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FATIGUE CRACK GROWTH IN A LOW-ALLOY
STEEL
by
Jeremiah Kitheka Musuva, B.Sc.(Eng.), M.Sc., D.I.C.
A thesis submitted for the award of the
degree of Doctor of Philosophy in the
University of London
October 1980
Department of Mechanical Engineering
Imperial College of Science and Technology
University of London t.
London S.W.7.
Dedicated to my wife
Christine
and to my children
Mwende
Muoka
Munini
1
2
ABSTRACT
A fracture mechanics approach was applied in the investigation
of fatigue crack growth characteristics in BS4360-50D low alloy
structural steel currently used for construction of North Sea oil
platforms.
Fatigue crack growth tests were performed on standard compact
specimens in both air and salt water environments. The tests covered
the full subcritical range of growth rates, from threshold to ductile
tearing at very high values of AK.
The effects of stress ratio, thickness, salt-water environment,
loading frequency, as well as transient effects under two-level block
loading, were investigated. In general, the crack growth rates increased
with increasing stress ratio and plate thickness. These effects were
most significant at low growth rates in the thinnest plate and at low
stress ratios. The crack closure concept was found to be satisfactory
in explaining the observed effects of stress ratio and thickness. A
model based on the effective stress intensity factor range was developed
to characterise low crack growth rates and thresholds.
Cyclic frequency had little influence on the growth rates at low
stress ratios. However, at high stress ratios, lower frequencies
caused higher growth rates. In salt water the decrease in frequency
caused an increase in the crack growth rates at both low and high stress
ratios.
Elastic-plastic crack growth behaviour under both monotonic and
cyclic loading was also investigated using the J-integral approach. The
onset of slow stable crack growth was found to be thickness dependent.
A double-mechanism model was developed to describe the crack growth
under elastic-plastic conditions.
3
Two-level block loading fatigue tests were performed to
investigate the transient effects. The crack growth acceleration was
recorded in Lo-Hi block loading, while the crack growth retarded in
Hi-Lo block loading. Crack closure effects, strain hardening of crack
tip material, as well as crack front changes were used to explain the
observed transient interactions.
The use of an A.C. crack microgauge for crack length measurement
was also investigated. It was found that the microgauge was suitable
for crack length measurements under both constant amplitude and
variable amplitude loading.
r
4
ACKNOWLEDGEMENTS
The author wishes to acknowledge the generous financial support
provided by the Commonwealth Scholarship Commission in the United
Kingdom during the period of this research work. The extended study-
leave granted by the University of Nairobi, Kenya, is gratefully
acknowledged; in this respect the contribution of Prof. P.M. Githinji
of the Department of Mechanical Engineering.is especially noted.
Special thanks and gratitude are extended to my supervisor,
Dr. J.C. Radon, for his extremely valuable advice, guidance and
encouragement during the course of the work.
The help received from my colleagues as well as from members of
the technical staff in the laboratories and in the department workshop
is highly appreciated.
The author also wishes to extend special thanks to all those who
were involved in discussions on this work and in particular
Dr. R.D. Hibberd and Dr. P. Oldroyd of this Department.
Special thanks also go to Mr. Potter of the drawing office for
arranging the preparation of the figures and to Mrs. Robertson for
typing the thesis.
Last, but by no means least, the author wishes to express his
gratitude to his wife and children for their understanding, encourage-
ment and patience during the course of this work.
CONTENTS
Abstract
Acknowledgements
Contents
Notation
List of Tables
List of Figures
CHAPTER ONE: Introduction 27
1.1 Reasons for the present work 28
1.1.1 Background 28
1.1.2 Objectives 29
1.2 Outline of the work 29
1.2.1 Crack growth behaviour at intermediate stress intensities 29
1.2.2 Threshold and low crack growth behaviour 29
1.2.3 Elastic-plastic crack growth
29
1.2.4 Load interaction effects
30
CHAPTER TWO: A Review of Aspects Relevant to Fatigue 31
Crack Growth
2.1 Introduction 32
2.2 The micromechanisms of fatigue crack growth 38
2.3 Application of fracture mechanics to fatigue crack growth 43
2.3.1 Introduction 43
2.3.2 The crack growth rate curve 43
2.3.3 Theoretical prediction of fatigue crack growth:
Crack growth laws 45
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4
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19
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2.4 The effect of stress ratio: Crack growth laws 47
2.5 The effect of thickness on fatigue crack growth 52
2.6 The effect of frequency on fatigue crack growth 54
2.7 The effect of environment on fatigue crack growth 55
2.7.1 Introduction 55
2.7.2 Crack growth behaviour in water or salt solutions 57
2.7.2.1 Frequency effect 57
2.7.2.2 Stress ratio effect 58
2.7.2.3 Waveform effect 58
2.7.2.4 Electro-chemical potential effects 59
2.7.3 Models for corrosion fatigue crack growth 60
2.7.3.1 Superposition models 60
2.7.3.2 Process competition models 62
2.7.3.3 The process interaction model 63
2.8 Thresholds of fatigue crack growth 64
2.8.1 Criteria for non-propagating cracks 64
2.8.2 Effect of stress ratio on AKth 65
2.8.3 Effect of specimen geometry and crack length on AKth 68
2.8.4 The effect of environment on AKth 70
2.8.5 The effect of microstructure on AKth 71
2.9 Elastic-plastic fatigue crack growth 72
2.10 Load history effects in variable-amplitude loading 75
2.10.1 General observations 75
2.10.2 Predictive methods for variable-amplitude loading 77
2.10.3 Effect of environment on load history effects 83 • Figures 2.1 to 2.4 84
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CHAPTER THREE: Experimental details and data processing
techniques 86
- 3.1 Material and test specimens 87
3.2 Testing machines 88
3.3 Crack length measurements during fatigue tests 90
3.3.1 Crack length measurement using A.C. potential
91
drop crack microgauge
3.3.1.1 Effect of crack front curvature on digital 93
meter reading
3.4 Processing of fatigue crack growth data 94
3.4.1 The graphical method 94
3.4.2 The finite difference or secant method 95
3.4.3 Modified difference method 95
3.4.4 The incremental polynomial method 95
3.4.5 The total polynomial method 96
3.4.6 General discussion of the data processing methods 96
3.5 Determination of the stress intensity factor 98
3.6 Measurement of crack growth during slow stable 100
cracking under cyclic and monotonic loading
3.6.1 Surface observations 100
3.6.2 Fracture surface measurements (or multiple specimen 101
method)
3.6.3 Unloading compliance method 101
3.6.4 Crack growth measurements during the present tests 102
3.7 Determination of J-integral from load-deflection curves 103
3.7.1 Monotonic J-integral 103
3.7.2 Cyclic J-integral, MJ 105
3.8 The resistance curve and the critical J for crack 106
initiation
3.9 Summary 107
Page
CHAPTER FOUR: Fatigue Crack Growth Behaviour of 121
BS4360-50D Steel in Mid-range Growth Rates
4.1 Introduction 122
4.2 Experimental procedure 122
4.3 Results 126
4.3.1 Effect of stress ratio (or mean stress) 126
4.3.2 The effect of thickness 129
4.3.3 The effect of frequency 129
4.3.4 Effect of salt-water environment 130
4.4 Discussion 131
4.4.1 The effects of stress ratio and frequency in air 131
4.4.2 The effect of thickness 139
4.4.3 The effect of salt-water environment 144
4.5 Conclusions 149
Figures 4.1 to 4.28 152
CHAPTER FIVE: Threshold and Near-threshold Fatigue Crack 180
Growth in BS4360-50D Steel
5.1 Introduction 181
5.2 Experimental Procedure 182
5.3 The results 184
5.4 Discussion of results 187
5.5 A simplified crack growth model 197
5.5.1 Introduction 197
5.5.2 Mechanical and fatigue properties of the material 199
5.5.3 Stress and strain distribution ahead of crack tip 199
5.5.4 The effective stress intensity factor range, AKeff 201
5.5.5 Fatigue crack growth process 202
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Chapter Five (continued)
5.5.6 Determination of AKc th 204
5.5.7 Evaluation of crack growth rates and AKc th 206
5.5.8 Discussion of the crack growth model 208
5.6 Conclusions 210
Table 5.1 213
Figures 5.1-5.22 214
CHAPTER SIX: Elastic-Plastic Crack Growth Behaviour in 236
BS4360-50D Steel
6.1 Introduction 23-7
6.2 Experimental procedure 238
6.2.1 Compliance calibration curve 239
6.2.2 The monotonic J-tests 240
6.2.3 The cyclic J-tests 241
6.3 Results 242
6.3.1 Monotonic J versus as resistance curve 242
6.3.2 Cyclic crack growth results 243
6.4 Discussion 244
6.5 Theoretical considerations 247
6.5.1 Introduction 247
6.5.2 Stress and strain fields 248
6.5.3 Fatigue crack growth process 248
6.5.4 Crack growth evaluation 252
6.5.5 Discussion of the double-mechanism model 253
6.6 Conclusions 255
Figures 6.1 to 6.15 257
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CHAPTER SEVEN: Load History Effects During Two-Level 271
Block Loading in BS4360-50D Steel
7.1 Introduction 272
7.2 Experimental procedure 273
7.3 The results 277
7.4 Discussion of results 280
7.4.1 Crack growth retardation behaviour in Hi-Lo 284
block loading
7.4.2 Crack growth acceleration behaviour in Lo-Hi block 286
loading
7.4.3 Modelling crack growth retardation and acceleration 289
7.5 Conclusions 290
Table 7.1 294
Figures 7.1 to 7.20' 295
CHAPTER EIGHT: General Conclusions and Proposals for 315
Future Work
8.1 Introduction 316
8.2 General conclusions 316
8.3 Proposals for future work 320
APPENDIX: Fundamental Concepts of Fracture Mechanics 321
A.1 Introduction 322
A.2 Linear elastic fracture mechanics 322
A.2.1 The energy balance approach to fracture 322
A.2.2 The stress intensity factor (K) approach 325
A.2.2.1 Elastic stress distribution at the crack tip 325
A.2.2.2 Fracture analysis using K-Linear elastic fracture 328
mechanics (LEFM)
11
Page
Appendix (continued)
A.3 Non-linear (yielding) fracture mechanics (YFM) 330
A.3.1 The Dugdale model and the crack opening displacement 330
A.3.2 The J-integral 331
A.3.2 Crack tip stress and strain fields under elastic- 334
plastic conditions
REFERENCES
338
LIST OF PUBLICATIONS 365
12
NOTATION
Symbol Meaning
A Area under the load-deflection curve
A0 Constant in crack growth rate expression
Al, A2, etc. Constants
As Cracked surface area
a Crack length
ac Critical value of a for unstable crack growth
of Final crack length
ai Crack length at the .
cycle
a0 Initial crack length
a Characteristic crack length at a notch
Aa Crack length increment
OaA Crack length increment affected by acceleration
during Lo-Hi block loading
AaDM Crack length increment corresponding to minimum
growth rates during crack growth retardation
AaDT Total La affected by crack growth retardation
Aai Crack length increment after the .
cycle.
B Specimen thickness
B1 Constant in the crack growth rate expressions
BO Constant
B' Constant
b Constant in the crack growth rate equation
b0,bl,b2 Constants
•
• 13
C, Cl, C2 ...
etc.
C'
Constants in the crack growth rate expressions
C e
CR J
CO Constant
C p
Crack growth retardation factor
CN1,CN2 Constants
D1 Constant in crack growth rate expression
d Grain length (or size)
da/dN Crack growth rate
da/dt Crack growth velocity
E, E' Young's Modulus
f Frequency
G Strain energy release rate
Gc Critical value of G for unstable crack growth
Glc Critical value of G for crack initiation
G' Shear modulus
H Specimen height
hk Average rise and fall in load cycle
IOn A function of strain hardening exponent, of A and
of state of stress
J A contour integral
J ,J Maximum and minimum values of J max min
Jc' Jlc Critical values of J for crack growth
J., J. Onset of stable ductile tearinglc 1
JR Value of J cur,,-
OJ Cyclic value of J
AJc th Critical range of the threshold value of OJ
AJ Effective value of MJ for crack growth
14
K Stress intensity factor
KI' KII, KIII Stress intensity factor for modes I, II and III
Kc Critical value of K for unstable crack growth
Klc Minimum value of Kc
KCL Level of K required to open the fatigue crack
Keff Effective stress intensity factor
Kt Theoretical elastic stress concentration factor
Kf Fatigue strength reduction factor
KISCC Threshold value of K in stress corrosion
K Value of K at the onset of plateau in corrosion fatigue
K , K , Maximum, minimum and mean values of K during max min
K load cycling mean
Kmax,eff' Effective values of K and K
Kmin,eff max min
Kmax,th' Values of K and K . at threshold
max min Kmin,th
mail' Kmax2 Values of K
max prior to and after load change in
, block loading
Kminl mint Values of Km
in prior to and after load change in
block loading
K Root mean square value of K rms
AK = K -KRange of K in a loading cycle max min
AKc Range of K above KCL necessary for crack growth
AKc th
Critical range of AK at threshold
AKeff Effective stress intensity factor range
AKth Threshold value of AK
AKOth Value of AKth at zero stress ratio (R = 0)
AKTen The tensile part of 6K during tension-compression
load cycling
s
15
OK s Root mean square of AK rm
AK1, AK2 Values of AK prior to and after load change in
block loading
k Constant
k Petch's constant y
!~ Constant
M Stress intensification factor
m, ml,m2... etc. Exponents in the crack growth rate
me, mR expressions
N Number of cycles
ND Number of cycles for retarded crack growth
Nf Number of cycles to failure
Ni Number of cycles at the 1 cycle
AN Increment of number of cycles
AND Value of AN for delayed crack growth
n Cyclic strain hardening exponent
P Applied load
P ,P Maximum and minimum values of P in a loading cycle max min
AP Range of applied load in a loading cycle
q • Notch sensitivity factor
R Stress ratio (i.e. K . /K or a . /a ) in min max min max
a loading cycle
Rc Stress ratio value corresponding to crack closure
Rc th Value of R corresponding to LKc th.
✓ Radial coordinate at the crack tip
rc Critical crack tip radius
ry Monotonic plastic zone size at the crack tip
ryc Cyclic plastic zone size at crack tip
ryi Current plastic zone size in the ith cycle after
load change
•
16
ryo Plastic zone size created by previous overload
s Arc length
sT Arc length on which tractions act
T Traction force
t time
U Potential energy
Up Pseudo potential energy
U1 Ratio of the effective to total values of AK
AU Change in potential energy
AUeS Change in elastic potential energy
AUps Change in plastic strain energy
AUs Change in energy to create new fracture surfaces
AUN Total accumulated potential energy after N-cycles
AU1 Potential energy in the first cycle
u Displacement
v Crack opening displacement (COD)
VD Digital voltage
VP Critical volume near the crack tip
W Specimen width measured from the loading points
w Strain energy density
x,y Cartesian coordinates
Y Geometric factor in the stress intensity factor
expression
a Constant of state of stress
a0 Constant
al Constant in the crack growth rate expressions
°'131Constants in the crack growth rate expressions
y Constant
Ys Specific free surface energy
• 17
Y Plastic fracture surface energy P
y0 Constant in crack growth rate expression
S Deflection at the loading points
S Fixed deflection Sm
OSel Range of linear elastic deflection at the loading
line in the load cycle
ASl Range of deflection at the loading points in the
first load cycle
OSN Range of deflection at loading points accumulated
after N cycles
e Strain
Ey Tensile yield strain corresponding to ay.
syc Cyclic yield strain corresponding to ayc
E e
Elastic strain
Ef True fracture strain approximated by the fatigue
ductility coefficient
E Plastic strain P
1e(x) Total strain range in the x direction
DeT(x) Tensile strain range in the x-direction
LEe Elastic strain range
AEp Plastic strain range
0 Angular coordinate at the crack tip
p Crack tip radius
P* Microstructural dimension
X Constant in the J-integral expression
Al Constant
✓ Poisson's ratio
o Applied normal stress level
acf Critical stress for fracture
of Fracture stress
18
ah Stress reduction due to hydrogen environment action
aN Nominal applied stress
a Root mean square of applied stress signal rms
a max ,a mm. n,amean Maximum, minimum and mean values of a in a loading
cycle
a Tensile yield stress y
ayc Cyclic yield stress
auts Ultimate tensile strength
aflow Mean flow stress = (a + a
uts )/2
da Stress range in a loading cycle
&a(x) Stress range in the x direction
Dae Stress range corresponding to fatigue limit
Shear stress
Yield stress in shear
Width of the intense zone of deformation at the
crack tip
ABBREVIATIONS
ASTM American Society for Testing and Materials
COD Crack opening displacement
Hi-Lo High-Low
LEFM Linear elastic fracture mechanics
Lo-Hi High-Low
YFM Yielding Fracture Mechanics
STP Special Technical Publication
rms Root mean square
19
LIST OF TABLES
Table 3.1. Steel BS4360-50D. (a) Chemical Composition (Weight);
(b) Mechanical Properties; (c) Fatigue Properties.
Table 3.2. List of Tension Component Correction Factors,
(Equation (3.12), for the Compact Specimen Geometry.
Table 5.1. Comparison between experimental and theoretical values
of LKc,th for various steels.
Table 7.1. Crack length increments affected by load interaction
effects.
•
20
LIST OF FIGURES
Figure 2.1. Schematic representation of the formation of a fatigue
striation.
Figure 2.2. Fatigue crack growth rate da/dN versus AK (schematic).
Figure 2.3. A typical fatigue crack growth curve showing
environmental effects.
Figure 2.4. Delay and transient acceleration phenomena.
Figure 3.1. Compact specimen geometry used in the experimental
programme.
Figure 3.2. Specimen preparation for the salt-water environment
tests.
Figure 3.3. Compact specimen modified for deflection measurement
at the loading line.
Figure 3.4. Positions of current and probe terminals for crack
length measurement and crack front configuration
estimation using the crack microgauge.
Figure 3.5. Experimental set-up showing the positions and fixing
of the current and probe terminals during the block
loading tests.
Figure 3.6. Variation of digital voltage with crack length at
various gain settings of the crack microgauge.
Figure 3.7. The effect of crack front curvature on the digital
voltage reading of the crack microgauge in the 50 mm
thick specimen.
Figure 3.8. Comparison of da/dN versus OK data calculated using
the secant and polynomial curve-fitting methods: Data
at R = 0.08 and 30 Hz.
Figure 3.9. Comparison of da/dN versus AK data calculated using
the secant and polynomial curve fitting methods: data
at R = 0.7 and 0.25 Hz.
•
21
Figure 3.10. Typical load versus load-point displacement curve
under monotonic loading: J definition.
Figure 3.11. Operational definition of 'OJ' after Dowling and
Begley [Ref. 150].
Figure 3.12. Operational definition of 'MJ' after 'N' loading
cycles, after Branco et al. [Ref. 152].
Figure 3.13. Schematic representation of the J-resistance curve
and changes of crack tip shape.
Figure 4.1. Crack length, a, versus number of cycles, N, for a
constant stress ratio fatigue test.
Figure 4.2. da/dN versus OK; B = 50 mm, 30 Hz, in air.
Figure 4.3. da/dN versus AK; B = 24 mm, 30 Hz, in air.
Figure 4.4. da/dN versus AK; B = 12 mm, 30 Hz, in air.
Figure 4.5. da/dN versus AK, at constant Kmean ; B = 24 mm,
30 Hz, in air.
Figure 4.6. da/dN versus QK, at constant K ; B = 12 mm, mean
30 Hz, in air.
Figure 4.7. da/dN versus AK; B = 24 mm, 0.25 Hz, in air.
Figure 4.8. da/dN versus AK; B = 12 mm, 0.25 Hz, in air.
Figure 4.9. Effect of negative stress ratio on da/dN versus
AKTen; B = 24 mm, at 1-10 Hz, in air.
Figure 4.10. Effect of negative stress ratio on da/dN versus LK;
B=12 mm, 0.12 Hz, in air.
Figure 4.11. da/dN versus OKTen; B = 12 mm, 0.12 Hz, in air.
Figure 4.12. Stress ratio effect on crack initiation and growth
from a notch; B = 24 mm.
Figure 4.13. R versus N at various values of as measured from the
notch (from Figure 4.12).
22
Figure 4.14. Effect of thickness on da/dN versus AK; R = 0.08,
30 Hz, in air.
Figure 4.15. Effect of thickness on da/dN versus AK; R = 0.7,
30 Hz, in air.
Figure 4.16. Effect of thickness on da/dN versus AK; 0.25 Hz,
in air.
Figure 4.17. Effect of frequency on da/dN versus AK; B = 24 mm,
in air.
Figure 4.18. Effect of frequency on da/dN versus AK; B = 12 mm,
in air.
Figure 4.19. Effect of frequency on da/dN versus AK; B = 24 mm,
R = 0.08, in air.
Figure 4.20. da/dN versus AK; B = 24 mm, 30 Hz, in 3.5% NaC1 solution.
Figure 4.21. da/dN versus AK; B = 24 mm, 0.25 Hz, in 3.5% NaC1
solution.
Figure 4.22. Effect of frequency on da/dN_versus AK; B = 24 mm,
R = 0.08, in 3.5% NaC1 solution.
Figure 4.23. Effect of frequency on da/dN versus AK; B = 24 mm,
R = 0.7, in 3.5% NaC1 solution.
Figure 4.24. da/dN versus AK; B = 24 mm, 30 Hz, in air - compared
with Scott & Silvester's [Ref. 110] results.
Figure 4.25. da/dN versus AK; B = 12 mm, 30 Hz, in air - compared
with Scott & Silvester's [Ref. 110] results.
Figure 4.26. da/dN versus AKeff; B= 12 mm and 24 mm, 0.25 Hz, in air.
Figure 4.27. da/dN versus AK; B = 24 mm, showing upper bounds for
salt-water data.
Figure 4.28. Correlation of da/dN versus AK data in 3.5% NaC1
solution using a superposition model.
23
Figure 5.1. Typical crack growth curves, a versus N, during
threshold tests.
Figure 5.2. Digital voltage change versus crack length increment
during a threshold test, compared with the prior
calibration curve.
Figure 5.3. da/dN versus AK; B = 12 mm, 30 Hz, in air.
Figure 5.4. da/dN versus AK; B = 24 mm, 30 Hz, in air.
Figure 5.5. da/dN versus AK; B = 50 mm, 30 Hz, in air.
Figure 5.6. Effect of thickness on da/dN versus AK; R = 0.08,
in air.
Figure 5.7. da/dN versus AK; B = 24 mm, in 3.5% NaC1 solution,
compared to air data.
Figure 5.8. Variation of AKth with R, in air.
Figure 5.9. Variation of AKth with thickness, in air.
Figure 5.10. da/dN versus AK data from "increasing R" test.
Figure 5.11. Variation of AK with R for different steels.
Figure 5.12. Positions for estimating KCL from da/dN versus AK
data; B = 12 mm.
