

SNECMA-CORBEIL - FRANCE

ABSTRACT: The mechanical behavior of the superalloy single crystal AMl has been investigated with monotonic (tensile and creep) and cyclic tests. The observed crystallographic anisotropy can be fairly well accounted for using Schmid criterion, as for as tensile mono­tonic mechanical behavior with low plastic strain is concerned, in a large temperature range (20-1100 0 C). Viscoplastic and cyclic aniso­tropy strongly enhance the singular behavior of <001> loading direction. Macro and microstructural investigations are needed to characterize the involved physical mechanisms and to model the mecha­nical behaviors. Nevertheless, an unidimensionnal cyclic viscoplastic unified law has been identified for <001> loading direction, and introduced into a F.E. computer code used for blades stressing.

1. INTRODUCTION Stressing of turbine blades operating in quite stringent thermo-mechanical environment requires the application of increasing complexi ty computer codes integrating the effects of plasticity, viscosity and cyclic loading. The high mechanical anisotropy of single-crystal superalloys complicates the predictions of durability of parts made from these materials which provide a significant gain in achievable gas temperature. Such anisotropy will not be satisfactorily considered until the strain mechanisms involved in particular creep and low cycle fatigue, which are the predominent damaging modes of these parts, are characterized (1). For this purpose, the effect of crystallographic orientation and of tempe­rature on the properties and mechanical behavior of a single-crystal superalloy (AM1) were investigated through laboratory tests in SNECMA.

Table 1

r-~--r----r----r----r----r----r----r-~--r----r----1 I Nl I Co I Cr I Mo I W I Ta I Al I Tl I C I Fe I I I I I I I I I I I I 1_~~~l_~~~l_~~~l_~~~l_~~_l_~~~l_~~~l_~~~l~~~O!l~~~~l

AMI Composition (weight %)

Heat Treatment 1300°C/3h/air + 1100 o C/5h/air + 870°C/16h/air

35

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 35-41. © 1990 Kluwer Academic Publishers.

36

2. EXPERIMENTAL PROCEDURES : This investigation covered the AMl single-crystal nickel base alloy, jOintly patented by several French Laboratories (2) and developed by SNECMA (3) to produce turbine blades for advanced engines. The composition range in weight percent is given in table 1, as the reference heat treatment. The monotonous and cyclic mechanical behaviors were characterized between the ambiant temperature and 1100 0 C in the elastiC, plastic (tensile) and creep ranges from tests made along basic orientations of the stereographic triangle :<001>, <011> and <111>. Test pieces were machined from single crystal bars cast in production conditions, i.e. identical to those for actual parts. These test specimens were submitted to the heat treatment applied to blades. This treatment was selected (4) because it gives the alloy an excellent resistance to high creep due to an homogeneous distribution of cuboid '(' precipitates of a size about 0,45 )Ull, while being compatible with the vapor phase aluminizing process developed (5) by SNECMA. In parallel with mechanical tests, the mechanisms were analyzed using classical X-ray Laue diffraction and optical micrography techniques.

3. TEST RESULTS 3.1. Tensile tests According to monotonous tensile test data, the best mechanical resistance was obtained when the stress was applied along <001> whatever the temperature (fig.l). Similar observations have been performed on CMS-X2 (6) and PWA1480 (7). The examination of test piece surfaces showed the operating slip systems: octahedral for stresses along <001> at all temperatures and along <011> up to approximately 800°C, cubic along <111> at all temperatures and along <011> above 800°C, as shown in figure 2. On other similar nickel based single crystals (6 to 8), octahedral slip system ~111S <011> have already been observed for <001> and <011> load axis, as cubic slip system(OOl\<Oll> for <111> load direction. At high temperature, for <011> tensile loading, cubic system activation has been unambiguously confirmed by microscopical investigation on spGcimen loaded at 950°C (fig.2b) coupled with X-rays Laue plane determination on a slice which has been cut parallel to observed slip lines. This slip system has already been evidenced in C.F.C. alloys (9). A noticeable strain rate sensitivity at 950°C is observed on this alloy (fig.3) along <001> and <111> load directions.

3.2. Creep tests In creep, the highest strength was consistently obtained along a tensile stress axis of the <001> type. One the other hand, the <111> direction was always the weakest, <011> being in an interme­diate situation with a strength near that of the <001> axis at 950°C (fig. 4). Quantitatively identical data were recorded with similar single crystal superalloys, e.g. PWA1480 or CMS-X2 (10). On the other hand, data obtained from first generation nickel base single crystals (MARM200, MARM 247) showed a different strength ranking of crystal orientations (11).

3.3. Cyclic tests In controlled strain LCF tests (Re = -1), the cyclic behavior anisotropy appears on mid-life cyclic stress-strain curves at 950°C (fig.5) : the <001> orientation exhibits a quite different strain hardening as compared to the other orientations. At 650°C, a plastic instability is observed (fig.6), especially for easy glide favourable orientations «011> and <123», in which case no cyclic curve has been plotted. Quite similar results have been recorded on CMS-X2 superalloy (12).

4. MECHANICAL BEHAVIOR ANALYSIS 4.1. Monotonic behavior The classical Schmid law (13) has been applied on the previously determined (83.1) slip systems. As shown in figure 7, an adequate global description of the tensile monotonic behavior, with limited plastiC strain (less than 2%), is provided by this criterion, assuming almost identical values of critical resolved shear stresses on cubic and octahedral planes. Coexistence of both activated systems at high temperature on <011> loaded specimens is then confirmed (as observed) on account of close Schmid factors for both glide planes (0,35 and 0,41). At low temperatures (up to 650·C) the fully plastic strain (up to 2%) is correlated to plastiC instability (see cyclic tests) due to a strong deformation heterogeneity (fig.2) : such a behavior is imputed to increasing propagation of shear bands (13). The influence of viscosity appears above BOOoC. The saturation stress at 950·C noticeably increases with strain rate, as usually observed in C.F.C. materials (13). Nevertheless, the <001> loading direction exhibits a much more important strain hardening than the other ones, due to probable multiple slip. In creep, discrepancies between <001> and <111> - <011> are observed (fig.4) at "low" (750°C) and high (1100·C) temperatures: along <001> an important primary creep deformation, followed by a flat steady creep period is recorded. At 950°C, the <011> direction shows the lowest secondary creep rate. The creep deformation mechanisms are not clearly determined (10). {1113 <011> and {1121<011> glide systems may operate, but a strong effect of y'morphologies (which change during creep) may influence the viscoplastic behavior.

4.2. Cyclic behavior Two types of cyclic behavior can be observed, depending on the test temperature: - purely plastiC behavior at 650·C, with a pronounced mechanical instability, reflecting a very sharply localized strain as for tensile tests. Anisotropy is hardly noticeable, except for the flow stress level which is always maximum for a load applied along <001>, - viscoplastic at 950·C, with a strong work-hardening anisotropy, the <001> direction being distinguished from the others by a very high degree of work hardening, as for monotonic tests.

37

38

The viscosity is enhanced by frequency change during testing: the largest stress amplitudes are recorded for the highest frequencies (fig.8), as classically observed. The strain hardening seems to be correlated to the crystallographic symetry of the loading axis: a load along <001> (8 slip systems) provides more strain hardening than along <011> (4 systems) and <111> (3 systems). However, these macroscopic observations and the activated microscopic phenomena remains to be correlated to each other.

5. MECHANICAL MODELING Apart from elastic and monotonous plastic changes limited to low distortions capable of being dealt with by simple models, the modeling of anisotropic cyclic viscoplastic behavior is much more difficult (14) : - application of a crystallographic model, where viscoplastic laws are assumed to apply to active systems, with interaction between these systems (15, 16), - use of a global mechanical concept, based on a thermodynamical approach such as the ONERA approach (15). The first model assumes that the involved mechanisms and their quantification are known in detail, which is not straight forward from the experimental point of view. The second model, already used for equiaxial grain alloys (17) and applied in a finite element code to turbine blade stressing (18), was identified for a blade airfoil stressed along the crystal growth axis <001>. The law is expressed as follows:

. I" ("' " cp:(=,a--X -R> wi\-h <l.L7::LA. ir u.70 , 0 if u.~o K

"I'I,K,R,CL,C · . .,...,,,,01 .. 1

X: inre<no.l ~t"re.ss

The coefficients nand k (viscosity), R (isotropic hardening) and a and c (kinematic hardening) have been identified at 650, 800, 950 and 1100 0 C. The application on tests specimens shows that the observed variations of stress and strain are quantitatively satis­factorily described (fig.9).

6. CONCLUSION The anisotropic mechanical behavior of the single crystal superal­loy AMl is qualitatively close to that of single crystals of a similar type (CMS-X2, PWA 1480, Rene N4). The strain mechanisms involved are fairly well identified from the cristallographic point of view in the case of a monotonous plastic loading, but need to be clarified for cyclic or viscoplastic loading. These elementary mechanisms must be known when developing accurate mechanical models describing the anisotropy ot these behaviors. However, the initial models developed by SNECMA are adequate to stress components in a satisfactory manner.

YS

ACKNOWLEDGEMENTS: This paper is published by permission of "Direction des Recherches, Etudes et Techniques" of the "Ministere de la Defense" who partially financed the study. REFERENCES : 1. R.L. Dreshfield, Metal Progress 8 (1986) 43 2. French Patent SNECMA, ON ERA , ECOLE DES MINES, IMPHY SA n083-20986(1983) 3. Y. Honnorat; Colloque "Alliages monocr istallins" Villard-de-Lans (1986) 4. E. Bachelet et G. Lamanthe Id 3 5. G. Gauje and R. Morbioli, Proceedings of AiME Conf. Atlanta (1983) o:-P. Caron and T. Khan, Proceedings of EUROPE ASM conference Par is (1987) 59 ~ Shah and D.N. Duhl; Superalloys 84 (1984) 105 8. R.V. Miner, R.C; Voigt, J. Gayda and T.P. Gabb, Met. Trans. 17a (1986) 491 9. J. Bonneville and B. EScaig, Acta Met. 27 (1979) 1477 10. P. Caron, Y. Ohta, Y.G. Nakagawa and T. Khan, Superalloys 88 (1988) 2'-5 11. R. Mc Kay, R.L. Dreshfield and R. Mayer, Superalloys 80 (1980) 385 12. D. Nouailhas and P. Poubanne, R.T. ONERA 68/1765RY (1988) 13. C.N. Reid, Deformation geometry fOr materialS scientists -Pergamon Press (1973) 14. E. Jordan, Proceedings of MECAMAT - Besan20n (1988) IV 239 15. G. Cailletaud, D. Nouailhas and P. Poubanne Id 14, IV 257 16. L.T. Dame and D.C. Stouffer, J. Appl. Mech. 55 (1988) 325 17. J.L. Chaboche, Int. J. Plasticity (1988) to be published 18. B. Dambrine and J.P. Mascarell - Rech. Aerosp.l (1988) 35

MPa

<111>

:::: :~;<:\ :L, 800 ~ .~. <001> <O~1> 600 ,

400

200 " e"c 100 600 700 800 900 1000 1100 1200

Fig. 1 : Yield stress of AM1

single crystal

a·20"C b·950"C

Fig. 2 : Slip lines on tensile

AM1 specimens

39

40

a (MPa)

fT = 1,110- 4 fT = 410- 4 r (0\=410- 5

.--- i, 810- 6

AMi· 950°C AMi·950"C AM1 - 950 Q C <001 > <011 > <111>

,p ! , , I II , !

00100 00200 00100 00200 00100 00200

Fig. 3 : Strain rate sensitivity of AM1 at 950°C (tensile loading)

f% E%

3 4 6 810 2 3 4 6 8 10~ 3 4 6 8 10 3 4 6 8102

e '" 750°C a = 750 MPa o = 1100°C (J = 120 MPa

1 0 f% _____ '" • • <111 > , .. /'" I I

/'" • ./ I .. •

/ ./ / • I / / <001 >

I • heures

3 4 6 810 3 4 6 8 102 3 4 6 B 103

(J = 950"C o = 240 MPa

Fig. 4: Creep curves and rupture lives of AM1 single crystal at 750, 950 and 1100°C

10001 T

500

(MPa)

T •

AM1 - 950"C - Cychc curves

• • /',

T ~~~;

loo_1~ ____________ " ______________ TI ____ • __ <_11_'_> ____ ~,,1 10- 5 10-4 10-3 10 2

Fig. 5 : Cyclic stress strain curves of AM1 at 950°C

600

- 600

1000

500

~c: (MPa)

t' .-

10- 5

AMi - 650"C <001:::.

10--4 10- 3

&. Cyclic - MonotOnic

1 10-2 10- 5

AMl - 650 c C <111>

.. ...------=----

1 10- 4

1 10- 3

1 10- 2

..lE pf~

1 10- 1

Fig. 6 : Cyclic ( ... ) and monotonic (~) stress strain curves of AM1 at 650·C

825

920 960

A-H--'F'-"--'CL.>-1000

985 975

825

910 <001 >

<111 >

f) '( Booce

<011 > o ) 950"C

Iso yield stress prediction Experimental yield stress Slip systems prediction

a (MPa)

- 0002

Fig. 7: Application of Schmid criterion to tensile monotonic behaviour

500

- 500

" (MPa)

__ 1'2 = 610-2 HZ ___ >'1 ::: 310- 3 HZ

.---, , , I

/ I

I 'I AMl <011>

950<>C 6.~T = 1,2%

'p

'-_ -':0-:0':c05:!--'~-CO:-CO~0-!0-'-..L-0:-.==00::'5 .... .n.'_--'-~-"~L05.L->-'--'0"".0J...00-'-' .... ''--'-, .... ~'-. .J.~o..J~'-J, L~-,-~...,~L05L' ..JI.....Ll -'0 ..... 0'-:10-'0 .... ILLI ..J~""OLJO-'~....JI

Fig. 8: Frequency influence on AM1 cyclic loops at 950·C (RE = - 1)

v1 = 310-3 HZ - v2 = 610-2 HZ

• Exp -Calc (J (MPa)

0 •

500 • AM1 <001 > AMl <001 >

950"C 950"C AMl <001 > .6.£r::: 1% !J.tT'" 1,7% 950"C

~'=OO33HZ v=OO33HZ ET = B 10-6

, J .p

0002 - 0 003 0003 001

Fig. 9 : Correlation between experimental (-) and calculated (-) stress strain curves at 950·C for cyclic and monotonic loadings

41

'p

THE CYCLIC DEFORMATION OF PWA 1480 SINGLE CRYSTALS AS A FUNCTION OF TEMPERATURE. STRAIN RATE AND ORIENTATION

Walter W. Milligan 1 and Stephen D. Antolovich 2

~ormerlY Graduate Student. Mechanical Properties Research Lab, School of Materials Engineering Georgia Institute of Technology Atlanta, GA 30332-0245

Currently Assistant Professor of Metallurgical Engineering, Michigan Technological University Houghton, Michigan 49931

~irector, Mechanical Properties Research Lab, Professor and Director, School of Materials Engineering, Georgia Institute of Technology Atlanta, GA 30332-0245

ABSTRACT: The cyclic deformation mechanisms of PWA 1480 are identified as a function of temperature and strain rate. It is shown that these mechanisms are vastly different as the test conditions change. At low temperatures shearing of precipitates, formation of faults, and cube cross slip are all important features. At high temperatures there is no asymmetry in deformation and the primary rate-controlling mechanism is climb. Existing constitutive models were evaluated in light of the physical findings. It was found that the Walker-Jordan model can be modified to incorporate the documented physical processes.

1. INTRODUCTION: PWA 1480 is a single crystal nickel-base superalloy which is used for turbine blades in gas turbine engines. Since the material is anisotropic in the single crystal form, neither the deformation behavior nor the consti tutive behavior of the allow are completely understood. The goals of this project were two fold: first, to further the understanding of the deformation behavior from a fundamental point of view; and second, to study current anisotropic constitutive models in light of this knowledge.

2. EXPERIMENTAL APPROACH: The alloy, described in detail elsewhere [1], is a single crystal nickel-base alloy and contains about 60% of the strengthening y' phase. Crystals with tensile axis orientations within 10· of <001> and within 10· of <123> were studied. Fully reversed, strain controlled LCF tests were conducted in the range from 20 to 1093°C, at

43

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 43-48. © 1990 Kluwer Academic Publishers.

44

strain rates of O. 5%/min and 50%/min. Deformation substructures were studied by transmission electron microscopy. The experimental details, as well as a complete set of results, are given elsewhere [2-4].

3. RESULTS AND DISCUSSION 3.1 Low Temperature Behavior

Below 800·C, the mechanical behavior of the allow is anisotropic. The material exhibits a sharp yield which is also a function of orientation. An example is shown in Figure 1, which contains plots of the yield strength in tension and compression for the two orientations. The <001> crystals were stronger in tension than compression, while the opposite was true for the <123> crystals. This behavior is perfectly compatible with the models of the monotonic strength of pure r' single crystals which are based on cube cross-slip [5].

Deformation of the alloy in this temperature range proceeded by shearing of the r' precipitates on {Ill} planes. However, it was found that the character of the dislocation debris on the active octahedral slip plane was a strong function of temperature. In particular, a high density of superlattice-intrinsic stacking faults (S-ISF's) and partial dislocations were observed after deformation at 20 and 200·C, while only unit dislocations were observed within the r' after deformation between 400 and 705°C. This reduction in the stacking fault density corresponded exactly with the unique reduction in the yield strength which is observed in this alloy in the range from 20 to 400·C. This observation has lead to a model for the monotonic yield strength of the alloy [2-4], based on a strengthening mechanism which is related to the presence of the stacking faults.

The observation of the cyclically stable tension­compression asymmetry also has profound implications for the models of the yield strength which are based on cube cross­slip. In particular. this observation requires that the cross-slip event must be ~eversible under a change in loading direction. This observation has been used as a critical test for the physical models of the cross-slip process (2-4 J • Using this criterion, it has been demonstrated that the rate­limi ting step during deformation cannot be a total cross-slip from the octahedral plane to the cube plane (2-4]. Instead, it was found that a limited cross-slip process, as proposed by Paidar et.al. [6~ is consistent with the reversibility requirement.

3.2 High Temperature Behavior At temperatures above 815 to 930°C (depending on the

strain rate), the behavior of the alloy is significantly simplified. At these temperatures, the material is isotropic in shear, does not exhibit a tension-compression asymmetry, and does not strain harden significantly. An example is shown in Figure 2, which contains plots of the yield strength in tension and compression at high temperature. No asymmetry is observed.

45

At high temperatures, deformation is accomplished by dislocation by-pass of the r' precipitates, as found earlier in the case of monotonic deformation [1]. Primary cube slip was not observed in our experiments.

3.3 Implications for Constitutive Modeling Selected anisotropic constitutive models were surveyed,

and a typical model was chosen to study the implications of our results on the model. Because of its generality, the Walker-Jordan [7] model was chosen for this purpose. The model is capable of handling all known sources of anisotropy in superalloy single crystals, as described briefly below.

The model is formulated from the crystallographic perspective, relating the plastic shear strain rates to the shear stresses via a unified viscoplastic flow rule. For example, on the octahedral slip systems, the plastic shear strain rate obeys a rule of the type

= • • • (1)

where t is the plastic strain rate, " is the shear stress, Q is the back stress state variable, and K is the drag stress state variable, all on the ith slip system. The effect of cube cross-slip is modeled in the evolution of the drag stress, which is a function of the constriction stress that promotes cross-slip:

f { t;, N, "c } • • • (2)

where N is the number of active slip planes and "c is the constriction stress which promotes cube cross-slip. The primary cube slip shear strain rate is calculated using a similar viscoplastic flow rule. The total plastic strain rate tensor is calculated by adding the contributions of the active cube and octahedral slip systems:

+ • . . (3)

This linear superposition of the plastic strains due to cube and octahedral slip systems has been used to model much of the anisotropy observed in these types of alloys.

Our results have shown that the Walker-Jordan model, in modified form, is capable of modeling the anisotropy and maintaining consistency with the observed deformation mechanisms (2-4). The major modifications are briefly outlined below.

First, our results (and the results of other studies)

46

have shown that the cube cross-slip mechanism is responsible for most of the observed anisotropy at low temperatures. Specifically, we observed significant anisotropy in strength and in the strain hardening behavior without the occurrence of primary cube slip.

Second, our results have shown that primary cube slip did not occur in near-<123> crystals at any temperature. An analysis of the data in the literature revealed that primary cube slip is alloy-dependent, occurring in PWA 1480 only in orientations very close to <111>.

Therefore, it was concluded that the constriction stress is responsible for most of the anisotropy observed at low temperatures, and the model should be modified to de­emphasize the primary cube sl ip (except near < 111 » and augment the effect of cube cross-slip. Additionally, boundary conditions were specified for different mechanisms, and several other minor suggestions were made. The full report should be consulted for the details [2,3].

4. CONCLUSIONS: PWA 1480 single crystals deform by several different mechanisms. At low temperatures, the r' precipitates are sheared on octahedral planes. The orientation dependence of the constriction stress which promotes cube cross-slip results in highly anisotropic behavior. This behavior can be incorporated into current anisotropic constitutive models.

At high temperatures, deformation occurs by r' by-pass. The homogeneous nature of this process results in isotropic mechanical properties. As a result, the constitutive equations which are necessary at high temperatures are also simplified.

5. ACKNOWLEDGEMENT: This research was funded by NASA-Lewis Research Center, under grant NAG3-503, which was monitored by Mr. Michael Verrilli and Dr. Robert C. Bill.

REFERENCES: 1. W.W. Milligan and S.D. Antolovich, Met,_ Tran~_~, 18A, 1987, p. 85. 2. W.W. Milligan, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA 1988. 3. W.W. Milligan and S.D. Antolovich, NASA CR-4215, 1988. 4. W.W. Milligan and S.D. Antolovich, submitted to ~et-,­

Tra!!~~ 5. D.P. Pope and S.S. Ezz, IntI. Metals_Rev~, 29, 1984, p. 136. 6. V. Paidar, D.P. Pope and V. Vitek, Acta~et~, 32, 1984, p. 435. 7. K.P. Walker and E. Jordan, In~_. of Fatigue_.Q.t_~_l1g_,_ MatIs. and Structyres, in press.

Figure 1.

500

eCce 0 0 c C ~l>l>l>.:l. l> .:l. .:l. .:l.

400

til 300 0.. :::;z ..

ii 200 f-

100

0= TeDilion A =- Compression

a a 5 10 15 20

Cy.cle

( a)

500

400

l>.:l. Co Co Co l> t.-

o 0

&: cCccc

300 :::;z

-M

• a 200 ...

100

a a

Octahedral (a) F40-1, <123>, ilr

p

o - TeIUllOIl

.:l. - Comprea8toll

5 10 15 20 Cycle

(b)

~IIiIX vs cycle. 705°C, 50lt/min. <001>, ilr = O.04lt. (b) F11, = 0.04lt. P

47

48

500

400

~ 300 ::::s

-.. . S

200 f-

aa~~e e 100

a-Tension l> ~ Compre •• iol1

0 0 5 10 15 20. 25-

Cycle

(a)

500

400

~ 300 ::::s

-.. . a 200 f-

aaaiillil

100

(J - TeDaioI1 l> - Compre.aiol1

0 0 5 10 15 20· 25

Cycle

(b)

Figure 2. Octahedral 1: max vs cycle. 927°C, 0.5lIUmin. ( a) F104-1, <001>, AT = 0.04%. (b) F13, <123>, = AT 0.05%. P

P

ANISOTROPIC MECHANICAL BEHAVIOR MODELING OF A NICKEL-BASE SINGLE CRYSTAL SUPERALLOY

P. Poubanne*

Office National d'Etudes et de Recherches Aerospatiales BP 72 - 92322 Chatillon, France

ABSTRACT: Various monotonic and cyclic tests are performed on the four crystallographic orientations <001>, <011>, <Ill> and <123> of the single crystal AMI at 950°C. Main features of the mechanical behavior are discussed below. Thanks to Transmission Electronic Microscope (TEM), microscopic mechanisms accountable for the plastic flow and hardening are investigated. Constitutive laws on slip systems may take into account main effects of these mechanisms through internal variables concept. Such a microphenomenological modeling can simulate uniaxial tests with good agreements.

1. INTRODUCTION: Progresses performed in the elaboration methods for new materials, allow the development of specific materials for given engineering applications. Thus, the use of nickel base single crystal superalloys for the realization of turbine blades, leads to the improvement of engine performances. Nevertheless, the design of engine components together with lifetime predictions need the development of constitutive equations accounting for the anisotropy inherent in the cristallographic structure of these materials.

2. EXPERIMENTAL PROCEDURES: Information on chemical composition, metallurgical microstructure or heat treatment on the material are repotted elsewhere (1). Monotonic tests are performed on a 20 mm cylindrical gage length, 3 mm in diameter samples, with a screw-driven testing machine at a prescribed displacement rate. Low cycle fatigue tests use a servo hydraulic machine, and 4.85 mm in diameter specimens are loaded under strain control Re = -1, with a triangular waveform-signal.

3. TEST RESULTS: Only the inelastic behavior is considered, so that experimental results are shown in a a - cP representation. The main results are: - mechanical behavior is very material orientation dependent; - at 950°C, it is also strongly strain-rate dependent, whatever be the orientation as shown in reference (1); - both monotonic and cyclic tests show the highest hardening of < 001 > orientation, compared to the others (see figures 1 and 2). We note the surprisingly low level of the <Ill> curve;

* Now at SNECMA - VILLAROCHE, France (DIvIsion Mlkanoque)

49

A. S. Krausz et al. (eds.!. Constitutive Laws of Plastic Deformation and Fracture, 49-55. © 1990 Kluwer Academic Publishers.

so

800 0 (MPa)

<001>

600 <011>

<123>

400 <111>

200

o 0.5 1.5 2

Figure 1. Comparison of the monotontc tensile curves for four crystallographic orientations on AM 1 at 950"C (displacement rate'" 0.2 mm/mn).

1.1012 (MPa)

600

500

400

300 I o <001>

Orientation x <011> * <111> + <123>

200

L 0 0.1

I.1Epl2 (%) I I

0:4 • 0.2 0.3

Figure 2. Comparison of the cye/lc hardening curves for four crystallographic orientation (period 30 s).

- a reversal loading after the first quarter of cycle reveals a very fast vanishing of the elastic domain, as show in figure 3. Therefore the hysteresis loops have a pronounced kinematic character. No noticeable mean stress is observed in the performed tests; - for a given strain amplitude, the stabilized hysteresis loop is reached very quickly (within 10 cycles): no significant cyclic hardening is observed.

1110MPa

Figure 3. Illustration of cyclic stress-stram curve of AM7 <723> at 950"C. (L1er= 7.95%, T =376 s). The initial elastic domam (a) vanishes to (b) at the first quarter cycle.

4. MICROSCOPICAL MECHANISMS: It is now well-known that a FCC single crystal flows by dislocations gliding on one or more of the 12 (= N) crystallographic octahedral slip systems (2), each of them being defined by Iig normal to the slip plane, and bg the slip direction (g= 1, N).

T~ TEM observations on tested specimens allow the determination of rig and bg of a given dislocation, which gives information on the activated slip system nature. Furthermore, since mechanisms which prevent slip are the cause of hardening, and conversely those which keep it soften the material, the observations may provide understanding in the global mechanical response, and can be summarized as follows: - plastic strain is accommodated by two slip systems iamilies: octahedral and cubic. Activation ofthe last one may explain the < 111 > specimens low level hardening. - dislocations are always inside the matrix ofy phase, which lies around the precipitates ofy' phase (see Figure 4). Therefore precipitates act as trap for dislocation slip, which causes hardening.

O.8l;!m

Figure 4. AMI <017> 950·C cycliC hardening. Dislocations are only In the

matrix of y phase.

Two main softening mechanisms are then observed: - OROWAN by-passing of the precipitates, which is possible only when the local resolved shear stress "{;g exceeds a critical value (see Figure 5) (3). A reversal loadin~ leads to a kinematic hardening effect, because reversal glide of dislocation is easier for line energy consideration (4); - cross-slip: if glide of a screw dislocation is disturbed by an obstacle, it can change of slip plane, to the cross-slip plane if this allows the dislocation to avoid the preCIpitate (see Figure 6) (3). This mechanism was observed in every specimen. It is probably the cause of elastic domain vanishing.

51

52

However, it occurs preferentially in rather far <001> oriented specimens, because cube planes may, in these conditions, be an additional cross-slip plane which has furthermore advantage not to cross the precipitate. This is not the case for <001> orientation. This accounts for the large differences between < 001> and other orientations inelastic behavior.

VB "'pp""' PreCIpitate of 'Y' phase

(2)

~=(§~ V M:~:"

Figure 5. II/urtration of the ORO WAN by·passing mechanism. 1 - AMI < 123> 950"C cyclic hardening.

2 - When dislocation by-passes precipitates, it increases its length from (1) to (2). A reversal load leads to a reversal dislocation glide from (2) to (I), which is easier for energetic reasons. This phenomenon induces

a kinematic effect.

Figure 6. II/urtration of the cross-slip mechanism I -AMI < 123> 950"Ccyclic hardening: cross-slip of the dislocation from the slip syrtem a:

f <011> (ITI)to the cross slip syrtem b: f <01 I> ('-i1).

...... --. /' "­, Obstacle ;'

......

2 . Screw dislocation (1) cross·slips on the cross-lip plane in order to avoid the obstacle {(2) and (3»). and then can cross·slip again on a parallel initial slip plane (4).

5. MICROPHENOMENOLOGICAL MODELING: The used model was proposed by Cailletaud (5). It introduces the crystallographic aspect of slip on both the octahedral and cube planes, as observed in experiments. The constitutive equations are expressed at the slip system level, in order to take into account the main effects of microscopical mechanisms.

The transit from macrosco'pical level to microscopical one, and conversely, is made classically by Schmid law (eqn. 1), and equation 2 (6):

(1)

N

(2) .p L Rg .g c'" = y

" iJ g=l

N = number of potential slip systems.

The constitutive law is then a relation between the resolved shear stress .g on the system g, and the viscoplastic shear strain rate yg on the same system. For each slip system, two hardening variables are used. The first one represents isotropic hardening (expansion of the elastic domain: rg), the second kinematic hardening (translation: xg). These two variables include respectively the cross-slip softening, and the reversal loading effects.

The evolution laws during the deformation process are given as follows:

dxg=c(a 4>(vg)dyg -x~d'll with 4>(v g)=4>:"+(1- 4>:"lexp(-o.vg) J J J J J

N

d~ = L h. H(g,s) Q. exp (- h. vS) Idys I with J J J

s=1

VSm = C I yB Idt

where j takes the value 1 or 2 respectively for octahedral and cubic slips, which means that two families ofmaterial coefficients are needed.

The important points are: - a non linear kinematic evolution, with accumulated strain history effect, is used for Xg; - the term H(g,s) of the "interaction matrix" allows to introduce the cross influence of the slip of the system s on the isotropic hardening of the system g. Diagonal coefficients describe the self-hardening which can be an hardening or softening effect according to the Qj value (Qj > 0 = hardening, Qj < 0 = softening). Cross-hardening effect is taken into account through h ( for weak interactions) and q (strong interactions) coefficients, which give finally the following 18 x 18 interaction matrix expression:

1 h h h 6 q h 6 q h q q 0 0 0 0 0 0 h 1 h q 6 q 6 q h 6 0 0 0 0 0 0 h h 1 q q

~ q q q 0 0 0 0 0 0 h

~ q 1 h h q q h

~ h 0 0 0 0 0 0

q ~

h 1 h q h ~

q ~

0 0 0 0 0 0

~ q h h 1 q ~ ~

q 0 0 0 0 0 0

~ q h

~ q 1 h ~

q 0 0 0 0 0 0 q

~ q

~ h 1 h q q 0 0 0 0 0 0

~ q ~ q h h 1 q

~ h 0 0 0 0 0 0

H(g,s) ~ q

~ q h ~

q 1 h 0 0 0 0 0 0 q

~ q

~ q

~ h 1 h 0 0 0 0 0 0

q q h q q q h h 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 h q q q q 0 0 0 0 0 0 0 0 0 0 0 0 h 1 q

~ q q 0 0 0 0 0 0 0 0 0 0 0 0 q q 1 q q 0 0 0 0 0 0 0 0 0 0 0 0 q q h 1 q q 0 0 0 0 0 0 0 0 0 0 0 0 q q q q 1 h 0 0 0 0 0 0 0 0 0 0 0 0 q q q q h 1

53

54

In a viscous framework, shear strain rate is given for each slip system by (5):

Material coefficients are then determined thanks to both experimental data and calculations.

According to single crystal AM! at 950°C, this model can simulate fairly well various uniaxial tests as shown below:

+++ : Tests -- : Calculation

<001>LlET=1.7% T=30s <011> LlET = 1 % T '" 30 s

T

<111> LlET = 1 % T = 30 s

+

• • 2000 MPa

-Ep = 0.02

<123> LlET=2% T=376s

+

Figure 7. Comparison between predIcted and experimental hysteresis loops for four crystallographIc OrientatIons of AMI at 950"C.

6. CONCLUSION: This micro phenomenological model, which takes into account main effects of inelastic strain microscopical mechanisms seems simple enough to be introduced in a F.E. code without too much difficulties, but is flexible enough to simulate successfully complex mechanical response of single crystal. Such modeling should be applicable in an industrial context, as SNECMA, in a new future.

ACKNOWLEDGEMENTS: This paper is a part of author's thesis (7) made at Office National d'Etudes et de Recherches Aerospatiales (ONERA) in cooperation with Ecole des Mines de Paris, with the financial support of Societe N ationale d'Etudes et de Construction de Moteurs d'A viation (SNECMA).

REFERENCES 1. J.Y. Guedou, Y. Honnorat, Mechanical behavior modeling of a Nickel­base single crystal superalloy. Proceedings of this Conference, Ottawa, 1989. 2. J.F. Nye, Proprietes physiques des cristaux. Dunod. 3. L.M. Brown and R.K. Ham, in Strengthening Methods in Crystals, edited by A. Kelly and R.B. Nicholson (Applied Sciences, 1971), p.9. 4. J. Friedel, Dislocations. Int. Series of Monographs and Solid State Ph~sics, Vol. 3, Pergamon Press, 1964. 5.. Cailletaud, Une approche micromecanique phenomenologique du comportement inelastique des metaux. These de Doctorat d'etat, Universite Paris 6,1987. 6. A. Zaoui, Comportement des materiaux. Cours de l'ENSTA. 7. P. Poubanne, Etude et modelisation du comportement mecanique d'un monocristal en superalliage pour aube de tur.bine. Nouvelle these, 1989 (To appear).

55

THE MECHANICAL PROCESSES OF THERMAL FATIGUE DEGRADATION IN IN-IOO SUPERALLOY

N.J. Marchand*, W. Dorner** and B. Ilschner***

* NSERC University Research Fellow, Depart. Genie Metallurgique, Ecole Poly technique , P.O. Box 6079, Station "A", Montreal, Canada, H3C 3A7.

** Project Engineer, Motoren and Turbinen Union (MTU) , D9000, Munich 50, Germany.

*** Professor, Departement des Materiaux, Ecole Poly technique Federale de Lausanne, CH-I007, Lausanne, Switzerland.

ABSTRACT: The effect of thermal fatigue history on the surface degrada­tion of IN-IOO nickel-base alloy was examined employing double-edge wedge specimens and a special induction heating procedure. . The respectives stress-strain histories, as determined from thermo-elasto­plastic finite element analyses, are presented. Depending upon strain history, two modes of surface degradation were observed: scalloping and through-thickness cracking of a uniform oxide layer. The degree of scalloping was shown to depend on the magnitude of compressive strain at the surface. Severe scalloping was observed after 3000 thermal cycles between peak strains of -0.48% at 1050· C and +0.08% at 400· C. More than 3000 cycles between peak strains of -0.24% at 1050· C and 0.23% at 400· C did not produced scalloping. The number of cycle to crack initiation was found to correlate with peak compressive strain. The findings are shown to be consistent with a mechanism for scallop initiation and growth involving cyclic oxide cracking and cyclic ratchetting. The implementation of the results in the operating and lifing procedures of commercial aircraft engines parts is discussed.

1. INTRODUCTION: During normal operation, turbine blade materials in aircraft turbine engines are subjected to severe cyclic thermal stresses in a highly oxidizing environment. This combination of factors lead to the development of fatigue cracks which must be controlled or reduced by proper material selection and blade design. The importance of oxidation in the high temperature fatigue process of superalloys has been recognized for some time and, in equiaxed alloys, various mecha­nisms and models [1-4) have been proposed to describe the oxidation­assisted grain boundary cracking process which dominates in these mate­rials. One of the key problem in setting up such models, is the assess-

57 A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 57-66. © 1990 Kluwer Academic Publishers.

58

ment of the degree of interaction between oxidation kinetics and cyclic straining.

At the present time crack initiation and oxidation resistances of mate­rials are estimated from simple isothermal low cycle fatigue tests which are carried-out in laboratory air. This is unfortunate because it has been known for years [5-6] that thermal cycling accelerates oxida­tion by some process of spalling of the protective oxide scales. In thermal cycling the spalling is induced in part by thermal expansion mismatch of the oxide and the substrate, and part from mechanical strains which result from the constraint to thermal expansion of the substrate. On the other hand, in isothermal LCF, the mechanical cycling alone drives the spalling. It remains to be demonstrated that the kinetics of combined oxidation and cycling by thermal and by isothermal straining are the same and that the isothermal LCF data can be safely extrapolated to thermal fatigue cycling.

The purpose of the present paper is to report the results of a quanti­tative study of surface degradation and cracking in thermal fatigue cycling. The influence of thermal fatigue on oxidation kinetics will be emphasized by examining surface degradation in an oxidizing environment in relation to the temperature and strain history experienced by the near surface volume.

2. MATERIAL AND EXPERIMENTAL PROCEDURE: The composition of the master heat used in this study was (in weight percent): 10.3Cr-14.7Co-3.l5Mo -5.68Al-4.6Ti-l.OlV-0.lSC-bal.Ni. The minor elements were 0.014 B, 0.060 Zr and less than 10 ppm each of S, 0 and N. The alloy was cast into plates 20 x 20 cm2 and 15 mm thick. The average grain size was 2.5 mm and the average secondary dendrite size about 0.15 mm. Both measure­ments were taken as the mean intercept length using quantitative metal­lography on a large number of specimens. Matrix dendrites were consti­tuted by a nickel-rich face-centered cubic matrix strengthened by about 0.60 volume fraction of ~' phase (Ni3Ti-Al). Massive eutectic ~'

nodules 20 to SO ~m in size were observed in interdendritic areas as well as MC carbides rich in titanium and molybdenum and about 50 ~m in size.

For the present study, double-edge wedge specimens were used with edge radii typical of trailing and leading edges of gas turbine airfoils. This allows modelling of the 5 to 8s heating and 6 to 30s cooling experienced by the edges of internally cooled gas turbine airfoils [7]. High frequency induction heating of the specimen periphery followed by high velocity forced-air cooling, was used to produce cyclic thermal strains in the near surface volume. Details of the induction-coil and cooling manifold designs as well as those regarding the installation, testing procedure and data recording, are given in references [8] and [9]. The magnitude of transient compressive strains, developed along the specimen periphery during heating, were controlled by varying the heating rate of the specimen. Transient tensile strains, developed dur­ing specimen cooldown, were controlled by adjusting the flow rate of cooling air to the specimen periphery [8-9].

To determine the effect of isothermal exposure on surface degradation,

59

and to provide a reference to compare with surface degradation obtained as a result of cyclic thermal strains, two specimens were isothermally oxidized in air for 50 hours at 1000°C. This time-length represents the sum of 3000 hold-times of 60s each used in the fatigue tests.

Specimen heating and cooling rates were systematically varied between temperature limits of 400 and 1000°C to determine the effect of strain history on surface degradation. Three thermal histories were studied all with minimum and maximum temperatures of 400 and 1000°C, and a 60s hold at 1000°C. The three types of thermal cyclic histories are labelled FHF (5s heating, 60s hold time, and l5s cooling), SHF (60s heating, 60s hold time, and l5s cooling), and FHS (5s heating, 60s hold time, and 60s cooling). The rapid heating of FHF and slow heating of SHF were included to examine how the magnitude of the compressive edge strains, encountered during specimen heat-up, affects crack initiation. The fast cooling of FHF and the slow cooling of FHS histories were included to show the effect of tensile strains on thermal fatigue cracking.

An advanced A.C. potential drop technique was used to monitor surface degradation and microcracking processes. All the details concerning the ACPD system, connection of the probes and calibration procedure, are given elsewhere [8-9]. For the particular geometry of specimen, the initiation of surface (or subsurface microcracks) about 20 ~m in length could be detected [9]. All the specimens experienced 3000 thermal cycles (either FHF, SHF or FHS).

After completion of testing, the specimens were examined using optical and SEM microscopy. After examination, the specimens were nickel-plated and then polished parallel to the their faces, down to the midspan plane. Photographs of the surface were taken every 120 ~m. This proce­dure demonstrated that subsurface cracks often initiate first (depend­ing on thermal cycle type) and that the ACPD system provided a more reliable definition of number of cycles to crack initiation than sur­face inspection (optical or SEM) [8]. The concentration of the major elements in or near the oxide layers were determined by micropobe ana­lysis using a Cambridge 250 Electron microprobe. Auger Electron Spec­troscopy (AES)-Perkin Elmer model CMA 590-was also used to quantify the chemical composition of the main elements.

3. STRESS AND STRAIN HISTORY OF TEST SPECIMENS: The stress-strain his-using thermo-elasto­

concerning the finite­inputs for the ana­

the finite element ana­histories at critical

tories for each thermal history were determined plastic finite element analyses. All the details element model, the methods of analysis, and the lyses, are given elsewhere [10]. The results of lyses are summarized in table 1 where the a-£-T locations for FHF, SHF, and FHS are presented.

The finite element analyses revealed [10] that ratchetting plastic strains caused the hysteresis loops to shift under cycling with FHF experiencing the greatest and SHF the least shifting. It is important to note that the peak compressive strains ranked (in increasing order) as FHF, FHS, and SHF, whereas the peak tensile strains ranked as SHF, FHF and FHS.

60

Table 1. Leading edge (critical location) heating and cooling times and peak surface strain history.

Tem~erature history (in s)

~ Heating time Cooling

FHF 5 SHF 60 FHS 5

4. RESULTS AND DISCUSSION: 4.1 Isothermal ex~osure

15 15 60

time

Peak strain (in %)

Com~ression Tension Range

-0.475 0.075 0.55 -0.235 0.225 0.46 -0.445 0.025 0.47

Observation of the isothermally exposed IN-100 specimens showed differ­ent forms of oxides. There were also oxide spikes which penetrated inward when MC carbides intersected the free surface. This phenomenon has been previously observed in many cobalt-and nickel base superalloys [3-5]. Oxide penetration at blocky l'eutectic were also observed, pre­sumably because interdendritic areas were present in the vicinity of the observed section.

Simple observations in the scanning electron microscope showed that the morphology of the oxide matrix layer is fairly complex. A first oxide layer, with polyhedric structure, was visible at the outer specimen surface. In the inner part of the oxide scale, there was a discontinu­ous layer of oxide and an adjacent layer of matrix which was depleted of l' precipitates. The oxide layer was about 10pm thick in the flat regions of the double-edge wedge specimen whereas it was about 30pm thick at the tip of the leading edge [8]. In both cases, the outer layer was about 2pm thick of (Ni, Co, Cr)O. There was also a thick inner layer (2 to 3pm) of discontinous Alz03. The matrix adjacent to the oxide layer was shown to be depleted in aluminum to a still larger distance. AES analyses showed peaks of Ni, Cr, and Ti which coincided with peaks of oxygen as well as intermediate peaks of Al ,.;hich coin­cides with another peak of oxygen. This suggests that the intermediate layer of oxides could be composed of NiCrZ04 and NiTi0 3 spinels, in agreement with other investigations [4-5,11]. The formation of oxides which are rich in Cr, Ti, and Al explains the depletion of these ele­ments in the adjacent matrix as well as the dissolution of l'precipitates which are rich in Ti and AI.

The fact that the mean thickness of the oxide layer was much larger at the leading edge than in the flat areas of the specimen, indicates that the oxidation kinetics in IN-lOO is strongly dependent on the state of stress. This is borne out the fact that among the principal sources of growth stress during isothermal exposure, specimen geometry (finite size and the resultant curvature) plays a key role [llJ. Hence, in systems where cations are mobile, growth stresses arise in the scale

61

because the scale must relax to maintain contact with the metal as the metal atoms cross the scale metal interface to diffuse outwards. If the scale cannot relax the compressive stresses generated in the oxide as it tries to follow the metal surface that retreats, voids do form at the scale metal interface as observed experimentally[8]. On a planar interface there are no forces restraining such relaxation, but at edges and corners it is not possible for the scale to relax in both, or all three, directions. The geometry of the scale in these regions is sta­bilized and resists such relaxation. In these geometrically stabilized regions, the scale must creeps to maintain contact with the metal at a rate that is determined by the oxidation rate of the metal. The adhe­sion between scale and metal is the maximum force that can be exerted to cause the scale to creep and maintain contact. All these factors indicate that extrapolating from results on large (or flat) pieces to small pieces or pieces of different geometry, may not be easy or feas­ible. Further, the oxidation kinetics must be studied in relation with the actual strain history experienced by the near surface volume if the results or to have any value for extrapolating to actual components and service conditions.

4.2 Thermally cycled specimens

Observation of thermally cycled specimens showed two distinct types of surface degradation obtained with thermal cycling: scalloping (some­times referred to as "rumpling" or "wrinkling"), which occurred for FHF cycling (rapid heating and cooling), and through-cracking of the oxide and depleted layers which occurred for SHF cycling (slow heating and rapid cooling). Figure 1 shows examples of these two modes of surface degradation. Associated with scalloping, numerous subsurface cracks were found (fig.lb). These cracks were identified as initiating below the surface because their faces were not oxidized or depleted in Al, Cr or Ti as determined by AES. About 1800 cycles were imposed before any subsurface crack could be detected. On the other hand, cracking of the oxide and depleted layers was detected after more than 2600 cycles for SHF cycling.

FHS cycling (rapid heating-slow cooling) did not produced scalloping. Instead, uniform oxidation of the surface accompanied with penetrating oxide spikes at MC carbides occurred as shown in fig.le. However, thermal cracking of MC carbides below the surface was observed as indi­cated in fig.lf. It appears that FHS thermal fatigue damage can be viewed as an average between FHF (subsurface cracking) and SHF (uniform oxidation) damages.

After 3000 thermal cycles, the average oxide thickness (at the leading edge) varied between 45pm for FHS (-0.445% :5 Ct < 0.025%), 82pm for FHF (-0.475% :5 Ct :5 0.075%), and l50pm for SHF (0.235% :5 Ct :5 0.225%) cyc­ling. This is to be compared with the 30pm thick oxide layers obtained under isothermal exposure. Detailed microprobe and AES analyses of FHF oxide and depleted layers, revealed that the variations in chemical composition of the same wavelength as the surface scallops were not observed. This result confirms that the observed dependence of scallop and oxidation depth on strain history is primarly mechanical rather than chemical in origin. Obviously, thermally-induced straining

62

greatly enhanced matrix oxidation kinetics.

4.3 Mechanical aspects of surface degradation: scalloping

Oxide breakdown by cracking or spallation and subsequent surface roug­hening by oxidation are believed to be likely precursors to scallop formation [5,12]. This is consistent with the SEM examinations of oxides formed on FHF specimens where oxide cracking and spallation along the leading edges were found [8]. As shown by Holmes et al., oxide cracking can occur during specimen cooldown, where tensile stresses develop in the oxide due to elastic expansion of the substrate [12]. Compressive shear-cracking (or spallation) of the surface oxide is thus possible during the initial stage of specimen heating where elastic substrate compression produces compression in the oxide. Fur­ther, initial oxide breakdown would be expected to occur first above oxide grain boundaries [4,11]. The fact that scalloping was not observed in specimens thermally cycled under SHF conditions, where sig­nificant tensile stresses develop, shows that tensile cracking of the oxide acting alone is not responsible for the initial roughening that is a precursor to scallop growth. By the same token, the absence of scalloping in the specimens thermally cycled under FHS conditions, where compressive cracking can occur, shows that compressive cracking acting alone is not responsible for surface roughening.

To summarize, it appears that oxide compression acting in conjunction with oxide tension, is responsible for the initial breakdown of the protective surface oxide. Once oxide breakdown has occured, surface roughening develops by the cyclic process of surface oxidation.

The absence of surface scalloping in the FHS thermally cycled specimen (negligible tensile substrate strain and high compressive oxide stress), suggests that although initial surface roughening can occur as a result of cyclic cracking of an oxide, tensile substrate straining is required for scallop growth. It should be noted that in spite of the fact that tensile or shear cracking of the oxide does not appear to be the mechanism directly responsible for scallop growth, strain intensi­fication at the root of a surface scallop is likely to enhance oxide cracking thus making a contribution to growth. It is likely that oxide cracking acts in parallel with other mechanisms to further scallop growth.

In analogy with the continued surface roughening mechanism proposed by McClintock [13] for cyclic plastic straining of materials exhibiting Bauschinger softening and linear strain hardening, kinematically irrev­ersible cyclic creep (KICC) is consistent with the experimental obser­vations. Hence, if scallop growth proceeds by KICC, any thickness dif­ference between two adjacent regions of a surface produces a strain difference between the two regions. This strain difference increases with continued cycling due to the kinematically irreversible Bau­schinger component of creep. The increasing strain difference further increases the area difference between the deforming regions of the sur­face. The most interesting feature of this mechanism is that the increased area difference expected for increasing strain amplitude is consistent with the correlation of average scallop depth with surface

63

strain range [12]. Further, according to this mechanism, a critical strain range level must be reached for cyclic cracking or spalling of oxides [4,11,14,15].

4.4 Mechanical aspects of surface degradation: cracking

For the rapidly cooled specimens oxide cracking along the leading edge occurred when the tensile strain reached 0.225%. No oxide cracking was observed when the tensile strain developed was considerably lower (£max = 0.08%). Examination of figure ld shows that the oxide cracks are lined with AI, Ti and Cr depleted ~' (point A). Thus, one is tempted to conclude from this result that early oxide cracking is unlikely and that cracking occurs after formation of the less ductile AI-poor ~'film. However, no evidence of cracking (as measured with the ACPD system) was found until 2600 cycles [9].

In summary, the critical strain for through oxide cracking lies in the range of 0.09 < £crit < 0.16%. Oxide cracking appears to occur late in the thermal fatigue life with subsequent depletion of Al along oxide grain-boundaries, resulting in the formation of AI-depleted ~'films.

5. CONCLUSIONS

1. The oxide layers chemical composition of thermally cycled specimens was relatively insensitive to strain history.

2. Surface degradation in oxidizing atmosphere depends critically upon the surface strain history. Two modes of degradation were observed: (1) scalloping, which occurs during rapid heating and fast cooling (~£t > 0.50%), and (2) oxide cracking, which occurs during rapid cooldown of the coating (£tensile > 0.10% to 0.16%).

3. The experimental results were shown to be consistent with a mecha­nism of surface scalloping involving initial surface roughening by tensile and compressive oxide cracking, followed by scallop growth due to kinematically irreversible cyclic creep.

4. For a maximum temperature of 1050°C, surface scalloping was not observed for a peak compressive surface strain greater than -0.40% and with a strain range of 0.47%. Thus, to prevent scalloping, (the result of compressive strains generated at the surface during rapid heating), it is necessary to maintain the peak compressive surface strain above -0.40%, with the strain range held below 0.50%.

5. Surface and coating degradation due to scalloping in aircraft gas turbines can be reduced by increasing the turbine to full power at a slower rate (e.g., in 20s rather than lOs).

6. Early oxide cracking can be precluded by limiting substrate tensile strains to less than 0.09% to 0.16% (for a maximum temperature of 1050°C). As strains are encountered on turbine cooldown after thrust reverse, the pratical solution is to reduce engine speed at a slower rate after thrust reverse (a practice that is not cur­rently employed by commercial aircraft companies but could be safely implemented.

64

REFERENCES:

1. L.F. Coffin, Metallurgical Transactions, 3 (1972), 1777-1788. 2. S.D. Anto10vich, R. Baur and S. Lin, in Superalloys 1980, The

Metallurgical Society of AIME, Warrendale, Pennsylvania, (1980), 605-614.

3. J. Reuchet and L. Remy, Metallurgical Transactions, 14A (1983), 141-149.

4. M. Reger and L. Remy, Metallurgical Transactions, 19A (1988), 2259-2268.

5. G.E. Wasilewski and R.A. Rapp, in The Superalloys, Wiley, New York (1972), 287-316.

6. L. Remy, F. Rezai-Aria, M. Fran90is, C. Herman, B. Dambrine, and Y. Honnorat, "Application of Isothermal Fatigue to the Study of Ther­mal Fatigue", Final Report, F6, COST-50 Round III (1984).

7. E.D. Thulin, D.C. Howe, and I.D. Singer, "Energy Efficient Engine High Performance Turbine Detailed Design Report", NASA Report CR-165608, Pratt & Whitney Group, East Hartford, CT, 1982.

8. W. Dorner, "Eine Untersuchung der Risswachstumsmechanismen bei Thermischer Ermudung", Dip10marbeit, Universitat Er1angen-Nurnberg (1987) .

9. N.J. Marchand, W. Dorner and B. I1schner, "A Novel Procedure to Study Crack Initiation and Growth in Thermal Fatigue Testing If, in Surface Crack Growth: Models. Experiments and Structures, Reno, Nevada, April 25th (1988). To appear in an ASTM STP.

10. N.J. Marchand, W. Dorner, and B. I1schner, in The Inelastic Behav­iour of Solids: Models and Utilization, MECAMAT, Besan90n, France (1988), 427-444.

11. N. Birks and G.H. Meir, Introduction of High Temperature Oxidation of Metals, Edward Arnold press, London, England (1983).

12. J.W. Holmes and F.A. McClintock, "The Chemical and Mechanical Processes of Thermal Fatigue Degradation of an Aluminide Coating", accepted for publication in Metallurgical Transactions (to appear).

13. F.A. McClintock, in Fracture of Solids, D.C. Drucker and J.J. Gil­man, Eds., (1963), 65-102.

14. A. Rahmel, Oxidation of Metals, 2 (1970), 120-132. 15. N.J. Marchand and W. Dorner, "The Mechanical Processes of Thermal

Fatigue Damage in Gas Turbine Blades", to appear in the Proc. of the Fourth Canadian Symposium on Aerospace Structures and Materi­als, Ottawa, May 15-16 (1989).

65

66

Figure 1 Micrographs showing surface degradation after thermal cyclic exposure (3000 cycles). FHF (a and b),SHF (c and d), and FHS (e and f).

INCLUSION OF DSA MODELING CAPABILITY IN UNIFIED VISCOPLASTICITY THEORIES, WITH APPLICATION TO INCONEL 718 AT 1100 Op

N.N. EI-Hefnawy*, M.S. Abdel-Kader* and A.M. Eleiche**

*Department of Mechanical Engineering, Military Technical College, Cairo, Egypt.

**Department of Mechanical Design and Production, Faculty of Engilleering, Cairo University, Guiza, Egypt.

ABSTRACT: One of the complex inelastic behavioural phenomena that struc­tural materials may exhibit when subjected to specific thermomechanical loading histories is Dynamic Strain Ageing (DSA). The modeling of DSA within the context of unified theories of viscoplasticity is studied in this paper. In particular, an algorithm is proposed to include DSA modeling capability in Chaboche's theory. In doing so, use is made of the so-called "correction function" proposed by other investigators. A rather general form of such a function is adopted, and included in the theory in three different ways. It is found that the simplest and most appropriate way to model DSA is to introduce the correction function into the initial value of the isotropic hardening stress, Do' As such, Do is made to change with the imposed strain rate within the range of negative strain-rate sensitivity, and is held constant otherwise.

The material parameters of the modified Chaboche theory are evalua­ted and revised for superalloy Inconel 718 at 1100 OF (593 OC). Predic­tive capabilities of the theory are examined in monotonic tension, creep and cyclic loading. Predicted responses are also compared with experi­mental data and found to agree fairly well. Comparisons with predictions of other theories are also performed, and are in favour of Chaboche's.

1. INTRODUCTION: Structural materials subjected to elevated service tem­peratures and different loading conditions usually exhibit many complex physical phenomena. One such phenomenon is Dynamic Strain Ageing (DSA). It is often manifested by an increase in flow stress, unstable inelastic flow (the well-known Portevin-Le Chatelier effect), negative strain-rate sensitivity, and a decrease in ductility (blue brittleness) (1). The strengthening effect can be advantageous, and is primarily attributed to the effects of impurities. At relatively low strain rates and strain levels, impurities can diffuse to dislocation sites and impede their mo­tion. Due to the time-hardening factor, lower strain rates would require higher flow stresses at the same strain level.

The need to model the inelastic behaviour of materials has stimula­ted rapid development of a large number of unified theories of viscoplas­ticity, e.g. (2-8). In particular, Chaboche's theory (2,9) can model a wide variety of inelastic behavioural aspects of homogeneous, initially­isotropic, strain-hardening (or softening), inelastically-incompressible

67

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 67-76. © 1990 Kluwer Academic Publishers.

68

materials under different thermomechanical conditions of loading. This theory, however, does not model strain ageing and the associated negati­ve strain-rate sensitivity; therefore, its applicability is generally limited to situations where DSA is not exhibited.

In Ref. (10), an attempt was made by James et al to account for ne­gative strain-rate sensitivity in the Bodner(3) and the Krieg et al (6) theories. In doing so, they adopted the Schmidt and Miller approach (5) in which DSA is modeled by a correction function Fc (also called solution strengthening function) of the form

-F2 (log ~ )2 Fl e Po (1)

This function is added to the drag stress, thus indicating that DSA effects have an isotropic nature. Contrarily to this, Walker's theory (4) assumes a directional character of DSA effects, and negative strain­rate sensitivity is modeled by adding to the back stress evolution equa­tion a correction term, Bc' of the form

Bc Fc 13 P , (2)

where 13 is the back stress and Fc is given in this case as . Fc

-F 1 log ~ 1 Fl e 2 Po (3)

In Eqs. (1) and (3), p (=1£"1) is the rate of inelastic strain accu­mulation, Fl is the maximum correction corresponding to p = p , and F2 is an exponent that controls the width of correction. 0

In the present paper, Chaboche's theory is extended to include nega­tive strain-rate-sensitivity-modeling capability. The extended form of the theory is then applied to describe the behaviour of Inconel 718 at 1100 of under monotonic, sustained and cyclic loading. Predicted respon­ses are compared with experimental data as well as with predictions of other theories, and relevant conclusions are drawn.

2. CONSTITUTIVE EQUATIONS: The one-dimensional form of Chaboche's theory is presented and used hereafter; its three-dimensional form can be found elsewhere (2,9). A basic assumption of the theory is the decomposition of the strain rate £ in case of infinitesimal deformations into elastic and inelastic components. Thus

£1 + E" (4)

The linear elastic strain rate s' is obtained from the time deriva­tive of Hooke's law, i.e.

(5)

where a is the stress and E is Young's modulus. The nonlinear inelastic strain rate E" is based on the normality hypothesis, and has the form

(£:) n Sgn(O-IB) , = { K

o

F>O (6)

F;;,O

where K and n are material parameters. F is the von Mises yield function

69

given by

F(O,IB,[) IO-IBI -[)(p), (7)

where IB is the kinematic hardening variable (also called back stress), p is the cumulative inelastic strain defined in terms of the inelastic strain rate by

p (8)

and D is the isotropic hardening variable (also called drag stress) asso­ciated with p.

The evolution of kinematic hardening is cast in a hardening/recovery format, and is given by

(9)

where c, a, y and m are material parameters. Equation (9) provides an effective tool by which anisotropic hardening and associated characteris­tics such as the Bauschinger effect can be appropriately modeled. The growth law of isotropic hardening has the form

-b (O-q) P (10)

where band q are material parameters. This form can be integrated to yield

iD (11)

where Do is the initial yield strength, taken to be constant in the ori­ginal theory. Nonetheless, it will be assumed to change with strain rate in subsequent development.

3. INCLUSION OF DSA MODELING CAPABILITY IN CHABOCHE'S THEORY: It can be seen from Eqs.(l) and (3) that a generic form of the correction function can be given by

F c (12)

In this form, Fl , p and F2 have the same definitions as in Eqs. (1) and (3). F3 is another ~arameter which, together with F2 , controls the width of correction; increasing F2 and F3 decreases the width (Fig. 1).

It has been stated above that previous efforts to model DSA have assumed it to have isotropic or anisotropic effects. Experiments aimed at determining whether DSA has an isotropic, anisotropic, or combined nature are yet to be designed and performed. In the present paper, all three possibilities will be examined.

3.1 Introducing Fe into Kinematic Hardening Growth Law Walker's correction term given by Eq. (2) is employed for this pur­

pose. Thus, Eq.(9) becomes

IS (13)

70

At relatively high strain rates, the thermal recovery term can be neglected, and Eq.(13) reduces to

IB (14)

Also, Eq.(6) can be combined with Eq.(7), specialized to monotonic loa­ding and inverted to give

(J IB + ° + K(E" ) lin (15)

It has been shown in Ref.(ll) that hardening is mostly kinematic du­ring monotonic loading, primarily because of the relatively high value of the ratio of the hardening exponents c and b (cf. Eqs.(9) and (10», and because of the small amount of inelastic strains that are likely to have accumulated. Therefore, it can be assumed that O~Oowhen the stress reaches its saturation value (Jsat. At the same instant E" tends to E and B reaches its maximum value, which can be found by solving the equation

(16)

Combining Eqs.(15) and (16), and substituting appropriate saturation values yields the expression

ca c a IEsat - c = -(J-s-a-t--=-::"';:D=-o---K-(-E-)""l-I:-n - c ( 17)

3.2 Introducing Fc into Isotropic Hardening Stress In a previous paper (11), Chaboche's theory was extended to model

rate-dependent initial yielding by postulating that the isotropic harde­ning variable D depends explicitly on the total strain rate E and the cumulative inelastic strain p, rather than on p alone, as postulated in the original theory. Moreover, the influence of E on U was assumed to be limited to its initial value. In the present paper, the same line of thought has been followed, and 00was assumed to take the form

- (F Ilog ~ I) F3 Do IDo + F c IDa + F e 2 • ( 18)

1 Po

where Do is the initial, rate-independent value of IDo. When the stress saturates at a relatively high strain rate, IE reaches its maximum value a (cf. Eq.(9». Accordingly, Eqs. (15) and (18) provide the expression

F c

• lin 0sat - a-Do - K(E) .

3.3 Introducing Fe into Kinematic and Isotropic Hardening

(19)

In this case, Eqs.(16) and (18) are considered simultaneously, and Fc is obtained, therefore, by solving the quadratic equation

~ + Do + Fc + K(E) lin. F +c c

(20)

4. DETERMINATION OF MATERIAL PARAMETERS: In its modified form, Chaboche's theory incorporates nine viscoplastic material parameters and the correc­tion function F , which are generally temperature dependent. In a pre­vious paper (12), the first nine parameters were evaluated from available experimental data of superalloy Inconel 718 at 1100 OF, employing the

71

algorithm of Ref. (13) . These values, however, needed to be revised upon modifying the theory to model negative strain-rate sensitivity as well, which is observed experimentally in the strain rate range of 10-3 to 10-5 s-l. A parametric study based on a trial and error procedure has been undertaken for this purpose. In effect, an optimal set of material para­meters has been obtained, and is listed together with the initial set in Table l.

Table 1. Initial and revised material parameters

a(MPa) c m yts-lMPa-m+l) n K(MPan,ls b q(MPa) lDo(MPa)

Initial 223 130 l.08 9.86E-7 6.27 1200 4.6 112 480 Revised 223 130 l.08 9.86E-7 5.27 1100 2.1 112 460

tion In addition (12) can be

Fl In(ln -

Fc

to these parameters, Fc needed to be identified. rewritten in the alternative form

) =

Equa-

(21)

which represents a straight line with a slope of F3 and an intercept of F3 In F2 . Thus, if a series of experimental tensile stress-strain curves at different strain rates spanning the range of negative sensitivity are available, a value of Fc can be calculated from Eqs.(17), (19) or (20) for each of these curves. The maximum of these values can be taken to be Fl and the corresponding strain rate to be Po' Furthermore, In(Fl/Fc ) can be plotted versus !log(p/po)! on a log-log scale and F2 as well as F3 evaluated, as shown in Fig. 2 for the case of Subsection 3.2. Data of Ref.(14) have been used for this purpose, and are listed in Table 2. In this table, each Os at value is the average of two experimental results. Calculated parameters are listed in Table 3 for the three cases treated.

Table 2. Experimental strain rates and saturation stresses in tension, and corresponding initial yield stresses and steady-state creep rates (14)

. (s-l) E l.002 E-3 3.127 E-4 9.926 E-5 3.054 E-5 7.253 E-6

°sat (MFa) 836 883 928 960 935

li?o (MFa) 475 542 636.8 685 690

Ess (s-l) 4.3985 E-5 2.5488 E-5 3.9067 E-6 l.8487 E-6 3.1434 E-7

Table 3. Parameters of correction function Fc

Fl F2 F3 p. (s-l) 0

Fc in IS 800 0.8 2.8 l.15 E-5 Fc in O?o 230 0.704 3.2 l.15 E-5 Fc in Band IDo 200 0.73 2.6 l.15 E-5

5. PREDICTIVE CAPABILITIES OF THE MODIFIED CHABOCHE THEORY: Following the determination of its material parameters, the modified Chaboche theo­ry has been used to predict the tensile stress-strain response of Inconel 718 at 1100 OF at six strain rates, four of which lie in the range of negative rate sensitivity. These predictions were compared with the

72

available experimental responses, as depicted in Figs.3-5 for the three cases considered. It is seen that introducing Fc into initial isotropic hardening stress appears to give the best results, and therefore further predictions will be limited to this case only. Introduction of Fc into kinematic hardening seems to represent the other extreme, and indicates that the present form of Fc is not the most appropriate one to be intro­duced into B. Note that the predicted responses at £=10-6 and 10-2 s-l, which lie outside the range of negative sensitivity, provided normal rate sensitivity, as would be expected.

In addition to monotonic loading, creep responses at different hold stresses were predicted. In each case, 00 had to be calculated from Eq. (lB). Based on the analogy between tensile and creep responses (15,16), the strain rates of the tensile tests of Table 2 have been used for this purpose. Values of Db as well as predicted steady-state creep rates are listed in the same table, whereas predicted responses are shown in Fig.6. Although higher hold stresses normally induce higher steady-state creep rates, the results obtained provided lower rates at higher stresses, with the exception of the response at Gh=935 MPa. These rates, however, lie within the strain rate range of negative sensitivity, and the results are therefore self-exp lanator1. Moreover, at Gh=935 MFa, a steady-state creep rate of 1.BO E-6 s- was obtained. This rate lies outside the range of negative sensitivity, which explains why this particular hold stress induced normal sensitivity. Experimental creep responses were not available to further confirm these arguments, and the need to acquire such data is addressed.

Figure 7 depicts the hysteresis loops of the first ten cycles of the response at s=±O.B% and £=3.054 E-5 s-l, employing the original and the modified forms of the theory. In addition, the stress amplitudes of these loops are compared with experimental values (10) in Fig. 8. It is clear from these figures that the modified form of the theory provides closer agreement with the experiment. The first cycle stress amplitudes of the original and modified theories were also compared with experimen­tal values at s=±O.B% for different strain rates, as shown in Fig. 9. Whereas the original form predicts higher stresses with increasing £, the modified form is capable of predicting lower stresses as £ increases within the range of negative sensitivity, and higher stresses otherwise. Note that the correction function is so identified that its value vani­shes at E ~ lE-3 s-l. Therefore, the modified and original forms provi­ded close stress values at this particular rate.

In addition to the responses of Fig. 9, the responses of the or~g~­nal and modified Bodner, and Krieg et al. theories, as well as those of the walker and the Schmidt and Miller theories are collected in Fig. 10. Detailed discussions concerning the modification of the Bodner and the Krieg et al. theories, as well as the determination of the material para­meters of the four theories can be found elsewhere (10,12). It is clear from Fig. 10 that the modified Chaboche theory offers the closest agree­ment with experiment.

6. CONCLUSIONS: Chaboche's unified theory of viscoplasticity has been modified to model negative strain-rate sensitivity associated with DSA. This has been accomplished by introducing a strain-rate dependent correc­tion function into the initial value of the isotropic stress, which vanishes outside the range of negative rate sensitivity. The material parameters of this modified Chaboche theory were evaluated and subsequen­tly revised for superalloy Inconel 71B at 1100 of from uniaxial data

73

available. Predictive capabilities of the modified theory were illustra­ted in monotonic tension, primary and secondary creep, and strain-contro­lled, fully-reversed cyclic loading. Predicted responses were compared to experimental data available and the agreement was found to be fairly good. The predictions were also compared with those of the original form of the theory as well as with predictions of other unified theories. These comparisons indicate that the modified Chaboche theory offers the greatest promise for successfully modeling DSA effects.

REFERENCES 1. A.M. Eleiche, C. Albertini and M. Montagnani, in Stainless Steels '87, The Institute of Metals, London, England, Book 426 (1988) 394-404. 2. J.L. Chaboche, Bulletin de l'Academie Polonaise des Sciences, Serie Sciences Techniques 25 (1977) 33-42. 3. S.R. Bodner, in Plasticity Today: Modeling, Methods and Applications, ed. by A. Sawczuk and G. Bianchi, Elsevier Applied Science Publ., Barking England (1984) 471-482. 4. K.P. Walker and D.A. Wilson, Proceedings of the Second Symposium on Nonlinear Constitutive Relations for High Temperature Applications (Cle­veland, Ohio) (1984) 5. C.G. Schmidt and A.K. Miller, Res. Mechanica 3 (1981) 109-129. 6. R.D. Krieg, J.C. Swearengen and R.W. Rhode, Inelastic Behaviour of Pressure Vessel and Piping Components, ed. by T.Y. Chang and E. Krempl, ASME, New York (1978) 15-27. 7. E. Krempl, ~ournal of the Mechanics and Physics of Solids 27 (1979) 363-375. 8. K.C. Valanis, Archives of Mechanics 27 (1975) 857-868. 9. J.L. Chaboche and G. Rousselier, ASME Journal of Pressure Vessels Technology 105 (1983) 153-158. 10. G.H. James, P.K. Imbrie, P.S. Hill, D.H. Allen and W.E. Haisler, ASME Journal of Engineering Materials and Technology 109 (1987) 130-139. 1]. M.S. Abdel-Kader, J. Eftis and D.L. Jones, Proceedings of SECTAM XIII (South Carolina) 17C4 (1986) 263-269. 12. N.N. EI-Hefnawy, M.S. Abdel-Kader and A.M. Eleiche, Proceedings of the Fourth Cairo University MDP Conf. (Cairo, Egypt) Suppl. Vol. (1988) 135-146. 13. M.S. Abdel-Kader and A.M. Eleiche, Proceedings of ~lliCAMAT Int. Semi­nar on the Inelastic Behaviour of Solids: Models and Utilization (Besan­con, France) (1988) 519-535. 14. P.K. Imbrie, G.H. James, P.S. Hill, D.H. Allen and W.E. Haisler, ASME Journal of Engineering Materials and Technology 110 (1988) 15. D.C. Stouffer, A Constitutive Representation for IN 100, Technical Report AFWAL-TR-4039 (1981). 16. S.L. Mannan and P. Rodriguez, High Temperature Materials and Proce­sses 6 (1984) 225-227.

74

F3 1 2

_____ 3

.3

.2

~ \\ \\ \\ \'

c. CC!--'--~-:3-""-..J4:-""'~5~-:---:':--:.8 110g(p /poll

,.0"'------------, !.o..--------------,

.9

F2=1

F3 1 2 3

8 3 ~ 110g (p/Pe ) I

F2 =2

F3 1 2 3

Fig. 1. Illustrating the response of Fc to changes in F2 and F3 • 2,--------------,---------------,

-3

EXPERIMENT( 14) REGRESSION LINE

1n{1n!J.) _ Fc

-4~-~~~--~~L--~--~--~ -1. 0 -. B -.6 -.4 -.2 0.0 .2 .4 .6 .8 1. 0

Inllog(C"; po) \

Fig. 2. Determination of the

parameters F2 and F3 in the correction function Fe •

1000r-----------------------___ ~ --,-- '/: -- Iii' // *//

/~ t)--IJ jI-/)' "

v, // ' tl' /~ . .,/~ .. /. ,

INCONEL 718

800

~ 600 (5

,:;/ v

1lf ~ U) 400

_____ 1.000E-2 _____ 3.054E-5 ___ • __ •• _ 9.926E-5 _. _____ 7.253E-6

____ 1.002E-3

, I.OOOE-S

EXP.