Figure 5.13. AK versus R curves for estimation of KCL; B = 12 mm.
Figure 5.14. Estimated values of KCL; B = 12 mm and B = 24 mm.
Figure 5.15. Measured values of KCL plotted against Kmax'
Figure 5.16. da/dN versus AKeff; B = 12 mm, in air.
Figure 5.17. da/dN versus LiKeff; B= 24 mm, in air.
Figure 5.18. Correlation of AKth with R using crack closure concept.
Figure 5.19. Crack tip deformation and crack growth during a fatigue
cycle (schematic).
Figure 5.20. da/dN versus AK at R = 0.7, in air; compared with
theoretical crack growth rates: BS4360-50D steel.
•
24
Figure 5.21 da/dN versus AK at R = 0.8, in air; compared with
theoretical crack growth rates: HT80 steel (Ref. 68].
Figure 5.22 da/dN versus AK at R = 0.72 in air; compared with
theoretical crack growth rates: C-Mn steel (Ref. 128].
Figure 6.1. Compliance calibration curve for the CS geometry used.
Figure 6.2. J-test using 10% unloading technique; B = 24 mm.
Figure 6.3. Typical hysteresis loops during cyclic loading.
Figure 6.4. J versus Ea (resistance curve) for B = 24 mm.
Figure 6.5. J versus pa for B = 12 mm, 24 mm and 50 mm.
Figure 6.6. Critical values of J versus specimen thickness.
Figure 6.7. Fracture surfaces showing extent of ductile tearing
during J-tests.
Figure 6.8. Elastic-plastic fatigue crack growth data; R = 0.08.
Figure 6.9. Cyclic and monotonic crack growth at high stress
intensities.
Figure 6.10. Crack tip stress and strain fields (schematic).
Figure 6.11. Crack tip deformations and crack growth under increasing
load.
Figure 6.12. Crack growth by monotonic loading.
Figure 6.13. Modelling of elastic-plastic crack growth in
BS4360-50D steel.
Figure 6.14. Experimental and predicted da/dN versus AJeff data:
BS4360-50D steel.
Figure 6.15. Experimental and predicted da/dN versus MJ data:
BS15 steel.
Figure 7.1. Apparatus used for the block-loading testing programme.
Figure 7.2. Calibration curve for the crack microgauge.
Figure 7.3. Calibration curve for the chart recorder.
Figure 7.4. The effect of stress on the digital voltage reading of
the crack microgauge.
•
25
Figure 7.5. as versus AN; crack length measurements using both a
microscope and the a.c. microgauge in Hi-Lo block
loading.
Figure 7.6. da/dN versus Aa; retarded growth rates detected using
both microscope and the a.c. microgauge.
Figure 7.7. La versus AN; crack length measurements using both a
microscope and the a.c. microgauge in Lo-Hi block
loading.
Figure 7.8. da/dN versus Aa; accelerated growth rates detected
using both the a.c. microgauge and a microscope.
Figure 7.9. da/dN versus Aa data for Hi-Lo block loading at R = 0.
Figure 7.10. da/dN versus åa data for Hi-Lo block loading R = 0.
Figure 7.11. da/dN versus Aa for Lo-Hi block loading, at R = 0 and
for Kmaxl = 15 MN/m3/2.
Figure 7.12. da/dN versus Aa for Lo-Hi block loading at R = 0 and
at various Kmaxl values.
Figure 7.13. da/dN versus to for Hi-Lo block loading at same AK
but different K mean'
Figure 7.14. da/dN versus Aa for Lo-Hi block loading at same AK but
different K mean'
Figure 7.15. da/dN versus Aa data; at constant Kmax and with changing
Kmin
from zero to -30 MN/m3/2 and back to zero.
Figure 7.16. Upper bound for the crack growth acceleration.
Figure 7.17. The effect of crack length on growth rates during block
loading test.
Figure 7.18. Variation of AaDM
with AK1.
Figure 7.19. Variation of AaDT
with Kmaxl'
Figure 7.20. Variation of AaA with max2/Knaxl during Lo-Hi block
loading.
•
Figure A.1.
Figure A.2.
Figure A.3.
Figure A.4.
Figure A.5.
Loading modes.
Kc versus specimen thickness (schematic).
The J-integral.
Evaluation of potential energy balance.
Contours around a narrow crack.
26
• 27
CHAPTER ONE
INTRODUCTION
28
1.1 REASONS FOR THE PRESENT WORK
1.1.1 Background:
The rapid expansion of the oil and gas exploration and development
programmes in the North Sea in recent years have led to a considerable
variety of offshore structural installations, and many more will be
built in the future. Such structures must be safely designed with
respect to a number of possible failure modes resulting from the forces
acting on them.
The forces acting on an offshore installation derive from a large
number of sources, but it has been possible to define four principal
loads, namely: dead, hydrostatic, environmental and imposed. Of these,
the cyclic loading produced by the wave action (environmental loading)
on the structures is now considered as a potential cause of failure by
fatigue, since in the North Sea persistent bad weather and heavy seas
occur for a large part of the year It is pertinent therefore to have
sufficient information on fatigue behaviour of structural steels used
for construction of structures in the North Sea.
These offshore structures, which are welded in construction,
contain crack-like flaws before installation or develop these during
the early part of their service life. Thus virtually all the life will
be taken up with fatigue crack growth. It is necessary therefore to
have adequate fatigue crack growth data for the life prediction of
these structures. Considerable effort is now being expended, in
different centres for offshore research throughout the United Kingdom,
in obtaining basic fatigue data under various loading and environmental
conditions relevant to the North Sea. The present work is a part of
this major programme.
29
1.1.2 Objectives:
The main objective of the present work was to generate fatigue
crack growth data for BS4360-50D low alloy structural steel used in the
North Sea. The aim was to investigate the effects on crack growth
rate of such variables as stress ratio (or mean stress), thickness,
frequency, load history as well as corrosive environment. In all the
investigations a fracture mechanics approach was used.
1.2 OUTLINE OF THE WORK
The work was carried out in four stages:
1.2.1 Crack growth behaviour at intermediate stress intensities:
The effects of stress ratio, thickness, frequency and salt water
environment were investigated. The aim was to obtain data by varying
the above factors over a wide range. The results showed a significant
influence of stress ratio and thickness at lower stress intensities
and this led to the second stage of the work.
1.2.2 Threshold and low crack growth behaviour:
Influence of stress ratio and thickness were the main factors
investigated. Thickness had not been investigated before at low stress
intensities.
1.2.3 Elastic-plastic fatigue crack growth:
Crack growth behaviour at high stress intensities was investigated.
The main aim was to obtain preliminary data on slow stable crack growth
under monotonic loading and to use these to predict elastic-plastic
fatigue crack growth.
30
1.2.4 Load interaction effects:
To investigate the transient effects observed after load changes
in simple variable-amplitude loading. In particular to measure crack
growth retardation and acceleration that occur in block loading.
Each of the above stages of work was supported by some theoretical
analysis of the data. In addition, a considerable amount of time was
spent on investigating the use of an AC potential drop microgauge for
crack length measurements, especially under variable amplitude loading.
31
•
CHAPTER 2
A REVIEW OF ASPECTS RELEVANT TO
FATIGUE CRACK GROWTH
32
2.1 INTRODUCTION
Investigations into structural failures have revealed that
components may fail in service by a number of different processes [1-4].
At normal temperatures and under static stresses fracture occurs either
because the normal tensile strength of the material is exceeded or
because stress concentration due to defects or design faults raises
the level of stress locally above the failure level. Two modes of
fracture can be identified for such failures, namely brittle or ductile
failure, but in neither case does the fracture type imply that the
component has or has not undergone a measurable amount of plastic
distortion.
Brittle failure is found not only in materials such as plastics
and ceramics but also in structural materials such as steels at
temperatures below the ductile/brittle transition or at high strain
rates. The fracture is usually characterised by a flat fracture surface
and occurs with little plastic deformation and at extremely high speeds.
Ductile failure, however, involves considerable localised plastic
deformation and the fracture surface is often characterised by a
dimpled surface.
Another type of failure usually found in materials which are
normally capable of undergoing plastic deformation is creep. Normally
creep occurs to a significant extent only at temperatures above
approximately half of the absolute melting temperature, which means
normal ambient temperatures for polymers but elevated temperatures for
most metals and alloys. Creep deformation occurs at stresses well
below the normal yield strength of the material and consists of a time-
dependent deformation, which may eventually lead to rupture of the
component or sufficient distortion to make the component unserviceable.
33
Fatigue failure is the major type of failure occurring in
structural components and is thought to be responsible for over 90%
of the failures which occur in service. Metal fatigue is a failure
phenomenon which has been recognised and studied since the middle of
the nineteenth century and now has a voluminous literature. The word
'fatigue' was then introduced to describe the progressive weakening of
a test piece or component subject to cyclic loading. Fatigue failure
involves cyclic stresses produced by fluctuating load conditions or
thermal cycling effects. Cyclic stresses well below the normal tensile
yield strength of the material can initiate surface cracks, usually at
points of stress concentration, and these cracks grow progressively
under the influence of tensile cyclic stresses. Ultimately the cross
sectional area of the sound material remaining cannot support the stresses
and catastrophic failure occurs by either brittle or ductile failure.
Some other types of failure involving environmental cracking such
as stress corrosion and corrosion fatigue also occur in service.
Failure under static stresses in a corrosive environment is usually
referred to as stress corrosion and involves the combined action of
corrosion and static stresses which, with time, generates cracking and
may lead to ultimate failure. Environmental cracking under cyclic
stresses on the other hand is known as corrosion fatigue. Here the
combined action of cyclic stress and corrosive attack enables a fatigue
crack to initiate and grow at a lower stress amplitude level than in a
test in air.
As has been mentioned above, fatigue failure is the main type of
failure occurring in service. Failure by fatigue has been subjected to
numerous investigations for the last century or so. However, its
application to structural failure has only received intensive studies
in the last three decades. Prior to this all investigations were
0
34
concerned with machine components since this was the area where most
fatigue failures were to be found.
Since the advent of new materials such as high strength alloy
steels and aluminium alloys, and new methods of manufacture and
fabrication, such as welding, structural fatigue failures have been
numerous. Designers are usually keen to exploit the strength or tough-
ness of a new material to the full, but unfortunately such static strength
properties are not always accompanied by corresponding improvement in
the fatigue resistance properties. Consequently design on the basis of
static strength properties has raised working stresses into the region
where fatigue is a major factor in the integrity of the structure.
The introduction of welding as a method of fabrication of structures
has resulted in severe fatigue problems. Defects within the weld and
stress concentrations due to the weld geometry are influencing factors
in fatigue failure. Such defects can take many forms, such as cracks
due to residual stresses in the weld region, lack of penetration,
inclusions and porosity and it is these which are harmful in terms of
fatigue. It is not therefore surprising that many laboratory fatigue
tests have shown that fatigue life of welded specimens is considerably
reduced. This is because the period normally required for the crack to
initiate is virtually absent and consequently the total fatigue life
is spent on crack growth.
The fatigue life of a component is determined by the sum of the
life required to initiate a fatigue crack and to grow the crack from a
subcritical dimension to the critical size before fracture occurs.
Consequently the fatigue life of a structure can be considered to be
composed of three phases: (1) fatigue crack initiation, (2) fatigue
crack growth and (3) final fracture. The contribution of fatigue
crack initiation and fatigue crack growth to the fatigue life of a
M
35
structural component depends on the type of structure and its intended
application. For example, the fatigue life of components designed for
infinite-life application, such as engine components, may be governed
by the fatigue crack initiation characteristics. On the other hand,
structural components that contain stress concentrations or initial
defects, such as welded structures,. may have their fatigue life primarily
dependent on the fatigue crack growth characteristics. Consequently,
the useful life of a structural component subjected to cyclic loading
can be determined only when the three stages in the life of the
component are evaluated individually and the cyclic behaviour in each
stage is thoroughly understood.
Conventional methods [5,6] used to design components against
fatigue provide a design fatigue curve, which characterises the basic
unnotched fatigue properties of the material, and a fatigue-strength
reduction factor. The fatigue-strength reduction factor incorporates
the effects of all the different parameters characteristic of the
specific structural component that make it more susceptible to fatigue
failure than the unnotched specimen, such as surface finish, stress
concentrations, defects, etc. The design curves are based on the
fatigue life prediction data in the form of cyclic stress versus life
curve (S-N curves) as determined from laboratory specimens. S-N curves
show an endurance limit, at least for ferrous materials. This limit
could be used to calculate a working stress, at which, theoretically,
the fatigue life should be infinite. Alternatively, if a shorter life
was acceptable, or if no fatigue limit occurred, then the working stress
could be calculated such that the required service life was achieved.
The application of S-N curves is not straight-forward as practical
components have stress concentrations, cannot be manufactured to the
same standards as test specimens and operate in adverse environments;
36
all of which will tend to influence the fatigue life.
The fatigue behaviour of materials presented in S-N curves refers
to fatigue under constant amplitude loading. In practice, however,
most structures are subjected to variable amplitude loading with the
variation in stress level following either a regular or random pattern.
The problem of fatigue life prediction under variable amplitude loading
has become known as cumulative damage. It involves the appreciation of
the amount of damage created during each and every cycle, and how that
damage may be summed and equated to a failure criterion.
The simplest empirical law for predicting cumulative damage was
postulated by Palmgren [7] and Miner [8] and is usually known as the
'linear cumulative damage law'. Later work has shown considerable
deviation of experimental data from the Palmgren-Miner law. Several
attempts have recently been made to improve cumulative damage predictions.
Among them is the 'double linear damage law' by Manson et al. [9] which
considers the damage process to consist of two distinctive stages which
are necessarily related to crack initiation and growth. A related
approach has recently been proposed by Miller and Zacharia [10] in an
attempt to predict the fatigue crack initiation life. Some of the
above cumulative damage laws and others reported recently [11] have been
compared with experimental data and their relative merits discussed [12].
As mentioned earlier, where weld defects and stress concentrations
occur, it is the crack growth phase of fatigue that will control the
life. Thus under those conditions, accurate life prediction can be
made on the basis of the time taken to grow a fatigue crack from an
initial defect to a critical size which causes unstable fracture. Thus
the essence of the integrity of a welded structure or joint is that a
crack should not grow, under the influence of cyclic loading, from the
initial defect to a critical size during the design life. The critical
size will depend on the criterion of failure, which in turn depends on
37
the structure and the material.
The prediction of fatigue life based on crack growth relies on
the ability to calculate the rate at which a crack will grow in a
given position in a structure. This is an extremely complex task
because the rate of growth of a crack is dependent on its size and
shape, the stress field around the crack, the environment and the
material into which the crack is growing. In recent years, research
into fatigue crack growth has shown fracture mechanics [13,14] a
satisfactory approach to fatigue crack growth. Fracture mechanics
provides the necessary descriptive and analytical framework for the
study of fracture processes. The basis of fracture mechanics is that
any flaw or defect, such as a fatigue crack, produces stress concentration
within a material and this in turn leads to the creation of intensified
stresses at the crack tip. Providing the nominal stress acting upon a
material is less than 0.8 of the yield strength, and also providing
that the region of plastic deformation ahead of a crack tip is small
relative to the total component size, the component can be considered
as a macroscopically elastic body. The state of stress ahead of the
crack is then determined by the linear elastic fracture mechanics [13].
In this case the stress intensity factor, K, can be used as a single
parameter to describe the elastic stress field in the vicinity of the
crack tip. On the other hand, under elastic-plastic conditions at
the crack tip (i.e. general yielding at the crack tip) other parameters
such as the J-integral have been used to characterise stresses and
strains at the crack tip. A detailed description of the fundamental
concepts of fracture mechanics is given in the Appendix.
•
2.2 THE MICROMECHANISMS OF FATIGUE CRACK GROWTH
The micromechanisms responsible for fatigue crack growth produce
average crack growth rates per cycle over a wide range. At low applied
stresses, this average growth rate approaches atomic dimensions of the
material whilst at high stress intensities growth rates of a millimetre
have been measured. Therefore a single mechanism is not responsible
for growth over such a wide range.
The growth of fatigue cracks that initiate in the persistent slip
bands in a shear mode have been called stage I propagation by Forsyth
[15], and it normally extends only a few tenths of a millimetre into the
specimen. The evidence in the literature suggests that the mechanisms
of slip controlled initiation and stage I growth are similar and are
affected by microstructure in similar ways [16]. In polycrystalline
materials, stage I growth involves many individual slip-band cracks
which eventually link up to form a dominant single crack at about the
time the propagation mode changes. The crack turns, at a certain crack
length, from the initial 45° to the stress axis to propagate along a
plane 90° to the stress axis. The tensile mode of growth is called
stage II propagation by Forsyth [15] and the change occurs when the
local stress concentration at the crack tip reaches a critical value.
The lower the stress or strain amplitude the more extensive is the
stage I process and the deeper stage I cracks penetrate the material.
When a crack grows from a notch it usually starts its growth as a
stage II crack. Transition from stage II to stage I has also been
reported and many non-propagating cracks, originally growing in stage II,
have been found to revert to stage I before effectively stopping.
Striations on stage I fracture surfaces have been observed [17] and
it is evident that reversed plastic deformation is necessary for the
38
39
growth of stage I cracks [18]. Kaplan and Laird [18] proposed two
possible mechanisms for stage I crack growth, namely, a plastic blunting
mechanism and an unzipping model which has recently been applied by Kuo
and Liu [19], in a finite element model of fatigue crack growth. This
unzipping mechanism assumes that no tensile stress acts across the
primary slip planes [20].
Because stage II cracking is more extensive than stage I, much more
work has been done in the tensile mode and the microstructural mechanisms
of stage II are clearer - as discussed in a recent review by Tomkins
[21]. The stage II fatigue crack growth process, as defined by Forsyth
[15] on the basis of fractography and metallography is seen as defining
the growth stage which produces fatigue striations (Figure 2.1). These
striations have proved to be of considerable diagnostic use when
investigating service failures. Forsyth and Ryder [22] demonstrated
that each striation corresponded to one cycle of crack advance and that
the striation spacing corresponded very closely to the average macro-
scopic cyclic growth rate, da/dN, of the fatigue crack. Forsyth [15]
also noted that stage II crack growth is macroscopically and to a large
extent microscopically normal to the maximum applied tensile stress.
However it is observed that local direction of growth may deviate from
the average direction and that other mechanisms may be occurring
concurrently. It can be suggested that each striation is produced by a
single cycle, however, the converse is not necessarily valid [16], since
the load levels may not be high enough to produce an increment of the
crack length. This point is probably more relevant to low stress cycling
and in random loading situations where low stress cycles are involved.
The simple mechanism of crack opening which produces stage II
crack growth in ductile materials was identified by Cottrell [23] for
plane strain conditions. This model shows that crack advance is
•
40
continuous and occurs by the injection of edge dislocations with
resulting shear displacements along lines at ±450 to the crack plane
ahead of the tip. As defined by Cottrell, for simple elastic-perfectly
plastic material, the crack advances by half the crack tip opening
displacement. Tomkins and Biggs [24] observed directly this advance
process in high strain fatigue for aluminium and noted that it led to
striation formation.
Pelloux [25] postulated and later Neumann [26] observed in copper
single crystals, that crack advance occurred by shearing on alternate
450 planes, thus maintaining a sharp crack tip. For simultaneous
operation of the planes, a blunt tip is initially formed but this would
sharpen on continued loading by tensile failure of the strained material
at the centre of the blunt tip. Wanhill [27] supported this mechanism
by pointing out that in an oxidising environment, fresh metal emerging
from the shear planes would readily oxidise while simultaneously under-
going continuous tensile straining. Fracture of the oxide would trigger
new shear bands, thus maintaining a sharp tip.
Laird [28] observed that crack tip blunting did occur at maximum
tensile strain with crack advance during resharpening of the tip under
the compressive part of the cycle by local instability. He observed
that the formation of striations was a result of localised shear
deformation. In this respect, striations can form during tensile
loading, as in stretch zones, and be accentuated by unloading on reversal
of the shear flow. As such, they do not necessarily represent the amount
of new crack surfaces created each cycle as they are a reflection of the
shear process which achieves both this and the accommodation of the
crack opening at the tip.
Forsyth et al. [29] distinguished between ductile and brittle
striations. Ductile striations consist of light and dark bands and
41
usually form on plateaux parallel to the general fracture plane, that is,
at 90° to the maximum tensile stress [30]. Brittle striations are
usually less regular in spacing than ductile striations and are formed
in association with river markings (characteristic of cleavage) on
very flat facets parallel to crystallographic planes. As their name
suggests they show very limited ductility. For both types of striations
the curvature is away from the origin of failure and large local variation
in spacing occurs as final fracture is approached in a constant load
amplitude test. The plateaux and facets on which striations lie are
usually at different levels joined by regions of ductile tearing [30].
McMillan and Pelloux [31] have shown that crack advance occurs
only during the loading half cycle and that the striation appearance
could be modified, for some aluminium alloys by unloading. They also
highlighted the importance of peak stress and load sequence effects on
striation appearance.
Crack growth mechanism by striation formation discussed above is
idealised for homogeneous ductile material. However, if the material
contains brittle inclusion particles [32] or embrittled grain boundaries
[33] static fracture processes at these second phase particles and
boundaries will certainly accelerate crack growth rate. These static
micro-fracture modes, frequently associated with stage III of crack
growth, include intergranular fracture, quasi-cleavage fracture and
micro-void coalescence as discussed below.
The final stage of crack growth before final fracture is called
stage III. This stage involves the addition of a new fracture mechanism
to stage II crack growth. When any crack in a material with some
capacity for plastic deformation is opened, some localised shear at the
crack tip will occur, but in addition, at large crack openings potential
crack nuclei ahead of the crack tip may be activated. These can range
42
from grains suitably oriented for cleavage to inclusions (or second
phase particles) to micro-voids and hydrogen clusters. All have the
effect of reducing material toughness or resistance to crack extension.
Crack extension is localised in the plane weakened by the additional
fracture damage, usually either the crack plane'for normal tensile
loading or the ±450 shear planes, and crack advance becomes discontin-
uous [231. It involves the extension of the main crack step by step
during a load increment, by plastic deformation of the ligament between
the main crack tip and the adjacent growing micro-void ahead of the tip.
The micro-void growth is confined to the crack tip region with most
growth and linkage seeming to occur in the 45° shear regions as well as
directly ahead of the tip [21]. This may sometimes result in growth of
the main crack in the slant plane. Growth prior and subsequent to this
rotation is usually termed 'plane strain' and 'plane stress' respectively.
For a given specimen thickness the transition has been related to a
critical growth rate and a critical stress intensity factor [34].
Substantial evidence is now available to suggest that the transition is
accompanied by a deceleration in crack growth rate [35] although this is
still disputed [34].