+ o •

OL--~--~~--~--~~~ 0.0 .2 .4 .S .8 1.0

STRAIN.' %

Pig. 3. Comparison of experimental (10) and predicted tensile stress­strain responses (Fe introduced into 13, Eg. (17)

1000 r-I-N-CO-N-E-L-7-1-8----------------=,...,

TeSS3"!: // /

800

/ ~,..::--I *....-:~ _­

L ~.~-, ~'r

~ 600 I o

~ en 400 I

II PREDICTIONS _ ~. ~~~=~ EXP.

200 ----- 3.054E-S __ ._._ •• _ 9.926E-S _._. ___ ._ 7.2S3E-S 0 _ _ _ _ _ 1.002E-3 *

i ,..,. I. 000E-6 0L---L--~----~ __ ~ __ ~ 0.0 .2 .4 .S .8 1.0

STRAIN.' %

Fig. 5. Comparison of experimental (10) and predicted tensile stress­strain responses (Fe introduced into B and Do' Eg. (20»

1000r-----------------------~

----­// -:;:;;~.

800

INCONEL 718 T-593"C

I. :::f'.~.::.--:­;::::.-...-'" -

~-1J'-; ..,.-1/ ' f~/r -

f'

~ SOO / u! 1:3 :: en 400 in", .. /(,-1) =.

200 I ------- 1.000E-2 ______ 3.054E-5 _. __ • ____ 9.92SE-S

I ----- 7.2S3E-6 0 _____ 1.002E-3 H

Ii . 1.000E-S

o~--~--~~--~--~--~ 0.0 .2 .4 .6 .8 1.0

STRAIN •• %

Fig. 4. Comparison of experimental (10) and predicted tensile stress­strain responses (Fe introduced into Do' Eg. (19» •

0.00 10 20 30 .0 50 60 70 80

TIME. MINUTES

Fig. 6. Predicted creep response at different hold stresses.

90

75

76

1000r-------------~------------_,

900

SOD

400

b

~ "'-200

-.9

INCONEL 718 T-593°C

STRAIN RATE

__ MODIFIED ORIGINAL

-.4 0.0 .4 STRAIN. g %

• B

Eig. 7. predict:d cyclic re~~onse at £=±0.8% and £=3.054E-5 s , employing the original and modified forms of Chaboche's theory.

1000

'<l 950 ! < 900

! !il 850

~ 800 . .

~ 750 . .; 700

!! 650 _ -11- _ ORIGINAL CHABOCliE

950

gOO

:<! 850

!

~ 800

750

I.l 700 d

~ 650

600

550

500 1

._---+::.._._._--_._-----...... __ ._---=::-

_ EXPERIHENT ______ MODIfIED CHABoct£

_ -f - ORIGINAL CHABOCI£

5 6 CYCLE NUMBER

10

Fig. 8. Comparison of experimental (10) and predicted stress amplitudes of the first ten cycles at £=±0.8% and S=3.054E-5 s-l.

950

'<l 900 ~

i sso

!il 900

~ 750 •

I 700 . .. 650 tJ !! 600

60ps.S -5.0 -4.5 -4.0 -3.S -3.0 -2.5 55PS.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 LOG (Strain Rat.a)

Pig. 9. Comparison of experimental (10) and predicted stress amplitu­des of the first cycle at £=±0.8% and different strain rates.

lOGCStrain Rata)

P·ig. 10. Comparison of experimental (10) and predicted stress amplitu­des of the first cycle at £=±0.8% and different strain rates for different theories.

CYCLIC DEFORMATION AND LIFE PREDICTION USING DAMAGE MECHANICS

A. Plumtree and G. Shen

Department of Mechanical Engineering University of Waterloo

Waterloo, Ontario N2L 3G1

ABSTRACT: Using damage mechanics a constitutive equation for fatigue dam­age accumulation has been developed which takes into account strain level ef­fects. In this manner, damage evolution over the entire life may be predicted accurately. It is shown that good estimates of fatigue life may be made when the model is applied to two level tests.

1. INTRODUCTION: An equation for fatigue damage evolution under strain control was proposed by Lemaitre and Plumtree [1]:

8l!. = _1_[1_ Dt" 8N p+l

(1)

where p is a temperature dependent material constant, iii = ~ is the normalized number of cycles (N is the number of cycles and Nj is the number of cycles of failure). The equation can be written in the integral form:

D = 1 - [1 - iiijim (2)

These are extensions of the equation developed by Chaboche [2] for stress control, which, in the differential form is expressed by:

8D O"m - ii (3

8N = [B(ii)(I- D)] f(D) (3)

where O"m is the maximum stress during the cycle and ii is the mean stress. Hence

77 A. S. Krausz et al. (eds.). Constitutive Laws of Plastic Deformation and Fracture, 77-85. @ 1990 Kluwer Academic Publishers.

(4)

78

1.0 .."....----------P-A-LM-G-R-EN--IIM-N-ER"""1

o.a :x:

Z 0.6 .......

:x: c 0.4

0.2

1.0

0.8

:x: Z 0.6 .......

~ 0.4

0.2

H-L

H-L

- PRESENT THEORY V,C EXPERIMENT [71

o INTERMEDIATE ANNEAL

L-H

Q2 0.4 0.6 0.8 1.0

nL/N L

(b)

(d)

Figure 1: Test data from reference NL = 16,000, b) NH = 1000, NL

[7] and present model: a) NH = 720, 200,000, c) NH = 1000, NL = 400,000,

d) NH = 900, NL = 700,000.

where a: is a stress dependent material constant and B, [3, A are temperature dependent material constants. Eq. (2) introduced the concept of nonlinear dam­age for strain controlled fatigue. Its application gave a much better prediction of fatigue life than that using Palmgren-Miner's linear damage rule. However, some problems cannot be solved by this model. For example:

a. Damage evolution appears to be strain dependent even when the normalized life is considered [3,4], yet according to Eq. (2) the damage variable D should be independent of strain when this normalizing procedure is carried out. Experimental and theoretical work by several researchers such as Ibrahim and Miller [5], and Kujawski and Ellyin [6J showed that linear damage summation was larger than unity for low to high strain tests (L -H) and less than unity for high strain followed by low strain tests (H - L), which can be seen in Fig. 1.

0.6

SYMBOL STRAIN AMPLITUDE. ~ 2

0.5 x 0.35'1'. 0 0.50% 0

• 0.50%

'" 0.75% 0.4 0 1.00%

0

W 0.3 <.!) « :E « 0 0.2

0.1

O.O~-~/><.&-M!!II~i:;;8L.iKJ~~ 04 05 0.6 0.7 0.8 0.9 1.0

NORMALISED LIFE, N/Nf

Figure 2: Damage Evolution Curves by Eq. (5)

h. Application of Eqn. (2) to express damage evolution over the entire life has presented some difficulties with the result that this equation has been modified [1]:

D = 1 _ (1 _ -:-:N,....---:-N:-* ) ,~, Nt-N*

(5)

where N* is a critical number of cycles below which damage or dam­age interaction may be regarded as being neligible (N* = O.5N, [11 and N* = O.6N, [3]). Fig. 2 gives results for an aluminum alloy under cyclic strain control [3,4]. Using Eq. (5) to correlate the experimental damage which was determined by monitoring cracked area and modulus changes, it becomes apparent that the experimental data lie under the theoretical curve for the early part of life and above the curve in the latter stages. This phenomenon may be explained by calculating the p value for each experimental damage point associated with a different life fraction and the corresponding values are given in Table 1. It is seen that p decreases monotonically with cycles from infinity to approach zero at failure. Obvi­ously these results conflict with the assumption that p is constant, as used in the derivation of Eq. (2).

c. The term N* appears to be strain dependent, as seen in Fig. 2.

79

80

Table 1: Coefficient p at Different Life Fractions (N/Nf ). Total Strain Amplitude = 1.0%

Modulus Change Cracked Area - N D=l-& p=~-1 D = hE: p=~-l N= Nf E In I-D A In I-D

0.039 0.013 2.04 0.000000 00

0.398 0.011 44.45 0.000064 7928 0.522 0.000 00 0.000289 2553 0.659 0.014 75.9 0.009970 109 0.789 0.022 70.4 0.013666 112 0.925 0.024 9.6 0.03725 77 0.972 0.231 12.7 0.232809 12.5 0.992 0.334 5.3 0.409126 3.85

This present work addresses the above points by modifying Eq. (2) to give a better correlation with experimental results. The model will be used to predict damage accumulation over the entire life as well as to predict the life of specimens subjected to two-level tests.

2. IDENTIFICATION OF MODEL:

2.1 Rationale for Variable p In extending Chaboche's model to strain controlled fatigue, Lemaitre and Plumtree [1] used the term f(D) = [1 - DtP to replace Eq. (4). Though it simplified the derivation, two of the four initial and final conditions were no longer met. These conditions for strain control are:

(a) N = 0, D = 0 (6)

(b) N = 1, D = 1 (7)

(c) N = 0, oJ? = 0 (8) oN (d) N- oD fin' I ( ) = 1, oN = a Ite va ue 9

Both Eq. (1) and (3) can meet conditions (a), (b), but only Eq. (3) can meet conditions (c), (d). Under strain control,

u - jj / -"'-- = K t!.fl '" = constant 1-D P

(10)

where t!.fp is the plastic strain amplitude and m is a material constant. Substi­tuting

(11)

and

f(D)IN=1 = [1- (1- D)~+1]'*m' ")IN=1 = 1

into Eq. (3), yields

oD I (K 1:!..f.1/m) ~ oN N=o = B(:) f(D)IN=o = 0

oD I (K 1:!..f.1/m) ~ oN N=1 = B(:) = a finite value

However, substituting:

and

into Eq. (1), yields

1

p+l

oI?l = _1_(I- Dtp l oN N=l p + 1 D=l

To meet conditions (c), (d), it requires:

and

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

For damage evolution to be a continuous process, p should be a variable, decreasing from infinity to zero with number of cycles, which would then agree with experimental results [3,4]. Consequently, damage D is a combination of power and exponential functions of N, rather than a power function only. It explains why the experimental data is not correlated well when equations which consider p constant are used, such as Eqs. (2) and (6).

2.2 Modification of Damage Evolution Equation: To describe the process of damage evolution accurately, a function of p = p(N) should be found. Since p is the function of normalized cycles N, the equation of damage evolution may be written as:

(21)

81

82

/ /

/

/ /

/ /

/

/ -// B(Nj+"Oj+,IH---'r-T

.psO,,/ 1/6' -y /1"'0

/ _ / -0

// A( N;,Ojl I 0 ,/ /1 / "tJ

// P s Pi+'""'v/ /~ / P'PI-Y /

// /r ~/ o /" ---" .".,./

o NORMALIZED LIFE. N

Figure 3: Schematic of damage evolution curve.

This is the superposition of damage evolution at constant P and that which results from the change of P with cycles. Substituting different values of P in Eq. (2), gives a family of curves, as shown in Fig. 3. If the specimen is in state A(Ni,Di) (N;,Di can be measured experimentally), it is possible to calculate the corresponding value of P from Eq. (2). Point A is found by the intersection of N = N; and the curve of P = Pi. After a cyclic increment dN, the specimen changes to damage state B (NHI,DH1 ). Similarly, point B lies on the curve of P = PHI. By joining AB, a segment of the damage evolution curve can be obtained. As shown in Fig. 3, the damage increment dD from state A to B consists of two parts: the increment along the damage curve P = PHI, dD I = !~ IpdN and that between two constant P damage curves, dD2 = ~~ Lvdp. Obviously, the whole damage curve consists of the points from constant P curves, each curve contributing one point. The equation of damage evolution in integral form has the similar form as that for constant P and can be written as:

-)--1..­D = 1 - (1 - N p(8)+1 (22)

Experimental results show that p(N) has a semi-logarithm relation with N:

lnp = aN + b or p = exp(aN + b) (23)

where a and b are coefficients. Equation (22) may be rewritten:

1

D = 1 - (1 - N) e41'Hb+1 (24)

Since p is a variable decreasing from infinity to zero and small values of p

correspond to larger amounts of D, it is possible to make conservative damage

13

.&l 12

I-Z LIJ II U lL. lL. LIJ

10 0 u

9 •

8L--L __ ~~~~~ __ J-~ 2.00 3.00 4.00 5.00

LOG Nr

Figure 4: Relationship between b and fatigue life, N,.

predictions by using the smallest value of p or the value of p at failure, ie, p = O. When substituted into Eq. (24):

(25)

Eq. {25} is the linear damage rule which may be regarded as a conservative simplification of the present model. Also, considering that engineering compo­nents should not be used to complete failure, substitution of a finite value of p at a given life fraction N modifies Eq. (24) so that it may be used to pre­dict the end of practical life. In this manner Eq. (24) becomes Eq. (2) which may be regarded as a conservative simplification of the present model based on engineering considerations.

3. VERIFICATION OF MODEL: Fatigue damage in aluminum alloy 6066-T6 at total strain amplitudes ranging from 0.20% to 1.00% is now considered. The experimental details may be found in references [3,4] where the cyclic strain controlled damage was defined as the ratio of the cracked area to the original cross sectional area of round, unnotched specimens.

Some of the experimental data are given in Fig. 2. The variation of p with N may be expressed by a In-linear relationship according to Eq. (23). The slope (a = -9.t) is strain independent whereas the coefficient b is strain dependent, given as

b = -1.60 log AEp/2 + 7.89 (26)

where AEp/2 is the plastic strain amplitude. Hence, the higher the strain, the

83

84

O.l0r:::==::::::-:=::-:--::-:-:::-:--~----, EXPERIMENTAL DATA

STRAIN AMPLITUDE

A 0.35 %

0.08 0 0.50% c 0.75 %

THEORETICAL CURVES

c

A

Figure 5: Damage accumulation obtained from cracked area measurements [3,4] and the present model.

lower the value of b, which is also fatigue life dependent since a given strain level corresponds to a particular fatigue life, N f . Fig. 4 shows this relationship expressed by

(27)

where Al and A2 are experimentally determined constants. Substituting Eq. (27) into Eq. (24) results in a generalized expression al­

lowing for different strain levels. Using this approach Fig. 5 gives the predicted damage evolution curves for various strain amplitudes imposed on the aluminum alloy. A very good correlation with the experimentally determined damage levels is seen.

Attention may now be turned to the two-stage loading tests carried out on steel [7] and shown in Fig. 1. By applying Eqs. (24) and (27) to the high-low and low-high fatigue data it is possible to develop predictive curves, which correspond to the different sequences. These are included in Fig. 1. The agreement between theoretical prediction and experimental data is good, particularly when it is realized that no allowance has been made for any material characteristics.

4. CONCLUSIONS:

a) Good agreement is obtained between measured and predicted fatigue dam-

age levels by modifying the Lemaitre-Plumtree equation for cyclic strain control. The coefficient p becomes a variable, decreasing monotonically with cycles from infinity at the start of cycling to zero at final fracture.

b) The damage variable D is best expressed as a combination of a power func­tion of N with an exponential function of N. In this manner damage accumulation over the entire life of a specimen is satisfactorily predicted. Both linear and non-linear damage evolution equations are conservative simplifications of the present model.

c) The effect of strain level on damage evolution is taken into account by the modified equation and accurate life predictions may be made by its appli­cation to two-level fatigue tests.

85

5. ACKNOWLEDGEMENTS: The authors would like to thank Mr. B.P.D. O'Conner for conducting the damage experiments. This work has been supported by the Natural Sciences and Engineering Research Council of Canada and the Govern­ment of the People's Republic of China. The authors express their thanks to Lorna Spencer for typing the manuscript.

6. REFERENCES:

1. J. Lemaitre and A. Plumtree, Journal of Engineering Materials and Tech­nology, Trans. ASME, 101, (1979),284-292.

2. J.L. Chaboche, Une loi differentielle d'endommagement de fatigue avec cumulation non lineaire. Revue Francaise de Mecanique No.50-51, (1974).

3. B.P.D. O'Conner and A. Plumtree, Fracture Mechanics: Nineteenth Sym­posium, STP 969, T.A. Cruse, Ed., ASTM, Philadelphia, (1988), 787-799.

4. B.P.D. O'Conner, M.A.Sc. Thesis, Fatigue Crack Growth and Damage Accumulation in an Aluminum Alloy. University of Waterloo, Ontario, (1985).

5. M.F.E. Ibrahim and K.J. Miller, Fatigue Engng Mater Struct, 2, (1980), 351-360.

6. D. Kujawski and F. Ellyin, Int. J. Fatigue, 14, (1984),82-88.

7. K.J. Miller and K.P. Zachariah, J. Strain Analysis, 12, (1977), pp. 262-270.

NON-LINEAR STRUCTURAL MODELING: INTERACTIONS BETWEEN PHYSICAL MECHANISMS AND CONTINUUM THEORIES

by

Norman J. Marchand

NSERC University Research Fellow, Department of Genie metallurgique, Ecole Poly technique , P.O. Box 6079, Station "A", Montreal, Canada H3C 3A7

ABSTRACT: A review of several viscoplastic theories that incorporate isotropic hardening, directional hardening as well as additional effects due to nonproportional loading is presented. The equations reviewed are generalization of the internal stress concept and use two internal state variables; namely the drag stress and the back stress variables. Theories that incorporate additional internal variables to model physical phenomena such as strain ageing have also been reviewed. Most of these constitutive equations employ isotropic evolutionary equations for the drag stress and nonlinear hardening rules for the back stress variable The physical arguments to justify this procedure are given. The similarities between a wide number of formulations of flow and evolutionary equations, as well as the essential differences between them are presented. Finally, it is shown how a careful exami­nation of experimental behaviors, in conjunction with micromechanical considerations based on the physics of deformation, leads to the formu­lation of a more general unified framework.

1. INTRODUCTION: The drive toward better performance and fuel econ­omy in structural components operating in high temperature environments has resulted in further increase in operating temperature. Assessment of component durability under complex non-proportional thermomechanical loadings, which lead to significant nonlinear inelastic deformation, requires detailed knowledge of the operating environment and the abil­ity to predict structural response for these loadings [1-2].

The prime objective in the development of constitutive models and per­forming inelastic analyses is not to determine the deformation response of structures perse, but rather, to assess their useful service lives.

87

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 87-99. © 1990 Kluwer Academic Publishers.

88

Results from inelastic analyses are necessary inputs to life assessing schemes or models [3-4]. Typically, theses models require quantities such as stress range, mean stress, inelastic strain including percenta­ge of creep and plasticity that make up the inelastic strain range. Mean stress relaxation and ratchetting strains are important as well. A flow chart taken from Walker [3], indicating where materials modeling fits into life prediction schemes is shown in figure 1. As can be seen, it is paramount that the predictive capability of viscoplastic models be capable of predicting these phenomena with reasonable accu­racy.

COUPONENT

R t§3 CRITICAL

CRACX JNmATION

S!TI!

VlSCOPWTIC CONSTInJTlVJ!

THEORY

rAmIFUL CYCLE TESTS t

USING COWPU'fED STlWN-TEWPERATURE -

mroRY A'1 CRACK sm

DESIGN VODIFlCATION

FIGURE 1 - Current Structural Technology is Limited by Inaccuracies in the Viscoplastic Constitutive Equation and Life Prediction Rules [3J.

Traditionally nonlinear analyses were intractable in engineering and simplified methods of analysis or computations based on a single cycle were performed. The cyclic stress-strain diagram provided a convenient means of performing a monotonic analysis giving rise to a stress state representative for midlife of a component [4-6]. The stresses and strains calculated at midlife provide an input to the life analysis under the assumption that they prevail throughout the lifetime of the component. While this method can be adequate for cyclic neutral materi­als, it can lead to highly inaccurate life predictions when there are cycle dependent property changes as observed in many engineering mate­rials [4-7]. Also, such analyses assume the existence of a typical cycle throughout the life of a structure and this is not realistic for most loading spectrums [8-9]. Furthermore, the history dependence of deformation may alter the deformation response which occur in cyclic hardened materials [10-11].

With the advent of ever increasing computing power, the capability of performing nonlinear analyses has also increase. This power, together

89

with the demand for predictability and reliability of performance, puts the emphasis on realistic and accurate models of the deformation beha­vior (i.e. the constitutive laws) and on the damage accumulation laws (i.e. life prediction methods). Accordingly, the area of constitutive modeling has been very active recently [12-58], specifically in regard to modeling time and temperature dependent phenomena.

The purpose of this paper is to present an up-date review of some of the most promising constitutive theories for structural modeling. The similarities as well as the major differences between the models will be highlighted. The physical significance (i.e. mechanisms underlying the phenomena) of the parameters used in the theories will be pointed out. Finally, in the case of the mathematically motivated theories, the physical origins of the mathematical concepts that are introduced will be examined to ensure that they are indeed appropriate components of the theories.

2. CONSTITUTIVE THEORIES-INTERNAL STATE VARIABLES CONCEPT In general, the constitutive equations for the mechanical behavior of materials are based on either one of the two following thermodynamical concepts:

(a) The present state of the material depends on the present values and the past history of the observable variables only (total strain, temperature, etc.).

(b) The present state of the material depends on the present values of both observable variables and a set of internal state variables.

In this paper the discussion will be restricted to models based on the second approach only because it is felt that changes and evolutions of microstructures (with deformation) are better described using a set of internal variables and proper evolution laws for these variables. As will be shown, this is particularly important in the case of non­proportional loadings. Examples of models based on the first approach can be found in ref. [12-16].

All theories develop within the classical framework of thermodynamics with internal variables explicitly assume the existence of a thermody­namic potential (e.g. free energy) from which the relations between state variables and associated thermodynamic forces are defined [56]. To allow the a priori verification of the second principle of thermody­namic, the existence of a dissipative potential which gives the evolu­tion equations for the internal state variables is postulated [56-57]. Most of the constitutive theories require the use of at least two internal variables in order to characterize the material behavior.

The wide number of formulations of flow and evolutionary equations for rate- and history-dependent plastic flow present some similarities and some essential differences which have been extensively reviewed by Walker [3], Chaboche and Rousselier [21], Chan et al. [19] and more recently by James et al. [47]. These equations are sometimes refered to as unified, since all aspect of inelastic behavior in both monotonic and cyclic regimes, such as creep, plastic flow, and stress relaxation, are treated together in representing the rate of the inelastic strain increment, ~p .• It should be noted however, that since the physical

lJ

90

phenomena in monotonic and cyclic regimes are different [59.60]. a single set of internal variables can hardly be expected to describe both regime.

Basically. the constitutive models [17-44] assume that the total strain rate can be divided into an elastic and inelastic components:

This equation is applicable decomposition is assumed equivalent to strain rates deformation [46]. i.e.

-e .p £ij - £ij + £ij (I-a)

only for small strains. but a similar to hold for the deformation rates (Uij).

if strains are small. in the case of large

(I-b)

Constitutive equations have been formulated either with or without the use of a yield criterion. Those without a yield criterion. such as Bodner's [26]. Walker's [18]. Miller's [24], Krempl's [29], Freed's [31], etc., do not contain an elastic regime and therefore give very small inelastic strain rates, (~p.), at low stress levels. On the other hand, in the theories with a ~ield criterion, such as Chaboche's [20], Cailletaud's [22], Delobelle's [33], Robinson's [34], Mroz's [36], Lee's [25], McDowell's [42], ~~. is zero until a function of the stress state, itself rate indepe~aent, reaches a prescribed value.

The essential features of these theories are (1) a flow rule and (2) evolutionary equations for the internal state variables. The flow rule or strain rate equation, gives the ~~j in terms of the deviatoric stresses Sij, the internal variables (e.g. K and aij 's). and the tem­perature T, 1. e.

'p 'p £ij = £ij(Sij' K, a ij , T) • (2)

On the other hand, evolutionary equations describe the rate of change of the internal variables due to stress, current structure, and temper­ature, i.e.

K(Sij' K, T) (3-a)

and Qij = Qij (Sij' a ij , T) • (3-b)

Here it is important to emphasize the conceptual framework for this class of models. Typically, two internal variables are chosen as the minimum set. One associated with kinematic hardening or deformation induced anisotropy often termed back stress and is given by a second order tensor a, which defines the center of the loading or yield locus, assumed undistorted. Physically, the back stress ~ is a resid­ual stress field embedded in the polycrystalline material at the crys­tallite or crystal-lattice level due to deformation of the agglomer­ation of the anisotropic crystallites and to dislocation pile-ups in the crystallites. This back stress a affects the magnitude of the sup­erimposed applied stress needed to-produce additional plastic flow and thus produces the Bauschinger effect, the type of anisotropy associated with kinematic hardening [48-49]. The other internal variable is asso-

91

ciated with isotropic hardening effects. In rate-dependent plasticity theories, it is included, most often, as a viscous drag stress (K), implying that the inherent resistance to dislocation glide is dependent upon the deformation history. These internal variables are thus associ­ated with microstructurally local stresses produced by dislocation rearrangements.

Flow Rules: About four basic forms have been identified, all ing plastic incompressibility (~~ = 0), and most of them from an associated flow rule, whic~ in turn is derived resolved shear stress rule for slip on every system within crystal [48-49).

satisfy­derived from the the poly-

The first form is the flow rule associated with a yield function and is expressed as:

where stress

.p af(Skl - °kl) £ij = as ..

l.J (4)

Skl and Okl are the deviatoric component of stress and back respectively and f( ) is the yield function or flow potential.

The second form is the well-known Prandtl-Reuss flow rule with the Von Mises criterion, i.e.

.p £ij = '\ Sij (5)

This equation is independent of the yield condition and can be a~plied for proportional loading. It states that the material response (c p .)

to stress is isotropic even though the plastic multiplier Al coula be stress history dependent. Since stress is directional, Al could have a directional character and therefore account for directional hardening effects. For nonproportional loading, the applicability of the Prandtl-Reuss flow law is questionable. At issue here is whether or not the inelastic strain rate vector is coaxial with the deviatoric stress vector under nonproportional paths. In fact, there are plenty of evidence showing the Prandtl-Reuss law to be inadequate for nonpro­potional loading paths [32,40-44).

The non-coaxility between the inelastic strain rate and deviatoric stress can be accounted for by using a generalized anisotropic Prandtl­Reuss flow law. This represents the third form of flow rule which can be written as

.p £ •• =

l.J (6)

The last flow rule form is obtained by introducing the kinematic vari­able of Prager [50) into the classical plasticity formulation to account for back stress hardening. In that case

(7)

where ~ij is called the effective stress. This equation represents the associated flow rule for the translated Von Mises yield surface. It

92

also account for the non-coaxility between;P and the deviatoric stress Sij associated with nonproportional loadih~s.

It should be noted that the last three forms can be derived from Eq.4 if they are associated with a flow potential. In general, the temper­ature-dependent functional relation between the strain rates, stresses and internal variables is of the form

>. >'(Sij, K, T) , (8-a) or

>. >'(Sij, K, Gi j , T) . (8-b)

For the isotropic second invariant flow rules, Eqs.5 and 7 can be squared to give respectively

(9)

(10)

The plastic multipliers, i.e. the >"s, are intended to represent the resistance of the inelastic state to plastic flow, e.g. via hardening and damage. The numerators in Eq.9 and 10 are often called equivalent strain rate (~p) and are related to the internal variables, the temperature T, and the deviatoric stress Sij, through a function which is either of exponential [17-19,26], power law (Norton's law) [20-22, 27, 30,32-34], or hyperbolic form [23-24]. This function is refered to as the equivalent strain rate equation and is formulated to model (explicitly or implicitly) the deformation processes that occur over the range of strain rate and temperature studied. For example, Lee and Zaverl [25] have proposed the following equivalent strain rate equation

~p (11)

where a is the equivalent stress, K is the drag stress or the maximum glide resistance, and £0' N, Q, D are material constants. In this particular equation, the first exponential term is used to represent the thermally activated motion of dislocations [49,51) whereas the sec­ond exponential term is related to diffusion controlled shear plastic flow [52-53].

Evolutionary Equations for Internal Variables: The general framework of the evolutionary equations for the internal variables is usually based on the Bailey-Orowan theory [53), which assumes that deformation

93

occurs under two simultaneously competing mechanisms: a hardening pro­cess which proceeds with deformation, and a recovery or softening pro­cess proceeding with time. The evolution rate of an internal variable is then the difference between the hardening rate and the recovery rate. Saturation of the drag stress or back stress variable is reached when the antagonistic effects of hardening and recovery cancel each other in the state variable evolutionary equation. For the drag stress one can write

(12)

where K is the evolution rate, h1(K) is the hardening function, and rl(K,T) is a static thermal recovery function. Other evolutionary forms for K, add a dynamic recovery term to equation (12) which is stress and temperature dependent [18-19, 21, 23-24, 26, 54]:

K = h1 (K) ~p - h2(aij ,K,T) ~p - r 1 (K,T) • (13)

The dynamic recovery function hz ( ) is dependent on the state of the internal variables aij and K, and of the temperature. This function is chosen to be positive for impact loads, and negative for thermal effects.

Perhaps the most important differences between the various theories is the treatment of the change in back stress aij associated with harden­ing. The general framework of these evolutionary equations follows the hardening and recovery formulation given by equation (12), i.e.

';ij = h 3(aij ) Mij - d(aij,T) Nij - r 2 (aij ,T) Vij + 9(aij ,T) TWij , (14)

where h3 (), d(), and r2( ) are the hardening, dynamic recovery, and static thermal recovery functions respectively. The function 9( ) represents the hardening or recovery associated with the rate of tem­perature change. The dynamic recovery term d(aij ,T) is viewed as essen­tial for proper multiaxial generalization [42]. Furthermore it has been established as necessary for proper correlation of experimental uniax­ial data [18,47]. The terms Mij , Nij , Vij , and Wij are the directional indexes of hz ( ), d( ), rz( ) and 9( ) respectively.

Chan et a1. [19] have pointed out that the choices of the directional indexes and the hardening and recovery functions vary substantially between theories. For example, some models utilize ~fj (the directional rule) while others employe the stress deviator Sij or the effective stress deviator (Sij - aij) as the indexes of Mij or Nij .

It is important to note that the evolution laws (13) and (14) are also refered to as nonlinear kinematic hardening rules.

3. CONSTITUTIVE THEORIES-MATHEMATICAL CONCEPTS: In parallel with the development of constitutive theories1 that provide a phenomenological

Here we refer to constitutive equations that utilize phenomenologi­cal internal state variables and nonlinear evolutionary equations for these variables resulting in nonlinear kinematic hardening.

94

(or physical) description of inelastic deformation, another class of mathematically motivated theories was developed [35-43]. These models, refered to as multiple-surface or two-surface theories, are formulated using general thermodynamics and mathematical considerations alone. All (or most) of them, can be viewed as practical simplification of the multiple loading surface theory initially proposed by Mroz [36]. This is usually achieved by analytically prescribing the variation of the plastic hardening modulus of the flow rate.

In the context of small strains and Von Mises form, the stress surfaces considered in these theories are:

1. a yielding surface which is defined by

2 f = J(~ - ~) - R = 0 , (15)

2. a bounding surface (or limit surface), which defines the limiting state of stress, i.e.

(16)

Here S is the deviatoric stress tensor, Q and ~ are the center of the yield and bounding surface and, Rand R* are the yield and bounding surface radii respectively. It should be noted that multiple surface strain space formulations have also been proposed [39,55-56] with mathematical forms analogous to Eqs. (15) and (16). However, in the following, we consider the more conventional stress space formulation only.

The bounding surface is a stress surface located outside the yield sur­face. Expansion of the bounding surface represents development of iso­tropic hardening while translation of the yield surface inside the bounding surface describes nonlinear kinematic hardening. Expansion of the yield surface and simultaneous translation of the bounding surface are also possible through a proper choice of the function R and the evolution law for ~*[35,42].

The rate-independent flow rules have (often) the following form [35-38, 40-43]:

~{ Sij (Sij - Qij) } (Sij - o;ij) if Sij(Sij - 0; ij ) > 0

L2 L2 .p

(17-a) e •. 1J

Si/Sij 0 if - 0; ij ) ~ 0

(17-b)

whereas other formulations have been proposed for rate-dependent plas­ticity. For instance, the viscoplastic flow rule introduced by McDowell

et al. [42) can be written as

1 [~2 - k)N exp {B[~2 H h2

'p £ •• ~J

o

95

if ~2 > k

(18)

In the above equations "H" is the hardening modulus, Sij is the devi­atoric stress rate, and k is the rate-independent yield surface. Note that the flow rule chosen by McDowell approximates a hyperbolic sine function often used for more accurate correlation of strain rate sen­sitivity over a wide range of inelastic strain rates. Essentially, this exponential flow potential results in a variable viscosity exponent, when viewed in terms of more conventional power law representations [35) .

The evolution laws for the internal variables R, R·, ~ and a· are usually of the form

(19-a)

(19-b)

(20-a)

~* A*~P 2

1/. (20-b)

and

[~ T T f/2 ~T (aij - a ij ) (aij - a .. ) •

~J (20-c)

Here f l ( ) and fz( ) are functions which depend on the accumulated inelastic strain and aT is a target point in stress space defining the image point configurationl . The functions f l ( ), fz( as well as ;T, 1/* are defined for each particular model [35-43,55-56). As ;enti~ned already, f I ( ), fz( ) as well as ;T, 1/* (and their corresponding evolutionary equations), may be ~hose~ such that linear and nonlinear kinematic hardening can be modeled. Furthermore, coupling between these variables have also been proposed [35,39,42-43] to account for nonproportional loading effects and thermal recovery phe­nomena (static and/or dynamic).

The reader is refer to Mroz [36] for a geometric interpretation of the image point in deviatoric stress space.

96

Although primarly developed on mathematical considerations alone, the equivalence2 between two surfaces Mroz-type theories and the so-called unified creep-inelasticity or state varible models has been established in length by Chaboche and Rousselier [20] and subsequently discussed by McDowell et al. [42-43]. Furthermore, this equivalence revealed that isotropic hardening may appear in both the flow rule and the kinematic hardening rule; its presence in the former affecting the inelastic strain rate and the rate of kinematic hardening in the latter [20].

With the foregoing equivalence between bounding surface theories and internal state variable models, McDowell et al. [42-43,57-58] have appealed to microstructural considerations to obtain physically based multiaxial and rate-dependent generalizations. They hence proposed iso­tropic and nonlinear kinematic hardening rules (see Eqs. 19 to 20) that comply with: (i) the presence of thermal and athermal obstacles to dis­location motion with different amplitudes and spatial periodicities, (ii) the rate-dependence of dislocation substructures, and (iii) the possibility of tensorial indices other than the total back stress ten­sor Q for dynamic recovery associated with dislocation cross-slip or climb around obstacles.

4.0 DISCUSSION AND CONCLUSIONS: It has been pointed out that several constitutive theories incorporate concepts taken from the physics of metal which are combined with general results of macroscopic tests and placed in format of continuum mechanics. In particular, some basic forms of constitutive theories were motivated by physical idea in the field of dislocation dynamics such as the non-requirement of a specific yield criterion and the functional form of the equation for inelastic straining [25,26,30-31]. Although thermodynamics has not had much influence on the specific forms of the proposed theory per se, it nev­ertheless, provides physically mandatory constraints [45-46,59].

An important aspect of most constitutive equations is that isotropic and directional hardenings are determined by separate internal state variables. Hence, isotropic hardening evolves with changes of the drag stress whereas directional hardening evolves with the evolution of Q.

In other word directional hardening does not result from anisotropic hardening of the drag stress variable. There are physical justification for maintaining the original isotropic hardening of the drag stress in the presence of directional hardening of the back stress The mechanisms responsible for directional hardening are essentially intragranular and planar. They influence dislocation motion in the slip planes and slip directions determined by the crystallographic nature of the material and the current stress state. Cross slip does not seem to be an impor­tant factor for directional hardening which is associated with rela­tively small plastic strains. As a consequence, directional hardening should affect the magnitude but not alter the direction of plastic straining from that obtained for slip on the active slip planes.

The review has shown that every model introduces some nonlinear evolu­tion of the kinematic tensor variable ~ (or back stress tensor). In a

2 For example, compared the form of Eqs. 14, and 20-a.

97

mathematical framework this is required to obtain an acceptable description of the translation of the yield surface (e.g. the concavity of the stress-strain loops under tensile-compressive loading) whereas, in a phenomenological framework, this is akin to the Bailey-Orowan hardening-recovery format that expresses the competition between the mechanisms of creation and annihilation of dislocations. This observa­tion is particularly important because it indicates that the formula­tion of stress-strain rate constitutive relations for materials can be structured only on the basis of the characteristics of the major mechanical phenomena involved and the constraints of continuum (thermo­dynamic) theory. Hence, if a mathematically motivated concept is intro­duced its physical origin need only to be examined to ensure that it is an appropriate component of the theory being developed. Similarly, for a physically motivated model its mathematical representation should be checked to ensure that it satisfies the requirements of continuum theory. For instance, objectivityl in the evolution for a is certainly a necessary requirement but, as pointed out by Prag~r [60] and Lee [61], it can be achieved in an infinite number of ways.

Since the various theories, considered as different in their general statements and point of departure, lead to similar descriptive possi­bilities and to quite identical flow rules, the differences between them seem to be more a question of presentation. Presentation in terms of explicit differential equations for a set of internal variables or from.a set of yield and bounding surfaces. Nevertheless the theories based on interval variables gives, in the author's opinion, the larger range of applicability if the following guidelines are followed:

1. use the simpliest model with a few internal variables; 2. if necessary add some complementary effects (e.g. change the char­

acteristic evolutionary functions) on the basis of physical obser­vations;

3. if a particular additional process is evident from experiments, try to introduce it through additional interval variables obeying par­ticular rules.

The considerations described above are particularly important for applications involving finite deformation (small strain) because the mathematical structure is then more complicated and expresses much more of the essence of the phenomenon. These considerations are even more significant when it is not possible to deduce in detail precise macros­copic constitutive relations from the analysis of the micromechanical phenomena involved and their interactions, and this includes some important structural materials. Nevertheless, basic analyses (which are less than complete) and deductive analysis of idealized materials are essential in suggesting the components of the type of phenomenological theories considered in this paper.

ACKNOWLEDGEMENTS: Financial support from the Natural Sciences and Engineering Research Council is gratefully acknowledged.

The principle of objectivity requires that the constitutive rela­tions must be frame-indifferent.

98

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FATIGUE CHARACTERISTICS OF SiCp-METAL MATRIX COMPOSITE

S.B. BINER

Ames Laboratories Iowa State University

Ames, IA 50011

ABSTRACT : The initiation and growth of cracks emanating from blunt notches in 6061-AI alloy reinforced with 25% particulate SiC metal matrix composite was investigated. To elucidate the role of aging condition of the matrix on the fatigue behavior studies were carried out at T6 and overaged conditions. The results indicated that the initiation of fatigue cracks are insensitive to the notch severity and to the aging condition of the matrix. The overaging heat treatment resulted in slower fatigue crack growth rates. The failure of the SiC particles during the fatigue process is given as the reason for the both observed initiation and crack growth characteristics.

INTRODUCTION Discontinuous fiber or particulate reinforced metal matrix composites (MMC) , apart from their high strength and stiffness, also have distinct advantages of being both machinable and producable by conventional metallurgical processes. Therefore, there is ever increasing potential in the use of MMCs in several engineering applications. In structural applications intended for these composite materials, resistance to fatigue failure is of significant importance. Experimental data on fatigue crack growth and fracture behavior have been reported recently(I-7). However, the role of geometric discontinui ties (e. g. notches, holes, sharp corners, etc ... ) on fatigue crack initiation and growth of short cracks emanating from stress concentrations have not been studied in detail. An accurate prediction of the growth rate of small cracks emanating from notches is particularly important for the MMCs. This is due to their inherently low fracture toughness (8,9) which yields very small fatigue crack growth for a critical crack length at which unstable fracture does occur.

In this study, the fatigue crack initiation and growth characteristics of cracks emanating from blunt notches in a particulate SiC reinforced aluminum MMC are investigated. The correlation of crack initiation and growth of short cracks with fracture mechanics parameters, and the role of aging treatment on the fatigue behavior are discussed.

101

A. S. Krausz et al. (eds.). Constitutive Laws of Plastic Deformation and Fracture. 101-107. © 1990 Kluwer Academic Publishers.

102

EXPERIMENTAL STUDIES and RESULTS: The MMC consist of 25% SiC particulate 606l-Al alloy in T6 condition and in the form of 1. 9mm thick sheet was received from DWA Inc. The material was tested in two metallurgical condition, at T6 (as-received) and overaged conditions. The overaging heat treatment was carried out by heating the as­received material at 177 °c for 100 hours. The variation in matrix hardness was measured with a diamond pyramid intenter under 5 gf load. After overaging heat treatment the hardness level was reduced from 118 to 84 (Vickers Dph) indicating a considerable loss in the matrix strength.

The single edge notch specimens(SEN) of 22mm width and l40mm in length were machined parallel to the rolling direction. During the studies, the fatigue crack initiation from five different notch geometries was investigated. The details of the experimental procedure and the full results are given in (10).

In Fig.l, the fatigue crack initiation data for the T6 and the overaged conditions are presented. In this figure, the number of cycles to initiate fatigue crack are correlated with Kt~a where Kt is the stress concentration factor and ~a is the net section stress.

The correlation of the crack growth rates with stress intensity factor range ~ is given in Fig.2. Although the slope of the da/dN curve is slightly higher for the overaged material, however, overall crack growth rates for a given ~K values are lower than the T6 condition. To predict the crack growth rates for the entire range of fatigue crack lengths, first, effective crack lengths were calculated from equation given below(ll) .

e-a(1.0-exp(-4.0(c/~)(1.0+a/Jap»)+c (1)

where a is the notch depth, p is the notch radius and c is the length of the fatigue crack emanating from the notch. These effective crack lengths than were used in Paris-Erdogan law together with predetermined material constants to predicted the crack growth rates. An example of such predictions and comparison with experiments is presented in Fig.3. As can be seen, the agreement is reasonably good.

The growing cracks from the notches remained relatively straight and planar on a macro scale. However, examinations at high magnifications indicated that crack paths are quite irregular on a micro scale. The decohered or broken SiC particles associated with the crack path often were seen. Also, it was seen that the crack path was through regions of the matrix having an absence of visible particles. Measurements of the normalized line fraction of the SiC particles associated with the crack paths gave, an average, slightly higher volume % of the particles than the measurements were made on other areas. The crack deflec tion and crack bridging from the decohered or fractured particles were often seen as given in Figs. 4 and 5.

DISCUSSION When the crack initiation data given in Fig.l are considered, the results are considerable'different from those commonly seen for the matrix alloy. Although the stress concentration factors

103

varied from 9.59 to 1.98, there is almost no influence of the notch geometry on the crack initiation. Also, the metallurgical condition of the matrix alloy did not have a significant effect on the crack initiation life. No direct observation could be made, however, the test results and metallographic examination of the crack paths, suggest that the nucleation of the fatigue cracks are associated with SiC particles. The nucleation event could take place by either the failure of the interface between the particles and matrix or the failure of the SiC particles at the notch stress field. For monotonic loading it is now well established that above failure mechanisms of carbide particles are essentially maximum principle stress controlled and the strength of the particles is inversely proportional to the square root of their dimension (12-14). Considering the inhomogenous size of particles, therefore, the probability of finding a crack nucleating particle increases with increasing sampling volume ahead of a stress riser. As the notch severity decreases, an increase in the number of cycles to nucleate a fatigue crack is usually expected. In the case of MMCs, it appears that the reduction in the stress concentration in blunt notches is compensated by the other competing factor which is the large sampling of particles (statistically an increase in possible nucleation sites). The other important significance of the crack initiation data presented here is the indication of the errors associated with the use of fatigue data based on the test results obtained from only smooth (notch-free) specimens. As can be clearly seen, during design or selection of these materials for components containing stress risers, the division of the fatigue strength by a stress concentration factor can be considerably in error and very misleading.

In Figs. 2 - 3 the observed crack growth rates for short cracks emanating from the notches are in the order of 10-4 to 10- 5 mm/cycle. These values are higher than the values usually measured for the matrix alloy at threshold values(10-6 to 10- 8 rom/cycle) (15 ,16). Also in the overaged composite, the growth rate of fatigue cracks for entire crack lengths are slower than the T6 condition. In the 2xxx and 7xxx Al alloys produced by powder metallurgy (P/M) , due to the very small grain size insignificant effect of aging conditions on fatigue crack growth has been observed(15,16). In the work of Christman et al. (6) on a whisker strengthened 2124 alloy produced by P/M technique, an insensitivity of fatigue crack growth to matrix aging heat treatment is also reported. However, they observed that the fatigue failure predominantly occurs within the matrix, even occasionally pulled whiskers were covered by the matrix material. Although the composite system studied in this work produced by P/M route, the observed different behavior than the typical fatigue characteristics of matrix alloy and whisker strengthened composite system is again assumed to be associated with the interaction of the SiC particles with the fatigue process. As discussed recently by Ritchie et.al.(7) the result of this interaction could have a series of mechanisms. Failure of the SiC particles ahead of the growing fatigue crack tip : i) can accelerate the crack growth due to sudden large crack

extensions.

104

ii) it can also decelerate crack growth by two mechanism. First, due to the crack bridging from uncracked ligaments left behind the crack tip. Second, crack closure induced by rough fracture surfaces resulting from the failure of large particles.

CONCLUSIONS : The initiation and growth rate of cracks emanating from blunt notches in metal matrix composite consist of 606l-Al alloy and 25% particulate SiC were studied for two aging conditions. The results indicate that : 1. The initiation of the fatigue cracks is insensitive to the notch

severity and to the aging condition of the matrix alloy. 2. The growth rate of fatigue cracks is found to be sensitive to the

matrix aging condition in this composite system. 3. Growth rates of short cracks emanating from notches can be

accurately described for this material by an effective stress intensity factor range ~Keff.

ACKNOWLEDGEMENTS The author would like to thank DWA Compos i te Specialties Inc. for providing the material. This work was carried out during the time period the author was an assistant professor at Bradley University Department of Manufacturing.

REFERENCES : 1. C.R. Crowe, R.A. Gray and D.F. Hasson; in "Proceedings 5th Int. Conf. on Composite Materials" Eds. W.Harrigan et. al. 1985, p843. 2. S.V. Nair, J.K. Tien and R.C. Bates; Int. Metall. Rev. 1985, vol.30, p.275. 3. W.A. Logsdon and P.K. Liaw; Eng. Fract. Mech. 1986, vol.24, p.737 4. S.S. Yane and G. Mayer; Maters. Sci. Eng. 1986, vol.82, p.45. 5. T.E. Steelman, A.D. Bakalyar and L. Konopka; Aluminum matrix composite structural design development, Tech. Report AFWAL-TR-86 1986 6. T. Christman and S. Suresh; Mater. Sci. Eng. 1988 vol.l02A p.2ll 7. J.K. Shang, W. Yu and R.O. Ritchie; Mater. Sci. Eng. 1988 vol.l02A, p.18l 8. D.L. Davidson; Metal. Trans. 1987 vol.18A, p.2ll5 9. A.L. Davidson; Southwest Research Inst. Technical Report, Report No. NR039-283 1984. 10. S.B. Biner to be published. 11. H. Jergus; Int. J. Fracture, 1978, vol.14, p.R113 12. S.B. Biner; Canadian Metall. Quarterly 1986, vol.24, p.155. 13. J.T. Barnby and E. Smith; Met. Sci. J. 1967, vol.l, p.56 14. T. Lin, A.G. Evans and R.O Ritchie; J. Mech. Phys. Solids 1986, vo1.34, p.477 15. S. Suresh, A.K. Vasudevan and P.E. Bretz; Metall. Trans. 1984 vo1.l5A, p.269 16. K. Minakawa, G.L. Evan and A.J. McEvilly; Metall. Trans. 1987 Vo1.17A, p.1787

105

6·0r-Vl <lJ • u • 0 >- • & & u 5·0- ___ ·_~~ ____ o ___ - -.-!?t'-- ~-- -~_&- - -

notch-I ·0 A &

Z • 0 • no tch-II 01 4·0..- • notch-III 0

• no tch-IV

• no tch-V

3.0 I I I I

1·8 2·2 2·6 3·0 log Kt ~(J ( MN m2 )

Fig. 1: Correlation of crack initiation data with stress concentration and net section stress. Open symbols are for overaged and solid symbols are for T6 conditions.

102

<lJ • u 103 f- .~~:;,~ >-u '-E E 104 ~ flY z 0 notch-I

""0

'05 ..- • '- 6- notch- II 0

""0 0 notch-III -6 I I I I 1 I t t

10 100 ( N / m m3/ 2 )

1000 t!.K

Fig. 2: Correlation of fatigue crack growth data with stress intensity factor range. Open symbols are for overaged and solid symbols are for T6 conditions.

106

102

C1J

u 103 >-u

..........

E E 104

z "0 105 ..........

0 "0

106 2 0 4 6 8 10

Crac k Leng th (mm)

Fig. 3: Comparison of the predicted (solid line) and observed crack growth rates. Notch depth 5 mm root radii 0.30 mm.

Fig. 4: Crack deflection and bridging from failed SiC particles.

107

ON CONSTITUTIVE RELATIONSHIPS FOR FATIGUE CRACK GROWTH

A. J. McEvily

Metallurgy Department and Institute of Materials Science University of Connecticut

Storrs, CT 06269

ABSTRACT: Consideration is given to various mechanisms of fatigue crack growth as a function of the ~K level, and a constitutive relation is discussed which incorporates these different mechanisms. The effects of crack closure are discussed as is also the effect of an overload as influenced by ~K level and thickness. The process referred to as anomalous crack growth is considered in terms of the development of crack closure in the wake of a newly formed fatigue crack. Additionally, the need for further research on environmental effects is indicated.

1. INTRODUCTION: One of the aims of research on fatigue crack growth behavior is to develop accurate, quantitative expressions for the prediction of the rate of fatigue crack growth based upon an understanding of the mechanisms involved in the growth process. Experience has shown that the fatigue crack growth process is primarily influenced by the mechanics of loading and unloading, with Young's modulus being the dominant material characteristic in the linear elastic range under constant amplitude loading conditions. Where variable amplitude loading is involved the yield strength also becomes a parameter of explicit importance, particularly in the process of crack growth rate retardation after an overload. The role of microstructure per se in fatigue crack growth is, perhaps somewhat surprisingly, only of secondary importance. The influence of microstructure is seen mainly through its effect on crack closure, on the tendency for planar glide and Mode II deformation, and on environmental interactions. The absence of a strong microstuctural effect on fatigue crack growth is useful in that it simplifies the development of constitutive relationships. In the following section a brief review of the nature of some of the constitutive relationships used in the analysis of fatigue crack growth will be given.

2.CONSTITUTIVE RELATIONSHIPS: The most widely used expression for the rate of fatigue crack growth is that of Paris and Erdogan (1),

(1)

109 A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 109-115. © 1990 Kluwer Academic Publishers.

110

where a is the crack length, N is the number of cycles, C and mare material constants and ~K is the range of the stress intensity factor, a parameter which serves as a correlating parameter. It has been observed that m can take on values from 2 to over 4, with m increasing with decrease in fracture toughness (2). However, Liu (3) has indicated that on the basis of dimensional analysis a value of 2 would be expected. A value of 2 is also expected on the basis of crack tip opening displacement considerations, and the following expression

(2)

where B is a constant which has been found to be inversely proportional to the square of the modulus (4), would therefore be expected to provide a representation of the dependency of the rate of fatigue crack growth on ~ K if the only factor involved were Mode I deformation involving crack tip blunting on loading and crack tip resharpening on unloading. Why then is the exponent m often found to be larger than 2? The answer appears to be that even in the absence of environmental effects other growth modes are involved in propagating a fatigue crack, and these other modes depend upon the ~K level. For example, a threshold level exists which must be exceeded for propagation to occur, and Mode II deformation is importantly involved in the near-threshold growth process. In addition, at high rates of crack growth the crack growth mechanism involves not only the plastic blunting and closure of the crack tip, but also the operation of static modes of separation which serve to accelerate the rate of crack advance. These static modes of separation entail the rupture of material within the crack-tip plastic zone, and their contribution to crack advance increases as the transition to complete overload failure occurs with increase in crack growth rate, i.e., the closer ~ax is to Kcc. Since Kmax is given as

~K Kmax=J-R (3)

where R is the ratio of minimum to maximum stress in a cycle, given ~K the higher the R level the more likely are the static of separation to become operative.

for a modes

As a result of near-threshold growth processes at low ~K and static modes of separation at high~, the crack growth range over which striation formation occurs is relatively small. The lower than expected rates based on Eq. 2 in the near-threshold region and the higher than expected rates rates in the terminal region result in a plot of da/dN vs. ~ in log-log coordinates of sigmoidal shape, which, if approximated by a straight line, will result in a value for m, the exponent in the Paris-Erdogan relation, higher than 2. However, this sigmoidal shape can be modeled over the entire range by the following equation which still incorporates the idea that an exponent of 2 is appropriate in representing the plastic crack-tip opening contribution:

da A 2 ~K.tT -dN = -E2 (~K.tT - ~K.fIIh) (I + K _ K (4)

cc max

In this expression~Keff is the difference between Kmax and Kop, where Kop is the stress intensity value at which the crack tip begins to open

111

on loading, l!.Keffth is the value of l!.Keff at the threshold level, and Kcc is the fracture toughness as determined under cyclic loading to failure. In the absence of crack closure, Kop is equal to Kmin, and there is no influence of mean stress on the rate of crack growth below the terminal range. A shake-down postulate has been advanced (5) which states that under steady-state conditions in the absence of crack closure the opening load and minimum load correspond, but that upon a change at constant l!.K from one mean stress level to another there will be a transition period involving a shake-down of the plastic deformation processes in the plastic zone at the crack tip in order to reestablish the steady-state conditions. During the shake-down a transient period in the rate of fatigue crack growth will occur. For example, upon reduction of mean stress there will be a period of crack retardation before the steady state rate of crack growth is attained.

An example of fatigue crack growth which is independent of mean stress and microstructure is given in Fig. 1 (6). Under the test conditions used, i.e., in vacuum at 538t, crack closure did not develop. When closure does develop there is an R dependency on the rate of crack growth as indicated in Fig. 2a (7). However when the results are corrected for this closure the R-effect is largely eliminated except at the higherl!.K levels where static modes of separation are operative, Fig. 2b. A comparison of the predictions of Eq. 4 with experimental results for a fine-grained aluminum alloy which did not develop closure is given in Fig. 3, with the constant A being equal to 9.4.

z " ';1

"

~

S-

f.: Vacuum

a Z 114 Cr ,M- 0 hIocI.8Cr

;I • 9Cr-2104o

; O~" R.o.05 o SoIiCI R.o.5

20 30 10050 100 200 AK (IoIPaYm)

Fig.l. The rate of fatigue crack growth of several steels at elevated temperature on vacuum as a function of l!.K (6).

3.PLANE STRESS: Under plane stress conditions, as Elber (8) has pointed out, once a correction for plasticity induced closure has been made, the rate of growth should also be independent of the mean stress level. However most fatigue crack growth processes are neither purely plane stress nor plane strain in nature. For example, when crack growth occurs in a plane specimen it is usually considered that a state of

112

plane stress exists at the surface of the specimen and that plane

.,->

'IS"

l~ t .... . .

rt'

.,. t t

~ .. -. .. . ..-... .. .

~ rr ..

. . .!aIL' • . ,.~

.' ...... , .:,.­

.: .-."", . ~: ..'

(a) (b)

00

Fig.2. Rate of fatigue crack growth in the aluminum alloy 7090-T6 as a function of R and (a) 6K. and (b) 6Keff (7).

1/1"

'. : ,;

D·I .... rr

t I D"

S .,.

D"

All __

Fig.3. Rate of fatigue crack growth in the aluminum alloy IN 9021 as a function of Rand 6K (7). Here 6K = 6Keff. no closure.

strain exists along the crack front in the interior of the specimen. It is of interest that little is understood about the mechanism of fatigue crack growth under plane stress conditions. Under mixed plane strain­plane stress conditions Davidson and Lankford (9) have observed by SEM that the growth process in plane stress is intermittent in nature, that is the crack does not advance in each cycle. While the crack tip is stationery however the tip grows progressively more blunt with cycling before advancing and repeating the progressive blunting process. One

113

interpretation of such behavior is that the sub-surface, plane-strain crack is advancing with each cycle, whereas the crack in the plane­stress surface region is not, but instead blunts to an increasing degree the more it lags behind the subsurface crack. Once the surface crack tip strain has reached a critical value the crack tears ahead to catch up with the subsurface crack, and the process is then repeated.

4. OVERLOAD EFFECTS: A single overload of sufficient magnitude can significantly retard the subsequent rate of crack growth. This effect has been most often studied in aluminum alloys and is known to be associated with the plane-stress overload plastic zone at the surface of the specimen. For example, the machining away of the surface layers after an overload can lead to the virtual elimination of the retardation effect (10). Therefore since the overload-retardation process is plane stress in nature one would expect to find that thickness plays a role in the process, and indeed this is the case. An example of effect of specimen thickness on the number of delay cycles spent in the retardation period following a 100% overload for the aluminum alloy 2024-T3 is given in Fig. 4 (11). This retardation occurs because of the development of increased crack closure in the wake of the tip of the fatigue crack as it penetrates the plastic zone associated with the overload. Within this zone the material has contracted laterally to a greater extent as compared to the lateral contraction at the baseline level. This additional contraction results in more material being present in subsurface planes immediately adjacent to the surface than would ordinarily be the case. As a result the extent of the region within which compressive stresses are formed on unloading is larger than that for the baseline level. As the fatigue crack advances through this region these compressive stresses are relaxed and an expansion of the formerly compressed material occurs. This expansion in turn leads to an increase in the closure level and a reduction in the rate of crack growth.

The excess closure due to the overload is given by (12)

KOL K~l li2 Excess Closure = E.C ... -2 - {I - exp[ - (--2 -) ]}

2frGy B (5)

and the number of delay cycles (defined as the total number of required to traverse the overload plastic zone minus the required in the absence of an overload) is given by (11)

K~L I I Nd =-2- x [ ] frGy A (~K - E.C. - ~Klh l (~K - ~Klh)2

(6)

cycles number

In deriving this expression it was assumed as a simplification that a constant level of closure and hence a constant rate of growth occurred throughout the plastic zone. The effect of thickness, B, is contained in the exponential term. A comparison of the number of delay cycles predicted by Eq. 6 with experimental results is given in Fig. 4. It is of particular interest that the U-shaped nature of the experimental results is predicted.

5. ANOMALOUS FATIGUE CRACK GROWTH: Crack closure can also be of particular importance with respect to the growth behavior of newly

114

lOGO

500

300

~ 2SO . io JOO

t .. ISO <.I

i 100

50

m4-T3CIM.I. 10ft0,U .2._ .3.~ ... --c....-

.01 0.1 0.1 0.' 1.0 O.Lh_z..~

Fig.4. The number of delay cycles after a 100% overload. Experimental results (11) compared to calculated.

Fig.5. Effect of closure devel­opment on fatigue crack growth in ger-1Ho alloy (12).

formed cracks. As the crack first forms there is no wake and no closure, but over a distance of the order of a millimeter crack closure will develop. We will be concerned with crack closure in the near threshold region, and in this region roughness-induced and oxide induced closure are of greatest importance. Plasticity induced closure has relatively little effect in this region. It is of interest that for steels, Hamberg et al. (13) have shown that the Kop level is related to the level of roughness through the following relationship:

(7)

This relationship has been generalized to include other alloys to the form (14)

Ko = 1.2_E- HI I3 P ESleel

(8)

These relations indicate that as the roughness approaches zero the closure level should also approach zero, and indeed this circumstance has been observed in the case of an aluminum alloy of sub-micron grain size (7). A similar correlation between roughness and Kop has also been observed by Kemper et a1. (15).

The following relationship between the value of Kop, a transient level in the wake of a newly formed crack, and Kopmax, the opening level associated with a macroscopic crack has been proposed (16):

(9)

where k is a material parameter of unit l/rnm, and 1 is the length of the newly formed crack in millimeters. This equation is useful in understanding the so-called anomalous growth of a crack from a stress raiser in Its early stages of growth. Experience has shown that in such

115

a case the observed rate of growth may be higher than that expected on the basis of macroscopic crack growth characteristics because closure is not fully developed in the case of a new crack and hence the rate of growth is higher since Keff is higher. The following equation gives the rate of crack growth including these transient effects:

k = Al [AK - (I - e-kl ) (Kopmax - Kmin) - AKellllY (0) dN

Eq. 10 can be used to predict crack growth behavior for various initial values of Keff under constant load amplitude conditions. The results of such calculations are shown in Fig. 5 (12). Between the initial 6.Keff line and the macroscopic 6K lines various forms of transitional behavior can occur depending upon the initial6.Keff level. As has been observed, a non-propagating crack forms at the lowest initial 6.Keff level. Consideration of closure development is also useful in understanding the basis for the fatigue notch-size effect (17).

6. CONCLUDING REMARKS: As indicated above it is possible to model some of the purely mechanical aspects of the fatigue crack growth process, and such modeling can provide a baseline for consideration of the modifying effects of the environment. Usually the environment exerts a deleterious influence in that the rate of growth increases on going from vacuum to an air environment. There are cases however where the air environment may prove beneficial as with steels at elevated temperatures where oxidation can result in an increase in closure and a reduction in crack growth rate in the near-threshold region. The development of constitutive relationships which include the effects of the environment is a challenge, and more work directed at understanding the detailed influence of the environment on the mechanisms of fatigue crack growth is needed for the development of such relationships.

ACKNOWLEDGEMENT: Appreciation is expressed for support of research to the U .. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science under Grant DE-FG02-84ER45109.

REFERENCES: 1. P.C. Paris and F. Erdogan, Trans. ASME, 85, (1963) 528. 2. R.O. Ritchie and J.F. Knott, Acta Met, 21 (1973) 639. 3. H.W. Liu, Trans. ASME, 83 (1961) 23. 4. R.J. Donahue et al., Int. J. Frac. Mech., 8 (1972) 209. S. A.J. McEvily, Fatigue '87, 3, (1987), EMAS, 1503. 6. H. Nakamura et al., Microstr.and Mech. Behavior, (1986) EMAS, 1503. 7. K. Minakawa et al., Met. Trans., 17A (1986) 1787. 8. W. Elber, Eng. Frac. Mech., 2 (1970) 37. 9. D.L. Davidson and J. Lankford, Fat.Engr.Mat.& Struct., 7 (1984) 29. 10. A.J. McEvily, Met. Sci. 11 (1977) 274. 11. R.S. Vecchio et al., Scripta Met., 17 (1983) 343. 12. A.J. McEvily and Z. Yang, Proc ICF7 (1989) Pergamon, to be pub. 13. K. Hamberg et al., Fatigue '87, 1 (1987) EMAS, 135. 14. A.J. McEvily and Z. Yang, Proc. ECF7 (1988), EMAS, to be pub. 15. H. Kemper et al., Eng. Frac. Mech., to be pub. 16. A.J. McEvily and K. Minakawa, Scrita Met., 18 (1984) 71. 17. A.J. McEvily, Eng. Fract. Mech., 28 (1987) 519.

Effect of Microstructyre on the short crack growth in AI-2Q24-UA and AI-8Q9Q-UA.

D.Downham (National Research Council,Ottawa,Canada),G.W.Lorimer and RPilkington (University of Manchester,U.K.)

Abstract The purpose of this study was to investigate the effect of microstructure on the short fatigue crack growth rates of a range of aluminium alloys. The two alloys chosen were AI-2Q24 based on AI-Cu-Mg and AI-8090 based on AI-Li-Cu-Mg-Zr. The alloys were fatigued in tension­tension where fatigue tests were interrupted at fractions of the expected fatigue life and specimens prepared for examination in the transmission electron microscope (TEM).Four-point bending tests were also carried out and fatigue crack growth rates measured. Material from the vicinity of the crack-tip was removed and specimens prepared for examination in the TEM. A catalogue of dislocation substructures has been obtained from these specimens From these substructures and the observations of other workers it has been possible to rationalise the short crack behaviour of the two alloys in terms of the slip distribution. 1.Q Introduction Previous electron optical studies of the fatigue process by Forsyth (1), Duquette and Swann (2) and Vogel et al. (3) were carried out in the long crack range either on pure materials or on single crystals. The present work concentrates on commercial alloys. The paper presented here forms part of an overall study of the effect of microstructure on both long and short crack initiation and growth in a range of commercial aluminium alloys. The two alloys studied are AI-2Q24 (AI-Cu-Mg) and AI-8090 (AI-U-Cu-Mg-Zr), both in the underaged condition.

The microstructures of the commercial aluminium alloys are complex, consisting of an AI-matrix and three main precipitate particle types:(i) inclusions, complicated insoluble compounds of alloying elements generally 1-20~m in size,(ii) dispersoids, also complex compounds of various alloying elements ranging from 50-500nm in size and thus only visible in the electron microscope and (iii) strengthening preCipitates, which are developed by solution heat-treatment and aging to give the alloy its required strength. The size of these particles is generally in the range of 5-50nm.

The present work concentrates on the effects of changes in the strengthening precipitates and dispersoid types on the short crack growth process. The investigation consists of two distinct sections: an investigation of pre-crack damage and an investigation of damage ahead of the advancing crack tip. 2.0 Experimental Prior to fatigue testing the materials were thoroughly characterised by measuring grain size,Q.2% Proof Stress, and U.T.S., and by investigating the initial microstructures and dislocation distributions in the TEM.

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A. S. Krausz et al, (eds.), Constitutive Laws of Plastic Deformation and Fracture, 117-123. © 1990 Kluwer Academic Publishers.

118

Specimens of the geometries outlined in figures 1 A and 1 B were prepared for tension­tension and four-point bending fatigue tests respectively. Tension-tension fatigue tests were carried out in an ESH servo-hydraulic fatigue machine with a maximum load capability of 100kN, at 60Hz and a load ratio of 0.25. Four-point bending tests were carried out on an Instron servo­hydraulic fatigue machine with a maximum load capability of 500kN, at 10Hz with a load ratio of 0.25. SIN data was collected for both tests and fatigue crack growth rate data was obtained from the four-point bending tests where possible. It should be noted that the primary aim of the fatigue tests was to produce short cracks for examination in the TEM and not for a rigorous fatigue crack growth rate measurement. 3.0 Results and Discussion Table 1 summarises the heat-treatment, grain size, 0.2% Proof Stress and U.T.S. results obtained on the AI-2024 and AI-8090 materials.

Figure (2) shows the fatigue crack growth rate measurements made on the AI-2024 and AI-8090 materials. Note that the AI-8090 material appears to have a lower short-crack fatigue growth rate than the AI-2024. Both of these short cracks arrested; this crack arrest phenomenon is likely due to the specimen geometry and the plastic zone around the notch ..

Figures (3)-(5) show the initial microstructures and dislocation distributions in the alloys. The strengthening precipitates in the AI-2024 material are G.P. zones which are not resolvable at the magnification of figure (3). The dispersoid particles are Cu-rich and Mn-rich , most probably CuAI2 and A120Cu2Mn3. Figure (3) also shows a high density of helical dislocations which form during quenching from the solution heat-treatment temperature. Figure (4) shows the high volume fraction of 0' contained in the AI-8090-UA material.and the AI3Zr dispersoid particles. Figure (5) shows a well developed subgrain structure with a relatively sparse dislocation density within the grain. 3.1 Pre-Crack Dislocation Damage AI-2024-UA Figures (6)-(8) show the accumulation of dislocation damage with increasing percentage of fatigue life. At 30%, figure (6) shows a low dislocation density with some dislocation-dispersoid particle interaction. By 70%, figure (7), there is sufficient dislocation-dispersoid interaction to generate dense dislocation tangles around dispersoids. At 90%, figure (8), networks of dislocations, delineated by dislocation tangles and dispersoid particles are well established throughout the specimen.

From the observations made on the AI-2024 material it is possible to determine how the dislocation damage accumulates. There is a large concentration of helical dislocations in the microstructure prior to fatigue cycling. During fatigue, dislocations are generated which move through the matrix and interact with the strengthening precipitates, dispersoid particles and helical dislocations. The G.P. zones are easily cut by the moving dislocations and do not form an effective barrier to the movement of dislocations. These moving dislocations also interact with the helices and dispersoids and their movement is impeded by this interaction, giving rise to the dense dislocation tangles and network structures observed in the latter stages of the fatigue life. Due to the repetitive interactions between moving dislocations, helices and dispersoids the resultant slip distribution in the AI-2024 material is homogeneous. No evidence of heterogeneous slip i.e. long straight dislocations or persistent slip bands (PSBs) was observed in any of the pre­crack damage specimens of AI-2024. The dislocation-dislocation and dislocation-dispersoid interactions during the fatigue process appear to inhibit reverse glide and promote cross-slip. 3.2 Crack-tip Damage AI-2024-UA Figures (9)-(11) show the dislocation distributions obtained from specimens taken in the Vicinity of the crack-tip.Two observations have been made on the crack-tip damage specimens which have not been made on the pre-crack damage specimens: the formation of PSBs and a cell-like network of helical dislocations.

119

Figure (9) shows the PSBs associated with small surface cracks observed in the AI-2024-UA material. PSBs might be expected to form in the AI-2024-UA material since the G.P. zones are easily cut by dislocations and therefore are more likely to promote heterogeneous deformation. It is interesting to note that the PSBs are not observed in the pre-crack damaged specimens even at the limit of the fatigue life. The pre-crack specimens were taken from bulk material whereas the crack-tip specimens were taken from the surface. Due to the microstructure of the AI-2024 material and the initial dislocation distribution the formation of PSBs may only be possible near the free surface since dislocations are able to flow to the surface allowing further dislocations to pass down the same slip plane, giving rise to more heterogeneous deformation. Vogel et al. (3) make a similar observation in their work on AI-Zn­Mg single crystals, claiming that PSBs originated near the free surface and progressed into the bulk material. In the bulk AI-2024 material microstructural barriers such as dispersoids or helices impede dislocation motion and cause dislocation pile-ups. Thus cross-slip is more likely than further slip along the initial slip plane.

The appearance of a cell or subgrain structure in the immediate vicinity of the crack tip has been observed by other workers, Grosskreutz and Shaw (4) and Wilkins and Smith (5). Figures (10) and (11) show the dislocation arrangement observed in the AI-2024 material; this arrangement consists of dislocation helices formed in an array which resembles a cell-like structure. The appearance of the helical dislocations is similar to that observed in the undeformed material; the two sets of helices may be distinguished by the pitch of the helix. The helices formed during the deformation process appear to have a smaller, tighter, pitch than those formed during quenching; this point is illustrated in figure (10). 3,3 Pre-crack Dislocation Damage AI-8090-UA The pre-crack dislocation damage in the AI-8090-UA material shows a heterogeneous slip distribution. Figures (12)-(13) taken from the specimen at the limit of the fatigue life show PSBs and straight dislocations. Observations indicate that these features develop early in the fatigue process and that heterogeneous deformation becomes more intense as fatigue continues.

The intense heterogeneous slip may be attributed to the microstructure of the material. The major strengthening precipitates of the AI-8090-UA material are the spherical, coherent d' particles whose main strengthening effect is their resistance to shearing, which is low. Since the coherency strains and interfacial energy between matrix and precipitate are also low then the major contribution to the resistance to shearing is the long range order of the preCipitates. This long range order is reduced when 8' particles are sheared, making further cutting more favorable. This results in co-planar slip and strain localisation.

The coherent AI3Zr dispersoid particles in AI-8090-UA are also capable of being sheared by dislocations as long as they are below the critical size for Orowan looping to occur. The measured size of the dispersoid particles in this material indicates that a proportion of the dispersoids are below this size.

The heterogeneous slip is further enhanced by the pronounced crystallographic texture in the material. Observations have shown that dislocations and PSBs easily traverse grain and subgrain boundaries. 3.4 Crack-tip Damage AI-8090-UA Figure (14) shows the dislocation substructure obtained in the vicinity of the crack tip, characterised by long, predominantly straight dislocations. It is clear that the craCk-tip damage is similar to the pre-crack damage and is dominated by heterogeneous deformation. In the AI-8090-UA material there are few microstructural features which may moderate the heterogeneous slip distribution. There is a sparse initial dislocation density and no evidence of dislocation helices as in the AI-2024.