Finally it is worthwhile to mention that the onset of the different
stages of crack growth mechanism during the fatigue crack growth life
will depend on the material. Indeed in some materials, especially a
brittle one, the middle stage (II), where the purely cyclic mechanism
predominates, may not occur at all. This is because a large proportion
of static fracture modes may be present throughout the growth of a
fatigue crack. The interactions between the various mechanisms means
that the occurrence of a simple predictive growth model is very
unlikely. This point will be highlighted in the next section where the
crack growth rate curve will be discussed and factors affecting the
crack growth rates examined in detail.
43
2.3 APPLICATION OF FRACTURE MECHANICS TO FATIGUE CRACK GROWTH
2.3.1 Introduction:
The prediction of fatigue crack growth life relies on the ability
to calculate the rate at which a crack will grow in a given location of
a structure. There are two main objectives in predicting fatigue crack
growth rates. Firstly it is desirable in engineering service to be
able to predict growth rates from a knowledge of stresses and other
geometrical dimensions, such as crack length, in order to predict
residual lives of the component. This information would be useful in
deciding on inspection intervals for certain structures on which non-
destructive inspections are carried out during their fatigue life.
Secondly it is useful to be able to predict which of a selection of
materials has the greatest resistance to fatigue crack growth and hence
predict which is most suited for a particular application. Such
information will only result from a thorough understanding of the crack
growth processes and how such processes may be characterised in terms
of some loading and geometric parameters.
2.3.2 The crack growth rate curve:
The fracture mechanics approach [Appendix] to fatigue crack growth
has been found particularly suitable and the method has been success-
fully applied in studies of cyclic crack growth behaviour in a wide
range of materials. It has been shown that the most suitable parameter
for characterising fatigue crack growth is the stress intensity factor,
K, which can be used to describe both the final fracture and the
preceding stable subcritical crack growth. However, the successful
application of growth data, analysed using the stress intensity factor,
depends on the solutions of K being available for the complicated
geometry of real structures.
44
Recent studies have also confirmed that the fatigue crack growth
rate, da/dN, is primarily controlled by the stress intensity factor
range, AK through the well-known Paris expression [36]
da dN
= CdKm (2.1)
where C and m are constants depending on a variety of factors, such as
material, loading conditions, temperature and environment. Experimental
investigations on a wide range of materials have reported values of m
lying in the range of 2 to 20. This range of values is found because
the expression (2.1) is not in general obeyed over a wide range of
growth rates. These investigations, however, have shown that
expression (2.1) adequately describes the fatigue behaviour in the mid-
range of growth rates between 10-5 to 10-3 mm/cycle, but overestimates
and underestimates the lower and higher growth rates respectively.
Consequently, the variation of the crack growth rates with the stress
intensity factor could be better described by a sigmoidal curve of
log da/dN versus log AK, as shown for steels in [37], Figure 2.2.
This sigmoidal variation of growth rates with AK has recently [37]
been rationalised in terms of primary crack growth mechanisms, in order
to explain the sensitivity of stress ratio, thickness, microstructure,
environment, etc. on the crack growth rates. It was shown that in
steels, for the mid-range of growth rates (regime II), failure generally
occurs by a transgranular ductile striation mechanism [25,38], and the
rate of crack growth is largely insensitive to stress ratio, thickness,
frequency, microstructure, etc. The large influence of stress ratio,
thickness, microstructure, environment, etc. may, however, predominate
in the high and low growth rate regimes (III and I respectively). At
high growth rates, the effects of the above factors could be ascribed
to the occurrence of superimposed static modes of fracture such as
45
cleavage, intergranular and micro-void coalescence accompanying or
replacing striation growth [39,40]. The mechanism of growth in the
low growth rates has been observed to be microstructurally sensitive
and also sensitive to stress ratio, involving the occurrence of induced
fracture modes, such as intergranular fracture in steels [41].
2.3.3 Theoretical prediction of fatigue crack growth: crack growth laws:
There are practical as well as theoretical reasons for attempting
to relate fatigue crack growth rate to other material properties via
analysis of theoretical models. A significant problem facing material
engineers is the selection of materials to resist fatigue crack growth
at intermediate and low growth rates. Consequently any guidelines
leading to the choice of most crack growth resistant material from a
group of candidates - short of conducting fatigue crack growth tests -
would be a great help to the engineer. From such analysis it would be
desirable to relate C and m in equation (2.1) to other known mechanical
and fatigue properties of the material.
Fatigue crack growth laws are numerous and a comprehensive
discussion of these is beyond the scope of the present investigation.
However, a brief review will be presented here to highlight the limi-
tations of theoretical models in predicting fatigue crack growth over
a wide range of growth rates.
Paris and Erdogan [36] made a critical review of crack growth
laws which were introduced before 1964 and they concluded that the laws
could be expressed in the form (2.1). They also presented experimental
data, spanning four orders of magnitude, which favoured an exponent
value m = 4. Head's Law [42], based on a mechanical model of rigid-
plastic work-hardening elements in the plastic zone and elastic elements
elsewhere had earlier predicted a value of m = 3. Fracture of the
46
element at the crack tip was assumed to occur when the ultimate tensile
stress was reached.
Using dimensional analysis together with the concept of total
hysteresis energy absorption to failure Liu [43] predicted a value of
m = 2.
Weertman [44] has applied the Bilby et al. [45] dislocation model
to fatigue-crack growth by imposing the criterion that fracture occurs
whenever accumulated plastic displacement achieves a critical value
which is a characteristic of the material. This approach and a similar
one by Rice [46] and McCartney [47] yields a value of m = 4.
Lardner [48] considered the changes of crack tip opening displace-
ment during cyclic loading and obtained a value of m = 2. Schwalbe [49]
has developed a theory of fatigue crack growth based on characteristic
crack tip displacement and obtained a value of m = 2.
Recent works [50-52] which use the Coffin-Manson low cycle fatigue
correlation as a failure criterion at the crack tip also predict a value
of m = 2, though using the same criterion Duggan [53] obtained a value
of m about 3.
Therefore from the above short review of the theoretical crack
growth laws it can be observed that most of these laws predict values
of m equal to 2 or 4. However, from experimental observations values
of m in the medium growth rates may vary between 2 and 8. This limitation
of theoretical growth laws has led many investigators to use empirical
laws for fatigue crack growth prediction.
The theoretical models for fatigue crack growth briefly discussed
above do not take into account the effects of such factors as stress
ratio, thickness, microstructure, environment, frequency, etc. These
factors are known to affect the fatigue crack growth rates as observed
in numerous experimental data. In the following sections a detailed
A
47
review of the effects of these factors will be presented with emphasis
on stress ratio, thickness, environment and frequency.
2.4 THE EFFECT OF STRESS RATIO-CRACK GROWTH LAWS
Fatigue crack growth models have been proposed to account for the
observed effect of stress ratio on crack growth rate. Two of the
earliest models were proposed almost simultaneously by Roberts and
Erdogan [54] and Forman et al. [55]. In both cases the original Paris
expression (2.1) was modified to include the influence of Kmax (and
thus of R) at high growth rates. From considerations of crack tip
plasticity, Roberts and Erdogan suggested an expression of the form:
da m m
dN C1~K 1 Kmax (2.2)
where m1 and m2 were both found to be 2. Forman et al. considered the
consequences of Kmax
approaching the fracture toughness, K c, of the
material and proposed a modification of the Paris expression to:
da C2 AK 3
dN {(1-R)Kc - AK)S
(2.3)
where R is unity. This last expression has been used successfully to
model growth behaviour in aluminium alloys [56-59] and high strength
steels [60]. However, in mild steel [61] and other alloy steels [62]
which are less sensitive to the stress ratio effect the expression
overestimates the effect of R.
Other growth models have been proposed to account for the behaviour
observed at low growth rates. In these models a threshold stress
intensity factor AKth, below which no growth occurs has also been
introduced. The earliest of these models was by Erdogan and Ratwani [63]
who included AKth into the Forman expression:
48
da (AK - AKth)m4
= dN C 3 {(l-R)Kc - AK} S (2.4)
Klesnil and Lukas [64] later proposed an expression of the form
da dN = C4 ET A
K }m5 - AK m5 ]
(1-R)Y th (2.5)
where the first term was a modified version of Erdogan expression and
the second accounted for the effect of AKth,y was a material constant.
Branco et al. [61] proposed another general expression to include
AKth and Kmax effects at low and high growth rates respectively of the
form:
2 da C5(1-R2)(Kmax - Kmax) dN K 2 2 - K
Kc max
(2.6)
Recently a generalised expression has been proposed and used for
random loading data correlation by Dover and Hibberd [65]:
dN = C6(AK)m6(Kmax Kmax,th)m7( Kmax )m8
max Kc - Kmax,th (2.7)
Concepts of the effective stress intensity factor range, AKeff,
have also been proposed. In these, the AK in the Paris expression is
replaced by AKeff. In particular, one of these is based on a crack
closure phenomenon originally advanced by Elber [66]. Elber detected
physical crack closure in specimens subjected to a tensile load. He
modified the Paris expression by replacing AK by an effective stress
intensity factor range:
•
da m
dN - C7(AKeff)
9 (2.8)
49 •
where
Keff = U1AK
and
_ Kmax - KCL U1
K - K . max min ( 2.9 )
KCL is the stress intensity factor at which the crack opens during
the tensile part of the load cycle assuming that damage does not occur
when the stress singularity is destroyed by closure. Elber proposed a
simple relationship between Ul and R for an aluminium alloy, 2024-T3,
he tested.
U1 = 0.5 + 0.4R (2.10)
Maddox et al. [62] recently attempted to develop a simple relationship
between U1 and R for some structural steels but, though fairly successful
for R < 0, the results for R > 0 were scattered. Subsequent studies
have revealed that crack closure is a complicated phenomenon and is
influenced by some additional mechanical and metallurgical factors in a
highly complex manner [67]. Recent results by Ohta et al. [68], who
measured strains in the proximity of the crack tip showed that damage
does occur even when the crack is closed. Thus these observations cast
some doubt on the understanding of the effects of crack closure based
on Elbers proposals.
Other forms of effective stress intensity factor of an empirical
nature have been suggested by Walker [69] and also by Sullivan and
Crooker [70]. In these models the LKeff
was defined in terms of max.
Walker's formulation is
AKeff = (1-R)
m10 max (2.11)
50
and was recently applied successfully by James [71] on Inconel 600 •
steel.
Sullivan and Crooker's [70] effective stress intensity factor was
of the form
AKeff = MAK (2.12)
where M is an intensification factor related to stress ratio. Since
OK = (1-R) max, they assumed that:
AKeff (1-bR)Kmax - 1717R AK.
and finally
da dN C8[ (1 RR)~K]m11
(2.13)
(2.14)
where mli was determined from R = 0 and b was evaluated empirically to
fit their data at different values of R, including negative R values.
Other crack growth models discussed above have neglected the effect
of the compressive part of a tension-compression loading cycle on crack
growth. Early work by Illq and McEvily [72] on aluminium alloys,
showed that the compressive part of the loading cycle did not contribute
to fatigue crack growth and this observation was used by Paris in
arriving at his growth law [36]. However, subsequent tests have
indicated that the compressive portion of the tensile-compressive
loading cycle may contribute to crack growth. So, for example, results
by Gurney [73] on mild steel subjected to alternating tension (R = -1)
showed that a part of the compressive portion of the loading cycle
appeared to contribute to propagating the crack. Results by Crooker [74] •
and also by Sullivan and Crooker [70] showed that the compressive portion
of tensile-compressive cycle contributed to crack growth, though to a
lesser degree than did the tensile portion.
51
Recently Maddox et al. [62] performed fatigue tests at stress
ratios ranging from R = -4 to R = 0.67 on centre cracked sheet specimens
of four structural steels. They based their stress intensity factor
range, AK, on the tensile part of the stress cycle, i.e. iKTen' and
their results of da/dN versus AKTen showed that minimum growth rates
occurred at R = 0 indicating that for R < 0 the compressive part of the
cycle contributed some damage and for R > 0 da/dN increased with R.
The results also showed that the growth rate was generally higher fdr
any given negative stress ratio than for a positive stress ratio of the
same magnitude, e.g. for R = -0.5 growth rates were higher than for
R = 0.5.
As a final comment, judging from the results reported in the
literature on stress ratio effect, fatigue crack growth in aluminium
alloys and high strength steels may be highly sensitive to mean stress
whilst in low alloy steels crack growth is not significantly influenced
by stress ratio. Recently, there have been a number of advances to
explain the stress ratio effect in mechanistic terms. Richards and
Lindley [37] have discussed extensively stress ratio effect and the
different fatigue crack growth mechanisms and concluded that crack
growth by a pure, ductile, striation-forming mechanism was insensitive
to stress ratio. Other growth mechanisms, such as microcleavage, void
coalescence or intergranular separation were found to be significantly
influenced by stress ratio.
•
52
2.5 THE EFFECT OF THICKNESS ON FATIGUE CRACK GROWTH
Most fatigue crack growth tests in the laboratory are performed
on fairly thin specimens. Data from these specimens could be relevant
to structures like those used in aircraft where thin material is
usually used for construction. However, for structures made from
heavy sections laboratory data could only be used for life predictions
if these were devoid of section or thickness effects.
There is little guidance in the current literature as to the real
effect of thickness on fatigue crack growth; what does exist is contra-
dictory. Despite this short-coming, design engineers performing growth
analysis, tend to extrapolate the growth rates data to thicker or thinner
sections as a routine engineering approximation. The wisdom of this
practice is doubtful judging from the conflicting results reported in
the literature on thickness effect on growth rates. Three types of
thickness effect have been reported in the literature: (a) no effect
of thickness [75-83]; (b) growth rates accelerated by increase in
thickness [84-89]; (c) growth rates accelerated by decrease in thickness
[90-93].
One of the earliest work on the effect of thickness on crack growth
rate was reported by Frost and Denton [75]. They tested mild steel
sheet specimens of thicknesses 2.5 mm, 7.6 mm and 25.4 mm. Their
results showed that although the fracture surfaces were different for
the three thicknesses tested, there was no difference in crack growth
rates noted. Later work by Hertzberg and Paris [76] on 2024-T3 aluminium
alloy for thickness ranging between 1.62 mm and 3.2 mm confirmed the
above result. More recent results by Clark [77] on ASTM A533-B steel
for a thickness range of 25.4 mm to 101.6 mm; Parry et al. [78] on
A514 base steel; Harn et al. [79] on 3% silicon ferritic steel in
thickness range of 1.52 mm to 12.7 mm; Griffith and Richards [80] on
a
53
low alloy steel for a thickness range of 1.5 mm to 20 mm and Bathias
et al. [81] all reported no appreciable effect of thickness on crack
growth rates. Recently Sullivan and Crooker [82,83] investigated the
effect of thickness on growth rates in two steels, i.e. Ni-Co-Mo-V-
steel and A516-60 pressure vessel steel of thicknesses varying between
6.25 mm and 50.8 mm and again reported no effect of thickness on crack
growth rates.
Work by Broek and Schjive [84] and Plumbridge [85] testing aluminium
alloys showed that higher growth rates were obtained with thicker
specimens. This result was later confirmed by Barsom et al. [86]
testing several high strength steels in 25.4 mm and 50.8 mm thick
specimens. Heiser and Mortimer [87] testing 4340 steel in 1.59 mm to
12.7 mm thicknesses supported the same result. Hudson and Newman [88]
tested two aluminium alloys 7075-T6 and 7178-T6 and found relatively
little effect on growth rates for the 7075-T6 alloy while in 7178-T6
alloy growth rates were higher in the-thicker specimens. Their specimen
thickness ranged between 1.3 mm and 12.7 mm. More recently results by
Kang and Liu [89] on aluminium alloy, 2024-T351, in two thicknesses,
1.25 mm and 6.25 mm, also confirmed accelerated growth rates with
increase in thickness.
Clark and Trout [90] observed faster growth rates in 25.4 mm thick
specimens than in 50.8 mm thick specimens of Ni-Mo-V medium strength
steel under essentially plane strain condition; an observation made
earlier by Hanna and Steigerward [91]. Jack and Price [92] later
supported the above result when testing mild steel specimens in thick-
nesses ranging between 1.27 mm and 22.86 mm.
A recent investigation by Dover and Boutle [93] on aluminium alloy,
BS2L71, in four thicknesses, 3 mm, 6 mm, 16 mm and 22 mm, reported both
accelerated and reduced growth rates with increase in thickness. Their
•
54
results showed that at high growth rates, an increase in thickness
caused an increase in growth rates whilst at low growth rates increase
in thickness caused decrease in growth rates.
A careful study of results reviewed above suggests that the thickness
effect could be a variable of material and loading conditions, such as
frequency, stress ratio and environment. The range of growth rates
investigated also seems to play an important role.
2.6 THE EFFECT OF FREQUENCY ON CRACK GROWTH
Evidence at present available suggests that the speed of cyclic
loading has little effect on the fatigue crack growth in mild environ-
ments over the range of frequencies between 4 to 100 Hz [94]. At higher
frequencies crack tip heating occurs causing relaxation of the residual
stresses as well as softening of the material resulting in a reduced
growth. At lower frequencies environmental effects due to water-vapour
in the air tend to increase growth rates.
Early tests by Schijve [95] showed that growth rates were about
30% faster at 0.33 Hz than at 33 Hz. He tested 2024-T3 aluminium sheet
specimens, 2 mm thick.
Later Hertzberg et al. [76] tested the same material as Schijve
and found that growth rates were higher at the lower frequency (0.5 Hz-
133 Hz).
Other results reported for mild steel, titanium and copper sheets
[96] showed only little influence of frequency. Recently Yokobori and
Sato [97] have reported growth rates decreasing with frequency and
proposed an empirical formula to correlate their data, i.e.
da m -a
dN = C9(aK) 12f (2,15)
55
where f is the frequency 0.016 Hz - 140 Hz and Al is a material
constant; Al = 0.08-0.09 for 2024-T3 aluminium alloy and Al = 0.12-0.14
for SM50 steel tested.
2.7 THE EFFECT OF ENVIRONMENT ON FATIGUE CRACK GROWTH
2.7.1 Introduction:
The effect of the environment on the fatigue strength of specimens
and components is generally known as corrosion fatigue. This definition
implies the presence of an environment that will influence the fatigue
properties of the material by chemical attack.
Although fatigue was first identified as a potential failure mode
in the middle of the nineteenth century, it was not until 1917 that
first corrosion fatigue observations were reported by Haigh [98].
Later an extensive review of corrosion fatigue was reported by Haigh [99]
in 1932. Generally metals which have a high resistance to chemical
attack from a particular corrosive environment exhibit higher corrosive
fatigue strength than those metals having a lower corrosion resistance.
The combined action of stress and corrosive attack enables a micro-crack
to develop at a lower stress level than in a test in air. In a corrosion
fatigue test, a much greater proportion of the total life is occupied by
macro-crack growth than in a comparative test in air. In sharply--
notched specimens or components the cracks form rapidly at the notch
root and the specimen life depends largely on the rate of crack growth
since crack initiation is reduced to a minimum [5].
The exact mechanism of corrosion fatigue is not clear since most
proposed models fail to account for all the experimental observations
reported. It is possible that no single mechanism is responsible for
the corrosion fatigue behaviour of all metal-environment systems.
•
56
Early investigators of corrosion fatigue [100] assumed that cracks
initiated in the bases of large hemispherical pits at the metal surface
created by the corrosion process. Cyclic stressing accelerated
corrosion pitting and caused the development of crevices or fissures.
However, Duquette [101] proposed a crack initiation mechanism based on
preferential local dissolution of the metal at emerging slip steps.
The dissolution unlocks otherwise blocked slip process accelerating the
extrusion-intrusion process which results in premature crack nucleation
and crack growth. Crack growth is accelerated by a corrosion created
notch and preferential dissolution of the base of a growing crack.
Evans [102] applying an electrochemical theory, described the corrosive
attack on a specimen in a wet environment as an electro-chemical process
in which the whole surface was considered to be divided between anodes
and cathodes, anodes being plastically deformed areas of the metal.
Other suggestions are that corrosion fatigue crack growth concerns
hydrogen embrittlement [103]. This form of corrosion fatigue mechanism
is encountered in steels and in titanium alloys due to atomic hydrogen
being absorbed from such sources as cathodic reduction during aqueous
corrosion, electroplating processes or from the dissolution of water
vapour at the metal surface usually at elevated temperature. Subsequently
this hydrogen diffuses to regions within the material where it can be
accommodated by producing less distortion to the crystal lattice and may
result in a build up of hydrogen at internal microcracks. Hydrogen
creates hydrostatic pressure within the cracks so that the stresses
acting upon the growing crack are increased resulting in accelerated
growth rates.
•
57
2.7.2 Crack growth behaviour in water or salt solutions:
The influence of aqueous solutions, such as salt water, on fatigue
crack growth, figure 2.3, has been the subject of many studies of
aluminium alloys [104,105]. On the other hand fatigue crack growth in
steels in water or salt solutions is more limited. Several factors
including frequency, stress ratio, waveform, etc. affect the crack
growth behaviour of steels in salt water. A brief review of most
important of these will be presented below.
2.7.2.1 Frequency effect:
The first systematic investigation' into the effect of environment
and loading variables was conducted by Barsom [106] on 12Ni-5Cr-3Mo
maraging steel in 3% NaC1 solution. His results showed that in air and
salt water at room temperature the crack growth rates could be
represented by
da dN - F(t)(AX)2 (2.16)
where F(t) is a function varying inversely with frequency, attaining a
constant minimum value for the frequency at which fatigue crack growth
is sufficiently fast to be unaffected by the environment. Data at
different frequencies plotted in terms of log da/dN versus log AK gave
parallel straight line graphs with the lowest frequency producing
highest growth rates.
However, data obtained by other workers [107-109] do not show a
tendency for the curves to be parallel as Barsom's did. Vosikovsky
[107] presented results on X-65 maraging steel, which showed high
frequency dependence at AK values above KISCC (the threshold for stress
corrosion cracking) but no significant effect at low AK values. Austen
and Walker [108] reported similar results for 835M30 steel which further
•
58
showed lower growth rates in NaC1 solution than in air at 64 Hz
frequency. Other interesting observations were made by Atkinson and
Lindley [109] in their study of two steels (En56C and A533B-1) over a
wide range of frequencies (0.001-10 Hz). Their results showed that for
a given AK, there was a critical frequency at which maximum environmental
attack occurred for both steels. This critical frequency increased with
increasing temperature and a value of 0..01 Hz was found for En56C steel
at room temperature and AK = 25 MN/m3/2.
2.7.2.2 Stress ratio effect:
The most significant factor affecting environmentally assisted
crack growth is the R ratio. In the water environments, fatigue crack
growth through almost the entire range of AK is significantly increased
at higher R ratios. Scott and Silvester [110] have obtained some
limited data in BS4360-50D steel immersed in sea water which showed that
high R ratios (0.7, 0.85) increase crack growth rates compared to low
R ratios (0.1) in a way which was not observed in tests conducted in
air. Similar observations have been reported recently for pressure
vessel steels, A533-B and A508 in water environments typical of water
cooled nuclear reactors [111,112]. However, other results on stainless
steels [113] show no effect of stress ratio in similar water reactor
environment. Hibberd and Dover [114] have reported increased growth
rates with increase in the mean stress in two alloy steels, Ql(N) and
HY100, tested in sea water under random loading.