120

By considering the microstructural information obtained it is possible to rationalise the short crack growth behaviour of the two alloys. It is clear that the lower crack growth rate of the AI-8090-UA material is related to the predominantly heterogeneous mode of deformation. Dislocations are free to glide along slip planes unimpeded during the rising part of the fatigue cycle; similarly they are free to experience reversed glide during the falling part of the load. Thus the number of dislocations which effectively contribute to crack growth is reduced. In the AI-2024-UA material there are sufficient microstructural barriers to prevent reversed glide from taking place; thus the to and fro motion of dislocations along the same glide plane is inhibited and cross-slip is promoted since it is energetically easier for the dislocations to Slip on another Slip plane than to overcome the microstructural barrier. Thus during each fatigue cycle the effective number of dislocations which contribute to the growth process is larger than the equivalent case for the AI-8090-UA material. Therefore the short fatigue crack growth rate for AI-2024-UA is greater than for AI-8090-UA. 4.0 Conclusions 4.1. AI-8090-UA exhibits a lower short crack growth rate than AI-2024-UA. 4.2. Whilst both materials contain shearable strengthening precipitates they exhibit different modes of deformation during fatigue. The AI-2024-UA material shows predominantly homogeneous deformation whilst the AI-8090-UA material shows predominantly heterogeneous deformation. 4.3. The difference in deformation mechanism may be explained by the difference in dispersoid type and the initial dislocation density in the material. The AI-2024-UA contains semi-coherent dispersoids which are effective in impeding dislocation motion. The dispersoids in AI-8090-UA are coherent and shearable below a certain size and are thus less effective in impeding dislocation motion. 4.4. The initially high helical dislocation density is also effective in impeding dislocation motion in the AI-2024-UA whereas the AI-8090-UA has a sparse initial dislocation denSity. References 1 P.J.E.Forsyth, Acta Metallurgica, 11, 1963,p.703 2. D.J.Duquelte and P.R.Swann, Acta Metallurgica,24,1976,p.241 3.W.Vogel,MWilhelm and V.Gerold, Acta Metallurgica,30,1982,p.21 4.J.C.Grosskreutz and G.G.Shaw, Acta Metallurgica,20,1972,p.523 5.MAWilkins and G.C.Smith, Acta Metallurgica, 18, 1970,p.1 035

Table 1: Heat-treatmenl,Grain size,0.2%Proof Stress and U.T.S. Result~

~Heat-treatment Grain size AI-2024-UA 1 h @ 4800C

water quench 1h@ 1900C

AI-8090-UA 1 h @ 5300C water quench 1h@ 1950C

0.2% proof Stress .u.r.s.. 233x35x471lm 250 MPa maximum 95x16x28llm

minimum 47x22x251lm

maximum 3x3x31lm

minimum

330MPa

420MPa

420MPa

1: 0 : ~75--~ r25~

5 ___ _ c:::r:::c:J __ ... 5 iE-Z5~

Figure 1A: Tension-tension fatigue test specimen geometry

{ SOI.AD

Figure1 B: Four-point bending fatigue test specimen geometry

-6." ,--_______________ ---,

log da/dN

-7.2'

-8.41 • ''''''Is HI-21l4-11ft

ftfJ "

\ • -'.n

~ o HI-I".-1Ift

-11.8'

-12." '--__ ...1... __ --'~ __ _'_ __ ___L __ ----' .... 1.21 2." 3.61 .... '.11

Figure 2: Comparison of short crack growth rates for AI-8090-UA and AI-2024-UA.

Figure 3: Initial microstructure and dislocation distribution in AI-2024-UA Electron beam direction <110>. x10000

Figure 4: Initial microstructure of AI-8090-UA showing 0' particles and AI3Zr dispersoids. Electron beam direction <100>. x36000

121

122

Figure 5: Subgrain boundary structure in AI-8090-UA. xi 0000

Figure 7: Dislocation network in AI· 2024-UA @ 70% of fatigue life showing dense tangles around dispersoids. Electron beam direction <11

Figure 9: PSBs ahead of crack-tip in AI-2024-UA. x17000.

Figure 6: Early network formation in AI-2024-UA @ 30% of fatigue life. Electron beam direction <110> g=002,5=0. x36000

Figure 8: Detail of di5per50id­dislocation interactions at 90% of fatigue life. Electron beam direction <110>.9=-111,5=0. x36000

Figure 10: Dislocation substructure in AI-2024-UA containing multiple short cracks. Electron beam direction <110> g=002,s=0. x36000

Figure 11 :Detail of dislocation network of figure 10. Electron beam direction <110>.x46000

Figure 12:PSBs cutting boundaries in AI-8090-UA @ 90% of fatigue life. x10000

Figure 13: Heterogeneous slip traces Figure 14:Dislocation substructure in AI-8090-UA. Electron beam direction in the vicinity of the crack-tip in <110>,g=002,s=0. x36000 AI-8090-UA. Electron beam direction

<1 00>,g=02-2,s=0. x28000

123

CYCLIC PLASTIC INSTABILITY IN PURE ALUMINUM AND ALUMINUM ALLOY 7075 T6: EFFECTS OF TEMPERATURE,

STRAIN RATE, AND WAVEFORM.

P. Li*, N.J. Marchand* and B. Ilschner**

* Respectively, Post-Doctoral Fellow and University Research Fellow, Departement de Genie metallurgique, Ecole Poly technique , P.O. Box 6079, Station A, Montreal, Quebec, Canada, H3C 3A7.

** Professor, Department of Materials, Swiss Federal Institute of Technology, Ch. de Bellerive 34, CH-I007, Lausanne, Switzerland.

ABSTRACT: Fully reversed strain-controlled push-pull tests were per­formed on polycrystalline specimens of commercially pure aluminum 1199 and aluminum alloy 7075 in order to investigate the influence of strain rate, waveform and temperature on cyclic plastic instability. The results show that shape instability is a strong function of temperature and of strain waveform, while the effect of strain rate is relatively small. Transmission electron microscopy (TEM) studies showed several dislocation substructures (either homogeneous or highly inhomogeneous ones) prevailing in the specimens that experienced geometrical insta­bility. No specific dislocation substructure was found associated with platic instability. The results of this study (mechanical and TEM observations) indicate that geometrical instability requires: (1) a critical amount of anisotropy in the yield behaviour of the material, and (2) the formation of macroscopic shear bands. No particular micro­structural softening event is needed. Accordingly, force-deformation constitutive relations to model the observed behaviours are suggested.

1. INTRODUCTION: Plastic instability or shape instability is a common phenomenon in materials deformed either monotonously or cyclically; the most simple example being the necking of a tensile specimen. Although several criteria have been proposed to rationalize the appearance of shape instability under monotonic loadings (such as those occurring in forming processes) [1-3], the case of instability resulting from cyclic

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A. S. Krausz et af. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 125-132. © 1990 Kluwer Academic Publishers.

126

loading is more complex. The fact is that an understanding of the physical mechanisms involved is lacking and this has precluded the development of adequate and physically motivated constitutive equa­tions.

Similar to monotonic loading, cyclic instability involves the develop­ment of macroscopic localized plastic zones [4-11]. However, contrary to the belief that asymmetric loadings or non-uniform geometries are required [5,8], this phenomenon has been shown to occur in uniform gauge section specimens under isothermal fully reversed deformation loadings [4,11-12]. Furthermore, the strain amplitude necessary to trigger shape changes has been found to be very small (~Et = 0.25%) compare to monotonic deformation [11-12].

Preliminary finite element (FE) modelling of geometrical instabilities has indicated that successful predictions can be obtained by consider­ing a small initial anisotropy in the yield function of the materials [13]. In agreement with the experimental observations [11], the FE results showed a strong influence of specimen geometry and of strain amplitude on shape instability. However, the good correlation (between experimental and predicted results) does not provide proof thant an initial anisotropy in the yield (or flow) behaviour is responsible for strain localization and shape instability. As indicated by Rice [14], other mechanical phenomena can permit to rationalize shape instability. It is still an open question whether localization occurs because of the progressive geometrical softening of the materials or because some other instability of the plastic flow process (i.e. micromechanism) first occurs.

The aim of this paper is to report detailed information pertaining to the effects of strain rate, waveform, and temperature on the occurrence of cyclic plastic instability, and to combine these results with micro­structural analyses. It was felt that the necessary and sufficient variables, required for adequate constitutive modelling, could be prop­erly identified by systematicaly varying these three parameters and by measuring their corresponding effect and microstructures on geometrical instability.

2. MATERIALS AND EXPERIMENTAL PROCEDURE: The materials selected for this study were aluminum 1199 (min. 99.99% AI) and aluminum alloy 7075-T6. Al-1199 was chosen because it is representative of pure high stacking-fault energy material (easy cross slip) and known to be proned to geometrical instability [6,7,11]. Al-7075 T6 was chosen because it is a low stacking-fault energy (planar slip) precipitation hardened material. Thus, these two materials covered a wide range of micro­structures and micromechanisms of cyclic plastic deformation [15-17]. It is expected that the conclusions drawn from this study could be applied to other FCC metal systems and alloys. The chemical composition of the materials, the final microstructure before testing, as well as the corresponding thermo-mechanical treat­ments, are presented in detail elsewhere [11-12,18].

Cylindrical specimens with a diameter of 10mm and a gauge length of 18rnrn (Al-7075) or 20rnrn were machined on a digital programmable lathe.

127

The final passes removed less than 0.12srnrn. All the specimens were mechanically polished with 600 grit emery paper and electropolished to remove a surface layer of at least 12s~m. The low cycle fatigue tests were performed in air at 20, 120, ISO, and 260 a C on a computer con­trolled servohydraulic testing system [12]. Fully reversed pull-push strain cycling was employed at a constant total strain amplitude (~£t/2) of ± 0.4% for Al-1199 and ± O.S% for Al-707s. Six different waveforms (triangular and trapezoid) were employed with the first half­cycle applied in compression. These waveforms are: symmetric fast­fast (f-f), symmetric slow-slow (s-s), slow-fast (s-f), fast-slow (f-s), fast-hold-fast (f-h-f), fast-fast-hold (f-f-h). The fast strain rate and slow strain rate were respectively 2 x 10-3sec- 1 and 2 x 10-5sec- 1 . The hold time period was 396s. A minimum of two tests were performed for each testing condition. Optical, scanning and transmission electron microscopy were used to examine the microstruc­tures of the specimens after testing.

3. RESULTS: Four specimens which exhibited shape changes during strain cycling are shown for examples in Figure 1. These geometry changes can be described in terms of center-necking, center barrelling, off-center necking, or off-center barrelling deformation mode. The occurrence of any of these modes depended on the applied testing conditions. For example, applying a tensile hold time (at maximum strain) induced cen­ter-barrelling, while a compressive hold time (at mLnLmum strain) induced center-necking as shown in Figure 1. The number of cycles at which either necking or barrelling was detected varied between 30 to some hundred cycles depending on the testing conditions. In order to provide a quantitative measurement of instability, a specimen geometry parameter (~) was introduced. This parameter was defined as

[D 2 max -

D o

D 2] min

·100

2

where Dmax , Dmin and Do are the maximum, minimum and original diameter of a deformed specimen.

The results pertaining to Al-1199 are summarized in Figures 2 and 3. As shown in Figure 2, plastic instability decreases with increasing tem­perature. Note that the ~'s at 20°C were measured after 120 cycles while the other measurements (at 120, 180 and 260°C) were carried-out after 2000 cycles. This was done so because the specimens tested at 20°C experienced drastic changes of geometry requiring test interrup­tions after a small number of cycles (~400). For the same reason, the ~'s of the specimens tested under s-f and f-s conditions were taken at N 200 while, for the other conditions, the ~'s were measured at N ~ 500.

Figure 3 shows that instability is a strong function of the imposed waveform. As expected, the asymmetric waveforms (s-f and f-s) induced the greatest amount of instability (for equal temperature and strain range), while symmetric strainings with no hold time were the most

128

stable. Figure 3 also indicates that the effect of strain rate (s-s and f-f) is relatively small. It should be mentioned at this point, that the ratio of the plastic strain amplitude over the total strain amplitude (i.e ~ep/~et) varied between 0.70 at 20°C to about 0.93 at 260°C. In other words, the plastic strain amplitudes (although small compared to monotonic loadings) were always larger than the elastic strain amplitudes.

The results pertaining to Al-7075 T6 showed instability to occur only in specimens tested at 260°C. At lower temperatures, macroscopic homogeneous cyclic deformation prevailed regardless of the waveshape and strain rates used for the testing. It should be pointed out that the ratios ~ep/~et were small (less than 3.0%) at 20 and 120°C, about 30% at 180°C, and more than 70% at 260°C. Comparing those ratios with the ones measured in Al-1199 indicates that a critical value of ~ep/~et (~O.30) is required to entail shape instability.

Optical and SEM observations of the fatigued Al-1199 specimens showed the activation of various deformation mechanisms with changing tem­peratures. At low temperatures (20 and 120°C) multiple slip systems were activated simultaneously or alternatively depending on the testing conditions [12,19]. At high temperatures (~ 180°C) grain boundary sli­ding (CBS) was observed and, as expected, increased with increasing temperature. At 260°C, slip bands, macroscopic shear bands and evi­dences of CBS were found in the fatigued specimens of Al-1199 and Al-7075 T6. In contrast with the observations made on Al-1199, the Al-7075 T6 specimens tested at low temperatures (T ~ 180°C) did not show trace of shear bands. Only slip bands, uniformly distributed on the gauge sections, were observed. The density of slip bands increased with temperature. These results indicate that shear bands formation is a necessary condition (but not sufficient) for geometrical instability to occur.

TEM observations of fatigued Al-1199 specimens tested at 20, 120 and ISO°C revealed the co-existence of three dislocation substructures. These are intense dislocation bands, dislocation cells, and subgrains [18]. This reflects the fact that three strain regimes prevailed in the specimens. The number of dislocation bands was found to decrease with increasing temperature and none could be found at 260°C [19]. Note that the absence of geometrical instability coincides with the absence of yield anisotropy, and with the disappearance of dislocation bands.

TEM observations of Al-7075 specimens fatigued at 20, 120 and 180°C showed microstructures consisting of intense dislocation bands imbedded in a matrix containing few dislocations pinned by second phase par­ticles [18-19]. The density of dislocation bands was also found to decrease with increasing temperature. At 260°C, a homogeneous network of dislocations pinned by second phase particles was observed in every fatigued specimen. This result is in contrast with the ones obtained with Al-1199 which showed greater amount of shape instability with increasing inhomogeneity of the substructure.

129

DISCUSSION: A review of the results indicates that there are three necessary and sufficient conditions for the occurrence of shape insta­bility. These conditions are: (1) a critical amount of anisotropy in the plastic yield function of the material, (2) a minimum amount of cyclic plastic deformation (~£p/~£t = 0.30) and (3) the formation of macroscopic shear bands. It should be emphasized that none of these conditions, taken alone or in combination of two, is sufficient to cause shape changes. The important data and arguments leading to the above conclusions are discussed in detail elsewhere [20].

One of the most important results of this study, is the fact that no specific dislocation substructure could be associated with shape insta­bility, shape changes occurring in either highly inhomogeneous sub­structures or in homogeneously deformed material. This is not to say that shape instability is totally independent of the detailed micro­structural events, but rather to emphasize the fact that shape changes are a direct consequence of an instability in the constitutive descrip­tion of a homogeneous deformation field. In other words, it is the macroscopic nature of plastic deformation that controls plastic insta­bility and not the occurrence of some particular microstructural sof­tening events. The main function of the detailed micromechanism of plastic deformation, besides controlling the kinetic aspects of insta­bility, is to determine whether microcracking will occur before or after instability. Hence, if significant microcracking develops early in the life (at inclusions, grain boundaries or slip bands) the bulk state of stress will be altered due to increasing compliance of the specimen. In that case, the bulk state of stress is controlled by the local fracture events (at the crack tip) rather than by the imposed deformation field. Crack propagation (controlled by localized fracture events) then becomes the most effective path for dissipating strain energy and supersedes bulk shear band formation and shape instability (controlled by bulk macromechanisms of plastic deformation).

Because shape changes can be viewed as the result of a critical insta­bility into the macroscopic deformation field, it follows that the for­mulation of force-deformation constitutive relations does not require a precise description of the micromechanical phenomena involved and their interactions. Hence, the constitutive relations need only to be struc­tured on the basis of the characteristics of the major mechanical phe­nomena and the constraints of continuum theory. At the limits, if a mathematically motivated concept is introduced, its physical origin needs only to be checked to ensure that it is an appropriate component of the theory being developed. Analogously for a physically motivated component its mathematical representation should be checked to ensure that it satisfies the requirement of continuum theory. In particular, it can be argued that the use of a quadratic expression for the yield condition and a different one for the flow potential (although it needs not to be) is physically justified and should result in good predic­tions of the yield stresses and strain localization during cyclic deformation [13,20].

CONCLUSIONS: Fully reversed strain-controlled push-pull tests were performed on pure aluminum Al-1199 and aluminum alloy 7075 to investi-

130

gate the influence of temperature, strain rate and waveform in promot­ing cyclic plastic instability. The main conclusions of this study can be summarized as follows:

1. Shape instability is a strong function of temperature and strain waveform. The effect of strain rate is relatively small in com­parison.

2. Several dislocation substructures, either homogeneous or highly inhomogeneous ones, were found in specimens that experienced shape instability. No specific dislocation substructures could be found associated with shape instability.

3. Shape instability occurred in specimens when all of the three fol­lowing macroscopic conditions were met: (a) a minimal amount of anisotropy in the yield behavior, (b) a minimal amount of plastic deformation (~£p/~£t > 0.3), and (c) formation of non­crystallographic shear bands.

4. The detailed micromechanisms of cyclic plastic deformation are important in the sense that they might lead to extensive micro­cracking before shape instability.

REFERENCES

1. C. Rossard, Rev. Metall., vol. 63, 1966, pp. 225-235 2. E.W. Hart, Acta Metall., vol. 15, 1967, pp. 351-355 3. J.D. Campbell, J. Mech. Phys. Solids, vol. 15, 1967, pp. 359-370 4. R.P. Skelton, Mater. Sci. & Engng., vol. 19, 1975, pp. 25-30 5. H.J. Westwood, in Fracture 1977, vol. 2, University Press of Water­

loo, 1977, pp. 755-765 6. L.F. Coffin, J. Basic Eng. (Trans. ASME) , vol. 82D, 1960, pp.

671-682 7. M.H. Raymond and L.F. Coffin, Acta Metall., vol. 11, 1963, pp.

8. 9.

10. 11.

12.

13.

14.

801-807 L.F. Coffin, ASTM, STP 612, 1976, pp. 227-238 K.D. Sheffler and G.S. Doble, ASTM, STP 520, 1973, pp. 491-499 K.D. Sheffler, ASTM, STP 612, 1976, pp. 214-226 P. Li, N.J. Marchand and B. Ilschner in Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials, edited by K.-T. Rie, Else­vier Applied Science, 1987, pp. 55-64 P. Li, Doctoral dissertation, Swiss Federal Institute of Technolo­gy-Lausanne, 1988, No. 735 N.J. Marchand, P. Li and B. Ilschner, in Int. Seminar on Inelastic Behaviour of letaud et aI, pub. by MECAMAT, Besan90n, J.R. Rice, in Theoritical and Applied Koiter, North-Holland Pub., p. 207, 1976

the Proc. of the Solids, edited by G. 1988, pp. 415-426 Mechanics, ed. by

Third Cail-

W.T.

15. H. Mughrabi, F. Ackermann and K. Herz, in Fatigue Mechanisms, ed. by J.T. Fong, ASTM STP 675, p. 69, ASTM, Philadelphia, 1979

16. H. Mughrabi, Proc. 7th Int. Conf. on the Strength of Metals and Alloys, Montreal, vol. 3, p. 1917, 1985

17. A. Korbel and M. Richert, Acta Metall., vol. 33, p. 1971, 1985 18. P. Li, N.J. Marchand and B. Ilschner, "Mechanisms of Crack' Initia­

tion and Growth in Low Cycle Fatigue of Aluminum Alloy T075-T6",

131

submitted to Mater. Sci. & Engng. 19. P. Li, N.J. Marchand and B. I1schner, "Dislocation Substructures

in Fatigued Aluminum and Aluminum Alloy 7075", to be published. 20. N.J. Marchand and P. Li, "The Mechanical Processes of Plastic

Localization and Shape Instability", to be published.

(a) S-f, N~200

Figure 1

(b) f-h-f, N~500

(c) f-f-h, N~500

Specimen shape after fatiguing:

(a), (b) and (c), Al-1l99, t.Et=0.8%, T~180°C

(d) Al-7075, t.Et~1.6%, T=260°C

(d) f-f, N~477

132

Figure 2

60

50

....... 40 ~

30 ~

20

10

0

Figure 3

30 AI-1199

T=20°C A&t=O.80% ~

&t=O.OO2 1/s »»»» »»»» »»»» »»»» »»»» »»»»

T=180°C - »»»» »»»» 20 »»»» »»»»

»»»~ »»»» »»»» »»»:~ »»»» »»»> »»»» »»»>

10

~ »»»» ~~~~~~~; »»»» »»»» ~~~~~~~? »»»» »»»» ~~~~~~~~ »»»» »»»» »»»> »»»» ~~~~~;~~