2.7.2.3 Waveform effect:
The effect of waveform on fatigue crack growth behaviour in corrosive •
environments have been studied and a number of investigations have shown
that the rise time during the loading cycle is a very critical factor
59
[108,109,115]; the lower the rise time the greater is the environment
enhancement. The damaging effect of the period during which the stress
is changing is the most significant, while the protective effect of the
stress free period and the maximum stress period decrease corrosion
fatigue damage. However, for some materials sensitive to stress corrosion
cracking, the maximum stress period causes increased growth rates.
Austen and Walker [108] reported data for 835M30 steel in NaC1
solution at different cyclic waveforms and showed that at R = 0.1 the
positive sawtooth waveform produced the highest growth rates while at
R = 0.8 the square wave produced the highest growth rates. Barsom [115]
reported similar results for 12Ni-5Cr-3Mo steel which showed that growth
rates for the sine and triangular waveforms were higher by a factor of
two than those for the square wave. No effect of waveform was observed
in air data.
2.7.2.4 Electro-chemical potential effects:
When a steel is immersed in seawater an equilibrium potential is
established between the component and a reference potential known as
the free corrosion potential. Under these conditions general corrosion
occurs. To prevent this, offshore structures are normally cathodically
protected using an impressed current system, sacrificial anodes, or
both.• In arduous conditions of a variety of kinds these systems are
likely to break down and this has prompted a large amount of investi-
gation into the effect of an applied potential on fatigue crack growth
behaviour.
Studies by Vosikovsky [107] on X-65 steel show that the growth
rates for the cathodic potential data were generally higher than those
at free corrosion. The maximum increase of growth rates above the air
data occurs at the lowest frequency and it is 50 times higher for
cathodic potential compared to 10 times for free corrosion. On the
60
other hand, there is also some evidence that cathodic potential can
reduce crack growth rates significantly in salt water to levels
observed in air [116]. Several results presented by Jaske et al. 11171
show that cathodic protection will significantly improve fatigue crack
initiation resistance of carbon steel in seawater. In some situations
crack growth accelerations are difficult to prevent because some degree
of negative potential is often necessary to reduce surface pitting or
crevice corrosion in ferrous alloys.
2.7.3 Models for corrosion fatigue crack growth:
In corrosive environments, crack growth may be due to either or
both of two mechanisms. In some cases true corrosion fatigue, which
is a joint action of both fatigue cycling and environmental aggression,
may be dominant. On the other hand, stress corrosion cracking may
occur due to electrochemical processes assisted by the local tensile
stresses. Three models have been put forward, namely: (i) the process
superposition models [118-120], (ii) process competition model [108],
and (iii) the process interaction model [122]. These will be discussed
briefly in turn.
2.7.3.1 Superposition models:
Studies by Wei and Landes [118] have led to the suggestion of a
single quantitative method for estimating the effect of aggressive
environments on fatigue crack growth in some ultra-high-strength steels.
This suggestion was based on the accumulation of experimental data and
fractographic observation, which seemed to indicate that the mechanism
for enhancement of fatigue crack growth is the same as that for true
stress corrosion under sustained loading. In the model, the rate of
fatigue crack growth in an aggressive environment is considered to be
K max (da) 1 f F(K)dK
dN SCC f( max Kmin) K . min
61
the sum of the rate of fatigue crack growth in an inert reference
environment and an environmental component of stress corrosion cracking
obtained in an identical aggressive environment. Wei and Landes obtained
(La) = (aa + r da K(t)dt ~N AN i 1 dN
(2.17)
where (Da/AN)c is the rate of fatigue crack growth in an aggressive
environment, (Aa/aN)i is the rate of fatigue crack growth in an inert
environment, and the integral term is the environment component computed
from stress corrosion cracking data obtained in the same aggressive
environment. However, corrosion fatigue may occur uelow KISCC, the
threshold for stress corrosion cracking.
A model based on that by Wei and Landes [118] was derived by
Gerbrich et al. [119] where the stress corrosion contribution on the
fatigue crack growth rate is given by
(2.18)
where f is the frequency of loading and F(K) is the stress corrosion
crack growth rate.
Recent fracture mechanics and surface chemistry based studied have
suggested that the rate of fatigue crack growth in aggressive environ-
ments can be regarded as the sum of the two components when the contri-
bution from stress corrosion cracking is negligible [120].
(da) _ (da). + (da) dN c dN i dN e
(2.19)
where (da/dN)i is the crack growth rate in an inert environment and
represents the contribution of 'pure' fatigue and (da/dN)e is the
environmental contribution or the corrosion fatigue component. Data on
62
aluminium alloys and high-strength steels indicate that (da/dN)e is
proportional to the amount of hydrogen produced and therefore the
extent of surface reaction during one load cycle [121].
2.7.3.2 Process competition models:
In this model, proposed by Austen and Walker [122] it is assumed
that the processes of stress corrosion and fatigue (or corrosion fatigue)
are mutually competitive and that the crack will propagate at the fastest
available rate pertinent to the prevailing stress intensity:
(i) When AK is below the threshold for growth, AKth,
da = 0 dN (2.20)
(ii) When AK is greater than AKth but less than that corresponding to
KISCC
da dN = C10(AK)m13 (2.21)
where C10
and m13 are constants from the air data.
(iii) When AK lies between values corresponding to KISCC
and the onset
of the plateau K
da dN = C11(AK)m14
where
1og[(āt) . 1 - log {C10[(1-R)KISCC]m13I mi4 log[(1-R)Kp] - log[(1-R)KISCC]
(2.22)
(2.23)
C11 (m13-m14)log[(1
-R)KISCC] + log C10 (2.24)
P
and f is the frequency.
63
(iv) When AK lies between values corresponding to K and the value
where the plateau rate is equal to the fatigue crack growth rate
da = (da) 1 dN dtp f
(v) When AK is less than the value corresponding to Kc
da dN = C10(AK)113
(2.25)
(2.26)
(vi) Finally when K reaches the value corresponding to Kc
da __ dN
= i.e. unstable fracture. (2.27)
2.7.3.3 The process interaction model:
Rhodes and Radon [122] have recently proposed a process interaction
model for corrosion fatigue. They argue that corrosion fatigue crack
growth may be represented by a superposition equation, modified to
allow for the interaction of the processes. If the stress intensity'
factor in each case is replaced by the effective stress intensity factor
range, IKeff,
adjusted to allow for the influence of the environment
(e.g. crack blunting, microbranching, etc.), and the effect of cycling'
on corrosion is replaced by a factor, y0, to account for the influence
of load cycling on the stress corrosion rate (e.g. due to strain rates,
etc.), then a process interaction model is formed. If the fatigue crack
growth rate in an inert environment is given by:
da dN = C
12(AK)m15 (2.28)
then, under corrosion-dominated corrosion fatigue conditions, they
proposed that-the growth rates would be given by
•
m 1/f a
dN C12(6Keff) 15 + A0~ Keff dt (2.29)
64
The constants A0 and a0 are derived from analysis of transient
conditions during variable load stress corrosion tests.
2.8 THRESHOLDS OF FATIGUE CRACK GROWTH
2.8.1 Criteria for non-propagating cracks:
The earliest criterion for a non-propagating crack was put forward
by Frost [123] in the form
M63a > A2 (2.30)
where A2 is an empirical material constant. Plots of &Q3a against log
endurance cycles on a single curve having the form of the conventional
S-N curve were made and the constant A2 was taken as the value of M3a
at which the curve become horizontal. A limitation of the above criterion
is the fact that the values of the parameter A2 have only been obtained
from experiments involving relatively small cracks.
The crack growth data using the Aa3a parameter were re-analysed
using the fracture mechanics approach, and AKth values obtained for
different materials [124]. A notable observation evident from these
results was the lower value of AKth found for shorter cracks (0.025-0.25 mm)
as compared with AKth obtained for longer cracks (0.5-5 mm).
There are two other physical models [125,126] which predict the
existence of a AKth which is expressed in the form
AKth = A317;17- 3 r af (2.31)
In the model of Weis and Lal [125] p* is taken as the limiting micro-
structural dimension over which a certain local critical fracture stress
of must be attained ahead of the crack tip for growth to occur, the
constant A3 being numerically equal to. The model of Sadananda and
Shaninian [126], on the'other hand, equates the threshold with the
65
minimum stress af to nucleate a dislocation at the crack tip, where p
is the minimum distance that the dislocation can exist away from the
tip (taken as the Burgers vector). In this form, equation (2.31) does
not predict any variation of AKth with mechanical, microstructural and
environmental factors, such as stress ratio, strength, etc. consistent
with the experimental data available.
Recently, Ritchie [127], assuming that the fracture stress of in
Weiss and Lal's expression was reduced by ah in the presence of a
hydrogen-producing environment, obtained an expression for AKth in the
form
AKth = /3Trp*/2 (af - ah) (2.32)
ah was evaluated in terms of yield strength, stress ratio and other
parameters of temperature, hydrogen concentration and the gas constant.
However, because of the uncertainty in the magnitude of various
parameters in the final expression for iKth, Ritchie has pointed out
that it is perhaps premature at this stage to utilise the expression in
a predictive capacity.
2.8.2 Effect of stress ratio on AKth'
Whereas little effect of R is usually observed in the midrange of
growth rates, low crack growth is generally sensitive to stress ratio.
Studies in a wide range of steels and non-ferrous materials, tested in
air at ambient temperature, indicate that the value of AKth is markedly
decreased, and that growth rates are increased as stress ratio is raised
from 0 to a high value.
One of the most widely quoted relationships of stress ratio, R,
and AKth is that by Klesnil and Lukas [641 in the form
•
and 2
AKth = AKOth [1-R2] (2.36)
66
AKth = AKOth
(1 - R)y (2.33)
where AKOth
is threshold at R = 0 and y is a material constant varying
from 0.5 to 1 for different steels. Values of y close to 1, indicating
a linear relationship, have been measured for most medium carbon
steels [128,129] and a cast steel [130] while values of y close to 0.5
for medium carbon martensitic steel [131] and intermediate values for
several low carbon steels [132] have been reported.
A linear relationship used earlier for HY130 steel [133] and for a
pressure vessel steel [134] has recently been applied successfully to
data for a wide range of steels [135]. This expression is in the form
AKth AK
0th - B0R (2.34)
It was found that values of both AKOth
and B0, which were considered as
material characteristics, varied over a rather wide range. Low-strength
steels with ferrite-pearlite microstructure generally exhibit higher
values of AKOth
and B0 than the higher strength martensitic steels.
Further expressions without material constants as those discussed
above have been suggested by McEvily [136] and Branco et al. [61].
These are in the form
1- 1 AK __ th AK .
Oth[ R
1+R ] (2.35)
•
respectively.
Recently, Davenport and Brook [137] suggested an expression
which also included the fracture toughness, Kc, and obtained a satisfactory
correlation for a 0.15% C, 1.5% Mn steel. Their expression was of the form:
67
AKth AKOth
Kc
K (1-R) 1/3 c max
K AKOth (2.37)
In addition to the above empirical expressions to correlate the
stress ratio effect on AKth, Schmidt and Paris (94] have attributed
the stress ratio effect on AKth to the crack closure phenomenon [66].
They argued that a similar process to that proposed by Elber could be
occurring in their tests but on a more microscopic scale since their
levels of AK and the corresponding plastic zone size, were much lower
than those employed by Elber. Since the reversed plastic zone sizes
produced by their AKth test levels were many times smaller than
individual grains, they speculated that micro-scale residual stresses,
within the grains induced a crack closure effect.
A stress intensity level to overcome this closure, KCL, was
postulated:
KCL = 1.212 a (2.38)
where
a is a constant value of microstress and d is the grain length in
direction of crack length.
Schmidt and Paris further argued that if the stress intensity factor
to open a crack is assumed constant, then there must be a portion of
applied AK above KCI,
AKeff' necessary to produce crack growth. If AKeff
is assumed constant with changes in R at constant growth rates then at
low R, if Kmin < KCL then
Kmax KCL + AKeff = constant. (2.39)
and therefore
AK = (KCL + AKeff)(1 - R) (2.40)
•
68
At the threshold AKeff = AKc th, and equation (2.40) above becomes
AKth - [KCL + AKeff][1-R] for K
min < KCL
(2.41)
and
AKth AKc,th for min KCL
The above relationships predict AKth versus R curves in which
AKth decreases linearly with increasing R at low stress ratios
(K KCL). and remains constant with further increase of R for
K. K' CL
2.8.3 Effect of specimen geometry and crack length on AKth'
To date no specific results have been reported on the effect of
specimen geometry, e.g. thickness, on AKth. It has usually been assumed
that the effects of thickness are those associated with changes in the
state of stress at the crack tip observed at high stress intensities.
In such instances, small differences in the growth rates between
specimens of different thicknesses can arise because of nominal yielding,
changes in fatigue-fracture mechanism near final fracture, and crack
closure effects. However, since at low stress intensities plane strain
conditions are prevalent, no distinct effect should be observed.
The influence of crack length, on the other hand, has been shown
to be particularly important. Early results [5] on notched specimens
showed that AKth was lower for short cracks than for long cracks. The
growth of small cracks under uniaxial cyclic tension, was studied by
Kitagawa and Takahashi [138] for HT80 steel. Their results showed that
whereas the condition for non-propagation of a surface crack could be
related to a constant AKth
for crack lengths in excess of around 1 mm,
below this crack size, a transition occurred in which the fatigue limit
stress became the threshold condition for growth.
7r(a+a' )
AKth Aa th (2.43)
69
El Haddad et al. [139] have attempted to rationalise the data by
Kitagawa and Takahashi [138] by defining the alternating stress intensity
AK at elastic stresses Aa in terms of
AK = Ac Ar(a+a') (2.42)
where a is the physical crack length and a' a constant characteristic of
a given material and material condition. Thus in terms of threshold
AKth the threshold stress range Aath is given by
r
such that as crack length becomes small, of the order a', then
A = AKth
ath
47- - Ace (2.44)
where Aae is the fatigue limit corrected for the appropriate R value.
Although these authors obtained good correlation with the experimental
data, there is no exact physical interpretation of the constant a'. It
should also be noted that the above tests involved very thin specimens
where the crack front for a short crack could be assumed straight and
thus crack length could be measured on the surface. However, under
plane strain conditions, i.e. thick specimens, crack fronts are known
to be curved and thus surface measurements may not produce reliable
results.
The other problem with short cracks is the plasticity effects
produced at notches. Thus it would appear that for short cracks the use
of the stress intensity factor, K, which is dependent on a linear elastic
stress field, may be invalid.
70
2.8.4 The effect of environment on AKth'
Data on the influence of environment on the threshold are most
comprehensive in aluminium alloys [105,140]. Tests performed on mild
steel immersed in brine, tap water and oil [5,141] showed some influence
of environment. Pook et al. [5,141] observed that for zero-tension
loading in mild steel at ambient temperature, the value of AKth was
reduced from 7.3 MN/m3/2 in SAE 30 oil to 6.0 MN/m3/2 in air and .
significantly reduced to 2 MN/m3J2 in brine. On the other hand, Paris
et al. [142,143] on low strength steels observed no change in AKth from
air to a distilled water environment, a result later confirmed by Branco
et al. [61] on mild steel in air and salt water environments.
Mantz and Weiss [144] tested 7050-T73651 aluminium alloy and D6aC
steel in room air, dry argon and wet argon. For the D6aC steel, room air
produced the lowest AKth while dry and wet argon produced identical but
higher OKth than in air. The effect in the aluminium alloy was, however,
more significant, i.e. AKth was much lower in air than in dry or wet
argon.
More recently, results on 13% Cr cast steel [130] showed that salt
water did not affect AKth at high stress ratio but caused a substantial
decrease on AKth at low stress ratios.
However, there are other results which show a significant decrease
in AKth when comparing crack growth data in vacuum to air. Cooke et al.
[131] obtained threshold data for EN24 steel and found an increase in
AKth from 5 to 7 MN/m3/2 at R = 0.1 and from 3 to 7 MN/m3/2 at R = 0.7
in vacuum as compared to air. Similar results have been reported by
Irving and Kurzfeld [145] who tested BS817M40 and SAE 4340 steels. More
recently, Kirby and Beevers [146] obtained other similar results for
three commercial aluminium alloys.
•
71
From the above results one is led to conclude that the environmental
effect on AKth lies in the nature of the reference environment used for
comparison. If dry argon is used as the reference environment, then
there is no significant effect on AKth - at least for steels. However,
when data in vacuum is compared to data in air then significant changes
in AKth are observed. It should be pointed out that rewelding effects
have been reported to occur for cracks in vacuum [147].
There is also an added complication in aggressive or wet corrosive
environments that AKth may be raised compared with more inert environments.
This may arise from the 'wedging' effects, where the presence of thick
corrosion products in the crack may prevent further access for environ-
mental species or from certain environments which promote crack branching.
These effects may also lead to crack tip blunting which may in some cases
cause crack arrest [140].
2.8.5 Effect of microstructure on AKth'
The effect of microstructure on threshold value for crack growth is
well documented elsewhere [127,147,148] and only a brief mention of
general trends will be made here. Data from martensitic steels show
that both thresholds and their stress ratio dependence decrease with
increasing strength, however, no significant change of AKth is observed
in these steels by increasing the grain size. On the other hand, data
from ferrite-pearlite steels show a great amount of scatter and cover a
relatively narrow range of strengths; thus the strength dependence of
threshold is not so clear. The effect of grain size is more pronounced
in low strength steels - OKth increases with grain size. In general at
high stress ratios the effect of strength on OKth is significantly reduced
in martensitic steels and it is in fact absent in ferrite-pearlite steels.
•
72
2.9 ELASTIC-PLASTIC FATIGUE CRACK GROWTH
Linear elastic fracture mechanics (LEFM) discussed in previous
sections assumes that the crack-tip deformations are small (small-scale
yielding). Thus there are significant limitations on the use of LEFM
under conditions of large plastic deformations. In such cases other
fracture mechanics techniques are employed.
In recent years several attempts have been made to extend fracture
mechanics techniques into areas of elastic-plastic fatigue crack growth.
One of these techniques is the J-integral concept [140] (Appendix).
However, the COD and other non-linear cyclic plastic strain methods
have also been employed.
Dowling and Begley [150] used the J-integral to describe fatigue
crack growth data in the elastic and plastic regions. They reasoned
that crack growth occurs during the loading and evaluated the increase
in J, that is &J, for the loading portion of cycles on elastic-plastic
specimens. Their tests were carried out on compact tension and centre
cracked specimens of A533-B steel. They then plotted the data in terms
of da/dN versus AJ and found that the crack growth rates were related
to AJ as
da dN - C13 (AJ)
m16 (2.45)
where C13 and m16 are material constants. The experiments were performed
under decreasing load conditions, a procedure which has recently been
used with success by Proctor [151].
Branco et al. [152] reported elastic-plastic crack growth data for
MS15 mild steel and obtained a similar correlation to that obtained by
Dowling and Begley (equation 2.45), although their MJ values were
determined under constant load cycling conditions.
The J-integral has also been applied in analysing crack initiation
•
73
and growth at notches. Lamba [153] showed that the J-integral, modified
for cyclic loading, OJ, yields the same form for the strain concentration
at a notch as the Neuber rule modified for the cyclic loading. They
therefore concluded that the J-integral could be used for life predictions
of both crack initiation and subsequent growth. Recently El Haddad
et al. [154] have applied the J-integral to the growth of short fatigue
cracks at notches, and obtained a good correlation.
Kaisand and Mowbray [155] have recently developed a model to
describe crack growth rate in a low-cycle fatigue test specimen (semi-
circular surface crack). The model involves a J-integral analysis and
a growth rate hypothesis in terms of M. Application of the J-integral
to high temperature fatigue has recently been reported [156].
The COD concept has received relatively little attention in appli-
cation to fatigue crack growth. Dover [157] reported results on crack
growth rates in mild steel under constant COD cycling, and found that
the crack growth rate was a power function of both the hysteresis loop
width and the overall crack tip opening displacement range.
Studies by Crooker and Lange [158] have shown that for plate bend
specimens containing an embedded surface crack, the crack growth rate
for both elastic and plastic strain cycling was described by
m
dN • C14(Ae) 17 (2.46)
where C14 and m17 are constants of the material and Ac is the total
strain range.
McEvily [159] studied the low cycle fatigue of copper and used a
strain concentration factor to describe his results in the form
m
dN C15 (Ae ✓ā)
18 (2.47)
•
74
where C15 and m18 are material constants. The parameter As)c was
termed the 'strain intensity factor'.
Solomon [160] performed low cycle fatigue crack growth tests in a
low carbon steel at constant plastic strain range, As , and obtained a
relationship
da m
dN C16 a(Ac ) 19 (2.48)
where C16 and m19 are material constants.
Lal and Weiss [161] proposed an expression for fatigue crack growth
at a notch of the form
m
dN = C17a(aN) 20 - 2 (2.49)
where C17 and m20 are material constants, p'~ is the microstructural unit
proposed by Neuber and aN
is the nominal applied stress.
Recently Skelton [162] reported a predictive expression for fatigue
crack growth in the form
da m m dN = C(Ac ) 21 a
22 18
P (2.50)
where C18, m21 and m22 are material constants, a is the crack length and
As is the plastic strain range.
•
75
2.10 LOAD HISTORY EFFECTS IN VARIABLE AMPLITUDE LOADING
2.10.1 General observations:
The load interaction effects on fatigue crack growth due to complex
variable-amplitude loading has been a subject of investigation for many
years. The inability to adequately predict these effects has prompted
recent interest in the study of the interaction due to simple load
patterns and its influence on fatigue crack growth [163]. More interest
has centred on interaction effects in tests with overloads or high-low
block loading which result in delay (Figure 2.4(a)) of fatigue crack
growth and in some instances complete crack arrest. However, relatively
less attention has focussed on the load sequences, such as low-high
block loading, which result in crack acceleration (Figure 2.4(b)).
The crack - growth retardation resulting from-a single overload,
multiple overloads or high-low block loading has been studied by several
investigators [163,164,170-175]. It is generally observed that the delay
period can be increased, in the case of overloads, by increasing the
magnitude of the overload, repeating the overload during the low level
cyclic loading, and applying blocks of overloads instead of single
overloads [163,164]. It is also observed that the retardation need not
immediately follow the overloads: in fact some further growth may be
required before the crack growth rate decreases. This delayed retardation
does not always occur and some authors have provided evidence of an
immediate decrease in growth rate after an overload [163,164]. Other
types of retardation such as lost retardation, crack arrest, etc. have
received extensive study by Bernard et al. [163].