~ »»»» »»»» ;~;;;;;; »»»» »»»» ;;;;;;;? »»»» »»»»

~~~~~~~~ »»»» »»»»

- »»»» »»»> »»»» »»»> »»»» »»»> »»»» »»»> »»»» »»»>

T=260°C »»»» »»»> »»»» »»»> »»»» »»»> »»»> »»»> o N=120 N=2000 N=2000

Effect of temperature on the sensitivity to cyclic plastic instability.

AI-1199 T=180°C

annealed A&t=O.80% &s=2X 1 (j5 S-1

2.5h at 1\1\1\/\ &f=2X1(j3 S-1 1\1\1\1\

3500C 1\1\1\1\ 1\,,1\/\ th=292 S I\I\AII 1\1\1\/\ 1\1\1\/\ 1\1\1\1\ 1\1\1\/\ AAI\/\ /\AAA AAAA AAAA A/\/\/\ AAA/\ A/\AA AAAA /\/1./1./\ AA/\/\ AAAA AA/I./I. AAA/I. /\/\/'/I. AA/l.A A/l.AA AAA/I. 1\/\1\/\

1\/\/1/\

N=200 N=200 N=500 N=500

Effect of waveform on the sensitivity to cyclic plastic instability.

PROCESSING DEFECTS AND THE FRACTURE OF CERAMICS AND DESIGNED CERAMIC/CERAMIC COMPOSITES

Patrick S. Nicholson

Ceramic Engineering Research Group Department of Materials Science and Engineering

McMaster University Hamilton, Ontario, Canada

ABSTRACT: 4.5 wt% Y 203-partially-stabilized Zr02 (YPSZ) four-point bend bars densified by dry-pressing/isostatic pressing/pressureless sintering were fractured and. several fracture origin types were identified. These defects were then classified via their response to various elimination techniques. An elliptical crack model was used to characterize the correlation between fracture stress and fracture origin size. Consistent results showed that an order-of­defect-severity existed, which enabled definition of a "fracture origin severity parameter" for the different origin types. The existence of a relative fracture-origin severity is related to the residual stress fields at the defect-matrix interfaces. The improved strength obtained by secondary processing to eliminate the fracture origin defects did not extend to high temperatures. Efforts were made to improve the strength and toughness of Y-PSZ by incorporation of Na-p-AI203 dispersed phase. The mechanical properties of these composites are reported.

KEYWORDS

flaw elimination; flaw-strength relationship; defect severity; interfacial coherence; ceramic­ceramic composites.

INTRODUCTION

The excellent mechanical properties of Y 203-partially-stabilized Zr02 (Y-PSZ) at ambient temperature has attracted extensive research (1-6). Elastic-brittle materials like Y-PSZ have fracture stresses correlatable with their fracture origin sizes. Several defect elimination techniques have been employed to remove or reduce the size of the defects reponsible for fracture (1,4,5,7,8). In a recent study Sung and Nicholson (9), a systematic improvement of the bend strength (MOR) of 4.5 wt% Y -PSZ was realized by conventional dry­pressing/isostatic-pressinglpressureless sintering combined with sedimentation and burnout techniques. The behaviour of several identified fracture origin types was studied over a wide range of fracture stresses and fracture origin sizes.

The relative severity ofthe defects in ceramics has been little studied. Previous investigators (10,11,12) showed there exists a severity of different defect types but quantitative analysis

133

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 133-146. © 1990 Kluwer Academic Publishers.

134

was not done. Attempts (11,13) have been made to correlate fracture stresses with fracture origin sizes through crack models and so identify the nature of the different types of defect. Various crack models of similar kind have been developed (14-16), refined (17,18) and applied (19,20) and a part-through-elliptical-crack model (16,17,20) is thought to be the most realistic.

The present study examines the different types of fracture origin in 4.5 wt% Y-PSZ and characterizes the correlation between fracture stress and fracture origin size to identify the severity of the fracture origin types.

Partially stabilized zirconia (PSZ) is a promlsmg ceramic material for heat engine applications (21). The transformation toughening process in this ceramic increases its fracture toughness to >9.0 MPa·mlf2 (22). At high temperatures, however, the fracture toughness decreases to that of fully stabilized zirconia ('" 2.0 MPa·m1l2) (23). Further deterioration is observed after prolonged exposure to high temperature or cyclic temperature variations. Additional modes of strain energy dissipation must therefore be found for PSZ at high temperature. This work also presents resistance-to-fracture results for 4.5 wt% Y 203-stabilized Zr02*/20 vol% ll-A1203 (mean particle size 120 11m) composites and strength results for YPSZl15 v/o ll-A1203 particles of 15-20 11m mean particle size.

DEFECT-CRACK ANALYSIS

A part-through elliptical-crack-extension model, first developed by Irwin (16) and later refined by Bansal (17), was used in the present study. This model, shown schematically in Fig. 1, calculates

a

~ ~

\ I \ I \ I

'--'

Fig. 1. Part-through elliptical crack extension model.

the stress intensity factor for an elliptical crack in the interior (or a semi-elliptical crack at the surface) of a finite plate subjected to a uniform tensile stress perpendicular to one of its elliptical axes. The plane-strain critical stress intensity factor for the opening mode (Klc) along the periphery of a crack is given by;

(la)

Yv'I_v2 _ K1C = <I> 0 f v' c for a ~ c

(lb)

where v is the Poisson's ratio, Of the fracture stress, c the semi-axis perpendicular to the applied stress, a is the other semi-axis and Y is a geometrical parameter describing the

135

position of a crack relative to the tensile surface. For a surface semi-elliptical crack, Y is taken as 1.94, and for a subsurface elliptical crack Y = 1.77. .p is a "shape parameter", describing the shape variation of the crack. It is an elliptical integral which varies with the ratio ale, i.e;

(2)

where e is an angle defined in Fig. 1. From equation (1) it is clear that the smaller crack dimension controls the fracture. If "a" denotes the smaller crack dimension, Equation (1) becomes;

yv'; K1C o . -- = --= = constant

f 4> 2 v'l-v

(3)

and a plot of Of vs "yv' aI.p" should give a curve characterized by a constant KIC value. yv' aI.p will be termed the "equivalent fracture origin size".

EXPERIMENTAL PROCEDURE

A. SECONDARY PROCESSING OF YPSZ

4.5 wt% Y 20a-partially-stabilized Zr02 submicron powder* was used as the starting material. Three processing routes were employed (Fig. 2). Route 1 was the as-received powder (without

I Route 1

4.5Wt'lo y-psZ

I Rcxie2

I

I Route 3

I f'o,Mjer disjl<1rsed in 2-propcnol

I

Powder disjl<1rsed inmothcnol (pH2....oterl

I SusJ,Sion

uitraso'lleotion

SusJ,Sion sedimentation

Supotk,ant centriliguol eonsol idOt ion

I PoM:Ier <lryirq

PowJMllling

Dry prossIng Dry/IsostatiC I prejing

Isostatic 9Jmout

pro~ng,-I ____ .-_____ ~_-_iSOSJI~~ pressonq I

I Sinler,,'9

l"ig. 2. Various processing routes for 4.5 wt% Y -PSZ.

136

powder beneficiation), Route 2 the as-received powder 2-propanol sedimented and Route 3, the as-received powder methanol (or pH-2 water) sedimented followed by a burnoutlre­isostatic pressing step. In Route 1 (the commonly used procedure), the powders were dry­pressed at 40 MPa and isostatically-pressed at 300 MPa. The compacted plates were pressurelessly sintered in air at 1400° C for 3 hours, cut into bars and ground and edge­chamfered to final dimensons of 25 X 2 Xl 5 mm. The finished bars were annealed at 1400°C for 0.5 hour to relieve surface residual stresses. Bars were broken in four-point-bend with an inner·.span of 10 mm and an outer span of 20 mm. The machine cross-head speed used was 0.2 mm/min. Test procedures conformed with American Standard MIL-STD-1942 (MR) (DOD, 1983). Both fracture surfaces of each bar were examined by optical microscope at 80 X to approximately locate the fracture origins and representative half specimens were then examined by SEM. Inclusion-type fracture origins were analyzed by EPMA to identify their composition. Route 2 involved dispersing and sedimenting the as-received powders in 2-propanol prior to pressing and sintering. The sedimentation technique (Rhodes, 1981; Parish, 1984) was introduced to eliminate powder agglomerates and inclusion· type defects. Route 3 introduced the additional burn-out step (Lange, 1986a) prior to final sintering The as­received powders were first sedimented in methanol (or pH-2 water), compacted and fired at 600°C for 24 hours. They were then re-isostatic pressed at 300 Mpa and sintered.

B. Na-j3-AI203-PSZ Composites

A large crystal aggregate of j3-A1203 was broken up into a powder containing particles from 20 to 120 11m in diameter. The 4.5 w/o% Y203-containing Zr03 powder (0.1 11m particle size) was mixed thoroughly with 20 vol% 13-1 Al203 in a solution containing 1 wt% PV A binder in ethyl alcohol. After the powder was dried, it was screened, dry-pressed, and isopressed into bars. The bars were sintered at 1500°C for 6 h and cooled to room temperature at a rate of 200°C/h.

Fig. 3. Type A fracture origin-fiber inclusion. Fig. 4. Type B fracture origin­agglomerate.

Fig, 5, Type C fracture origin-iron inclusion, Fig, 6, Type D fracture origin­alumina inclusion.

Fig. 7. Type E fracture origin-pore.

137

The resistance-to-fracture test was performed on four-point-bend (bending span 25.4 and 12.7 mm), chevron-notched bars. Maximum fracture stability was assured by utilizing a slow, constant cross-head displacement rate (0.05 mrnls) machine and a high-stiffness (10 MN/m) piezoresistive load cell. The displacement of the sapphire loading rolls was measured by LVDT with a practical resolution of =-0.1 11m. The compliance of the testing system between

\38

the knife edges ('" 0.0411mlN) was corrected for in the analysis. The specimens were cycled through multiple loading/stable fracture/unloading stages (24,25).

RESULTS AND DISCUSSION

The bend strengths of Route-l synthesized YPSZ samples were 650-1020 MPa, average 880 MPa. Five types of fracture origin were identified, i.e. fiber inclusions, agglomerates, iron inclusions, alumina inclusions and pores (Figs. 3-7). Fiber inclusions (Fig. 3) are evidenced by regularly-shaped elongated pores. They are left by fiber-shaped inclusions after burn-off. Agglomerates (Fig. 4) appear as aggregates of large grains centered in a cavity surrounded by smaller grains. The large grains and surrounding porosity result from the differential sintering rate of the agglomerate and the matrix (23). Iron inclusions (Fig. 5) show as round or elliptical dark regions. EPMA detected iron. Iron oxide impregnated Zr02 grains sinter abnormally generating a porous region. Alumina inclusions (Fig. 6) are large regions of irregular shape oriented parallel to the surface. EPMA detected aluminum. Pores (Fig. 7) are irregular-shaped cavities left after pressing or developed during sintering. The

1500

\.\. o fiber InclUSion .. agglomerate • iron InclustOn • alumina Inclusion

1400 A pore

1300

1200 ~\: -; .. ~ \ :I .

':1100 . . '" G

~

~ ,

'\0 0;1000 \., ~ ~ \. ... 900

I..

)~ 800

" . "

700

GOO 200 300 400 500 600 700 800

EQI,II..,.lent Fracture Origin DImension ~(m'I2.103)

Fig. 8. Of - YVa/<I> plot for 4.5% Y-PSZ.

1500

1400

1300

1200

-;; ~ 1100

b 1000

900

800

700

600 0

J

j, J.;. j 1}

I !I: . ." 'l l J .: J." t ./ "~ dO I ) tl ° .I .0

" • • 1

11 · o tiber i nclu810n .. agglomerate • I ron InCIUliOn • alumu. Inclusion 4 pore

100 ZOO 300 400 500

1 / Vi"" (m"")

Fig. 9. Of - lIYVa/<I> plot for 4.5 wt% Y-PSZ.

139

bend strengths for Route-2-synthesized samples were 740-1280 MPa, average 995 MPa. The agglomerates and iron inclusions were reduced in size by sedimentation but the fiber inclusions were unaffected. The resulting strengths for Route-3 bars were 950-1470 MPa, average 1150 MPa. This procedure further reduced agglomerates in size and the iron inclusions were totally eliminated. The fiber inclusions were removed by burnoff-reisopress procedures. Detailed experimental results are described elsewhere (9).

The experimental data were plotted as fracture stress (Of) vs. calculated equivalent-fracture­origin size (Yvalcll) (Fig. 8). The fracture origin types follow the part-through crack extension model (Equation (3)). The agglomerate-type fracture origins deviate below an equivalent size -4 X 10-3 m l !2 (fracture stress -900 MPa) and finally merge with the curve for iron inclusions and pores (these fall on one curve).

The fracture origin types differ with respect to their responsibility for fracture, i.e. fiber inclusions are the most severe, then agglomerates, iron inclusions and pores and finally alumina inclusions. The part-through crack model assumes a pore with a traction-free periphery. Therefore the fiber inclusion and the agglomerate fracture origins are more "severe" than predicted by the pore-based model and, the iron inclusion and pore fracture origins are as predicted. The alumina inclusion fracture origins are less severe than predicted. This fact and the relative coherency of the AI203-PSZ interfaces, led to the exploration of a j3-AI203 particulate dispersion to high-temperature-toughen the YPSZ.

The relative severity of the type of fracture origin is thought to be related to their residual stress fields. For agglomerates, the increased local green density causes differential sintering and tensile residual stress build-up in the local matrix. Incoherency should relieve this stress (26) but the peripheral matrix will also press to a higher green density so a"shell" of predensitied and grain-grown matrix will develop around the agglomerate. This "densified" matrix prohibits complete stress relief and a residual tensile stress will superimpose on any appliec;l stress causing fracture at lower stresses than predicted ie., an agglomerate fracture origih will be more severe than a pore. Similar reasoning holds for a fiber inclusion. By resisting the forming pressure during dry and isostatic pressing, a denser packing of matrix powder results local to the fiber. During sintering this denser layer sinters faster resulting in local residual stresses similar to those of the agglomerate. This denser layer is thicker than for the agglomerate and residual tensile stresses are unrelieved, i.e. a more severe condition results than for an agglomerate (as observed). Iron melts and diffuses into the Zr02 particles so no residual stresses develop. Porous unsintered Zr02 particles inside this defect act like a pore and exhibit a similar severity (as observed). An alumina inclusion will develop a residual compressive radial stress and a residual tensile tangential stress around its periphery due to its lower thermal expansion (8-9 X 10-6t'C vs. 10-llX10- 6t'C for Zr02) (the induced tangential tensile stress may promote the t~m transformation of local Zr02 grains and the 3% expansion involved would change the peripheral stress state to compression). The superposition of this stress on the applied stress results in a higher fracture stress than predicted by the pore crack model. An alumina inclusion thus has a severity less than a pore. The size-effect of the agglomerate fracture origin severity is explained by the magnitude of the residual stresses. When agglomerates exceed -4X 10-3 ml12 equivalent size (corresponding to a fracture stress of -900 MPa) the residual stresses are further relieved due to microcrack formation as predicted on exceeding a critical defect size (27). Deleterious solid defects have a commonality of incoherent interfaces. a-A1203 defects have coherent interfaces with YPSZ and have a beneficial effect with respect to pores. This was the rationale for the inclusion of ll-A120a particles of narrow particle size range in the YPSZ matrix, i.e., to explore the possbiIity of maintaining strength at acceptable levels yet increase toughness and high temperature performance.

If equation (3) is rearranged, i.e.;

140

( KIC ) 1

of = Vl_vZ YVa/cI> (4)

and OfVS llYValcp is plotted for the various fracture origin types, the resultant line has slope of KIC/Vl-vZ. Setting v = 0.25 for the tetragonal ZrOz, the "apparent" KIC value (K*IC) for each fracture origin type can be obtained. Spch plots are shown in Fig. 9, and give the following K*IC values: 3.2±0.1 MPa m I/2 for fiber inclusions, 3.4 ± 0.1 MPa m1l2 for agglomerates, 4.4±0.1 MPaml12 for iron inclu~ions and pores and 6.7±0.1 MPa m 1l2 for alumina inclusions. These values also reflect the relative severity of the fracture origin types. The microindentation technique (28) was used to \measure the actual KIC of the Zr02 matrix and gave KIC = 6.9 ± 0.3 MPa mII2. Comparison Iwith the theoretical value calculated from the part-through crack extension model for pores gave a ratio of "" 1.55. This result is close to that of a previous study (20).

The existence of a differential severity of fracture origin types means the crack model requires correction. A "fracture origin severity parameter" (X) is therefore defined as;

YV;' K = 0 . X . -- (for plane stress)

IC f cI> (5a)

KIC YVa -= = 0 . X· -- (forplanestrain) VI_v2 f cI>

(5b)

and X = 1.03 for the alumina inclusions, 1.57 for the pores and iron inclusions, 2.03 for the agglomerates and 2.16 for the fiber inclusions. These values predict fracture stresses when the type, dimension and position of a defect is known (by NDE for example).

The decreased severity of the Al20a inclusions is of interest for ceramic/ceramic composite design. A residual compressive stress is developed around an alumina inclusion and the stress fields in the tensile direction (x-direction) along a section of the inclusion-matrix interface and in the matrix can be calculated using ~a = 2XIO- S rC, ~T = 1000°C, Em = 220 GPa, Ei = 400 GPa and Vrn = Vi = 0.25, 0ie = 490 MPa and Oa = 350 MPa. Now Of = 900 MPa to 1300 MPa so X = 0.61 to 0.73, average 0.67. This calculated X value agrees well with the experimental value (1.03/1.55 = 0.66).

A persistent probelm with ceramic/ceramic composites is the action of the dispersed particles as the critical strength-determing flaws. This is the problem in spite of their toughening role via crack-deflection, crack-surface closure and crack-tip shielding. This deleterious effect is reduced if the particulate-matrix interface is strong and the particle-size and associated residual stresses minimized.

To this end, the toughening potential of Na-/3-AI203 particles for the YPSZ matrix was investigated. The interfacial reaction is known to produce a-A1203 (29). The load­unload/deflection curves for YPSZ at 25°C are shown in Figure lOCal.

PSZ behaves as a brittle, ideal elastic materiaL Some permanent deformation (""0.3 11m) was obtained in the first cycle, possibly due to specimen surface indentation by the loading rollers. The last two cycles of the series gave a nonelastic displacement of = 2.0 11m due to the length of the main crack approaching 100% the specimen width.

The PSZI/3-AI203 composite (Fig. lOeb)) exhibited a large amount of irreversible deformation, unusual for a ceramic materiaL As the crack extends, an increasing share of the energy input

141

is dissipated nonelastically. The different loading and unloading paths are indirect evidence ofthe formation of microcrack zones in the vicinity ofthe main crack.

The energy, We, responsible for the main crack extension, i.e., the compliance increase, was obtained by locating the origins of the loading and unloading paths at the same point.

The resistance to crack extension (Re) for pure YPSZ decreases from'" 50 to '" 40 J/m2 as full fracture is approached. The nonelastic energy dissipation rate (Rn) oscillates around 5 J/m2

and decreases toward zero at full fracture.

PSZ 2S'C

10 15

DISPlACEMENT,D(~m)

Fig. 10 Load-displacement diagrams for (a) pure PSZ and (b) PSZ/{l-AI203.

142

In the initial stages of fracture of the composite (A S 10%) the Re similar to that of pure PSZ ("" 50 J/m2), wheras the Rn of the composite ("" 50 J/m2 is 1 order of magnitude larger ("" 5 J/m2). Subsequently Rn increases to values> 500 J/m2 . Small crack length changes in the final stages of the experiment (most of the energy input being dissipated irreversibly) result in a large scatter in the calculated Rn and Re values. The total energy consumption to drive the crack through the composite is 1 order of magnitude larger than that for pure PSZ.

PiN) PSZ2.1JOOC

I i

30 I \

20

10

o

o

I, 1;\1 , 1\

I

!

I

n !~

l!~

1 I / )

I I y I ! )11 ,/ )/

200

Figure l1(a) Load-displacement diagram for pure PSZ at 1300°C.

P(N)

20

10

o

1\ II~ : I Ir\ I I II I

I I i

60

Z4A120B 1300 c

80 U(mV) 100

Figure l1(b) Load-displacement (plotted as milli-volts potential drop) diagram for PSZ/Il­Alz03 at 1300°C.

143

Three-point bend fracture tests of20 composite samples gave an average strength of 127 Mpa with a standard deviation of 19 MPa. A Weibull distribution plot of the fracture probability (Pr) against stress suggests that the strength is controlled by two different flaw populations (20). The high value of the Weibull modulus at low strengths (M = 43) results in a 0.1% fracture probability at 100-MPa stress. The lower strength of the composites is explained by the (3-AI203 particles being the critical flaws. Their narrow particle-size-distribution explains the high Weibull Modulus for the lower-strength samples. Fracture likely initiates from surface (3-AI203 particles whereas the higher-strength, lower Weibull Modulus samples fracture from natural flaws.

Fig. 12. Crack (bottom of figure) intercepts a a (3-AI203 particle in PSZ at 1300°C

Fig. 13. Improved PSN(3-AI203 interface (bar = 1 ].lm)

1300° results for YPSZ and YPSZl120 ].lm-Na-(3-AI203 composites are shown in Figure 11 (a), (b). The latter results report crack-growth in terms of increasing electrical-potential drop as the crack separafes the oxygen-ion-conducting YPSZ matrix. Deflection and potential drop are directly related.

The nature of crack/(3-AI203 particle interaction and the particle-matrix interface are illustrated in Figure 12 (31). The (3-AI203 particle with its a-A1203 halo resides astride the traversed crack faces exerting closure forces thereon. The crack has clearly been deflected on encountering the particle. The interface is partially coherent, explaining the role of the particles as critical flaws resulting in low strengths. The fracture energy for YPSZ at 1300° drops from 2000 J/m2 to < 500/m2 as the fracture area increases. The values for the YPSZlNa­(3-AI203 composite are 3800 J/m2 to 700 J/m2 .

144

Two avenues have been explored to re-establish the degraded strength of the YPSZJNa-j}­Al203 composites. The j}--AI203 structure can be ion-exchanged and introduction of larger ions causes crystal expansion. Ion-exchange of the N a-j}-AI203 surface particles of the composite to K-j}--AI203 causes -1 % expansion and the composite strength increases from 550 to 650 MPa for 15 v/0-25 !lm j}-AI203 particles(32).

The second approach seeks to improve the interfacial coherency of the fI-A1203 in the YPSZ matrix. Optimization of the secondary processing of YPSZ powder allows reduction of sintering temperatures by -100°C. This reduction, together with the incorporation of smaller fI-A1203 particles (20-30 !lm average size), has led to markedly improved interfaces (Fig. 13) and room temperature strengths (127±19MPa up to 612±50MPa) (32). The increased standard deviation of the latter suggests natural flaws are strength-controlling rather then the Na-fl-AI203 particles. Work is continuing on the 1300°C strength and toughness.

CONCLUSIONS

As a result of the work here reported the fracture origin types and their severity in Y-PSZ have been identified and removed by secondary processing. The relative severity of the fracture origin types is explained by the associated residual stresses and a "fracture origin severity parameter" (X) defined. The comparatively benign action of a-A1203 defects led to the incorporation of Na-fl-AI203 particles in YPSZ to improve the latter's high temperature mechanical properties. The strength-degradation effect of these toughening particles has been reduced by developing coherent particle/matrix interfaces and by ion-exchange-induced compressive surface stresses in the composites on potassium for sodium ion-exchange of the surface fI-A1203 particles.

REFERENCES

1. F.F. Lange, B.I. Davis and and E. Wright. Processing-related Fracture Origins: IV. Elimination of Voids produced by Organic Inclusions. J. Am. Ceram. Soc., 69, (1986) 66-69.

2. M. Matsui, T. Soma and I. Oda. Effect of Microstructure on the Strength of Y -TZP Components. In N. Claussen (Ed.), Advances in Ceramics, Vol. 12, Science and Technology of Zirconia II. The Am. Ceramic Soc., (1984) pp. 371-381.

3. T. Masaki, Mechanical Properties of Toughened Zr02-Y 203 Ceramics. J. Am. Ceram. Soc., 69, (1986) 638-640.

4. K. Tsukuma and M. Shimada. Hot Isostatic Pressing of Y 20a-Partially-Stabilized Zirconia. J. Am. Ceram. Soc., 64, (1985) 310-313.

5. K. Tsukuma and K. Veda. Strength and Fracture Toughness of Isostatically Hot­Pressed Composites of Al203 and Y 203-Partially-Stabilized Zirconia. J. Am. Ceram. Soc., 68, (1985) C-4-C-5.

6. K. Tsukuma and K. Veda. High-Temperature Strength and Fracture Toughness of Y 203-Partially-Stabilized Zr02/A1203 composites. J. Am. Ceram. Soc., 68, (1985) C-56-C-58.

7. W.H. Rhodes. Agglomerate and Particle size Effect on Sintering Yttria-Stabilized Zirconia. J. Am. Ceram. Soc., 64, (1981) 19-22.

145

8. H. Taguchi, Y. Takahashi and H. Miyamoto. Effect of Milling on Slip Casting of Partially Stabilized Zirconia. J. Am. Ceram. Soc., 68, (1985) C-264-C-265.

9. J. Sung, and P.S. Nicholson. Strength Improvement of Yttria-Partially-Stabilized Zirconia by Flaw Elimination. J. Am. Ceram. Soc., (71) (1987) 788-95.

10. A.G. Evans, G.S. Kino, P.T. Khuri-Yakub and B.R Tittmann Failure Prediction in Structural Ceramics. Material Evaluation, 35, (1977) 85-96.

11. A.G. Evans, M.E. Meyer, K.W. Fertig, B.I. Davis and H.R. Baumgartner. Probabilistic Models for Direct Initiated Fracture in Ceramics. J. Nond. Eval., 1, (1980) 111-122.

12. F.F. Lange. Advanced Processing of Ceramics: Controlling Flaw Populations. In P.S. Nicholson (Ed.), Transactions of the Canadian University-Industry Council on Advanced Ceramics, 2nd Workshop, (1986) pp 1-29.

13. H.P. Kirchner, RM. Gruver and W.A. Sotter. Characteristics of Flaws at Fracture Origins and Fracture Stress-Flaws Size Relations in Various Ceramics. Mater. Sci. and Eng., 22, (1986) 147-156.

14. F.I. Barratta. Stress Intensity Factor Estimates for a Peripherally Cracked Spherical Void and a Hemispherical Surface Pit. J. Am. Ceram. Soc., 61, (1978) 490-493.

15. A.G. Evans. and G. Tappin Effects of Microstructure on the Stress to Propagate Inherent Flaws. Proc. Br. Ceram. Soc., 20, (1972) 275-297.

16. G.R Irwin. Crack-Extension Force for a Part-Through Crack in a Plate. -L...A!mL Mech., 29, (1962) 651-654.

17. G.K .Bansal. Effect of Flaw Shape on Strength of Ceramics. J. Am. Ceram. Soc., 59, (1976) 87-88.

18. A.G. Evans, D.R Biswas and RM. Fulrath Some Effects of Cavities on the Fracture of Ceramics: II. Spherical Cavities. J. Am. Ceram. Soc., 62, (1979) 101-106.

19. J.J. Mecholsky, Jr., S.W. Freiman and RW. Rice. Fracture Surface Analysis of Ceramics. J. Mater. Sci., ll, (1976) 1310-1319.

20. S.G. Seshadri and M. Srinivasan. Estimation of Fracture Toughness by Intrinsic Flaw Fractography for Sintered alpha Silicon Carbide. J. Am. Ceram. Soc., 64, (1981) C-69-C-71.

21. L.J. Schioler. Workshop Studies Ceramic Engines - Current Status and Future, Bull. Am. Ceram. Soc., 64, (2) (1985) 269-270.

22. M.V. Swain, RH. Hannink, and RC. Garvie. The Influence of Precipitate Size and Temperature on the Fracture Toughness of Calcia-and Magnesia-Partially-Stabilized Zirconia, (1983) pp.339-55 in Fracture Mechanics of Ceramics, Vol. 6. Edited by RC. Bradt, A.G. Evans, D.P.H. Hasselman, and F.F. Lange. Plenum Press, New York.

23. L. Li and RF. Pabst. High Temperature Fracture Toughness Measurements and Aging Processes ofPSZ, (1983), pp.371-82 in Fracture Mechanics of Ceramics, Vol. 6. Edited by RC. Brandt, A.G. Evans, D.P.H. Hasselman and F.F. Lang, Plenum Press, New York.

146

24. M.V. Swain. R. Curve Behaviour of Magnesia Partilly Stabilized Zirconia and its Significance to Thermal Shock, (1983), pp. 355-70 in Fracture Mechanics of Ceramics, Vol. 6. Edited by R.C. Brandt, A.G. Evans, D.P.H. Hasselman and F.F. Lang, Plenum Press, N ew York.

25. M. Sakai, K. Urashima, and M. Inagaki. Energy Principle of Elastic-Plastic Fracture and its Application to the Fracture Mechanics of a Polycrystalline Graphite, J. Am. Ceram. Soc., 66 (12) (1983), 868-74.

26. F.F. Lange, and M. Metcalfe. Process-Related Origins: II. Agglomerate Motion and Cracklike Internal Surfaces caused by Differential Sintering. J. Am. Ceram. Sec., 66, (1983) 398-406.

27. D.J. Green. Critical Microstructure for Micocracking in A120a-Zr02 Composites. !l Am. Ceram. Soc., 65, (1982) 610-614.

28. K. Niihara, R. Morena and D.P.H. Hasselman. Evaluation of KIC of Brittie Solids by the Indentation Method with Low Crack-to-Indent Ratios. J. Mater. Sci. Lett., 1, (1982) 13-16.

29. S.J. Glass, P.S. Nicholson and C.B. Clark.Characterization of Interfacial Relations Between j3-AI20a and Y20a-Partially-Stabilized Zr02, J. Am. Ceram. Soc., 68 (7) , (1985) C-176.

30. C.A. Johnson. Fracture Statistics of Multiple Flaw Distributions, (1983), pp. 365-86 in Fracture Mechanics of Ceramics, Vol. 6. Edited by R.C. Brandt, A.G. Evans, D.P.H. Hasselman and F.F. Lang, Plenum Press, New York.

31. T.B. Troczynski and P.S. Nicholson. "The Resistance to Fracture ofPSZ and PSZ-Na­{3-AI20a Composite at 1300°C", Am. Ceram. Soc. Bull. 65 (5) (1986) 772-75.

32. 'W.Q. Gou and P.S. Nicholson (to be published).

INDENTATION CREEP IN SEMI-BRITTLE MATERIALS

N.M. Everitt and S.G. Roberts

Department of Metallurgy and Science of Materials, University of Oxford, Parks Road, Oxford OXI 3PH.

ABSTRACT: For brittle solids microhardness testing may be the only means of producing plastic flow over a wide temperature range. Indentation creep is a possible method for assessing the time­dependent deformation of brittle materials, but the controlling mechanisms at lower temperatures are not well understood. Single crystal germanium is a good model material for a study of such behaviour, and Knoop indentation was carried out on n-doped material in the <110> and <100> orientations at temperatures between 200 C and 7000 C with load times of 20 to 600 seconds. The results can be qualitatively explained by the different amount of slip interaction occurring beneath indentations at the two orientations. In the <110> orientation dislocations can move out into new undeformed material as the size of the indentations increases. The creep process is therefore controlled by the ease with which diverging slip can take place, and the creep exponent increases throughout the temperature range studied. However beneath <100> indentations the slip systems activated are continually becoming locked as creep takes place. Thus whilst the creep exponent initially increases with temperature, it becomes dominated by the rate at which recovery can take place and levels off to an approximately constant value.

1. INTRODUCTION

The static indentation test is a rapid and convenient method of assessment of material behaviour. For hard brittle solids the largely compressive stress field produced by microhardness testing may be the only means of producing controlled plastic flow over a wide temperature range. Hence short-time hardness testing has been used as a quick and convenient guide to the in-service performance of brittle materials. Generally hardness decreases slowly with increasing temperature until half the absolute melting temperature (Tm) is reached. Above this temperature there is a rapid drop in hardness with rising temperature (see reviewl ). However, indentation creep, the continuing penetration of the indenter when left under full load, may be a better test for situations where sustained mechanical loading

147

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 147-153. © 1990 Kluwer Academic Publishers.

148

occurs2. This reveals the resistance of the material to time-dependent plastic deformation.

Two or three regimes of indentation creep behaviour may be identified. High temperature creep occurs in all crystalline materials at or above 1/2 Tm, and this appears to be dominated by conventional creep mechanisms based on dislocation glide and climb. Activation energies derived from hardness data by Atkins and Tabor3 are consistent with mechanisms based on self diffusion.

"Anomalous creep" where the creep rate decreases with increasing temperature is observed at low temperatures for some non-metals and is probably due to chemo-mechanical effects4 •

Brooke~, Hooper and MorganS identify a third regime « 1/3 Tm), which has been observed in some metals (eg, Mo and V) and in some ceramic solids (eg. MgO and A1203). Here creep accelerates with temperature but activation energies are not characteristic of conventional creep.

The correlation between "conventional" and indentation creep data is not as direct as might be supposed since there are considerable differences between the stress conditions of the two types of test. Conventional creep tests are generally uniaxial tensile or compressive tests where stress is constant or increases with time. Indentation creep is always compressive with very high initial stresses which decrease with time. Additional aspects include the increase in the volume of deformed material with time (since new undeformed material is affected as the indentation size increases), and the non-homogenous indentation stress field.

2. THIS STUDY

In order to understand the mechanism(s) of indentation creep at lower temperatures, simple model systems are needed. Single crystal germanium is a material with well defined slip geometry and well characterised stress/temperature dependence of dislocation velocities. Furthermore, the effect of doping on dislocation velocity can be utilised to investigate the dynamics of the creep processes without changing the crystallographic and material constraints. Indentation testing on single crystals is complicated by "hardness anisotropy"6. However this effect can be turned to advantage, as certain orientations of indenter/crystallography produce very simple deformation patterns which may be (qualitatively) understood in terms of the indentation stress field7 • Although other authors have looked at the variation of hardness and anisotropy with temperature, surprisingly little work has been done on the variation of the anisotropy with indentation time at a given temperature. Kollenberg8 compared the prismatic and basal planes in sapphire at 500 and 7S0oC and found the creep rate to be dependent on crystallographic orientation as well as temperature, but did not attempt to analyse the mechanisms involved.

For diamond cubic materials two types of slip may be distinguished: (i) those with the slip direction parallel to the surface ("rosette slip") and (ii) those with the slip direction inclined to the surface ("inclined slip"). An earlier paper 9 uses a

resolved stress model to predict possible slip geometries with various

149

indenter configurations. When the indenter is aligned along <110> (the softest orientation) the expected slip geometry consists of inclined slip of two types: converging slip beneath the indenter which forms locks, and diverging slip on free running-planes outside the indenter (figure 1a). Rosette slip might also be present in the near-surface regions and is shown on figure 1a. Along <100>, inclined slip is predicted beneath the indenter on intersecting planes (causing dislocation locks to be formed), whilst rosette slip is expected in the near-surface regions (see figure 1b - rosette slip planes are omitted for clarity). Experiments lO show slip patterns in accord with these predictions.

(b) Figure 1. Slip geometry beneath Knoop indentations

a) <110> b) <100> orientation.

In this study Knoop indentation was carried out on {001} single crystal germanium in the <110> and <100> orientations with load times between 20 and 600 seconds at temperatures between 20 and 700oC.In addition to recording the actual size of the indentations, the extent and spatial arrangement of the dislocation activity around and beneath the indentations were examined. This characterisation provides the basis for an understanding of the physical processes underlying creep and anisotropy in semi-brittle materials.

3. EXPERIMENTAL PROCEDURE

The germanium used was n-doped single crystal material. The crystal was aligned using Laue X-ray photography, and then cut parallel to the {001} planes using a high speed diamond saw. The 0.5mm thick slices were lapped with alumina slurry before polishing with diamond paste. A final "Syton" (Monsanto 2360) polish was used to remove any residual surface damage.

The results described here were obtained from Knoop indentation tests in a Stanton Redcroft high-temperature microhardness machine. The sample and indenter are independently heated to give the required temperature to ± 30 e. Tests were carried out under vacuum using 50g

150

loads. Indentation lengths and other dimensions were measured on an optical microscope with a curtain micrometer using Nomarski interference contrast and magnifications between 200 and 1000 times. Each quoted result is the mean of at least 5 readings.

In order to examine the dislocation array beneath indentations a sectioning method was developed. The indented specimen was placed between two rectangular glass pieces with the line of indentations paralle 1 to a long edge, and the "sandwich" gl ued together with epoxy resin. The sandwich was then lapped and polished perpendicular to the specimen surface. The polished surfaces were etched (HF/HN03/CH3COOH/ I2 reagent)ll to reveal the dislocations.

4. EXPERIMENTAL RESULTS AND DISCUSSION

4.1 Hardness measurements

d (IJm)

200

150

<> -

100 ~40(t'C

50 ~300OC

_ - 201f'C

0 0 200 400 600

time (s)

H (GPa)

4

0 0

- <> (100)

----<>- (110)

+-

200 400 time (s)

H(GPa)

600 200 400 time (s)

(c) Figure 2. Variation of (a) Knoop diagonal, (b) Knoop hardness with

time in the range 2000 C to 700oC,

(a) (b)

(c) Expanded version of the lower portion of (b).

Figure 2 shows the variation of Knoop diagonal, and hence hardness, with time over the temperature range concerned for the <110> and <100> orientations. (The 5000 C data are omitted from figure la because a different load was used). Room temperature hardness is 87 CPa (865 Kg/mm2 ) with no measurable anisotropy. In general the hardness decreased slowly with temperature until over 2000 C when it started to decrease more rapidly; some creep was evident even at 200oC. By 3500 C anisotropy was present at longer indentation times and 20 second indentations showed anisotropy at 400oC. Above 4000 C the <100> orientation was always harder than the <110> orientation under the same conditions. The data at 7000 C suggest that at very high homogolous temperatures anisotropy decreases with time.

Expressions similar to equation (1) are commonly used for analysing creep data (eg. Atkins and Tabor 3 ):

Ht = C . exp(-Q/RT) . t- l / n (1)

151

where H is the hardness, t the indentation time, Q an activation energy, C and n are constants. Thus a plot of log hardness against log time should give a straight line whose gradient is -l/n. n may be regarded as a creep exponent, but may not be directly comparable to the stress component derived from conventional "steady state creep" over the whole temperature range concerned 12 • Linear regression analysis was performed and the results are shown in table 1. (Work using a displacement transducer found that the indenter took 9 seconds to reach the sample's surface, and this was taken into account in the indentation "dwell time" values used. Using different values for the "loading time" did not significantly alter the values for 1, second hardness and n).

Table 1. Results from linear regression on log hardness/ log time data

Temp. (OC)

200 300 400 500 700

T/Tm

0.39 0.47 0.56 0.64 0.80

H at <110>

1 sec n (GPa)

10.6 9.5 7.9 3.8 1.7 5.9 0.9 8.2 0.35 10.5

<100> H at 1 sec n

(GPa)

10.0 13.2 7.9 4.1 2.4 6.5 1.2 7.9 0.58 6.8

From the slip geometries described in section 2 (figure 1), we might expect different creep characteristics between the <110> and <100> orientations, and the variations in n with temperature in table 1 are consistent with this. Thus at higher temperatures the <100> creep is basically controlled by the ease with which recovery can take place releasing the dislocation locks beneath the indenter, and n is roughly constant. However for the <110> orientation, n increases throughout the temperature range suggesting different controlling mechanisms. (The apparently anomalous values of n at 2000 C probably reflect the relatively large errors in hardness resulting from the small indentation diagonal measurements).

4.2 Dislocation arrays

The two indenter orientations showed characteristic dislocation arrays (see figure 3). At 3500 C the most striking aspect of the surface arrays is the long "rosette arms" which emanatee from the i1lllllediate proximity of the indenter impression, but as the temperature increased, the density of dislocation activity obscured these and no consistent pattern of rosette length was observed. However the hardness is probably controlled not by these surface effects but by the slip occurring beneath the indenter.

The dislocation array beneath an indentation at a particular temperature and orientation retains its geometrical similarity but increases in size as the indentation time (and the size of the indentation) increase (compare figures 3a and 3b). Away from the massively deformed indenter impression there is surprisingly little dislocation activity. The etch lines which appear in sections (i) and (ii) suggest some deformation by micro-twinning IO • The dislocation

152

pattern for <100> is consistant with dislocation loops travelling away from the indentation on planes diverging from the central pyramid until they intersect with another such plane (figure 3, sections(ii) and (iii), a and b). The converging inclined slip planes forming a central pyramid do not appear until very close to the indentation (figure 3, section (i), a and b). Rosette slip is visible on the right hand side of the 60 second indentation (labeled "R"). In the <llO> orientation the most obvious dislocation activity occurs on diverging inclined planes which extend a long way into the crystal (figure 3c).

SURFACE

ARRAYS

(;) SECTION

B.NEATH

CENTRE

{ii} SECTION

aENE.~ TH

TIP

(lid SECTION

AWAY FROM

TIP

(0)

Figure 3. Typical dislocation arrays second indentations along <100>, (c)

5. CONCLUSIONS AND CURRENT WORK

(b) {el

at 350oc. (a) and (b) 20 and 60 60 second indentation along <110~

Indentation creep in n-doped germanium takes place at temperatures as low as 2000 C for both <110> and <100> orientations. Anisotropy increases with time, first becoming apparent in the longer time indentations at 300oC, <100> being harder than <110> although by 7000 C there were indications that hardness values might converge at very high homologous temperatures. Sectioning experiments have shown that the majority of the dislocation activity beneath indentations takes place on slip planes inclined away from the main indenter axis for both orientations.

These results can be qualitatively explained by the different

153

amount of slip interaction occurring beneath indentations at the two orientations chosen. In the <110> orientation dislocations can move out on diverging inclined slip planes into new undeformed material as the size of the indentations increases. The creep process is thus at least partially controlled by the ease with which the slip process can take place, and the creep exponent increases throughout the temperature range studied. (In germanium, dislocation mobility is limited at lower temperatures but increases rapidly with temperature). However beneath <100> indentations the slip systems activated are continually becoming locked as creep takes place. Thus whilst the creep exponenent initially increases with temperature, it becomes dominated by the rate at which recovery can take place and levels off to an approximately constant value.

The dislocation rosettes and other dislocation arrays on the surface of the samples are indicative of the dislocation mobility at that temperature, but it is the motion of dislocations beneath the indenter which determines the indentation creep characteristics. Work is continuing on construction of dynamic models which consider the resolved shear stress from the indenter and the stresses within the dislocation arrays in order to estimate the slip systems' response to time for the different orientations. These models will then be matched to experimental data in germanium and other materials. (Such an approach, based on dislocation glide, was first proposed by Gerk13 but took no account of the crystallographic nature of slip).

6. REFERENCES

1. A.G. Atkins, in The Science of Hardness Testing and Its Research Applications, J.H. W;;tbrook andl[. Conrad (eds.), American Society for Metals (1973) 223-240. 2. J.L. Henshall, G.M. Carter and R.M. Hooper, Proceedings of the European Mechanics Colloquium, Leicester UK, 1988. 3. A.G. Atkins and D. Tabor, Proceedings of the Royal Society (London), A292 (1966) 441-459. 4. W.W. Walker, in The Science of Hardness Testing and ~ Research Applications, J.H. Westbrook and H. Conrad (eds.), American Society for Metals (1973) 258-273. 5. C.A. Brookes, R.M. Hooper and J.E. Morgan in European Applied Research Reports, Hj Matzke (ed.), Harwood Academic, 7 (1987) 1127-1145. 6. C.A. Brookes, J.B. O'Neill and B.A. Redfern, Proceedings of the Royal Society (London), A322 (1971) 73-88. -----7. S.G. Roberts, P.D. Warren and P.B. Hirsch, Journal of Materials Research, 1 (1986) 162-176. --8. W. Kollenberg, Journal of Materials Science Letters, 7 (1988) 1076-1077. --9. S.G. Roberts, P.D. Warren and P.B. Hirsch, Materials Science and Engineering, A105/106 (1988) 19-28. 10. C-D. Qin and S.G. Roberts, Proceedings of the Materials Research Society Fall Meeting, Boston USA, 1988. 11. B. Tuck, Journal of Materials Science, 10 (1975) 321-339. 12. P.M. Sargent and M.F. Ashby, Cambridge University Engineering Department Report (1989) CUED/C-Mats./TR.145. 13. A.P. Gerk, Philosophical Magazine, 32 (1975) 355-365.

ON THE FRACTURE BEHAVIOR OF ROCK SALT

U. Hunsche

Federal Institute for Geosciences and Natural Resources (BGR) Stilleweg 2, 0-3000 Hannover, FRG

ABSTRACT: Research is conducted by the BGR on the creep and failure of natural salt rock in order to describe its thermomechanical behav­ior. A true triaxial test rig for cubic samples has been used for 240 tests on 8 different types of rock salt. The effects of varying mean stress, load geometry, and temperature on failure strength and resid­ual strength are determined in load-controlled quasi-static tests. A failure criterion in the sense of a conservative stability limit has been developed to describe the influence of the above-mentioned vari­ables on strength and residual strength of rock salt of the Gorleben salt dome. Distinct differences in strength exist between the rock salt types. Moreover, thin anhydrite layers slightly increase strength. Fracture patterns, i.e. angles and width of fracture planes, and sample damage correlate with load geometry.

1. INTRODUCTION: Knowledge of the thermomechanical properties of rock salt is important for the dimensioning of a permanent repository for radioactive wastes in a salt dome and for the necessary safety analysis, as well as for the desig~ of mines and caverns. Therefore, research is conducted by the BGR 011 the creep and failure behavior of natural rock salt. Equations have been derived which describe strength and creep under various conditions (1, 2, 3, 4). They are needed for finite element model calculations.

The failure strength of rocks is a function of the three prin­cipal stresses, temperature, and, to varying degrees, the rate of stress change or rate of deformation. Full information on the strength of a rock can be obtained with a true triaxial test rig, which permits independent control of the three principal stresses al , a2 , and a3' temperature of specimen, and control of the rate of stress change or rate of deformation. Such tests on cubic specimens have been described for rock salt (3, 4, 5, 6). Descriptions for other kinds of rock are cited in (7). True triaxial experiments can also be carried out using hollow cylinders (7).

The failure experiments reported in this paper were conducted on the BGR true triaxial test rig for cubic samples. The results compare well with Karman-type compression and extension tests (3, 6).

155

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 155-163. © 1990 Kluwer Academic Publishers.

156

Because of its practical importance, the main objective of this research has been to develop a strength criterion for rock salt from the Gorleben salt dome in Lower Saxony (FRG), which is a candidate for a radioactive waste repository. For safety reasons, it was decided to determine a conservative estimate of the strength in terms of stress, called the stability limit (Fig. 1), and also to describe this limit by a physically plausible empirical equation. Previous failure equations were based on averaged test results and are not conservative. Moreover, they do not include the influence of the intermediate principal stress (or the load geometry) and are mostly restricted to compression tests (e.g. 3, 8, 9, 10).

Because the strength of rock salt is only slightly influenced by deformation rate (11), and thus the total strain to failure, this in­fluence has been neglected in the equation. Failure does not occur in tests at low deformation rates and with confining pressure, because the material merely creeps under this condition.

Another objective of the tests is to investigate the differences in failure behavior of different rock salt types. Therefore, standard test series have been carried out on eight types of salt totaling 240 specimens.

Another objective is to study the fracture patterns and to cor­relate them with the experimental conditions. In addition, it is of interest to investigate the influence of thin anhydrite layers.

b)

Figure 1: Representation of the stability limit (a) in the al , a2 , a3 coordinate system and (b) in the octahedral plane

2. THE TEST RIG: A detailed description of the apparatus is given in (4). Briefly, the BGR true triaxial test rig consists of a rigid frame in which six double-acting pistons are arranged opposite each other about the center of the frame. A load is applied to the cubic speci­mens via square steel plates (platens), which can be heated.

A schematic diagram of the apparatus is shown in Figure 2. The most important specifications are as follows:

- Maxinrum force (per axis): 2000 kN - Sample size (edge length), standard: 57.5 rom

maximum: 200 rom - Temperature maxinrum: 400°C - Friction of piston: < 1 % of load - Independent load or deformation control in the 3 axes - Loading rates in standard tests,

hydrostatic phase: 7.6 MPa,lmin deviatoric phase: 15.1 - 32.2 MPa/min

- Deformation rate at failure, standard test: - 0.007 S-l

- Digital data collection, standard interval: 1 s - Samples are lubricated with paraffin wax or graphite (at

elevated temperatures)

TRUE TRIAXIAL PRESS

Figure 2: Schematic diagram of the true triaxial press of the BGR

157

A large number of experiments have been conducted to determine the influence of sample size in true triaxial tests. The main results are (i) that the sample size has no influence on the measured strength and (ii) that the ratio (K) of the edge length of the sample to that of the platens has a considerable influence, lower K yielding higher strength. This effect is important in compression and negligible in extension tests. A ratio of K = 1.15 proved to be a good compromise and was chosen for the experiments.

3. EXPERI~ PROCEDURES: In a true triaxial test, the three prin­cipal stresses, or three equivalent independent variables, describe the stress state completely. In this paper, octahedral normal stress ao ' octahedral shear stress "to' and the Lode parameter (load geometry) m are used (Table 1). Three invariants of the stress or stress devia­tion can also be taken instead; this is often useful for numerical model calculations.

158

Table 1: List of variables

t (a, + a, + a,) octahedral normal stress

'to -I 3' [(a, - a,)2 + (a2 - a,)' + (a, - a,)2)'/2 octahedral shear

stress

'tOB '

m - I 8, > S2 > s,

m(9 - m2 ) (3 + m' ,.n ( 3 )'/2 (J) - ---y-; of

failure strength. residual strength

Lode parameter. load geometry

cOIlPonents of stress deviation

invariant of stress geometry

Z" and 3" invariants of stress deviation

The Lode parameter m corresponds to a specific direction of ~ in the octahedral plane (Fig. 1). The following values for the Lode parameter correspond to the following types of experiments

m - -1: compression test~ m - +1: extension test; m = 0: torsion test. All tests are load controlled, the stress is first applied hydro­

statically, i.e. the same stresses are produced on all three axes. When the desired ao level has been reached, the three principal stresses are changed linearly with time so that ao is held constant (deviatoric phase). This is done until the octahedral shear stress reaches the strength of the sample and fracturing occurs (Fig. 3). Fracture strength ~OB is defined as the maximum of ~ in a test.

It should be noticed that ~ does not go to zero on failure; the fractured sample has a considerable residual strength ~R' This is determined by controlled deloading after failure and increasing the load a second time (Fig. 3).

10

10

ooIMPc!)_

Figure 3: Course of compression tests from series 8; T = 25°C; distance between successive dots: 1 s

159

A standard test series is carried out on each rock salt type. It usually consists of 29 tests at various values of ao and m, including four uniaxial tests and five tests at elevated temperatures between 60 DC and 260 DC (only m = -1).

4. RESULTS: The eight series of rock salt samples from three deep boreholes in the Gorleben salt dome are represented in Table 2. The results of the first six series have already been published (4). Series 7 and 8 are new.

Table 2: Eight test series of rock salt from the Gorleben salt dome. A number of samples contained anhydrite layers several mm thick.

Series Borehole Depth (m) Stratigraphy Tests Anhydrite 1 5002 766 - 775 z30SU 30 no 2 5002 463 - 476 z30SU 31 no 3 5002 790 - 801 z30S0 15 no

16 yes 4 5002 443 - 457 z30S0 20 no

22 yes 5 5001 845 - 857 z30S0 20 yes 6 5001 863 - 875 z30SU 29 no 7 1005 477 - 489 z2HS2 22 no

518 - 519 6 no 8 5001 730 - 738 z3BK/BD 22 no

821 - 823 7 no

Examples of tests of series 8 are shown in Figure 3. The results of the compression tests at room temperature are summarized in Figure 4 for all eight series. This graph demonstrates that the strength of rock salt increases nonlinearily with increasing ao ' It also shows that there are distinct differences between the different types of rock salt: The strength TOB varies up to 4 MPa for samples without anhydrite, series 8 exhibits the highest strengths and series 1 and 2 the lowest. The samples with anhydrite exhibit a little higher strength: Series 5 has the highest, in series 3 and 4 the samples with anhydrite show up to 2 MPa higher strength than those without. This is in agreement with the observation that the layers acted as fracture planes in only a few cases. Usually there was no visible interaction between anhydrite layers and fracture planes.

The results of all the tests at room temperature are shown in Figure 5, grouped according to the value of m for the respective test. It is important to note that the extension tests (m = +1) exhibit the lowest strengths, averaging about 30 % less than the compression tests in each series. This effect decreases at high ao ' however.

The residual strength of the different types of salt varies in a manner similar to the variation in failure strength, as already shown in (4). The stability limit for residual strength for all values of m is quite similar to the stability limit for the initial strength mea­sured in extension tests. This somewhat surprising result has also been observed in experiments on cylindrical samples. Thus, load geom­etry does not appear to influence residual strength.

160

e . 9 i- . ~~ l>

.s • % • ~ 0

~ '. )( 4iI~ o~

~o

t:1 0 ----~- ---- -- .. _---- -, ..

B • 0"

• 0

10

, containing

" anhydrite

Ii serIes 1 : 710m 2 1t70m 0

I l • 800 m a • ---~ -- ~-------- 4.450m 0 +

~ 5 850m . 0+

6 815m I>

1 480 m a 8 135m B

!i.ymbols with Q dot uniaxial tests.

50

---- 6 0 [MPal

Figure 4: Failure strength for series 1 to 8; compression tests; T - 25 DC

10 20 30 40

~ 30 6 [Ii

t-'

t 20

10~----~~~------~-------

10 20 30

0

<> 0

~~m=-l o a ~

=.1

m = ... 1,0 • biaxial

m ...... 1 c m •• 0,5 I>

m ,.:to 0

m • -0,5 " m .. -1 <>

0 0

m = - \0 • uniaxIal

40 50

-----< .. - °0 [M Pal

Figure 5: Failure strength for series 1 to 8, grouped according to the value of m; T = 25 DC. The lines represent the equa­tion for the stability limits at m - -1 and m = +1.

The results of experiments at elevated temperatures show that strength decreases significantly only above 100 DC and that the dif­ferences between the salt types remain (see 4).

Deformation to failure ranges from 2.5 % for uniaxial tests to 30 % at high ao • These results compare well with the corresponding

results (deformation rate about 0.007 S-l) of deformation-controlled Karman-type tests with constant confining pressure, conducted at the BGR.

5. EQUATIONS FOR STRENGTH AND RESIDUAL STRENGTH: The experiments clearly show that the strength of rock salt is a function of 410 , m, and T. Thus, the development of a formula for the stability limit of rock salt was based on the following relationship:

TaB ( 410 ,m,T) - f( 410 ) g(m) h(T). (1)

161

Because the development of the equation has already been discussed in (4), here the results are given in short. It has to be stressed, that, in spite of the addition of two new series, it was not necessary to make any change in the parameters.

It has to be noted that this formula was fit to the enpirical data not by regression but by fitting it to the lower limits of strength (i.e. to the weakest salt) for different values of m. The fit is shown for m = -1 and m - +1 in Figure 5.

The results are (see Table 1 for variables): f(41o) - a + b(41o/cr*)p, (2)

where a - 0, b - 2.7 MPa, p - 0.65, and 41* - 1 MPa.

g(m) - 2k/[ (1+k)+(1-k) .J,.], (3)

where k - 0.74. Thus: g(~l) - 1; g(m-+l) - k. Jl for 20 DC < T < 100 DC

h(T) - \l _ c(T - 100 DC) for 100 DC < T < 260 DC, (4)

where c - 0.002 K-1. Formula (2) is an extended nonlinear Drucker-Prager formulation,

which has also been used in (8). To be conservative, a - 0 was chosen, i.e. no shear or tensile strength is allowed.

Formula (3), which describes the influence of load geometry, was given by (12) and is suitable due to its smooth and convex curvature, which is desired for physical plausibility and favorable numerical behavior in finite element calculations. The value of k is the average quotient of the extension and compression stability limits.

The range of validity of the equation for the stability limit, equation (1), is

and 20 DC < T < 260 DC

s = { 7 to 35 MPa o \20 to 35 MPa

for m = -1 for m - +1.

Below the lower limit of validity, real tension stresses are necessary for failure; such tests have not been carried out. The uniaxial tension strength is about 2 MPa. CUrves for the formula for TOB are shown in Figure 6, including the limits of validity.

162

The residual strength can be approximated by simply calculating the strength at m - +1:

(5)

...... m.~l

10

10 20

01 bl

Figure 6: Plots of the equation for the stability limit of rock salt at T < 100°C: (a) in a graph of 'to vs. 0"0 and (b) in the octahedral plane. Dotted lines show the limits of validity.

6. FRACTURE PATTERNS: After failure, the patterns of the fracture planes were inspected. They are well defined in all samples, except those tested above 150°C. Their shape and deformation are distinct. The fractures are clearly transgranular, at least at low tempera­tures.

The specimens, which were loaded near the extension load path (m > 0.75) exhibited fracture planes nearly perpendicular to the extension direction. The angles relative to the highest principal stress direction ranged from 0° (biaxal tests) to 25°. Samples tested in compression (m < -0.75) yield angles between 25 and 35° relative to the highest principal stress direction. Torsion tests (m = 0) yield angles between 15° and 25°. These results compare well with those obtained by Brace (13) from different types of dog-bene-shaped rock samples. The tests also verify that all failure planes are parallel to the direction of intermediate stress (0"2) (7).

Samples that failed in extension tests exhibit sharp and narrow fracture planes « 1 mm wide), the rest of the material shows little damage, i.e. it did not whiten much during the test. In compression the fracture planes are wider (1 - 2 mm) and the material shows much more damage and subsequently a higher volume increase at failure. At m = 0 the appearance is close to that at m = +1.

7. CONCWSIONS: An equation for a conservative stability limit as the lower boundary of failure strength and residual strength have been developed on the basis of 240 true triaxial failure tests on specimens of eight types of rock salt from the Gorleben salt dome. They can be used in conjunction with a flow rule in finite element model calculations for the identification of potential failure zones in a salt mine.

It has to be noted that there are distinct differences in strength between the rock salt types. The patterns of the fracture planes and the amount of damage are primarily correlated with the load geometry. These results need further explanation. Rather than being decreased by the anhydrite layers, the strength is slightly increased.

OWing to the type of tests, the investigation is valid only under the condition that the three principal stresses are in compression. In practlce, however, most fractures occur at low mean stresses in mine pillars or drifts with an extensional load geometry. This pre­sumes real tension stress in one direction. Until corresponding tests have been carried out, the conservative assumption has to be made that the material does not bear any tension stresses.

ACKNCMLEDGEMENTS: The investigations were supported by the Federal Ministry of Research and Technology (BMFT). We thank C. Caninenberg and his team for the careful performance of the experiments.

REFERENCES: 1. H. Albrecht and U. Hunsche, Fortschr. Miner. 58 (1980) 212-247. 2. U. Hunsche, in The Mechanical Behavior of Salt, Trans Tech

Publications, Clausthal (1984) 159-167. 3. N. Diekmann, U. Hunsche and D. Meister, Z. dt. geol. Ges. 137

(1986) 29-56. -4. U. Hunsche and H. Albrecht, Engineering Fracture Mechanics (1989)

(in press). 5. S. Serata, S. Sakurai and T. Adachi, in Basic and Applied Rock

Mechanics, The Am. Inst. Mining, Metallurgical Engineers Inc., N.Y. (1972) 431-473.

6. U. Hunsche, in The Mechanical Behavior of Salt, Trans Tech PUb­lications, Clausthal (1984) 169-179.

7. M. S. Patterson, Experimental Rock Deformation-The Brittle Field, Springer-verlag, Berlin (1978) 40-41.

8. M. Wallner, Proc. of Waste Management 86, V.II, Tucson (1986) 145-151.

9. W. Menzel and W. Schreiner, Neue Bergbautechnik 5 (1975) 669-676. 10. F. D. Hansen, K. D. Mellegard and P. E. Senseny,-in The Mechani­

cal Behavior of Salt, Trans Tech Publications, Clausthal (1984) 71-83.

11. M. Wallner, Proc. 5th Int. Congr. on Rock Mechanics, Melbourne 1983, V.II, Rotterdam (1983) D9-D15.

12. O. C. Zienkiewicz and G. N. Pande, in Finite Elements in Geo­mechanics, John Wiley & Sons, London (1977) 179-190.

13. W. F. Brace, in State of Stress in the Earth's Crust, Proc. of the Int. Conf., Santa Monica (USA) 1963, American Elsevier, N.Y. (1964) 111-178.

163

AN EXAMINATION OF CONSTITUTIVE LAWS BY HIGH TEMPERATURE CREEP OF ENGINEERING MATERIALS

K. Maruyama *, C. Tanaka + and H. Oikawa * * Department of Materials Science, Faculty of Engineering,

Tohoku University, Sendai 980, Japan. + Environmental Performance Division, National Research

Institute for Metals, Nakameguro, Meguro-ku, Tokyo 153, Japan.

ABSTRACT: Creep of type 316 stainless steel, strengthened by thermal­ly unstable precipitates, was analyzed by two different creep con­cepts; One assumes creep to be a steady state process characterized by minimum creep rate, and the other supposes that two relaxation pro­cesses, strain hardening and weakening due to microstructural degra­dation, take place simultaneously during creep. In the former the mini­mum creep rate cannot provide correct information on creep process, but in the latter a rate constant for the relaxation processes gives a rational conclusion that creep is a diffusion-controlled process. When microstructural degradation severely occurs like engineering materi­als, creep deformation should be analyzed as a relaxation type phenom­enon rather than a steady state process.

1. INTRODUCTION: It is generally postulated that elevated temperature creep is a steady state process and microstructure is stable at least up to the end of secondary creep stage. Tertiary creep (creep acceler­ation) is a consequence of mechanical instability and usually ignored in steady state creep concept. The following equation [1] is a widely used constitutive equation based on the concept:

E = E i + A{1-exp(-Bt)} + Est, (1)

where E i is the instantaneous strain upon loading, the second term describes primary creep (A and B are numerical constants), and E B is the steady state creep rate characterizing creep deformation.

Engineering materials are usually strengthened by precipitates. They are thermally unstable and coarsen during creep. This coarsening (microstructural degradation) causes weakening of the materials and results in creep acceleration. As the coarsening starts at the beginning of creep, there should not be a steady state in creep of such materials. On this basis, Evans and Wilshire [2] have proposed the (J projection concept. The following equation proposed by Maruyama et al. [3] is one of constitutive equations based on the (J projection concept:

E = Eo + A{l-exp(-a t)} + B{exp(a t)-l},

165

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 165-171. © 1990 Kluwer Academic Publishers.

(2)

166

16Cr-13Ni-3Mo -t!p/~-973K / A,t

7

/w,!,~ ~ fI 923 K ,(f8 , ~47~O-._-,,'rP A (/a073K-

<~ l / :~{rf=/

0' A

11 / /1

10-0·4 0·5 0·6 0·8 1·0 1·5 2·0

a/E, 10-3

(a)

Figure 1 (a) Stress and (b) temperature dependence of minimum creep rates i 18 of type 316 stainless steel. Creep stress 15 is normalized by Young's modulus E. Qc: activation energy for E 18.

where c 0, A, B and a are numerical constants. This equation supposes that creep is a relaxation phenomenon characterized by the rate con­stant a, and creep deceleration due to strain hardening (second term) and acceleration due to microstructural degradation (last term) take place simultaneously over the entire creep stages. Secondary creep is achieved by the dynamic balance between the two relaxation processes.

There are essential differences in interpretation of creep be­tween Equations (1) and (2); Creep acceleration really starts at the onset of tertiary creep in the former, whereas it actually initiates at the beginning of creep in the latter. Creep is a steady state process characterized by E s in the former, but a relaxation process charac­terized by a in the latter. This paper aims at examining which creep concept is more useful to describe creep of engineering materials. For the examination used is creep data of 16Cr13Ni3Mo (type 316) stainless steel solution-annealed at 1363K for 3.6ks; The chemical composition is 16.2Cr, 13.3Ni, 2.7Mo, 0.4Si, 1.6Mn and 0.09C in mass%. Some results have been reported in Reference [4].

2. ANALYSIS AS A STEADY STATE PROCESS: Minimum creep rates Em of 16Cr13Ni3Mo steel are plotted against creep stress and reciprocal temperature in Figure 1. Creep stress 15 is normalized by Young's modulus E, in order to compensate the temperature dependence of E. Values of E recommended in AS ME Code [5] are used in this paper: E =152, 146 and 140MPa at 873, 923 and 973K, respectively. E 18 is usually considered to be the steady state creep rate E. in Equation (1). Em falls quickly below a critical stress 15 c; 15 cI E = 1.0, 0.8 and 0.7 xlo-a, respectively, at 873, 923 and 973K.

Structural materials are used at stresses lower than yield strength Sy; in the case of AS ME Code, lower than 90% of Sy which

16~---r----r----r----.----'. 15~---,---...,...----.--r----'

16Cr-13Ni-3Mo steel I ~

14

16Cr-13Ni-3Mo steel D 873 K -+----11---_+_­o 973K N

I o ..... 10

o 923K. alE =6.6j 10- .. /~

12~---+----+----+--~-

o 973K. aIE=6.7xl0-1 OJ

I. .-0

Col 5 ~0~0.0~0''''014c 10~---~--+----+-- ."o~o I 0-- I I . ..oP

o~~~ N

b ...... 8 ..,

0·2 0·4 0·6 t /tr (a)

0.8 1.0

Figure 2 Representative creep curves. (a) Stress dependence and (b) temperature dependence. Time is normalized by time to rupture tr. In (b), i. at 973K is converted to those at 923K using the activation energy for lattice self diffusion Qn.

167

corresponds to (j IE = 6.8x10-4 for type 316 stainless steel at tempera­tures from 873 to 973K [5]. In addition to this limitation, i. must be lower than 2.8x10-u /s (1% strain for 105 hours) for a use at elevated temperatures where creep takes place during a service. As seen in Figure 1 (a), the steel is to be used below (j c in engineering plants, and what practically should be known is creep deformation below (j c. Therefore, this paper focuses on creep in this regime.

Creep is known to be a diffusion-controlled phenomenon, and it has been established on single phase metals and alloys that E. is express as

E s = Eo (0" / E)n D and D = Do exp(-QnIRT), (3)

where E 0 and Do are constants, and R is the gas constant. n is the stress exponent and usually takes 3-5. D is the lattice diffusion constant whose activation energy is Qn. In Figure l(a), values of n are 7.2 above (j c and as high as 20 below (j c. The n value below (j c is quite high as compared with those reported on single phase materials. It follows from Equation (3) that activation energy Q for E. should be equal to Qn, but the Q value (663kJ/mol) obtained from the slope of the line for a IE = 6.5xlO-4 (a <ad is twice as large as Qn (287kJ Imol [6]) reported on austenitic stainless steel. These findings point out that Equation (3) does not hold for i III of engineering materials strengthened by thermally unstable precipitates.

168

Representative creep curves of 16Cr13Ni3Mo steel are given in Figure 2. Time is normalized by time to rup­ture tr • As seen in Figure 2(a), the creep curve shape does not signif­icantly change above <t c (results at 873K), while it substantially changes below <t c (results at 973K). Stress dependence of initial creep rate ii, an average rate over 0 to 2% of tr, is shown in Figure 3. E i does not exhibit a high n value even below <t c. Two creep curves under the same stress at different temperatures are compared in Figure 2 (b). Creep rates at 973K are converted to those at 923K using Qo for lattice self diffusion. At the beginning of creep, the creep rates at the two temperatures coincide with each other, indicating that Equation (3) holds at this stage. At the mInImum creep rates, however,

'I III

16cr-13Ni-3M0t;~4'-973K l'

7 • Ej ee o Em - r-'1

e a_-lf -~~

9 -/-- .'--

e ~ VO

0 0'

1 II

10-1 0-4 0·50-6 0·8 1·0 1·5

a/E, 10-3

Figure 3 Stress dependence of initial creep rate E j.

those values differ SUbstantially. These facts indicate that, for engineering materials in which microstructural degradation severely alters their original strength, minimum creep rate is a temporary value attained by the dynamic balance between strain hardening and weakening due to the degradation. It cannot stand for overall creep behavior. This is an essential difference from what observed in single phase materials in which microstructural degradation is not severe.

As demonstrated above, minimum creep rate of engineering material is not the steady state creep rate to be used in Equation (1). An analysis of creep based on the steady state concept is inappropriate for such engineering materials.

3. ANALYSIS AS A RELAXATION PROCESS: Equation (2) has four param­eters, e 0, A, B and a. These parameters were determined so that t::,. 2 takes the minimum.

t::,.2 = ~[ek - cO - A{1-exp(-atk)} - B{exp(atk)-l}]2, (4) k

where c k and tk are strain and time of the kth datum point on a creep curve. For more details, refer [3, 7] for parameter determination procedure and [8] for obtained results.

Figure 4 shows how properly Equation (2) represents creep curves. From Equation (2), creep rate is given by

c = Aa exp(-a t) + Ba exp(a t). (5)

Creep strain and creep rate were calculated with Equations (2) and (5) using the parameters determined by the above procedure. Equation (2) well represents the measured data, unless mechanical instability be­comes significant near rupture. Equation (5) indicates that, when In E is plotted against t, datum points approach asymptotic lines,

169

InE = InA a -a t and lni = lnBa +a t, respectively, near t =0 and t =tr. Slope of the asymptotic lines offers a and intersections at t = 0 give A and B. When B is small (B = 6.0xlO-5 for Figure 4(b» pronounced primary and tertiary creep stages appear (see Figure 4(a», whereas creep curve becomes linear (see Figure 2(b» when B is large (B = 8.9xlO-3 for Figure 4(c».

In Figure 5, rate constant a in Equation (2) is plotted against stress and reciprocal temperature. The functional form of Equation (2) resembles the well known formula representing relaxation phenomena. In the case of relaxation analysis, rate constant gives information on rate-determining process. The stress dependence of a is expressed by a power law at lower stresses and increases at high stresses (power law breakdown). Its stress exponent is 6 in the low stress regime, being close to the stress exponents for E s reported on single phase metals and alloys. The activation energy Qa. for a in this regime (Figure 5(b» is 300kJ/mol, and agrees with Qo. Similar to E s, there­fore, a can be expressed as

a = ao (0' /E)II D, (6)

where a 0 is a constant. When creep data of engineering materials are analyzed by Equation (2), one can successfully arrive at a reasonable conclusion that creep is rate-controlled by diffusion.

a is related to E II by

. Em

o

= 2aJ AB •

50 t. Ms

100

(7)

150 6"'--~-'-.....----.--'---r-----"'--nI

16Cr-13Ni-3Mo steel I . 923 K. alE: S· S x 10-4 8 ! t=Eo+A(I-exp(-(lt)!~rexp«(lt)-I~ -, ]

<D ® I

5

N 4 ,T b !® -.31---+--+---+---+-1-1-Col

I 2 1---+--+---+---9 .

<D

,-' I --r-~-J./i 00 ·-O~· 0.4· 0·6 0·8 1·0

tltr (a)

Figure 4 Comparison of measured creep data (open circles) and curves calculated with Equations (2) and (5). (a) and (b) are strain and rate of the same creep curve, and creep curve for (c) is shown in Figure 2 (b).

10·13L--....J....--:-'-~:-'-::---:-L:----..J o 0·2 0·4 0.6 0.8 1.0 tltr

(b)

16Cr-13Ni-3Mo steel 973 K. a/E:6.7xl0·4

7 j J J

,,)~lJ,.,.J~· ... ·t~ -.:;..-.--:'- I

c-il> I I m-r--r---0·2 0·4 0.6 0·8 1.0

t Itr (c)

170

16Cr-13Ni-3Mo steel -+-__ "-0 alE = 9 x 10-4

: 1~' fof:fol 10- 7

0;--10-8 ':-::---:-':-::----.J':-::---'!--.O> "~___l

1·00 1.05 1·10 1.15 1·20 r-1 , 1O-3K-1

(b)

Figure 5 (a) Stress and (b) temperature dependence of rate constant a in Equation (2).

Table 1 Activation energies for lattice self diffusion, Qn, for minimum creep rate, Q::, and for rate constant a, Qa.'

Base element QD Alloying Qc Qa. (kJ/mol] element [kJ/mol] [kJ/mol]

Fe (BCC) 340 0.4Cr Mo V C 550 346 1Cr Mo V C 354 353 9CrMoVNbC 350

12Cr Mo V Nb C 500 352 2.2Cr Mo C 373 361 2.5Cr Mo C 420 352

Fe (FCC) 295 16Cr 13Ni Mo C 663 300 10'"" stress 420 high stress

Al (FCC) 140 2.3Li Cu Mg Zr 255 161 10'"" stress (hot extruded) 187 high stress 2.4Li Cu Mg Zr 326 146 10'"" stress (cold rolled) 175 high stress

Zn (HCP) 95 pure Zn 100 95

When the shape of creep curves, in other words A and B, does not sig­nificantly change with stress and temperature, stress and temperature dependence of i m is close to that of a [9,10] and Equation (3) holds for Em. In 16Cr13Ni3Mo steel, on the other hand, B lowers substantial­ly with lowering temperature as mentioned above and decreases quickly with decreasing stress as expected from the drastic change in the creep curve shape shown in Figure 2(a). The decrease in B results in t}1e high apparent stress exponent and the high activation energy for c m.

Table 1 is a summary of activation energies Q, for i m and Qa. for a obtained in various materials. In some cases, Q, takes substantially high values as compared with Qn, whereas Q a. agrees with Qn in all the cases.

The four parameters in Equation (2) reveal simple stress and

171

temperature dependence as the example on a shows, and their values under a long term service condition of engineering plants can be easi­ly evaluated by short term tests. It has been proved that creep curves ten times longer than the longest test duration could be accurately predicted [7]. A rupture parameter derived from Equation (2) [111 has been confirmed to be useful in long term rupture life prediction [7]. The 8 projection concept, therefore, is successful not only in fundamental understanding of creep deformation but also in prediction of long term behavior. This concept is more relevant than the steady state concept for engineering materials strengthened by thermally unstable precipitates.

4. CONCLUSION: Engineering materials are usually strengthened by thermally unstable precipitates. In such materials, secondary creep is apparently achieved by the dynamic balance between strain hardening and weakening due to microstructural degradation, and minimum creep rate cannot offer correct information on creep process. On the other hand, when creep is analyzed as a relaxation process, Equation (2) always gives a rational conclusion which represents the physical real­ity that creep is a diffusion-controlled process. Therefore, creep of engineering materials is better to be analyzed as a relaxation type phenomenon rather than a steady state process.

ACKNOWLEDGEMENT: The creep data used in the present study was obtained in National Research Institute for Metals, Japan. Part of this study was supported by Yazaki Memorial Foundation for Science and Technology and a Grant-in-Aid for Fusion Research (No.63055005) from The Ministry of Education, Science and Culture, Japan.

REFERENCES: 1. F. Garofalo, in Fundamentals of Creep and Creep-Rupture in Metals,

The Macmillan Co., New York (1965) Chapter 2. 2. R. W. Evans and B. Wilshire, in Creep of Metals and Alloys, The

Institute of Metals, London (1985) Chapter 6. 3. K. Maruyama, C. Harada and H. Oikawa, Journal of the Society of

Materials Science Japan 34(1985) 1289-1295. 4. T. Kawada, S. Yokoi, C. Tanaka, Y. Monma and N. Shinya, Trans­

actions of the Iron and Steel Institute of Japan 11(1971) 167-175. 5. Case N-47 of ASME Boiler and Pressure Vessel Code, Version N-47-

21, The American Society of Mechanical Engineers, New York (1981). 6. B. Million, J. Ruzickova and J. Vrestal, Materials Science and

Engineering 72(1985) 85-100. 7. K. Maruyama, C. Tanaka and H. Oikawa, ASME Pressure Vessels and

Piping 141(1988) 77-83; Transactions of the ASME Journal of Pressure Vessel Technology in Press.

8. K. Maruyama, C. Tanaka and H. Oikawa, to be published. 9. K. Maruyama and H. Oikawa, Proceedings of the Third International

Conference on Creep and Fracture of Engineering Materials and Structures, edited by B. Wilshire and R.W. Evans, The Institute of Metals, London (1987) 815-828.

10. K. Maruyama and H. Oikawa, Transactions of the Japan Institute of Metals 28(1987) 291-298.

11. K. Maruyama and H. Oikawa, Transactions of the ASME Journal of Pressure Vessel Technology 109(1987) 142-146.

A CREEP CONSTITUTIVE EQUATION OF A SINGLE CRYSTAL NICKEL-BASED SUPERALLOY UNDER <001> UNIAXIAL LOADING

M. Ma1dini and V. Lupinc

CNR-ITM, Via Induno 10, I 20092 Cinise1lo Balsamo (MI), Italy

ABSTRACT: The formalism of continuum damage mechanics is utilized in a two state variable physically oriented description of creep behaviour in the single crystal superalloy SRR 99 crept at constant load and temperature in the 750 - 950 0 C range. The adopted description is excellent at 850 - 950 0 C while it is less satisfactory at 750 o C.

1. INTRODUCTION: Superalloys for gas turbines are continually being developed to increase thrust, operating efficiency and durability. Recently single crystal rotating blades with excellent high temperature creep and low cycle fatigue resistance in the < 001> crystalline directions have appeared. For a correct design of these components the knowledge of stress rupture behaviour is not always sufficient and the creep strain vs. time relation must also be considered. Thus physically meaningful constitutive equations that describe complete creep curves and have extrapolative capabilities are needed for advanced design.

After the original continuum damage mechanics (CDM) approach by Kacanov (1) and Rabotnov (2), and the extensions of Hult (3) and Leckie and Hayhurst (4), National Physical Laboratory and Cambridge University researchers proposed a physically oriented CDM description of creep behaviour, i. e.: Dyson and McLean (5) showed that tertiary creep in superalloys is mainly due to strain rather than time dependent instability. The results by Basso et al. (6) are basically consistent wi th that conclusion although these authors do not exclude that, for special conditions, some influence of particle growth on creep strain rate can occur. Ashby and Dyson (7) examined the various mechanisms that can contribute to creep damage during the tertiary creep and Dyson and Gibbons (8) adopted a two damage variable description of tertiary creep so as to take into account both intrinsic strain instability and cavitation in superalloys. Ion et al. (9) proposed a two state veriab1e description of primary and tertiary creep and applied it in a personal computer package able to calculate the accumulation of strain vs. time

173

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 173-179. © 1990 Kluwer Academic Publishers.

174

at extrapolated testing conditions and at

temperature. variable load and

2. CREEP MODEL: Creep curves of SRR 99 superalloy at 850 - 950°C exhibit a short and small primary and the majority of these curves is tertiary creep. This behaviour justifies the semplification adopted by the authors in a previous paper (10) where the creep curves were described by an instantaneous strain followed by tertiary creep. The principal objective of the present study is to describe complete creep curves, including primary creep, so as to extend the description of creep behaviour of SRR 99 from 850 - 950°C to lower temperature, 750°C, where primary creep can not be neglected.

In multi-phase engineering alloys pr~~~~~~~~~E can be caused by stress redistributions between the various heterogeneities in the material, e. g. between soft primary dendritic arms and hard inter­dendrite-arm portions of material and, on a smaller scale, between soft matrix and hard particles. Ion et al. (9) have proposed the following coupled differential equations for decelerating primary creep:

~. (1 - s) ~ (1)

s He - Rs

where the internal variable, s, is considered to be directly related to the development of an internal stress, ~. is the initial strain rate, H

1 a strain hardening parameter and R a recovery parameter. These equations represent the McVetty-Garofalo (11,12) type of primary.

During !~~!~~~~~eeE of the alloy examined only microstructure variations (e.g. mobile dislocations density) can contribute to damage since oxidation effects can be neglected and cavitation is not activated at all. To describe the tertiary creep induced by this kind of damage Ion et a1. (9) took into account two different sets of equations:

E E. (1 w) E w

+ E. e (2) 1 and ~ .

CE Vi C~ w

where w is, in both cases, a damage variable proportional to the creep strain and C is a constant. In (10) both sets of these equations were checked with tertiary creep of SRR 99 in the 850 - 950°C range and the best fit was obtained using the linear dependence of strain rate on w.

To describe the ~~mpl~!~_~~~~E_~~~ve primary and tertiary creep mechanisms are combined in an interactive mode" by multiplying Eq.s 1 and the first set of Eq.s 2 in the following two state variable set of coupled differential equations:

~ ~. (1 - s) (1 + w) 1

S H~ - Rs(l + w) (3)

W CE!(l - s)

175

These equations are similar to those proposed earlier (9), but recovery

here is driven by the internal stress variable, s, multi,lied by the

damage factor, 1 + w, that is considered to be related to mobile

dislocation density, and w is proportional to the strain rate cleared

of the primary creep mechanism contribution, since during primary creep

anelastic rather than plastic phenomena can predominate. The Eq. s 3

proposed for constant stress and temperature creep have the following

analytical solution:

bt -ate - 1) bt

'" = M1 - e J + B(e - 1) (4)

where A = H~~/(H~. + R)2 controls

a = (H~. + R)/C~. ~nd b = C~. both l l J,.

creep kinetics, while B = R/C(H",. l

the primary creep amplitude,

contribute to determine the primary

+ R) and b determine the tertiary

creep behaviour. The curvature of primary creep of this description can be larger than that possible with the McVetty-Garofalo relation

(11,12), that has often been found insufficient (13,14). The Eq. 4 can also deal with sigmoidal primary creep, i.e. when C > H

The two sets of parameters (A, B, a and b of Eq. 4 and ~., C, H and R of Eq.s 3) are mathematically equivalent, but denote di1-ferent

approaches to studying creep. Our choice here is to characterize the

set of parameters of Eq. s 3 al though ~., and to some extent also H, l

seem to be particularly depending on the first points of the creep

curve that are critically sensi ti ve to experimental errors due to

loading procedure (e.g. non-axiality of the load).

Since the single crystal studied in this paper is a relatively

ductile superalloy, the extension of the validity of Eq. s 3 from

constant stress to constant load creep should take into account the

stress dependence of the parameters appearing in that equations. To

allow for the external loss of section due to homogeneous tensile

elongation of the constant load creep, as a first step only the

dependence of c. on stress has been taken into account. The l

interpolation of the creep curves with Eq. 4 revealed a Norton type • n

stress dependence for the parameter "'i cr. Thus Eq.s 3 were changed into the following:

~ = E.en "'(l - s)(1 + w) l

S = H ~ - Rs (1 + w)

,; = Ce/(l - s)

All strains in this paper are true strains.

( 5)

3. MATERIAL AND EXPERIMENTAL TECHNIQUES: Alloy SRR 99, designed for

single crystal blades in advanced aeronautic gas turbines, has the following nominal composition (wt %): 0.015 C, 8.5 Cr, 5.5 AI, 2.2 Ti, 5.0 Co, 9.5 W, 0.25 Mo, 2.8 Ta, balance Ni. The creep specimens,

supplied by FIAT Aviazione SpA, Turin, were cast to shape, partially

176

homogenized at temperatures slightly lower than solidus temperature and

finally aged. Coolig following homogenization treatment and ageing at

870°C produced cuboidal 'i' particles of 0.3 to 0.5 pm size. The hardening phase occupied two thirds of the volume of the alloy. The

specimens had cylindrical symmetry of 5.6 gauge diameter and 28 mm

gauge length. Deviations of < 001> crystalline direction from the

specimen axis were within 12°. The creep tests were performed in three

different laboratories.

4. RESULTS AND DISCUSSION: The constant load and temperature creep

tests studied in this work were performed at 750, 850 and 950°C at nominal stresses between 140 and 790 MFa at CSM SpA, Rome, at NEI-IRD

Co Ltd, Newcastle upon Tyne and at our laboratory (Table 1).

Firstly, every single E vs. t creep curve was fitted up to t =

0.7t using Eq.s 5 with the imposed value of the parameter n = 8. This r

is the average value, at the three studied temperatures, of the stress

sensi ti vi ty of the parameter €., These interpolations produced the values of the four parameters :L E. ,C, Hand R at different nominal

:L stress, 0, and temperature, T. In fitting data the final 0.3t portion

r of each creep curve was not considered to avoid the influence of final

fracture mechanisms and the localized reduction of area, Fig. 1.

In Fig. 2 logC plotted vs. stress exhibits, in 'the 850 - 950°C range, a linear relation while no temperature dependence can be

defined. This behaviour suggests the following simple description: C = Dexp(-do), utilized also for the tests at 750°C where the dispersion of

the data does not allow to find any relation. The double-logarithmic

plots of E. and R vs. stress, Fig.s 3 and 4, show a Norton type 1

15,------------r------.--.

~ 10

5

850·C 400MPa

o~~~~~~~~~~ a 500 1000 1500

Time (h)

TABLE 1. Creep Test Parameters

T(OC) (MPa) t (h) E (%) r r

790 375 19

760 570 14

750 730 840 13 710 1173 13

680 1764 15

490 248 21 850

400 800 18

220 379 --195 838 32

950 180 1302 --140 5008 --

Fig. 1. Two examples of creep curves, obtained by integrating Eq.s 5 using only open points, extrapolated to 0.15 strain.

104r--------------------------,

u

10-1 "--'----1 __ -'---'-__ -'---'--__ '------"'-----'--'

o 200 400 600 800 1000 Stress (MPa)

ig. 2. stress dependence of

parameter C.

10- 1

10-2 ,,·1"1 .<: cr: 10-3 ~

'" Q; E 10-4 ::tl <1l

D-

10-5

10-6 100 200 400 600 1000

Stress (MPa)

Fig. 4. Stress dependence of parameter R.

177

10- 2

-7 l , 10-3

5 ·W ~ 10-4 '" -I Q; E ::tl <1l

10-5 D-

10-6

100 200 400 600 1000 Stress (MPa)

Fig. 3. Stress dependence of ini­

tial strain rate parameter.

103

\ I 850·C 0 0

Q;

~ '" 102 E ::tl <1l

D-

10 100 200 400 600 1000

Stress (MPa)

Fig. 5. Stress dependence of

parameter H.

178

behaviour with a strong T influence; these parameters increase when a and/or T increase, while the parameter H, Fig. 5, generally behaves in the opposite way, i.e. it decreases when a and/or T increase.

The interpolated values of the parameters, represented in Fig.s

2-5, were used to nUirerically integrate the Eq. s 5, obtaining the calculah'd creep curves plotted together with the experimental points

in Fig.s G-8. The arrows in these graphs indicate the upper bound of experimental data (0.7t ) used for interpolation.

Apparently the cal~ulated creep curves at 950 and 850°C, Fig.s 6 and 7, excellently describe experimental behaviour, in particular if it is kept in mind that only 70 % of each creep curve was used for the calculation of the parameters. The same model is less satisfactory when creep curves at 750°C are examined: actually, the values of parameter C are strongly dispersed at 750°C and the straight line of the log£ vs. £

curve at 750°C (Fig. 9) indicates that an exponential, eW, rather than linear, 1 + w, tertiary creep damage function could be more ap­propriate; the upward convex curve at 950°C in the same figure confirms that linear damage law is preferable for creep behaviour at higher

temperatures.

5. CONCLUSION: The mechanistically oriented continuum damage mechanics creep model adopted to describe the creep behaviour of SRR 99 single crystal superalloy produced encouraging results, although further work is necessary, possibly on wider data bases, to check extrapolative capabilities.

REFERENCES:

1. L.M. Kacanov, !zv~~!ja_~~ad~mi~~~~~~~~~k SSSR 8 (1958) 26. 2. Y.N. Rabotnov, ~~Z~~~~!UTA~,Stanford (Ed.s Hetenyi and Vincenti)

Springer (1969) 342.

3. J. Hult, Q~_IoP~~~_~~~Epl~~~Co~!~~~~~_~~~hani~ (Ed.s Zeman and Ziegler), Springer (1974) 137.

4. F.A. Leckie and D.R. Hayhurst, ~~!~_~~!~~~~ 25 (1977) 1059. 5. B.F. Dyson and M. McLean, ~~!~_~etal~~ 31 (1983) 17.

6. S. Basso, V. Lupinc and M. Maldini, ~~z~~~nt~Conf~Z~~~~~E' Tokyo, JSME, IMechE, ASME, ASTM, (1986) 483.

7. M.F. Ashby and B.F. Dyson, ~~Z~~_!CF_~,New Delhi, Pergamon (1984) 3 8. B.F. Dyson and T.B. Gibbons, ~~!~_Me!~~~~ 35 (1987) 2355. 9. J.C. Ion et al., NP~~port_DM~_~~~~, April (1986).

10. M. Maldini and V. Lupinc, ~~~~pta_~~!~~~~ 22 (1988) 1737. 11. P.G. McVetty, ~~~~~~ngng~ 56 (1934) 149.

12. F. Garofalo, in ~undame~!~~~-z!_~~eeE-~~~~~~~E-R~E!~re_~~_~et~~~, Macmillan (1965).

13. D. Sidey and B. Wilshire, ~~!~~_~~~~~ 3 (1969) 56. 14. F. Gabrielli and V. Lupinc, in ~!~eng!~_z~~!~~~_~~~_~llZYs Proc.

ICSMA 5, Aachen, Ed.s Haasen et a1., Pergamon, Vol. 2 (1979) 485.

15~~~------------------.,

~ 10

5

195MPa nOMPa

180MPa

140MPa to.

1000 2000 3000 4000 5000 Time (h)

F~g. 6. Comparison of calculated

creep curves with experi­

mental data at 950 o C.

;;'! 10

500

o o

750'C

1000 1500 Time (h)

2000

Fig. 8. Comparison of calculated

creep curves with experi­

mental data at 750°C.

179

200 400 600 Time (h)

800 1000

Fig. 7. Comparison of calculated

creep curves with experi­

mental data at 850°C.

10-2 ~--------------,

1: 10- 3 0 7~5~ia ., -;;; a:

Fig. 9. Two examples of strain rate

vs. strain creep curves.

A Creep Constitutive Model of Dislocation Thermal Activation

C. D. Liu, Y. F. Han and M. G. Yan

Institute of Aeronautical Materials, Beijing, China 100095

ABSTRACT: A new creep constitutive model of dislocation thermal activation has been established in this investigation based on the creep test results, i.e., the creep rate first decreased with creep deformation until a minimum creep rate reached, and then increased with further creep deformation. The constant stress and stress change creep tests of Inconel 718 has been carried out in the stress range of 491 to 856 MPa at 650· C. The experimental results have shown that this model can characterize the creep deformation behaviour of Inconel 718 under the creep test condition used in present investigation.

1. INTRODUCTION: In order to improve the safety reliability and endurance of hot section structures, it is necessary to analyse stress and strain of components and to develop high temperature constitutive relation. Creep constitutive model is basic and essential for high temperature structure components. In 1926, Bailey proposed an internal yield stress to describe creep deformation behaviour. This idea had been improved by Orowan (1) who introduced the concept of back stress.Since then,many other internal variables have been widely used to characterize material behaviours to set up constitutive relation (2-7). The basic problems relative to constitutive equation are as follows: 1). physical background of internal variables, 2).evolution of internal variables, 3). relationship between inelastic strain and internal variables. Therefore, some internal variables has been studied and the creep constitutive equation to describe all creep stages for both constant stress creep and stress change creep have been established in this investigation.

2. EXPERIMENTS: The material used in this study was Inconel718 superalloy.Creep tests have been conducted at 650· C.Temperature has been controlled within_±lo C. Creep deformation has been measured by a high temperature extensometer and a displacement

digitizer with accuracy up to 0.3%. The dimension of creep specimens was <I>8x100 rom. All creep tests has been performed by a DTS-5 creep test machine.

2.1 Cree.p Deformation Behaviour of Inconel 718: Constant stress creep tests under different stress levels have been carried out and the creep curves of the alloy have shown in Fig.la. Experimental results indicate that the creep deformation of the superalloy

181

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 181-187. © 1990 Kluwer Academic Publishers.

182

5r---------------------~

El.

3

o 100 200 hr

Fig.1a Creep Curves for different stress of Inconel 718

~ 10')

" cr c. '" '" '-u

10'- L-___ -'----___ -"-___ ....J

o 2 3 Creep Strain E 10.2

Fig. 1 b Creep rate vs. creep strain ofInconel 718

Inconel 718 under present experimental stress levels of 491 to 856 MPa can be divided into two stages, Le., transition creep where the creep rate decreases with creep strain and secondary creep where the creep rate increase with creep strain, as shown in Fig. 1 b. It has been observed that there exists a minimum creep rate between transition creep and secondary creep for each specimen. After reaching the minimum rate, the creep rate increases with creep strain quicldy and then increases slowly, Le., asymptotically to a limitation value till rupture. It is also noticed that the transition creep strain is only one percent of total creep strain and can be almost neglected.

2.2 Creep Degradation: The fact that creep rate increases with creep strain observed in the present study, as shown in Fig.1b, suggests that a creep softening process takes place during the secondary creep of Inconel 718, which may be due to the creep resistance degradation caused by the change of material structure, such as, the growth of second phase particles, the formation and the aggregation of micro-voids and the growth of subgrains.

3. CREEP CONSTITUTIVE MODEL OF INCONEL 718: The resistance of dislocation motion depends on the size and the interspace of second phase particles which distributes inhomogeneously in alloy. Therefore, there exists a minimum resistance, Rm, for a certain microstructure condition in a material. When applied stress is larger than minimum resistance, Rm, a dislocation can overcome the resistance and move. The value of Rm increases during plastic deformation where the interaction of dislocation occurs, and hence a material is hardened. When the applied stress keeps constant, plastic flow will stop due to the plastic deformation hardening during loading process. However, at the temperature above absolute zero the thermal fluctuation exists in any material, therefore the thermal activation will make the inhibited dislocation to move, leading to inelastic deformation, i.e., creep deformation occurs.

3.1 Thermal Activation of Creep Deformation: During a creep test a dislocation

possesses the energy, U(cr), where C1 is an applied stress. On the other hand, there exists a resistance due to obstacles or a potential barrier, U(po), which should be surmounted for a dislocation to move. When the value of U(po) is greater than U(cr), no plastic flow occurs. However, creep deformation will take place because of thermal activation.

183

According to statistical thennodynamics, the probability of a dislocation possessing a thennal energy, E, at a moment is

p = l/kT exp(-E/kT) (1)

Therefore, a dislocation possesses the energy arising from both thermal activation, E, and the applied stress, U(a). When the value of E+U(a) is equal to or larger than barrier potential U(po), i.e.,

E + U(a) ~ U(po) or (2)

E ~ U(po) - U(a)

The dislocation can surmount the obstacles. Hence, the probability of a dislocation moving forward is

Pf = pf{ E ~ (U(po) - U(a) }

= f; l/kTexp(-E/kT) dE (3)

Where Er = U(po) - U(a). When the thennal energy of a dislocation E is greater than the

value of U(po)+U(a), there exists the probability pb for the dislocation to move back, i.e.,

Pb = pb{ E ~ (U(po) + UCa)) } (4)

Therefore, the net probability for a dislocation to surmount obstacles to yield creep deformation is as follows:

(5)

It is reasonable to assume the relationship of energy with stress or resistance as

UCa) = B a

U(po) = B R (6)

where B is a material constant. In general, the probability of resistance or potential barriers obeys nonnal distribution, i.e.,

o R<Rm feR) = ( (7)

2/(j-121to. ) exp{ -(R-Rm)2 no. 2} R>=Rm

Creep rate at high temperature can be expressed as

~= f~f (R)bpdR (8)

184

where b is Burgers vector. As a consequence of above derivation. the creep constitutive equation of thermal activation for the dislocation can be obtained as follows:

e = A exp( -B Rm/kT) (sh(BO'/kT)} (9)

3.2 Evolution of Resistance: As the mentioned above. the creep degradation process has been observed by experimental result in this study. This can be considered as the degradation of resistance or potential barrier due to obstacles. A proposed nonlinear weakening rule to describe resistance degradation can be expressed as

dRm = cO' de - rRm de (10)

Under constant condition. the evolution equation of the resistance can be obtained by integrating equation (10).

Rm = clr 0' - (c/r 0' - Rmo) exp(-re) (11)

Where Rmo is a initial resistance to the dislocation and can be considered as the initial yield stress of a material.

4. VERIFICATION OF CREEP CONSTITUTIVE MODEL: 4.1 Constant Stress Crer.p: A creep constitutive model based on the thermal activation of dislocation has been established in this investigation, by which the deformation behaviour of Inconel 718 during creep weakening stage in the stress range of 491 to 856 MPa can be successfully explained and described. In this model, the relationship between the creep

rate e and the applied stress 0', is approximately linear at low stress level, and the power law relationship will approach as stress increases. In order to verify the dislocation thermal activation creep (DTA) model, creep tests of Inconel 718 under various stress levels at 650' C have been carried out and the results are shown in Fig.3. Fig.3 shows that the DTA creep constitutive model has a good agreement with the experimental results. To further verify the model, Powell's experimental result, as shown in Fig.4. has been used in this study, which also agree with the DTA model.

Combining equation (9) and (11), the relationship between creep rate and stress as well as creep strain can be established during creep softening process. And then correlating this relationship with creep test data of Inconel 718 under four different stresses, all the parameters in the DTA creep model can be obtained. i.e., the creep constitutive model is given by

. e =5.65xl0-7 exp(-0.0195 Rm) sh(0.0195 0')

(12)

Rm = 0.446 0' - (0.4460' -491) exp(-1.636 e)

It is interesting to notice that initial yield stress Rmo can be obtained by correlating the DTA creep model with the four creep group data and this obtained value of Rmo is in agreement with yield stress measured from tensile tests.

· .c e

·w

soo Stress

Fig. 2 DT A model vs. creep results ofInconel718

.c e

• \.OJ

0.

'" ... ~ 10-4

E :> E "c 1:

IO-S

185

~6~~ ______ ~ ______ -=~~~

SO 70 90 110 Stress MPa

Fig.3 DTA model vs. Powell's creep result

It is easy to describe a curve using one mathematical function, however it is very difficult to correlate a group of curves using one function. The fact that the present DT A creep model can correlate four groups of creep data suggests that this model can characterize the creep deformation process, as shown in Fig.5.

4.2 Stress Change Creep Tests: In order to further prove DTA creep model, a stress change test has been designed, as shown in Fig.6, and the comparison of DTA model with experimental result of creep rate e is also shown in Fig.6. It can observed that DTA model can successfully predict the stress change creep behaviours.

5. CONCLUSION: A new creep constitutive model based on dislocation thermal activation has be established by using statistical thermodynamics and the nonlinear creep degradation rule. This model can successfully describe the creep deformation behaviour during weakening stage of Inconel 718 alloy and for both constant stress creep and stress change creep in the applied stress range of 491 to 856 MPa.

186

1~1.----------------------------------.

• 859 MPQ • 781 MPQ • 703 HPQ • 640 MPQ

2 3 4 5 Creep S~rQin E 10-2

Fig. 4 DT A creep model vs. est. results of four groups

Fig.5 Comparison of DT A model prediction with stress change creep test data

187

REFERENCES: L R. W. Evans and B. Wilshire, Creep of Metal and Alloys,(l985) 114-156. 2. A. Miller, in Transaction of ASEM, Journal of Engineering Materials and

Technology, (1977) 97-105. 3. T.C. Lowe and A.K. Miller, in Transaction of ASME, Journal of Engineering

Materials and Technology, (1984) Vol. 106,337-342. 4. T.C. Lowe and A.K. miller, in Transaction of AS ME, Journal of Engineering

Materials and Technology, (1986) Vol. 108,365-373. 5. E.W. Hart, in Transaction of AS ME, Journal of Engineering Materials and

Technology, (1976) 193-201. 6. TJ. Delph, in Transaction of ASME, Journal of ASME, Journal of Engineering

Materials and Technology, (1980) Vo1.102, 327-336. 7 T.L. Chaboache, in International Journal of plasticity, (1986) Vol.2, 149-185. 8. D.J.Powell, Proceeding of the Second International Conference on "Creep and

Fracture of Engineering Materials and Structures", Part II Edited by B. Wilshire and D.RJ. Owen, (1982)

DETERMINING A CONSTITUTIVE EQUATION FOR CREEP OF A WOOD'S METAL MODEL MATERIAL

Mark Belchukl, Dan WaW,and John Dryden2

IDepartment of Engineering Materials, University of Windsor Windsor, Ontario, Canada N9B 3P4

2Department of Materials Engineering, University of Western Ontario London, Ontario, Canada N6A 5B9

ABSTRACT: Having chosen Wood's metal (melting point 68' C) as part of a model material for creep of two-phase materials, it is necessary to determine satisfactory constitutive laws which describe its macroscopic behaviour. A number of constitu­tive equations for creep of Wood's metal alloy are compared with creep data obtain­ed at 35' C (TIT m = 0.90) for stresses up to 20 MPa.

1. INTRODUCTION: Considerable progress has been made in understanding the creep of homogeneous single phase materials. There are however other materials, of engineering significance, whose creep behaviour has not been characterized with clarity. Of particular interest to the present study are materials with a microstruc­ture which comprises hard crystals surrounded by a soft continuous boundary phase.

An example of such a material is liquid phase sintered ceramics which, under moderate temperatures, experience softening of the amorphous grain boundary phase leading to creep dominated by grain boundary separation. The creep deform­ation in these materials is then generally governed by the properties of the boundary phase and its volume fraction.

Interest in the creep of Wood's metal alloy is the result of a search for a mater­ial that behaves as a Newtonian fluid during slow deformation near room tempera­ture and whose viscosity is a weak function of pressure~ A material of this nature is required for a model material for the study of creep in materials with soft boundary phases (1).

In this context, Wood's metal alloy becomes attractive as the second phase material in a model material consisting of rigid hardened steel blocks surrounded by the Wood's metal second phase. If the results of the model material tests are to be properly analyzed, then the creep properties of the Wood's metal alloy must be appropriately characterized.

2. EXPERIMENTAL PROCEDURE AND DATA ANALYSIS

2.1 Specimen Preparation: The Wood's metal alloy was prepared by melting to­gether, bismuth, lead, tin, and cadmium in proportions of 50.0%,26.7%, 13.3%, and 10.0% respectively. All constituents were of 99.99% purity. The resulting homo­geneous mixture was then cast into glass tubing at 100' C and allowed to air cool to room temperature. Since Wood's metal expands upon solidification, the glass tub­ing was easily shattered and discarded. Several sizes of tubing were used to obtain sample diameters varying from 4 mm to 8.5 mm. Cylindrical compressive creep samples were cut to two diameter lengths using a cutoff tool in a high-speed lathe.

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A. S. Krausz et aL (eds.), Constitutive Laws of Plastic Deformation and Fracture, 189-195. © 1990 Kluwer Academic Publishers.

190

Metallographic preparation for the scanning electron microscope consisted of polishing using standard methods as well as etching with a 2:2:2:5 acetic acid: nitric acid: 30% hydrogen peroxide: distilled water etchant for some specimens.

2.2 Testing: As a compromise between the problems of buckling and barrelling of specimens in compression, a height to diameter ratio of 2: 1 was used. A further effort to reduce barrelling through friction reduction was accomplished by the appli­cation of a thin film of multi-purpose grease at the specimen/platen interface.

Strain-time data was obtained in air at 308 K and at 314 K using a constant load compression creep tester. The sample gauge length was measured using a lin­ear variable displacement transducer (LVDT) at specified time intervals throughout each test. The test temperature was computer controlled with an accuracy to ±0.1 K making use of electrically heated forced air within the test chamber.

2.3 Data Analysis: The strain (() versus time (t) data from each test was run through a programme developed by Evans and Wilshire (2) to estimate creep curve parameters (a, b, c, d) for six different equations as follows:

c

E = a(1- exp(-bt)

E = at1/ 3 bt

E = a(1- exp(-bt» + c(exp(dt) -1)

E = at 1/ 3 + bt + ct3

E = a(1 - exp(-bt) + ct

E = at 1/ 3 + bt2/ 3 + ct

Wood's Metal Creep at 35°C

0.00011\-----------------,

-0.005

-0.010

-0.015

·2 -0.020 (j) o ,,=3.54-MPa

t. ,,=7.21 MPa -0.025

-0.030

-0.035

-0.040 +--I-----+---!---t--.-+--t----+---i o 50 100 1 50 200 250 300 350 400

Time (min)

Figure 1. Typical Strain-Time Curves for Compressive Wood's Metal Creep

(1)

(2)

(3)

(4)

(5)

(6)

From the above six equations, the one which best represented the data was chosen on the basis of the smallest standard error as defined in the programme. The

191

f. vs t equation of best fit was then differentiated to give an equation for creep strain rate with time. An example of some typical test results for two stress levels appear in Figure l.

The strain rate data points in Figure 2 are obtained by central difference num-erical differentiation of the (-t data. Maximum creep strain rate ((max) is then

determined from the second derivative of the best fit equation. The value of (max

was assumed to be the steady-state strain rate ((s) at the initial applied stress (0-) level of the test.

Each test analysis yielded a point on a (max vs (J plot which was fit to three different typical steady-state creep rate equations:

ts = A exp(B(J)

ts=A [sinh(Bu)]n

(7)

(8)

(9)

where A is a function of microstructure and temperature, B is a constant indepen­dent of stress over an appropriate range of conditions, and n is a constant whose magnitude can vary from unity to 40 depending on the material and the test condi­tions.

11-2.0E-6 " .......

o 15-4.0E-6 a:: c .~

U)-6.0E-6

Wood's Metal Creep Rate at 35°C

o u=3.54MPa 'rnax=-3.02E -7 sec-1

" u=7.21MPa 'rnax=-2.57E-6 sec-1

-B.OE -6 -t----+----4 o 50 1 00 1 50 200 250 300 350 400

TIme (min)

Figure 2. Typical Strain Rate versus Time Curves for Wood's Metal Creep

The temperature dependence of the steady-state creep rate can be generally represented by an Arrhenius equation

fs=Aexp(-QclRT) (10)

where A is taken to be an empirical constant, R is the gas constant, Qc is the activa­tion energy for creep, and T is absolute temperature. For a small temperature change A is constant and the value of Qc can be determined by substitution and rearrangement of equation (10) as follows:

192

(11)

where (s and (s are steady-state strain rate values collected at temperatures Tl 1 2

and T2 respectively at the same stress level. A limited amount of test data was collected at 314 K and it was used in con­

junction with the data collected at 308 K to determine an appropriate activation

energy for creep. The values of fs and fs were replaced by functions of stress 1 2

according to the power law equation (7) and then an average activation energy was calculated by dividing and integrating equation (11) over the stress range of 3 to 7 MPa.

3. DISCUSSION AND CONCLUSION: Micrographs of Wood's metal alloy using the scanning electron microscope show that the microstructure is composed of at least four phases. From the back scatter electron image shown in Figure 3 the large light coloured phase was found to be

Figure 3. Scanning Electron Micrograph of Wood's Metal Alloy

composed exclusively of bismuth. The small dark phase in the lower part of the micrograph is tin and the light spots within it are bismuth. The large needle-like phase which runs across the entire micrograph is found to be cadmium. This corres­ponds with binary phase diagrams of cadmium with bismuth, lead, and tin that show cadmium to have a low solubility in these elements. The light phase with the

193

dark spots is a eutectic structure primarily composed of bismuth and lead with a small amount of tin and cadmium. Figure 4 is a secondary electron image at 40X magnification of an etched Wood's metal specimen. This micrograph helps to show the size and distribution of the needle-like cadmium phase. It is suspected that these needle-like cadmium phases form shear bands which may lead to premature fracture in some specimens.

Figure 4. Scanning Electron Micrograph of an Etched Wood's Metal Specimen

It has been shown that the activation energy for self-diffusion is approximately equal to the activation energy for high temperature creep (3). The experimentally determined activation energy for creep of Wood's metal alloy (Qc = 88.0 kJ Imol) is close to the self-diffusion values for all of the individual element components. This merely shows that the experimentally determined value may have some credibility. Since the data collected at 41° C was obtained within a stress range of 3 to 7 MPa, then the experimentally determined activation energy can only be expected to apply in this stress range.

Even though the activation energy for high temperature creep is approximately equal to the activation energy for self-diffusion suggesting the creep deformation is related to the diffusion of the constituent atoms, it is not possible to identify a creep mechanism from this fact alone (4).

In trying to determine a creep mechanism much attention has been directed toward the value of the creep exponent (n) in equation (7). It has been found to be about 5 in pure metals and some alloys and about 3 in other alloys (5). Creep mechanisms have been classified into Class I (n ~ 3) and Class II (n ~ 5) alloys. In pure metals the creep mechanism is dominated by dislocation cell structure forma­tion and recovery. Class I alloys (e.g. AI-Mg alloys) have been found to creep ac-

194

cording to a viscous glide process in which solute effects lead to uniformly distribut­ed dislocations. Creep in Class II alloys (e.g. Cu-AI alloys) has been observed as a combination of recovery as in pure metals and viscous glide of solute dragging dislo­cations.

The determination of a particular classification for an alloy is the result of a considerable amount of testing coupled with detailed transmission electron micro­scopy of dislocation structures. In addition, multi-phase alloys such as Wood's metal further complicate classification attempts, since the potential exists that creep behaviour other than those mentioned above are present. These obstacles have lead to examinations that go beyond observing dislocation structure and deter­mining stress exponents. The shape of the primary creep curve, stress change exper­iments and the use of effective stress over applied stress in determining stress expon­ents have all been used to classify creep in alloys (6).

Determination of a creep mechanism( s) for this alloy is not the goal of the present work. The purpose is to find a constitutive equation that satisfactorily represents the creep rate over an appropriate stress range at a particular tempera­ture.

Curve fitting of the fs vs u data to equations (7) through (9) was accomplished using a Gauss nonlinear curve fitting programme. Table 1 below gives the value of the constants and the standard deviation for each equation fit. It can be seen from the standard deviation that the hyperbolic sine equation (9) provides the best fit of the data.

~ o

" ~ LOE-4

.! ell LOE-6 I ~ "

Steady-State Strain Rate versus Stress

o Data Points - Power Law

g I

LOE -7 +--""'--+-----<>--__ ......... --+-if------+-----f

2 10

Stress (MPa)

Figure 5. Log-Log Plot of Stress versus Steady-State Creep Rate

It is common to present creep curve fits on log-log plots that give a straight line as in Figure 5. However in comparing the fit of the three equations it is more useful to display them on one ~raph as in Figure 6. This plot shows a rather poor fit by the power law of equation (7) at the higher stress levels and an even worse fit by the exponential equation (8) throughout the data range. The equation which fits best is the hyperbolic sine equation (9) which reduces to a power law at low stresses and an exponential at high stress levels (7). The power law equation (7) has been traditionally used to represent creep rate data primarily due to its simplicity in obtaining a fit. The ready access to computers have now made curve fitting of more

195

Table 1. Coefficient Values for Steady-State Creep Curves

Equation A B n Standard Error

power law (7) -6. 114E-9 3.095 8.077E-6

exponential (8) -9.803E-8 0.3385 1. 874E-5

sinh (9) -6.415E-5 4.563E-2 2.926 3.706E-6

complex equations such as the hyperbolic sine equation (9) particularly easy. Per­haps the time has come to replace the power law equation with the more represent­ative hyperbolic sine equation in some cases.

Steady-State Strain Rate versus Stress

~ o.Ota3Il1E1!l~~;o,.:.;~ "­'::::"-2.0E-5 .2l o '" -4.0E-5 c 'E en -6.0E-5

<D

:8 -8.0E-5 Ul I -g -1.0E-4

"

o Data Points -- Power Low .... Exponential - Hyperbolic Sine

.,:. >C

" , 0·.', . , .. , . , , ,

en -1.2E-4 +--+--l--+---+-t-----f---t-+--+--------1f--lJ 2 4 6 8 10 12 14 16 18 20 22

Stress (MPo)

Figure 6. Curve Fits of Steady-State Strain Rate versus Stress

REFERENCES: 1. M.A. Belchuk, D.F. Watt and J.R. Dryden, Modeling Creep in Materials with Soft Boundary Phases, presented at the 27th Annual Conference of Metallurgists, Montreal, 28-31 August (1988). 2. R.W. Evans, and B. Wilshire, Creep of Metals and Alloys, The Institute of Met­als, London, England, (1985) 274-294. 3. O.D. Sherby, and P.M. Burke, Prog. Mater. Sci., 13 (1968) 325. 4. D.O. Northwood, and LO. Smith, Metals Forum, 4 (1985) 237-249. 5. J.E. Bird, A.K. Mukerjee and J.E. Dorn, Quantitative Relation Between Proper­ties and Microstructure Edited by D.G. Brandon and A. Rosen, Israel Univ. Press, (1969) 255. 6. T.G. Langdon, Proc. 6th Int. Conf. on Strength of Metals and Alloys Edited by R.C. Gifkins, Pergamon Press, 3 (1982) 1105. 7. F. Garofalo, Fundamentals of Creep and Creep Rupture in Metals, MacMillan, New York, (1965).

"DISLOCATION CRACK-TIP INTERACTIONS: INFLUENCE ON SUB CRITICAL CRACK GROWTH"

W. W. Gerberich, T. J. Foecke and M. Lii Chemical Engineering and Materials Science, University of Minnesota

Minneapolis, Minnesota 55455-0132, USA

ABSTRACT

A number of hypotheses on environmentally-induced crack growth involve a two-step process of dislocation rearrangement and brittle crack advance. This ranges from film rupture/creep to microcleavage/plasticity blunting as might be associated with stress corrosion cracking and hydro­gen embrittlement. What are the continuum descriptions of stress and strain appropriate to such processes? Are continuum plasticity descriptions adequate near the crack tip or are more detailed "continuum-elastic-dislocation" descriptions necessary to understand crack-tip stability. Applica­tions of computer simulations to parallel dislocation arrays at crack tips in anisotropic crystals will be presented. Comparison to dislocation nucleation in NaCI and discontinuous crack growth observations in Fe-3wt%Si single crystals (H2 cracking on the {001}) will be made.

1. INTRODUCTION

Crack instability/stability can be examined at two levels, the global and the local. In terms of brittle fracture, one might argue that a material is either very brittle (no plasticity) and therefore below the ductile-brittle transition or is elastic-plastic in the global sense and therefore above the ductile-brittle transition. But it is well known that considerable plasticity can occur prior to brittle fracture.(1-2) In fact, local energy criteria in terms of effective surface energies and dislocation pile-up approaches have a long history.(S-5) However, these energy-balance criteria are often not very specific and do not address why a material suddenly will cleave. An exception might be a critical size carbide fracturing which then triggers cleavage in the surrounding materiaL<s-5) But even here, the initial event of the carbide particle cleaving is not well described. Moreover, in relatively clean materials where such fracture nuclei may not exist, cleavage can still occur by still undefined processes. Superimposed upon this local ductile/brittle process can be an environment which produces sub critical crack growth at applied stress intensities as low as ten percent of the fracture toughness. Here then, one has an effective surface energy that depends on both local plasticity as well as adsorption, absorption or reaction of some embrittling species such as hydrogen. In this sub critical regime, how does a crack initiate at such low (presumably) stress or strain levels? Why does it arrest and then grow again or is it continuous on a local scale?

To address such fundamental issues in two material systems, NaCI and Fe-3wt% single crystals, we are using the approach outlined in, Figure 1. ,For example, using transmission microscopy and back-scattered electron channeling techniques, we are evaluating both dislocation substructures and local strain distributions to within one micron of the crack tip. The question is whether or not local distributions deviate substantially from continuum predictions and what impact this might have on

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A. S. Krausz et al. (eds.), Constitutive Laws of Plastic De/ormation and Fracture, 197-205. © 1990 Kluwer Academic Publishers.

198

failure criteria. The next two units address the macroscopic properties of threshold stress intensity or sustained-load growth via mechanical testing and through computational simulations. It is this latter aspect which will be emphasized.

Mal'ls Fracture Computational Characterization Mechanics Mechanics

TEM SEM

Ep(r,8)

«- » ----(=-»----MTS Cray II

{Rp, Ls, _L's} Ojj (r,e)

Fig. 1. Approach to resolving fundamental issues in crack-tip stability problems.

While most are familiar with tight binding or embedded atom techniques for atomistic level and finite element or boundary element methods for continuum level modeling, it is that in between area which we believe to be the important one. It is precisely this intermediate region which contains such microstructural features as dislocations and grain boundaries if polycrystalline. As such, we will discuss some current findings on strain distributions and dislocation substructures as to how these represent the inputs required for the computational programs. For the most meaningful simulations, one should have the ability to match up with continuum mechanics at the one extreme and atomistics at the other. While we can achieve the former, the latter is beyond the scope of this paper.

In the present study we present some initial experimental results on dislocation emission and lo­cal plastic strain distributions. In addition, how these are proposed to affect the instability/stability question is addressed. Then the computational approach, as guided by experimental observation, will analyze the stress distribution in the critical microstructural region using a discretized disloca­tion model. Finally, how this all affects hydrogen-induced subcritical crack growth on the cleavage planes of Fe-3wt%Si single crystal with be hypothesized.