Crack extensions caused by overloads themselves are larger than
expected from constant-amplitude tests. This phenomenon is also observed
in low-high block loading. Investigations into this aspect of load
r
76
interaction effects causing crack acceleration are rare [163,164].
One reason for this lack of interest is that this acceleration is not
easily detected from macroscopic crack growth observations since it
occurs over a relatively small portion of the crack increase. More
accurate measurements of the crack increment and fractographic analysis
do reveal locally accelerated crack growth.
Negative overloads have been reported to have a relatively small
detrimental effect on crack growth [163,165]. However, a negative over-
load added immediately after positive overloads can significantly reduce
the crack growth delay of the latter sequence. Stephen et al. [163]
investigated the effect of negative R ratio cycling following overloads
and observed that as negative R ratio was increased, retardation decreased.
They observed that compressive-tension overloads caused slightly less
retardation than the tension overload applied independently. However,
compressive overloads following tension overloads were extremely
detrimental resulting in virtually no delay periods. They concluded
that negative R ratio loading following overloads must be considered in
any mathematical model involving crack growth life predictions.
In program tests, similar sequence effects are found as observed in
step loading. Specifically, a retarded crack growth after a drop of the
stress' amplitude is clearly observed. In tests with a low-high-low
sequence, the crack growth rate in the descending part is lower than in
the ascending part of the program [163].
The effect of high loads in program tests is similar to the effect
of overloads in constant-amplitude loading. In flight-simulation tests,
it is well established that the application of rarely occurring very
high loads can decrease the crack growth significantly. Reduction of
these high loads to lower levels gives faster growth rates [163,166].
During a complex load sequence, it is more difficult to observe the
local interaction effects separately. The overall effect, however, can
77
easily be deduced from macroscopic measurements of the crack growth.
These observations and others obtained from fractographic techniques
can reveal the nature of interactions during a complex load sequence.
Generally, crack growth rates derived from crack growth measurements
are usually found to be significantly lower than those predicted from
constant amplitude cumulative damage approach. Apparently, retardation
effects dominate the acceleration effects in complex loading sequences.
Nowack [167] studied crack growth under random loading with a
constant root-mean-square of the applied stress signal (cams), but a
step wise change in mean stress. He observed that a decrease of mean
stress caused retarded crack growth.
Recently Hibberd and Dover [168] observed that random loading
introduces interaction effects into crack growth, which result in a
decrease in growth rates with increasing clipping ratio at medium to
low stress intensities and the opposite at high stress intensities.
2.10.2 Predictive methods for variable-amplitude loading:
Realistic crack growth predictions for variable amplitude crack
growth still remain difficult to come by, because development of the
essential mechanisms and the influencing factors of the cycle by cycle
crack extension is still incomplete. Among the mechanisms accounting
for load interaction effects are: crack tip blunting [169,170], crack
front incompatibility [171,172], residual stresses at the crack tip
[163,166, 173-177] and crack closure [66,163,178]. Various analytical
models have been suggested to predict crack growth under variable
amplitude loading. Usually such models are based on one of the mechanisms
mentioned above, though regression analysis based on empirical trends
have also been proposed [163,179-181].
•
(da) = C (5.11) dN retarded p dN baseline
78
The two best known models proposed to explain crack growth
retardation induced by overloads are the Wheeler and Willenberg models.
Both models use the concept of residual stresses in the plastic zone,
but with different interpretations. Wheeler [173] introduces a crack
growth reduction factor, C,
da = C F(AK) dN p (2.51)
where F(OK) is the usual crack growth function. The retardation factor,
C , is given as
C = ( ryi )Q p ry0 - Dai (2.52)
where ryi is the current plastic zone in the ith cycle under consideration,
Lai is the crack length increment after overload and r y0
is the plastic
zone size created by previous overload, and Q is an empirical constant.
There is retardation as long as the current plastic zone size is
contained within a previously generated plastic zone. If ryi > (ryŌ aai)
the reduction factor, C, is equal to one. The retarded crack growth rate
can be determined from the baseline (constant amplitude) crack growth
rate as
(2.53)
On the other hand Willenborg's model [174] makes use of an effective
stress intensity factor. The maximum and minimum stress intensity factors
are reduced by the same amount, the overload does change AK but causes
only a reduction of the cycle ratio, R, as long as min,eff > 0. When
min,eff < 0, it is set at zero. In that case R = 0 and LK = Kmax,eff'
The two models do not account for (i) the reduction of the
retardation effect by negative overloads, (ii) the difference in retarda-
tion caused by single and multiple overloads, (iii) the delayed
79
retardation (iv) or crack acceleration, all of which are phenomena
observed experimentally. Both models have been checked by various
authors [163,166,175,176,182/3] against experimental data, but a
systematic agreement with test results was not always found. However,
some recent applications have proved successful. Glinka [176] has
applied the Wheeler model to the analysis of fatigue crack growth in
low alloy steel weldments, and obtained good correlation when used in
conjunction with the Forman expression [55]. Broek and Smith [166]
have used both Wheeler and Willenborg models for a large amount of data
from 'flight-by-flight' loading and obtained good correlation. Druce
et al. [175] have used the same residual stresses concept as Willenborg
and obtained good correlation.
The crack tip blunting mechanism has also been used to explain
interaction effects [169,170]. This mechanism makes load interaction
effects dependent on the tip radius of a growing crack rather than on
the amount of crack growth after changing the cyclic load. Recently
Brown and Wearman [170] explained the retardation effects in terms of
residual stresses and crack tip blunting. They also observed that
growth would reach a minimum value when the compressive residual stress
dominates the effect of increased strain from the overload.
Crack tip incompatibility has also been used to explain the inter-
action effects by some authors [171,172]. This implies that there is a
mismatch between the low amplitude loading and the crack front orientation
after the overload. Bathias and Vancon [172] explained crack growth
retardation in terms of crack disorientation. Low-intensity overloads
produce a temporary disorientation of the crack and a growth retardation
similar at mid-depth and near the edges of the test specimen. The highest
overloads produce a static tear at depth while the crack remains arrested
near edges by the significant plastic deformation which develops under
•
80
plane stress conditions. The crack so disturbed starts again at the
surface in the ligaments limiting the tear.
The crack closure concept [66] has acquired considerable popularity
as a viable explanation of the interaction effects [66,163,164]. Using
this concept, a qualitative explanation has been given for crack
retardation after an overload or blocks of overload and crack acceleration
in a high-low load sequence.[66,163,164]. Regression models based on
crack closure have also been proposed [178,181]. However, recent results
on physical crack closure measurements during tensile overload tests
[170] showed that crack closure may not be the best mechanism to explain
interaction effects.
Recently a mechanism based on deformations caused by the propagating
crack has been proposed and checked with experimental data by Nowack
et al. [163,184]. They argued that sequence effects mainly occur due to
interaction of the plastic zones of the respective loads in the
subsequent load cycles. Of predominant importance in this case are the
Kmax
controlled plastic zones. The residual deformations built up within
the area of the Kmax
-controlled plastic zones hinder the formation of a
stable condition corresponding to constant-amplitude loading. They
observed that crack closure, cyclic strain hardening of crack tip
material, strain hardening along the plastic zone, blunting of the
crack tip, incompatible crack front, etc., all become simultaneously
active.
Based on a close correlation between the variations in the displace-
ment per cycle, Av, measured in the crack tip plastic zone area and the
crack propagation behaviour under variable amplitude loading, Nowack
et al. [184] suggested an expression for the crack growth per cycle, pa,
for arbitrary loading histories
Da = CO(w)k (2.54)
•
81
where C0 and k are material constants. They assumed a two-dimensional
stress field and also an ideal elastic-plastic material behaviour. An
experimental displacement analysis where a special grid technique was
applied to measure displacements showed that the measurement results
coincide well with the predictions of the stresses and displacements by
continuum mechanics. This continuum mechanics model was used successfully
to explain both crack retardation and acceleration phenomena observed
experimentally.
It appears that the observations of tests with more complex load
histories are qualitatively in good agreement with the results of simpler
type of loading. The agreement concerns effects of high loads, delays
and other sequence effects. However, accelerations and delays in a
complex load history are very sensitive to the sequence of the various
loads [163]. Therefore it is not surprising that most of the existing
models for random loading do not incorporate these factors quantitatively.
These models are of a predictive nature depending on experimental trends.
Barsom 1185] studied the fatigue crack growth under variable
amplitude loadings such as occur in actual bridges and found that the
growth rates can be represented by the expression
da m
dN = C19 (~Krms) 23
(2.55)
where 1K s is the root-mean-square stress intensity factor range and rm
C19 and m23 are material constants.
In a more recent work Kitagawa et al. [186] have also obtained good
correlation for their fatigue crack growth data using the K value. rms
Their investigation on a low alloy steel was carried out under both •
constant amplitude and random loading with different power spectra,
under stationary random fully reversed tension-compression loading.
•
82
However, it is known that K is wave-form dependent. Hibberd rms
and Dover [168] have shown that random load fatigue crack growth can be
satisfactorily analysed using a stress intensity factor analysis based
on Kh in conjunction with a growth rate based on the average frequency
of rises and falls in K. Kh is defined as
Kh = m24 hk24 (2.56)
where hk is the average rise and fall in stress intensity factor and
where, for a given material m24 is the slope of the da/dN versus Kh
curve.
A concept based on the crack closure phenomenon has recently been
proposed by Elber [163]. Elber proposed to replace random-load spectra
with an equivalent constant-amplitude loading in both analysis and
tests. The maximum load and the crack-opening load in the constant-
amplitude loading are chosen to be equal to those for the spectrum so
that both crack growth mode and the crack length at failure are equivalent
to those under the random-load spectra. The number of cycles of constant-
amplitude loading is chosen so that the amount of crack growth is equal
to that due to a given sequence or block of the random spectrum loading.
The concept was tested experimentally after predicting the equivalent
number of constant-amplitude cycles for six different random-load
sequences. The agreement between the predictions and test data was
good.
•
83
2.10.3. Effect of environment on load history effects
The effect of environment on fatigue crack growth under constant
amplitude loading was discussed in section 2.7, and it was observed
that corrosive environments can enhance the.crack growth rate. Much
less work on the contribution of environment to interaction effects
under variable-amplitude has been reported.
During their fatigue crack growth tests on BS4360-50D steel in air
and seawater, Scott and Silvester [110] observed accelerated crack
growth rates when the stress amplitude or stress ratio was increased.
Crack growth rate then stabilized over 5 mm to 7 mm of crack extension.
The calculated plastic zone size for the maximum stress intensity factor
at the time of load change was 1.2 mm which is considerably less than
the distance over which the apparent load history effect on growth rate
persisted. The crack acceleration observed in seawater was much larger
than that in air.
Shih and Wei [163] testing Ti-6A1-4V alloy in 3.5% NaC1 solution
observed that crack growth delay may be increased by 40 times through
increasing the hold time from zero to 15 minutes, the specific amount
depends on the overload ratio and the value of the low-amplitude fatigue
load. Other tests by Buck et al. [163] on 7075-T651 alloy showed that
humidity strongly decreased the delay due to an overload.
Recent results by Ranganathan et al. [187] on 2024-T351 alloy in
air and vacuum showed that the crack length affected by the overload was
one and a half times greater in vacuum than in air. The number of delay
cycles in vacuum was 5-10 times larger than that in air. They argued
that since the application of the overload modifies the residual stress
distribution in a similar manner independent of the environment, then
the difference in crack growth rates obtained could be due to inherent
differences in the crack growth behaviour in the two environments considered.
1 &K
REGlTMEI I REC 1ME IL.
TIME
TIME
84
CRACK ADVANCE r"--"i PER CYCLE i 1 1
-i sT E5
I .
I I I -- __ _I_ y mss
CROCK gRovy'Z'H P ECT1O141
Figure 2.1. Schematic representation of the formation of a fatigue striation.
LOCO aK
•
I/ I
— STRESS
Figure 2.2. Fatigue crack growth rate da/dN versus AK (schematic).
dN
a) DELAY b) AGCELERA7toNI:
85 •
1
Figure 2.3. A typical fatigue crack growth curve showing environmental effects.
Figure 2.4. Delay and transient acceleration phenomena.
86 •
CHAPTER THREE
EXPERIMENTAL DETAILS AND DATA PROCESSING
TECHNIQUES
87
3.1 MATERIAL AND TEST SPECIMENS
The material investigated was a low alloy structural steel,
BS4360-50D, and was available in hot-rolled plates of thicknesses 12 mm,
24 mm and 50 mm. The chemical, mechanical and fatigue properties of
the material are given in Table 3.1. No additional heat treatment was
performed on either the plate or the machined specimens.
The specimens used in the programme were of standard compact
specimen (CS) geometry of dimensions as shown in Figure 3.1. The
specimens were manufactured from the supplied plates so that the fatigue
crack propagated perpendicular to the rolling direction. The CS geometry
was found to be satisfactory for the testing machines used and the results
were sufficiently repeatable. The specimens were provided with sharp
notches to give fast crack initiation at moderate applied loads. The
surface of one side of each specimen (and both sides for 50 mm
specimens) was -polished sufficiently to facilitate viewing of the crack
tip.
The specimens used for tests in salt water were specially prepared
as illustrated in Figure 3.2.
To facilitate optical observation of the crack during the salt water
environment tests, the polished side of the specimen was coated with a
transparent acetone-based varnish. The varnish ensured that the area in
the vicinity of the crack remained uncorroded and thus made optical
crack viewing possible. The fluid was collected via channels made
from thin aluminium plate and stuck around the specimen with Araldite•
adhesive. The adhesive was strong enough to hold the channels and yet
flexible enough not to affect the specimen compliance. A transparent
perspex strip was stuck to the channel on the side where optical viewing
of the crack was made. The fluid was introduced into the notch of the
88
specimen via a thin plastic tube and flow rate was controlled by
gravity feeding and was adjusted by means of a screw clip. The plastic
tube was placed so that the fluid would drop as close as possible to the
tip of the notch.
Specimens used for the elastic-plastic tests were also specially
prepared to facilitate measurement of deflections along the loading
line. These were provided with suitable knife edges as shown in
Figure 3.3. However, for some specimens the deflection measurement was
made at the front surface of the specimen and appropriate knife edges
were made for this purpose. These were fixed onto the specimen using
screws (two screws for each knife edge).
3.2 TESTING MACHINES
The tests in the present investigation were carried out using an
Instron TT-C testing machine, two Avery hydraulic fatigue machines, a
Dowty Electro-Hydraulic fatigue machine, and a Mayes Servo-Hydraulic
fatigue machine.
The Instron TT-C testing machine has a maximum load capacity of
45 kN and was capable of cycling up to 1 Hz depending on the applied
load. This machine was used for testing at 0.25 Hz for the 12 mm and
24 mm thick specimens and especially for those tests at negative stress
ratio. Special grips were provided to permit reversed cycling.
The Avery hydraulic fatigue machines have a load capacity of 250 kN
and 500 kN respectively and were capable of cycling up to 1 Hz. The
tests performed in these machines were those at low frequencies (0.25 Hz)
both in air and salt water environments.
The Dowty fatigue machine has a range of frequencies from 0.06 up
to 100 Hz, depending on the applied load. The machine has three load
cells of maximum load capacities of 15 kN, 30 kN and 60 kN respectively.
89
The Mayes servo-hydraulic machine has a range of frequencies from
0.01 to 150 Hz and again the choice of frequency depends on the applied
load. The machine has three load cells of maximum load capacities of
2.5 kN, 25 kN and 250 kN respectively. Both the Mayes and the Dowty
machines were used mainly for the high frequency tests (i.e. those at
30 Hz).
In all these machines, with exception of the Instron, it is possible
to adjust the load while the test is in progress and automatic switch-off
is provided when specimen failure occurs, that is when specimen deflection
exceeds a certain amount. The Dowty and the Mayes are both provided with
facilities for choosing the shape of the input load control waveform
(sinusoidal, triangular, or rectangular).
All the machines were provided with suitable specimen grips for
holding the compact specimens used in the testing. The grips were
designed to hold both the 12 mm and 24 mm specimens. Larger grips were
made for the 50 mm thick specimens which were tested only in the bigger
Avery machine and the Mayes machine, these machines having the load
capacities required for this thickness.
The Dowty and the Mayes machines both have facilities for feeding
the load signal into an external recorder and also both can take external
load signals.
The Dowty and the Mayes have digital voltage displays of the applied
loads (enabling mean, maximum and minimum loads to be observed). The
Dowty load display reads 10 volts at maximum load for the respective
load cell while that for the Mayes load reads in percentage of the load
cell capacity. The two-decimal-place load displays enable these machines
to sense load changes of less than 0.06 kN for the least sensitive load
cell used.
The load calibration of all the machines used was checked and found
within allowable limits. Some check fatigue crack growth tests were
90
performed to ensure that all the machines used gave similar results
under identical testing conditions.
3.3 CRACK LENGTH MEASUREMENT DURING FATIGUE TESTS
There are several different methods which can be used to monitor
crack length during a fatigue test. The most commonly used methods are
the optical and the potential drop techniques. The optical method
measures the crack length on the surface of the specimen but cannot
detect the crack front configuration in the interior of the specimen.
On the other hand, potential drop techniques can be arranged to measure
the average crack length over the whole crack front [188]. In the present
investigation both the optical and the potential drop techniques were
used.
Measurements of crack length were made using microscopes mounted on
travelling micrometric stages. These microscopes could measure crack
increments of less than 0.02 mm. For the 12 mm and 24 mm specimens
crack measurements were made onone side of the specimen since the crack
front curvature was small. However, for the 50 mm-thick specimens the
crack front curvature was sometimes substantial, also non-uniform across
the specimen thickness, and one side of the crack front leading the other.
Thus crack length measurements were made on both sides of the specimen
and after each test the specimen was opened up and the measured crack
length corrected for curvature of the crack front, as recommended in •
ASTM standards [189].
At a later stage of the investigation an AC potential drop crack
microgauge [1901 was acquired for crack length measurements. This
instrument and its performance are briefly discussed below.
91
3.3.1 Crack length measurement using A.C. potential drop crack microgauge:
The crack microgauge is designed to detect and measure the depth
of cracks or other discontinuities in metallic components. It can be
used on both steels and aluminium alloys with or without prior
calibration depending on the application and the accuracy required.
The principle upon which the instrument relies is that a steady alternating
current (A.C.) field, 2 amps, is generated to lie approximately normal
to the defect under examination. Due to the type of field generated,
the current path is limited to the outer skin of sample under test.
When a suitable measurement probe is placed on the sample under test,
a very accurate measurement of the voltage drop between the probe
electrodes is made. If a fatigue crack is present in the sample, the
A.C. current will follow the surface of the metal around the crack. By
interpretation of the potential drop across the crack, either directly
from theory or using prior calibration, a measure of the crack length
or depth can be determined.
A digital meter, which will read from zero to ±1999 is provided.
Coarse and fine gain control adjustments can be set to give the most
convenient reading on the digital panel meter depending on the accuracy
required. The instrument also has an output facility to enable recording
of the probe voltage onto a chart recorder. This facility was used in
the present investigation during the block-loading testing programme.
The current terminals (supplied by RS Components Ltd, London, and
of type 4BA insulated terminal - 10A) were screwed into 4BA taped holes,
drilled symmetrically on the front face of the specimen, a distance of
20 mm apart, Figure 3.4. This position was found most suitable for the
geometry used and gave consistent and repeatable results. The current
terminals were connected to the current leads using insulated bunch pin
plugs, also supplied by RS Components Ltd., London. The current lead,
92
cut from a twin type 6.2 mm diameter screened cable with twisted cores
for hum reduction, was connected to the crack microgauge using a two-way
cable socket. The probe terminals, made from a soft iron wire, were
spot-welded onto the face of one side of the specimen, 10 mm apart and
equidistant from the crack line. A spot welding machine, made by
Unitec Corporation, U.S.A., model 1-048-03-02 and provided with 1.6 mm
(1/16 in) copper electrodes, was used for the spot-welding process. The
machine was set at 60 watt/seconds for the best welding results. Three
separate positions, Figure 3.4, along the crack direction were used for
different tests depending upon the accuracy of the crack lengLri-measurement
required. Position P1P1 was used during the initial set-up of the
instrument. Position P2P2 was used for the threshold tests where large
potential drop changes for a relatively short crack length increase were
required. Position P3P3 were used for the block loading tests where
fairly high potential drop changes over a relatively long crack length
increase were required. In each case the probe terminal wires were
insulated using PVC sheaths and stuck along the specimen surface using
Araldite adhesive, as shown in Figure 3.5. This was to prevent any
movements of the wires during the fatigue test where such movements had
been found to affect the digital meter reading. The probe terminal wires
were connected to the probe lead via a 5A polythene terminal block. The
probe lead, cut from a single core 2 mm diameter screened cable, was
connected to the crack microgauge using a four-pin plug.
The variation of the digital voltage with crack length is linear
and increasing the gain results in corresponding increase in the
instrument's sensitivity without affecting the linearity. This property
is illustrated in Figure 3.6.
For each set of probe position and for each specimen thickness
calibration curves were produced using both a fine saw cut as a crack
and also using a propagating fatigue crack. The results from different
•
93
specimens of same thickness and same current and probe terminal
positions were found adequately repeatable.
3.3.1.1 Effect of crack front curvature on digital meter reading:
The effect of crack front curvature on the digital meter reading
was investigated to see whether this reading needed any adjustment for
crack front curvature similar to that used in the case of the surface
crack length measurements. This investigation was carried out in two
series of tests. In the first series, the probe terminals were
interchanged between four different positions on either side of a
specimen with a growing fatigue crack. Readings were taken with the
probe terminals at position P1P1 (see Figure 3.4), on corresponding
positions on the other side of specimen denoted by P1
tP1' and then on
opposite but alternative positions, P1'P1 and P1P1'. The digital meter
readings from these four positions for same crack length were found the
same. The crack length deduced from these readings was found equal to
that obtained from surface crack measurements corrected for crack front
curvature. That is, the digital meter reading could be assumed to
give the average crack length over the entire crack front at least for
the current and probe terminal positions tested.
The above observation was confirmed with another series of tests.
In this series, a special probe, made with spring-loaded terminals, was
used to measure the crack length at different positions across the
specimen thickness (see Figure 3.4). These positions were 5 mm apart
starting from one surface of the specimen. First calibration curves
were obtained using a specimen with a straight saw cut. Using a specimen
with a growing fatigue crack estimates of crack length were made, for
different probe positions across the specimen thickness, from the
individual calibration curves. After each crack length reading the crack
front profile was marked by changing the applied load. At the end of the
•
94
test the specimen was opened up and the crack front profiles,
corresponding to the crack length measurements, estimated from the
fracture surface. Throughout the test AK was kept constant, at about
15 MN/m3/2 and at R = 0.1. The results are shown in Figure 3.7, where
it is observed that the digital meter reading always seemed to give the
average crack length whatever the position of the probe terminals across
the specimen thickness.