II. CRACK TIP STABILITY, DISLOCATION EMISSION AND STRAIN FIELDS

More than a decade ago Rice and Thomson(7) proposed a concept for crack stability in simple, crystalline solids based on dislocation emission. The idea was that if the Griffith criterion were achieved first the crack would become unstable and if dislocation emission occurred first the crack would be stable. The respective local stress intensities for these two processes would be

kIG = y'2ErG/(1 - 112) (la)

kJe = J.Lb/sinOcos;v'2ll"ro(1- II) (lb)

where E, J1. are the tensile and shear moduli, II is Poisson's ratio rG is the surface energy, b is the Burgers vector, 0 is the angle between the crack and emission planes and ro is the core cut-off radius on the order of b in magnitude. As an example, consider iron. Since b ~ ro , this gives

(lc)

For the slip plane intersecting along the crack front, one could have cracking on the (001) cleavage plane with:

(i) (001)[110] growth with {1l2}(1lI) slip; (ii) (001)[010] growth with {011}(1lI) slip

199

These would give 35.26° and 45° traces for crack-tip emission on a slip plane which emerges at the free side surface. Using 0 = 35.26°, 45° in Eq. (lc) with E, i and b given by 1.32 x 105

MPa, 0.3 and 2.48 x 10lom, one finds k1e = 0.83 and 0.70 MPa-ml / 2 for (i) and (ii) respectively. With iG ~ 2 x 1O-6MN/m, kIG is equal to 0.76 MPa-ml / 2 , midway between the two kle values. This implies that case (i) should be brittle and case (ii) ductile. In fact, case (i) was observed to emit dislocations along {II2} but case (ii) was not observed to occur. Both (i) and (ii) appeared to be ductile well beyond kIG with applied stress intensities greater than 50 MPa-ml/ 2• These observations initially appeared to negate the criteria expressed by Eqs. (1). However, it is also known that emitted dislocations will be shielding dislocations, exerting a back force on the crack tip, effectively reducing the local kI. This allows the applied KI to increase without exceeding the cleavage condition. Eventually, however, the local kI may exceed the Griffith criterion for the shielded crack. This condition is given by(8).

k1c = 20"fv'2iiTr sinOcos( 0 /2) ; kIC ~ kIG (2)

where O"f is the lattice friction stress and rf is the distance from the crack tip to the shielding dislocation. Note that rf may be limited by microstructural units such as the grain size or secondary slip planes. Thus the question of stability/instability really resides with Eqs. (1) and (2), i.e. with the emitting or the microstructural condition. Both of these points, that Eqs. (1) are not well posed or that Eq. (2) may be possible under special circumstances, prompted a closer look at these.

First consider dislocation emission. A number of tests have been accomplished on NaCI crys­tals oriented in the [001] with dislocation emission on (011)[011] and (011)[011) planes. This was prompted by knowing all of the parameters in Eq. (1), by the relative ease of accessibility and cleavability of such crystals and by their optical properties which included transparency as well as birefringence. The latter was used effectively to ascertain even when subsurface dislocation sources were activated. This proved to be important since as many as eight isolated sources through the thickness of 5mm thick samples were identified prior to dislocations emerging at the free surface. The following parameters are appropriate:

E[OOl] = 4.07 X 104 MPa b = 3.988 x lO-lOm ra ~ 7.0 x lO-lOm

l/ = 0.248 0=45°

Inserted into Eq. (lb) these give k1e = 0.200 MPa-ml / 2• In eight separate tests, it was observed that k~BS = 0.067 ± 0.003 MPa-ml / 2, a very reproducible value at room temperature. Thus, it appears as though the experimental observation is a full factor of three below the theoretical value. Very recently, Thomson(9) reviewed approximately ten different investigations and, except for one, all found k1e to be a factor of three to five less than predicted. Thus, Eqs. (lb) and (lc) are still not well posed and if the Fe-Si single crystals of the present investigation behave as other materials, it is not surprising why fracture did not initiate at low stress intensities since k~BS ~ kIG.

Although this clarifies the initial stability, what about higher stress intensity levels? After slowly loading a number of the Fe-Si crystals to high stress intensities in the range of 30 to 40 MPa_ml / 2,

their strain fields were examined in detaii.ClO,ll) For sustained- load cracking in one atmosphere hydrogen at room temperature, the strain distributions as determined by electron channeling are shown in Figure 2. Strains approaching unity were found at 10j.tm from the crack tip(lO) agreeing with previous studies which demonstrated such strains at the surface of a similar sample.<ll) The solid lines are theoretical predictions for a growing crack in an elastic-plastic material with strain hardening'<l2) By extrapolating the lower plane strain curve, it can be shown that strains of 0.4 could result at one j.tm. Nevertheless, such samples would undergo brittle fracture with a very

200

small increase in strain rate. Since such strains were achieved, why should there be any instability associated with cleavage as opposed to ductile fracture? During the course of the strain-distribution investigation, it was found that plastic deformation was inhomogeneous and anisotropic near the crack tipPO) In fact, it appeared as though sharp slip bands penetrated through the multiple­slip process that characterized large scale plasticity at these high stress intensity levels. Thus, local dislocation shielding could still be playing a role in stability/instability even at large KI . Such sharp, anisotropic slip produced bands on the order of 30-40jlm long, as found by electron channeling.(lO) It was essential to use such a result in the computational modeling scheme.

III. COMPUTATIONAL PROCEDURE

Although much as been written and reviewed about dislocation shielding(13), it is useful to review the concept briefly. When dislocations are emitted from the crack tip, they produce both an interaction force with the crack-tip stress field and an image force with the free surface. De­pending on how the dislocations are arranged, various types of shielding problems may be solved using the equilibrium of forces. Several such approaches, as illustrated in Figure 3, are a single superdislocation,(14) a spread out array of mini-super dislocations(15) and a large superdislocation with a small series of mini-superdislocations located near the crack tip'p6) This superdislocation is generally used to represent far-field plasticity while the mini-super dislocation can represent near­field plasticity that is inhomogeneous and anisotropic. In all of these equilibrium must be maintained, and thus forces produced from a number of sources must be considered. The force on a dislocation associated with the external stress, T k1 , must be considered. Similarly, we have stress interactions arising from (') the dislocations themselves, (") the external stress and the crack, (III) the crack­dislocation interactions, and ("") the image force term. These five major stress contributions may be summed to give the stress tensor:

k= 1,2,3 1= 1,2 (3)

Except for the image force term, Atkinson and Clements(14) had worked out the complete procedure for finding equilibrium of a single superdislocation at a crack tip contained in an anisotropic elastic solid. These terms are identical to those given before(14) except for the image term which is:

(4)

Note that this term just considers the pair wise interaction of the image with itself.

With K", the positions of the dislocations, Z", the point of interest where the stress state is to be evaluated and T22 the applied stress for uniaxial tension as used in this simulation, the stress field could be determined defining the relationship between d?) and the Burgers vector, bi l ) by:

(5)

The matrices B, L, M and P are functions of the elastic constants, Cll, C12 and C44 for cubic crystal, being 24.2, 14.65 and 11.2 in l04MPa units for iron. Additional theoretical details14 and the matrices specific to Fe-3wt%Si(16) may be found elsewhere. This equilibrium solution requires that the friction stress, such as the Peirerl's barrier, which controls dislocation motion be invariant. In addition, the total shear stress acting on the dislocation in the slip plane in the specified slip direction must be equal to this intrinsic friction stress, giving

(6)

with nkl the slip plane normal, and bl being the slip vector.

1.0

~'o i:",·

c: 0100 o ~.

0 'E 0

iii x position 0

0 .... 130nl1nl ~ b. 16501lm \l 0 03 a: ... 18SIlllnl

"- V 2140pm

'" 0.010 X posilion o 640pm IJ 740l'm OSOOl'm .90011f11

o 001 ___ L......1.._LLJ_LU'---------~ ____ L _____ . 10 100 1000 10.000

Distance from the t=raclure Surface, Y, (Jim)

Fig. 2. Plastic strain distributions about slowly growing cracks in Fe-3wt%Si crystals. For a plastic strip of about the same plastic zone size, the distribution is indepen­dent of the X position at which channeling patterns were taken. Solid curves are theoretical predictions for plane strain (lower) and plane stress (upper) conditions.

Fig. 3. Superdislocation configurations used in the com­putational simulations: (a) single superdislocation, (b) an array of equally spaced mini- superdislocations, (c) modified simulation with a combination of discretized mini- and superdislocations.

201

r Londlng DIrecllon

(a)

f Loading Direction

(b)

r Loading Direction v

Lx

(c)

Some previous results of this simulation scheme are shown in Figs. 4 and 5. First, by only considering (7~1 from Eq. (3), the elasticity solution for the stress field around a crack tip in iron for KI = 22.6 MPa_m1/2 is shown in Fig. 4(a). This was based upon T22 = 127 MPa with a crack length (2a) of O.Olm embedded in an infinite plate and agrees with analytical solutions. If now 500 mini-superdislocations of strength, 80b, are spread out over a 300/Lm long slip band as depicted in Fig. 3(b), the stress contours in Fig. 4(b) start to deviate from elasticity at about Imm from the crack tip. This is more clearly seen in Figure 5(b) with complete shielding being predicted at about 2/Lm from the crack tip. For comparison, the original formulation, using a single superdislocation as posed by Atkinson and Clements,(14) is shown in Figure 5(a). Here, complete shielding is predicted at about 100 /Lm from the crack tip. While both of these solutions satisfy equilibrium and Eqs. (4)­(6), both are unrealistic from a physical view point. The first in Fig. 5( a) predicts a negligible stress at distances from the crack tip well beyond the one pm distance where it is known that hydrogen embrittIement produces discontinuous c1eavage.(1S) The second was found later to require a friction stress that was negative close to the crack tip. The next section describes a more physically-based model.

IV. CURRENT RESULTS AND IMPLICATIONS TO STABILITY

Subsequent to the initial results(i5,17) additional experimental and theoretical insight led to the model depicted in Fig. 3(c). First, the relatively few dislocation near the crack tip allows the

202

Contours of Normal Stress fJyy (GPA)

~ : I §. §

~ ~ ti ~ '" u §

5 2 u: Q) ~ <.l 0 C 0 '" ~ Oi C g

§+-~--~~--~~--~~--~-r~ 0.000 200 400 600 800 1 0001 200 1 400 1 600 1 800 2.000

Distance From Crack Tip (mm)

~ .s a. i= .... <.l E (J

E e u.. Q) <.l C

'" Oi 0

Fig. 4. Elastic and elastic-plastic contours of the Uyy stress fields (courtesy of Scripta Metallurgica, Reference 15).

Fig. 5. Normal stress distribution for models (a) and (b) of Figure 3, curves 1 and 2, respectively.

Contours of Normal Stress fJyy (GPA)

g~------~----~~-----------, g ,00

g ~ g g 0

1iI

o 000 200 400 600 800 1 000 1 200 1 400 1 600 1 800 2.000

Distance From Crack Tip (mm)

QOO

'2 BOO

e= 700 • 0

... 600 1~

... b 500

1:l 400

'" .!: 300 til

g 200

... 100 <:> Z

-1~o-7 10- 6 to· 5 10. 4 10. 3 1~-2'''7~ '," 100

Dislance from crack tip (m)

fixed position requirement to be relaxed in terms of available computational time. This in turn allows the dislocations to position themselves such ~hat an invariant friction stress satisfied Eq. (6). This small array of mini-superdislocations was prompted by two results. First, it was found that a sharp array of dislocations was emitted out of the crack tip even after a relatively large plastic zone had already formed. (18) Second, this anisotropic, imhomogeneous slip was found below the fracture surface at a KJ value approaching 40 MPa-m 1/2. This was quantified by serially etching the fracture surface array and following the line broadening of selected area electron channeling patterns,<IO) The anisotropy in the patterns, indicative of specific slip planes being activated, demonstrated that such emitted slip bands only extended to about 30 to 40llm in length. As the present study was directed toward evaluating a hydrogen embrittlement mechanisms, the associated threshold stress intensity of 16 MPa-ml / 2 was initially utilized. Based upon the slip band scaling with KJ 2 , an initial band length on the order of (16/40)2 30j.lm or about 51lm was selected. The third motivating factor was that Lin and Thomson(13) had shown that a single dislocation near the crack tip coupled with a large super dislocation in the far field could represent the shielding aspects, quite well. Thus, an array of five mini-superdislocation, each with a strength of about 35b, and a superdislocation of strength, 10,600 b, were chosen to represent the shielded crack at KJ = 16 MPa-m l / 2. See the schematic in Fig. 31(c), the details of which are given in Table 1.

The stresses normal to the crack plane, U yy , are shown in Fig. 6 for KJ increasing from 2 to 16 MPa-ml / 2 . Three important points emerge here. First, very close to the crack tip, it is shielded from

203

10 5

""' .. rf"" "" ~ 10 4 •• 0 . .. .. DD~ .. • ...... &il • ;>, •• ..... - • 1:>;>' 0 • .......

• • ••• 103 0 .... ! a

• • til

~ 102 • K.2 0 • K.4 Z a K .18

10 1 10 10 lcr 9 1 cr 8 1 cr 7

Distance away from crack tip (m)

Fig. 6. Normal stress distributions for models (a) and (b) of Figure 3.

Fig. 7 Sustained crack growth on (001)(100) in an Fe-3wt.% single crystal in 1 atmosphere of hydrogen at room temperature showing parallel, 1ILm spaced arrest marks. Microscopic growth is in (110).

large stresses. Second, very large stresses exist even for the lowest KJ values but these are removed to greater distances which increase as KJ increases. For example, the stress maximum is at about 1 nm for KJ = 2MPa-ml / 2 but is at 200 nm for KJ = 16MPa-m1/ 2• Third, the region of high stress or the breadth of the high stress peak increases with increasing KJ. Keeping in mind that Fig. 6 is

204

a log scale, the volume of highly stressed material is increasing as to about KJ4. As these stresses are on the order of 26,000 MPa, it does not take a large perturbation to produce cleavage since this approaches the theoretical fracture stress of 30,000 MPa estimated for {100} cleavage of iron.(20,21) The significance of these results on instability occurs at two levels, on the final instability and on sub critical crack growth. As the Kr value increases, the region of very high stress moves further out into the microstructure. Very possibly, this region becomes unshielded as the scale of the far field dislocation substructure becomes commensurate with the scale of the shielding array. It has been observed that a macroscopic fracture toughness, as Kr -> KIC, has been achieved in this material at values in the range of 50 MPa_m1/2 and yet cleavage still occurs. At the other extreme, we have been able to hydrogen-induce slow crack growth under sustained loading. Here, the crack velocity is on the order of 5 x lO-sm/s. This will occur at Kr values only slightly greater than threshold which has been determined to be 16 MPa_m1/ 2,(16) Nevertheless, even at these low Kr values, growth occurs by an intermittent cleavage process as shown in Fig. 7. The crack is growing in the macroscopic [100) growth direction and {112} slip can intersect with the crack front providing shielding. Although the simulation isn't directly applicable in terms of crystallography, taking the trace of the stress tensor, O'h == O'il/3, is found to be 24,300 MPa. With the typical type of concentration model,(22,23) this will produce such a large lattice dilation that concentrations will approach unity. It is not unrealistic to anticipate such high concentrations coupled with stresses approaching theoretical to nucleate cleavage. As indicated in Figure 9, this would occur in the submicron region near the stress maximum at stress intensities of 16 MPa_m1/ 2 near threshold.

Thus, the crack nucleates at about 20 nm and runs to 1000 nm where it either arrests due to the microstructure, the lack of hydrogen enrichment or both. With regard to the latter, it is seen in Fig. 6 that the size of the region under very high stresses and thus high hydrogen is at least 100 nm in size and perhaps this is sufficient to sustain cracking over a 1000 nm region considering the dynamics alone.(24) The extent of cracking and arrest is a subject needing considerably more attention.

CONCLUSIONS

1. Dislocation emission in NaCI occurs at an emission stress intensity value, kle, about 1/3 of theoreti cal.

2. Although theoretical values of the local, kre, and Griffith, kIG, stress intensities are commensu­rate, sub critical growth intermittently occurs in Fe-3%Si(H2) to very high applied KJ values.

3. Large scale inhomogeneous, anisotropic plasticity shields the crack tip in Fe-3%Si crystals.

4. A computational, discretized dislocation model for an anisotropic elastic solid provides insight. The crack is always on the verge of brittle fracture: -at stresses approaching theoretical; -at 10's of nanometers in front of the tip.

ACKNOWLEDGEMENTS

The authors are grateful to the staff of the Super Computer Center of the University of Min­nesota for use of their facilities and for support from the Corrosion Center and the Department of Energy, Basic Energy Sciences, Materials Science Division, grants DE-FG02-88ER 45337 and DE-FG02-84ER 45141.

205

References

1. N.J. Petch, Phil. Mag. 3 (1958) p. 1089.

2. J.F. Knott, J. Meeh. Phys. Solids, 15 (1967) p. 97.

3. A.N. Stroh, Advances Phys., 6 (1957) p. 418.

4. A.H. Cottrell, Trans. AIME, 212 (1958) p. 192.

5. E. Smith, Proc. Conf. Phys. Basis of Yield and Fract., p. 36, lnst. Phys. Soc., Oxford (1966).

6. E.A. Almond, D.H. Timbres and J.D. Embury, Fracture 1969, p. 253 (ed. by P.L. Pratt) Chap-man and Hall, London (1969).

7. J.R. Rice and R. Thomson, Phil. Mag., 29 (1974) p. 73.

8. I.-H. Lin and R. Thomson, Acta Metall., 34 (1986) p. 187.

9. R. Thomson "Dislocations and Cracks" in Third International Con! on Fundamentals of Frac­ture, Irsee, Germany (June 1989).

10. S.H. Chen, Y. Katz and W.W. Gerberich, "Crack tip strain fields and fracture microplasticity in hydrogen induced cracking of Fe-3w%Si single crystals," submitted for publication (1989).

11. W.W. Gerberich, D.L. Davidson and M. Kaczorowski "Experimental and theoretical strain distributions for stationary and growing cracks," J. M echo Phys. Solids, accepted for publication (1989).

12. Y.-C. Gao and K.-C. Hwang, Advances in Fracture Research, p. 669, 5th Int. Conf. on Fracture, Cannes, France (1981).

13. I.-H. Lin and R. Thomson, Acta Metall., 34 (1986) p. 187.

14. C. Atkinson and D.L. Clements, Acta Metall., 21 (1973) p. 55.

15. M. Lii and W.W. Gerberich, Scripta Metall., 22 (1988) p. 1779.

16. X. Chen, T. Foecke, M. Lii, Y. Katz and W.W. Gerberich, "The role of stress state on hydrogen cracking in Fe-Si single crystals," in press, Engr. Fract. Meeh., (1989).

17. X. Chen and W.W. Gerberich, Scripta Metall., 22 (1988) p. 245.

18. M. Lii, T. Foecke, X. Chen, W. Zielinski and W.W. Gerberich, Mat. Sci. and Engng. A, 113 (1989) p. 327.

19. K. Maeda and S. Fujita, Scripta Metall., 23 (1989) p. 383.

20. A. Kelly, Strong Solids, Oxford Press, London (1966).

21. A. Kelly, W.R. Tyson and A.H. Cottrell, Phil. Mag., 15 (1967) p. 567.

22. J.C.M. Li, R.A. Oriani and 1.S. Darken, Z. Physik Chern., 49 (1966) p. 271.

23. J.P. Hirth, Metall. Trans. A, 11A (1980) p. 861.

24. K. Sieradzki, in Chemistry and Physics of Fracture, p. 219, R.M. Latanision and R.H. Jones, eds., Martinus Nijhoff, Boston (1987).

THE EFFECT OF QUENCHING PROCEDURES ON MICROSTRUCTURES AND

TOUGHNESS O:F TEMPERED 4Cr5l\ioSiV1 (AISI H13) STEEL

Y.L. Yang and X.Z. Feng

Department of Metals and Technology, Harbin Institute of Technology, Harbin, China

ARCTRACT: The effect of quenching procedures on both me­chanical properties and microstructures of tempered 4Cr5Mo­SiV1 (AISI H13) were studied. Commercial quenching (1050oC/ oil), high temperature quenching (1160 oC/oil) and double quenching (1160 oC/oil, 720 0 C tempering, 10500 C/oil) were ewployed. As compared with C.Q., the H.Q. leads to improve plane fracture tOl)ghness,K1c,and hardness,HRC,with loss in impact toughness,Ak,while D.Q. leads to obtain best fracture toughness with no loss in impact toughness. The role of mi­crostructures in the change of mechanical properties will be discussed.

1. INTRODUCTION: It is well known that K1c of alloy struc­tural steels were improved by high temperature quenching (1) while the Ak decreased. Recently, much interest has been shown in the improvement in K1c with elimination of decrease in Ak. This study was to apply the understanding of related studies to a high stregth steel in an attempt to impart higher K1c with no loss in Ak at hardness exceed­

ing BRC 52 (corresponding to yeild stren~th about 1500Mpa).

2. EXPERIMENTAL MATERIALS AND METHODS: The commercial 4Cr5-MoSiV1 steel used in this study had following composition (in wt pct).

207

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 207-213. © 1990 Kluwer Academic Publishers.

208

C Cr Mo Si v IVIn S P

0.39 5.35 1.39 0.99 0.95 0.40 0.03 0.03

The quenching procedures employed in this study are shown in Table 1. The quenched specimens were tempered

twice for 90 minutes at 350, 520, 600°0 respectively.

Table

Name

Oommercial quench­ing

High temperature quenching

Double 1uenching

Quenching procedures

Procedures

Austenitized for 15 min. at 10500 C quenched into room temperature oil

Austenitized for 15 min. at 11600 C quenched into room temperature oil

Austenitized for 15 min. at 1160 0 C quenched into room temperatgre oil, tempered for 90 min. at 72g 0, austen­itized for 15 min. at 1050 C quenched into room temperature oil

The mechanical properties were determined through plane fracture toughness tests using three point bend specimens in accordance with ASTI\1 E399-72 and impact tests using mesnager notch specimens at room temperature. Micro­structures were examined using both optical microscopy and thin foil transmission electron microscopy. Alloy content in matrix were determined using electron probe X-ray micro­analyzer. Retained austenite were determined using rotation anode X-ray differatometer.

3. RESULTS AND DISCUSSION: The room temperature mechanical properties are listed in Table 2. From these data it is evident that raising the austenitizing temperature from 1050 0 C to 1160°0 has clear improvement in K1c but loss in Ak. Double quenching leads to obtain highest K1c with no loss in Ak.

~"he results of fatigue crack growth tests are shown in Fig.1 and 2. It is evident that double quenching leads

to obtain highest resistance to fatigue crack growth on the condition of 350°0 tempering.

209

Table 2 Mechanical properties of 4Cr5MoSiV1*

Quenching Tempering temperature (oC) procedure

350 520 600 HRC K1c Ak HRC K1c Ak HRC K1c Ak

1050oC/oil 51 52 52 54 35 37 50 35 31 1160oC/oil 54 68 4? 58 37 18 56 39 12

Double 52 73 50 56 40 34 54 42 29 quenching

*K1c (MN.m- 3/2 ); Ak ( -2) J.cm

1r; 1-1 050oC/ oil 10-4

1-1050oC/oil 14 2-1160 oC/oil . 2-1160 oC/oil

()

£: 3-double :>, 3-D.Q.

-.512 ()

quenching '--

til 1 ~ 10- 5 -......-

10 !=1 'd

8 '--350°C til T. '0

6 10-6 0 10 20 30 40 50 60 8 10 12 15 20

N (x104cycles) AK (TiU'J.m- 3/ 2 )

Fig.1 a-N curves Fig.2 da/dn-flK curves

The microstructure features of tempered 4Cr5I,';oSiV1 are shown in Table 3, Table 4. The photographs of micro­structures are shown in Fig.3, Fig.4, Fig.5.

Table 3 },icrostructure features of 4Cr5MoSi Vi

i.iuenchine; Grain Twin in Alloy content in Dndissol-procedure size lath mar- matrix (wt %) ved car-

ASTM No tensite Cr Mo V bide sise

1050 oC/oil 9 20% 4.72 0.61 ()

1.09 2000 A 1160oC/oil 6.5 < 5% 5.15

0

1.22 0.79 1300 A Double 8 <510 5.35 1. 38 0.81 830 A quengching

210

Table 4

Quenching procedure

10500 C/oil 1160 0 C/oil

Retained austenite in tempered 4Cr5MoSiV1 (vol %)

Tempering temperature (oe)

350 520 600 4.5 2.7 1.2 6.7 3.3 1.3

Double quenching 9.4

Fig.3 Transmissbon electron micrographs of 4Cr5rr:oSiV1 tempered at 350 C: bright field of lath martensite (a) and twin in lath (b) quenched at 1050 0 C; bright field of lath martensite and retained austenite (c) and dark field of austenite reflection (d) quenched by double quenching.

3.1 Effect of quenching procedure on toughness The explanation for the improvement in K1c resulted

by high temperature quenching or double quenching may be following:

(a) Tb.e si7:e and volum fractoin of undissolved car­bide were decreased (Table 3). It increases the space

211

between carbide particles (2). ~) The decrease of twin in martensite (Table 3). As

the crack propagates in dislocation martensite, the energy is required more than that propagates in twinned martensite.

(c) The quantity of retained austenite between mar­tensite lath were increased (Table 4 and Fig.3c).

Impact toughness is sensitive to grain size (3), and therefore the Ak were decreased with raising austenitizing temper.ature and the double quenching leads to retain a high

level of Ak (Table 2 and 3).

3.2 Effect of tempering temperature on toughness The 350°C tempering leads to obtain highest K1c as

compared with 520 and 6000 C tempering (Table 2). The cause of highest K1c caused by 350 0 C tempering may be following:

(a) After 350°C tempering, the thick films of retain­ed austenite between martensite lath doesn't resolved.

(b) After 350°C tempering, the E carbides were pre­cipitated in martensite (Fig.4b) which are coherently bond wtth martensite. The [, carbide can be deformed with matrix, and therefore it can't deteriorated plastisity of steel.

Fig.4 TranSmli38J.u" elecdron micrographys of 4Cr5fvloSiV1 double quenched and 350'C tempered: bright field of twin in lath (a); bright field of £ carbide in lath (b).

At 520 0 C tempering, The l"fo2C carbides were precipi ta­ted on dislocation lines in martensite lath, it leads to locking of dislocation (Fig.5a). In addition, the retained austenite were resolved and the carbide chain were precipi­tated between martensite lath (Fig.5c), and therefore the

212

plastisity of steel was deteriorated obviously.

}<'ig. 5, Transmi8~i8n ele?tron l?icrographs of 4Cr5T:'oSiIT1 q.uencned at 1160 0: br1.ght £1.eld (a) and dark held (b) of N020 carbide precipitated on dislocation lines in lath nnd bright field of carbide shain precipitated between lath (c) in steel tempered at '320\ 0; bright field of carbides pre­c1pi tated on pibor austeni te grain boundary in Gteel tempered at 600 0 (d).

Fig.6 Scanning electron micrographs showing fracture sur­faces of impact toughness speciBens:austen~tized at 1160°0 and tempered at: 350°0 (a), 520 0 (b), 650 0 (c).

At high temperature tempering, the coagulation of carbides were occured, the dislocation were free from lock-

213

ing,and therefore the plastisity of steel was better than that tempered at 520°C and the decrease in K1c was elimi­nated. In addition, at high temperature tempering, the carbides were precipitated on prior austenite grain boun­dary (Fig.5d), and therefore there were intergranular frac­ture on the fracture surface of impact toughness specimens quenched at 1160 0 c and tempered at 600 and 650 0 C (B'ig.6).

4. CONCLUSIONS: The use of double quenching and 350 0 C tem­pering for 4Cr5JV!oSiV1 steel leads to improve the fracture toughness and resistance to fatigue crack growth with no loss in impact toughness and hardness. Microstructural observation revealed that the double quenching resulted in increase of alloy and carbon content in matrix, increase of retained austenite, decrease of twin in martensite, decrea­se of size and volum fraction of undissolved carbide and elimination of growth of austenite grains. Either of these microstrnctural features by itself could enchance the tonghness of steel.

5. REFERENCES 1. ',,'I.E. Wood, R.A. Clark, V.F. Zackay and E.R. Parker, Metallurgical Transactions 5 (1974) 1663-1670. 2. Birkle, Wei, Pellissier, Transaction ASK 59 (1966) 981. 3. R.O. Ritchie, B. Francis and 'd.L. Server, Metallurgical Transactions A 7A (1976) 831-838.

FRACTURE TOUGHNESS MODELING FOR MATERIALS WITH COMPLEX MICROSTRUCTURE

Asher A. Rubinstein

Department of Mechanical Engineering Tulane University

New Orleans, LA 70118

ABSTRACT: Several physical aspects of the toughness characterization of materials with complex microstructure are discussed. The considered materials are assumed to consist of a homogeneous matrix with multiple implantations (inclusions or microcracks) aimed to enhance the toughening process. Primary attention is given to a case of a quasi-statically growing crack, rather than to a stationary crack, interacting with material microstructural components. The toughness increase due to a shielding mechanism, and the possibility of applying this approach to the case of a growing crack, are specifically discussed. The distribution of the material components is assumed spacious enough that averaging of the elastic properties will not lead to accurate results. The stability approach to the shielding mechanism and its applicability to other toughening mechanisms are discussed. Special attention is given to the role of a crack pattern in the toughening process and to the possibility of controlling the crack path trajectory formation.

1. INTRODUCTION: The fundamental ideas of fracture mechanics are based on consideration of a crack in a homogeneous material. General theory cannot be directly applied to many modern materials which are characterized by a homogeneous matrix with multiple implantations aimed to enhance the fracture toughness. Even application of methods of linear fracture mechanics has difficulty in this respect. The typical examples are ceramics reinforced by particles or fibers, or toughened by microcracking. The spacing, size and mechanical properties of these implantations are such, that any averaging of mechanical properties, or using the effective elastic properties of the system, does not predict with sufficient accuracy the effective material toughness. The mechanisms governing the toughening processes are based on the shielding effect or entrapment of the crack by ductile particles or by particles which have undergone a phase transformation accompanied by particle dilation. The result of the shielding effect can be characterized by the ratio S of the local stress intensity factor KO to the stress intensity factor expected without interference by the presence of inclusions or microcracks (applied stress intensity factor). In cases when the size of the

215

A. S. Krausz et af. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 215-221. © 1990 Kluwer Academic Publishers.

216

inclusions and their spacing are small relative to the typical size of the propagating crack, the applied value may be interpreted as a remote value XOO for the case of small scale analysis l . If this ratio S-Ko /XOO is smaller than one, the shielding takes place; if it is greater than one, the material is weakened by the presence of the particles or microcracks. The analysis of the shielding ratio, strictly speaking, is applicable to a stationary crack with a specific positioning of particles or microcracks (defects). This data, although very useful, cannot be directly applied to the material characterization from the point of view of material resistance to crack growth. The shielding value strongly depends on the particular geometry of the crack tip and the defect positions with respect to each other (see ref. [1-3]); this makes it an instantaneous characteristic of the potential crack extension rather than a continuous material parameter. A typical shielding ratio variation versus a microdefect position is depicted in Figure 1. (ref. [1]), where a is a coordinate of the center of a microdefect given as a complex number.

1.3

<.> .9

< .8 .... .6

en .5 z ... z

.1 en en .0 ... '" ~-.14

-.3

• CRACK • HOLE

SOLID LINE BROKEN LINE

• INCLUSION (.-2.0)

--------~-------- /_-----8

.00 32.00 64.00 96.00 128.00 160.00 POS I r I ON ANGLE !DEG.)

Figure 1. Variation of the shielding ratio versus the angular position of the defect.

Additionally, the information obtained from the shielding ratio is limited to the expectation of a rectilinear crack path pattern. However, even in the case of Mode I remote loading, the local stress intensity factor KO consists of two components corresponding to Mode I and Mode II; they represent, respectively, real and imaginary parts of KO in the case when methods of complex variables are employed. Thus, expectation of the rectilinear crack path is not very realistic, and the reliability of predictions based on shielding analysis only, generally speaking, is not sufficient. The variations in the local values of the stress intensity factors due to crack path deflection, as shown in [4], may significantly alter the final value of KO.

The desired toughening criterion has to take into account the distribution of the material structural components on microscale, and the effect of the possible curvilinear crack path traj ectory. The material toughness cannot be judged on the basis of the instantaneous shielding values since these may vary significantly during crack growth. To develop the appropriate characteristics of the toughening mechanism, one has to deal with a combination of effects based on

1 The small scale analysis is a matter of convenience rather than necessity; it makes it possible to obtain results applicable to the material properties and then to use it for different geometries.

217

distributions of microstructural components (microdefects) and the possibility of the curvilinear crack path trajectory.

The possible direction of the development of toughening process model is outlined in the following sections, along with some results that can be applied to the quantitative characterization of material toughness.

2. SHIELDING ACTION ON A GROWING CRACK: Several useful results of the shielding analysis of a stationary crack can be transferred to the case of a quasi-static growing crack. As was mentioned above, the intensity of the interaction of a stationary crack with a defect (or microstructural component) strongly depends on the position of the defect relative to a crack tip. From the analysis of the stationary crack one can find positions corresponding to the best shielding and to obtain a complete profile of the shielding ratio during the crack propagation. This data may serve as a basis for the determination of a mathematical expectation of the shielding ratio. Having the value of the expected shielding, one can consider the possibility of the crack being trapped as it passes the area of a high shielding ratio. If the result shows that a crack may be arrested in the vicinity of the defect, the probable shielding ratio may be interpreted as a value characterizing the material crack growth resistance. However, this result will not be practically reliable. Naturally, the practical crack path will deviate from the perfect geometry considered in the analysis. It is necessary to consider the probability of measurable area of the crack entrapment, thus introducing consideration of the

a)

~

--~Ip MODE I

- Infinite crock 08 array

---Two cracks 07 (p/..l=05)

------ Smgle crack

06

05

04

03

02

MODE II

b)

- Infinite crock 08 array

--- Two croCkS (p/b:O 5)

------- Single crack

07

06

Figure 2. Shielding ratio for a crack propagating t.hrough a stack of cracks.

a) Mode I case, b) Mode II case.

stability of the shielding effect. The relative variation of the shielding ratio with respect to the variation of the crack tip position may be used as the criterion of shielding stability. An example of the physical situation where the stability of shielding is important was described in [5], where the idea of shielding stability was introduced. The crack passing through the transverse array of

218

cracks (stack of cracks) was considered in [5J. Results show that when the crack tip is located between the microcracks in the stack, the shielding effect may be very strong; S may drop to 0.3. The high shielding effect was observed in cases of both Mode I and Mode II remote loading, with the difference being that in the case of Mode I, the shielding takes place during a reasonably extended length, and in the case of Mode II, this region is very short. This data is given in Figure 2. The natural conclusion is that material with that kind of microcrack arrays will show higher toughness under Mode I loading and will break more easily under Mode II loading; and this is not due to the difference in magnitude of the shielding ratio.

Another important piece of information that may be obtained from the stationary crack analysis is the condition of the local structure with respect to possible alteration of the crack path trajectory. Exactly how to do that is not clear at this time; however, the res"u.lting effect of the crack path deflection is available, and is presented in the subsequent section, followed by the data regarding mechanics of the crack path formation.

3. CRACK PATH DEFLECTION TOUGHENING: The toughening effect of the crack path deflection is based on three factors [4J; the change in actual crack path length, the crack tip orientation with respect to the applied load, and the crack path trajectory shape.

The elongation of a crack path increases the integrated material resistance over the increment of crack growth. The measure of the change can be taken as a ratio of actual toughness, integrated over the crack path, to an integrated toughness of equivalent rectilinear crack path; the latter is a conventional, or measured, toughness. Remarkably, when the value of the amplitude is small relative to the wave length of the sinusoidal path, the resulting ratio is approximately equal to the ratio of crack path lengths. Thus, the actual toughness is equal to conventional toughness multiplied by the ratio of the conventional to actual crack length. This result was obtained in reference [4J for amplitude to wave length ratios up to 2.5. However, the variation of the local toughness value along the crack path may be very high. Thus again, the practically important value may turn out not to be the resulting toughness but the local current value at which the crack may get arrested. For the crack to be arrested, it has to be a crack path interval of sufficient length with a low shielding ratio value. The stability principle of the shielding process, discussed above, has to be used again to determine effective material toughness.

During a curvilinear crack path formation, the orientation of the crack tip changes with respect to the main direction. The effect of this change materializes in the change of local values of the stress intensity factor. The data obtained in [4] helps to identify the severity of this change in the case of a crack located in a uniform stress field (a case of a homogeneous material). As the direction of the crack tip changes, the local fracture mode becomes a combination of two modes and, therefore, it is better described by us ing the energy release rate. With the change of the crack tip direction, the local energy release rate cannot be obtained by means of conservation integrals. The local value of the energy release rate, depending on the fracture mode, may become higher or lower than the value corresponding to an initial direction. There are always directions

219

with significantly reduced energy release rates. The aim of the material composition designer is to make sure the crack tip will change direction toward the direction of potential crack arrest.

The shape of the crack path is important from another point of view as well. The wavy pattern isolates the crack tip region so the crack tip experiences reduced applied load. Practically, it acts as a micro crack in the vicinity of a macrocrack. This effect, analyzed in the case of a sinusoidal crack path trajectory, is illustrated in Figure 3.

2 ;i <!J'

r:3 ::>

~ g 9

1.35

1.20

1.05

0.90

0.75

0.60

0.45

0.30

0.15

0.00

-0.15 0.00

.~ ---­~KI(T'IPl/KHAPPU ._-----OrTIPl/O(APU

KllrTlPl/KHAPPU

.................. ...... _--------- ------ .. _-------------_ .. _---------_ .... _-_ ...... _ .. _-_.

1.00 2.00 3.00 4.00 5.00 AMPLITUDE OF THE CRACK PATH PAITERN (AIL)

Figure 3. Crack tip values of the local stress intensity factors

and energy release rate for a sinusoidal crack.

4. CRACK PATH FORMATION MECHANISM: There are several hypotheses explaining the crack path alternations. Briefly they may be summarized as follows. A crack propagates in the direction of Mode I, that is zero Krr direction; a crack propagates in the direction of maximal energy release rate; the direction of maximal shear stress; and maximal hoop stress direction. The resulting crack path deflection angles predicted by these theories are close to each other but sufficiently different to produce totally different crack paths. To obtain a more realistic criterion or to select the best one from those available, a detailed investigation of the actual crack path trajectories developed in heterogeneous stress field was conducted in [6]. The experimental data, reported in reference [7], was analyzed by employing an accurate numerical procedure. The experiment was conducted on single notch plates under a uniform tension. In the vicinity of a potential crack path, the circular hole was drilled, thus creating heterogeneous stress field and forcing the crack to change its direction. The subjects of the analysis in [6] were a history of development of the main fracture mechanics parameters along the crack path trajectories, and the variations of these parameters with slight perturbations of the crack path direction.

The investigation [6] concludes that crack path defection is controlled by strictly local variations of the fracture mechanics parameters. None of these exhibit any optimal behavior along the crack path trajectory when examined from the point of view of the system as a whole; for example, the energy release rate does not necessarily increase as the crack extends. On the other hand, these parameters

220

exhibit to a degree an optimum state with respect to local perturbations of the crack path directions. Actually, the best direction for the crack growth, judging by the direction of maximal energy release rate, is a few degrees off the experimentally observed path. This phenomenon is observed in examining the other criteria, mentioned above. The reason, as appears from the observed data, is explained in [6] by the presence of the nonlinear zone propagating along the crack. This zone serves as a buffer in absorbing the stress field changes as the crack path develops, thus creating delay in response to the outside stress field demand. The critical factor in the change of the crack growth direction is the energy release due to the crack advance gradient taken with respect to angular perturbation, 8G/8a. This gradient may be associated with asymmetrical development of the nonlinear zone along the crack, and, as concluded in [6], the material properties determine the critical magnitude of this gradient in order for the crack to change its orientation. In Figure 4. the crack surrounded by the nonlinear zone and inclined toward the hole is illustrated.

f f , f " f f f f f f f f f f f f f " f

nonlinear zooe o + - [~l

" " " " " , " " " " " , Figure 4. The crack tip region in the vicinity of a hole.

The value of the critical energy release gradient may be associated with the stability of the crack path. Any non-zero value of this gradient, in combination with an applied stress field, determines the crack extension increment in the direction which is less favorable from the point of view of the energy release per crack advance. The initial crack path deflection may be stimulated by any microscopical inhomogenuity confronting the crack. The length of this crack extension increment will determine how soon the crack will return to the original path, if ever.

5. CONCLUDING REMARKS: Several aspects of fracture toughness modeling were described above. Each section basically represents a separate part of a total picture. The specific areas discussed are the stability of a shielding mechanism for a growing crack; the crack path deflection effect on the resulting toughness; and the crack path formation mechanism.

The view that was emphasized above is the methodological approach to the model development. It is clear that the deterministic models cannot provide a complete description of the toughening process of

221

modern material, but can serve as a basis for a probabilistic model or as a test model for statistical analysis.

The stability phenomenon has a significant role during the crack growth process and in the determination of the crack path. Methods of stability analysis may serve as a tool for converting the data obtained from stationary crack analysis to crack growth process data.

The development of methods capable of crack path prediction is an important direction for future development of new models and material composition design.

ACKNOWLEDGEMENT: The work reported here was supported by NASA Lewis Research Center under Grant NAG 3-967 and by the Institute for Computational Mechanics in Propulsion at NASA Lewis Research Center.

REFERENCES

1. A.A. Rubinstein, "Macro crack - Microdefect Interaction." Journal of Applied Mechanics, Vol. 53, pp. 505-510 (1986) 2. L.R.F. Rose, "Microcrack Interaction with a Main Crack." International Journal of Fracture, Vol. 31, pp. 233-242 (1986) 3. M. Kachanov and E. Montagut, "Interaction of a crack with certain microcrack arrays." Engineering Fracture Mechanics, Vol. 25, pp. 625-636 (1986) 4. A.A. Rubinstein, "Crack Path Effect on Material Toughness." Submitted for publication. 5. A.A. Rubinstein and H.C. Choi, "Macrocrack Interaction with a Transverse Array of Parallel Microcracks," International Journal of Fracture, Vol. 36, pp. 15-26 (1988) 6. A.A. Rubinstein, "Mechanics of the Crack Path Formation," Submitted for publication. 7, A. Chudnovsky, K, Chaoui, A. Moet, "Curvilinear Crack Layer Propagation." Journal of Material Science Letters, Vol. 6 (1987) pp. 1033-1038

THERMAL ACTIVATION AND BRITTLE FAILURE OF STRUCTURAL STEELS

by

Bernard Faucher and W.R. Tyson

Metals Technology Laboratories, CANMET, Energy Mines and Resources, Ottawa, Ontario KIA OGI

ABSTRACT: Parameters characterizing the thermally activated temperature and strain rate dependence of the lower yield stress are obtained for an Arctic grade steel. These parameters are used in an expression based on a micro-mechanical weakest link model describing the probabili ty of failure to yield the temperature and strain rate dependence of brittle fracture toughness.

1. INTRODUCTION: Brittle failure of structural steels occurs at low temperature when a critical condition is reached, i.e. the Griffith stress is attained at a micro-crack within the material. Micromechanical models have established the probability of brittle failure as a function of thickness, stress intensity factor, yield stress and work-hardening rate (1-4). For example, Tyson and Marandet (1) obtained the cumulative probabili ty of failure by cleavage, iii, as a function of the stress intensity factor, K, from:

In[l/(1-ili) j=C B (J m(N-1)/2 Km (1) y

where C is a material constant, B is the specimen thickness, (J and N are, respectively, the yield stress and the stress exponent for the material and m is a parameter that is found experimentally to be close to 4. In the temperature range (below room temperature) where structural steels are susceptible to brittle failure, their yield strength is strongly temperature dependent, as a result of thermally activated dislocation movement (5). Thus, since fracture toughness depends on yield stress (Eq.(I», it should be possible to express the probability of fracture in terms of thermal activation.

In this work, previous experimental results (6,7) will be used, first to obtain the thermally activated parameters associated with yield stress, then to demonstrate their application to the brittle failure of a structural steel.

2. EXPERIMENTAL RESULTS: All the tests were carried out on a micro-alloyed steel that was experimentally developed by the Algoma Steel Corporation to meet the requirements of Lloyd's LT60 specifications for the Arctic (6,7).

223

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 223-228. © 1990 Kluwer Academic Publishers.

224

I~r-------~-------r------~------~

N()()

200

°O~------~IO--------2~O------~30------~40

STRAIN (0/0)

Figure 1. Typical stress-strain curves at a strain rate of 0.002s·'.

The lower yield strength was obtained from the minimum stress value in the Luders range over a wide range of temperatures and strain rates (Fig. I), and the stress exponent was shown to be almost independent of temperature and strain rate (6). Fracture toughness could not be measured by either the plane strain fracture toughness, K1c ' because of excessive plasticity, or by the critical value of the J-integral, J 1c '

because there was no ductile tearing before the occurrence of cleavage. Therefore, fracture toughness was measured with KJ=Y[EJc/(I-v2)], where J c is the value of J at the occurrence of cleavage, and E and v are, respectively, Young's modulus and Poisson's ratio for the steel. The results (7) were all obtained at temperatures at which specimens failed by cleavage without any indication of ductile tearing.

3. THERMALLY ACTIVATED YIELD STRENGTH: Previous results (6) have shown that between 400 and 700 MPa, the yield stress depends on temperature and strain rate according to the relation:

rry=905.9 - 0.0942 T In( 2.7xlOB / e} (2)

where rry is expressed in MPa, T in K and E in s·'. During the development of Luders strain, the stress remains essentially constant as does the density of mobile dislocations. Thus, the deformation kinetics is similar to creep conditions, and Eq.(2) corresponds to thermal activation in the forward direction only, over a single energy barrier with linear stress dependence of the apparent activation energy (5). The fit of Eq.(2) to experimental data is shown in Fig. 2.

At high temperature, when the applied stress is only slightly larger than the long range internal stress, Eq.(2) is inadequate because it does not take into account backward activation. This has been done in an elementary way in the present work by assuming the energy barrier for plastic flow to be symmetrical. In this case, the thermally activated

1000r-----~~----------------__.

~ rn 800

~ Cl ~600 >-

~ 3400

o £-0.00002/S} .. ~=O.002/8 Tension

.. E -0.2/ .. o £ -201" Compre" .. i 0

• £ -0.0021 s Compress i 0 o ~/y = 905.9 - 0.0942 T In (2.7 x 108,t)

~

o;t, 01 o

2000 2000 4000 6000 8000 10000

T 2.n (£iE )(K)

Figure 2. Lower yield stress as a function of the rate parameter.

plastic strain rate is expressed as (5):

225

E=EO exp(-~G#/kT) sinh[V(u-ui)/kT] (3)

where Eo is the pre-exponential factor, ~G# the free energy of activation, k Boltzmann's constant, V the activation volume, and ui the long range internal stress. The parameters have been obtained by least squares fit of previous results (6) to Eq.(3) with a steepest descent method and the initial parameters from Eq.(2). The analysis yielded the values Eo=4.52x109s-l, ~G#=0.53eV (51 kJ/mol), V=1.54xlO- 28 m3 (=8 b3 , where b is Burgers vector) and ui=367 MPa. This analysis of the temperature and strain rate dependence of the lower yield stress is not the best method of obtaining activation parameters. Therefore, no attempt will be made to refine it by considering, for example, an asymmetric energy barrier. Nevertheless, the values obtained are physically meaningful, and can be associated with a point defect energy barrier such as double kink spreading in the Peierls-Nabarro mechanism (5). From these values and Eq.(3), the lower yield stress is given by:

uy = 367 + 0.0898 T sinh- 1 [ 2.21 10-10 E exp(6138/T)] (4)

where the stress is expressed in MPa, the temperature in K, and the rate in s-1. The curves plotted in Fig.3 are in excellent agreement with the experimental results.

226

~ • Q.

!

'" '" W z :t t!l => 0 ...

.. n. ~

:!: w g: tfl 500 0 -' W

>-450

400

350 100 150 200

TEMPERATURE (Kl

250

o

300

Figure 3. Temperature and strain rate dependence of the lower yield strength. The curves have been calculated with Eq.(4).

500 500

.1 1 • I

400 I • 400 •

I 300 ~

200

1 00 ~

(a) (b)

Figure 4. Comparison of experimental results with the probability of failure calculated at 5, 50 (dashed curve) and 95% from Eqs.~4) and (5) for 20 mm thick specimens at strain rates of a)1.25xlO- Is and b)O.125/s.

227

4. THERMAL ACTIVATION AND BRITTLE FRACTURE TOUGHNESS. The bri ttle fracture toughness can be obtained from Eq.(l). It has been shown previously (7) how to obtain the various parameters from a set of experimental results. For the steel studied, the fracture toughness was found to be given by:

In K =40.35+0.248ln(50/B)-5.74In~y+O.248Inln[1/(1-~)1 (5)

where K is expressed in MPatm, B in mm and ~y, in MPa. The yield stress was obtained from the plastic strain rate at the elastic-plastic boundary. This strain rate was estimated to be E~O.5A/W for the standard three­point-bend specimen geometry (6), where A is the load point displacement rate, and W the width of the specimen. The yield stress in Eq.(5) was obtained from Fig.2 (6), but it can now be replaced by its value from Eq.(4), giving the analytical temperature and strain rate dependence of brittle fracture toughness.

Figure 4 shows the agreement between the predictions from Eqs.(4) and (5) and the experimental results; it becomes excellent as the strain rate decreases.

5. DISCUSSION. The results of this analysis indicate that, for the grade of steel studied, the increase of brittle fracture toughness with temperature is essentially attributable to the strong temperature dependence of yield stress. When the conditions are such that the yield stress becomes almost independent of temperature (i.e. at low strain rates and rela t i vely high temperatures), the frac ture toughness is also expected to be temperature independent as a direct consequence of Eq.(l). This is the case in Fig.4 for the lowest strain rate and probability of failure. In practice, this high-temperature plateau of brittle fracture toughness is not normally observed experimentally, because of a change of failure mode to ductile fracture, which is not considered in this study.

Some caution has to be exercised in this type of analysis. Because the temperature dependence of the fracture toughness is obtained from Eq.(l) through the yield stress, the stress exponent in Eq.(l) should be independent of the rate and temperature. This was nearly verified for the steel used, but in other circumstances, the analysis would need to be modified accordingly. Also, it has been assumed that the temperature is uniform throughout the fracture test piece, Le. that the deformation rate is low enough that adiabatic heating is negligible. However, it has been observed that under a sufficiently high rate of deformation, the temperature at the crack tip could rise significantly, invalidating the analysis.

6. CONCLUSIONS: Because brittle fracture of structural steels occurs under conditions where the yield stress is thermally activated, the toughness of these steels is strongly temperature dependent.

The probabili ty of failure can be described wi th the same activation parameters as the yield stress, in good agreement with experimental results.

ACKNOWLEDGEMENTS: The results reported here have been obtained as a part of MTL's work on steels and standards for Arctic applications, within the Offshore Structures program supported by the Panel on Energy Research and Development (PERD).

228

REFERENCES: 1. Y.R. Tyson and B. Marandet, in Fracture Mechanics: Eighteenth Symposium, ASTM STP 945, D.T. Read and R.P. Reed, Eds., American Society for Testing and Materials, Philadelphia, 1987, pp. 2. A.G. Evans, Metallurgical Transactions 14A (1983) 1349-1355. 3. F.M. Beremin, Metallurgical Transactions 14A (1983) 2277-2287. 4. K. Wallin, Engineering Fracture Mechanics 19 (1984) 1085-1093. 5. A.S. Krausz and H. Eyring, Deformation Kinetics, John Wiley & Sons, New-York (1975). 6. B. Faucher, K.C. Wang and R. Bouchard, in Proceedings of the International Symposium on Fracture Mechanics, Y.R. Tyson and B. Mukherjee, Eds., Pergamon Press, New York, 1988, pp.133-144. 7. B. Faucher and Y.R. Tyson, in Strength of Metals and Alloys, P.O. Kettunen, T.K. Lepist~ and M.E. Lehtonen Eds., Pergamon Press, Oxford, 1988, pp.l077-1082.

THE USE OF ELASTIC-PLASTIC STRESS FIELDS TO DESCRIBE MIXED MODE 1/11 BRITTLE FRACTURE IN STEEL

T.M. Maccagno* and J.F. Knott

Department of Materials Science and Metallurgy University of Cambridge

*Current address:

Pembroke Street Cambridge, U.K. CB2 3QZ

Structures and Materials Laboratory National Aeronautical Establishment National Research Council of Canada

Ottawa, Canada, K1A OR6

ABSTRACT: Fracture by transgranular cleavage in two steels subjected to mixed mode IIII loading has been investigated using edge cracked bend bar specimens loaded in anti-symmetric and symmetric four point bend configurations. Both En3B mild steel and lCr-1Mo-1/3V structural steel were tested at -196°C. The results were found to agree with predictions made according to a maximum tangential tensile stress criterion based on the HRR elastic-plastic stress field. The finding is consistent with the present understanding of the fracture micro-mechanisms of these steels.

1. INTRODUCTION: In the special case of mode I loading, fracture toughness methodology is well established. However, cracks in real structures are more likely to be subjected to some combination of the 3 loading modes, and it is important to develop an understanding of cracking behaviour in these 'mixed mode' situations. The present paper addresses the problem of low temperature brittle fracture in low and medium strength steels throughout the range from mode I (opening) to mode II (sliding).

Mixed mode 1/11 fracture behaviour of brittle materials, such as the glassy polymer PMMA, can be described (1) by a maximum tangential tensile stress criterion based on the crack tip stress field equations for a linear elastic material developed by Irwin (2) and Williams (3). A similar criterion might apply to brittle fracture of metals by trans granular cleavage, a mechanism which is generally accepted to be tensile stress controlled (4). However, even under conditions which promote cleavage, predictions for mixed mode 1/11 fracture of metals based on the linear elastic field do not appear to be adequate (5). Presumably, this is a consequence of local crack tip plasticity invalidating the use of a strictly linear elastic field.

Hutchinson (6) presents crack tip field equations for plastic materials which exhibit a yield stress (Jo' corresponding yield strain Eo, and then undergo deformation according to the power law relation:

229

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 229-235. © 1990 Kluwer Academic Publishers.

elastic­at a

plastic

230

( 1 )

where n is the strain hardening exponent, and ex is a material constant. If n = 1, Eqn. (1) reduces to linear elastic behaviour, and if n = 00, the behaviour is elastic-perfectly plastic. If one uses the polar coordinates r,e to denote the position ahead of the crack, the stresses at that position can be described in terms of the stress component ~ij (where i,j = r,e), and the elastic-plastic field equations take the form:

(2)

where fij(e) are functions of e. The coefficient Km gives the magnitude of the field and can be thought of as a plastic stress intensity factor. Rice and Rosengren (7) simultaneously developed a similar expression and consequently Eqn. (2) is known as the 'HRR' stress field. The full details for Km and f ij (e) in the case of mode I loading are given in (6) and (7), Hutchinson (8) provides the details for mode II, and Shih (9) provides the complete description throughout the range from mode I to mode II.

In the present study mixed mode 1/11 testing was carried out on En3B mild steel and 1Cr-1Mo-1/3V structural steel at -196°C to promote catastrophic brittle fracture by transgranular cleavage. The results are compared with the predictions of a maximum tangential tensile stress based on the HRR elastic-plastic field.

2. EXPERIMENTAL DETAILS

2.1 Material: An equiaxed grain structure of about 70 pm grain size was obtained in En3B mild steel (composition in wt%: .11 C, .30 Mn, .07 Si, .01 S, .02 P) by heat treating specimen blanks at 1200 0 C for 4 hours followed by furnace cooling to room temperature. 1Cr-1Mo-1/3V (.15 C, .60 M , .21 Si, .03 S, .02 P, .24 Ni, 1.00 Cr, .95 Mo, .32 V) steel with an equiaxed grain structure of about 400

JllD. grain size was stress relieved at 650°C for 5 hours prior to being machined into specimen blanks.

2.2 Testing: Mixed mode 1/11 testing was carried out using anti­symmetric four point loading developed by Gao et ale (10) to obtain mode II and mixed mode 1/11, and using conventional symmetric four point loading to obtain mode I. These bending arrangements are illustrated in Fig. 1, together with the associated shear force and bending moment diagrams. The mode I and mode II stress intensity factors, KI and KII, are given by:

K,= M~y,(~). KII=~YII(b) (3) B\J2 \J B\J 2

where a is the crack length, W is the width of the specimen, B is the thickness, and YI(~) and YII(~) are appropriate calibration functions. A complete description of the loading arrangements, specimen geometry, and testing procedure can be found in (5).

P

/t;~1 1-2S-+S~

2P/3 Pia I I

I I I I

Pia 2Pla

~~

~M.SOPI3

"<:::;:::0'

ANTI-SYMMETRIC LOADING

SPECIMEN

APPLIED LOADS

SHEAR FORCE

BENDING MOMENT

P

~ ~S+~

Pl2 PI2 I

I I

I I PI2 PI2

Q.o n 0

M=SP/2

SYMMETRIC LOADING

I

Fig. 1 Loading arrangements for mixed mode IIII testing.

3. RESULTS

MODEll COMPONENT INCREASING

Fig. 2 Mixed mode IIII specimens of CrMoV steel.

l

3.1 Fracture Angle: For both materials, introduction of a mode II component to the loading caused the crack to extend at an angle to the original crack plane (Fig. 2). Furthermore, this angle increased with increasing mode II. The angle between the original crack plane and the initial portion of the subsequent crack path was measured for each specimen using a microscope fitted with a rotating stage.

3.2 Fracture Load: The load versus time trace for each specimen of CrMoV steel was linear all the way to the instant of catastrophic propagation of the crack and there was no evidence that stable crack growth occurred prior to this instant. Inspection of the fracture surfaces indicated that crack propagation was by cleavage. The load at fracture, Pf, was used to calculate the mode I and mode II stress intensity factors at failure, KIf and KIIf, according to Eqn. 3. The complete calculations for each specimen can be found in (5).

In the case of En3B mild steel, the load versus time trace was linear for most specimens, but deviation from linearity was exhibited by specimens in the near mode II regime where the applied load achieved a value of greater than about 50 kN. With each of these specimens, non-linearity corresponded with the advent of plastic deformation under the load points. It was judged that the requirements to ensure that K is a valid crack tip characterizing parameter were not met in these cases, and the testing was halted before fracture occurred. For all other En3B specimens the behaviour was similar to that exhibited by the CrMoV specimens and KIf and Kllf were calculated in a similar manner.

231

232

At this point it is appropriate to introduce a convention used when discussing mixed mode situations. For the case of a central crack in a large plate loaded by a remote uniaxial tensile stress oriented at an angle ~ with respect to the axis of the crack, the ratio of KI to KII can be related to ~ by (1,5):

KI -=tanf3 (4) K"

and it is apparent that the factor tan~ provides a convenient measure of the mode I to mode II ratio regardless of the testing configuration employed. Hence for situations which do not actually use the inclined crack configuration:

-I( KI) f3 eq = tan K" (5)

where ~eq is the 'equivalent crack angle'.

75

! 60

w .... ~ 45

'" w 0: :::> 30 t; ~ IL

15

'MILD STEEL

-CrMoV

°9~0-~7~5~~60~~475-~3~0~~15~~ (MODE I) (MODE II)

EQUIVALENT CRACK ANGLE ~eq (deg)

Fig. 3 Experimental values of 90 versus ~eq compared with prediction according to maximum (Tee criterion.

4. ANALYSIS

~ " J

1.5

. A .

1i 1.ot----'"-------------------i

~ ~ j

~ l 'MILD STEEL 1 0.5,

I .,-, °9:-c0--c7=5-~60~~475 -~3=O- 15

(MODE I) (MODE II) EQUIVALENT CRACK ANGLE ~.q (deg)

Fig. 4 Experimental values of «(Teec at ~eq)/«(Teec at ~eq = 90°) versus ~eq compared with (Tee criterion based on linear elastic field (n = 1).

4.1 Fracture Angle: The fracture angle, 90 , versus the relative amount of Mode I to mode II loading, ~eq, for all specimens is plotted in Fig. 3. Also plotted in this figure is the prediction for 90 versus ~eq according to the maximum tangential tensile stress «(Tee) criterion first proposed by Erdogan and Sih (11). It is apparent that the experimental points conform closely to the theoretical prediction.

4.2 Load at Fracture: The maximum O"ee hypothesis maintains that fracture occurs when the value of the tangential tensile stress, O"ee, attains a critical value, O"eec, which is constant for the ma terial. This means that O"eec measured under mode I conditions, (O"eec at ~eq = 900), should be equal to O"eec measured under any combination of mode I and mode II, (O"eec at ~eq), and therefore the ratio of (O"eec at ~e9) to (O"eec at ~eq = 90 0 ) should be equal to unity for all combinat~ons of mode I and mode II.

The value of (O"eec at ~eq) for each specimen has been calculated using the linear elastic stress field and the results are shown Fig. 4 (Note: this procedure is described more fully in (1,5». The results do not agree very well with the prediction according to a maximum O"ee criterion based on the linear elastic stress field.

In order to determine whether a maximum O"ee cri terion based on the HRR elastic-plastic field is more appropriate, the work of Shih (9) is applied. In his analysis, Shih distinguishes between the region in the immediate vicinity of the crack tip (the 'near field'), and radial distances from the crack tip which are large compared with the extent of the plastic zone (the 'far field'). If KI and KII characterize the situation in the far field, Shih suggests that the relative amount of mode I to mode II can be given by the 'far mixity parameter', Me, where:

M =tan - x-e _1(KI) I K" 90 0

(6)

It is apparent that Me = ~eq/900 in the terminology used in the present work up to this point, and that Me = 1 for mode I, and Me = a for mode II. Shih also defines a 'near field mixi ty parameter', MP, to characterize the relative amount of mode I to mode II in the near field. The relationship between MP and Me depends upon the Hutchinson strain hardening exponent n (see Eqn. 1), and is presented in Fig. 5.

1.0 I" ,c"n~2"

o.sl n=5~=3 ~ I n=13

0.6 n=oo MP

0.4

0.2

o _-.1 __ .J_:_1 _ _.~ o 0.2 0.4 0.6 0.8 1.0

Me

Fig. 5 Near field mixity parameter MP versus far field mixity parameter Me (after (9».

1.4

1.2 .

o.s n=5 n=3 I

0.6 ~(\.:~~ I -~.:::::---- n=oo

0.4 ---L--'~. ,---,--~ o 0.2 0.4 0.6 O.S 1.0

MP , Fig. 6 Product

( tr •• C8,MP) )

of (,.CMP- I»)"'x '.CMP)

a .. co,MP - I) (9 ).

Following the work of Hutchinson, the stress component O"ee can be given in terms of the polar co-ordinates r,9 as:

, <fee = <foKmr -~a=e.(e, MO) (7)

233

234

where cree(e,MP) is a dimensionless function of e and MP given by Shih. It can be shown (5) that the ratio of (crasc at ~eq) to (craec at ~eq = 90 0 ) can be calculated from:

(8)

where the subscripts m and I are introduced to identify quanti ties calculated at ~eq and at ~eq = 90 0 respectively, and In (MP) is determined numerically. The second and third terms of this expression have been determined by Shih, and are reproduced in Fig. 6. Thus Eqn. (8) becomes:

(9)

and a maximum O"ee hypothesis incorporating the HRR elastic-plastic field can be checked with experimental results for KIf and Kllf' Fig. 7 shows this calculation when carried out for the present results on 1Cr-1Mo-1/3V steel and En3B mild steel, when n = 2. The agreement between experimental points and the theoretical prediction is very good.

Fig. 7

1.5

~ " J

1;j

iii 1.0!--~.-~ A.t

J .' 1;j

1 0.5

6 MILD STEEL

• CrMoV

o . __ I 90 75 60 45 30 15

(MODE I) (MODE II) EQUIVALENT CRACK ANGLE ~.q (deg)

Experimental values of (O"eec at ~eq)/(O"eec at ~eq = 90 0 ) versus ~eq compared with maximum O"ee

criterion based on HRR field with n = 2.

4.3 Discussion: A maximum O"ee criterion based on the HRR elastic­plastic field equations provides a very good description for cleavage fracture of these two steels in mixed mode 1/11 loading. This conclusion is consistent with the generally accepted micro­mechanical view that transgranular cleavage is a fracture process controlled by the attainment of a critical stress, after a

sufficient amount of strain has nucleated micro-cracks ahead of the crack tip (4).

A similar conclusion was reached in (1) where it. was found that a maximum ~ee criterion also applies to mixed mode 1/11 brittle fracture of PMMA. This is interesting because the specific details of the crack propagation mechanism for PMMA (i.e. rapid craze formation and breakdown) are very different from those for steel (i. e. transgranular cleavage). However, in both cases the micro­scale fracture processes appear to be stress controlled and, therefore, it seems reasonable that tensile stress criteria should apply. By the same token, it would be surprising if a stress criterion provided a good description for a strain controlled crack propagation process such as void coalescence.

From an engineering point of view, the finding that mixed mode cleavage fracture in steel is well described by a tensile stress criterion is very useful. Cleavage is a particularly notorious type of fracture mechanism in steel structures, and the results of this study provide a clearer understanding of cleavage in more complex and, therefore, more realistic stress fields.

5. ACKNOWLEDGEMENTS: Support for one of the authors (TMM) by the Edmonton Churchill Scholarship Foundation is gratefully acknowledged. The authors also wish to thank Prof. D. Hull of the Uni versi ty of Cambridge for provision of research facilities, and Dr. J-P. Immarigeon of NAE for reviewing the manuscript.

REFERENCES:

1. T.M. Maccagno, J.F. Knott (1989), "The fracture behaviour of PMMA in mixed modes I and II," to appear in Engineering Fracture Mechanics. 2. G.R. Irwin (1957), Transactions ASME v. 79 - Journal of Applied Mechanics v. 24, 361-364 3. M.L. Williams (1957), Transactions ASME v. 79 - Journal of Applied Mechanics v. 24, 109-114. 4. T. Lin, A.G. Evans, R.O. Ritchie (1987), Metallurgical Transactions A, v. 18A, 641-651. 5. T.M. Maccagno (1987), "Fracture in mixed modes I and II," Ph.D. thesis, Dept. Materials Science and Metallurgy, Univ. of Cambridge. 6. J.W. Hutchinson (1968), Journal of the Mechanics and Physics of Solids, v. 16, 13-31 • 7. J.R. Rice, G.F. Rosengren (1968), ibid., 1-12. 8. J.W. Hutchinson (1968), ibid., 337-347. 9. C. F. Shih (1974), "Small scale yielding analysis of mixed mode plane strain crack problems," in ASTM STP 560, American Society for Testing of Materials, 187-210. 10. H. Gao, Z. Wang, C. Yang, A. Zhou, (1979), Acta Metallurgica Sinica, v. 15, 380-391 (in Chinese). 11. F. Erdogan, G.C. Sih (1963), Transactions ASME - Journal of Basic Engineering, v. 85, 519-527.

235

MICROSTRUCTURE AND FRACTURE CHARACTERISTIC OF ALUMINIUM - ZINC - TIT ANIU M ALLOYS

* A.M. ELSheikh

Department of Metallurgy, Faculty of Engineering, Cairo University, Cairo, Egypt.

ABSTRACT: The microstructure is one of the significant factors that influence the mechanical properties of the material. The microstructural characteristics of heat treated and untreated Al-Zn-Ti alloys were stuided with optical microscopy, TEM, and SEM fractography. The results indicate that a proper treatment causes a significant improvement in the toughness of the material and decrease the tendency of the brittle fracture at low temperatures. An increase in both tensile strength and yield stress is also recognised. These findings are mainly related to the well-distributed pre­cipitates, the elimination of precipitate free zone (PFZ) and the elimination of discontinuous intergranular precipitates.

1. INTRODUCTION: In recent years numerous investigations have been made on the effect of a third element such as magnesium or copper on precipitation behaviour in aluminium base-zinc alloy systems 0-3). And much of this work has been done on research grade alloys using a wide variety of investigative techniques such as TEM and small angle X-ray scattering. Studies on commercial grade materials have generally been restricted to a correlation of heat treatment and/or thermomechanical treatment (If-5) to the resulting mechanical properties.

The investigation to be described studied the effects of systematic variations in heat-treatment of Zn-AI-Ti alloys with respect to: tensile strength, yield stress and impact strength. In addition, thin-foil electron microscopy was employed to correlate heat-treatment with microstructure, and microstructure with properties.

*--------------------------------------------------------------------------------------Now is a visiting Assoc. Prof., Engineering Department, The American

University in Cairo, Cairo, Egypt.

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A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 237-243. © 1990 Kluwer Academic Publishers.

238

2. EXPERIMENTAL MATERIALS AND METHODS: High purity aluminium (99.996%) and zinc (99.99%) were used in preparing the alloys to avoid possible complications due to the presence of minor alloying elements. Titanium was added in form of a master alloy containing AI-IO% Ti.

The high-purity elements were melted in an alumina crucible in an electric furnace to minimize gas pick-up. The proper amounts of master alloy were wrapped in aluminium foils and added directly to the molten material. The melts were stirred with an alumina rod to ensure complete homogenization and were cast without being degassed. The nominal com­positions of the as-cast alloys are presented in Table 1.

Table I. Nominal compositions of the as-cast alloys (wt%)

Alloy No Zn Ti Fe Si Al

60.05 0.12 0.10 Balance

2 60.10 0.10 0.10 " 3 60.20 0.10 0.08 "

Castings were done into a steel mould having a slab cavity of 145 x 145 x 15 mm. The mould consisted of seven cooling channels each of 10 mm diameter and running water was introduced through the channels 15 minutes before and during pouring of the molten material.

Flat specimens of 40 mm guage length, 10 mm width and 5 mm thickness were machined for tensile testing. Standard notched specimes for Charpy impact test were also machined from the as-cast slabs.

Table 2, summerizes the heat treatments applied to the present alloys. The associated heat treatments are indicated by different codes.

Code

A

B

C

D

E

F

G

H

Table 2. Heat treatments applied to the alloys

Alloy No

1,2,3

1,2,3

1,2,3

1,2,3

1,2,3

1,2,3

1,2,3

1,2,3

1,2,3

Solution treatment temperature

3000 C

3000 C

3000 C

3000 C

3000 C

3000 C

3000 C

3000 C

\

Quenching Primary temperature aging

50 C 50° C, 4h

400 C 50° C, 4h

5° C 500 C, 4h

40° C 50° C, 4h

50 C 50° C,20h

400 C 50° C,20h

5° C 50° C,20h

40° C 500 C,20h

Secondary aging

1000 C,36h

1000 C,36h

100° C,36h

100° C,36h

The topography of fracture surfaces was examined in a scanning electron microscope to reveal the features of the fracture under different testing conditions.

Thin foils for electron microscopy examination were prepared from some of the tested specimens to reveal the substructures that develped by different heat treatments. The foils were prepared by electrochemical polishing in a solution of 2.0 pct perchloric acid in ethanol.

3. RESULTS AND DISCUSSION:

The results of tensile tests are presented in Table 3.

Table 3. Results of tensile tests

Codet 2 2 UTS (Kg/mm ) 0.2% Y.S. (Kg/mm )

Alloy No. -+ 2 3 2 3

A 35.2 43.8 48.5 31.8 36.3 38.4

B 37.4 46.1 49.2 32.1 38.5 40.2

C 34.2 43.1 47.8 31.8 34.2 35.7

D 34.8 43.6 46.3 31.9 34.5 35.7

E 33.1 42.7 46.2 31.2 33.7 34.8

F 35.1 44.6 47.1 33.2 37.2 38.4

G 34.8 41.2 45.4 31.7 34.2 35.4

H 33.3 41.9 44.7 31.5 33.6 34.2

32.7 42.2 46.8 31.2 33.8 35.9

Preliminary experiments indicated that variation in the duration of solution treatment had a negligible effect on the behaviour of the material.

With reference to Table 3 and Figs. 1 and 2, one can notice that the ultimate tensile strength (UTS), and yield stress (Y.S.) are increased by rapid quench - primary aging (code B). This is mainly attributed to the fine precipitates formed in the matrix, as well as on the grain boun­daries as shown in Fig. 3 a. On the other hand, a slow quench - primary aging (code C) causes a noticable loss in mechanical properties. This can be ascribed to the fomation of relatively coarse precipitates as shown in Fig. 3 b. With reference to Fig. 3 b, one can also notice that slow quench - primary aging leads to the formation of a narrow precipitate free zone (PFZ). Previous investigations on AI-Zn-Mg system (4-6) indicated that the presence of the PFZ was responsible for the remarkable loss in the mechanical properties of the material. Prolonged primary aging at 50° C (codes F and G), is seen to cause more decrease in both UTS and Y.S. This loss in tensile strength is mainly attributed to the formation of the coarse and non-coherent precipitates shown in Fig. 3 C.

239

240

U.T.S. 2 (Kg/mm)

50 -

40 -

30 -

20 -

10 -

..... r-

123123123123123123123123123

0C!J0CEJ0000CD Figure 1. Results of ultimate tensile strength (UTS)

0.2% V.S.

(Kg/mJ)

50

40 -

30

20 -

10 -

..... r- ..... r- r- r-

r-0-

123 123 123 123 123 123 123 123 123

000~ITJ0000 Figure 2. Results of 0.2% yield stress (Y.S)

( a )

-Figure 3.Mlcrostructure resulting from ( a ) rapid quench­primary aging (code I),

(b) slow quench primary aging (code C), and ( c ) primary aging for 20 h at 50° C <Code G).

( a )

( C .. )

( b)

Figure 4. Microstructure resulting from secondary aging at 1000 C for 36 h (a) and for 72 h (b).

241

242

The data presented under codes D, E, H and I in Table 3 were obtained after secondary aging at 100 0 C for 36 h. These data, when compared with those of primary aging, show that secondary aging reduces the strength of the material. The microstructure shown in Fig. 4 a, was taken from a sperimen after secondary aging at 100 0 C for 36 h (code D). If a comparison is made between the micrographs shown in Fig. 3 a and Fig. 4 a, one can notice that secondary aging produces a significant coarsening of the precipitate as well as the formation of discontiounous intergrammlar precipitates leading to a remarkable loss in the strength. In fact, few experiments in the present work ( the tensile results are not included), showed that an extension of the duration time of secondary aging lead to greater strength losses. Our findings indicate that the width of the PFZ is affected more strongly by secondary aging as shown in Fig. 4 b. Again, the great loss in tensile properties can be ascribed to the formation of wide PFZ in this case.

Table 4 summerizes the impact energy data for alloy No.2 subjected to different heat treatments. It is clearly shown that rapid quench -primary aging enhances the impact strength of the material. The improve­ment in impact strength is maily caused by the formation of the fine­uniformly distributed precipitate in this case (Fig. 3 a). On the other hand, formation of the coarse secondary precipitate as a result of slow quench - secondary aging (e.g. code 1) leads to a remarkable loss in impact strength. In fact, our findings indicate that the alloy having the structure shown in Fig. 4 b exhibits the lowest impact strength. This suggests that the impact strength of the material is greatly affected by the presence of the wide PFZ.

Table 4 Impact strength for alloy No.2

Code Impact strength (Kg. m)

A 7.1

B 7.3

C 6.8

D 5.8

E 5.5

F 6.1

G 6.0

H 5.3

5.1

The impact fractographs taken from alloy No 2 (code B) and (code I) are presented in Fig. 5 a and Fig. 5 b respectively. The fractograph shown in Fig. 5 a exhibits typical ductile fracture features as indicated by the appearance of dimples. On the other hand, the fractograph presented in Fig. 5 b shows both cleavage and dimple appearance indicating a "mixed­mode" fracture in this case.

(a) (b)

Figure 5. Impact fractographs of alloy No.2 treated as indicated by code-B (a) and Code I (b)

4-. CONCLUSIONS: The ultimate tensile strength (UTS), yield stress (Y.S), and impact strength of AI-Zn-Ti alloys can be improved by rapid quench -primary aging. This is mainly attributed to the formation of well-distributed fine precipitates on the matrix as well as on the grain boundaries.

Slow quench primary aging and secondary aging rsult in the formation of relatively coarse precipitates leading to a noticable loss in mechanical properties.

Prolonged secondary aging leads to the formation of a wide pre­cipitate free zone (PFZ) that causes a more decrease in both UTS and Y.S.

ACKNOWLEDGEMENTS: The help of Dr. Burchard at Gemeinschaftslabor fUr Electronemikroskopies, Aachen, West kGermany is hereby acknowledged. Thanks to Miss Moamena Saleh of the American University in Cairo for typing this manusscript.

REFERENCES: 1. J.M. Chen, T.S. Sun, R. K. Viswanadham, and J.A.S. Green, Metal­lurgical Transactions 8 A (1977) 1935-194-0. 2. L.M. Brown, Proceedings of the 5thlnternational Conference on the Strength of Metals and Alloys (Aachen), ATB Metallurgie (1979) 73-77. 3. M. Conser va and P. Fiorini, Metallurgical Transactions, 4- A (1973) 857-862. 4-. D.A. Porter and K.E. Easterling, in Phase Transformations in Metals and Alloys 1984-, Van Nostrand Reinhold, Berkshire, UK (1984-) 303-309. 5. J.S. Santner, Metallurgical Transactions 12 A (1981) 1823-1825. 6. D. H. St. John and L.M. Hogan, Journal of Materials Science 19 (1984-) 939-94-4-.

243

THREE-DIMENSIONAL ASPECTS OF mE FRACTURE PROCESS ZONE AND CAUSTICS

T. W. Webb, D.A. Meynt, and E. C. Aifantis

Michigan Technological University, Houghton, MI, 49931, U.S.A.

ABSTRACT: The method of caustics has been applied to the study of hydrogen-assisted cracking (HAC). On the basis of two rather unexpected observations concerning the stress intensity (K) vs. caustic diameter (D) dependence and crack tunnelling, we elaborate on two specific issues related to through the thickness or three-dimensional (3D) effects pertaining to the definItion of the fracture process zone (PZ) and the caustics interpretation.

1. INTRODUCTION: The PZ is commonly defined as the small region ahead of the crack tip (usually embedded within the plastic zone) where the microscopic fracture processes occur and a local loss of material stability takes place. The concept of PZ is central to the modelling of discontinuous subcritical cracking. The PZ length is usually identified with the crack jump occurring when a critical PZ configuration is achieved.

Recently, Neimitz and Aifantis [1-4] critically reviewed the previous work on the subject and presented a unifying theoretical PZ model for HAC in elastoplastic materials. Their model provided a detailed description of the PZ in the plane strain regime and considered the influence of hydrogen (H) on the PZ configuration during HAC.

On the experimental side, Meyn et al [5] used the method of caustics to measure the localized deformation at the crack tip and the influence of H on the PZ during HAC. Two unexpected observations were made suggesting that the PZ is nonuniform along the crack front and that the standard interpretation of the method of caustics is inadequate for originating curves close to the crack tip. The purpose of this paper is to elaborate on the theoretical implications of these observations and discuss 3D crack tip aspects related to both topics of PZ and caustics.

2. PZ AND RELATED MODELLING DURING HAC: Neimitz and Aifantis [1] defined the PZ as a 2D region embedded within the plastic zone and extended from the blunted crack tip to the point of maximum opening stress. Its size, shape and stress state were Permanent address: Naval Research Laboratory, Washington, D.C., 20375, U.S.A.

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A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 245-251. © 1990 Kluwer Academic Publishers.

246

completely determined by the applied stress and three material parameters (v, m, 11). By cancelling the plastic HRR singularity at the tip of the PZ, balancing energies at the crack tip, and defining the crack opening angle () in terms of the PZ geometry, the following three equations were derived for the calculation of v, m, and 11

Amw + BmW-I + C = 0 (2.1)

11 = md ~ d )

D m + (1 -

D

v = i [-U- - 1) , where ~ and d are determined from elastoplastic fracture mechanics,

n W = 1I(n + 1) with n denoting the work hardening exponent, and (A, B, C) are material coefficients. The above PZ model was further modified to include the effect of the environment in the case of HAC. This was done [2] by introducing scaling parameters Al - A4 as

follows

A = rH/ r ,A = JH/ J ,A3= mH/m ,A4= I1H/l1 , (2.2) I p p 2

where the PZ length r and crack tip opening displacement J are p I

related through v [r = J(- + 2v)]. By determining these scaling p 2

parameters in connection with the loading and HAC characteristics and identifying the length of the crack jump with the length of PZ (Lla = r), the characteristic "knee-shape" crack tip velocity (V) vs

p stress intensity factor (K) curve was obtained.

The above results were applied [3] to describe the onset of instability (K ) during HAC. It was found that

Ib

m -I m (2.3)

where the subscript c denotes critical conditions. On assuming, in particular, stress-assisted H transport until the amount of H within the PZ reaches a critical value, an alternative relation for A:b was

deduced in the form [3] C lUX

A th=I+{~ -1 [ 1 [[_0 ]-1] +1){ E ) (24) I Z (J Pd+v·) kv'P (31t(2v + !)d (J (~-1)' .

y c 2 n y

where the new parameters IX, p, k, v· are constants, C denotes o

ground H concentration, a is the yield stress, and P is the y

external pressure. On comparing (2.3) and (2.4) a P vs Ktb

relationship is obtained, in agreement with experimental data [3]. For describing the progress of HAC beyond threshold [4], an

average crack velocity V was defined as the ratio LlaiLlt, where LIt is a "waiting time" or the time interval between crack jumps. Assuming that fracture occurs when the mean H concentration within the PZ

achieves a critical value C , the following relation for the crack c

velocity is obtained [4]

pV H + 1i S nV D

V = 2wC (2.5) c

where w is a PZ shape factor whose value depends on m , p is the concentration of hydrogen atoms at the crack tip, V H denotes the

mean hydrogen velocity, n is the production of dislocations at the crack tip, V D is the mean dislocation velocity and 1i S represents

the number of hydrogen atoms that leave the dislocation core per unit length. By expressing V H and V D in terms of K, theoretical

relations for V vs K were derived in agreement with experimental data [4].

The above considerations completely neglect the extent of PZ through the thickness of the specimen. This aspect becomes important in the case of crack tunnelling where the 3D character of the PZ needs be considered. This point is further substantiated in Section 3 with the aid of caustics observations. A preliminary discussion on 3D aspects related to the PZ and caustics is given in Section 4.

3. CAUSTICS AND RELATED OBSERVATIONS DURING HAC:The method of (reflected) caustics can be used to obtain an optical representation of the nonuniform out-of-plane deformation of the crack tip. In the case where the caustic originates from a curve outside of the plastic zone, the elastic stress-intensity (K) vs the caustic diameter (D) formula is given by [6]

EOS 12

K = 10. 71z vt (3.1) o

where E and v are the usual elastic constants, t is the specimen thickness, and z denotes the distance between the screen and

o

specimen. If the caustic originates from a curve in the plastic zone whose stress state is governed by the HRR field, then the J-integral (J) vs the caustic diameter (0) formula is given by [7]

S 2 0+1 30+2

J = a oay [ E ]-0- 0-0 (3.2) E aa z t '

y 0

where a and S are constants, a denotes the yield stress and n is o y

the work hardening exponent. Strictly speaking, the above relations should hold for thin plane stress specimens where 30 or through the thickness effects are not important. In the case where the caustic originates in the PZ, there is not an appropriate formula in the literature to relate mechanical and optical quantities.

When the method of caustics was applied to study HAC [5], it was experimentally found that below threshold the relationship between K and D is not given by equation (3.1) or (3.2). Instead, it approximately obeys a linear relationship as shown in Figure 1.

247

248

This suggests that standard caustics interpretations are inadequate for caustics originating close to the crack tip, where 3D effects dominate. Figure 1 also shows a deviation from linearity in the K-D dependence when the applied load is near threshold. This "anomalous" enlargement of the caustic was attributed to internal crack propagation (tunnelling). Figure 2 depicts a macrophotograph of a typical fracture surface for a 4340 steel specimen subjected to HAC [5]. Thus, if the PZ fracture mechanism is applicable to the case of tunnelling, it follows that the nonuniform cracking across the thickness suggests a 3D rather than 2D character of the PZ.

4. 3D ASPECTS OF THE PZ AND CAUSTICS: Motivated by the previous experimental findings, we provide in this section a preliminary discussion pertaining to 3D aspects of the PZ and caustics.

4.1 3D Character of PZ during Tunnelling under HAC Conditions

To model the nonuniform growth of crack (tunnelling) during HAC, we assume that initially (at threshold) the PZ can be viewed as a minute semi-elliptical effective crack along the straight front of the main crack before growth. The direction and size of the subsequent crack growth increments (determined by the PZ length) depends on the local stress field and H concentration at a specific point ahead of the crack. In particular, we assume, as in [8], that fracture occurs when the equilibrium H concentration reaches a critical value at a critical distance (r '" PZ length) ahead of

p

the semi-elliptical crack. As there is no analytical solution for the local stress field (K) of a semi-elliptical crack, we assume that the "blunted" front of the main crack can be approximated as a free surface such that the semi-elliptical crack can be treated as a "surface" crack. Then we can use the well-known relation [9]

K = (0' + HO') / .JI; F , t b '<

(4.1)

where (0', 0') are the tensile and bending stresses applied at the I b

boundary of the specimen, a is the crack depth, and (H, Q, F) are specified geometric factors.

Now the equilibrium H distribution in front of the crack is given by [8J

p = p [1 + PAK]a, o rr

where a, P and A are constants. On assuming crack jumps when p = p at r = r then

c p

r = p2 A2K2

P [(p cf P ) 1 I a_I ] 2 .

(4.2)

further [8] that the

(4.3)

Unlike the case of a straight crack, we assume here that the boundary concentration p varies within a surface layer in front of

o the main crack as follows

(4.4)

where p is the concentration at the outer surface of th e layer (in os

direct contact with the environment), P. is the concentration at 01

the inner surface material), x is the surface layer.

of the layer (in direct contact with the depth coordinate, and d the thickness of the

On the basis of (4.3) and (4.4) we conclude that the PZ is nonuniform along the elliptical crack front. The new crack configuration is determined by incorporating the crack growth increments r to the initial crack. Then by approximating the new

p

configuration as a semi-ellipse and repeating the process again and again we can predict initial crack growth patterns for HAC, as shown in Figure 3.

4.2 Finite Thickness Effects on Caustics

To model the experimentally observed K vs D relationship w~ffh departs from the classical plane stress elastic formula (K '" D ), the through thickness or 3D variation of the stress field should be taken into account. Strictly speaking and independently of specimen thickness, plane strain conditions (e = ° ,a =1= 0) begin to

zz zz dominate as the crack tip is approached. In HAC studies, where relatively thick specimens are required, this fact becomes important, especially in connection with the interpretation of caustics originating very close to the crack tip (PZ region). However, 3D solutions for crack problems are not available in the literature for direct application to caustics interpretations. To overcome this difficulty, a short-cut to the 3D problem is proposed by incorporating a simple modification to the plane stress elastic mode I analysis. We assume that the in-plane stress components are given as before by the 2D expressions, while the out-of-plane stress component is modified as follows [10]

a = 2Kv g(rlt, z/t)cosO, g(r/t zIt) = {o plane stre~s (4.5) zz -- Z ' 1 plane stram

v'21tf The mapping equations for the light rays are given by

-3/2 X = r cos9 + Cr p(r/t) cos 39 -2Cr-1I2ap(r/t) cosO cos9 ,

z-- Or z -3/2

Y = r sinO + Cr p(r/t) sin 39 -2Cr- 1I2ap(rlt) sinO cosO z-- or Z

Kz v t12 o and p(rlt) = 2 J {I - g (r/t, Zit)} dz

v'mE where C

o

(4.6)

In the case where g(rlt, zit) = 0, equations (4.6) reduce to the standard mapping equations. The initial curve r of the caustic is

o then given by a polynomial in 1'-112 of the form

o

249

250

a r-5 + b r-4 + c r- 3 + d r- 2 + e r- 1I2 + 1 = 0 , (4.7) o 0 0 0 0

where (a,b,c,d,e) are known functions of (r,O). Again, when g(rlt, zit) = 0 equation (4.7) reduces to the standard equation for the initial curve. By solving equation (4.7) for r = r and inserting

o the result into equation (4.6) we obtain the appropriate caustics equation.

As the exact form of g(r/t, zit) is unknown, simplifying assumptions for the component a through the thickness [l1J can be

zz made. For example, if we assume that g is uniform in the radial direction, while it varies through the thickness such that to simulate plane strain in the center of the specimen and plane stress near the surface (within a layer of thickness equal to the size of the plastic zone along the crack axis) we obtain [11]

nEa 2 1/3 5/6

K = [21.4hz v) D . o

(4.8)

We see that the power law dependence on D is 5/6 rather than 512 and equation (4.8) compares reasonably well with the linear D vs K relationship shown in Figure 4.

REFERENCES: 1. A. Neimitz and E.C. Aifantis, Engineering Fracture Mechanics 26 (1987) 491-503. 2. A. Neimitz and E.C. Aifantis, Engineering Fracture Mechanics 26 (1987) 505-518. 3. A. Neimitz and E.C. Aifantis, Engineering Fracture Mechanics 31 (1988) 9-18. 4. A. Neimitz and E.C. Aifantis, Engineering Fracture Mechanics 31 (1988) 19-25. 5. D.A. Meyn, T.W. Webb and E.C. Aifantis, Engineering Fracture Mechanics (in press). 6. P.S. Theocaris, Journal of Applied Mechanics 37 (1970) 409-415. 7. A.J. Rosakis and L.B. Freund, Journal of Engineering Materials and Technology 104 (1982) 115-120. 8. D.J. Unger and E.C. Aifantis, Acta Mechanica 47 (1983) 117-151. 9. J.C. Newman, Jr. and I.S. Raju, Engineering Fracture Mechanics 15 (1981) 185-192. 10. A.J. Rosakis and K. Ravi-Chandar, International Journal of Solids and Structures 22 (1986) 121-134. 11. T.W. Webb, E.E. Gdoutos and E.C. Aifantis, (Preprint).

o EXPERIMENTAL (NOT PROPAGATING)

II EXPERIMENTAL (PAOPAGA TlNG)

o~~ __ -L __ J-__ L-~ __ ~ __ ~~ o w ro ~ ~ ~ ro ~

STRESS INTENSITY FACTOR (MPa.Jiiij

Figure 1. D-K curve [5].

Figure 2. Macrophotograph of fracture surface [5].

CRACK GROWlH PATIERN

0.5

Figure 3.

1.5 2 L5 Z (mm)

Crack growth pattern during HAC.

~ 3 ... ,. '" is " 2 (;; ::>

'" "

o ,,,,,.,, ...

10

,,;"'-- -EON (JII

---0-- EXPERIMENTAL .~,

,. (CAL_RATION CUAVE)

;, ...... ----lEON 4 .... 1. 1~174

20 30 40 50 00 70 00

STRESS INTENSITY FACTOR (MP.Jiii)

Figure 4. Theoretical D-K curve [l1J.

251

ON THE BEHAVIOR AND THE MODELIZATION OF AN AUSTENITIC STAINLESS STEEL 17-12 Mo-SPH AT INTERMEDIATE TEMPERATURE

DESCRIPTION OF DISLOCATION-POINT DEFECT INTERACTIONS

P. Delobelle Laboratoire de Mecanique Appliquee - URA 279 CNRS

Fac. Sci. Techn. - Route de Gray - 25030 Besan~on Cedex - France

ABSTRACT: This article shows the diversity of monotonic and cyclic mechanical proporties of an austenitic stainless steel in the intermediate temperature range of 20 S T S 700·C. The phenomena of strong interactions between dislocations and point defect configurations are brought to the fore. The incorporation of the phenomenological modelization of these interactions into a unified viscoplastic model developed elsewhere allows most of the observed phenomena to be accounted for.

1. INTRODUCTION: The industrial materials used in certain components of modern installations (the nuclear industry in the present case) are often submitted to mechanical and thermal loadings and the knowledge of the behavior and the development of the anisothermic laws proves to be indispensable for the prediction of the life duration of the installations. However, before performing and modelizing real anisothermic tests, where the thermal and the mechanical loadings evolve cyclicaly and simultaneously, the identification and the phenomenological modelization for different isotherms of the physical mechanisms taking part in the strain, is necessary. This article presents the results relative to this first step, in the case of an austenitic stainless steel.

2. EXPERIMENTAL METHODS : The specimen of stainless steel are obtained from slabs cut out of 30 mm thick plates and hyper-quenched from l200·C. The weight composition of this low carbon steel, with controlled nitrogen content (French specificity) is given in Table I.

Table I : jeight COmprsitiOj of ,he strel CiS I P I Si Rn I Ni Cr Mo N B I Co I Cu I sO.03 sO.OOl sO.02l 0.44 1.084 12.3 17.54 2.47 0.075 0.001 0.15 0.175

No thermal treatments are performed after the machining of the specimen. Different machines have been used to perform the tests, notably hydraulic (tensile) machines, electrodynamic (tensile­torsion) machines controlled by computer and creep machines (tensile­torsion creep tests at constant stress) [1].

3. EXPERIMENTAL RESULTS AND DISCUSSION: 3.1. General presentation of the mechanical properties:

Some experimental results relative to the monotonic, cyclic and

253

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 253-261. © 1990 Kluwer Academic Publishers.

254

viscous properties of the material are successively presented. Fig. 1 shows the evolution of the flow stress uec for a constant strain rate, £~z = 6.6 10. 4 s·', and different strain levels, as a function of the temperature. A plateau appears between 300 and 550°C. Fig. 2 represents, for a constant shear strain rate, £~e = 4.6 10. 5 s·', and different amplitudes of imposed cyclic strain (~£T /2), the evolution of the stabilized stress (~u~!ab/2) as a functionZ~f the temperature. The behavior is very different from that observed in Fig. 1 and a significant peak is seen, having a maximum around 550°C and whose width is contained between 200 and 600°C, that is, approximately the zone corresponding to the plateau of the preceeding figure. Note that such a maximum has already been observed for a 304 type stainless steel [2].

"

o tests ---model

100 200 300 400 500 600 700 8 !tel

~ Variation of the flow stress uec with the temperature. Tests and modelization

~ : Evolution of the stabilized cyclic stress (~u~:t/2) as a function of the temperature for different strain levels. Tests and

modelization

In addition, Fig. 3 shows that this maximum corresponds to a maximum of the cyclic hardening, the first quarter of the cycle being a decreasing function of the temperature.

~ : Amplitude variation of the cyclic hardening with the temperature Fig. 4 : Effect of the strain rate ____ , on the stabilized stress for different isotherms. Tests and modelization

'5

10

-'5

20

'5

-'I _10

In terms of strain rate effects (in a ratio of 103 , 1.38 10. 3 ~ £!e ~ 1.38 10. 6 s·'), Fig. 4 shows that the sensitivity coefficient of the stabilized stress (~u~~ab/2) to the strain rate can be positive or negative depending of the temperature of the test. The negative value

255

is observed in the intermediate temperature domain (300 ~ T ~ 500°C) and corresponds exactly to that of the peak mentioned earlier. At low temperatures, this coefficient is positive, which is due to a cold viscosity effect. For higher temperatures, T ~ 600°C, the sensitivity is naturally positive since the thermally activated recovery effects lead to an increase of the viscosity with the temperature.

Ln[B,,]

·14

.15

.16

.17

.19

.20

.21

~ : Evolution of the viscous component with the strain and the temperature. Tests and simulations

13

12

~ : Evolution of the axial ratchet (2D)t rate with the temperature for different numbers of cycles 10

~ : Creep tests under different stress ~ levels and several temperatures. Tests and modelization.

The different properties presented above are summarized in Table II.

Table II : Summary of mechanical properties 50 KID B> 200 250 300 3SO t (h)

- Cold creep (Fig.4,5,7)

- ratchet (lD and 2D) (Fig.6)

- low cyclic hardening (Fig.2,3)

- positive sen­sitivity of the stress to the strain ra­tes (Fig.4)

20 < T < 200°C

- strain nearly time independent (Fig.5,7)

- neglitible ratchet­ting lD and 2D) (Fig. )

- significant cyclic hardening (Fig.2,3)

- negative sensitivity of the stress to the strain rates (Fig.4)

200 < T < 550°C

- high temperature thermal­ly activated creep (Fig.4,5,7)

- thermally activated rat­chet (lD and 2D) (Fig.2,3)

- moderate cyclic hardening with time effect (Fig.2,3)

- positive sensitivity of the stress to the strain rates (Fig.4)

T ~ 550°C

256

As for the viscous properties, the viscous component Uv (uv = u -a, where u is the applied stress and a the sum of the contributions of the different internal stresses) is measured as a function of the temperature by the method of inverse relaxation [3] (strain dip test technique), during a test conducted at a constant strain rate ET = 6.6 10'4 s-1. Fig. 5 shows, for different strains e, that Uv increases with £ and passes through a minimum around 300·C. This observation is confirmed by that of Fig. 6 which represents the evolution of the axial strain rate (cumulated strain) from 2D tensile-torsion ratchetting for a constant number of cycles Nc ' as a function of the temperature. For a large number of cycles Nc ' a minimum appears around 300·C [4]. Fig. 7 presents the results of several creep tests for different temperatures and stresses and corroborates the results presented in Fig. 4 to 6 cold and hot viscosity effects, loss of viscosity at intermediate temperature.

3.2. Strain mechanisms at intermediate temperatures : In order to correlate a physical parameter to the loss of

viscosity associated with the maximum of cyclic hardening and with the increase of the flow stress under monotonic loading, measurements of the evolution of the flow threshold have been performed, quantified by the parameter AUr after relaxation during a time tv' The principle of the method is represented by the diagram in Fig. 8. Fig. 9 shows that if, T > 550·C then ~r is negative, which corresponds to a time recovery and which leads to a thermally activated steady creep. For T < 550·C, 8Ur is positive, which results from a time hardening but vanishes with the strain. Note that instability of the type Portevin-Le Chatelier appear on the stress-strain curves, in the transition zone 8Ur < O. With the help of a kinetic study of the evolution of 8Ur (8Ur

f(t» and the determination of the relaxation times, it seems possible to correlate the supplementary hardening with short distance interactions (of the Snoek-Schoeck type [5,6,7]) dislocations-pairs or triplets of point defects (in interstitial positions C, N, B or substitutional positions, Mn, Ni,Cr, Mo) in the solid solutions F.C.C.

., 100 200 300

·2 ·3

f' ·4 .. • 4 ·7 + 8 ·8 .12

Fig. 8 : Diagram of the principle of the -~ yield point return method used to quan­tify the static aging Fig. 9 : Evolution of 8~r with the tem­perature and the strain for tv - 48 h

~(O) ---~- -1~1'--___ J: :v,.1ax.

I L .. --i ~':---q,) ~ &,.0

T),525"c

Thus following the physics of the interaction [6,7], it is

possible to attribute a scalar component yCi) to each interaction that:

yC i )

co (") Wei) rSN = r 1 (O)exp----

SN kT

257

such

(1) •

In these equations: pCi) is a constant, r~~) the thermal~1 activated relaxation time intrinsic to the interaction considered, y~ th~ efficien~y func~ion corre~ponding to a Gaussian distribution [7], rCl)(O) WC1 } yCl)Sat TC1 } T are constants which are directly

SN ' , 00, 'i measurable by experiments. Eq. (1) predicts a first order time hardening vanishinf exponentially with the strain. The steady regime corresponds to yCi = 0 :

(2) .

yCi)st is a decreasing function of the strain rate and leads to the possibility of describing the negative value of the sensibility exponent of the stress to the strain rate. In the domain 20-700°C it has been possible to count five interactions «i)=5), which contribute in an additive manner to the global hardening of the material [8,9] and the identification of the set of coefficients has been realized [10]. It can be notice that an austenitic stainless steel with manganese [11] has a completely different thermomechanical behavior than 17-12 SPH, in this intermediate temperature range. The increase of the carbon and nitrogen contents and the replacement of nickel by manganese strongly affects the whole of interactions [11]. These interactions are closely tied to the spatial redistribution of the point defect clusters in the stress field of the dislocations and thus are a function of the distribution and the density of these dislocations (Fig. 9). It remains then to relate the interaction spectrum to the scalar variables Y and y+ which quantify the evolution of the density of the dislocations (see Table III). This is done by writing

y* Y (l+-reYe)

y*+ = y+ (l+-r;Ye) with Ye

'" yC i ) ':'c i ) '" yCi)sat Uc i) co

} (3) ,

where Ye is the total fraction of reoriented atoms at short distances and -re , -r; two distribution constants. For higher temperature, T > 600°C, the interstitial atoms migrate to long distances to precipitate in a heterogeneous manner along the grain boundary [12,13]. A formulation has been proposed [10] but space does not allow its development in this paper.

4. MODELIZATION : Table III presents the tridimensional anisothermic version of a unified viscoplastic model developed elsewhere for a single isotherm [14-16]. Without entering into the details of the formulation of this model, its different components are reviewed together with their associated microstructural parameters. Several

258

physical properties appear in the viscoplastic state equation on which depend the strain rate, namely: the critical velocities of the Luders bands, the variation of the stacking fault energy with the temperature g(E t ) [17] and the effective diffusion coefficient D ff [18]. This adimensional equation is identified over the domain 20-7~0·C.

Three tensorial variables g, g" g2 appear in this kinematic model which are associated with the internal stresses induced by the interactions at different distances between the mobile dislocations and the substructure. There are thus three interaction distances, respectively short, medium and long. The scalar variables Y and y+ essentially depend on the accumulated plastic deformation and represent the variations of the density of the free dislocations and the substructure dislocations with the strain and temperature. Note that these variables condition the increase of the asymtotic states of the kinematic variables.

The partial memorization of the pre-strains is realized by the introduction of non-hardening surfaces [19-20] and reflects the stability, in relation to the strain, of some developed substructures such as ; twinned zones, subgrain boundaries, tilt boundaries, ... The loss of stability, with respect to time, of these substructures is described by the presence of the recovery term in the yariables g, Y and y+. It was shown above that the variables y(l) describe the phenomena of dislocations-point defect interactions.

Fig. I, 2, 4, 5 and 7 compare the predictions of the model with respect to the experimental results. The agreement is satisfactory and one of the new notable possibilities of the model is the description (Fig. 4) of the strain rate effect on the stabilized cycle ; the sensitivity exponent being either positive or negative depending on the temperature. Except for the ID and 2D ratchetting, the mechanical characteristics of this steel between 20 and 700·C, summarized in table II, can be described with the help of this model. As far as the ID and 2D ratchet phenomena are concerned, they are directly related to the non-linear kinematical terms and are overestimated by the present model. The solution to this problem has not yet been found though several improvements have been made [4] [21].

5. CONCLUSION The identification of the dislocation-point defect interactions phenomena and the associated phenomenological formulation, when integrated into the context of a unified viscoplastic model, allow the description of a large variety of the observed behavior of this steel between 20 and 700·C.

ACKNOWLEDGEMENTS This study has been financed by Electricite de France (E.D.F.) under contract n· 3K0665.

REFERENCES [1) P. Delobelle and D. Varchon, Rev. Phys. Appl., 18 (1983) 667 [2) N. Ohno, Y. Takahashi and K. kuwabara, J. Eng. Mat. Techn., to

appear (1989) [3) A.A. Solomon, Rev. of Sci. Inst., 40 (1969) 1025 [4) P. Delobelle, J. Nucl. Mat., to appear (1989) [5) G. Schoeck and A. Seeger, Acta Met., 7 (1959) 469 [6) A.S. Nowick and B.S. Berry, in "Anelastic Relaxation in Cristalline

Solids", Academic Press, (1972) 176

259

[7] A.K. Miller and O.D. Sherby, Acta Met., 26 (1978) 289 [8] R.E. Reed-Hill, S.C. Park and L.P. Beckerman, Acta Met., 31, (1983)

1715 [9] P. Delobelle, D. Varchon and C. Oytana, Met. Trans. ,1617 (1985) 361 [10] P. De1obel1e, E.D.F. report, to be published (1986) [11] R. Billa and P. De1obel1e, IX Brazilian Congress of Mech. Eng., 7-

11 Dec. F1orianopo1is Brazil, 2 (1987) 1133 [12] B. Weiss and R. Stickler, Met. Trans., 3A (1972) 851 [13] M.J. De1eury, Thesis Paris, France (1980) [14] P. De1obe11e and C. Oytana, Nucl. Eng. Design, 83 (1984) 333 [15] P. De1obe1le and C. Oytana, J. Nuc1. Mat., 139 (1986) 204 [16] P. De1obe11e and C. Oytana, J. Press. Vessel Techn., Part I and

II, 109, (1987) 449 and 455 [17] L. Remy and A. Pineau, Mat. Sci. Eng., 36 (1978) 47 [18] J.R. Spingarn, D.M. Barnett and W.D. Nix, Acta Met.,27 (1979) 1549 [19] N. Ohno, J. Appl. Mech., 49 (1982) 721 [20] J.L. Chaboche, K. Dang-Van and G. Cordier, SMIRT V, August 13-21,

West Berlin, Germany (1979) Div. L, L1/13 [21] J.L. Chaboche and D. Nouai1has, to appear in J. Eng. Mat. Techn.

(1989)

Table III : General equations of the tridimensional anisothermic version of the model

I. The state equation 2

a-a a-a

sinh ( ___ )n(T)

N(e,T)

n(T) T-T

1 + f3 0 exp-f31 ( __ e)2 Tf -T

A---kT

g(Et ) where

(1)

(2)

(3)

(4)

(5)

* f3o ' f3 1 , TE , Te , E~, Eo' Ee aO and A are constants. TF , G, b, k, Dv,De , E~ and Vs are respectively the melting point of the alloy, its shear modulus, the Burgers vector, the Bo1tzman's constant, the diffusion coefficients respectively in volume and in the pipe, the stacking fault energy and the critical velocity of Lliders bands.

II. The kinematic variables, g, gl and g2 : a ..

+ • -: - Ml MO 'J } "".J. ~ p (2/3 (Y+Y )£ .. -("' .. -"', .. )£)-Ro(T)(sh f3 2 ("') ) (-) • m " ' J 1 J 1 J ... -?'lij ~ PI (2/3 Y ;ij-(""ij-"'~ij)£) '" "'2 i· ~ P2 ( 2/3 Y £ i· - "'2 i· £), with the initial vaiues : J ",(0)~"'1 (0)~"'2 (O)~O.

b.Ho

(6a), (6b) , (6c) ,

Rm(T) ~ Rmo exp - -, Pm,Pl,P2,f32 ,Mo ,M1 ,Rm and b.Ho are constants. kT

III. The isotropic variables, Y and y+ :

260

y = b(ysat(T)-Y)[T-R(T~IY-Yo IlOsign(Y-Yo)]' with yeO) - ao+~oexp-(T/To) , ysat (T) is a variable given by the equation (13),

DeffGb R(T)= r----- , b. lO' aO' ~O' To and r are constants.

kT

y+ =b + cr a t+ (T) -y+) [it -R+ (T) IY+ -Y! Il 0 si (y+ -Y~)] with : y+ (0)=0 and ysat+ (T)=as.;.tQ +~s;t~xp_ (T/To)~

DeffGb R+ (T)=r+-----

kT

q+ is given by equation (11) and b+, Lo,aS!t+ ,~~t+,To and r+ constants.

are

Yo = Max (Y-(1/3)(Ro/R)1/l0(Sh P3 (Max a)Ml)MO/lO ]

Y~ = Max (Y+-(Ra /R+)1/l0(Sh P3 (Max a)Ml)MO/lO] (9)

and Ro = P4Rm'

IV. The non-hardening surfaces and the memorization parameters

G = £-E - q SO) G+ = "£ - q+ S 0

] (10)

(11)

~ij = ~2 [(l-~)R(G)<n n*>n* I + (l-R(G» n* gq r] (12)

R(.), <.>, nand n* are defined in the notation summary.

ysat = bsat (ysat (T) _ ysat) q with yS,ll,t (T) = a ~t + ~O exp - (T/Tol ' ysat(O) = (a~at + ~o exp - (T/To) ) (l+~ Y ) sat sat bS a t d Tee a"" ,aO ' , ~O an 0 are constants.

yc is given by equation (15).

V. The scalar variables y(i) : y~)_y(i)

_pC i )y( il; ____ _ r( i)

y(i) , with i E [1,5],

}

w( i )SN T-T( i)

r( i) = r( i ) (0) SN SN

exp -- and y(~= y(~sat kT

exp _ ( _____ ) 2

The different parameters are defined in the text. Redefinition of the new isotropic variables y* and y*+ :

Ti

(13)

y* = Y (1 + ~c Yc) and

Y*+ y+ (1 + ~: Yc) with Yc

If ~ = ~+ 0 then (15) reduce Y* c+ c

and ~c = O.

VI. EX2ression of the viscous com2onent or * £

'<' Y( ; ) ""( ; )

: ao

a v a-a N Argsinh (- (_)n)1/ n with £, N

C(T)CY+O.5Y+) . * nO NCT) where C(T)=(£,/co ) , * £0 and no are two constants.

SUMMARY OF NOTATION: T: Temperature X: notation for a second rank tensor (X;j)

261

} (15).

On take ~c f 0

(16)

(17) •

a tj components of the applied stress tensor a j j at j = a; j - (0; j/3) au : components of the deviatoric stresses

tensor a;j' a(j : components of the kinematical stresses tensor and deviatoric

tensor a,a,a-a a (3/2 a;jafj)'/2, a

equivalent stresses (3/2 Von-Mises

components of the strain tensor I = (2/3 £ .. £ . . )1/ 2 equivalent strain

'! 1 J , 1 2 ~ = (2/j (£;j-l;j)(C;j-lij»

rate : second invariant of the

tensor ~ , n = ~2

I

= k3 and n * ajj-a jj

and * £;j-I;j

.!2 n - 0 a-a <x> x H <x> with H(x) 1 if x ~ 0 and H(x) = 0 if x < 0

APPLICATIONS OF A THEORY OF MOBILE DISLOCATION DENSITY TO THE STUDY OF RATE-SENSITIVE DEFORMATION

Thomas H. Alden

Department of Metals and Materials Engineering University of British Columbia

Vancouver, British Columbia, V6T IW5

ABSTRACT: A new theory of mobile dislocation density is applied to the problem of the relation between transient changes in stress and strain rate in iron. With a knowledge of the constants relating dis­location velocity to stress, the stress sensitivity of strain rate can be calculated; alternatively, it is possible to obtain the ve1ocity­stress equation from stress sensitivity data.

1. INTRODUCTION: Orowan's equation for the rate of inelastic strain by the movement of dislocations,.

€=Pmbv, (1)

failed to excite wide interest until the first experimental measure­ments of the dislocation velocity were reported by Johnston and Gilman (1). It then became the focus of intense experimental and theoretical study. The methods and equations which evolved, which came to be called dislocation dynamics, were the basis of a decade of great progress in the understanding of the deformation properties of metals and alloys (2-5).

Despite such progress, however, the problem of the mobile dislo­cation density remained, in these years, largely unsolved. A decade later, Pharr and Nix (6) noted that while the dislocation velocity equations among materials were extreme in their differences, the mechanical properties showed a much lesser variation. They suggested that this paradox could be resolved if differences in the mobile density were understood, and developed a model involving the stress­dependent release of dislocations from a network (6). In its application to problems such as the stress-strain curve and the stress sensitivity of the deformation rate, these authors achieved considerable success.

My contribution to the solution of this problem is based on the experience of deformation experiments done in a soft tensile machine (7), in which direct control is exerted over the stress and the rate of stress increase, rather than over extension and extension rate. I was then inclined (8) to attend to microstructural studies in which the measured (total) dislocation density has been linked to

263

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 263-272. © 1990 Kluwer Academic Publishers.

264

stress (9-11) rather than strain, (1,5,6)

cr = cr* + aGbP~/2. (2)

cr* is an elastic limit stress. The other symbols have their usual meaning. Thus if the stress is increased, dislocation sources will operate so that Eq. (2) is satisfied. It remains to be assumed that the newly created dislocations are mobile and the basis for a theory of mobile density is established.

One cannot discuss deformation by slip without considering the problem of strain hardening. We must know, first, at What rate mobile dislocations are trapped to become, instead, part of the "obstacle structure" and second, once trapped, by what mechanism and with What (quantitative) effect they interact with the remaining mobile dislocations to further restrict their mobility.

In the present study, the trapping descriptively, by asserting that mobile path", r o ' which describes the trapping strain hardening,

problem is treated only dislocations have a "mean free rate (8,12). For linear

(3 )

where T* is a constant. (I am inclined, without solid evidence, to ascribe this trapping to the formation of stable attractive junctions between intersecting dislocations (13». Then the total differential equation for the mobile dislocation density is

(4 )

The first term in this equation is the time derivative of Eq. (2).

The limitation of mobility may be described with reference to the characteristic power-law equation which relates stress to dislocation velocity, (1)

(5)

If the strain hardening interaction is over long range, then the increment to the strength, with strain, is to the rate-independent (static) part. The remaining (dynamic) component of strength arises from other sources, for example impurities. In Eq. (5), the stress must be replaced by the smaller, and declining, effective stress cre ; the velocity tends to fall. (This tendency may be overcome, of course, if the external stress is rising). Alternatively, if strain hardening involves short range interaction, there will be an increase in the dynamic strength, which may be described (5,14) by all increase of "0' In theoretical study of deformation involving chang'E!s of deformation rate, it is necessary to identify the relative magnitudes of static and dynamic strain hardening. If we assume that strain hardening interactions are similar in iron and aluminum (barely extended dislocations), then experimental evidence (15) suggests that for iron at room temperature, the dynamic component is about 10%.

265

2. EXPERIMENTAL MATERIALS AND METHODS: I apply the theory to two separate experimental studies of iron polycrystals. Michalak (16) used a zone refined iron of purity 99.97% and carried out rate-change tests in a hard tensile machine. He varied both the base crosshead speed and the speed change ratio to obtain an extensive data set. In my own work, the material was coarse-grained (35 ~m) titanium doped iron, described as "interstital free" or IF iron, kindly supplied by Armco Corporation. Deformation was realized by the continuous application of various stress rates; periodic, abrupt stress decreases were then imposed and the change of strain rate measured.

3. RESULTS AND DISCUSSION: The first task (17) was to attempt the calculation of stress sensitivity in a metal in which the constants of dislocation velocity are already known. This is a test of the predictive capabilities of the theory and was done for the zone­refined iron using velocity data obtained by Turner and Vreeland (18). The complimentary problem, that of obtaining the velocity constants from stress sensitivity data, is more difficult, but has been success­ful in the titanium doped iron (19).

3.1 Hard Machine Stress Sensitivity in Zone-refined Iron. In order to apply the theory to Michalak's results, it is necessary to know, for his iron, the drag constant and the exponent, Eq. (5). There are values obtained from direct observation of dislocation positions by x-ray topography (18); they are n = 2.8 and ~o = 3.0 E08 Pa at room temperature. Unfortunately, the iron used in these measurements is of a different composition; as well, the values are given for edge dislocations whereas the deformation rate is mainly controlled by the slower screws (20). These problems are solvable but only with somewhat arbitrary choices (17). The value of n is retained, but ~o is adjusted to equal 5.0 EO§ Pa. In order to fit the stress-strain curve at moderate strain, T is given the value 280 (Fig. 1).

80,--.--,---,--,-----..,----.----,---,

IRON t/' &'"

~60 /

:a 40 ../

~ I // - THEORETICAL CURVE

20 II" '" EXf'ERlMENTAL POINTS

oL-~_L~_L-~_L~--J

o ~ ~ ~ ~ STRAIN (PERCENTI

Figure 1: Stress-strain curve for zone-refined iron

The computer generates numerical solutions to Eqs. (1-5) as the time variable is increased in small increments. The principal mechanical variable is the stress rate, Eq. (4). In the hard testing machine simulation, it is determined indirectly by the crosshead speed, the characteristics of the deformation, and the elastic properties of the machine and specimen (17,21). The theory predicts that if the crosshead speed is abruptly increased, the stress also rises abruptly, (Fig. 2). The general appearance of this theoretical curve closely approximates what is observed experimentally. The value

266

of the stress change is determined graphically using a short extrapolation of the upper strain hardening curve; in this case the speed was increased by a factor of five at a strain of 2.0% and the measured stress increase is 5.69 MPa (Fig. 2).

~~~~~~~~~

J4

.,/~ = 50

132 ',"66'1Ots =569_

~:Jl t;;

28

IRON

19 21 STRAIN 1%1

23

Figure 2: Theoretical stress versus strain curve for iron on abrupt increase of crosshead speed.

Simulations were done for three crosshead speed ratios, in correspondence with the measurements reported by Michalak (16). The results are in good agreement with experiment, (Fig. 3). Worthy of note is that the calculated (and measured) stress change is nearly strain independent. Since a relative strain rate change follows primarily from a change in the relative effective stress, Eq. (5), this result indicates that the effective stress is approximately constant despite the increase of strain, a conclusion also reached by Michalak. At the initial strain rate of 1.66 E-04/s, speed ratios of 2.0, 5.0 and 10 give stress changes of about 2.0, 6.0 and 9.0 MPa respectively. Comparison with values obtained for other materials shows that these stress increases are quite large, a result which follows from the small stress-velocity exponent in combination with a quite large effective stress in iron.

10

:-- • • i2/fit~10 • 4", • 166 II 'Kf'l_

~. l>

.6~tcSO '"

• 0 '" ~TICAL POINTS -- EXPERIMENTAL CURVES

-""" -v- u u 0 ." -~

IRa'! E-2/€, = 20

o o 002 004 006 ooe 01 012 014 016

STRAIN

Figure 3: Stress change on change of crosshead speed

There is another feature of the behaviour of iron which the theory suggests is linked to small n value, and that is the strong dependence of the stress sensitivity of strain rate on base crosshead speed, (Fig. 4). Equation (6), on which an analysis (6) of this

oL-~ __ -L __ -L __ ~ __ L-~ __ -L __ ~

o ~ ~ ~ ~ ~ ~ ~ " STRAIN

Figure 4: Stress sensitivity at two crosshead speeds.

effect is based, is derived from Eqs. (1) and (5):

* n d.tncre d.tnPm

n 'CfIii"(1 + 'CfIii"(1 (6 )

Using the theory, we calculate that at the lower rate, 3.33 E-05/s, both the*effective stress and the mobile density are considerably reduced. The consequence is that it requires a considerably smaller (relative) increase of stress to achieve a given fractional increase of strain rate. This result is shown in Fig. (5).

oL-~ __ -L __ -L __ -L __ ~ __ ~~LJ

o 20 40 60 80 10 12 14 STRAIN I%l

Figure 5: The two components of stress sensitivity as calculated by the theory for zone-refined iron.

3.2 Measurement and Analysis of Soft Machine Stress Sensitivity. In the measurement of the stress sensitivity of strain rate in a soft machine, the stress rises continuously at a fixed rate (both before and after the stress drop) and, at some selected point, the stress is abruptly lowered by a selected value. The extension is measured both

267

*Since in this case, the strain rate is lower by a factor 5, the point may seem trivial, Eq. (1). However, if n is large, say 10 or more, a small decrease in the effective stress may be sufficient to account for virtually the entire decrease of strain rate.

268

before and after the drop and the extension rate is calculated by the digital recorder, Fig. 6.

Figure 6: Measured extension and extension rate. The stress is 127 MPa, stress rate 1.0 MPa/s and stress drop 2.0 MPa.

The problem of obtaining the drag constant and velocity exponent from stress sensitivity data is more challenging than the reverse procedure for which the velocity function is known, as described above. Indeed, this task cannot be accomplished directly, but involves an attempt to fit experimental data by means of a number of trial and error calculations, in which various values of the velocity constants are employed. However, we have been successful in this endeavor using soft tensile machine data obtained for the IF iron. At the same time, the theory allows the determination of the effective stress; the quantitative success of this determination has been confirmed by an additional experimental test.

The study of IF iron indicates that the deformatio~ rate is stress sensitive, even near the yield stress, with an n value near 30. In my initial attempts to understand this result, I assumed that n was large, perhaps 15 or more. We soon observed however, in agreement with the observation of Michalak, that the stress sensitivity was strongly dependent on the base stress rate (or, roughly, strain rate). As indicated above, such behavior is incompatable with large n; the slope of the theoretical curve in Fig. 7 is much too shallow. In

3.0

~' ~':~~'100pa ~. 45 fLm

"-0-. -20 MPo

.II. EXPERIMENTAl ~ paNTS ~""

- THEORETICAl ClJRVE I to L-_..L __ ...l...-_-'-_--' __ --'-'

01 02 05 10 2.0 5.0 STRESS RATE IMPoIS)

Figure 7: Determination of stress-velocity constants by means of a theoretical fit to stress sensitivity data.

fact, a successful fit to these data requires that n 3.6 E09 (Fig. 7).

2.0 and ']; o

269

If we attempt to fit the strain versus stress curve using these values and an adjustable value for the mean free path, ro = 4.5 ~m, then the theoretical material is much too soft. Specifically, the yield strength at 1.0 MPa/s is about 60 MPa whereas the dynamic strength is calculated to be only 21 MPa. The nearly 40 MPa of static strength cannot be the result of strain hardening because, at yield, the strain is too small. Thus we conclude that there is a source of static strength in the IF iron which is not present in the zone-refined iron, perhaps as a result of residual titanium impurity or precipitates of titanium carbo-nitrides. Its value, as a shear strength, is 11 MPa, Fig. 8. (The Taylor factor used for the poly­crystalline iron is 2.75).

10

80 .o. EXPERIMENTAL PONTS .; 8D

THECIlETICAL amES • 1;;

_60 h! 6D@

'$. ~ ;;!; 40 4D~ : L ,/ n=20 t;; 7. / -';'=36'I)'Po

20 ••• /" r.=45 Jim 20 II'" i! ·1.1.10' Po

0

50 m 150 200 250 :m STRESS IMPaI

Figure 8: Strain and strain rate versus stress in IF iron. The material parameters, from which the theoretical curves are obtained, are listed.

With these material parameters in hand, Fig. 8, it is possible to predict several other features of the sample response at and after a sudden decrease of stress. Fig. 9 shows the effect on the strain rate ratio of the magnitude of the stress decrease. (E l is measured before the stress drop and E2 after). The open points are predicted, the filled point fitted, from Fig. 7. This relationship is exponential.

STRESS OEREASE IMPaI

Figure 9: Strain rate ratio for various stress rates and stress decrements.

270

The reason is that on decrease of stress (not on increase), only the velocity is changed, not the mobile density. This conclusion is a consequence of the model of mobile density, Eq. (4), in which an increase of stress injects additional mobile dislocations, but a decrease of stress does not extract them. It leads to a further prediction concerning the recovery of strain rate over time, following the stress drop. For the time period equal to ~a/a, dislocation sources do not operate, but movement and trapping of dislocations continues. Thus, for this period, the mobile density declines and the recovery of strain rate is slow. This "delay time" is observed and can be seen in Fig. 6. (It ends, and the strain rate begins to recover more rapidly, when the stress again reaches its prior maximum value). Also, in Fig. 10, are plotted points from a similar experiment at lower stress rate along with the theoretical curve. This figure may be taken as evidence for the validity of the physical ideas on which the theory of mobile density are based.

'~10t ~A .,;'08

I ::y.-' ~=-= ~ 02 .;.= 02 MPo/s ~ CT=l29MPo

o lICT=-20MPo

o ~ ro ~ ~ ~ ~

TIME's'

Figure 10: Prediction of strain rate recovery.

If the material constants are known, the theory allows the calculation of the dynamic strength (equal in magnitude to the effective stress) for any applied stress rate. The results are shown in Fig. 11 along with the curve of velocity versus effective stress,

IF RIO 'T'20'C).~: ~ n • 2.0 to· J6'o'Po 6-.so

0; -12' l>EIJIET1:AI. F'IlNTS NIl UNE /0 ~: ~~

/ ~:2l,

/

0 ~~~9,;=1s

a.12D ......

u<W~~2D~--~~~~Q---2~--~~~~ EFFECTNE STRESS ,_,

Figure 11: Velocity versus effective stress; the calculated effective stress values are shown as a function of applied stress rate.

Eq. (5). (This curve has been obtained easily in comparison to prior methods which have required direct observation of dislocations). The effective (shear) stress varies from 2.8 MPa at a stress rate of 0.04 MPa/s to 19 MPa at 17.5 MPa/s; the corresponding range of velocity is about 5.0 E-D7 to 3.0 E-D5 m/s.

271

Having calculated these values of effective stress, it was decided to attempt a "strain transient dip test", used in the analysis of high temperature creep behavior by W.D. Nix and his co-workers. At a = 0.2 MPa/s, a stress drop of 4.6 MPa should reduce the strain rate to zero. Within the resolution of the measurement, this is, in fact, what happens (Fig. 12).

'T/AI,E"

Figure 12: Change of extension rate on reduction of effective stress (as calculated) to zero.

4. CONCLUSIONS: A theory, in which the mobile dislocation density is determined by a competition between stress rate dependent injection and velocity dependent trapping, has been shown to be successful in the analysis of prior measurements of the stress sensitivity of strain rate in iron. It is also shown that the constants of dislocation velocity may be determined from the results of stress change experiments. This method is much easier than prior efforts based on direct observation of dislocation movement. Other features of the theory, taken from prior investigation by a number of scientists, tend to be reinforced by its success. An example is the idea that the strength of iron has just two components, dynamic and static, and that the contribution of strain hardening is almost entirely to the static component. The dynamic strength, described as a velocity-dependent drag on moving dislocations, then arises from the shorter range interaction with impurities or with the lattice.

ACKNOWLEDGEMENT: The author is appreciative of assistance given over many years by W.D. Nix and J.C. Gibe1ing. Financial support has been provided by the Natural Sciences and Engineering Research Council of the Government of Canada.

272

REFERENCES: 1. W.G. Johnston and J.J. Gilman, Journal of Applied Physics, 30

(1959) 129-144. 2. A.S. Argon, Materials Science and Engineering, 3 (1968/69) 24-32. 3. W.D. Nix and R.A. Menzes, Annual Review Materials Science, 1

(1971) 313-346. 4. A.R. Rosenfield et al eds., Dislocation of Dynamics, McGraw-Hill,

New York NY (1968). 5. J.J. Gilman, Micromechanics of Flow in Solids, McGraw-Hill, New

York NY (1969). 6. G.M. Pharr and W.D. Nix, Acta Metallurgica, 27 (1979) 433-444. 7. T.H. Alden, Metallurgical Transactions, l6A (1985) 375-392. 8. T.H. Alden, Metallurgical Transactions, l8A (1987) 51-62. 9. A.S. Keh, Philosophical Magazine, 12 (1965) 9-30. 10. J.D. Livingston, Acta Metallurgica, 10 (1962) 215-239. 11. J.E. Bailey and P.B. Hirsch, Philosophical Magazine, 5 (1960) 485-

497. 12. T.H. Alden, Metallurgical Transactions, 18A (1987) 811-826. 13. J. Friedel, Dislocations, Addison-Wesley, Reading (1964). 14. D.F. Stein, in Microplasticity, ed. McMahon, Interscience, NY

(1968) 141-158. 15. A.H. Cottrell and R.J. Stokes, Proceedings Royal Society A, 233,

(1955) 17. 16. J.T. Michalak, Acta Metallurgica, 13 (1965) 213-222. 17. T.H. Alden, Acta Metallurgica, 37 (1989) in press. 18. A.P.L. Turner and T. Vreeland, Jr., Acta Metallurgica, 18 (1970)

1225-1235. 19. T.H. Alden, Metallurgical Transactions, 20A (1989) in press. 20. C.J. McMahon, Jr., in Microplasticity, ed. McMahon, Wiley, New

York (1968) 121-140. 21. J.H. Holbrook, R.W. Rohde and J.C. Swearengen, in Mechanical

Testing for Deformation Model Development, Rohde, Swearengen eds., ASTM STP 765 (1982) ASTM, Philadelphia, PA, 80-101.

THERMODYNAMICALLY CONSISTENT CONSTITUTIVE LAWS IN PLASTICITY

INCLUDING DAMAGE

Th. Lehmann

Institut fUr Mechanik, Ruhr-Universitat Bochum

D-4630 Bochum, Germany

ABSTRACT: A rather general frame for the formulation of thermodynamically

consistent phenomenological theories of large non-isothermal deformations of

solid bodies including damage processes is given. Particularly the consequen­

ces resulting from energy and entropy balance are discussed. This will be

demonstrated particularly with respect to plastic deformations including

damage processes by arising micro-voids.

1. INTRODUCTION: A phenomenological description of thermo-mechanical

processes of solid bodies may be based on the following two fundamental

assumptions:

(I) The body can be considered as a classical continuum even if the

body is damaged by micro-defects.

(II) On the adopted level of process description the thermodynamical

state of each material element is determined uniquely by the ac­

tual values of a set of (external and internall thermodynamical

var'lables even if the body is not in thermodynamical equilibrium

(principle of local thermodynamical state).

Assumption (J I 1 means that we are dealing on the adopted (macro- llevel of

process description in a so-called large state space. Thermodynamical state

variables introduced in this way result from certain averaging procedures in

space and time. As a consequence we have to observe that besides

[1] classical reversible processes representing a sequence of equili­

brium states governed by thermodynamical state functions (energy

potentials) and

[2J classical irreversible processes characterized essentially by non­

equilibrium states and dynamical relaxation functions (dissipation

potentials 1

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274

we also have to take into account

[3J non-diss ipative processes appearing as a sequence of equilibrium

states but not governed by state equations (in contradiction to

[1 J) and

[4J dissipative processes appearing as a sequence of constraint

equilibrium states (in contradiction to [2JJ.

The different kinds of thermodynamical processes have to be specified within

the constitutve law. The corresponding decomposition of the evolution of

energy and entropy plays an important role in the formulation of the consitu­

tive law.

2. THERMODYNAMICAL FOUNDATIONS: The first law of thermodynamics

states

u = w - + r

with the following notations:

u: specific internal energy,

p: mass density,

q i: energy flux,

r: specific energy sources.

(1)

All quantities are related to the actual configuration and Ii means he covariant

derivative. The rate of work has to be decomposed according to

w=wtwtw (2) (r) (h) (d)

The rate of revers ible work w can be defined uniquely by introducing an ac-(r) *

companying reference state (with temperature T) defined by a fictitous unloading

process with frozen internal variables (1) The rate of immediately dissipated

work";" corresponds to thermodynamical processes of kind [2J and [4J with (d)

the condition

W ?- 0 (d)

according to the second law of thermodynamics, whereas

w ~ 0 (h)

(3)

(4)

represents the rate of work interacting with changes of internal material

structure in (at least partially) non-dissipative processes of kind [3J.

According to our fundamental assumption (II) u must be expressible as a

function of (extensive) state variables, i.e.

where

u = u( E ki ,s, a, o:~, w) (r)

E i . reversible strain (r)k' s specific entropy

i a, O:k' w : (representative) set of internal state variables.

(5)

By a double Legendre transformation we introduce the specific free enthalpy

(Gibbs function)

!J; = u -i k i i

o sk E. - Ts = !J;(Sk' T, a, O:k' w). Q (rll

(6)

i where the (weighted) stress s k and the (absolute) temperature T represent

the respective conjugated external (intensive) state variables, and a super­

scribed 0 relates to the initial state.

From (6) we derive in known manner

k i i the thermal state equation: E. (s k ' T, a, 0: k' w)

(rll

h I · t t t' I T i) 841_ t e caorlc s ae equa Ion: s (Sk' ,a, O:k' h) = - 8T

(7)

(8)

Furthermore we derive from (1), (6), (7) and (8) a balance equation for

energy concerning the interaction between external energy supply and changes

of internal state. In the special case when these changes are only due to

applied mechanical work this balance equation reads (see (2),(3))

w - T ~ =s~_ a + ~jI-- ~ 'kl + lltJl_ (,) (hl I) a < i S w

o O:k (9)

where YJ denotes the rate of dissipation involved in this interaction. The to-

tal entropy production is in this case

Ts (dl

w -(dl

1 i . - q T I. + T YJ ;, O. P I

(10)

275

276

3. SOME GENERAL REMARKS ON THE FORMULATION OF CONSTITUTIVE

LAWS IN PLASTICITY INCLUDING DAMAGE PROCESSES: The relations

(9) and (10) represent serious restictions for the formulation of constitutive

laws for inelastic deformations. The constitutive law consists of

1 . state function 4J 2. evolution law for inelastic deformations or several evolution laws

for different mechanisms, respectively,

3. evolution laws for internal variables,

4. laws determining the different parts of wand 11, (d)

5. laws for energy fluxes.

Only if we disregard energy fluxes apart from heat the evolution laws 2. to

4. represent a system of ordinary differential equations of first order with

time as independent variable. Otherwise they represent a system of partial

differential equations of first order in space and time (for more details see

(4)).

Thermodynamically consistent formulations for plastic and viscoplastic defor­

mations without damage processes are treateo in several papers taking into

account different yield mechanisms (see, for instance, (2), (4), (5)). When

we want to include damage processes by micro-voids we may define the

actual damage state by a scalar valued damage parameter w which describes

the volume fraction of voids in the unloaded reference state (marked by * ) according to

o * W :: 1 - cJ~ :: 1 - 2.. (0.; h) .; 1)

dV 9 (11)

The corresponding damage rate is defined by

w = (1 - w) (12)

where d r denotes the rate of volume changes due to nucleation and growth (v)r

of voids.

Concerning the influence of damage by voids we may distinguish four cases:

(A) The specific free enthalpy of the matrix material is not influenced by

existing voids. This does not mean that the behaviour of the material is un­

affected by voids since the weighted Cauchy stress tensor s ik -becomes

(13)

where oik denotes the Cauchy stress. In this case, however, W does not

represent a thermodynamical state variable but only a thermomechanical pro­

cess variable and consequently the damage rate belongs to the evolution laws

of inelastic deformations.

(8) The specific free enthalpy can be decomposed according to

ljI(s~,T, a, !X ik , w) = 1jI*(s~, T, w) + 1jI** (T, a, !X~). (14)

This means on the one hand that the elastic behaviour of the matrix material

is not influenced by the hardening state and on the other hand that the har­

dening state of the matrix material is not influenced by existing voids. This

does not exclude that the yield condition depends on the damage state since

W represents a thermodynamical state variable. Therefore the damage rate

belongs to the evolution laws of internal variables which leads to serious

restrictions resulting from the balance equations (9) and (10), This case is

discussed in some more details in (3).

(Cl The specific free enthalpy can be decomposed according to

ljI(s'k,T, a, !X~, wl (15)

1jI*(s~,T,w)+ 1jI**{T,a, !X~,w).

This means that the elastic behaviour of the matrix material remains un­

affected by the hardening state, whereas the hardening state itself depends

on the damage state, too. This leads to further restrictions in the formulation

of the corresponding evolution laws.

(D) In the most general case a decomposition of the specific free enthalpy

is not possible any more.

277

278

4. SOME ADDITIONAL REMARKS: Even if we restrict ouselves to damage by

micro-voids we have to suppose that the influence of damage is not only gi­

ven by the volume fraction of voids. In a more detailed investigtion we shall

find that also the pattern of void distribution influences the material behaviour.

This leads to the consequence that the state of damage by micro-voids has

to be described by more than one scalar valued internal state variable.

The situation becomes more complex when we include damage processes due

to evolution of micro-cracks and micro-shearbands which lead to anisotropic

damage states. Some remarks concerning anisotropic damage can be found

in (3) and in the literature quoted there. Many questions. however. are still

open in such cases.

REFERENCES:

1. Th. Lehmann. in Constitutive Laws and Microstructure ( D.R. Axelr"ad and

W. Muschik. eds.l Springel"-Verlag. Berlin etc. (1988) 27-42

2. Th. Lehmann. to be published in European Journal of Mechanics.

3. Th. Lehmann. to be published in Acta Mechanica.

4. Th. Lehmann. Lecture Notes CISM-Course "Internal Variables in Thermo­

dynamics and Continuum Mechanics" LJdine (1988)

5. Th. Lehmann. in Constitutive Laws for Engineering Materials (C.s. Desai.

E. Krempl. P.D. Kiousis. T. Kundu eds.) Elsevier New York/Amsterdam/

London (1987). 173-184

COMMENTS ON MODELING PLASTIC DEFORMATION OF LOW CARBON STEEL

JERZY T. PINDERA

Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI

ABSTRACT: Some characteristic features of elastic-plastic deformation states occurring in specimens made of low carbon steel are presented. On this basis the issue of physical admissibility and testability of constitutive material relations is briefly discussed.

1. INTRODUCTION: The term constitutive equation usually denotes an analytical relation between the parameters of a mathematical model which was developed to represent a particular physical process. In mechanics a particular case of this notion is used, called a constitutive material equation, which usually denotes a relation between stresses and strains and their spatial and temporal derivatives. Consequently it is assumed that the constitutive equation defines the material. It is usually assumed that during the process of elastic and plastic deformations occurring in metallic materials the deviations from the thermal equilibrium are small, and thus they are neglected, with infrequent exceptions. As a result, the common constitutive material equations, which do not contain the thermodynamic (or dynamic) terms are inherently unable to reliably simulate the real irreversible deformation states which occur in real materials and structures under influence of real loads. The thermoelastic reversible effect occurring within the range of reversible deformations may increase locally the macroscopic stresses by an insignificant amount at typical environmental conditions, Fig. 1 (1). However, the heat generated in a typical cylindrical round tensile specimen made of a low carbon steel may locally increase the temperature of a specimen by 100°C, or more, which locally generates thermal stresses comparable with the yield stress. The question is what is the reliability of the analytical predictions, and of the evaluated experimental data, when the inherent thermal processes are ignored both in the mathematical models of responses of materials, and in the theory of the experiment. A more general problem is what are the coupled explanatory and predictive powers of the constitutive equations which concentrate on the phenomenological, descriptive simulation of the experimental data. This reduces to the basic question whether the phenomenological approach, without a satisfactory insight into the involved mechanisms of deformation, can yield

279

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280

reliable predictions, that is whether such an approach satisfies the theoretical requirements of the scientific modeling of reality (2).

0.3

0.2

0.1

(6 Til = 6 T (E) = - T (aT ICE) 6o-kk at E=Ei<t-to>o

Materia I : stress proof steel o-y = 700 N mm- 2

Fig. 1 Example of a thermoelastic, reversible, response of a steel.

2. ISSUES OF INTEREST: The empirical evidence presented in this paper pertains to the following interrelated issues:

- rational and quantifiable definition of the plastic deformation and of the related notion of the yield stress;

- testability of the model represented by a constitutive equation. Evidently only the predictions of models simulating real materials are testable, whereas the predictions of models representing hypothetical materials can only be assessed with regard to the range of their practical usefulness;

- reliability of information evaluated on the basis of experimental data, which depends on reliability of the theory of the experiment;

- theoretical admissibility of the model represented by the constitutive equation, that is the physical and the mathematical admissibility;

- rational assessment of the explanatory descriptive and predictive powers of the models represented by the constitutive equations.

3. EXPERIMENTAL EVIDENCE: The major issue of a reliable evaluation of experimental data on elastic and plastic responses of metals is illustrated by Fig. 2 (3), which represents a typical tensile test of round specimens. Evidently, the evaluation of the experimental data represented by the stress-strain diagram, or even the existence of certain material parameters, depends on the chosen theoretical model of the experiment. Thus the tensile stress-strain curve does not directly represent the response of the tested material, but it rather represents the response (quasi-static and dynamic) of the dynamic system consisting of the specimen and the testing machine, strongly influenced by the local, transient temperature fields. Stress state in the necking region is strongly three-dimensional, not uniaxial anymore, and is influenced by local thermal stresses, often comparable

281

with the yield stress. Thus, the yield stress evaluated from a simple tensile test of a round specimen is, in fact, a quantity related to a three-dimensional, mechanical-thermal stress state.

Mathematical models of tensile test based on.

I' j-lo ---I~o 'j p -- t---~-- ot---$;:t - p

necking AD

(0) Speculative (very simplified) model: (b) Basic physical model.

P II Lo P P ilL 0-= A;; , E=~ for 011 values of Lo o-=To or 0-= AlE)' E 'To = Ello/O)

displacement moss gage

Fig. 2

p plastic matenal behavior

0- (property of material)

I influence of necking

- (o;,)u' (o-Y)I-material constants

I i near material behavior -(property of material)

tn-I E

T speCimen = const

To~-----------------

Example of incompatible of a simple tensile predictions.

p

E

response of system specimen a loading

related to A (E), slress undefined

machine

I ~ * '-v---' influence of chosen mathematical model of strain (La /0; A)

quasI-linear response influenced ~ by strain rate and thermoelastic effect

defined -- IIla,E

I / I ,

I/\depends on (la/D) V +---

.~.::.-=--- ..... ...... " ........ ,

vibration of the system

lila

thermoelaslic 8 plastic effects

phenomenological and physical models test, and of their incompatible

Problems related to satisfactory, quantifiable definitions for plastic deformation, the yield stress, or the boundary between elastic and plastic regions are illustrated by Fig. 3. It presents·the state of the surface of a previously polished steel specimen after plastic deformation. This phenomenon was used to determine the plastically deformed regions and their boundaries, some examples of which are presented below (4).

282

Fig. 3 One of the physical measures of dislocations and slips, caus ing development of irregularities.

for plastic deformation: density within and across the grains, Luders lines and surface

A typical testing of predictions of a constitutive material equation representing a phenomenological model is depicted in Fig. 4. As an object of evaluation the case of a rectangular prismatic beam was chosen. The beam, made of a low carbon steel was loaded by three forces. The particular issue was the development and growth of plastically deformed regions, determined by the elastic-plastic boundary. A classical analytical solution presented by Sokolovski (5) is given on the left, together with the analytically predicted elastic-plastic boundaries corresponding to two load levels. A sample of the empirical evidence pertaining to the same problem is given at the top right, and the evaluated empirical results, representing two characteristic deformation states, are given at the right bottom of the Figure. Evidently, the analytical solution which rigorously represents the chosen mathematical model is at variance with the actual elastic-plastic responses of a real steel.

The problem whether a deterministic approach is capable to yield satisfactorily reliable and accurate predictions is illustrated by Fig. 5. The tested specimens, made of a low carbon steel, are represented by a sketch, top left. The experimental arrangement, which allows simultaneous recordings of the developing plastic deformations at both faces of the tested specimens, is gIven at bottom left. The elongation-load diagram in the centre of the Figure gives information on the overall state of deformation at the time of particular recordings. Samples of recordings of plastic regions (light regions) are given at the right. Two pieces of information are immediately given by the recordings. At first, the deformation state is noticeably inhomogeneous; it consists of randomly distributed plastic and elastic regions, which indicates the presence of strong, local fields of residual stresses. Secondly, the observed plastic deformations indicate the presence of two at least mechanisms of the plastic deformations, with some obvious implications. Also, it appears th~t the local (microscopic) and the global (macroscopic) plastic deformations are influenced by different mechanisms.

Prediction of a classical

phenomenological model

h 2 z3_ 3h 2 = 2 -

f P

(x-f) -Po

283

Empirical evtdence

Empirical results

Fig. 4 Testing the predictions of a phenomenological model of elastic-plastic deformation. Tested specimen: low carbon steel beam under three poin~ load. Task: determination of boundaries of plastically deformed region.

flat tensile specimen,law carbon steel

!-f---F-f--+ ~ I_ f -I

scheme of recordings

: I: \ I I

I " \ : I ~ f specimen \ I I I

\ \: I

" I,/! / \ I I

\ I, " \ II,

I" recorder I,

tensile diagram

P [I03NJ

1706 1955

Recordings of plastic regions

point 4

Fig. 5 Testing some assumptions underlying the models used to construct constitutive equations. Tested specimen: flat tensile bar, made of a low carbon steel. Task: determination of patterns of development of plastically deformed regions.

284

4. SUMMARY: The presented evidence shows that at the present state of knowledge the classical phenomenological methodology used to construct mathematical models of hypothetical materials is not able to predict elastic-plastic responses of at least some real materials, within a sufficiently wide range of variation of the geometric and physical parameters. Consequently, the physical approach is indispensable to satisfy not only the intellectual requirements but also the actual needs of modern technology. A purely descriptive power of a constitutive equation, which may satisfactorily reproduce the actual responses of the tested standardized specimens within a prescribed range of variation of only some involved parameters, is not a reliable theoretical basis for extrapolation and generalization. It appears that the requirement of the testability of a constitutive material equation should encompass an assessment of its interrelated explanatory, descriptive, and predictive powers. Analogous comments pertain to the theories of experimental testing and to the validity of the evaluated results.

In particular, one may conclude that the phenomenological materials equations which are developed without understanding of the real involved physical processes and thus are based on various simple assumptions such as the assumption of a single mechanism of deformation, are inherently unable to yield reliable predictions outside the already experimentally explored ranges of variation of physical parameters. Consequently, such equations should be tested very carefully before they are used in technology. It appears that a single mechanism of an inelastic deformation of real materials is an exception rather than a rule, as illustrated in (6), or (7). Analogously, one may conclude that a constitutive equation without thermodynamic terms represents only a first approximation of the actual process of plastic deformation (8).

REFERENCES: 1. J.T. Pindera, P. Straka, and M.F. Tschinke, VDI-Berichte No. 312 (1978) 579-584. 2. J. T. Pindera, in New Physical Trends in Experimental Mechanics, Springer-Verlag, New York (1981) 199-327. 3. J.T. Pindera, Transactions of the CSME 11 (1987) 125-138. 4. J.T. Pindera, Rozprawy Inzynierskie (Engineering Transactions) 5 (1857) 31-47. 5. V.V. Sokolovski, Theory of Plasticity, G.J.T.-T.L., Moskva, Leningrad (1950) 6. F.A. Mohamed, and M.S. Soliman, Materials Science and Engineering 53 (1982) 185-190. 7. J.T. Pindera and P. Straka, Rheologica Acta 13 (1974 (338-351 8. U. Blix, Mittelungen aus dem Institut fur Mechanik, Ruhr­Universitat Bochum, No. 40 (1983).

MODELING OF PLASTIC DEFORMATION OF METALS AT MEDIUM AND HIGH STRAIN RATES WITH TWO INTERNAL

STATE VARIABLES

J.R. Klepaczko

Metz University, Faculty of Sciences Laboratory of Physics and Mechanics of Materials

lle du Saulcy, 57045 METZ CEDEX, France

ABSTRACT : This study presents a consistent approach to the kinetics of plastic behavior of metals and alloys with FCC, BCC and HCP latti­ces. The modeling of rate-sensi ti ve strain hardening, temperature and rate sensitivity is rigorously performed in terms of kinetics of dis­location glide and annihilation. Consequently, the formalism has been applied in which the thermal activation analysis is used for both the kinetics of glide and kinetics of microstructural evolution (1,2).

Evolution of microstructure is characterized in the formalism by two state variables, i.e. the mean immobile dislocation density p. and the mean mobile dislocation density p • The most proper choile of equations for the glide kinetics and thW, kinetics for structural evo­lution has been proposed. Numerical results are shown for the case of polycrystalline aluminum. The formalism enables for exact modeling of transient phenomena. So called short transients as well as long tran­sients in strain hardening behavior after an abrupt change of strain rate or temperature can be exactly predicted.

1. INTRODUCTION One-parameter models based on evolution of mean disloca tion density p have proved theirs usefulness in consti tuti ve modeling of ra te and temperature effects. They can predict more or less exactly the long transients of strain hardening associated with evolution of the internal stress '~. Since some reviews on one-parame­ter models are at present availaole (4,9) no attempt is offered to discuss this subject.

Recently, at temps of a two-state-variable approach to model strain hardening have been published (10,13). The introduction of a second state variable provides more flexibility in modeling and in the present case permits for an exact description of the short stress transients. The so called short and long transients are demonstrated schematically in Fig.1. The short transients in the form shown in Fig.la are typical for an abrupt strain rate and temperature changes. They are associated with the instantaneous reaction of microstructure to abrupt changes of rand T ; r is the strain rate in shear, (r = tgy), and T is the absolute temperature via changes of the effective

285

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286

b

Figure 1. Schematic representation of sh~rt (a) and long (b) transients in FCC metals after an abrupt change of ~ or T.

stress ,* (thermal component of flow stress). The long transients are associated with an evolution of strain hardening and the relaxation of the internal stress , • It is believed that the short stress transients are caused by flow k~netics and more specifically by changes in the mobile disloca tion density P (14), whereas the long transients are caused by mul tiplica tion of lJlisloca tions and recovery kinetics, i. e. by evolution of the immobile dislocation density Pi (15).

2. THE FORMALISM : A unified theoretical concept is employed in which rate-sensitive strain hardening, temperature and rate-sensitivity is rigorously treated in terms of kinetics of dislocation multiplication, glide and annihilation. The formalism differs of that proposed earlier by Mecking and Kocks (7) and employed recently to high strain rate plasticity in (8).

The notion is adopted that plastic deformation in shear is the fundamental mode of metal plasticity and appropriate Taylor factors should be employed to find macroscopic quanti ties C3, 4). However, in this study the macroscopic quanti ties are used as discussed in (1). The flow stress in shear , at constant structure is given to a good approximation by

where '11 and ,* are respectively the internal and effective stress components. The assumption of additivness (1) implies the existence of two sets of obstacles opposing the dislocation movements. The first set of obstacles associated with '~ are supposed to be strong obstacles to dislocation motion like cell walls, grain walls, twins, etc. The secondary defects, while more numerous, like forest dislocations, Peierls barriers, second phase particles, are supposed to be weak obs­tacles and they can be overcome by moving dislocations with assistance of thermal vibration of crystalline lattice and the effective stress ,*. The kinetics of defect (dislocation) movements interrelates at constant microstructure, characterized by j state variables s., the instantaneous value of effective stress ,* and the instantaneous diastic strain ratetP = drP/dt

287

fl G. (T,,*,s.) r (T'Sj) [-

1 J ] (2 ) = v. exp kT 1

or after inversion ,* f* { s. [h(f,T)], r, J

T } (3)

where v. is the frequency factor, ~G. is the free energy of activation, T is th1e absolute temperature and R is Bol tzmann cons tant. The sub­script i indicates the i-th, so far unspecified, thermally activated micromechanism of plastic deformation. Generally flG depends on the effective stress ,* in a non-linear manner (16).

As it has been discussed previously (1 and 17) , the internal stress, must be also rate and temperature dependent via dynamic re­covery p~ocesses, i.e. relaxation of long range internal stresses due to dislocation annihilation and rearrangements of obstacles to dislo­cation motion. Thus, the internal stress is

, Il

f { s. [h (f, T)] } Il J

(4)

It is assumed here that recovery processes leading to a structural evolution may be thermally activated. It is the source of temperature and strain-rate dependence of 'Il. Both, strain rate and t.emperature enter into eqs (3) and (4) in the functional form, since r (r p) and T( r P ) are defined deformation histories. The total flow stress , at constant structure is given by introduction of (3) and (4) into (1)

(')STR = fll { S}hcf,T)J}STR + f* {Sj[h(f,T)],f,T}STR (5)

where h(f,T) indicates that the internal state variables s. do depend on history of f and T defined more.exact.1Y asr(r P) and T(JrP), it is also assumed in eqs (1) to (5) that r p ~ r. Consequently, plastic res­ponse of a material is divided into two logic steps : the flow stress , depends on the current structure, defined by s. state variables and the current values of f and T, next the structureJevolves with plastic strain r P. This is the fundamental assumption of the model.

Both components of stress, eq. (5), are wri t ten for a current state characterized by s. state variables. Since the microstructure undergoes an evolution, a~d the state of microstructure is defined by s. variables, the state evolution is assumed in the form of a set of j dffferential equations of the first order

ds. _J

dr P k = 1 ••• j ( 6)

Solution of the set (6) provides current values of s. to be introduced into eq.(5). Thus, the flow stress can be calculated for any deforma­tion history.

3. IDENTIFICATION OF MICROSTRUCTURE : Flow stress in polycrystalline metals and alloys can be related to characteristic spacing of obstacles to dislocation motion associated with a particular microstructure (16). The following four will be assumed as a satisfactory choice. Thus, the microstructure will be characterized by the mean distance L between forest dislocations, the mean value of a dislocation cell d, the mean value of a grain diameter D and the mean distance between twins fl. Each of those obstacles to dislocation motion will contribute

288

to the total value of the internal stress T can be written ~

T ~

( b) 6 11 L Ci. -j AJ. j

A general expression for

(7)

where ~ is the shear modulus, Ci.. are constants which characterize dis­location/obstacle strength, b i-t the modulus of Burgers vector, A. is the generalized characteristic spacing (L, d, D, tI) and 6 is an expo~ent which is equal 1 for one-dimensional characteristic spacing (L) and 6 = 1/2 for two-dimensional spacings (d, D, tI). The mean spacing L between2 disloca tions is related to immobile dislocation density p. by p. = L- • It is important to note that all characteristic spacings~can b~ determined from microscopic observa tions. It is also obvious that L, d, D and tI undergo an evolution during plastic deformation. Relation (7) can be written in the explicit form

.J ( b)6 (b)1/2 (b) 1 /2 (8) T~ = Ci.l~b Pi + Ci.2 d(Pi ) + Ci.3~ D + Ci.4~ "i.

The first three terms in eq.(8) are related respectively to disloca­tion/dislocation interaction, evolution of subgrain and the effect of grain diameter - so called the Hall-Petch term (9). It is well known that at low temperatures and at very high strain rates some metals and alloys produce deformation twins. Thus, the fourth term in eq. (8) accounts for twin formation as an dislocation obstacle. There is a numerous experimental evidence that beginning of certain strain level, usually r :::l 0.08, dislocations form cells with linear dimension d. Generally, evolution of d ( P .) intensifies strain hardening with a larger effect at small strain\;. The exponent 6 in the second term of eq.(8) has a dual nature, 6 = 1 for cells and for small subgrains with large misorientations, and 6 = 1/2 for "ideal" subgrains in thermally recovered metals (18).

Another state variable, considered in this analysis, is the mo­bile dislocation density p • Athough in the FCC lattices p does not undergo a very large change~ it plays an important role in ~escription of the short transients (14). In BCC lattices an evolution of p is important at the begining of deformation when the yield drop 'fs a consequence of this condition (2,20,23).

4. EXPERIMENTAL EVIDENCE OF SHORT TRANSIENTS : An ample experimental evidence of short as well as long transients in FCC metals has been demonstrated in the review paper (23). Additional evidences are provi­ded in (14) and (24). The stress-strain behavior of copper after large increment in strain rate is shown in Fig.2. The short transients are evident within few percent of deformation increment.