Thus from the two series of tests on the effect of crack front
curvature on crack length measurement using the crack microgauge, it was
concluded that for the CS geometry the crack microgauge measures the
average crack length across the entire crack front.
3.4 PROCESSING OF FATIGUE CRACK GROWTH DATA
During fatigue crack growth tests, measurements of crack length, a,
are made at appropriate number of cycles, N. Various techniques have
been used to analyse such data in order to estimate the fatigue crack
growth rates. A summary and brief description of some of these
techniques is presented below.
3.4.1 The graphical method:
This method involves drawing the best curve through the a versus N
data and estimating the crack growth rate, da/dN, by measuring the slope
of the tangent to this curve at the corresponding experimental point,
(a.,N.). This visual appreciation of the data suffers from errors due
to the individual's subjective choice of the 'best' fit and also the
best tangent through the point for the determination of da/dN.
s
95
3.4.2 The finite difference or secant method:
This technique involves calculating the slope of the straight line
connecting two successive data points, (ai,N.) and (a. ,Ni+1), i.e. 1 1+1
da __ ai +l ai dN N. -N. (3.1)
This method can result in erratic gradients at times due to the inherent
scatter in the experimental fatigue data. It should, however, be noted
that when a large number of data points are available, which are taken
at reasonably suitable intervals of ai and Ni, the scatter in da/dN data
can be considerably reduced using the secant method in conjunction with
the graphical method as was observed in the present investigation.
3.4.3 Modified difference method:
This method belongs to a class of finite difference techniques used
to estimate derivatives of an'unknown function by the derivatives of
either an nth order polynomial passing through a number of adjacent
pivotal points or by Taylor's series expansion of the unknown functions.
This technique uses sets of successive data points over the range
a. < a < a. to calculate da/dN at the mid-point, ai.
3.4.4 The incremental polynomial method:
This method for estimating da/dN involves fitting a second order
polynomial (parabola) to sets of seven successive data points. The form
of the equation for the local fit is [191]
N-C N-C a = b0 +bl( I) + b2( C NI)2
N2 N 2 (3.2)
•
where b0, b1 and b2 are regression parameters which are determined by the
96
least square criterion over the range ai_3 < a < ai+3. The parameter
CNI (N1_3 + Ni+3)/2 and CN2 = (Ni+3 - N1-3)12 are used to scale the
input data; thus avoiding numerical difficulties in determining the
regression parameters. The rate of crack growth is obtained from the
derivative of equation (3.2) which is given by the expression
dab
dN Cl + 2b2(Ni - C N1)/CNl
N2 (3.3)
3.4.5 The total polynomial method:
This method involves fitting a high order polynomial to selected
data points (typically 20 to 70 points) covering the entire range of
data. The curve fitting model is of the general form
N a = E B.F.(N)
i=1 (3.4)
where the fitting functions F.(N) are power functions whose exponents
are determined by the min-max criterion, that is, so as to minimise the
maximum deviation between the observed and fitted values of crack length.
The regression parameters, Bj, are subsequently determined by the least
squares criterion. The rate of crack growth at each data point is
estimated from the derivative of the above expression (equation (3.4)).
3.4.6 General discussion of the data processing methods:
The scatter of the crack growth rate data obtained using the
various techniques discussed above will depend on the degree of variability
in the raw experimental data. If, for example, the data points lie in a
smooth curve then the secant method may prove satisfactory. If, however,
the experimental data are subjected to great amount of scatter a curve
fitting method will be preferred.
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97
Davies and Fedderson 1192] attempted to represent the experimental
data using polynomial curve fitting but observed that the fitted curves
oscillated about the experimental data in such a way that the gradients
estimated by differentiation•were seriously in error. They also
observed that the secant method in their case produced better results.
McCartney and Cooper [193] have developed a least squares method
of estimating growth rates by making use of a spline function (piece-wise
continuous polynomial). Though they obtained comparatively good results
still oscillations about some data points were observed.
Recently Wei et al. [194] statistically compared crack growth rate
data obtained using finite difference method and some incremental
polynomial techniques. They observed that the incremental polynomial
methods generally introduced a systematic bias into the da/dN data.
However, in practical sense, the bias may not be significant in terms of
probable errors in predicted fatigue lines. The finite difference
(secant) method and the parabolic curve-fitting procedure (Simpson's
rule) did not show significant bias, although these methods produced
greater scatter. The secant and the incremental polynomial methods are
recommended by the ASTM standards [189].
In the present investigation a polynomial curve fitting procedure
using least squares technique was developed during the initial stages.
The method was found satisfactory for test data with sufficient experimental
points and limited amount of scatter. However, raw data with considerable
amount of scatter produced oscillations of the crack growth rates about
some data points. The method was also found unsuitable for use on
experimental data where sudden load changes were made during the test,
for example, in step-down tests at low crack growth rates.
The amount of scatter in da/dN data is also influenced by the
interval of crack length measurements 1194]. Although large measurement
98
intervals might produce acceptable variation in da/dN data, it could
severely limit the number of observations made on a single specimen and
affect the precision of computing AK. Thus, the choice of measurement
interval is important and depends on a realistic assessment of measure-
ment precision, which should be established experimentally for individual
situations. In the present tests it was established that measurement
intervals of the order of 0.5 mm produced satisfactory results of da/dN
calculated using either the secant method or the polynomial curve fitting
technique. An example of such data is shown in Figure 3.8. It was also
observed that the scatter in the raw data could be significantly reduced
by first smoothing the data graphically before calculating the da/dN
using the secant method. The effect of initial graphical smoothing of
data is illustrated in Figure 3.9. As seen in this figure the resulting
data is more suitable for comparison of crack growth rates under different
loading conditions. Thus, it was decided that this latter procedure,
that is, secant method for raw data with little or no scatter, or
graphical smoothing followed by secant method, to determine da/dN, would
be used in the present investigation. The polynomial curve fitting
procedure was abandoned since it could not be used for processing data
obtained under different loading conditions.
3.5 DETERMINATION OF THE STRESS INTENSITY FACTOR
The crack growth results, in the present investigation, have been
analysed using the fracture mechanics approach. Thus, it was necessary
to establish an expression for the stress intensity factor, K, for the
particular geometry.
The specimens used were of standard CS geometry whose K expression
is [195]
•
f(a/W) = 2+a/3/2 {0.886 + 4.64(a/W) - 13.32(a/W)2 + 14.72(a/W)3 (1-a/W)
- 5.6(a/W)4} (3.8)
99
K = _Er f(a/W) (3.5) BW 2
where P is the applied load, B is the specimen thickness, W is the specimen
width, a is the crack length and f(a/W) is a measure of the compliance
of the specimen. For a constant value of specimen height, H, to
specimen width, W, the function f(a/W) can be expressed in a polynomial
form as a function of the dimensionless crack length (a/W). The compact
specimens have H/W = 1.2 and the f(a/W) can be expressed as [195].
f(a/W) = 29.6 (a/W)k - 185.5(a/W)3/2 + 655.7(a/W)5/2 - 1017(a/W)7/2 +
638.9(a/W)9/2
(3.6)
for 0.3 < a/W < 0.7.
Expression (3.6) covers a limited range of crack length, however,
calibrations covering a much broader range have recently been developed.
Wilson [196] has obtained an expression for f(a/W) for lengths in the
region a/W > 0.8 of the form
f(a/W) = 2(1 + a/W) + 1
1 - (a/W)3/2 2(1 - a/W)1/2 (3.7)
A K-calibration expression valid for a/W > 0.2 has recently been given
by SNawley [197] where f(a/W) is given by
100
3.6 MEASUREMENT OF CRACK GROWTH DURING SLOW STABLE CRACKING UNDER
CYCLIC AND MONOTONIC LOADING
Various experimental methods have been developed to measure crack
growth by stable tearing. However, none of the techniques provides
a complete solution to the problem for all materials and specimen
geometries. However, some of the techniques that have been employed to
date have recently been reviewed by Pisarski and Garwood [198] and only
a brief discussion of those relevant to the present work will be made
below.
3.6.1 Surface observations
This is a simple procedure and involves crack length monitoring by
surface measurements of the crack tip, either as viewed directly through
a travelling microscope, or from a set of photographs taken at specific
intervals during the test. The method suffers from certain flaws,
primarily because it is purely a surface measurement and the behaviour
of the entire crack front is not monitored. For very thin sheet
specimens, where fracture is generally by a 45° shear mechanism or for
deeply side grooved specimens, this method can be practicable. However,
in thicker specimens initiation occurs in the central part of the
testpiece and,as the crack propagates, large 45° shear lips develop
towards the side surfaces while the centre of the crack remains flat.
Here, surface measurements do not give any indication of initiation and
overall crack growth and produce misleading estimates of the amount of
tear. This method was not found suitable for the present investigation.
101
3.6.2 Fracture surface measurements (or Multiple Specimen Method):
This is the most accurate and widely used method. It consists of
loading a specimen to obtain certain amount of ductile crack extension.
The specimen is then either heat tinted and broken open or it is
fractured by a different failure mechanism, such as fatigue or cleavage.
The amount of ductile crack extension is clearly visible on the fracture
surface and an average of nine measurements along the crack front is
normally used. A disadvantage of this technique is its cost, since
a whole series of specimens must be used to develop one crack growth,
åa, versus J curve (JR-curve).
3.6.3 Unloading compliance method
This method consists of periodic partial unloading of the specimen
after the commencement of ductile tearing. From the measurement of
unloading slope and the use of a calibration curve relating slope to the
dimensionless crack length (a/W) the current crack length can be
estimated. The amount of unloading is normally limited to 10% [199]
However, recently de Castro et al [200] have shown that the 10% unloading
could be replaced by 30% unloading with sufficient reliability; the
latter being more suitable for drawing straight lines through the
unloading curve for estimation of the slope. Thus it is possible to
obtain a crack resistance curve from a single testpiece. Prediction of
crack length using this method can be subject to some errors especially
in the detection of crack length increments near initiation where greater
accuracy is required [198].
Other methods of crack growth monitoring during stable ductile
tearing include D.C. and A.C. potential drop techniques, ultrasonics,
acoustic emission, etc. It is clear from studies of methods that, with
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102
the exception of the multiple specimen technique, none of the other
techniques has reached a sufficiently advanced stage to allow them
to be used without prior calibration.
3.6.4 Crack growth measurement during present tests:
During the present elastic-plastic tests the crack extensions by
stable ductile tearing were monitored using the unloading compliance
method, though in some tests stable crack growth was measured optically
on the fracture surface as required in the multiple specimen method.
A calibration curve was first obtained using the same equipment and
loading specimen grips. The displacement of the knife edges was recorded
with a clip gauge for a range of crack length values obtained by fatigue
cracking. A single calibration curve obtained for all the specimen thick
nesses was found to compare satisfactorily with the theoretical calibra-
tion curve recently derived by Saxena and Hudak [201] in the form of a
compliance polynomial for C, specimens, for a/W > 0.2, i.e.
(BEPLL) _ (
l+a/W)2 {2.1630 + 12.219(a/W) - 20.065(a/W)2 - 0.9925(a/W)3
+ 20.609(a/W)4 - 9.9314(a/W)5} (3.9)
The clip gauge used was a Welwyn type (supplied by Welwyn Strain
Measurement Ltd., Basingstoke, England), model 101 which is based on the
British Standard - BS5447-1977 - entitled "Plane strain fracture toughness
(K1C) of metallic materials". The clip gauge consists of two parallel
cantilever beams machined from titanium alloy and incorporating precision
knife edges at the tips. High resistance precision foil strain gauges
are mounted on the beams to form an open Wheatstone Bridge and five lead
wires are provided to permit the use of an intermediate balancing facility
103
between clip and readout instrument. The clip gauge has a deflection
range of 4 to 12.5 mm and was first calibrated with a micrometer reading
up to 0.000254 mm. The load signal, obtained directly from the machine,
was used together with the clip gauge signal obtained through a carrier
amplifier (type SE 423/1, made by S.E. Labs Ltd., Feltham, Middlesex,
England), to plot load-deflection curves on an X-Y plotter (made by
Bryans Ltd., Mitcham, England). The areas under the load-deflection
curves were measured, for the J-integral determination (see section 3.7),
using a planimeter (type ALLBRIT, made by Cooke, Troughton & Simms Ltd,
England).
3.7 DETERMINATION OF J-INTEGRAL FROM LOAD-DEFLECTION CURVES
3.7.1 Monotonic J-integral:
In the last few years, the experimental techniques for the deter-
mination of the J-integral have substantially improved. Begley and
Landes [202] attempted an experimental determination of J based on the
energy interpretation expressed as
J 1 dU B da
(3.10)
where B is the specimen thickness, U the potential energy and a the
crack length. This technique, known as the compliance method, is slow
and tedious and has now been superseded by a much simpler procedure [203]
based on an approximate formula for calculating J from load-displacement
curves as proposed by Rice et al. [204], namely:
J 2A B(W - a)
(3.11)
where W is the specimen width, and A is the area under the load-
displacement curve at a displacement measured along the loading axis,
103
between clip and readout instrument. The clip gauge has a deflection
range of 4 to 12.5 mm and was first calibrated with a micrometer reading
up to 0.000254 mm. The load signal, obtained directly from the machine,
was used together with the clip gauge signal obtained through a carrier
amplifier (type SE 423/1, made by S.E. Labs Ltd., Feltham, Middlesex,
England), to plot load-deflection curves on an X-Y plotter (made by
Bryans Ltd., Mitcham, England). The areas under the load-deflection
curves were measured, for the J-integral determination (see section 3.7),
using a planimeter (type ALLBRIT, made by Cooke, Troughton & Simms Ltd,
England).
3.7 DETERMINATION OF J-INTEGRAL FROM LOAD-DEFLECTION CURVES
3.7.1 Monotonic J-integral:
In the last few years, the experimental techniques for the deter-
mination of the J-integral have substantially improved. Begley and
Landes [202] attempted an experimental determination of J based on the
energy interpretation expressed as
- - 1 dU J B da
(3.10)
where B is the specimen thickness, U the potential energy and a the
crack length. This technique, known as the compliance method, is slow
and tedious and has now been superseded by a much simpler procedure [203]
based on an approximate formula for calculating J from load-displacement
curves as proposed by Rice et al. [204], namely:
J - 2A B(14 - a) (3.11)
where W is the specimen width, and A is the area under the load-
displacement curve at a displacement measured along the loading axis,
104
Figure 3.10. This expression is applicable to a body containing a deep
crack, subjected to pure bending and is suitable for use in such cases
as compact and three or four point bend test specimens. Further
analysis [205-208] has shown that equation (3.11) is slightly inaccurate
for compact specimens over the a/W range normally used in fracture
toughness tests and the improved form is
J - AA B(W - a) (3.12)
where A takes into account a correction for the tension component in
loading.
Merkle and Corten [205] reported values of A varying from 2.55 at
a/W = 0.3 to 2.31 at a/W = 0.7.
Srawley [206] demonstrated that, in general, the value of A was
related to the work done on the test specimen at fixed displacements,
m, by
3(lndm) - 8(ln(W - a)) (3.13)
Hickerson, Jr. [207] has experimented on seven alloys and found
that the value of A for compact tension specimens was material dependent.
However, he also found that for 0.6 < a/W < 0.9, the measured values
of A appeared independent of crack length, suggesting that a single
value of A = 2.18 would be sufficient.
Recently, Clark and Landes [208] have evaluated values of A for
compact specimens and these are tabulated in Table 3,2. These values
have been used in the present investigation for the estimation of J,
though a value of A = 2.18 as suggested by Hickerson [207] for a/W > 0.6,
was used for the initial data analysis.
105
3.7.2 Cyclic J-integral, AJ:
As the deformation theory of plasticity, on which the J-integral
concept is based, does not directly account for plasticity effects
observed on unloading, there may be some doubt concerning the applica-
bility of the J-integral to cyclic loading [203]. However, recent
applications of elastic-plastic fatigue crack growth [150-154] have
shown encouraging results.
Dowling and Begley [150] determined cyclic J values from areas under
load versus deflection curves during the loading half of the cycle, and
subtracted the effect of crack closure, as shown-in Figure 3.11. The
operational definition of AJ illustrated is related to the Rice et al.
[204] approximations of equation (3.11) and therefore the factor 2 should
be replaced by X as in equation (3.11).
Branco et al. [152] considered the load-deflection diagram for a
cracked specimen loaded in tension, Figure 3.12, above the yield point.
Because only loading conditions have been considered, the energy inter-
pretation of the J-integral can be extended to cyclic loading of an
elastic-plastic material to define a cyclic AJ value after N-loading
cycles as:
OJ = 1 d(iU) B da
(3.14)
where dU = AUN - AU1. AU1 is the potential energy of the first cycle
(area OABHO, Figure 3.12) and WN
is the total accumulated potential
energy after N-cycles (area OECHO, Figure 3.12). Equation (3.14) is
equivalent to equation (3.10) for monotonic loading and therefore does
not take account of the effect unloading might have on the material
properties.
106
From the above expression,
AJ - - 1 {d(ouN) d(oUl)
B da da (3.15)
d(MU1) and the term da
is clearly equal to zero because AU1 depends on the
initial crack length and not on the current crack length. Clearly from
Figure 3.12 DUN can be expressed as:
AUN = MSN P A2 Sel (3.16)
where AdN
is the total deflection HC at maximum load and ASel
is the
elastic loading deflection ED. Substituting equation (3.16) into
equation (3.15) leads finally to a generalised equation valid for any
specimen geometry, i.e.
d(Od ) d(AS ) DJ = - 1 N
AP OP el da 2 da (3.17)
Thus, we may compute AJ for any specimen geometry provided that the
deflection or extension at the loading points is known under load
cycling conditions.
3.8 THE RESISTANCE CURVE AND THE CRITICAL J FOR CRACK INITIATION
To establish a crack initiation measurement point under dominant
slow-stable crack growth, the procedure proposed by Begley and Landes
[202] is used. J-integral values are plotted against the corresponding
measured crack growth values, Aa, in the form of a resistance curve
(R-curve). The critical value of J, JC , for initiation of slow-stable
crack growth is taken as the intersection of the crack tip blunting
line, defined by da = J/20flow
(where oflow
is the average value of
yield strength and the ultimate tensile strength, to allow for work
hardening), with the resistance curve. The crack tip blunting line takes
into account the crack tip geometry changes that occur before the actual
107
material separation, as shown schematically in Figure 3.13. To
establish a sufficiently reliable R-curve, and thus a Jc value, at
least four to six points (obtained from either a single specimen or
from as many specimens) are required.
The above procedure is based on the assumption that the stretch
zone width is approximately equal to the extension of the plane of the
starter crack due to the stretch zone and that J = aflow
. COD which
then follows that J = aflow . 2
aa. There is no justification given for
this assumption since in most cases J = 2aflow . COD. This point will
be discussed in more detail in chapter six.
3.9 SUMMARY
From the preceding discussion on experimental details and data
processing techniques the following summary of major details concerning
the present work can be made:
1. Compact specimens (CS) are used in all the tests carried out in the
present investigation. This geometry is found to be satisfactory
for the testing machines used and the results obtained using this
specimen geometry are sufficiently repeatable.
2. Five different machines are used in the present test programme.
These machines are found to give similar results under identical
testing conditions.
3. Both the optical and the a.c. potential drop techniques are used to
measure crack length during the fatigue crack growth testing. The
a.c. crack microgauge is found to be the most suitable technique
under the present test conditions. For the CS geometry, this
instrument measures the average crack length across the entire
•
108
crack front and therefore it is not necessary to correct the
measured crack length for crack front curvature as is the case
with the optical method. In addition, the crack microgauge has
facilities for varying the instrument's sensitivity to crack length
increments depending on the accuracy required for the particular
test. Using a suitable chart recorder, the crack microgauge can
also be used to monitor the crack length continuously during a
fatigue test.
4. The secant method, sometimes used in conjunction with a graphical
data smoothing technique, is found to give the best crack growth
rate (da/dN) results under the present test conditions. This
method is simple to apply and also the resulting data, which retains
the scatter inherent in fatigue crack growth behaviour of engineering
material, is more suitable for ,comparison of the crack growth rates
under different loading and environmental conditions.
S. Crack extension by fatigue and also by stable ductile tearing during
the elastic-plastic tests is monitored using the unloading
compliance technique. However, in some tests, measurements of
stable crack growth are made optically on the fracture surface.
A Welwyn clip gauge is used for deflection measurements during
the elastic-plastic tests.
6. Standard expressions for the CS geometry are used for the deter-
mination of the stress intensity factor, K, and of the J-integral
in the present investigation.
•
109
TABLE 3.1
STEEL BS4360-50D
(a) Chemical Composition (Weight)
Element C Si Mn P S Cr Mo Ni Al Cu Nb
% 0.180 0.36 1.40 0.018 0.003 0.11 0.020 0.095 0,035 0.16 0,039
(b) Mechanical Properties
Direction
Longitudinal Transverse Through thickness
Yield strength (MN/m2) 386 382 388
Ultimate tensile strength (MN/m2) 560 551 563
elongation (GL equal 3") 30 29 18
reduction in area 74 68 61
(c) Fatigue Properties
Cyclic yield strength, ayc
Strain hardening exponent, n
True fracture strain (or fatigue ductility of coefficient),
415 MN/m2
0.19
1.35
•
110
TABLE 3.2
LIST OF TENSION COMPONENT CORRECTION FACTORS, A (EQUATION (3.12),
FOR THE COMPACT SPECIMEN GEOMETRY (SEE NOTE BELOW)
a/W A a/W A
.45 2.2896 .63 2.1966
.46 2.2848 .64 2.1912
.47 2.2800 .65 2.1858
.48 2.2750 .66 2.1804
.49 2.270 .67 2.1748
.50 2.2650 .68 2.1694
.51 2.2598 .69 2.1640
.52 2.2548 .70 2.1586
.53 2.2496 .71 2.1532
.54 2.2444 .72 2.1476
.55 2.2392 .73 2.1422
.56 2.2340 .74 2.1368
.57 2.2286 .75 2.1314
.58 2.2234 .76 2.1260
.59 2.2180 .77 2.1206
.60 2.2126 .78 2.115
.61 2.2074 .79 2.1096
.62 2.2020 .80 2.1042
NOTE: The tensile correction is obtained from a modified Merkle-
Corten derivation [Refs 205 and 208]
= 2[(1+a) / (1+a2)]
a = [(2 a/b)2 + 2(2a/b)+2]1/2 - (2 a/b + 1)
b = W-a
•
111 •
D+MEN5ICW A 8 C D E F Ci ii R W
StZE("tm } 5 ,
O!Z12 52.4
o "13''° 3'0 20 0 Z18 47.7 95.4 6.35 79•5
Figure 3.1. Compact specimen geometry used in the experimental programme..
112 •
C,RAV1TY FEED
\
SCREW CLIP
,
Figure 3.2. Specimen preparation for the salt-water environment tests.