5. EVOLUTION OF IMMOBILE DISLOCATION DENSITY : Common to all recent microstructure - related models is that the set s. of internal state variables is reduced to a single structure paratneter which can be identified with the mean dislocation density p, (1,8). Since not very high strain rates are analyzed the generation of twining will not be discussed here. A general form of the differential equation for evolu­tion of immobile dislocation density Pi can be written as, (3,4)

289

dP i dr = Mg (pi,I-> - Ma (Pi,f,T) (9)

where the difference M ff = M -M is the effective coefficient of dis­location multiplicatiog. OncegM ~f is known one can predict the current value of plastic strain r P ~ r tnat will be accumulated in any process of deformation by integrating eq.(9).

COPPER 50r--------,--------,---------r-------,

-iii a. (:,

b_ (f) (f) 40 W a: f-(f)

W -l (jj Z 30 W f-W :::> a: f-

Ej =2.23)(10-°.-, ~r=6.2e)( 10; .-' Ej =0.1445 T =293°K

20~ ______ ~ ________ ~ ________ ~ ______ --"

0.10 0.15 0.20 0.25 0.30 TRUE TENSILE STRAIN,.,

Figure 2. Demonstration of short and long transients in eu after chan­ge in strain rate, after (24).

Since the mobile dislocation density is much lower than fast increasing density of immobile dislocations p., eq.(9) can be applied directly as describing evolution of p .• A simple equation for evolution of p. has been applied in present cal~ula tions <3,4,15)

l

M. (f) - k l 0

• -2m T ( r) 0 r

o ( 10)

. where k is the non-dimensional annihilation factor, r is the frequen-cy fact~r of annihilation, m is the absolute rate J'ensitivity asso­ciated with dislocations. TheOgeneration rate of immobile dislocations M.~h can ~e related.to the mean free path of dislocation storage Atr) as M.(r) = l/bA(r). The mean free path is diminishing when r is increasin~ (3,4). Thus, the explicit form of the internal stress ~ (r) can be found after integration of (10) and introduction of the r~sult into (8). For f = const. and T const. the analytic solution is available with initial conditions Pi = Pio at r = 0

Evolution of the subgrain dip) in eq.(8) has been discussed elsewhere (3), contribution of D and "'- to ~ is neglected in this analysis.

J.I

6. EVOLUTION OF MOBILE DISLOCATION DENSITY : It is well known that determination of the mobile dislocation density pm at different condi-

290

tions of plastic flow is a difficult problem. All methods are indirect and they are based on kinetic relations or on ultrasonic attenuation. Those indirect methods have provided evidence that P slightly increa­ses as a function of stress, and definite increases mare observed when strain rate is increased. One of those results is shown in Fig.4 where the evolution of Pm has been determined in Cu as a function of strain ra tli by an ultrasonic method, (26). ThUt a li~ear r;ela tion -fm :1 Pm + cr \S found for Cg wit~ P = 2.57x10. l/cm at r = lxl0 sana (ap /ar)r = 5.876xl0 s/cm ;me = (ap tart is the rate sensitivity of p ~eneration. On the other hand m\f!cham.cal tests indicate that the p\f!rsistence of short transients is limited to a few percent of defor­mation. After careful analysis of experimental data the following re­lation for evolution of Pm has been derived

dPm 1]1 ""d1' = (Pm - Pmo ) (r - 1112 ) ( 12)

where 1]1 is constant characterizing activity of dislocation sources, 1]2 is probability of immobilization, P is the initial mobile dislo­cation density from which integrationmostarts, r is counted as an increment. The main difference between eq.(10) and eq.(12) lies that the latter must be applied incrementally. General solution of eq.(12) is

( 13)

2.(+9 r--------------------,

:J .5(+9 COPPER < SHEAR STRAIN 0.017 E u ~

/"

..i 1. (+9 /"

/"

Ul H

;-

CI /" ;-

/" /" Pi .*/" ~5.(+7 /" *;-

.,- *--0

v;-

0 100 200 300 400 500

STRAIN RATE [sA-I]

Figure 3. Changes of P (r) for Cu determined at r = 0.017 by ultrasonic attenuation, data afte~ (25) transferred into rand r.

Introducing back (13) to (12) the explicit form for differential evo­lution equation of Pm is obtained

( 14)

where g(r,T) is the influence function of rand T on multiplication activity. In general, g-function can be specified as

291

. g(f,T) N(r,T) I(r,T)

where N is the number of dislocation sources per 1 cm2 and I is acti­vity of sources. The other possib~lity is that the activity of sources can be taken into account by 111 (f, T). This problem can be solved by more careful analysis of experimental data and it will not be discussed here. Relations (12), (13) and (14) have transient character, i.e. at the begining of incremental deformation p reached at fm = 111/112 and finally for larg~ f,

p , next a maximum is If~ = p . In literature f-'" mo

of thermal activation strain rate analyses it is commonly assumed that p = const. and consequently dp Idf = O. For this case all short tran­sTents are automatically elimrnated. Analysis of experimental data with short transients for FCC metals has indicated the region of f , 0.01 < f < 0.08, thus for example if it is assumed that f = 0.05 tWe relationmbetween 111 and.11 is obtained as 111 = f 11 and ifl = 0.0511 2 • The rate coefficient g(f,~) can be estimated for ct from Fig.2. Using genera~ solution (13) and experimentally found linear relation p - p = Cf one obtains

m mo

. g(f,T) ( 15)

f

For constant strain g increases linearly with r. Analyses of experimen­tal data indicate that 112 :::: 40 and con~equently 11 1 '5 2. ~ith those values of 111 and ~2 and tne val~e (ap laf~ = 5.876xl0 slcm the rate coefficien t is g (f , T) = 38.97 f. Int)Xloducing normalization by p and •• rno fo the final form g(f,T) for RT and f 0.017

g(r,T) .

0.1506 p (J:....) mo fo

Thus, the final form of solution of the evolution equation f~r Pm is

( 16)

with A = 0.1506 for Cu at RT and f = 0.017.

7. RESULTS OF NUMERICAL SIMULATION FOR AL : An numerical simulation has been performed for polycrystalline aluminum. Simulations of short transients is continuation of earlier calculations with evolution of the mean total dislocation density p at different strain rates and temperatures (3,4,9,15). All constants and parameters which enter into equations (2), (8), (10) and (11) have been given in the papers cited above. The set of two differential equations for structural evolution p. and p have been integrated by the 4-th order Runge-Kutta progedur2. Similar ~onstants as for Cu have been assumed, i.e. p = 5xl0 llcm , 111 = 2.2; 112 = 44 (f = 0.05), A = 0.1, in additf6h, the8initi~1 immobile dislocation de~sity p. was taken as p. = 2.5xl0 llcm, consequently the initial fractio~of = pip. has \~lue f = 0.02.

o mo LO 0

292

Figure .4. rate~l r 1 200s .

150

125 " '" D. L 100

Ul Ul 75 w ~ I-Ul 50 ~ a: w I 25 Ul

0 0

N I

( I. E+S E 0

RLUMINUM ui 8.E+8 z w '" -.i 5.E+8 tn ~

'" aJ 4.E+8 0 :>:

lL 0 2.E+8 f-z W :>: w '" u z .2

INCREMENT OF SHERR STRRIN

Numeri3a~1 sifllulation o! ~yolution for p at dif\erent strain = lxl0 s , r 6 = lxl0 s , increments IBetween r 5 and r 1 :

.05

RLUMINUM 150 ,------_____ --,

RLUMINUM 125

100 11=2E-4 [l/sJ 12=3E+2 [I /s J

75

50 T=523 K

11=2E-4 U/sJ 25 12=3E+2 [1/sJ

.1 . 15 .2 .05 .1 . 15 .2

SHEAR STRAIN SHEAR STRAIN

Figure 5. Numerical ~imu~a tions '?.f4 '.( r) curv~f at different T with jumps in strain rate r ; r1 = 2xl0 ,r2 = 3xl0 •

Results of numerical calculations of evolution for p at different values of r '2rE!:.lshow~ in Fi~.~.l The region of higher stmrain rates is covered, lxl0 s < r < lxl0 s ,where the short transients are more likely to occur. Results of numerical simulation of or( r) curves at different temperatures along with incremental changes of r are shown in Fig. 5. Many other numerical results have been obtained during the course of this project. They show a good agreement with experimental

293

results for FCC metals. The upper and lower yield limits in BCC latti­ces can also be exactly simulated numerically using eq.(12). Simulation of HCP lattices are in progress.

REFERENCES : 1. J.R. Klepaczko, Mater. Sci. Engng. 18 (1975) 121. 2. J.R. Klepaczko, A Model for Yielding and Flow of Iron and BCC Me­

tals Based on Thermal Activation, Tech. Rep. Brown Univ. DMR-79-23257/ 132, Providence (1981). 3. J.R. Klepaczko, in Constitutive Relations and Their Physical Ba­

sis, Proc. 8-th Rise Symp., Roskilde (1987) 387. ~ J.R. Klepaczko, in Proc. Int. Conf. on Mech. and Phys. Behaviour of Materials Under Dynamic Loading, Les editions de physique, Les Ulis (1988) C3-553. 5. Y. Estrin and H. Mecking, Acta Metall. 32 (1984) 57. 6. H. Mecking and Y. Estrin, in Constitutive Relations and Their Phy-

sical Basis, Proc. 8-th Rise Symp. Roskilde (1987). 7. H. Mecking and U.F. Kocks, Acta Metall. 29 (1981) 1865. 8. P.S. Follansbee and U.F. Kocks, Acta Metall. 36 (1988) 81. 9. J. R. Klepaczko, in Impact : Effects of Fast Transient Loadings,

A.A. Balkema, Rotterdam (1988) 3. 10. F.B. Printz and A.S. Argon, Acta Metall. 32 (1984) 1021. 11. G. Gottstein and A.S. Argon, Acta Metall. 35 (1987) 1261. 12. G. Ananthakrishna and D. Sahoo, J. Phys. D : Appl. Phys. 14 (1981) 2081. 13. W.D. Nix, J.C. Gibeling and D.A. Hughes, Met. Trans. 16A (1985) 2215. 14. H. Neuhauser, in Dislocation in Solids, Vol.6, North Holland, Ams­terdam (1983) 408. 15. J.R. Klepaczko, in Impact Loading and Dynamic Behaviour of Mate­rials, Vol.2, DGM Informationsgesellschaft Verlag, Oberursel (1988) 823. 16. U.F. Kocks, A.S. Argon and M.F. Ashby, Thermodynamics and Kinetics of Slip, Pergamon Press, Oxford (1975). 17. J.R. Klepaczko and C.Y. Chiem, J. Mech. Phys. Solids 34 (1986) 29. 18. F .R.N. Nabarro, in Strength of Metals and Alloys, Proc. ICSMA-7, Vol.3, Pergamon Press (1986) 1667. 19. M.A. Crimp, B.C. Smith and D.E. Mikkola, Materials Sci. Engng. 96 (1987) 27. 20. J.R. Klepaczko, in High Energy Rate Fabrication - 1984, ASME, N.Y. (1984) 45. 21. A.S. Krausz and M.L. Aggarwal, Mater. Sci. Engng. 48 (1981) 271. 22. J.R. Klepaczko, A. Rouxel and C.Y. Chiem, in Proc. Int. Conf. on Mech. and Phys. Behaviour of Materials Under Dynamic Loading, Les edi­tions de physique, Les Ulis (1985) C ; (in French). 23. J.R. Klepaczko, R.A. Frantz and J. Duffy, Engng. Trans. 25 (1977) 3. 24. J.R. Klepaczko, Strain Rate Incremental Tests on Copper, Tech. Rep. Brown Univ. GK-40213/6, Providence (1974). 25. J. Shioiri and K. Satoh, in Mechanical Properties at High Rates of Strain 1979, The Institute of Physics, Bristol (1979) 121.

APPLICATION OF CONTINUUM SLIP APPROACHES TO VISCOPLASTICITY

David L. McDowell Georgia Institute of Technology

Atlanta, GA 30332-0405

John C. Moosbrugger Clarkson University Potsdam, NY 13676

ABSTRACT: The link between continuum slip theory and macroscopic, phenomenological models for viscoplasticity is explored. In particular, the microstructural origins of kinematic/isotropic hardening and rate-dependence are examined within the framework of the continuum slip theory of Rice and the single slip theory of Bammann and Aifantis for dislocation glide. The microstructural bases for existing forms of state variable viscoplasticity become apparent, as do limitations on their primitive assumptions. A form for rate­dependent evolution of backstress proposed recently by the authors is derived.

1. COMMON PHENOMENOLOGICAL ASSUMPTlONS·HARDENING AND RATE-DEPENDENCE: Most phenomenological models assume the existence of a tensorial internal stress a and a scalar drag stress It. These internal variables serve the roles of kinematic and isotropic hardening, respectively. For viscoplastic theories, the isotropic hardening variable typically attenuates the overstress II a' - a' II in a nonlinear flow rule. As such, these models interpret isotropic hardening as affecting drag on dislocations rather than the level of directional internal stress. However, recent investigations (1-3) indicate that multiaxial nonproportional loading experiments are best correlated by interpreting these effects as an increasing rate and saturation level of kinematic hardening, at least for some FCC metals.

It has been determined that the evolution equation(s) of the kinematic internal variable(s) is (are) most appropriately written with a hardening/dynamic recovery format. A mechanistic interpretation, heuristically applicable at the continuum slip level, has been given for microstructural evolution leading to strain hardening with dynamic recovery (4-6). The dynamic recovery term is typically strain rate-dependent. Kocks (5-6) has proposed a description for the saturation level of the strength of obstacles to dislocation glide which has the same form as that proposed for dynamic recovery theories based on cross-slip (7-9).

295

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture. 295-303. © 1990 Kluwer Academic Publishers.

296

The distinction between thermal and athermal barriers to dislocation motion appears consistently as a fundamental premise in the materials science literature on thermally activated dislocation motion (10). In spite of this, few phenomenological internal variable theories recognize the importance of the decomposition of backstress in correspondence with the different size scales and activation energies associated with dislocation/barrier interactions.

At this point it is noted that conventional state variable viscoplasticity models do not typically reflect rate-dependence and "isotropic" hardening in internal stress evolution.

2. CONTINUUM SLIP APPROACH: Rice (11) presented a continuum slip model of crystal plasticity as an example of his internal variable theory. In his treatment of the continuum slip theory, Rice sets forth a concise and unambiguous definition of the component of stress producing forces on dislocations causing crystallographic slip. This is defined to be the shear stress 1CJ. which is the thermodynamic conjugate to 1CJ. where 1CJ. is the rate of shearing in a smooth continuous sense on slip system a. Here, the total shears . .,a are not internal state variables since the state of the crystal is not necessarily independent of the order of application of the increments b"fa (where 1CJ. =: b"fCJ. ib~. The rate of working per unit reference volume on slip system a is 01 = 1CJ.1 ~ 0 which (within a weighting factor of reciprocal temperature) is also the rate of specific entropy production due to microstructural rearrangements associated with slip. This dissipation inequality arises by virtue of the dependence of the free energy

ex on the shears "f .

The precise kinematical framework adopted by Rice (11) is important in the mechanical interpretation of thermodynamic conjugate force 1a. However, the details are not particularly germane here. The principal result of interest is

1a = IFol a,CJ. CJ.(F ) -0 ~ A - P -0' 8, "f (1)

(no sum on CJ.) where IFool denotes the determinant of the thermoelastic deformation gradient FO-in the reference state and 8 is the temperature. The nominal shear stress acting on slip system a (local stress resolved into the slip plane with unit normal nCJ. and into the slip direction va) is ~a and).a is the elastic stretch ratio in the slip-direction. The quantity pa, which we will refer to as the slip system backstress, is defined to be the change in the free energy density ¢ per unit of shear rearrangement "fa when 8 and F 0 are held fixed. The dependence of free energy ¢ on "f denotes dependence on the slipped state throughout the crystal and in general includes history dependence of slip. We may in principle carry these findings to the intermediate unstressed reference state without loss of generality. Clearly, slip system backstress results from stored energy due to slip.

Comparisons may be drawn between Rice's continuum slip framework and other continuum models based on internal variables. The developments of Chaboche (12) are representative of the latter class of models. Although these

297

approaches are very different in terms of viewpoint, they yield very similar results. In Rice's development, the backstresses emerge naturally as result of dislocation rearrangement and the dissipation inequality is straightforward.

Although Rice's framework is very general, a more detailed study of specific dislocation mechanisms requires specificity of the slip system hardening laws and volume averaging schemes. A recently proposed theory based on single slip is a most useful tool to explore details.

3. A SINGLE SLIP APPROACH: The single slip theory of Bammann and Aifantis (13-14) with later developments by Aifantis (15-17) lends itself to relatively straightforward connection of microstructural events with parameters in phenomenological equations. The approach may be viewed as a mixture theory. Using this framework, we will investigate rate-dependence and decomposition of internal stress from a more fundamental viewpoint, restricting consideration to isothermal behavior. Dislocation climb effects will be neglected.

The proper starting point, following Aifantis (17), is the behavior of a single slip system. Introducing the notion of dislocated (excited) and lattice (normal) states, we may write the mass and momentum balance equations for the dislocated state as

v • Ct = f -i -i

(2)

where Pi is the dislocation density corresponding to the ilh dislocation family or population, ,li is the corresponding dislocation mass flux and ci and !i represent effective mass and momentum exchange between dislocated and lattice states. Here, Pi denotes the effective mass density of dislocations as defined elsewhere (13-14) and differs only by a constant from the usual metallurgical definition. It is noted that the components of Cti pertain, in general, to periodic elastic dislocation interaction stresses at different size scales. Examples are dislocation interaction with grain and subgrain boundary pileups, superdislocations, cell walls, ladder and vein substructures, forest dislocations, etc. Typically, these interactions occur across a continuum of sub-grain size scales. We may draw direct analogy between 2:Cti and pct in Rice's development, considering the latter to correspond to the shear resolution of the former onto the ct slip system. The Pi may denote both immobile and mobile dislocation families. Function ci represents generation, immobilization or annihilation of the ilh dislocation species, and fi includes lattice friction, viscous drag, the Peach-Koehler force, etc. -

The backstress Cti reflects elastic dislocation interactions. We may decompose the appliedstress ~ into a lattice component and ~i as per

L a = a + 2: Ct. i -1

(3)

298

Defining the orientation tensor M in the usual way as M = (n~v + ven)/2 where v and n are unit vectors in the slip and slip plane normal directions, respectively, we may write the resolved shear stress as 1 = aL:M. For quasi-static equilibrium, V· a = O. Following Aifantis (17), we may adopt constitutive equations for dislocation glide of the form

a. = t . M + t . n3 n -1 m1 - n1 - -

(4)

(5)

for the ith dislocation family where l:1]; = 1, c; = ~;(Pk,jk,l]k1) and fc is the climb force. The dislocation flux in the slip direction is given by j; = L·~. The function F; of dislocation flux in Equation (5) represents nonlinear viscous drag force, which effectively includes overcoming of short range barriers via thermal activation. In (5), €; represents lattice resistance which arises from such sources as the Peierls force, solute atom interactions, etc. and l]j1 assists dislocation motion, i.e. Peach-Koehler force.

The representation of internal stress a j in Equation (4) warrants comment. The traceless component tmjM is augmented by a dyadic component in the slip plane normal direction. This component, which is not traceless, represents dislocation interactions along the slip plane with dislocations moving on intersecting slip systems, constraints of grain boundaries or substructures on slip, dislocation dipoles, etc. which are not accurately reflected by the traceless component. In fact, the influence of latent hardening (18) of secondary slip systems would introduced through this term. From the work of Weng (19), for example, it is known that both grain boundary constraint and latent hardening influence the modeling of deformation-induced yield surface anisotropy.

3.1 Size Scale Invariance: Neglecting the divergence terms in Equations (2) for the case of macroscopic behavior, we may adopt a form of evolution of Pj' i.e.

(6)

Also, f. = 0, where f. may now be viewed as a "smeared" or averaged dislocation _1 _ _1 th

resisting force for the i family. We must take care through the size scale transition to recognize that the values of coefficients may change to reflect the averaging process even though the same symbols will be used herein for both micro scale and macroscale.

3.2 Dissipation Inequality: As assumed by Aifantis, we may express the plastic strain as E P = l:-y.(pj,Q)M = M where Qj is the mean free path for the ith species and 'Y is the shea; for the (~,~ slip system. Differentiating, J?P = l:j.yj~ + L:j'Yj1!.

299

Dropping the 1M term as second order under the assumption of small strain and rotation leads to-E,P = 1M.

Identifying r-with r'fl in Equation (1) and assuming that only the "inelastic" part of the free energy depends on the shears (11), the specific entropy production rate for the single slip case under isothermal conditions is given by r1 ~ 0, which may be expressed as

(a - ~ a.) : EP ~ 0 - 1-1

(7)

using the preceding developments. Clearly, specification of cj and jj are constrained by the dissipation inequality.

3.3 Polycrystalline Behavior Based on Single Slip: For the generalization of M to the macroscopic polycrystalline case, we maximize the specific entropy production rate r1 for a given 1. Following Aifantis (17), we may solve a constrained optimization problem to maximize the entropy production rate and determine that r = I la'-2:a.' I l/zl/2 = (J ,)1/2 and M = 1/2(a'-2:a.')/(J ,)1/2 where a' and a' _ _I 2 _ _ _I 2 __

represent deviatoric quantities. Note that the same symbols are used at the microscopic and macroscopic levels, although the macroscopic interpretation of M is that of the directional index of inelastic strain rate. It should be noted that a rigorous connection between this maximization procedure based on single slip and the multislip case is not clear; statistical mechanics could be of some use in this respect.

Summing the relation fi = 0 over i leads to r = 2:Mi + FjG) for the glide component. If we assume2:FjU) = F(Aj), then r = If. + F(Aj) which may be inverted as

1 -1 j = - F (r-If.)

A

where If. assumes the role of a macroscopic friction or drag stress.

(8)

The assumption which led to Equation (8) is a very important one which is implicit in overstress theories of viscoplasticity. It is tantamount to assuming a unique macroscopic overstress versus dislocation flux relation. The product Aj represents the fraction of the total dislocation flux j associated with dislocation families overcoming rate-limiting short range barriers. If the total dislocation flux is associated with these particular families, or essentially so, and the ratio of mobile dislocation densities at long and short range barriers remains constant then A may be essentially constant provided the function F suitably reflects the dominant viscous drag mechanism(s).

Often, only one dislocation family is engaged in the rate limiting step with a dominant set of rate limiting short range obstacles so that the assumption which led to Equation (8) (for a given temperature) is not unreasonable.

300

Using the macroscopic generalization for 1 and M and the chain rule of differentiation, -

[2:A.(P.)c. i 1 1 1

(9)

where p="i.Pj, A(Pj)=a'Y/apj, B(p)=S'Y/se and e is the average mean free path of mobile dislocations. The second term corresponds to the flow rule of many unified creep-plasticity or state variable theories, and is also the flow rule adopted by Moosbrugger and McDowell (2,20-21) with p-l left unspecified. Drag stress If,

may depend on dislocation density of any or all families. The first term in (9) accounts for dislocation annihilation and generation

processes as well as transfer of dislocations from mobile to immobile families (and vice versa) associated with changes in straining direction, dislocation interactions, etc. This term is intimately related to transient dislocation rearrangement processes. The second term obviously expresses rate-dependence.

3.4 Hardening/Dynamic Recovery Backstress Evolution: We may formally differentiate Equation (4) to give

[ t. t. t 'J a. = ~ - ~ ~ EP + -1 • t.·-

'Y nl 'Y (10)

for the single slip case where Equation (4) and E,P = 1M have been used to eliminate M and n@n. Obviously, the dynamic recovery term (tn/tn;)ai is operative only when dislocation backstress normal to the slip plane may be relieved by mechanisms such as cross slip.

We next assume the simple evolution equations

i . = - h.(j)k.(p)t . 1 nl 1 1 nl

i . = k.(p)c.(p) 'Y m1 1 1

(11 )

For simplicity, we have made the evolution of tni and tmi dependent on total dislocation flux j and density p to reflect interactions of different families. Equation (11.2) is a general nonlinear hardening form for tmi. In writing Equation (11.1), we assume that the backstress component tnil1®n is recovering dynamically. One could include a direct hardening term in (11.1) with the proviso that the recovery rate dominates the hardening rate over the range of strain/time considered. Substituting into Equation (10),

a. = k.(p)[c.(p) + h.(j)T(p)] EP - h.(j)k.(p)a. 'Y -1 1 1 1 - 1 1 -1

(12)

301

where T(p) = f k;.(p)cj(p)(a-y/ap)dp; cj(p) is generally a decreasing function of p such that lim~., cj(p) = O. Such a condition on cj(p) also leads to decreasing rate­dependence of the saturation level of backstress eli with increasing p. It is noted that the dependence on dislocation flux appears in the dynamic recovery term as in the aforementioned approach of Estrin and Mecking (14). This is due to the admission of hi(j) in the evolution rule for tni which may be associated physically with rate-dependence of cross slip.

We may rewrite Equation (12) and use the preceding macroscopic generalization (section (iii» to write

a. = K.CP)HCiJ72'-K:)( [C. CP)H- 1CiJ72'-K:) + v.CP)]n - el. ) IIEYII (13) -1 1 J -2 1 J -2 1 - -1 -

where n = (a'_"i.el/)/2(J/)1/2. Dependence on dislocation flux has been replaced by overstress de-pendence by making use of Equation (8). Also, note that the functions ~(p), Ci(p), and Vi(p) are not identified directly with those in Equation (12) owing to the polycrystalline generalization.

The form is directly analogous to that recently proposed by Moosbrugger and McDowell (2,20-21), interpreted therein as a rate-dependent bounding surface theory with decomposed backstress. In their work, rate-sensitivity was apportioned between the function H·I and the flow rule. The saturation level of internal stress is dependent on the inelastic strain rate. The function C may not depend on p for materials which exhibit no change of rate sensitivity with isotropic hardening. Dislocation density dependence refers to isotropic hardening; yep) introduces isotropic hardening into the saturation level of backstress. The only substantial difference between the formulation in Refs. (2,20-21) and Equation (13) is the rate dependent coefficient ~(p) H which arises due to the rate-dependence of cross slip.

It is noted that rate-independent forms for O!i may be obtained by neglecting h(j) in Equation (11.1) which results in the Armstrong-Frederick hardening/dynamic recovery rule for O!j. Such forms, however, are inconsistent with the notion of thermally-activated cross slip.

4. CONCLUSIONS: The following conclusions are drawn:

1. A continuum slip framework leads to a succinct, physically appealing statement of the dissipation inequality when the free energy associated with inelastic processes is assumed to depend on the shears. 2. Decomposition of internal stress into components corresponding to different dislocation families has been carried out using the single slip framework proposed by Bammann and Aifantis. It is seen that the flow rule contains distinct terms reflecting dislocation generation/annihilation and the overstress versus dislocation flux relationship. Typical assumptions regarding the dominance of short range size scales in this latter relationship are invoked to write a relationship between total dislocation flux and overstress. Backstress evolution

302

equations for the various families have been derived which follow an Armstrong­Frederick form. Rate-dependence of thermally activated dislocation cross slip leads to rate-dependent backstress evolution in agreement with the materials science literature. The connection with recent, more phenomenological proposals for a rate-dependent bounding surface formulation is made.

ACKNOWLEDGEMENTS: D.L. McDowell would like to acknowledge the U. S. National Science Foundation (NSF PYIA MSM-8552714, NSF MSM-8601889) for support of this work.

REFERENCES:

1. Moosbrugger, J.e. and McDowell, D.L., "On a Class of Kinematic Hardening Rules for Nonproportional Cyclic Plasticity", ASME Journal of Engineering Materials and Technology. Vol. 111, 1989, pp. 87-98.

2. Moosbrugger, J.c., A Rate-Dependent Bounding Surface Model for Nonproportional Cyclic Viscoplasticity, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1988.

3. McDowell, D.L., "An Evaluation of Recent Developments in Hardening and Flow Rules for Nonproportional Cyclic Plasticity", ASME Journal of Applied Mechanics, Vol. 54, 1987, pp. 323-334.

4. Estrin, Y., and Mecking, H., "A Vnified Phenomenological Description of Work Hardening and Creep Based on One-Parameter Models", Acta Metallurgica, Vol. 32,1984, pp. 57-70.

5. Mecking, H., and Kocks, V.F., "Kinetics of Flow and Strain Hardening", Acta Metallurgica, Vol. 29, 1981, pp. 1865-1875.

6. Kocks, V.F., "Laws for Work-Hardening and Low-Temperature Creep", ASME Journal of Engineering Materials and Technology, Vol. 98, 1976, pp. 76-85.

7. Haasen, P., Philosophical Magazine, Vol. 3, 1958, p. 384. 8. Shoeck, G., and Seeger, A, Defects in Crystalline Solids, Physical Society,

London, 1955. 9. Follansbee, P.S. and Kocks, V.F., "A Constitutive Description of the

Deformation of Copper Based on the V se of the Mechanical Threshold Stress as an Internal State Variable", Acta Metallurgica, Vol. 36, No.1, 1988, pp.81-93.

10. Conrad, H., deMeester, B., Yin, C. and Doner, M., "Thermally Activated Deformation of Crystalline Solids," Rate Processes in Plastic Deformation of Metals, ASM, 1975, pp. 175-226.

11. Rice, J.R., "Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity", Journal of the Mechanics and Physics of Solids, Vol. 19, 1971, pp. 433-455.

12. Chaboche, J.L., Description Thermodynamique et Phenomologique de la Viscoplasticite Cyclique avec Endommagement, These, Vniversite Paris 6, 1978.

303

13. Bammann, D.J. and Aifantis, B.C., "On a Proposal for a Continuum with Microstructure," Acta Mechanica, 45,1982, pp. 91-121.

14. Bammann, D.J. and Aifantis, B.C., "On the Perfect Lattice-Dislocated State Interaction," Proc. International Symposium on Mechanical Behavior of Structured Media, Carleton Univeristy, Ontario, Canada, May, 1981.

15. Aifantis, B.e., "On the Microstructural Origin of Certain Inelastic Models," ASMB Journal of Engineering Materials and Technology. Vol. 106, 1984, p. 326.

16. Aifantis, B.C., "On the Structure of Single Slip and Its Implications for Inelasticity," Chapter 17 in Large Deformations of Solids, Eds. Gittus, Zarka and Nemat-Nasser, Elsevier, 1986, pp. 283-325.

17. Aifantis, E.C., "The Physics of Plastic Deformation," International Journal of Plasticity, Vol. 3, 1987, pp. 211-247.

18. Asaro, R.J., "Crystal Plasticity", ASME Journal of Applied Mechanics, Vol. 50, 1983, pp. 921-934.

19. Weng, G.J., "Anisotropic Hardening in Single Crystals and the Plasticity of Polycrystals," International Journal of Plasticity, Vol. 3,1987, pp. 315-339.

20. McDowell, D.L. and Moosbrugger, J.C., ''Bounding Surface Interpretation of Rate-Dependent Metallic Behavior under Nonproportional Loading," Proc. Int. Seminar on the Inelastic Behaviour of Solids Models and Utilisation, Besancon, France, Aug. 30 - Sept. I, 1988.

21. Moosbrugger, J.e. and McDowell, D.L., "A Rate-Dependent Bounding Surface Model with a Generalized Image Point for Cyclic Nonproportional Viscoplasticity," under revision for publication in the Journal of the Mechanics and Physics of Solids, 1989.

CONSTITUTIVE LAWS PERTAINING TO ELECTROPLASTICITY IN METALS

H. Conrad, W. D. Cao and A. F. Sprecher

Materials Science and Engineering Department, North Carolina State University, Raleigh, NC 27695-7907 USA

ABSTRACT: A constitutive equation is given for the effect of an electric current on the plastic strain rate in metals in the quasi-static regime. The equation includes an electron wind assisting the applied stress in helping dislocations overcome obstacles, and possible effects of the current on the obstacle strength, activation area and pre-exponential in the Arrhenius rate equation. Experimental results on the electroplastic effect in Cu were in reasonable accord with predicted behavior. Two effects of an electric current were identified: (a) an electron wind and (b) an effect on the obstacle strength and/or the activation area, and possibly the pre­exponential factor.

1. INTRODUCTION: It is generally recognized that the conduction electrons in a metal exert a drag on moving dislocations, which becomes especially significant at low temperatures and high dislocation velocities (v > 10 cm/s) [1,2]. Less well known is that upon application of an electric potential the drift of the conduction electrons exert a push on dislocations, which becomes significant at high current densities and lower dislocation velocities (v < 1 cm/s) [3J. This latter phenomenon, which was first reported by Troitskii in 1969 [4], is termed the electroplastic effect (EPE).

To avoid excessive Joule heating the EPE has mainly been studied by determining the increase in plastic strain rate which occurs upon application of high density (103 - 106 Ncm 2) electric current pulses of - 100 Ils duration during the plastic deformation of a metal in constant strain rate, creep and stress relaxation tests. To arrive at the contribution of the drift electron­dislocation interaction to the increase in strain rate, special attention was given to eliminating the side, or indirect, effects of the current pulses, e.g., Joule heating, pinch, magnetostrictive and skin effects [5-7].

The increase in plastic strain rate which results upon application of high-density electric current pulses during the plastic deformation of a number of metals is reviewed in [3]. The strain rate increased about three

305

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 305-31l. © 1990 Kluwer Academic Publishers.

306

orders of magnitude for each order of magnitude increase in current density j above a critical value of - 103 A/cm2. Support that the EPE reflects a drift electron-dislocation interaction is provided by its dependence on the polarity of the current [8-10].

This paper deals with the constitutive equations which apply to the EPE in polycrystalline Cu at low homologous temperatures (77 - 300K) and quasi-static strain rates (10-4 -100s-1), taking into account the nature of the drift electron-dislocation interactions which may occur.

2. THEORETICAL CONSIDERATIONS:

2.1 Electron Wind An electric current may influence the mobility of dislocations through

the electron wind force exerted by the drift of the conduction electrons. Theoretical treatments of the electron wind force are of two types [3]: (a) those based on considerations of specific dislocation resistivity [11-14] and (b) those based on Fermi momentum considerations [15-17]. The former yield for the electron wind force Few per unit dislocation length

(1 )

and the latter

(2)

where (Po/No) is the specific dislocation resistivity, Po the resistivity due to dislocations, No the dislocation density, e is the electron charge, ne the conduction electron density, j the current density, b the Burgers vector, PF the Fermi momentum and v the dislocation velocity. a. ... 0.25 - 1.0 depending on the details of the model and the Fermi surface. Nabarro [18] has pointed out that in a free-electron model Eqs. 1 and 2 are equivalent within a factor of the order of unity, since b PF "'1i/(poINo) '" 41i1nee2 and ve = jene, where 1i is Plancks constant and ve is the electron drift velocity. Additional parameters of interest which relate to Few are the electron wind force coefficient Kew (= Few/j) and the electron wind push coefficient Bew (= Few/ve). These are obtained directly from Eqs. 1 and 2 by making the appropriate substitutions.

The values of Kew and Bew derived for Cu using available values for the physical parameters in Eqs. 1 and 2 are given in Table 1. It is here seen that by taking a. = 1, Eq. 2 yields values for Kew and Bew which are similar in magnitude to those obtained from Eq. 1. However, worthy of note is that the value of Bew so derived is about an order of magnitude larger than the

generally accepted values for the electron drag coefficient Be = v/'C*b [1,2,19],

where 1.* (= 1. - 1.11 ) is the effective applied stress, 1. the total applied stress

and 't~ the long-range internal stress. The difference results mainly from the

value of the constant a in Eq. 2, being taken to be 0.1 for Be [2], compared to 1.0 for Bew [13, present].

2.2 Constitutive EQuations The plastic strain rate range in which EPE measurements have been

carried out to-date is within that where thermally-activated motion of dislocations applies. The effect of an electric current on the plastic strain rate E may then be given by [3]

{ -' r AG'-A''t'b]} (A~F ) NO,mbsv eXP1 kT . 2 cosh ¥

e./e= { J} J * * * - ' AG -A 't b

ND,mbsv expi kT ]

(3)

where ND m is the mobile dislocation density, s the average distance moved

per successful thermal fluctuation, v' the frequency of vibration of the dislocation segment I' involved in the activation event, AG' (= AH' - TAS') the Gibbs free energy of activation, A' = I'x' the activation area, x' the activation distance and kT its usual meaning. The subscript j refers to the values during application of the current; no subscript indicates values prior to application of the current. The hyperbolic cosine function reduces to an exponential for (A'lewlkT) > 1 and to (AjFe/kT)2/2 for (A'yFew/kT) < 1.

For comparison with experimental measurements, it is desirable to take the logarithm of both sides of Eq. 3 and rearrange to give for A'Few/kT > 1

[ ( ) (AG' -AG') (A; - A')'t'b] A~F

In(e /E) = In E o./E 0 - j kT + kT + ~;w (3a)

where Eo = ND,mbs v*. For the case A'j '" A', Eq. 3a reduces to . . . A

In(E /E) = aU) + kT F fNJ (4)

where a(j) includes ali terms within the brackets on the RHS of Eq. 3a. Hence, a plot of In (E/E) vs A'/kT should yield a straight line whose slope

gives the magnitude of Few' and the intercept the sum of the terms in the brackets.

307

308

3. EXPERIMENTAL:

3.1 Procedure To check the validity of the theoretical considerations presented above,

EPE studies were carried out on Cu at 17 and 278K. The test specimens were cut from 0.5 mm dia. polycrystalline AESAR 99.9% Cu wire purchased from Johnson Mattley Inc. and annealed at 600°C in vacuum to give a grain size of 18 11m and resistivity at 300K of 1.67 llQ-cm. High density (2.5 x 105

- 5.5 x 105 Ncm2 ) electric current pulses of 120 Ils duration were applied during tensile deformation of the wire specimens at a strain rate E = 1.7 x 10-4s-1 and the resulting drop in flow stress measured. At 298K the specimens were cooled by air from a laboratory blower; at 17K they were immersed in liquid nitrogen. The maximum temperature rise resulting from a current pulse was 5K. Details regarding the experimental procedure and analytical methods employed to determine the value of Ej of Eq. 3 from the drop in flow stress are given in Refs. 6 and 7.

The activation area A* was determined in the present tests by making cyclic 5:1 changes in strain rate during the tensile test and employing the relation

• a~G - . A b = - = M kT (a In ElaO")T at (5)

where M (= 3) is the Taylor orientation factor and 0" the applied tensile stress. A* decreased significantly with strain, in keeping with the intersection of forest dislocations as the rate controlling dislocation glide mechanism in Cu at low homologous temperatures.

3.2 Results Fig. 1 presents a semilog plot of EiE vs A*b as a function of the current

density for the tests at 17K. The resulting straight lines are in accord with Eq. 4, assuming A*Few/kT > 1. Similar behavior occurred at 298K. The values of Few derived from the slopes of the straight lines in Fig. 1 are plotted vs j in Fig. 2. Also included are the values for 298K. The straight lines through the origin are in keeping with Eqs. 1 and 2. Values of Kew and Bew derived therefrom are included in Table 1. It is here seen that the experimental values are in reasonable accord with theoretical predictions (especially at 17K), but show a stronger temperatures dependence than predicted. Also worthy of mention is that the present results at 300K are in good accord with those obtained previously [6] on another source of Cu and by another person.

309

The intercepts on the ordinate in Fig. 1 give the values of aU) in Eq. 4. Log-log plots of a vs j (Fig. 3) yield straight lines with a slope of ~ 2, giving

(6)

with n = 1.6 and ic,a = 1.8 x 103 Ncm2 at 77K and n = 2.2 and ie, a = 1.8 x 104

Ncm2 at 300K.

10'r--r---,..---r--r--r--r---,

la' '="0 --'---:---:---4.l..---!:---!::----,I

Fig. 1. S~mi1og plot of sj/s vs A"b as a function of current density for EPE tests on polycrystalline Cu at 77K.

10'

10'

10'

I

10' I I

I I

I I

I 10"

10' 10' 10' 10'

J (A/em')

0.5 r--,---,---,---,---..---,

04 ::--

z

Fig. 2.

Copper (99.9%)

4

Fe vs j for EPE tests onWpolycrystalline Cu at 77 and 300K.

Fig. 3. Log-log plot of the constant a vs j for EPE tests on polycrystalline Cu at 77 and 300K.

310

TABLE 1. Theoretical and experimental values for Kew and Bew for Cu. Units for Kew are 10-8

(dyn/cm)/(Alcm2) and for Beware 10-4 dyn-s/cm2.

Theoretical

K (1)_ ew - Bew (1)=

T(K) (poiNoHene) (po'NoHene)2

77 2.65 ± 0.35 3.53 ± 0.47 298 2.88 ± 0.35 3.83 ± 0.47

NOTES: (1) Based on Eqn. 1 of text. (2) Based on Eqn. 2 of text.

Experimental

Kew (2)= Bew (2) =

Kew= Bew= bpFne bpFne Fewli Kewene

2.22 2.96 2.9 3.9 2.22 2.96 5.4 7.2

poiND is from Refs. 20-22; ne = 8.33 x 1022 cm-3 is from Ref. 11; PF = (2 EFme)1/2, where EF =

h2/2me (3 112)2/3 ne2/3 with h = Plancks constant and me = electron rest mass.

By assuming A*Few/kT < 1, similar behavior was found as for A*Few/kT > 1; however the derived values for the various parameters were 10-30% larger.

4. DISCUSSION: The experimental data on the EPE in Cu are in reasonable accord with Eqs. 1-3 for the effects of an electric current on the plastic strain rate of metals. A discrepancy however exists in the temperature dependence of the electron wind force Few, the observed dependence being larger than predicted. This could result if the EPE does not simply reflect an electron-dislocation interaction, but includes an electron-phonon-dislocation interaction as well. Another possibility is that the quantity (A*j - A*) of 3a is non-zero, but increases with A*.

The effect of i on the parameter a U) resides in one or more of the terms in the brackets on the RHS of Eq. 3a. The fact that ic,a increases with temperature suggests that one or both of the terms which contain 1fT (i.e. L1G* or A*) may be affected by the current.

Finally, noteworthy is that the electron wind push coefficient Bew is about an order of magnitude larger than the generally accepted value for the election drag coefficient Be. This may result from the fact that the drag coefficient applies to dislocations moving at high speeds whereas the push coefficient derived here pertains to dislocations temporarily held up at obstacles.

Additional work is underway to resolve the questions raised above.

5. ACKNOWLEDGEMENT: The authors gratefully acknowledge support of this research by the U. S. Army Research Office under Grant No. DAAL03-86K-0015.

6. REFERENCES

1. J. M. Galligan et ai, Scripta Met 18 643-672 (1984). 2. V. I. Aishitz and V. L. Indenbom, in Dislocations in Solids, Chap. 34

ed by F. R. N. Nabarro, Eisevier Publishers, B. V. (1986) p. 43. 3. H. Conrad and A. F. Sprecher, ibid. Chap. 43 (1989) in print. 4. O. A. Troitskii, Zh. Eksp. Teor. Fiz. Pisma 10 18 (1969). 5. K. Okazaki, M. Kagawa and H. Conrad, Mat. Sci. Engr. 45109 (1980).

6. A. F. Sprecher, S. L. Mannan and H. Conrad, Acta Metall. 34 1145 (1986).

7. W. D. Cao, A. F. Sprecher and H. Conrad, "Measurement of the Electroplastic Effect in Nb", submitted to Jnl. Phys. E: Sci. Instr. (1988).

8. O. A. Troitskii, Prob. Prochnosti, NO.7 14 (1975). 9. L. B, Zuev, V. E. Gromov, V. F. Kurilov and L. I. Gurevich, Dokl. Akad.

Nauk SSSR 239 84 (1978). 10. Vu. I. Boiko, Va E. Geguzin and Yu. I. Klinchik, Zh. Eksp. Teor. Fiz.

Pisma 30168 (1979); 81 2175 (1981). 11. R. A. Brown, J. Phys. F: Metal Phys. 7 1269, 1283 (1977). 12. F. R. N. Nabarro, Theory of Dislocations, Clarendon Press, Oxford

(1967) p. 529. 13. A. M. Roshchupkin, V. E. Miloshenko and V. E. Kalinn, Fiz. Tverd. Tela

21 909 (1979). 14. H. Conrad in Ref. 3. 15. V. Va. Kravchenko, Zh. Eksp. Teor. Fiz. 51 1676 (1966). 16. M. I. Kagonov, V. Va Kravchenko and V. D. Natskik, Usp. Fiz. Nauk 111

655 (1973). 17. K. M. Klimov, G. O. Shnyrev and I. I. Mouikov, Dolk Akad. Nauk SSSR

219 (2) 323 (Nov. 1974). 18. F. R. N. Nabarro, private communication (1988). 19. A. V. Granato, Scripta Met. 18663 (1984). 20. Z. S. Bazinski, J. S. Dugdale and Howie, Phil. Mag. 8 1989 (1963). 21. V. F. Gantmakher and G. I. Kulesko, Sov. Phys. JETP 40 1158 (1975). 22. G. I. Kulesko, Sov. Phys. JETP 48 85 (1978).

311

PLASTIC DEFORMATION AND FRACTURE OF CONTINUOUSLY CAST 5083 ALUMINUM ALLOY INGOT

* ** *** *** T.TAKAAI ,A.DAITOH ,Y.NAKAMURA and Y.NAKAYAMA

* Facult.y of Engineering, Yamanashi Universit.y (Kofu, Japan) ** Kisaradzu National College of Technology (Kisaradzu, Japan) *** Graduate Student of Yamanashi University (Kofu, Japan)

ABSTRACT The experimental data relative to tensile properties and Charpy

impact fracture characteristics obtained in the present work pertains to two kinds of continuously cast 5083 aluminum alloy ingot specimens, which were sub,iected and not subjected to the homogenizing treatment. Thermally activated deformation in tensile test was also discussed mainly from view points of temperature and strain rate dependence of the tensile properties at temperatures ranging mainly from 77 to 300 K applying strain rate change methods.

Results indicate that slight differences between the homogenized and non-homogenized specimens are found and that thermal components of flow stress was not so sensitive to changes in process technology or structural varlations.

Activation energy for the deformation of the homogenized specimen showed slightly higher values as compared to that of the non­homogenized specimens and tending to be higher values for frequency factors during the thermally activated deformation process for the homogenized specimens than those of the non-homogenized specimens.

1. INTRODUCTION

In many factory and plants, continuous casting methods of aluminum alloy have been very widely applied as widespread production procedur'(' for various kinds of aluminum and aluminum alloy products and carried out extensively in world wide aluminum alloy production factories. However,various fundamental property including physical and mechanical properties of the continuously cast ingot slab have not been so systematically investigated.

The fracture and flow stress, elongation to fracture and Charpy impact fracture characteristics etc are very important mechanical properties that describe the response of the continuously cast ingot to various kinds of applied external stresses. In this study, these various mechanical properties thoroughly discussed with the object achieving an effective further workillg of the continuously cast ingot.

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A. S. Krausz et at. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 313-319. © 1990 Kluwer Academic Publishers.

314

2. EXPERIMENTAL SPECIMENS AND EXPERIMENTAL METHOD 2.1 Experimental specimens 2.1.1 Continuous cast of 5083 aluminum allQx

Experimental specimens applied in the present work were prepared and cut off from the continuously cast aluminum slab kindly presented by Fukaya plant of SKY Aluminum Co. Ltd., Japan. The presented ingot slab size was 400 mm in thickness and 1,300 mm in width. Specimens cut off positions are as shown in Fig. 1.

~---Dlr,ctlDn of

L /J ./ 1-------'"

Two kinds of specimens were chosen respectively from the continuous cast ingot slab which was subjected to homogenization treatment ( hereafter, this is referred to as specimen H ) and other continuous cast ingot slab which was as cast condition(hereafter, this is referred to as specimen NH ).

Fig.1 Continuous cast ingot slab and test speci men cut-off positions.

2.1.2 Homogenizing treatment of the continuously cast 5083 aluminum _alloy ingot

Homogenizing treatment of the continuous cast 5083 alloy ingot has been carried out in soaking pit at 793 K for 10 hours. While, Non­homogenized continuously cast specimens were chosen as continuously cast state.

2.2 Experimental methods

2.2.1 Tensile test Tensile tests were achieved applying an lnstron type tensile

testing apparatus and selectively changing the cross head speed. Accordingly, initial tensile straiy rates wer~4 s~-l:ect.ed in accordance with the test specimens as 1.9x10 to 1.9xlO (s ).

Tensile tests at cryogenic temperature conditions were carried out in a cryostatt vessel filled with liquefied nitrogen or with ethyl alcohol cooled by liquefied nitrogen. Thermally activated deformation mechanisms were discussed applying strain rate change method with the consideration of the temperature dependences of the various tensile properties.

2.2.2 Charpy impact test Charpy impact tests were carried out at test temperatures

ranging from 473 to 7i K applying an instrumented Charpy impact testing apparatus having a capacity of 10 kgf. Load-displacement,load­time and displacement-time curves were recorded or under consideration

315

respectively for all of the test specimens.

~. RESULTS AND DISCUSSION

3.1 Microstructures of the continuouslx_cafLt_!)083 alumiuum allQ.Y jB.@.t

The grain size of the specimens Hand NH are nearly in same order with each other. Apparent distinctions of the micro-structures, however, were observed between the specimens H and the specimens NH.

Respective microstructures of the specimens Hand NH were analyzed quantitatively using X-ray microanalyzer( hereafter, this is referred to as XMA ) in detail. According to the analyzed results by XMA, magnesium concentration within the crystal grains of each specimen H and NH were characterized by remarkable difference. In other words, magnesium concentration within the crystal grains of the specimen II were higher than those of the Nil specimens by about 2.5 times or more, probably due to variation of precipitation and solutioning of intermetallic compounds and magnesium into the crystal grains of the respective specimens Hand NH.

It was also found that magnesium enriched phases was found preferably in grain boundary regions of the crystal grains of the specimen NH. Mg 2Si system intermetallic compounds were found selectively in speClmen H.

3.2 Temperature dependence of tensile properties

NH

U1 U1

Temperature dependence of the tensile properties of the specimens and the specimens H 1S summarized in Fig.2. As shown in Fig.2,

400

Test temperature, K yield stress or 0.2 % proof 77 173 273 373 473 stress of the specimen NH was

~\J I I I 0<::'0 - Specllletl H ;}AlI -- SpeclJletl NH -

\ -••• Rolled piate

~\ - I -

higher by about 10 to 15% as compared to that of the specimen H and showing almost the same level value with that of the rolled plate of Lhe 5083 alloy.

, '-

~300 Vi ~ ~ ', .... _ Tenslie strength.

r-- ---_ :i} --~ a - - -~_

It is considered that the existence of the magnesium enriched phases are closely related to the raising of the 0.2 % proof stress of the specimens NH by reducing substantially the distance a dislocation moves before it is locked.

,'" -~-- -, _. -41-___ ~ O()L9X1G-'sec-' <II 't} <::''''-l.9X10-'sec-' ......... CI OClL9xlO-'sec-' ~

A

i~' J ~ ~-:.0a:;,.CI ~~e~_s~~~.

LO~i~$~- --l~~~

zoo

100 02 04 05 03 0.1

Homologous temperature' T ITm

Fig.2 Temperature dependences of tensile properties of the specimens NH and specimens H.

316

In contrast to the yield stress, the athermal stress of the specimens NH Nas, for example, about 216 MFa, while, that of the specimens H was about 225 MPa for plastic strain of 5 % and the temperature Tc at which the thermal part of the flow stress varies from its full value to nil was about 200 K for the specimens NH and slightly lower for the specimens H. The proof stress increased from about 140 MPa at lower strai n f'ates to about 155 MPa at 77 ~4 for the spe~ime!:!lH increasing the strain rate from about 1.9 x 10 to 1.9 x 10 (s ). The effect of strain rate on the yield stress and tensile strength, however, were not so remarkable except at cryogenic temperature conditions for the yield stress and at higher temperature conditions for the tensile strength.

3.3 :Lhermal ty_~ti vated deformation Thermall y act i vated deformation of the specimens NH and H has

been studied in view of effective stress, activation energy for deformation and activation volume , which is relative to the work done by external applied stress, at cryogenic t.emperatures ranging from 300 K to 77 K (1),(2),(3),(4).

The effective stress T * is shown in Fig.3 regarding with the llomologous temperature. As shown in Fig.3, no distinctive differences of the effective stress T* , given by To - TG (TA; applied stress, T(,; athermal stress ), Nere observed betNeen the specimen NH and the specimen ll. The temperature Tc increased slightly as the plastic strain increased for both the specimens NH and H corresponding to diminution of the athermal part. Slightly higher Tc values, however,

30

0.1

Test temperature . K 77 173 2 3 273

\

~ Plastic strain D· .. yielding 0 .. • 5%

02

() .. ·lD% () .. ·15% ~"'2D%

Homologous temperature 03

T/Tm Fig.3 Variations of the effective stress wit.h test

temperature for the specimens NH and H.

I,ere observed for the specimen H as compared to those of the specimen

317

NH with regard to the respective plastic strain probably due to the increase of the magnesium concentration by homogenizing treatment within the crystal grains. Activation energies for the tensile flow were obtained as a function test temperature applying the equations (1) and (2).

y = v . exp {- H (T *, T) /k T }

H(T*,T)=-kT2(alny/a Th· (3 T* /3 T)i-

(1)

(2 )

The activation energy concerned with the plastic deformation of the specimens NH and H were not suhstantially different each other. The activation energy for the specimen NH at the temperature Tc was about 0.25 eV and about 0.27 eV for the specimen H which was sUbjected to the homogenizing treatment. These values were rather very low as compared to those of commercial 1050 aluminum plate obtained by one of the present authors(5). The values v of the frequency of vibration of the dislocation segment involved in the thermal activation process, derived from the slopes of the plogs ap£iying _lquation (1), were variously estimated in range from 10 to 10 (sec ), tending to be higher for the specimen H than those of the specimen NH.

Activation volume which i.s relative to the work done by the external stress when a dislocation overcomes the thermal barrier is obtained applying the following equation (3) and are shown in Fig.4 with respect to the effective stress for tensile flow.

v * = k T ( () In Y /3 T)T (3)

As is shown in Fig.4, the activation volume for the specimen Hand NH are not so remarkably different each other and the values were rapidly increased below 5 MPa of the effective stress. These values, however, were rather lower as compared to those of obtained by Basinski et al(3) and one of the present authors(5) for the commercial pure aluminum specimens. The work done by the applied stress are represented as the product of the effective stress T * and aet i vation

'1l it'-

>. 1000 Q.o

E ::l -0 >

§ 100

iii >

OPlastic strain 5% f)Plastic strain 10% (tPlastic strain 15% .Plastic strain 20%

~ I

i ~ Specimen H Specimen Nil

\ , . ~~~ ~

-"Qo I °If)-~I I -f)r~-i

10 20 30 0 10 20 30 Effective shear stress r*', MPa

Fig.4 Effect of the effective stress on the activation volume for tensile flow of the specimens NIl and H.

318

volumev*/b 3 , that is, -(l',J'/b 3 • These values were also discussed for hoth the specimens NH and H regarding with the test temperatures.

According to Cottrell and Stokes(6), the ratio of the flow stress at two temperatures is almost independent of plastic strain after the first few percent of plastic deformation, that is,O'Tl/O'T2=K. This has been found to be approximately Lhe case for the continuously cast 5083 aluminum alloy ingot of the present experiment. The ratios 0'273/0'1n, (j"2<"3/G223were confirmed to be constant for the plastic strains of 2 to ] 0 %, the ratioG 273/ G,'1, however, was not the case and was decreased linearly with the increase of the plastic strain in the present work. The specimens NH showed always higher values of the ratio than those of the specimens H.

3.4 Charpy impact fractllre characteristics of continuous cast 5083 aluminum alloy ing~21

Charpy impact fracture characteristics of both the homogenized and non-homogenized specimens are summarized in Fig.5 with respect to the test temperature. As contrasted with the tensile flow properties, the absorbed energies of the specimen NH showed distinctive variations to the specimen H. The absorbed energies of the specimen NH were decreased linearly with the decrease of the test temperature and markedly embrittled below about 200 K as like as the specimens H which were work hardened by cold rolling.

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() / of 18% reduction)

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75

Homologous temperature T I Tm

Fig.5 Absorbed energy versus test temperature ['elation for the continuous cast ingot and rolled plate of the 5083 aluminum alloy.

Embrittlement of the specimen NH at cryogenic temperatur'e r'ange is considered that probably owed to the existence of the magnesium enriched phases in grain boundary regions. While, the marked depression of the absorbed ener'gies of the str'ain hardened specimen H is considered to be that the existence of the residual 8tre8s or strain field in the neighborhood of the intermetalJic compounds and grain boundary regions facilitat.es the propagation of the crack during

319

fracture by Charpy impact testing(7}.

4. CONCLUSIONS

From the present studies, the following conclusions were obtained; (1) Yield stresses of the non-homogenized, as cast state specimen were higher by about 10 to 15 % as compared to those of the homogenized specimens and showing almost the same level value with the rolled plate of the 5083 alloy. (2) No distinctive differences of the effective stress were observed between the specimens NH and Specimens H. (3) The athermal stress of the specimens H was higher by about 5 % as compared to that of the specimens NH for the plastic strain of 5 to 15 % and the temperature Tc was increased slightly with the increase of the plastic strain for both the specimens NH and H corresponding to the diminution of the athermal part of the flow stress. (4) The activation volume for the specimens NH and II are not so remarkably different each other and the values of the respective specimens are lower as compared to those of the values obtained for the commercial pure aluminum plate. (5) The absorbed energies for the specimens NH and H in Charpy impact fracture were markedly different each other. The absorbed energies of the non-homogenized specimens NH were decreased linearly with the decrease of the test temperature for the test temperatures from 473 to 77 K. (6) The ratios of flow stress at two temperatures were constant in case of the continuously cast ingot for the plastic deformation of 2 to 10 %, however, was not the case. The specimens NH values than those of the specimens H.

REFERENCES

1. H.Conrad, Mater. Sci. Eng., vol.6, (1970) 265. 2. A.Seeger, Phils. Mag., vol.l, (1956) 651.

(j273/(jj73, (j273/J223

5083 aluminum alloy the ratio J 27,/ (j77 ,

showed always higher

3. Z.S. Basinski, Acta Metall., vol.5, (1957) 684. 4. H.Conrad and H.Weidersich, Acta Metall., vol.B, (1960) 128. 5. T.Takaai,T.Tatsumi and T.Tamura, Proceedings of ICSMA(7) ,Montreal ,

Canada, 12-16 August, (1985). 6. A.H.Cottrell and R.J. Stokes, Proc. Roy. Soc., 233A (1955) 17. 7. K.Welpman et aI, Z.Metallkde., Bd.71, (1980) 7.

I"UlELLIN3 l1-E IIIFllEJ\[E £F MHNJSTRt..J::ll.ROL I!ll-£MI:E\EITY []\I Hla-t ~n.R:: IEFl:R'¥\TI(J\I ?'N) FRPCn.R::

D. S. Wilkinsc:n Depa~tment of Mate~ials Science and Enginee~ing

McMaste~ Unive~sity

Hamilton, Onta~io, Canada

AEISlRPCT: The effect of mic~ost~uctu~al inhomogeneity on mechanical behaviou~ is explo~ed th~ough the use of two examples. In the fi~st, the use self-consistent analysis is used to dete~mine the of a non-unifo~m

g~ain size dist~ibution on the c~eep ~esistance of a mate~ial exhibiting both dislocation and di ffusion c~eep. In the second, models fo~ the effect of an inhomogeneous dist~ibution of g~ain bounda~y c~eep cavi ties, and fo~ the effect of continuous cavity nucleation and coalescence a~e ~eviewed. Finally, the use of Di~ichlet tessellation methods fo~ cha~acte~izing

mic~ost~uctu~al inhomogeneity is p~esented b~iefly.

1. INTRIJIl..I::TI(J\I: All enginee~ing mate~ials a~e mic~ost~uctu~ally

inhomogeneous to some deg~ee. It is the~efo~e ~easonable to inqui~e into the extent to which non-unifo~mity affects mechanical p~ope~ties. In this pape~, the effect on both defo~mation and f~actu~e behaviou~ at elevated tempe~atu~es will be discussed. This is clea~ly a ~athe~ b~oad topic, and an exhaustive ~eview is beyond the scope of this pape~. Instead, a numbe~ of selected examples will be p~esented, in an attempt to show that mic~ost~uctu~al inhomogeneity does indeed playa significant ~ole in the mechanical behaviou~ of mate~ials ope~ating unde~ c~eep conditions. Mo~eove~, it will become clea~ that methods do exist o~ can be developed which enable the effect of inhomogeneity to be inco~po~ated into constitutive equations fo~ flow and f~actu~e.

Defo~mation and f~actu~e depend ~athe~ di ffe~entl y on inhomogeneity. Defo~mation is a coope~ative p~ocess to which all pa~ts of a st~uctu~e

cont~ibute. Fo~ this ~eason it is often ~easonable to ~elate the ~esistance to defo~mation to global mic~ost~ctu~al p~ope~ties such as ave~age g~ain size o~ pa~ticle volume f~action. F~actu~e on the othe~ hand, need involve only a small pa~t of the st~uctu~e. It is the~efo~e much mo~e sensitive to ext~eme values in the cont~olling mic~ost~uctu~al pa~amete~s. It is only because the ext~eme values of a dist~ibution tend to scale as the mean that some success has been achieved in co~~elating

321

A. S. Krausz et al. (eds.), ConstitUJive Laws of Plastic Deformation and Fracture, 321-331. © 1990 Kluwer Academic Publishers.

322 2

f~actu~e behaviou~ with global pa~amete~s.

2. IlEFl:R'1ATICN: The c~eep ~esistance of hete~ogeneous mate~ials has been t~eated by a numbe~ of di ffe~ent methods. Fo~ example, bound theo~ems [see e.g., Chen and A~gon, 1979] have been used to estimate the limits on c~eep behaviou~. This type of solution wo~ks well when the two (o~ mo~e) phases which make up the mic~ost~uctu~e have simila~ c~eep ~esistance. Howeve~ this is ~a~ely the case. Thus the ~ange between uppe~ and lowe~

bounds is usua 11 y too la~ge to p~oduce useful ~esul ts. Pnothe~ , mo~e

accu~ate app~oach is based on self-consistent modelling. The application of this method to c~eep has been discussed in some detail by Chen and A~gon [1979]. It d~aws on the classical analysis of inclusions fi~st

developed by Eshelby [1957]. In self-consistent analysis a set of p~oblems is solved fo~ the ~esponse of an inclusion of one the phases embedded in a continuum mat~ix with the ave~age p~ope~ties of the composite being studied. This is done in tu~ fo~ each phase in the system. The p~ope~ties of the continuum which yield a sel f-consistent analysis of the system a~e the~eby obtained. This type of analysis assumes that scale of the composite is sufficient that continuum p~ope~ties can be used to desc~ibe the ~esponse of each phase. Chen and A~gon have analyzed the ~esponse of a hete~ogeneous two-phase solid in which each phase exhibits powe~-law c~eep behaviou~ but with diffe~ent pa~amete~s. It is also of inte~est to study the p~ope~ties of a single phase alloy with a dist~ibution in g~ain size, defo~ming by both powe~-law c~eep and by diffusion c~eep. Powe~-law (o~ dislocation) c~eep is, to a ~easonable app~oximation, independent of g~ain size. Oiffusion c~eep cont~olled by g~ain bounda~y c~eep (which is most applicable at the lowe~ homologous tempe~atu~es associated with se~vice and also with supe~plastic fo~ming) depends on g~ain size as 0-3• A simplified constitutive equation fo~ such a mate~ial

e = F sn + G(D) s

whe~e e is the st~ain ~ate and s the applied st~ess, while n, F and G a~e mate~ials pa~amete~s, G being g~ain size dependent. Apa~t f~om this change in consti tutive ~elationship, the t~eatment now follows that of Chen and A~gon [1979]. The p~oblem is t~eated as one in non-linea~

elasticity using the analogy between elasticity and c~eep in which st~ains a~e ~eplaced by st~ain ~ates. We t~eat the simplest p~oblem only in which the mate~ial consists of g~ains of only two sizes - 01 and 02. The volume f~action occupied by each g~ain size is fl and f2' ~espectively. In each phase it follows that

de/dsj = F n stl + G(OI)

We wish to dete~mine the ove~all inc~ement of st~ain ~ate de, in ~esponse to an inc~ement of st~ess ds. Fo~ an isot~opic the ~esult is

de = ds / 3M

whe~e M is the c~eep shea~ modulus (o~ viscosity). This same ~elationship applies to each individual g~ain size, but with M ~eplaced by MI and ~

323

for each phase. Using the self-consistent scheme and the Eshelby solutions we arrive at a final set of solutions, equivalent to equations (15) and (16) in Chen and Argon [1979J. If these equations are now solved for the case in which the volume fraction of small grains (of size D1) dominate, then the following solution is obtained for the creep resistance:

1 { Mz - (3-50-(2-50 - ± 6 Ml

This calculation indicates how the strain rate changes in a material with bi-modal grain size during the transi tion from

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power-law creep to di ffusion creep. Prev ious a t tempts to analyze such behaviour have been based on the bound developed by assuming equal strains in both types of grains [Ghosh and Raj, 1981 J • Using that assumption a sigmoida 1 shaped s

Figure 1 The strain rate sensitivity as a function of stress for different values of grain size ratio and volume fraction f of the smaller grains.

- e curve is found in the transition regime. This has been used to explain the region of high rate sensitivity found during superplastic flow. However, using this more accurate method of analysis, for all reasonable values of grain size and volume, no significant sigmoidal behaviour is observed (see Fig. 1). For example, a modestly sigmoidal curve is achieved if a large volume (301.) of large grains (lOx larger than the matrix grain size). This suggests that grain size distribution is not the cause of superplastic behaviour as suggested by Ghosh and Raj.

3. FRPCTl.I1E: In brittle materials, fracture is an extreme value problem. Failure occurs when the stress on the largest flaw exceeds the critical value for catastrophic crack propagation. The strength in such materials is determined by resistance to flaw propagation (the toughness), and by the distribution of flaw sizes. The role of microstructural inhomogeneity in this case is very clear, and has been modelled extensively using the techniques of fracture mechanics and Weibull analysis. For this reason we will not deal with this problem here. Of more interest is the process whereby fracture occurs as a result of gradual damage accumulation in a structure leading to the development of a critical flaw. This process is also affected by inhomogeneities, but in a more subtle way than for fast

324

fracture from pre-existing flaws.

The dominant mode of failure in structural materials at elevated temperatures is grain boundary cavitation. In this process, grain boundary cavities nucleate and grow, and eventually coalesce to form a microcrack. This may then propagate by means of enhanced cavitation in the field of high stress surrounding the crack. It may also link up with other growing microcracks. Eventually a flaw of sufficient size for fast fracture results. It is well known that cavities are inhomogeneously distributed on grain boundaries. Some boundaries cavitate readily while others are more resistant. This may be due differences in local chemistry or differences in structure with grain boundary orientation. Along a given boundary facet the distribution of cavities is also non-uniform. Moreover, the density of cavities tends to increase with time under load, at a rate which is usually proportional to the strain rate. This is generally thought to be due to continuous nucleation, al though it may also be due in part to a distribution in growth rate for cavities below the size at which they become visible (about 0.2 !-1m). Reliable models of creep cavitation capable of predicting lifetime must include these features. That is, both the effect of a spatial and a temporal distribution in cavity density need to be modelled.

3.1 Cavity Growth Rates: The problem of cavitation occurring on only a fraction of available grain boundaries has modelled extensively (Dyson, 1976, 1979; Rice, 1981; Tvergaard, 1984; Anderson and Rice, 1985). In this process, load is shed from those boundaries which cavitate to those which do not. Cavity growth is thus "constrained" by the rate of deformation in the non-cavitating regions of the material. As shown by Riedel [1985), the cavitated facet in this case behaves mechanically like a crack. Moreover the model distinguishes only two types of grain boundary - cavitating and non-cavitating. In reality, all boundaries cavitate, but to varying degrees. An alternative approach is to treat the case of a grain boundary plane on which the distribution of cavities is non-uniform. This plane can be thought of as a single facet or as many connected facets running perpendicular to the tensile axis. The growth of cavities with different distributions can then be studied. This has been done [Wilkinson, 1988) for several different distributions of cavities as shown in Fig. 2. The highly clustered distribution represents the case in which a wide discrepancy exists between cavitating and non­cavitating regions, comparable to that analyzed in the constrained growth models of the Dyson-type.

The rate of cavity growth in such an inhomogeneous distribution can be modelled by adapting models previously developed for uniform arrays of cavities (e.g. see Cocks and Ashby [1982)). The details are presented elsewhere [Wilkinson, 1988). The main requirement is to determine from the local environment, the area of grain boundary associated with each cavity. This is done using the Dirichlet tessellation, which ascribes to each cavity, the region of grain boundary which is closer to that cavity than to any other. The cavities will naturally grow at different rates. However, they are coupled by the need to maintain a uni form rate of separation between the two crystals above and below the grain boundary. This is accomplished through a redistribution of load across the boundary

325

Figure 2: Four sets of point patterns and their associated Dirichlet tessellations. They range from random to highly clustered. The method used to generate these patterns is described elsewhere [Burger, 1986; Wilkinson, 1988].

subject to the condition of mechanical equilibrium. It is possible to develop analytical solutions for the growth rate of each cavity in an array for void growth controlled by grain boundary diffusion, surface diffusion and power-law creep. The global void growth rate can also be calculated, defined as the rate of increase of total cavitated area fraction on the grain boundary. This has been done for a series of cavity distributions for the case of small voids all of the same size. The distributions used are illustrated in Fig. 2.

The ratio of global void growth rates for these distributions to that of a uniform void distribution at the same global area fraction have also been calculated. The resul ts show a large decrease in global void growth rate with increased clustering of the void growth distribution, such that for the most clustered distribution shown in Fig. 2, the global void

326

gn:llillth rates are smaller than those for a uniform distribution by a factor of 3 to 5, depending on the void growth mechanism. Grain boundary diffusion controlled void growth is the mechanism most sensitive to void clustering, as might be expected.

3.2 Continuous Cavity flUcleatic:n and Coalescence: We now turn our attention to the effect of cavity nucleation and coalescence. Both of these processes change the densi ty of voids on a grain boundary as a function of time. In consequence, the spacing between voids changes and so does the rate of void growth. It is therefore clear that the processes of cavity nucleation, growth and coalescence are interdependent.

The three processes can only be treated in a model based on an arbitrary spatial void distribution by means of a simulation model. Such a model would introduce an initial arbitrary void distribution. Voids would then be nuc leated in either a random or pre-determined pattern. Stress redistribution would be calculated according to the scheme outlined in the previous section. The appropriate mechanism for void growth would be chosen for each void separately. Coalescence would follow as a natural consequence of void growth as nearby voids impinge upon one another. Suc h a simulation is possible, al though it would be complex. It has not yet been attempted. There have been some early attempts by Fariborz et a1. [1985, 1986). However, their work is one dimensional in nature, and restricts all voids to grow by a

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10-3 n = 5 g b d

10-4 L_ ________ '---=-________ '---=-________ -'-____ -'-----' 10-4 10-3 10-2 01 0.5

Figure 3: The cavity area densi ty and void growth rate plotted as a function of the void area fraction. Both functions are normalized bY. their values at a cavitated area fraction of 10-4s-1•

327

grain boundary controlled mechanism.

An alternative approach [Wilkinson, 1987a,b] is to consider the effect of a random void distribution (as opposed to uniform). Because of the presence of a distribution of void spacing voids will coalesce continuously. Moreover, continuous void nucleation can be superimposed on the problem in a simple manner. We therefore start by considering a grain boundary containing a Poisson distribution of voids of denSity r . We use the simple empirical equation proposed by Dyson [1983J for the° rate of void nucleation proportional to strain rate. Thus

r=r+Be n 0

gives the density of cavity sites. A simple geometrical calculation shows that if coalescence occurs when grain boundary impingement occurs, and if all cavi ties are assumed to grow at the same rate then a simple relationship for the cavity density results, namely

r = B e - 2 f (B e + r ) o

50

40

n 30 ~, 20

10

28

--- Experimental ---- BDC mode 1

15 ._-- SON mode 1

m

10

SD model --------1

8D model --------l O~~--~--~--~~~~

600 BOO T (0 C) 1000 1200

Figure 4: The experimental I y determined stress exponent for transverse creep in MA754 is compared the predictions of various models.