T i- 7t____ f ___
i 1
PZ P3 --e— -•-- i E
E Oc
P3
-0 1,
SPECIAL PRpBE w1TH SPRwNCi-LOADED TERMINALS SPECIMEi`I
.113
UNITS : mm
Figure 3.3. Compact specimen modified for deflection measurement at the loading line.
I-I CURRENT TERMINALS P P, P2 1:22 * P3 P3- PROBE TERMNALS
(PI P: 15 5AME A9 P P, BUT IS ON OTHER SlDE OF gPECIMEN )
Figure 3.4. Positions of current and probe terminals for crack length measurement and crack front configuration estimation using. the crack microgauge.
•
Current terminals
Probe terminals
Insulation and
Adhesive
114
i
Figure 3.5. Experimental set-up showing the positions and fixing of the current and probe terminals during the block loading tests.
A
■ k/ i A
i 411/
I /7
i 0 j•
046 A
0
/ •A)IVo
~~~®
f +A+ 04.4
R=0.08 B=12mm
O
• O
115
600
400
w 350
O 300
J I-- 3 250 ō
200
150
100
50
0 20 24 28 32 36 40 44 48 CRACK LENGTH , a, mm
Figure 3.6. Variation of digital voltage with crack length at various gain settings of the crack microgauge.
550
500
450
ACTUAL CRACK FRONiT CONIFtCNRAfOt4
EG11MA.TED CRACK FRoF4T u5tticf AG M%cROC1AUCCE ,000 4g5,ouo ‘2I2,OOO 1,374;000 GYCL%S
~670,000
NOTCH
t4:O 454000 2,o46, 000
26 28 30 32 34 - 36 CRACK LEP TH , 0 , ("w")
Figure 3.7. The effect of crack front curvature on the digital voltage reading of the crack microgauge in the 50 mm thick specimen.
•
117 •
• DATA CALCULATED, FROM eAPERiMENTAL POINTS, USINCI SECANT METHOD .
O DATA COMPUTED Us1NCI POLYNOM IAL. CURVE - FITiINCI , METHOD.
1 I I I I II I 20dK` CMN40. , GO 50 100 10
Figure 3.8. Comparison of da/dN versus AK data calculated using the secant and polynomial curve-fitting methods: Data at R ='0.08 and 30 Hz.
a 0 •
0 DATA FROM E)CPERJ -4EP4TAL PO t NITS
CAL.cU LATED u5tNG SECAIQT M ETHO D .
0 DA`CA FQot MA?QuALLY SMOOTHED CuQv . CALcuLATED uStt44C.i SECAt"1Y METHov,
5 I I I 11, 1 I I
0 0 0
118
10 20 40 60 SO too
AK., (MN /ml
Figure 3.9. Comparison of da/dN versus QK data calculated using the secant and polynomial curve fitting methods: data at R = 0.7 and 0.25 Hz.
Unloading line -0 d 0 J
J =XA B(W-a)
A J = X (hatched area) B (W-a)
E
deflection, A b
estimated closure point 1 1 1 1
119
Load- Point Displacement 6
Figure 3.10. Typical load versus load-point displacement curve under monotonic loading: J definition.
Figure 3.11. Operational definition of 'OJ' after Dowling and Begley [Ref. 150].
0
H
G
AU
_1 d(AU) S do..
1
g OEFLECT1ONI
0 Q 5
Figure 3.12. Operational definition of 'LJ' after 'N' loading cycles, after Branco et al. [Ref. 152].
CRACK GROWTH LINE
CRACK TIP BLUNTING LINE (..)/ 20-flow )
CRACK GROWTH A a
Figure 3.13. Schematic representation of the J-resistance curve and changes of crack tip shape.
120
121
•
. CHAPTER FOUR
FATIGUE CRACK GROWTH BEHAVIOUR OF BS4360-50D
STEEL IN MID-RANGE GROWTH RATES
122
4.1 INTRODUCTION
Many studies have confirmed that the fatigue crack growth rate,
da/dN, is primarily controlled by the stress intensity factor range, AK,
through the well-known Paris expression [36]. It has also been
established (as observed in the review made in Chapter 2) that certain
mechanical and metallurgical, as well as environmental factors affect
fatigue crack growth rate, the role of stress ratio, thickness, frequency
and environment being of some significance.
The effects of stress ratio, thickness, frequency and salt water
environment on fatigue crack growth in BS4360-50D steel were studied in
the mid-range of crack growth rates and the results of this study are
presented and discussed in this chapter. A critical analysis of results
reported in the literature seems to indicate that the effect of stress
ratio on the crack growth rate may be influenced not only by the
structure of the material but also by at least three factors: cyclic
frequency, size of the specimen, and the environment. The test conditions
in the present investigation were therefore varied to ensure that
comprehensive results were obtained.
4.2 EXPERIMENTAL PROCEDURE
Details of material, specimen geometry and preparation as well as
testing machines, have been discussed in the previous chapter. Thus,
only the main experimental aspects peculiar to the test results presented
in this chapter, will be discussed in this section.
Fatigue crack growth tests were performed on compact specimens of
the three thicknesses, that is, 12 mm, 24 mm and 50 mm. The tests were
performed in laboratory air and, in the case of some of the 24 mm thick
specimens, in 3.5% sodium chloride solution. All tests were performed
under ambient conditions (i.e. at room temperature of 21 °C and about 50%
•
123
relative humidity). Five different machines were used for the tests.
For the tests at 30 Hz and positive stress ratios the Dowty and the
Mayes machines were used. Tests at 0.25 Hz in air and in salt water
were performed in the two Avery machines though some of these tests were
performed in the Dowty machine. Tests at negative stress ratios were
performed in the Instron and the Dowty machines, using special grips
enabling reversed cycling. As pointed out earlier (section 3.2) some
check results were made to ensure that all the machines gave similar
results under identical testing conditions.
To investigate the effect of positive stress ratio, for the three
thicknesses, tests were performed at R equal to 0.08 and 0.7 at a cyclic
frequency of 30 Hz. Additional tests for the 12 mm and 24 mm thicknesses
were performed at R = 0.5 and 30 Hz to confirm the experimental trend.
A few tests at constants Kmean
of 20 MN/m3/2 and 40 MN/m3/2 were performed
at 30 Hz to investigate the effect of Kmean
However, in these tests,
a continuous adjustment of the applied load was necessary to maintain
K constant and this could have some effect on the results. mean
To investigate the effect of frequency some other tests were
performed at a cyclic frequency of 0.25 Hz and at stress ratios of 0.08
and 0.7 for both the 12 mm and 24 mm thick specimens. For the 24 mm
specimens additional tests were performed at 0.06 Hz and 0.02 Hz both
at R = 0.08.
To investigate the effect of negative stress ratios, a second series
of tests was arranged. The specimens (12 mm and 24 mm thick specimens
only) were precracked to equal values of crack lengths. A constant
maximum load limit, Pmax,:
was chosen for all the specimens while the
minimum load limit, P min
, was varied from specimen to specimen in order
to obtain the required negative stress ratio (for example R = -0.17,-0.7
and -2). Thus, in these tests, the only variable was the negative part
of the cycle. Results from these tests showed a tendency of the negative
•
124
stress ratio to have significant effect on growth rates for shorter
cracks growing from the notch. It was therefore decided to perform
further tests on the 24 mm thick specimens starting each test right
from initiation at the notch. All the specimens had the notches cut
using the same tool and therefore all were of same geometry.
The effect of thickness could be deduced from the results obtained
from the above tests and therefore no further tests in this case were
necessary.
The effect of salt water environment was investigated only for the
24 mm thick plate. Corrosion fatigue tests were performed at stress
ratios, R, of 0.08, 0.24 and 0.7, and at frequencies of 30 Hz, 3 Hz
and 0.25 Hz and 0.06 Hz using a sinusoidal waveform. The fluid was
introduced.into the specimen at a flow rate of approximately 1.5 ml/min.
Stress corrosion tests were also performed but these did not produce
any reliable results. No stress corrosion seemed to occur in this
material.
In all the above tests the specimens were prepared as was discussed
in the previous chapter. The specimens were then pre-cracked at a
suitably higher frequency using a low load range giving a Kmax not
exceeding 17 MN/m3/2. Subsequently the fatigue crack was propagated
at the required loading conditions (i.e. stress ratio, frequency, etc.)
for some 2 mm to 3 mm before the actual test readings were started.
This process ensured that uniform growth at the particular load range
was attained and thus eliminated the influence of short cracks at the
original notch or that of load histories, on fatigue crack growth rate.
All the tests were performed at fixed load limits, i.e. constant R-ratio
allowing the crack to grow from its original length to the final length
required. For some tests at the lower frequencies (0.25 Hz) and growth
•
rates below 2 x 10-5 mm/cycle, step-up or step-down tests were performed
125
to speed up the investigation. For each step the value of OK was
adjusted upwards or downwards by not more than 10% and the crack allowed
to grow at those fixed load limits by more than the existing plastic
zone size, r, calculated from
1 Kmax 2 r -
3w ( ay ) (4.1)
before the first reading was taken. At least two readings of crack
length and a corresponding number of cycles were then taken for each
step. This process ensured the elimination of load history effects
to which this material was found sensitive.
Crack length measurements were made using microscopes mounted on
travelling micrometric stages as discussed previously. The experimental
data were recorded in terms of crack length, a, and the corresponding
number of cycles, N. A typical plot of a versus N in the present data
is shown in Fig. 4.1. To calculate the crack growth rate, da/dN, from
the experimental data a combination of graphical curve-fitting and
finite difference (secant) methods was used. When the readings showed
little scatter of the da/dN data, the raw experimental readings were
used to calculate the growth rates. However, when the scatter was
substantial, as at low frequencies and in salt water tests, smooth curves
were fitted (graphically) to the experimental data and points on these
curves used to calculate da/dN using the secant method. This process
reduced the scatter in raw data (for example see Figure 3.9). It was
also observed that scatter was much reduced by taking readings at crack
increments of between 0.5 mm and 1 mm and therefore these were used for
all subsequent calculations of da/dN.
An attempt was made to apply the polynomial curve-fitting method
using the least squares method (section 3.4.6). A computer programme
was developed from which da/dN and corresponding AK values could be
calculated. This procedure was very satisfactory, when a large number of
126
experimental points was available or when the scatter was minimal (as in
high frequency constant-R tests). However, this method was unsuitable
for short tests and for other results containing a considerable amount
of scatter. Therefore, for consistency of results all the data reported
here were analysed using only the secant method.
The stress intensity factor, K, was calculated using the appropriate
standard expressions for CS geometry, i.e. using equations (3.5), (3.7)
and (3.8). The crack length, a, used to calculate K is the average of
the two successive readings used to calculate the corresponding da/dN.
4.3 RESULTS
The experimental results are plotted in terms of da/dN versus the
corresponding AK values for all the tests.
4.3.1 Effect of stress ratio (or mean stress):
Figure 4.2 shows the results for 50 mm thick plate tested at 30 Hz
using stress ratios, R, of 0.08 and 0.7. These data, within experimental
scatter, show no effect of stress ratio on the crack growth rates.
The results for the 24 mm thick specimens tested at the same
frequency of 30 Hz and applying stress ratios, R, of 0.08, 0.5 and 0.7,
are shown in Figure 4.3. Again these data, like those for 50 mm in
Figure 4.2, show no significant effect of stress ratio. However, there
is some tendency towards a small stress ratio effect at lower growth
rates.
Figure 4.4 shows the results for 12 mm thick specimens also tested
at 30 Hz and at stress ratios, R, of 0.08, 0.5 and 0.7. These results
suggest a stronger effect of R than in the previous ones on 50 mm and
24 mm thick plates. The stress ratio effect seems to increase considerably
•
• 127
at lower growth rates. However, at higher growth rates (about 2 x 10-4
mm/cycle) the stress ratio effect seems to disappear altogether.
Figure 4.5 shows results for the 24 mm thick specimens obtained
at constant Kmean
values of 20 MN/m3/2 and 40 MN/m3/2 compared with the
data at constant R-ratio of 0.08. Again as observed in Figure 4.3
there is no effect of mean stress on crack growth rates. Similar K mean
data for the 12 mm thick plate are shown in Figure 4.6. However, these
data were obtained by linear interpolation of the data in Figure 4.4
using the relationship
Korean 2 (1 + R)/(1 - R).AK
(4 .2)
The results obtained at a cyclic frequency of 0.25 Hz for the 12 mm
and 24 mm thick specimens are shown in Figures 4.7 and 4.8 respectively.
These results show an increased effect of stress ratio at the lower
frequency. When compared with the corresponding data at 30 Hz they
also show that only growth rates at high stress ratios are significantly
affected by the change in frequency.
Figures 4.9 to 4.12 show the results at negative stress ratios for
the 12 mm and 24 mm thick specimens. Figure 4.9 shows the results for
24 mm thickness plotted using only the tensile portion of the load
cycle. These results show that at lower growth rates the compressive
portion of the cycle has no significant contribution to cyclic crack
growth. However, at higher growth rates (> 10-4
mm/cycle) the results
show that a part of the compressive cycle does contribute to crack growth.
Figure 4.10 for the 12 mm thickness is plotted, taking into account
a complete compression-tension load cycle, while in Figure 4.11, also for
the 12 mm thickness, only the tensile portion of the cycle was used.
Again there seems to be a contribution of the compressive portion of the
cycle to fatigue crack growth. However, unlike the results for 24 mm
128
thick specimens, the contribution of the compressive portion of the
cycle seems to decrease at higher values of AK.
Figure 4.12 shows results for crack initiation and growth at a notch
tested at both positive and negative stress ratios. In all the tests the
load limits were continuously adjusted after every 0.25 mm of crack
increment so as to maintain a constant AKTen of 15 MN/m3/2. The time
to initiate the fatigue crack (to about 0.1 mm) was found to be longest
at R = 0.0, where 126,000 cycles were required. For the test at R = 0.4
it took about 50,000 cycles to initiate the crack. For both R = 0.0
and R = 0.4 the crack subsequently propagated at increasing growth rate,
being faster for the R = 0.4, from growth rates of 10-6 mm/cycle and
stabilised at about 2 x 10-5 mm/cycle after 1 mm and after 3.8 mm
respectively. On the other hand, for the tests at R-ratios of -1 and
-2, the number of cycles for initiation of the crack was considerably
reduced; being about 2,000 cycles and less than 500 cycles respectively.
The subsequent crack growth rates 3 x 10-5 mm/cycle for R = -1 and
2 x 10-4 mm/cycle for R = -2 but these decreased with crack length to
stabilise at 2 x 10-5 and 3 x 10
-5 mm/cycle respectively. In Figure 4.13
R is plotted against N for various values of crack increments, Aa,
measured from the notch. It can be seen that the compressive load is
very damaging during initiation and growth of short fatigue cracks at a
notch. At high negative stress ratios the initiation period is reduced
to very few cycles while the immediate crack growth rate is comparatively
high.
It was also observed that applying a compre ,,,,sive load, prior to
precracking of the specimen under tension-tension loading, considerably
reduced the initiation period. For example, precracking at R = 0.0
and at AK = 15 MN/m3/2 initiation period was reduced from 126,000 cycles
to about 60,000 cycles by initially applying a compressive load equivalent
to K of -15MN/m3/2.
129
4.3.2 The effect of thickness:
In order to evaluate the effect of thickness on fatigue crack
growth rates, the best-fit lines through the results in Figures 4.2 to
4.4 are plotted in Figures 4.14 and 4.15. Similarly, the results from
Figure 4.7 and 4.8 are plotted in Figure 4.15. It may be observed from
Figures 4.14 and 4.16 at R = 0.08, that, for a given value of AK, the
fatigue crack growth rates are higher for the thicker material up to
approximately 2 x 10-4
mm/cycle growth rates. However, above this value
the crack propagates faster in the thinner material. Figures 4.15 and
4.16 show that the effect of thickness at R = 0.7 is considerably
reduced. In fact for the three thicknesses the results at R = 0.7 all
seem to lie in a narrow scatter band.
4.3.3 The effect of frequency:
Similarly, the effect of frequency can be evaluated by drawing the
best-fit lines through the results in Figures 4.3 and 4.7 shown in
Figure 4.17 for the 24 mm thick plate and through results in Figures 4.4
and 4.8 shown in Figure 4.18 for the 12 mm thick plate. Crack growth
rates do not seem to be affected by frequency change from 30 Hz to 0.25
at R = 0.08 apart from a region at lower growth rates. However, at
R = 0.7 the growth rates are higher at the lower frequency (0.25 Hz),
and here the influence is appreciable. These results show that the
growth rates may be doubled by changing the frequency from 30 Hz to
0.25 Hz at R = 0.7.
Figure 4.19 shows additional results obtained at R = 0.08 but at
the much lower frequencies of 0.06 Hz and 0.02 Hz plotted with those at
0.25 Hz and 30 Hz for the 24 mm thick plate. The results at 0.02 Hz
show a significant influence of frequency even at this low stress ratio.
130
4.3.4 Effect of salt-water environment:
Figure 4.20 shows the results of tests conducted in salt-water
solution at 30 Hz and at three stress ratios (i.e. at 0.08, 0.24 and
0.7). These results are compared with those in air. This figure shows
that, like the results obtained in air at the same frequency, stress
ratio has no significant effect on crack growth rates in salt-water
environment at high frequency. However, growth rates are lower in salt
water for AK less than 30 MN/m3/2 whereas above this value the growth
rates in salt water are higher than those in air.
Figure 4.21 shows the results for tests performed at the lower
frequency of 0.25 Hz and at two stress ratios of 0.08 and 0.7 in salt-
water solution. The results are again compared with those obtained in
air. This figure clearly shows that crack growth rates are much higher
in salt water than in air at both low and high stress ratios. There is
not much difference in growth rates at both the R-ratios of 0.08 and 0.7
in salt water but these are between three and four times higher than
those in air.
Figure 4.22 shows the results of tests performed at R = 0.08 and
at various frequencies. It can be observed from this figure that growth
rates increase with decreasing frequency. Although static load
environmental tests performed on this material do not seem to show any
crack growth by stress corrosion, it is interesting to note the small
plateaux in the corrosion fatigue curves at both the 0.06 Hz and
0.25 Hz frequencies. The results in this figure compare well with
similar results obtained in air which are shown in Figure 4.19.
Figure 4.23 shows the-results of tests performed at R = 0.7 and at
0.25 and 30 Hz frequencies. These results show that reducing frequency
from 30 Hz to 0.25 Hz can result in an increase of growth rates in salt-
water solution by one order of magnitude.
•
131
Figures 4.24 and 4.25 shows the present results at 30 Hz in air
compared with the mean line through Scott and Silvester's [110] results
performed in air for 25.4 mm thick specimens of same material. These authors
used testing frequencies varying between 1 Hz and 10 Hz. Their results
compare very well with the present results with the exception of a
higher AK exponent of 3.23 obtained by these authors compared to that
.of 3.0 obtained in the present results. However, growth rates at
R = 0.08 in the 12 mm thick specimens are overestimated (by a factor of 5)
by the mean line through Scott and Silvester's data, i.e.
da dN -
2.88 x 10-9
(AK)323 (4.3)
4.4 DISCUSSION
4.4.1 The effects of stress ratio and frequency in air:
Stress ratio, R, has frequently been used [68-74] to investigate
the effect of mean stress on fatigue crack growth. Tests at constant
mean stress intensity factor, Kmean
, have been carried out [61,65] for
the same purpose. The results of one form (constant R-ratio) can
easily be transformed into the other (constant Kmean
), provided adequate
experimental results are available. This transformatiōn can be
accomplished using equation (4.2). It will be realised that constant
R-ratio tests are easier to perform on the specimen geometry used in
this study; they are also more appropriate in life calculations for
non-redundant structures where the stress ratio, R, remains constant
with increasing crack length.
In analysing the results, attempts were made to evaluate the growth
rate data using a range of models found in the literature for comparison
purposes. The Paris model [36], equation (2.1), was found to be
satisfactory when applied to the results from both the 24 mm and 50 mm
132
thick specimens tested at 30 Hz. These data were found to fit an
expression of the form
da dN =
6.71 x 10-9 AK30 (4.4)
where da/dN is in mm/cycle and AK is in MN/m3/2. From these results it
is apparent that the stress intensity factor range, AK, is the dominant
variable in determining the rate of crack growth in this material. The
stress ratio independence of growth rates is in agreement with the
Richards and Lindley [39] characterisation of the ductile, striation-
forming growth mechanism.
For other results (for example those at 0.25 Hz), where the effect
of stress ratio was larger, the Paris expression was not suitable and
the correlation of the data was unsatisfactory. On the other hand, the
growth model proposed by Forman at al. [55] was found to over-estimate
the stress ratio effect substantially, particularly for tests at higher
frequencies (i.e. those at 30 Hz), where the influence of R ratio id-
small. The same problem was encountered when applying the model
proposed by Branco at al. [61]. The generalised model proposed by Dover
and Hibberd [65] was found satisfactory. The empirical constants in the
model could be varied to accommodate the present data.
However, a common problem with the above models is the choice of a
suitable fracture toughness, Kc, value. Values of Kmax
recorded at the
moment of sudden fracture of the specimen during fatigue were found to
vary not only with thickness, stress ratio and frequency but also with
crack length. For example, values of max
at fracture for the 12 mm,
24 mm and 50 mm thicknesses were found to average about 115 MN/m3/2,
95 MN/m3/2 and 85 MN/m3/2 respectively. Values of Kc determined from
R-curves were also found dependent on thickness and crack length; being
largest for the thinnest plate. Thus, use of a single value of Kc in
the present analysis was difficult to justify.
•
133
An attempt was made to correlate data using crack closure models
[66]. The effective stress intensity factor range, <eff,
based on
crack closure, is not easily determined. This is because the stress
intensity factor for crack closure, KCL, as originally proposed by
Elber [66], is a function of a number of mechanical and metallurgical `
factors [67]; it is also dependent on the testpiece, as recently
reported by Dover and Boutle [93]. Maddox et al. [62] have shown that
for various steels there exists no single expression for AKeff
based on
crack closure. Consequently, this value has to be evaluated not only
for different loading conditions but also for different specimens. Results
of tests performed at negative stress ratios seem to indicate that fatigue
damage may occur, even when the crack is closed [68]. However, although
the above observations cast doubt on the use of an effective stress
intensity factor range based exclusively on the crack closure concept,
crack closure effects on crack growth cannot be discounted altogether,
especially at low stress intensities (for example see Chapter Five).
Empirically derived effective stress intensity factor range, AKeff3
which normalises data at a series of R value to the growth rates at
some particular value of R (e.g. R = 0) was successfully used to
correlate the present data. Models by Sullivan and Crooker [70] and
Walker [69], which were discussed in chapter two, appear to be
particularly suitable. The empirical constants in the respective models
are easily determinable from experimental data under specific test
conditions and can be evaluated for both positive and negative values of
R. The Sullivan and Crooker model was found more suitable for correlating
the present results and this can be expressed in the form (Equation 2.14)
da __ C (1 - bR)mll AK11 dN 8 1-R (4.5)
•
134
The constants m11 and C8 are normally determined for R = 0 and b is then
determined for any other value of R. If the material is insensitive
to stress ratio then a value of b equal to unity is obtainable.