Note that r is the actual density of cavities present after coalescence and is therefore in general less than r n, the density of cavity sites. The first term in this equation represents the rate of cavity nuc leation; whi Ie the second and third represent coalescence due to the deformation-nucleated and the initially present cavities respectively. This equation needs to be solved simultaneously with a mechanistic void growth rate equation. When this is done a modified void growth equation which inc ludes nuc leation and coalescence effects is found. This has been done numerically for several different void growth mechanisms [Wilkinson, 1987b]. For example, Fig. 3 shows the void growth rate and the void density as a function of void area fraction for grain boundary diffusion controlled void growth. It is clear that as the stress increases increased deformation leads to a greater dominance of cavity nucleation. At low stresses very little nucleation takes place during the life of the material. Instead, the cavi ty density is dominated by the coalescence of pre-existing cavities. At an intermediate stress (s/so=3), nucleation and coalescence are balanced and r remains approximately constant. Thus, in considering Fig. 3b, the effect of a time-dependent void densi ty can be seen by comparing the curves with that for s/so=3. It is clear that both nucleation and coalescence

change the void growth rates substantially. As a result one would expect

328

Table I Effect of Void Distrirutien en the Global Void GrOlllth Rate

Cluster G.b. Diffusion cre:? Surface Diffusion parameter f G=10-4 f -10-3 f -10-4 f -10-3

6- fG=10 G- G----------------------------------------------------------------

random 0.756 0.747 0.857 0.828 0.828 0.719 0.709 0.834 O.SOl 0.800

0.7 0.602 0.589 0.755 0.709 0.709 0.601 0.589 0.746 0.703 0.702

0.5 0.436 0.422 0.606 0.514 0.514 0.399 0.385 0.570 0.551 0.551

0.3 0.208 0.197 0.337 0.321 0.321 0.220 0.209 0.372 0.294 0.294

---------------------------------------------------------------

the time to fai lure to be al tered. Indeed, since coalescence and nucleation depend in different way on stress and strain rate than do void growth even the stress and temperature dependence of the failure time will be affected. This is best seen by considering the limiting cases in which the void density is controlled either by nucleation events or by coalescence. In these cases, an analytical solution for the time to failure can be obtained. l'1ixed stress and temperature dependencies do indeed result as illustrated in Table I. Both the stress exponent and the activation energy for time to failure are predicted to be weighted averages of those for creep and either grain boundary or surface diffusion (depending on the void growth mechanism). The predicted stress exponent is evaluated for different n values in the Table. It is clear that the stress exponent cannot be used in most cases to determine the mechanism responsible for failure.

We have attempted nonetheless to compare the predictions of the models with a range of experimental data - with some success. For example, data on the creep fracture of pure metals containing implanted bubbles compare favourably with the models for coalescence dominated fracture [Wilkinson, 1987bJ. l'1ore conclusively, we have compared the measured stress exponent for creep fracture in 1'1A754 tested transverse to the rolling direction, with the models. This is shown in Fig. 4.

The striking feature of this data is the large value of the exponents. The Creep exponent ranges from about 45 to 18 with increasing temperature. That for creep fracture ranges from about 17 to 7 over the same range. These exponents are clearly too small to be explained as Monkman-Grant behaviour, which requires the same exponent for creep and fracture. They are much too large for the simple models of cavity growth controlled

329

failure which are all in the range of 1 to 3. Two of the models based on time-dependent void density do fit the data very well, as shown in Fig. 4. These assume, in one case, grain boundary diffusion controlled cavity growth with rapid nucleation leading to coalescence-dominated void densi ty; and in the other case, surface diffusion controlled cavity growth, with continuous void nucleation. Further work, involving detailed microstructural studies, is required to determine the correct mechanism. However, it is c lear that only a model of the type developed here is capable of explaining the measured stress exponents.

The results indicate clearly that void growth is constrained due to stress redistribution. Moreover, there is a wide range in the growth rate of individual voids due to their local environment, with more isolated voids growing more quickly as load is shed onto them. It is quite likely in fact that for the most isolated voids the increase in stress wi 11 be sufficient to cause a change of mechanism, from grain boundary diffusion to power-law creep for example. This is consistent with previous models for constrained growth in which non-cavitating regions are assumed to deform by power-law creep. The advantage of the present model is that it allows one to calculate the rate of damage accumulation in the stress­bearing regions.

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DISTRIBUTION • RANDOM • CLUSTERED-O.3 • CLUSTERED-O.5 • CLUSTERED-O.7

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o 0.5 I. CLUSTERING INTERACTION DISTANCE

(NORMAL! ZED)

Figure 5 An analysis of clustering in a series of generated point patterns which varying degrees of imposed clustering.

4. AS1:ES9t::NT (F MI~ II\I-l:MIE\EITY: In applying models related to the effect of microstructural inhomogeneity on deformation and fracture it is necessary to have tools available for quantitative assessing microstructures. Such tools have been developed in recent years, at least insofar as the analysis of point patterns generated from metallographic sections is concerned. Methods developed trus far have

330

been based either- on distance appr-oaches (in which the distr-ibution of near-est neighbour- distances is analyzed), or- on cell methods (in which the section is divided into a space filling ar-r-ay of cells, one per- featur-e). One of the most attr-active methods is based on the Dir-ichlet tessellation, intr-oduced in the pr-evious section. We have analyzed the applicability of this method to the char-acter-ization of distr-ibutions by compar-ing tessellations constr-ucted for-m a pseudo-r-andom point patter-n compar-ed wi th mor-e per-iodic and mor-e cluster-ed (see Fig. 2) patter-ns [Bur-ger-, 1986; Bur-ger-, 1988]. The sensitivity of var-ious statistical par-ameter-s to the degr-ee of c luster-ing or- or-der-ing has been measur-ed and the r-esul ts ar-e summar-ized in Table II. This shows that, for- example, the standar-d

1. i) ii)

2. i) ii)

3. i) ii)

4. i) ii)

5. i) ii)

Table II

Sensitivity of Tessellation Parameters to the Degree of Periodicity or Clustering in a Distribution

NEAREST-NEIGHBOUR DISTANCE aver-age value standar-d deviation NEAR-NE I GHBOUR D I STANCE aver-age value standar-d deviation I\LJMBER OF SIDES PER CELL aver-age value standar-d deviation CELL AREA aver-age value standar-d deviation CELL ASPECT RATIO aver-age value standar-d deviation

RELATIVE SENSITIVITY TO PERIODICITY CLUSTERING

WEAK WEAK WEAK WEAK

VERY WEAK VERY WEAK STRONG WEP¥,

NO SIGNIFICANT DEPENDENCE WEAK MI LDL Y STRONG

NO DEPENDENCE WEAK STRONG

WEAK STRONG WEAK STRONG

deviation of the Dir-ichlet cell ar-ea is str-ongly sensitive to the degr-ee of cluster-ing, while other- par-ameter-s, such as the standar-d deviation of the near--neighbour- spacings is mor-e sensitive to the degr-ee of or-der-ing in mor-e per-iodic str-uctur-es. It should be noted that these par-ameter-s do not fully descr-ibe the degr-ee of inhomogeneity in the micr-ostr-uctur-e. For­example, they tell nothing about the scale on which or-der-ing occur-s, which is of impor-tance in consider-ing the r-elationship between par-ticle cluster-ing and mechanical pr-oper-ties. Infor-mation of this type can be obtained however-, by extending the analysis. We have used the near-est­neighbour- distr-ibution to this. The fr-action of par-ticles which r-eside wi thin an ar-bi tr-ar-y distance of another- par-tic Ie (called the cluster­inter-action distance) nor-malized by the aver-age near-est neighbour- spacing, is plotted as a function of that distance in Fig. 5. This indicates that the number- of particles residing in clusters increases mor-e rapidly for

331

a cluste~ed dist~ibution than fo~ a ~andom point patte~n. Mb~eove~, the inte~action distance ~equi~ed to p~oduce significant cluste~ing (say 501. of pa~ticles in cluste~s) can be dete~mined.

Techniques such as these can be used, in combination with quantitative mic~oscopy, to evaluate both the deg~ee of inhomogeneity in the mic~ost~uctu~e of enginee~ing mate~ials, and the inhomogeneous dist~ibution of damage in c~ept mate~ials. By using these techniques it ought to be possible to test and the~eby ~efine a numbe~ of theo~ies such as those discussed above.

5. SUMMARY: This pape~ has attempted to give some flavou~ of the kind of theo~etical app~oaches which can be used to add~ess the effect of mic~ost~uctu~al inhomogeneity on mechanical p~ope~ties. In pa~ticula~, the use of self-consistent analysis to study the c~eep of mate~ials with non-unifo~m g~ain size has been p~esented. In addition, models fo~ the effect of a tempo~ally and spatially inhomogeneous density of g~ain

bounda~y c~eep cavities have been ~eviewed. Finally, methods fo~

dete~mining the deg~ee of inhomogeneity in ~eal mic~ost~uctu~es have been p~esented.

Ande~son. P. M. and Rice, J. R. (1985), 33, 409. Bu~ge~, G. (1986), Ph.D. thesis, McMaste~ Unive~sity, Canada. Bu~ge~, G., Koken, K., Wilkinson, D. S. and Embu~y, J. D. (1988), in Advances in Phase T~ansitions (J. D. Embu~y and G. R. Pu~dy, eds.; Pe~gamon P~ess).

Chen, I. W. and A~gon, A. S. (1979), Acta Metall., 27, 785. Cocks, A. C. F. and Ashby, M. F. (1982), P~og. Mate~. Sci., 27, 189. Dyson, B. F. (1976), l"Ietal Sci. J., 10, 349. Dyson, B. F. (1979), Can. Metall. Qua~t., 18,31. Dyson, B. F. (1983), Sc~ipta Metall., 17, 31. Eshelby, J. D. (1975), P~oc. Roy. Soc., A241, 376. Fa~ibo~z, S. J., Ha~low, D. J. and Delph, T. J., (1985), Acta Metall., 33, 1. Fa~ibo~z, S. J., Hadow, D. J. and Delph, T. J., (1986), Acta Metall., 34, 1433. Ghosh, A. K. and Raj. R. (1981), Acta l"letall., 2'3, 607. Rice, J. R. (1981), Acta Metall., 2'3, 675. Riedel, H. (1985), Zeit. Metallk., 76, 669. Tve~gaa~d, V. (1984), J.I'1ech. Phys. Sol., 32, 373. Wilkinson, D. S. (1987a), Acta Metall., 35, 1251. Wilkinson, D. S. (1987b), Acta Metall., 35, 2791. Wilkinson, D. S. (1988) Acta Metall., 36, 2055.

CONSTITUTIVE EQUATIONS FOR STRENGTH AND FAILURE AT ELEVATED

TEMPERATURES AND STRAIN RATES IN AUSTENITIC STAINLESS STEELS

N.D. Ryan and H.J.McQueen Mechanical Engineering, Concordia University

Montreal, Canada H3G 1M8

ABSTRACT: Constitutive equations for strength and for ductility of austenit~c stainles~lsteels were derived from torsion data between 900 and 1200 C (0.1-5 s ).

1. INTRODUCTION AND TECHNIQUES: The hot workabil ity of austenitic stainless steels has been reviewed including the effects of alloying and impurity elements (1-4). The present paper will give particular emphasis to the strength and fracture constitutive equations determined for four good quality alloys, 301, 304, 316, and 317, in both the continuous-cast (C) and homogenized worked (W) conditions (Table 1); their behavior in various aspects has been reported earlier (3-21). The torsion testing (5'f,8,11,13,16) was carried out in the range 900-12000 C and 0.1 to 5 s- with control and data acquisition by microcomputer; the data was transformed to equivalent stress and strain in the normal manner (3,5,6,11,13,16).

The flow curves exhibited the stress peak, flow softening and steady state regime (5-20) characteristic of dynamic recrystallization (DIDO (1-19,23-25). Optical microscopy revealed that DRX commenced shortly before the peak and occurred repeatedly maintaining almost equiaxed grains to very high strains (8-11,14,16,18,20). Electron microscopy of thin foils showed that the DRX grains contained a dynamically recovered (DRV) substructure similar to that present before the peak (9-11,14,16,26). The sizes of both the DRV subgrains and the DRX grains were inversely related to the peak stress and to the logarithm of Z, a temperature (T) compensated strain rate i: (9-11,14,16,26). The worked alloys exhibited true fracture strains from 4 to 18, the mechanism being intergranular fissuration impeded by DRX and DRV (1-6,8,11,13,16-18,20,23,25,27). For 301W, 304W, 316Wand 317W, the strength and its Z dependence increased, but the grain and subgrain sizes and ductility decreased with rising solute in the order given (6-20). For the cast alloys, the peak stress and strain and the ductility were strongly dependent on the density and distribution of a-ferrite which induced strain concentrations in the a-phase leading to enhanced DRX nucleation but also to accelerated interphase cracking (3,5,6,18,19,20).

This paper will examine the constitutive equations relating to

333

A. S. Krausz et at. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 333-340. © 1990 Kluwer Academic Publishers.

334

TABLE 1: COMPOSITIONS AND SUMMARY OF CONSTITlTI'IVE PARAMETERS

Alloy Commposltlon X Met. Solute X Q kJ/mol nT', K C Cr Me Nl expo mean expo mean· expo mean Mean

301W .110 17.120.20 7.92 27.24 26.29

304C .069 18.31 0.08 8.68 29.51 304W .062 18.28 0.28 8.27 29.02 30.30

316C .017 16.922.76 12.42 34.60 316W .010 16.402.73 12.05 33.67 34.27

399 378 (4) 4.4

407 4.5 393 410 (40) 4.6

402 4.5 454 460 (20) 4.5

317C .035 18.60 3.22 13.88 37.87 508 4.0

4.3 1403

4.3 1486

4.3 1522

317W .035 18.603.22 13.88 37.87 37.12 496 503 (5) 4.5 4.3 1486

• Number of reports referenced.

the peak stress, to dynamic recrystallization, to the saturation stress arising from DRV and to failure. The peak strength will be a brief summary but will also include mean parameter values derived from 70 reports in the literature. The failure discussion will include the analyses proposed by Gittins and Sellars (27,28) and by Elfmark (29).

2. PEAK STRESS: SINH .AND ARRHENIUS RELATIONS: The peak stress ~ and c p decline as T rises, c decreases and solute increases but the former is raised and the latter reduced by o-ferrite segregated in solidification (5,6,8,20). The following relationships similar to these in creep have been examined:

n • A' 0' P = C exp (Q~IRT = Z' (1)

A" exp (3 O'p = £ exp (Q~w/RT) = Zoo (2) n .

A (sinh O:O'p) = c exp (QH/RT) = Z (3)

where A', A", A, n, 0:, {3 ~ o:n, R = (8.31 kJ/mol oK), Q' , Q" ,and QHW are constants. The power law (Eqn 1) was found suitabr~ atHMigh T, low e but broke down for 0' > 100 MPa (12,13,16). The exponential law, Eqn 2, was found satisfact6ry for the stronger as-cast material (5,6,19) but for the worked material broke down ~or 0' < 100 MPa (6,9,12,13,17). The sinh law (Eqn 3 with 0: = 0.012 MPa- ) wa~ found to fit the data for as-cast and worked alloys (6-13,15-17,20-22). The constants in Eqn 3 for all materials and in Eqn 2 for as-cast metal are listed in Table 1. These relationships are shown in a plot of 0' vs Z (Fig. 1); the reorganization of the data into a single line facilitates interpolation and extrapolation (12,13,17,20).

A search of the literature uncovered data for 70 alloys which permitted the determination of mean values of the constants for average compositions of each alloy (Table 1) (12,13,16,20). In addition, the activation energy was found to depend linearly on the total metallic so 1 ute (12): +

QHW = 25 - 13.5 (wt% solute) (4)

An inverted form of Eqn 3 has been proposed by Tanaka (30). -1 •

0' = (1/0:) sinh (c exp {Q/RT [(liT) - (liT' )]}) (5)

where the constant T' (=Q IR In A) is a constant for the material being about 0.8T (Table ~y (12,13,16). Finally the data was corrected for deformationar heating which results in greater T rise at higher £

335

and lower T where flow stress is higher. This rotates lines in graphs of log (sinh aq) vs (liT) to higher slopes increasing the activation energies by about 22% (12,13,16,20)' Lastly, Eqn 3 has been applied successfully to multistage, declining T tests, however, with changes in the constants (22).

The above analysis permits calculation of peak untested heats since QHW can be calculated by Eqn 4, T' constant and n averages 4.3 (12). The peak stress can be following equation to calculate stress-strain curves for (31,32) .

O'IO' = [(e/e ) exp (l-e/e )c p p p

stresses for is an alloy used in the any condition

(6)

where c is a constant which varies with Z and composition. The calculation depends on knowledge of e which was found to increase linearly with O' (31,32). For alloy 304, it was aOs95esta~1\~~ed from data on 40 allo*s that c varies linearly with (D . Z' ) where D is the original grain Pslze (12,14). This analy£ical system permits o • calculation of the stress at any T, e or e up to e p beyond which a plateau is assumed as an approximation (12,13,20).

3. DYNAMIC RECRYSTALLIZATION, STEADY STATE STRESS: The initiation of DRX, O' and e 1s most easily found from the inflection in 6-O' curves (as ex~lainedcin the next section) which starts the downturn to O'p(6-O') (10,13-16). From the critical stress, the crit~cal strain can be evaluated and the time to the start of DRX (e Ie) (14,33). The conclusion (98%) of the first wave of DRX is consideFed to be at the end of work softening and the start of steady state at O' and e. The two time values can be used in the Avrami analysis to aetermin~ the progress of DRX along the stress strain curve (14,16).

The value of O' can also be employed in the formula: s n'.

As (sinh aO's) = e exp (QORx/RT ) = ZDRX (7)

from which one can evaluate Q (Fig. 2) (14,16,29). From the values tabulated (Table I), it c~xbe seen that Q is considerably less than Q because the work softening is much ~~Jater at high Z as a result ofH~he migrations of the DRX grain boundaries removing a much finer (stronger) substructure than they do at low Z (14,16). The value of O' is not suitable for processing force calculations as is O'; howe~er, it is used in the Elfmark fracture analysis discussed later:

4. SATURATION STRESS, DYNAMIC RECOVERY: The flow curves for all alloys have been converted into 6-O' graphs where 9 is the rate of strain hardening (dO'/de) (7-13,15-17). At constant T and ~, 9 initially decreases linearly with rising O' from a commom value 9 0 (at 0'=0). A curved section is related to ~ubgraln formation, and a lower segment which extends to saturation O' (9=0) resulting from DRV alone. As T iQcreases and ~ decreases, 9 decreases more rapidly with rising O' and

;fe~:n~o:~:~l:~c~h~h~~~:~ b:;~e~~di~:ti~:t~~ru;~~nc~~ ~RVdefi~cti~: downwards to reach e at O' as a result of DRX.

The hypothesis 8f K06ks and Mecking (33,34), that the softening in high T deformation is related to a series of DRV mechanisms changing with T, leads to the equations:

i:li: = exp (-r In [O'* IO'*]lRT) = (O'*I/ )r/RT (8) o ~ s s ~

336

Q .. LJ .tU..OY. IWlMOL.

31S

317 ,.'

<) 301. "" .t o 304. 4.1 liS A 3"_ .u 4!4 a 311'. 4.S 4It

Z~ ! j go!

• ~ a . g 60 3 0 ... ~

50

40

30

. 3

~.Z

"

-.2

-3

-- 0 3O'''s

=g =. ---0 _ ...

a. -1.% 110·' :\IPO-I

tOllll" (Es"E.:J/E

~ • STRESS AT 99% OAX

• STEAm STAT! STAIESS 0Q .. " ACTMTlON ENERGY '0" QRX

Q=-s; ·2.l"_~LO'l£l

1100 1000 900 , , 0.14 0.11 o.az

liT 110', IIC'"!

Fig. 1. Confirmation of Eqn 3; breakdown of Eqns 1 & 2 Fig. 2. Derivation of QHW

where r and Co can be evaluated ~rom plots of c vs log (~;o/~;). ~so (~t OK) was found to be 7.5 X 10 MPa for 304 from linear Plots of log ~ vs T (8,13). The activation enthalpy AH decreases from 427 to 305 kl/mol for 304C and from 392 to 274 kJ/mol for 304W as T falls and c rises (13). The values for the other steels are in the process of derivation for later publication (36). The AH values agree with the Q values at high T where the dominant DRV mechanism is dislocation cr~mb. The variation of AH across the range studied in the Kocks-Mecking analysis is quite different from the constant QHW associated with the single mechanism assumed in the Arrhenius analysis. The facility of using the Z parameter is not available in this analysis.

5. HOT FRACTURE BEHAVIOR. As reported earlier for these steels, a marked increase in hot ductility is caused by either a rise in T or a decrease in c; but for 301W and 304W an increase in c is effective (Fig. 3) (3-6,8,11,13,16-20). The temperature increase raises ductility principally because it enhances DRX which slows fissure propagation and secondarily DRV which reduces stress concentrations at sliding grain boundaries GB (1-4,19,23,25,27). The rise in e causes a beneficial reduction in proportion of grain boundary GB sliding and increase in the rate of DRX, but a damaging increase in stress concentration and retardation in DRX nucleation which speeds crack nucleation. Which of these dominates depends on the alloy and the temperature range.

6. FRACTURE CONSTITUTIVE RELATIONSHIP; GITTINS-SELLARS ANALYSIS: As in creep, the stress concentrations inducing W-cracking rise with the general stress level. Since the sinh relationship is valid for

337

20~--~----~---r--~----~--'

18

16

~ 10

2

0,03

= g ;g~: PRESENT

--- 0 304W McCUEE1I ET AL. _. 304C RYAN ET AL.

<)

W WORKED CAS-CAST 'C

1200

--0-- 8-900

~ --------- - 0- - - -1000

• • • 1000

,I .3 I 10

STRAIN RATE. E. ,"

FRACTURE (lHW Qf

kJ/MOL ft, 393 5.Z 454 6.8

Fig. 3. Fracture strain vs e Fig. 4. Derivation of Qf from Eqn 9.

correlating strength data for hot working, the following strain constitutive equation is used for time to fracture t f (23,27,28):

-n t f = Ao(sinh «erf ) f exp (Q/RT) (9)

It is pos~ible to obtain a Qf from a Ptot of log "~f vs liT at constant e (Fig. 4), because the lines are parallel in a plot of log t f vs sinh «erf (Fig; 5). The difference in nr and Qfvalues from nand QHW is a measure of E effects above. As with strength data, the use of the T-compensated parameter Zf(28):

-n Zf = t f exp (-Q/RT) = Ao(sinh "~f) f (10)

permits the correlation of data for different T and ~ on a single line. In consideration of triple point cracking, the time to fracture is dependent on the size of DRX grains as described by the equation (28) :

• 1/2 tl = [r8n (l-v)I/lDsl (11)

where r is the fractured surface energy, v is the Poisson ratio, /l is the shear modulus and 0 ,the steady state DRX grain diameter, is taken as the length of the sl~ding grain boundary. Substitution of Eqn 3 into Eqn 11 gives the following equation:

1/2 -1/2 -n t f = [r81t (l-v~//ll _~s A' [sinh "~pl exp (QH\/ RT) (12)

~~~h:n~i~~t~ ~~~m~~~~neth!tSr ~1~~rN;~~W/lWh:~~.~s(~~ m;~~re~p~o ~~ o

1200 e), and v =0.31, then the predicted value t f exp (-QlRT) = 3.6 X

338

", -0304W 5.2

100 ~\-- ---A 316W 6.8

... :I ;:

FRACT~\\ PREMATURELY \

\

~---........--\ --.; -::- ..:::- - _ }E .-t -- --::: 0,10

\ \

t, :I A,IT)lSINH<I."f f"f 0.5 ft,:I STRESS EXPONENT

d • 1.2.10.1 MPo'l .c

- 0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 LOG SINH ClO'f

0.8 0.9

30 40 50 65 80 100 120 150 175 200 225 240 EQUIVALENT FRACTURE FLOW STRESS. Of • MPo

Fig. 5. Time to fracture vs sinh a~f

10-20s as compared with t~e experl!"fntal value of 4.4 X 10-18s. For the low Z condition (1200 C~175 s ) where D equals ~9~m , the calculated value is 2.8 X 10 s as compared witfi 6.5 X 10 s for the experimentally measured one. Unfortunately, the fracture times determined by this equation are 2 to 3 orders of magnitude lower than those measured experimentally. The reason appears to be the intermittant growth of cracks which are stopped as the grain boundary migrates away from them during DRX. Each time DRX occurs, cracks are effectively blunted and stress concentrations must build up by sliding on a newly captured GB before the crack can propagate (25,28).

7. ELFMARK DUCTILITY CONSTITUTIVE EQUATION BASED ON RATE OF DRX: According to Elfmark (29), the temperature compensated time for DRX WORX is given by the following expression:

WORX = tORX exp (-QORX/RT) (13)

and is plotted against Z on a log-log scale in Fig. 6. This function draws the data for all c8~Aitions and alloys into a single line. The following function satisfies the present alloys (16,17):

W 0.0024 Z-O.8 (14) ORX ORX

which compares with a power of 0.9 found by Elfmark. The W function is the inverse of the stress one, hence high ductility is associated with high W values but with low ~ values. The ductility for 316W is finall~Rfelated to WORX in Fig. 7 where the resulting line indicates a unique relationship:

e Z-1 = 7 X 10 4 W,·37 (15) f ORX ORX

The slope of 1.37 agrees well with 1.33 determined by Elfmark (29).

8. CONCLUSION: The hot strength , the DRX behavior, and the time to fracture of r stainless steels can be expressed by the hyperbolic sine

339

~ .. -t. ",10 ::I ;:: Q

'" l-e .. s .. 10

i ~ 10-"

~

~ '" -t' I- 10

T!MPt!RATURE COII'£NSATED TIME II FOR 99%DIIX ....... ·CE.- EcI/E

- __ " ... EXPC-O ... /RTI -0.0024Z·O,,. • 0.0024CEEXP[ -a. .. IRT] 1--

ZlFOR INITIATION _ ...... 1.' E./E --- ..... ·,.EXPC-O ..... IITI

• O.OOSCEEXP[-O_/RT) ,-.'"

(II C21 0... Q ... _~/MOL WMOL o 301 301 371

o 304 291 352 356 360

~ ~ ~ ~ ~ ~ ~ ~ ~ ZENER HOLl.OMAN PARAMETER.Z.s·'

.. ,::.. 631ftZMLJ?_

o ID4 .1 I.D ....-

.. z..-:. ?ala·....:,

w • • '. DPI-a,.II1T1

Z.· «DPla •• /1ITI

-IJ -IZ ·U -10 LaCl WI .. c· t .. !XIIt-a_'IT" ...

Fig. 6. T-compensated time vs ZORX Fig. 7. ZORX corrected (; f VS WORX

-f

and Arrhenius functions with different constants. The strength constitutive equation accurately predicts the power requirements for hot rolling, with the activation energy being a function of the metallic solute. The Kocks-Mecking hypothesis which reflects the changing DRV mechanism in the activation enthalpy is verified. The intermi ttant growth of cracks d\ie to DRX is the feature needed to raise the Gittins-Sellars fracture time to the experimental values. In consideration of this mechanism, Elfmark's analysis based on DRX kinetics accurately predicts the fracture strain.

REFERENCES: 1. W.J. McG. Tegart and A. Gittins, in Hot Deformation of Austenite, Metallurgical Society of AI ME, Warrendale, Pa (1977) 1-46. 2. B. Ahlblom and R. Sandstrom, International Metallurgical Reviews 27 (1982) 1-27. 3. H. J. McQueen and N.D. Ryan, Stainless Steels '84, Institute of Metals, London (1985) 50-61. 4. H. J. McQueen, R. A. Petkovic, H. Weiss, and L. G. Hinton, in Hot Deformation of Austenite, Metallurgical Society of AIME, Warrendale, Pa (1977) 113-139. 5. N.D. Ryan and H.J. McQueen, and J.J. Jonas, Canadian Metallurgical Quarterly, 22 (1983) 369-378. 6. N.D. Ryan and H.J. McQueen, in New Developments in Stainless Steel Technology, A. Soc. for Metals, Metals Park, OH (1985) (293-304). 7. N.D. Ryan and H.J. McQueen, in Strength of Metals and Alloys, ICSMA 7, Pergamon Press, Oxford (1986) 935-940. 8. N.D. Ryan and H.J. McQueen, in Plasticity and Resistance to Metal Deformation, Ferrous Metall. Inst., Niksic, Yugoslavia (1986) 11-26.

340

9. H.J. McQueen, N.D. Ryan, and E. Evangelista, Materials Science and Engineering 81 (1986) 259-272. 10. H.J. McQueen, N.D. Ryan, E. Evangelista, in Annealing Processes, Recovery, Recrystallization, and Grain Growth, Riso National Laboratories, Roskilde, Denmark (1986) 527-534. 11. E. Evangelista, N.D. Ryan, and H.J. McQueen, Metallurgical Science and Technology 5 (1987) 50-58. 12. N.D. Ryan and H.J. McQueen, in Stainless Steels '87, Institute of Metals, London (1987) 498-507. 13. N.D. Ryan and H.J. McQueen, High T Technology (1989) in press. 14. N. D. Ryan and H. J. McQueen, Canadian Metallurgical Quarterly 28 (1989) . 15. N.D. Ryan and H.J. McQueen, Czechoslovakian Journal of Physics B40 (1989) in press. 16. N.D. Ryan and H.J. McQueen, Journal of Mechanical Metalworking Technology 15 (1989) in press. 17. N.D. Ryan and H.J. McQueen, in Strength of Metals and Alloys, ICSMA 8, Pergamon Press, Oxford (1988) 1323-1330. 18. H.J. McQueen and N.D. Ryan, Microstructural Science 18 (1988). 19. N.D. Ryan and H.J. McQueen, Journal of Mechanical Working Technology 12 (1986) 279-296, 323-349. 20. H.J. McQueen, E. Evangelista, N.D. Ryan, in Innovation for Quality, Bologna, Italy (1988) 215-234. 21. N.D. Ryan and H.J. McQueen, in The Science and Technology of Flat Rolling, IRSID, Mazieres-Le Metz, France (1987) F17.1-9. 22. N.D. Ryan and H.J. McQueen, in Physical Simulation of Welding and Hot Forming, CANMET, Ottawa (1988) in press. 23. C.M. Sellars and W.J.McG.Tegart International Metallurgical Reviews 11 (1912) 1-24. 24. H. J. McQueen and J. J. Jonas, Journal of Applied Metalworking 3 (1984) 233-241. 25. H. J. McQueen and J. J. Jonas, Journal of Applied Metal working 3 (1984) 410-420. 26. L. Fritzmeier, M.J. Luton, and H.J. McQueen, in Strength of Metals and Alloys, ICSMA 5, Pergamon Press, Oxford (1919) 95-100. 21. H.J. McQueen, J. Sankar, and S. Fulop, in Meohanical Behavior of Materials, ICM3, Pergamon Press, Oxford (1979) 615-684. 28. A. Gittins and C.M. Sellars, Metal Science 6 (1972) 118-122. 29. J. Elfmark, Czechoslovakian Journal of Physics B35 (1985) 269-214. 30. K. Tanaka, T. Nakamura, Y. Hoshida and S. Hara, Res Mechanica 12 (1986) 41-57. 31. A. Cingara, L. St. Germain and H.J. McQueen, in Processing Microstructure and Properities of HSLA Steels, Metallurgical Society of AIME, Warrendale, Pa (1988) in press. 32. H.J. McQueen, A. Cingara, N.D. Ryan, Journal of Mechanical Working Technology 15 (1989) in press. 33. U.F. Kocks, Journal of Engineering Materials Technology 98 (1916) 76-85. 34. H. Mecking, in Dislocation Modelling of Physical Systems, Pergamon Press, Oxford (1981) 197-211. 35. N.D. Ryan and H.J. McQueen, Acta Metallurgica 37 (1989) in press.

MODELING OF FLOW BEHAVIOR OF THE NICKEL BASE SUPERALLOY NK17CDAT AT ISOTHERMAL FORGING CONDITIONS

Y. COMBRES and Ch. LEVAILLANT

Ecole Nationale Superieure des Mines de PARIS, CEMEF Sophia-Antipolis, 06565 VALBONNE Cedex, FRANCE

ABSTRACT: The dependence between the equivalent stress, isothermal forging conditions and microstructure for the nickel base superalloy NK17CDAT has been studied by means of torsion experiments. Two sets of constitutive relationships are proposed based on two different ideas of what should conveniently represent the microstructure. The first one assumes that it should be the average grain size only and the second one that it should be the whole distribution. In the second case, some useful deformation mechanism maps are proposed.

1. INTRODUCTION: Isothermal forging conditions (temperature T, strain rate ; and forging time t) are conditioning not only the materials flow during working but also their final microstructure. Since the superplastic behavior is very sensitive to the later, its intensive use requires the understanding of the interrelationships between the rheological data (i. e. the equivalent stress 0') and the microstructural ones (assumed to be gathered in (S).

2. MATERIAL AND METHODS: The alloy investigated here is a nickel base superalloy which grade is NK17CDAT (i.e. approximately 17% Co, 15% Cr, 5% Mo, 4% AI, 3.5% Ti). Its as-received state is displaying an initial average grain size of 3.7 /Lm. Torsion 4 tests2 a:e1 performed in the 1085-1115 'C temperature, 10- -10- s strain rate and 0-2 strain ranges after (1). At the end of each experiment, the sample is water quenched and polished areas are etched in Kalling reagent in order to allow microstructural investigations.

341

A. S. Krausz et al. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 341-347. © 1990 Kluwer Academic Publishers.

342

3. EXPERIMENTAL RESULTS: Figures 1 and 2 present the stress-strain curves. The flow strength of this alloy either continuously increases (low strain rates) or increases first, takes a peak value then decreases (large strain rates) and, finally, in the two cases, reaches a kind of steady state.

The apparent strain rate sensitivity exponent m, determined at constant € and calculated in a lna-ln f plot using linear regression (correlation coefficient >.99), is found to continuously decrease when increases. On the other hand, the strain rate sensitivity exponent ms ' determined for constant structure using f. step tests, is found to remain somehow constant at the classical value of 0.5.

These evolutions are to be closely related to the ones of the the average grain size D get after microstructural investigations. Effectively, after grain growth due to annealing before testing (1), grain refinement is encountered for low strain rates whereas grain coarsening occurs at large rate of strains (figures 3 and 4).

4. FUNDAMENTALS OF THE MODELING: In the general case, the microstructure of an alloy should not be summarized to only one or two parameters. In fact it looks like a collection of features (1).

The list of the parameters includes some species related to the general morphology of the grains to know the average grain size D, the standard deviation s and the corresponding distribution function f(D) and the shape of the grains through the grain aspect ratio GAR evolutions. The crystallographic texture, the relevant orientation distribution functions ODF, the dislocation activity and the one of the vacancies, the nature of the phases and their respective proportion %X, the microstructural mechanisms such as dynamic recovery (DR), dynamic recrystallization (DRX), the continuous recrystallization (CRX) or the grain boundary sliding (GBS) can be taken into consideration.

Because of this large number of parameters, the notation IS} has been chosen to represent the material microstructure. Generally speaking, the microstructure influence in the superplastic flow is made using D. Few studies hav"e been performed using the couple (D,s)(2,3), the GAR(4)or the %X(S). As shown by the experimental results of the NK17CDAT, the superplasticity occurence depends, at least at the first order, on the grain size. Two hypotheses can thus be made: the first one is to assume that {S}={D} and the second one is to set (S}={D,s). They are basically different and their implication in the general understanding of the superplasticity is not the same.

5. ASSUMING THAT {S)-{D}: The use of such an assumption is to idealize the actual microstructure and replace it by a group of unique sized grains. Some useful parameters can

100

0,1

0--0- 0

NKI7CDAT

T=1085 C o 10-'8-1 • 3.10-'8-1 A 10-' 8-1 I 3.10-'8-1

0/ '0 --0 0 ~o-o-

0,5 I,D

8TRAIN

1,5 2,0

Figure 1_ (a, €) curves at 1085 ·C for the NK17CDAT

\0 NKI7CDAT

T=I085 C

T1V T - T;tVi

- ~Y-l--l:o ~-1\ 1 0 IO-'S-I

10

~ • 3.10-' 8-1 A 10-' 8-1 I 3.10-'8-1

10' 10' 10' TI~lE lsI

Figure 3. (D,t) curves

10'

at 1085 ·C for the NK17CDAT

343

100

E ~ u.J N Uj :z: ::;: Q: Q

NKI7CDAT

T= \liS C

o 10-' 8-1 • 3.10-'8-1 A 10-' 8-1 I 3.10-'8-1

°-0-°-0-0_0 __ 0_0_0_

10

0,1 0,5 1,0 8TRAIN

1,5

Figure 2 _ (a , €) curves

2,0

at 1115 ·C for the NK17CDAT

10

5

10 10' 10' TIME lsI

o 10-'8-\ • 3.10-'8-1 l 10-' 8-1 I 3.10-' S-I

10'

Figure 4_ (D, t) curves

10'

at 1115 • C for the NK17CDAT

344

then be determined such as the strain rate sensitivity exponent ms (eq. (1)), the grain size sensitivity exponent Ps (eq. (2)) and a material constant only depending on temperature ao. So, the stress is expressed by eq. (3).

(a Ina /a In E)T ,D (1)

Ps - (alna/alnD)T, i (2)

a = ao. Ems.DPs (3)

The apparent strain rate sensitivity m can therefore be deduced from eq. (3) using:

m = ms + ps.( alnD/ alnnT, l (4)

The consistency of such an assumption is showed with figures 5 and 6 only at 1085 • C but it can be extended for 1115 C as well. The prediction is fairly good.

T I I

NKI7CDAT 40 T=1085 C -

• 3.10"S-1 ~ 30 ,,4 IO"S-I os .. ~.~ j 3.1O"S-1 ill ./ --~ "" 20./" A~ tn ;( ~'

/.4/" ,_.::'::-:-. ,,' ___ ~J.~

10 1;.0 .... ~,).--. _ ,..<-1/ MODEL ____ _

EXPERIEtiCE __

-

-

I I I

0,1 0,5 I,D 1,5 2,0 STRAIN

Figure 5. {S}-{O} consis-tency on (a, l) curve

~!ODEL ___ _

0,1 EXPERIENCE __

0,1 0,5 1,0 STRAIN

NKI7CDAT

T= 1085 C

1,5

Figure 6. {S}={D} consis-tency on (m, l) curvE'

2,0

6. ASSUMING THAT {S}=(D,s}: It consists in replacing the actual microstructure by a group of ranked grain classes with their own proportion given by the distribution function f. The actual distributions are found to be best fitted using a logarithmo-normal distribution. Considering a class D, this whole class can either chose a superplastic (SP) behavior if D is small enough or a classical behavior (NSP) if D is very large. Some constitutive relationships can describe the D class behavior:

either a = k Sp , iml.DPl

or a -

(5)

(6)

345

With such a view, a transition grain size Dt can be determined for which eq. (5) and eq. (6) give the same value of €. For D<Dt, and for a given applied stress, eq. (5) leads to greater value of f than eq. (6); the SP behavior can be established; the reverse tendency is encountered when D>Dt. The analytical form of Dt can be easily determined by equaling the two pre-cited equations.

Dt = A(T). € (m2-ml)/pl (7)

The location of Dt( €) in the space (D, €) gives a kind of frontier below which the material displays a superplastic behavior characterized by grain boundary sliding (SP) and above which. classical dislocation creep (NSP) is dominant. This provides a useful deformation mechanism map. Such an approach can be applied in the case of the NK17CDAT assuming that two superplastic behaviors compete together, to know the already successfully applied Gittus (6) one (ml=.5, pl=l) and the Coble (7) one (ml=l, pl-3) whereas the creep behavior is characterized by m2=.25.The figure 7 presents the corresponding deformation mechanism map.

20

E -= UJ N 10 en z ::c c.: <.;>

10-' 10- 3

STRAIN RATE 10-'

Figure 7. Deformation Mechanism Map in case of Gittus and dis­location creep models

is somehow T investigated temperature range. After the alloy will display in majority behavior and will keep it because of

The frontier D=Dt

grain refinement.

independent in the thermal treatment, some superplastic the strain induced

Assuming that the strain rates are homogeneous in all the grain classes (i. e. SP and NSP), the equivalent stress can be expressed as the sum of the two parts of the

346

material by the following integrative form (1):

Dt a = k Sp . eml . J feD) .DPl.dD

o

+'" 'm2 J + k NSP . f. f(D).dD

Dt

The (S)-(D,s) except at the Coble

figures 8 and 9 enlighten model. The Gittus model 1115 C and f -2 for the

the consistency is found to be

low strain rates one is better.

so ,/ SO

/" /

/ , £=1,0 ~,

, ,

/ , ,

10 ,/" 10 ;E

~IOOEL .. " EXPERIENCE =-:::.= Cl-

;g '/ / ;g

ill SO ,~./

c.: ':/ t;:; /

/" f·O ~'

10 /, 10

(8)

of the suitable

for which

Figure 8. sistency

(S)=(D,s) at 1085 'C

con- Figure 9. sistency

(S}=(D,s) at 1115 'C

con-

7. DISCUSSION AND CONCLUSIONS: The choice (S)=(D) leads to some good fitting of the stress evolutions (especially for the continuously increasing curves) and the one of the apparent strain rate sensitivity exponent m. Such an approach is a semi empirical one and requires the test of several initially different microstructures to get the grain size sensitivity exponent Ps and the use of Estep tests to have the value of the strain rate sensitivity

exponent ms' Moreover, the value of the cannot be related to special microstructural of the mean effect due to the choice of the size as the relevant structural parameter.

347

couple (ms'ps) model because average grain

The hypothesis {S }={D, s} appears to be more convenient since the microstructure is presented by the grain size distribution. The stress values are correctly predicted and so are the experimental m ones (1). Such an idea needs the a priori choice of superplastic behaviors characterized by (mi,Pi) and classical ones given by mj values. Its integrative formulation allows to distinguish two parts in the material obtained by comparison with a transition grain size Dt. This particular value in the distribution, only depending on the imposed thermomechanical conditions (T,e), expresses the competition between the superplastic and classical behaviors. This competition can be understood by mapping the Dt values in a (D,E) space. Below Dt, the material displays grain boundary sliding. Such deformation mechanism maps appear attractive though they do not predict grain refinement or grain coarsening as shown in (3). They nevertheless seem useful on a practical point of view for the isothermal forging operations for which the technicians can easily appreciate the further behavior of the material only using a comparison between the microstructure used by them and Dt.

REFERENCES: 1. Y. Combres, Ph.D. Thesis, Etude du comportement superplastique des alliages de titane TA6V et de nickel NK17CDAT au moyen de l'essai de torsion, Ecole des mines de Paris, (1988). 2. A.K. Ghosh and R. Raj, Acta Metall., 34 (1986) 447. 3. A.K. Koul and J. -Po Immarigeon, Acta Metall., 35 (1987) 1719. 4. C.H. Hamilton, Proceedings of Superplastic Forming, Ed. S.P. Agrawal, (1984) 13. 5. Y. Honorat, Annales des Mines, Feb., (1985) 83. 6. J.H. Gittus, Trans. ASME, 99, (1977) 244. 7. R.L. Coble, J. Appl. Phys., 34, (1963) 1679.

The authors gratefully acknowledge the SNECMA company for alloy p rep a rat i on, ' too 1 in g 0 f s p e c i men s and f r u i t f u 1 discussions.

CONSTITUTIVE LAW FOR CALCULATING PLASTIC DEFORMATIONS DURING CZ SILICON CRYSTAL GROWTH

c. T. Tsail , V. K. Mathewl, T. S. Grosl, O. W. Dillon, Jr.4, and R. J. De AngelisS

IResearch Scientist, Department of Aeronautics and Astronautics, Air Force Institute of Technology Wright-Patterson, AFB, Ohio 45433.

2Graduate Student, ~fessor, ~fessor, Department of Metallurgical Engineering ~fessor, Department of Engineering Mechanics University of Kentucky, Lexington, KY 40506.

ABSTRACT: During the growth of CZ silicon crystals, dislocation motion and generation are induced by the thermal stresses which arise from rapid cooling from the solidification temperature. These defects limit the fraction of acceptable silicon device chips obtained from a crystal. Current semiconductor technology is restrained by the lack of a predictive material model that can reliably calculate the dislocation motion and multiplication in CZ material at temperature close to the melting point. The concepts of Haasen and Sumino are modified to predict axial tensile results in CZ silicon up to 1300 °C. This modified formulation is the basis of the constitutive model presented.

1. INTRODUCTION: Large diameter CZ crystals having a low dislocation density is a goal of many

silicon crystal grower. However, high dislocation densities are created in large diameter crystals. The dislocation densities are generated by the increased magnitude of thermal stresses encountered in the larger crystals. A constitutive equation is, therefore, needed to understand how the dislocation density is related to the thermal stresses, temperature, strain rates, and impurity concentration. The theory of viscoplasticity, which has the dislocation density as an internal variable, provides the mathematical formalism to handle the wide range of strain rates and temperatures which arise in growing crystals. Experimental data by Sumino et al. [1-5] shows that the stress-strain and dislocation density-strain characteristics of single crystal silicon are very sensitive to temperature, strain rate, the value of the initial dislocation density and the concentration of impurities such as oxygen, nitrogen, carbon and phosphorus. An accurate constitutive model which can completely acount for the effects of all of these parameters from room temperature to the melting point (i.e., 1412 0c) is not yet available.

The major objectives of this work are to modify the concepts of Haasen and Sumino in order to extend the useful temperature range for their deformation model to 1300 °c for CZ silicon. The modified deformation model is incorporated into a generalized 3-dimensional constitutive equation capable of predicting dislocation densities encounted during the growth of CZ ingots. The model developed here is improved over the earlier version because it includes the high temperature tension test data of CZ silicon and it has been further modified to include the effects of oxygen as a function of temperature.

349

A. S. Krausz et at. (eds.), Constitutive Laws of Plastic Deformation and Fracture, 349-355. © 1990 Kluwer Academic Publishers.

350

2. THE EXISTING HAASEN·SUMINO MODEL: The original Haasen·Sumino deformation model is based on mechanical testing

data collected, mostly on float zone silicon, in the temperature range from 700 to 1000 °c. The stress-strain and dislocation density-strain characteristics of this model depend on the temperature, strain rate and mobile dislocation density. The plastic strain rate er and the mobile dislocation density generation rate Nm, which is assumed to be proportional to the area of slip plane traversed in the crystal, are given by [6-7]

er=Nmvb (1) and

Nm = BNm v. (2) The dislocation velocity has been experimentally determined to be a function of stress up to 1000 °c according to the relation

v = V o('t. - D'{N:l (3) where

v 0 = ( Bo )exp( ~~ ) = koexp( ~~ ). ~

The parameter ~ which characterizes the multiplication rate in the above model is assumed to be given by [8]

~ A _ r.:T A Gb u = K'teff = K(,t. - D-v Nm), where D = T' (4)

and 'Ceff is the effective stress, 'to the applied stress, b the magnitude of Burgers vector which here is taken to be 3.8x1O-IO m, Nm the density of moving dislocations, ~ is a parameter characterizing the interaction between dislocations which is 3.3, Q is the Peierls potential and taken to be 2.17 eV, k is Boltzman's constant and taken to be 8.617x1O-s eV/oK, T is the absolute temperature, Bo is the dislocation mobility, experimentally determined to be 4.3x104 m/sec, K and A. and p are constants and reported by Sumino, to be 3. Ix 10-4 m!Newton, 1.0 and 1.1 respectively, and 'to is assumed equal to 107 Newton/m2. The shear modulus G is E/2(1+ v), where E is Young's modulus which is given by Hartzell [9] as E(T) = 1.7x1011 - 2.771x104x(Tl Pascals. When 'to S D..JN;., the dislocation velocity v is zero which means that no dislocations are generated then as well. In order to im~ motion to the dislocations, the applied stress 'to has to exceed the barrier stress DV Nm set up by the existing forest dislocations.

3. THE MODIFIED MODEL: The tensile stress-strain data for both FZ and CZ silicon have recently been

determined [10,11] in the temperature range from 900 to 1300 °c. The value of the constant K and ~ calculated from the experimental data are 2.4x10-4 mlNewton and 1.0. Although these are not significantly different in magnitude from the constants reported by Sumino, the value of ~ has a strong affect on the stress at the lower yield point. However in this high temperature range interactions between point defect such as impurity atoms and dislocations can occur. The oxygen concentration in CZ silicon of about 38 ppm will impede the motion of dislocations at temperatures above 1000 °c. The oxygen atoms affect motion by introducing a drag stress on the dislocation line. The value of the drag stress depends on the concentration of oxygen in solid solution, strain rate, temperature, and energy of binding the impurity to the dislocation line. However, an estimate of the drag-stress is introduced into the formulation in the expression for the dislocation velocity, and eq. (3) becomes

351

v = V o(t. - DfN: - td)P, (5) where t" is the drag stress due to the interaction of impurity atoms with dislocations. Prior to dislocation motion, the applied stress has to overcome the internal stress fields that are set up by other dislocations and by the impurity atoms which increase the resistance of the crystal to dislocation motion.

The drag-stress due to the oxygen content in silicon, can be assumed to be

td = F(T)xCo , (6) Po

where Po is a constant (4x1017 atoms/cm'), and Co is the equilibrium concentration of oxygen which is given by 1.53x102Ixexp(-1.03/kT) atoms/em' [20]. The function F(T) is obtained from the experimental data. At low temperature F(T) is assumed to be the stress below which dislocations cease to move because of the formation of a solute atmosphere. In macroscopic measurements there exists three regions of different strain rate dependence. The high strain rate region corresponds to dislocation breakaway from solute atmospheres, the intermediate plateau region corresponds to unstable breakaway and repinning by solute atmosphere, and the low strain rate region corresponds to drag of stable solute atmospheres [12,13]. The value of F(T) has been determined from the CZ tensile measurements [14] to be 3 MPa at 1100 °c and decreases to 1.81 MPa at 1300 °C. For the calculations presented below, F(T) is taken to be 3 MPa for temperatures below 1300 °c and 1.81 MPa for temperatures 1300 °c and higher.

The response function for the one dimensional tensile test of silicon is

t= ; +t"I, (7)

where t is the total strain rate, a is the stress rate, and tp1 is given by eq. (1). From eqs. (2) and (7), the stress-strain and dislocation density-strain curves, are easily calculated [15].

4. RESULTS AND DISCUSSIONS: Initially a comparison of the modified tensile model with experimental data will be

presented and discussed. Stress-strain and dislocation density-strain curves obtained by evaluating eqs. (2) and (7) depend on the values of the apparent machine stiffness associated with the experiments. The values obtained from the load-time/data are 833 MPa for 900 °c, 371 MPa for 1100 °c, and 108 MPa for 1300 °C. This data supports the results of Hockett and Gillis [16] who observed that machine stiffness is a variable quantity which generally tends to increase with applied load and in tension at a constant load can have a variation of 30 percent or greater.

Typical stress-strain curves obtained by prescribing the temperature and the initial dislocation density No, at the various total strain rate indicated, are shown in Figs. 1-3. The magnitude of the upper yield stress, the lower yield stress and the magnitude of the stress drop from the upper to the lower yield stress decreased as the temperature increases. These results of the simulation are within 2 MPa of the experimental data, as can be seen from the partial data sets presented in the figures. This agreement is very good. considering the sensitive nature of the tensile measurements and the idealization inherent in the model. The Haasen-Sumino model was originally developed to describe the yield region. For a prescribed temperature and No. the magnitude of the upper yield stress. the lower yield stress and the magnitude of the stress drop from the upper to the lower yield stress decreases as the total strain rate

352

decreases as shown in Figs. 1 and 2. Dislocation density-strain curves obtained by prescribing the temperature and the initial dislocation density, at the various total strain rate indicated are shown in Figs. 4-6. When the temperature is near the melt point (i.e. 1300 0c), there are no stress drops, and the dislocation density-strain curve appears to be a straight line, and the stress-strain and the dislocation density-strain behaviors are not affected by the total strain rate as shown in Figs. 3 and 6. From Eq. (1), the plastic strain rate is a product of the dislocation density and the dislocation velocity. The lower dislocation velocities at the lower temperatures require a higher density of dislocations to accommodate the strain rates while at the higher temperature the deformation process is controlled by the magnitude of the back-stress. Dislocation densities in specimens subjected to 50 % shear deformation range from 107 to 108cm-2

[14]. These values are close to the calculated density in the yield region. Even though the final strains in both cases are significantly different, major changes in the dislocation density do not occur during deformation past the yield region.

At 900 DC, silicon can be subjected to a large degree of deformation before the onset of multiple slip and better agreement is observed between the calculated and experimental results. At 1100 DC, the crystal deform in multiple slip at very low strain and a significant deviation between the calculated and experimental curves are observed beyond the yield point.

Since ingot modelling utilizes only the very early stages of deformation behavior, good agreement between the calculated and observed values in the low strain is all that is required. In the CZ ingot growth modeling, the machine stiffness is set equal to the modulus of silicon, E(T), mentioned above.

-; "-! I/) I/) III II: l-I/)

30 T .too·c .... 1000 c",-z 0-.44 XIO- 4 ,.

0-214lCIO·41$ 9 K-I 44 x 10-511

Z!S 6-2.14 x 10-SII

ZO

I!S

10

!S

o~----~----~----~----~ 0.00 o.oz 0.04

STRAIN

0.06 0.08

II

1

I D

! !S .. .. 4 III II: :;; 3

2

D

i

o~ __ ~~ __ ~~ ____ ~ ____ ~ 0_00 0.02 0.04 0.01 0.011

STRAIN

Fig. 1. Stress-strain curves for the modif- Fig. 2. Stress-strain curves for the modif-ied model and the experimental data ied model and the experimental data at 900° C. at 1100° C.

7 T '1300'C N.' 1000 em'z

6

2

0.02 0.04

STRAIN

0.06 0.08

Fig. 3. Stress-strain curves for the modified model and the experimental data at 13000 C.

10 T-IIOO·C

• N. -1000 '11'1- 2 ,. 8

! -2 7

.. I ~ 0; z 5 ... <> z 2

4

!c 3 u 0 ...

2 .. Ci

0 0.00 0.08

STRAIN

Fig. 5. Dislocation density-strain curves for the modified model at 1100° C.

15 T •• oo·c .. _1000,.-1

~ '0

~ 25 .. !:: ., 20 z .. <> z 15 0

ii .., 10 0 ... fI)

is

S.OO 0.06 ooe STRAIN

Fig. 4. Dislocation density-strain curves for the modified model at 900° C.

4.0

3.5

~ 3.0

~ .. 2.5 !:: .. z

2.0 ... <> z 0 1.5 ., c .., 9 1.0 !! 0

0.5

0.0 0.00

T-1300·C Ho-'OOO tlft-2

0.02 0.04

STRAIN

0.01 o.oe

Fig. 6. Dislocation density-strain curves for the modified model at 13000 C.

5. THE APPLICATION OF THE MODIFIED MODEL:

353

In order to calculate the material behavior of crystalline solids during the growth process, a three dimensional response is assumed by generalizing the modified one­dimensional model. Silicon is assumed to be isotropic in both its elastic and plastic responses by Dillon, et al. [15,17]. Recent work by Kim, et al' [18] in the web type crystal has indicated that the isotropic assumption and a crystallographic analysis lead

354

to remarkably close results. Therefore we assume that the material model for silicon is [15,17]

(8)

where .:2.

bkoNmekT ( ~ - n.JN: - 'td f tl;! = fSji, and f = ,(9) ~

and v is the Poisson's ratio, ex is the coefficient of thermal expension, and £'1j' crij' 5ij' and ~I are the components of the total strain rate, stress rate, Kronecker delta function and plastic strain rate tensor, respectively. J2 is the second invariant of the deviatoric stress tensor dermed by J2 = SjjSjj /2. The Sjj are the components of deviatoric stress tensor which are Sjj = O"jj - O"kk5jj /3 and f is viscosity. The rate of generation of the dislocation density is taken to be [15,17]

..:2. Nm = KkoNmekT ( ~ - n..J"N: - 'td f( ~ - nfN:)\ (10)

and Nm = ~I = 0, if ~ - nfN: - 'td ::;; O. The thermal stress field and the dislocation density are coupled by eq. (10). We assume that this model is applicable from the end of the experimental test data, 1300 DC, to the melting point, 1412 DC. This implies that the dislocation mechanisms do not change in the extrapolation temperature range.

The above modified model is now applied to the growth of a 20 cm long and 10 em diameter silicon crystal with an initial dislocation density of 0.001 cm·2 along the solid­melt interface. The temperature changes from the melting point along the solid-melt interface to 1262 °c at the far end [19]. The effective stress contour plot are obtained by the method in [19]. are shown in Fig. 7. The effective stress is the greatest in the initial centemeter near the edge. These are due to the high thermal gradients existing in this region. The predicted dislocation densities have also been calculated and are shown in Fig. 8 illustrate that they increase rapidly to a maximum in the first 4 cm of growth and thereafter is nearly constant. This also indicates that most dislocations are generated near the edge.

E u

20.0 r-------------"

_________ /0.72 MPa

11.9 ----__ ............ ,,,,""' ........

............ -~---' ------. -'---.......... -..... ~.4 "-

36 \

12.9 t

0.0 ~=3 ... 6_-.L--:-= __ ~..:;;~..u:.-:~ 0.0 1.8 3.2

Yleml

20.0~

19

69

! 4.9

3.1

1.6

0.0 ..., 0.0 y

210 tnrl I I I I : ; ; r 170 Ii!; I : I! i 130 : I i II I III 85 I::; I I II I 43--+i1'IIII,j I ------41". 250===t' :a'tu±' U: ! i ~:g: 'I" I! 380:==t+:' H';:j:lt:!=i:!+l 420 ,I

\:'..-:'~ -------

3.1

Y leM'

I

4.2 4.1

Fig. 7. Effective stress contour plot for a Fig. 8. Dislocation density contour plot for 20 cm long 5 cm in radius CZ crystal. a 20 cm long 5 cm in radius CZ crystal.

355

6. CONCLUSIONS: A modified constitutive equation that contains dislocation motion and

multiplication, and the effects of temperature on oxygen concentration to near the melting point is formulated. The equation is then used to calculate the dislocation density in a CZ crystal during growth. The dislocations are mostly generated near the solid-melt interface at the edge of the ingots. Models of this type are essential for improving CZ ingot quality, because dislocation generation takes place at very high temperature and is very sensitive to thermal gradients.

REFERENCES: 1. 1. Yonenaga and K. Sumino, "Dislocation Dynamics in the Plastic Deformation of

Silicon Crystals, 1. Experiments", Physi. Stat. Sol. (a), Vol. 50 (1978), pp. 685. 2. I. Yonenaga, K. Sumino, and K. Hoshi, "Mechanical Strength of Silicon Crystals

as a Function of Dxygen Concentration", J.Appl. Phys., Vol. 56(8), Oct., 1984. 3. M. Imai and K. Sumino, "In Situ X-Ray Topographic Study of the Dislocation

Moblility in High-Purity and Impurity-Doped Silicon Crystals", Phil. Mag. A, Vol. 47(4), 1983, pp. 599-621.

4. K. Sumino and M. Imai, "Interaction of Dislocations with Impurities in Silicon Crystals Studies by in Situ X-Ray Topography",Phil. Mag. A, Vol. 47(5), 1983, pp. 753-766.

5. K. Sumino, I. Yonenaga, and M. Imai, "Effects on Nitrogen on Dislocation Behavior and Mechanical Strength in Silicon Crystals", J. Appl. Phys., Vol. 54(9), Sept. 1984, pp. 5016-5020.

6. P. Haasen, "Zur Plastischen Verformuny Von Germanium und InSb", Z. Phys., Vol. 167, 1962, pp. 461-467.

7. H. Alexander and P. Haasen, "Dislocations and Plastic Flow in the Diamond Structure", Solid State Phys., Vol. 22,1968, pp. 28-156.

8. M. Suezawa, K. Sumino, and I. Yonenaga, "Dislocation Dynamics in the Plastic Deformation of Silicon Crystals, II. Theoretical Analysis of Experimental Results", Physi. Stat. Sol. (a), Vol. 51,1979, pp. 217-226.

9. R. Hartzell, personal communication, Texas Instruments, Dallas, Texas, 1984. 10. T. S. Gross, V. K. Mathews, R. J. De Angelis, and K. Okazaki, "Dynamic Strain

Aging in CZ Grown Silicon Single Crystals", Submitted to Matis. Sci. & Eng .. 11. V. K. Mathews and T. S. Gross, "Deformation Characteristics of Float Zone Grown

Silicon Single Crystals at Elevated Temperatures", to be submitted to Phil. Mag .. 12. H. Yoshinaga and S. Morozumi, "The Solute Atmosphere Round A Moving

Dislocation and Its Dragging Stress", Phil. Mag., 23,1971, pp. 1367. 13. H. Yoshinaga and S. Morozumi, "A Portevin-Le Chatelier Effect Expected From

Solute Atmosphere Dragging",Phil. Mag., 23,1971, pp. 1351. 14. V. K. Mathews, "Deformation Characteristics of Silicon Single Crystals at

Elevated temperatures", Ph.D. Dissertation, University of Kentucky, 1988. 15. C. T. Tsai, ''Thermal Viscoplastic Stress and Buckling Analysis of Silicon

Ribbon", Ph.D. dissertation, University of Kentucky, 1985. 16. J. E. Hockett and P. P. Gillis, "Mechanical Testing Machine Stiffness-Part 1-

Theory and Calculations", Int. J. Mech. Sci., 1971, Vol. 13, pp.251-264. 17. D.W. Dillon, Jr., C.T. Tsai, and R.J. De Angelis, "Dislocation Dynamics During

the Growth of Silicon Ribbon",J.Appl. Phys. 60(5), Sept. 1986, pp. 1784-1792. 18. Young K. Kim, R.I. DeAngelis, C.T. Tsai, and D.W. Dillon, Jr., "Dislocation

Motion and Multiplication During the Growth of Silicon Ribbon", Acta metall. Vol. 35, No.8, pp. 2091-2099,1987.

19. C.T. Tsai, D.W. Dillon, Jr., and R.J. De Angelis, "Dislocation Dynamics During the Czochralski Growth of Silicon", Mat. Res. Soc. Symp. Proc. Vol. 104, 1988, pp. 125-128.

20. R. A. Craven, SemiconductorSUicon 1981, The Electrochem. Soc., 1981, Pennington, N. J., pp. 154.

CONSTITUTIVE RELATIONS FOR DEFORMATION AND FAILURE OF FAST REACfOR CLADDING TUBES

I J Ford and J R Matthews

Theoretical Physics Division Harwell Laboratory

UK Atomic Energy Authority Didcot, Oxon, OXII ORA

ABSTRACf: We use constitutive relations suitable for porous materials to describe the plastic yielding and eventual failure, in tensile tests, of irradiated cladding materials used in Fast Reactor fuel pins. The failure mechanism consists of the growth and coalescence, under deformation, of the neutron-induced voids produced in the clad under irradiation. We draw attention to a possible swelling dependence of the macroscopic yield stress of the cladding, and also the loss of ductility in AISI 316 steel above swellings of about 20%.

1. INTRODUCTION: The cladding material of a Fast Reactor fuel pin is subjected to high temperatures and severe neutron irradiation damage over periods of years, conditions not encountered in other engineering applications. In this environment, the material is deformed by neutron-induced void swelling and irradiation-driven creep, and is also subjected to chemical attack by the various reactive species produced by the nuclear fission processes occurring in the fuel. The material must maintain sufficient strength and integrity during this period.

A number of steels have been developed with, to a greater or lesser extent, the required properties under operational conditions, thermal creep setting the limits to their endurance. However, it is important to understand the mechanisms which might cause the cladding material to fail in a hypothetical severe reactor accident, as the clad constitutes the first line of defence against release of fuel and fission products into the coolant. In such a situation, the cladding tube would experience a raised internal stress, due either to fission gas pressurisation, fuel vaporisation or mechanical interaction with solid fuel pellets. Also, the temperature would rise rapidly, due either to loss of coolant flow, or an increased fission rate. Within the range of accident conditions there are a number of failure mechanisms [1]. The intergranular cavity growth mode, controlled by thermal creep, largely determines failure by coalescence of these cavities at very high accident temperatures. At lower temperatures, trans granular failure modes of the irradiated steel dominate. The fracture surface is found, variously, to exhibit cleavage­like facets (channel fracture) or a dimpled appearance. This second mode is given attention in this paper. We seek to describe this failure mechanism as a plastic flow instability in the ligaments of material between the neutron-induced voids present in the clad.

2. CONSTITUTIVE RELATIONS: A number of authors have considered the effect of porosity on the macroscopic plastic behaviour of materials [2-4]. The

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A. S. Krausz et ai. (eds.). Constitutive Laws of Plastic Dtiformation and Fracture, 357-363. © 1990 Kluwer Academic Publishers.

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macroscopic constitutive equations describing the plasticity are due to Ourson [2] with subsequent modification, upon detailed numerical testing, by Tvergaard [3]. The

plastic strain rate tensor tPij is given by

.p • dt/> Gj = A­

o(Jij ,

where O'ij is the macroscopic stress tensor and

(1)

(2)

The plastic potential <I> acts as a yield condition: no flow occurs if <1><0, whilst <I> = 0

during flow. CTe is the von Mises' stress given by (3/2 Sij Sij)l/2 where Sij = CTij - 1/3

CTkk. The local yield stress of the matrix between the voids is represented by CTy. The function f* of the fractional porosity f is given here by

f* = f; f < fc = 0.15

( f -fc )2 =f+ fF-fc (fu-fF);f>fc=0.15

f < fF = 0.25 (3)

where fu = l/ql' The parameters ql and q2 are equal to 1.5 and 1.0 respectively. This formalism has been used with some success in describing the process of failure of porous substances (defined by the condition f* = fu, when the macroscopic yield stress . vanishes). The flow rate parameter A can be expressed in terms of known quantities, and the evolution of f is described by a combination of pore growth and nucleation controlled by stress or strain criteria [6].

In the application to which we wish to put this formalism, however, we make use of the understanding of neutron-induced void swelling which has been gained over several years [7]. There exist well characterised correlations for the swelling of cladding materials in a wide range of reactor conditions. We ignore stress- and strain­induced pore nucleation and consider the following porosity evolution equation:

. .p. f = (1- f) Gi + S,

(4) .

where S is the swelling rate. In order to calculate the evolution of the macroscopic yield stress CTey.which is the value of CTe to make <1>=0, it is necessary to find an

expression for cry . This would describe the changes in the strength of the matrix material due to reactor conditions. Simplified mechanistic considerations [8] suggest the following:

(5) where

(6) The hardness parameter a is modelled on the dislocation density in the material, .

but is taken to represent the contributions to material strength due to dislocations. D is the dose rate in dpa/s, modified by a factor of (1 +f) in an attempt to account for the increased resistance to dislocation motion due to very small voids. A similar contribution to O"y due to small scale voids is considered by Garner et al [9]. Ds is the self diffusion constant of the steel, and the term where it appears in eq. (6) represents . damage recovery. The other two terms in the expression for a represent the evolution, and saturation of, radiation damage hardening. 0"0 and A are temperature

dependent constants whose values are found from unirradiated strength data, using a equal to the cold work strain. a, band c are constants; b is found from cold work recovery data and a and c are parameters that are fitted to data for the yield stress after irradiation, as follows.

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The criterion for yield, <1>=0 can be used to evaluate O"ey after a given temperature and irradiation history using the above evolution equations. There is data available for AISI 316 steel [10] for comparison with the model, such as in Figure 1. One feature of our results is that O"ey eventually falls with the dose, due to the rise in f. This is essentially due to the stress-raising action of the voids on the ligaments between them.

3. TENSILE TESTING: The value of the above approach lies in the ability, through eq.(I), to follow the evolution of plastic strain during a tensile test. The model has the facility to predict failure by the criterion f* = I/qI . What is more, there can

develop an instability in the plastic flow described by these constitutive equations which serves as an equivalent failure criterion. During a uniaxial tensile test the evolution equations are:

¢=o

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O"li

0;,. <ry -L

<r11

. (1 - f) 2g<r/O' L f

<rl~ (2<r11 /<r/ + 1 h g) . . <r11 L

1- rf*' (1-f)g (7)

360

where

g 3ql q2 f* . h q2 0i I -- sm--O"y 2ay

(8)

f*' is df*/df, G' is dG/dO"y and E = G(O") is the uniaxial stress-strain relation for the

matrix material (here, the form of eq.(5) is used with a=E). The instability in the

plastic flow occurs when Yf*'(I-f)g --> 1 due to the evolution of the various parameters.

We have used the above formalism to attempt to account for the experiments of Dupouy et al [11], where the yield stresses, ultimate tensile stresses (UTS) and swellings were measured for irradiated AISI 316 cladding from Phenix fuel pins. We used the actual swellings and yield stresses for cladding at various points along the pin as initial data, and calculated the ultimate tensile stresses numerically for tests at room temperature. The results are shown in Figure 2 for two AISI 316 clad pins irradiated to 90 and 94 dpaF respectively. The dips in measured UTS follow the observed peaks in fractional swelling, as can be seen from the information in table 2. Where points are shown for the 94 dpaF case but none for the 90 dpaF case, the model failed to produce fracture up to stresses over 1000 MPa. Presumably, a different transgranular mode, such as channel fracture, would control failure in those cases. It is encouraging that the calculations show a qualitative agreement, especially in relation to the dip in failure stress at about 550 mm from the bottom of the fuel column (BFC) for the 94 dpaF example. The failure strains are not well reproduced, being up to 50% in our calculations for the higher failure stresses, whilst the observed values were typically less than 10%. However, the failure strains in our calculations should really be interpreted as local strains at the failure position rather than the observed uniform elongations. With this in mind, the similarity in shape of the calculated and observed failure strains is a further encouraging feature of the model. To relate the calculations to an actual failure strain would require consideration of the inhomogeneity of the tested material. Our calculations draw attention to a loss of ductility with swelling. The point at 550 mm from BFC in the 94 dpaF case had a swelling of about 20% and zero failure strain. Our calculations indeed suggest that instability and fracture are reached very early on ater yielding for this case.

The model can easily be extended to describe tube rupture tests, where an internal pressurisation produces a biaxial stress state in the cladding. There is a considerable amount of data from such tests involving irradiated cladding for comparison. We intend to assess our model against this data.

4. CONCLUSIONS: The growth of voids in cladding due to neutron irradiation alters the constitutive relations describing plastic deformation of the material. A simple model shows that the swelling has a minor effect on the yield stress of the material; the introduction of a slow decrease of this parameter with radiation dose. In contrast, the

plastic flow following yield is quite affected by the fractional porosity of the material, giving rise to a correlation of low ultimate tensile stress with high swelling. At a swelling of about 20% the model predicts failure at the yield point, in agreement with experimental results [11].

Table 1: Parameters used in the calculations for AISI 316 steel

0"0 = 328.5 - 0.214T MPa A = 935 - 0.319T MPa Ds = 3.7 x 10-5 exp(-33615mm2s-1 (T in Kelvin) a = 0.04 dpa(NRTt 1 b = 7 x 1O-15m-2

c = 0.055 dpa(NRTt l

D = 9.4 x 10-7 dpa(NRT)s-1 (Figure 1)

Table 2: Comparison of calculations with the data of ref U 11

Case Position from Swelling/% O"eyIMPa O'UTsIMPa £p 05*kc /MPa eiALC

BFC/mm

------------------------90dpaF 225 8 420 560 1 679 58

430 10 300 530 5 580 45 550 4 260 620 9 700 0.5 270 700 17

94dpaF 225 9 420 630 3 647 50 430 14 300 370 0.5 430 16 550 20 260 340 0 260 0 700 8 270 440 2 638 61

ACKNOWLEDGEMENT: This work was funded as part of the UKAEA's Programme of Underlying Research.

REFERENCES: 1. J.R. Matthews, T. Preusser, Nuclear Engineering and Design, 101 (1987) 281. 2. AL. Gurson, Journal of Engineering Materials and Technology, 99 (1977) 2. 3. V. Tvergaard, International Journal of Fracture, 11 (1981) 389,.l.8. (1982) 237. 4. M. Saje, J. Pan, A Needleman, International Journal of Fracture, 19 (1982) 163. 5. V. Tvergaard, A Needleman, Acta Metallurgica, 32 (1984) 157. 6. A Needleman, J.R. Rice in "Mechanics of Sheet Metal Forming" (eds. D.P. Koistinen et al), Plenum Publishing Corporation, (1978) 237. 7. See for instance, R. Bullough, S.M. Murphy, M.H. Wood, in "Dimensional Stability and Mechanical Behaviour of Irradiated Metals and Alloys", BNES Conference. Brighton. UK, (1983) Vol.1, p43. 8. J.R. Matthews, Harwell Report AERE-R11564, 1985. 9. F.A.Garner, M.L. Hamilton, N.F. Panayotou, G.D. Johnson, HEDL Report HEDL-SA-2518-FP. 10. R.L. Fish, N.S. Cannon, G.L. Wire, in "Effects of Radiation on Structural Materials", ASTM STP 683, Eds. J.A. Sprague and D. Kramer, ASTM, 1979, p450. 11. I.M. Dupouy, J.P. Sagot, J.L. Boutard, ibid. ref. [7], p157.

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