The above expression is most suitable when the experimental data
show crack growth curves, at different stress ratios, R, which run
parallel to one another, (that is, different values of R-ratio having
same exponent of AK), as in Figures 4.7 and 4.8 of the present results.
Thus, when the exponent of AK varies with stress ratio then the
equation (4.5) is not altogether satisfactory. Only average values of b
can be useu over the range of growth rates considered for the particular
case. This is so with the data shown in Figure 4.4 for the 12 mm thick
specimens tested at 30 Hz, where at lower growth rates (< 2 x 10-5 mm/
cycle) the slopes of data at R = 0.08 and at R = 0.7 are significantly
different. A similar observation is made for crack growth results at
negative stress ratios shown in Figure 4.9.
The results presented above will be discussed in terms of
environmental and crack closure effects. It is proposed that at low
stress ratios (including negative R-ratios) crack closure effects are
important and in addition, at low frequencies environmental effects are
also important.
It is thus proposed to modify the Sullivan and Crooker [70] model
to include the effect of crack closure and also the effect of environment,
both of which have been observed in the present air results. To include
the effect of crack closure AK in equation (4.5) is replaced by AKeff
such that
AKeff
= U1AK (4.6)
where U1is the proportion of AK for which the crack is closed. Equation
(4.6) neglects any environmental effects. The constant b is then
•
• 135
associated with effects of stress ratio as a result of changes in K max
resulting in additional crack growth such as the case with environmental
effects. Thus equation (4.5) becomes
da - = C (----)me (AK )me
dN e 1-R eff (4 .7)
The present results obtained at stress ratios of 0.5 and 0.7 and
at 30 Hz can be regarded as not influenced by either crack closure or
environment. Therefore the value of b can be assumed to be equal to
unity and AK equal to RKeff. Thus
da m
dN - = Ce(AKeff) e
(4.8)
and values of Ce and me can be determined from the experimental data for
all the three thicknesses and at R = 0.5 and 0.7. Regression analysis
of these data gives values of Ce = 7.6 x 10-9 and me = 3.07. It is
therefore proposed that the rest of the data should be normalised to
growth rates predicted by equation (4.8).
The data obtained at 0.25 Hz, shown in Figures 4.7 and 4.8, were
correlated using equations (4.7) and (4.8) and these are shown in
Figure 4.26. Values of b obtained in the present results are tabulated
below for R > 0.
t
• Thickness Frequency b
12 mm 30 Hz 1
12 mm 0.25 Hz 0.87
24 mm 30 Hz 1
24 mm 0.25 Hz 0.9
50mm 30 Hz - 1
136
For the negative stress ratio data, Figure 4.11, an average value
of b equal to 1.4, for the 12 mm thick specimens, was estimated.
Similarly data for the 24 mm specimens at R = -0.7 and R = -2,
Figure 4.9, seemed to indicate a value of b = 1.9. A discussion on the
effect of crack closure and environment on the present data will follow
below.
As discussed previously, crack closure phenomenon has often been
used to explain the observed effect of stress ratio on fatigue crack
growth rate. According to this concept, at low stress ratios, the crack
closure stress intensity factor, KCL, is above the minimum applied stress
intensity factor, Km. , and thus the effective stress intensity factor
range responsible for crack growth is substantially reduced. At high
stress ratios, the crack remains open for the whole or for a larger
part of the load cycle and thus crack closure effect becomes less
pronounced. Following this argument it would seem that crack closure is
negligible for the 24 mm and 50 mm thick specimens tested at 30 Hz even
at the lower stress ratio of 0.08 since these data fall into a narrow
band. According to Elber's argument that crack closure was primarily due
to plastic deformations at the crack tip, it would seem reasonable to
expect crack closure effects to increase as AK increases. However, this
is not the case with the results for 24 mm and 50 mm thick plates
(Figures 4.2 and 4.3), unless, despite the increase of crack closure
stress, KCL
remains < Km. .
Considering now the results obtained on the 12 mm thick specimens
at 30 Hz (Figure 4.4), it appears that the effect of stress ratio is
diminishing with increasing growth rates. If crack closure arguments are
used exclusively to explain this data it would appear that the closure
stress intensity factor, KCL, decreases with AK (by comparing data at
R = 0.7 and R = 0.08). This should not be the case since crack closure
•
137
due to plastic deformations is more likely to increase at high stress
intensities. It has also been observed that there is no influence of
stress ratio on growth rates for steels tested in vacuum [131] and this
observation seems to confirm that crack closure is not the only factor
influencing the stress ratio effect. However, it should be noted that
re-welding of the crack tip has been observed to occur in vacuum and
this complicated the crack tip deformation characteristics in vacuum
compared to those in air.
It is possible that the effect of stress ratio on the growth rates
observed at lower frequencies (0.25 Hz) in air could be connected with
environmental effects. It is known that hydrogen in moist air can cause
embrittlement at the crack tip, thus leading to increased growth rates.
Hydrogen atoms in steel can diffuse to the regions of maximum hydrostatic
tension ahead of the crack tip, thereby lowering the cohesive strength
of the lattice [103]. This process, causing faster growth rates, is
more significant in high strength steels than in low alloy steels.
Higher fatigue crack growth rates could thus be caused by the
increased stress ratio, since the higher value of Kmax
creates a large
stress gradient to promote hydrogen diffusion. Higher crack growth rates
at high testing frequencies (such as 30 Hz) would not provide favourable
conditions for hydrogen to diffuse to the crack tip in time to cause
embrittlement, since the crack is advancing at too high a velocity.
However, at low crack velocities, that is at low growth rates and at low
test frequencies, the environmental effects are maximised. This argument
is supported by comparing Figure 4.3 with Figure 4.7 and Figure 4.4 with
Figure 4.8. It also seems that the plate thickness could influence the
transport of hydrogen from moist air to the crack tip. The thicker the
material, the less the effect of environment at a certain crack velocity,
since it takes more time for hydrogen to permeate along the whole width
138
of the crack front. Comparison of Figures 4.7 and 4.8 would support
this proposition.
It could also be argued that, at low stress ratios, the crack is
closed during the greater part of the cycle. Since closure is primarily
a surface effect, it may reduce the rate of hydrogen diffusion into the
crack front. This conclusion is supported by the very small effect of
frequency observed for both the 12 mm and 24 mm thicknesses at a stress
ratio of 0.08 (Figures 4.17 and 4.18). However, at high stress ratios,
the crack is fully open for a larger part of the cycle and the
environmental effects are also maximised. Environmental effects are
confirmed by the behaviour observed in Figure 4.19 at the lower frequency
of 0.02 Hz; characteristic of crack growth in aggressive environments.
Thus, with the absence of stress ratio influence on growth rates for
steels tested in vacuum (131] and a reduced stress ratio effect observed
in the present tests at higher growth rates and at a high testing
frequency, the environmental conditions would seem to provide a good
explanation of the stress ratio effect on growth rates observed at low
frequencies. The crack closure phenomenon would then be only of
secondary influence in these results.
Further, the effect of negative stress ratio especially for short
cracks at notches, could not adequately be explained in terms of crack
closure alone. The present results show that the fatigue damage does
occur even below zero applied stress, Figures 4.9 and 4.11. It would
appear that, during the application of the compressive load, the crack
surfaces are highly compressed, especially in the region close to the
crack tip. This would result in a sharpened crack. This process would
cause increased stress intensity at the crack tip leading to accelerated
growth. However, not all the compressive load is used to collapse the
crack surfaces; some of it is dissipated in reducing and even reversing
•
139
the residual stresses in the vicinity of the crack tip remaining from
the preceding tensile cycle. Some of the applied compressive load is
wasted in pressing the fractured surfaces together and in this respect
the length of the fatigue crack from the notch and also the amount of
opening of the crack tip from the previous tensile cycle is important.
For short fatigue cracks the compressive load would cause more damage
than for long cracks. This is supported by the present results shown
in Figures 4.12 and 4.13. These results show that compressive loads are
very important in fatigue crack growth behaviour of short cracks originating
from a notch. It is, thus, concluded that the crack growth models should
include the effect of compressive-tensile cyclic loading, since the
compressive portion of the cycle, applied to the crack tip, is found to
cause fatigue damage.
4.4.2 The effect of thickness:
The results of the present investigation show a considerable effect
of thickness on crack growth rates especially at low growth rates and at
low stress ratios. From these results it was observed that, for the
same loading conditions and for all stress ratios considered, the crack
will propagate faster in the thicker plates at growth rates below
2 x 10-4 mm/cycle. A similar increase of growth rates with increasing
thickness was reported by Barsom et al. t86] and Kang and Liu (89].
However, in the tests at higher growth rates, i.e. above 2 x 10-4 mm/cycle,
the situation changes distinctly and the effect of thickness is only
marginal. The region between 2 x 10-4 and 7 x 10-4 mm/cycle may be
described as transitional. Further, at high growth rates (i.e. above
7 x 10-4 mm/cycles) for this type of steel and at low R values such as
0.08) lower growth rates were recorded in the thicker plates. At high
stress ratios, such as 0.7, the crack growth rate always appears to be
•
140
higher in the thicker plate (see Figures 4.15 and 4.16). However, the
effect of thickness at high stress ratios appears much reduced compared
to that observed at low stress ratios. These results suggest that
stress ratio, R, as well as the stress range (or AK) may substantially
affect the relationship of thickness and crack growth. Comparison of
results at negative stress ratios seems to indicate that the effect of
thickness also increases with increasing IRS at least for the range of
negative R-ratios tested.
The effect of thickness on fatigue crack growth, where reported, has
often been associated with the state of stress and strain at the crack
tip, i.e. whether plane strain or plane stress conditions [87,88,90].
It is generally found that the direction of crack growth in a uniaxially
loaded sheet specimen of some materials, such as zinc, is normal to the
loading direction with the fracture surface in a plane through the
thickness at 90° to the plane of the specimen. However, in other
materials, such as steels, copper alloys and aluminium alloys [5], a
transition to growth at a plane through the thickness inclined at
approximately 45° to the plane of the specimen usually takes place after
an initial period of 90° crack growth. This change in fracture has been
associated with a change in plane strain to plane stress [209,210].
It has also been proposed [209] that this change occurs when the size
of the crack tip plastic zone, equation (4.1), (which is a measure of
state of stress) reaches a certain proportion of the specimen thickness.
However, some attempts to correlate the transition with plastic zone
size have proved inconclusive [76,921. While the effect of thickness has
been investigated specially, it would be expected from the above argument
that it could be deduced from observations on the transition in the
fracture plane. However, there is disagreement in the literature about
the effect of this transition on crack growth rates. Frost and Dugdale
•
141
[211] found no effect of transition on the crack growth rates in mild
steel and an aluminium alloy, while McEvily and Johnson [210] and
Swanson and Cicci [212] found evidence of lower growth rates after
transition in aluminium alloys. In contrast Liu [209] and Frost [213]
found that growth rates increased when transition took place. Further,
the effect of thickness observed at low growth rates cannot be explained
by plane strain/stress transition arguments and thus a better explanation
is required.
The relationship between thickness and fracture toughness reported
by Johnson and Radon [214], could provide an alternative explanation of
the thickness effect. The implication here is that, as Kmax
tends to
approach the Kc value, a 'steady-state' cracking might occur and cause
an accelerated crack growth. This slow stable process is related to the
well known tearing of the material. For specimens, where mixed mode or
plane strain conditions prevail at`the crack tip, as in the present tests,
the thicker material has a lower value of K. Here, the role of stable
cracking would be more significant and lead to higher growth rates; a
prediction supported by the present results especially those at high
stress ratios where Kmax
approaches the K c. However, this argument does
not explain the pronounced effect of thickness observed in the present
results at low stress intensities.
Kang and Liu [89] have advanced another explanation of thickness
effect on crack growth. During their investigation they observed that as
soon as a particular value of Kmax
was exceeded, the calculated strain
values were underestimated compared to the experimentally measured strains.
They also noted that the condition of 'plane strain' at the crack tip
was controlled by the plastic zone size relative to the plate thickness.
They concluded that localised necking which increased crack tip strains
led to accelerated fatigue damage, and thus faster growth. This localised
•
142
necking is likely to occur first in the thinner material under similar
loading conditions. The decreasing effect of thickness at higher growth
rates observed in the present results seems to concur with the Kang and
Liu's observations. However, localised necking effects do not satisfactorily
explain the behaviour observed at lower growth rates in the present results
where these effects are insignificant.
Another well known approach is that of crack closure.[66] and this
has frequently been used to explain behaviour at low growth rates. It is
believed to be caused by residual stresses, mainly due to plastic
deformation close to the crack tip. These stresses are more significant
in the thinner material, where large plastic deformations take place,
resulting in slower growth in the thinner material. This phenomenon
would be of significance at medium and high stress intensities where
plastic deformation is substantial. However, at low stress intensities
plastic deformation effects are small and crack closure arising from
them should not be sufficient to explain the thickness effects observed
in the present results.
Dover and Boutle [93] made crack closure measurements during fatigue
crack growth tests on aluminium alloy specimens of different thicknesses.
They observed that under plane strain conditions, crack closure stress
decreased as the crack length increased. They argued that crack closure
which occurred was probably not related to events at the crack tip but
due to the residual stresses present in the plate. The magnitude of the
closure stress and possibly those residual stresses, seemed to depend on
the plate thickness in such a way that thickest plates had the largest
values of closure stress. The thickness effect observed in the present •
results can be explained in terms of crack closure effects due to residual
stresses already in the plate as observed by Dover and Boutle. However,
unlike the results obtained by these authors, the present results seem
• 143.
to indicate that crack closure stress had the largest values in the
thinnest plate. Preliminary crack closure measurements made during the
present investigation seem to confirm this tendency. These results will
be presented in the next chapter (i.e. Chapter 5).
The residual stresses in the plate could be introduced during
manufacturing processes of the plate and also during subsequent machining
operations on the specimens. The residual stresses due to the latter
would produce more significant effects on the thinner than the thicker
specimens. The effects would be more pronounced at low stress intensities
where the applied Kmin
would be less than the crack closure stress
intensity factor, KCl. This in effect would reduce the effective stress
intensity factor responsible for crack growth. On the other hand at
higher stress intensities and at higher stress ratios the applied R min
would be greater than or of the order of KCL and thus the effective stress
intensity factor range would be equal to or of the order of the applied AK.
It is therefore proposed that the effect of thickness observed
in the present investigation at low and intermediate growth rates could
be explained in terms of crack closure due to residual stresses in the
specimen and also in terms of 'steady-load' cracking which favours the
thicker specimens as explained above. However, at high growth rates the
localised necking takes place at the crack tip and since this necking
occurs earlier in the thinner material, growth is accelerated.and thus
the higher growth rates are found in the thinner material at high values
of AK. The interaction between crack closure, 'steady-load' cracking,
localised necking and. environmental effects observed at low frequencies
would be difficult to quantify. This justifies the use of an empirical
expression to correlate the present data.
144
4.4.3 The effect of salt-water environment:
The present results have shown that salt water environment has an
influence on the fatigue crack growth in the BS4360-50D steel investigated.
The magnitude of this influence seems to depend on the loading conditions
such as stress ratio and frequency.
The results obtained at high frequency (i.e. at 30 Hz) showed
decreased growth rates for lower AK (< 30 MN/m3/2) and higher growth
rates for higher AK values compared to air results. These effects have
been observed and predicted by several investigators, especially the
acceleration effect at high AK values. However, the deceleration effect
of the environment, though previously observed, has not been clearly
defined. This effect could be explained in terms of solution chemistry
and the mechanical interaction between the fluid and the crack tip geometry.
The entry-and exit momentum of the fluid is important as explained
by Hartt et al. [215]. It is possible that, due to the nature of the
channels used to collect the fluid, the slow flow rate, and the high
frequency, all combine to force a rapid mixing of the crack-tip and bulk
solutions. This could have a two-fold effect. Firstly, the difference
in pH between the solution at the crack tip and the bulk solution could
be less than that observed when the bulk solution is replaced more rapidly.
Secondly, the possibility of the 'used' fluid and corrosion products
re-entering the crack tip region is increased. As a result, corrosion
products could deposit at the crack tip and cause blunting of the crack
tip which in turn reduces the growth rates. The 'wedging effect' of the
solution and corrosion products which decreases the effective stress
intensity factor (and thus the growth rates) was observed by Plumbridge
[216] and could be a possible cause of deceleration in growth rates at
low AK values.
145
The oxygen content of the solution is also important. The slow
flow rate used in the present investigation could result in the solution
at the crack tip becoming relatively de-aerated and its oxygen content
dropping to low levels. This would result in deceleration of the crack
growth rates as observed by Scott and Silvester [110] even at low
frequencies. Using a much higher flow rate (1 litre/min) Scott and
Silvester observed no influence of sea-water environment above the
frequency of 1 Hz.
At low frequencies, the salt-water environment seems to change the
crack growth rates significantly. This effect seems to depend on the
applied Kmax
as shown in Figure 4.21 where at low AK values the growth
rates at R = 0.7 are much higher than those at R = 0.08 but at high AK
values, where Kmax
is high in both cases, though not of same order of
magnitude, the growth rates at the two stress ratios are the same. From
Figures 4.21 and 4.22 it seems that the increase of stress ratio and
decrease in frequency result in an overall behaviour such that the growth
rates increase is 'saturated' at some upper bound, as shown in Figure 4.27.
The over all upper bound may be represented by.an expression of the form
dN - 2.11 x 10-7 (AK)2.63 (4.8)
This expression predicts growth rates five times higher in salt-water
than in air at the lower frequency (that is, at 0.25 Hz) and about 10
times higher than those obtained in air at 30 Hz.
The above observation is similar to that made by Scott and Silvester
[110], who tested the same material in sea-water at different stress
ratios, at 0.1 Hz and applied a range of electro-chemical potentials.
They obtained an upper bound expression which predicted slightly higher
growth rates (by a factor of 1.1) as compared to equation (4.8).
The crack growth behaviour in aluminium alloys tested in salt-water
has been explained quantitatively [217] by assuming an anodic dissolution
•
146
mechanism under stress corrosion cracking conditions, which causes micro-
branching (thus inhibiting mechanical crack growth processes) but which
is accelerated by load cycling, as fluctuations in crack opening displace-
ments influence local mass transfer kinetics. This mechanism accounts for
the stress corrosion characteristics observed, even at high growth rates,
in corrosion fatigue tests, and also provides an explanation for the
retardation of crack growth at high stress intensities (formation of
'plateaux') when a corrosive environment is introduced. In such cases,
the microbranching is so severe that crack growth for corrosion fatigue
may be slower than either that for fatigue in air or that for true stress
corrosion cracking. No stress corrosion characteristics have been
identified below the static value of KISCC
in corrosion fatigue tests
on aluminium alloys.
The appearance of a 'plateau' at very low frequencies in corrosion
fatigue data in steels, as in the present results, indicates that a
time-dependent process is making significant contribution to crack growth,
even though no such growth occurs under static load of a similar magnitude.
Stress corrosion in high strength steels is generally agreed to be
dominated by hydrogen embrittlement [103] and scanning electron fracto-
graphy of a specimen cracked in the 'plateau' region shows extensive
secondary cracking, penetrating into the crack sides. This appears to
have the same effect as the corrosive microbranching observed in aluminium
alloys, reducing the stress intensity factor and retarding the crack
growth.
Models for correlating and predicting corrosion fatigue crack growth
data were reviewed in section 2.7.3. The process competion
model [108] was found difficult to apply in the present investigation
because of the piece-wise analyses involved and also because it requires
information on the stress corrosion cracking characteristics of the
• 147
material (including the KISCC
value). This information is not available
at present for BS4360-50D steel.
The superposition model, proposed by Rhodes and Radon [122],
modified to allow for the processes to interact appears (as seen from
their report) to give a good correlation of the corrosion fatigue data
when reliable stress corrosion cracking data are available. However,
when stress corrosion cracking data are not available, as at present,
the model reduces to a curve-fitting expression of the form of a super-
position model. It was, therefore, decided to use a simple superposition
model of the form of equation (2.19), where (da/dN)e can be written as
4
da = B1 a1 (dN)e f (Keff)
__ da _ m15 (dN)c C12(~Keff)
(4.9)
where f is the frequency in Hz, and B1 and S1 are empirical constants
obtained by plotting (da/dN)e versus Keff on a log-log basis for a
particular frequency. Thus
m15
$ (dN)c = C12
(AK eff) 15 + f (Keff) 1
(4.10)
To apply equation (4.10) to the present data the constants B1 and
sl were evaluated from the corrosion fatigue data at 0.25 Hz. The mean
• line through the crack growth data at 30 Hz, Figure 4.3, was used to
derive the constants C12 and m15 giving C12 = 6.71 x 10-9 and m15 = 3.0.
It was assumed that Keff Oef
' Clearly B and $1 do not remain constant
over the whole range of growth rates. However, three regions were
distinctly identified where B1 and a1
could be assumed constant, and
their values were estimated. These regions were:
• 148
(i) Keff < 20.6 MN/m3/2.
B1 = 6.11 x 10-16
and S1
— 8.2
(ii)
(iii)
20.6 < Keff < 35.3 MN/m3/2:
B1 = 1.44 x 10-9,
Keff > 35.3 MN/m3/2
B1 = 1.36 x 10-2,
01
01
=
=
3.4.
-1.09.
Crack growth rates predicted from equation (4.10), using
the above values of C12' m15' B1
and S1
for 0.06 Hz, 0.25 Hz, and 3 Hz
are plotted in Figure 4.28. It is seen from this figure that growth
rates at 0.06 Hz are overestimated by the above procedure. On the other
hand when data at 0.06 Hz was used to derive the constants B1 and al,
these were found to underestimate the growth rates at 0.25 Hz. This
behaviour is thought to be due to the fact that the above curve-fitting
procedure does not take into account the 'saturation' of Kmax and
frequency effects which hās been observed before [109,110] and also in
the present results. It is, therefore, concluded that care must be taken
when using curve fitting procedures for prediction of corrosion fatigue
crack growth.
149
4.5 CONCLUSIONS
• The linear elastic fracture mechanics approach was used to
characterise the effects of stress ratio, thickness, frequency and
salt-water environment upon the fatigue crack growth rates in BS4360-50D
steel tested in ambient conditions and at 21 °C. From the results
obtained in the medium growth rates (2 x 10-6 < dN < 2 x -3
it was concluded that:
1. Stress ratio, thickness, frequency and salt water environment may
substantially affect the fatigue crack growth rate. However, these
effects are not of a cumulative nature.
2. In general, the fatigue crack growth rate increases with increasing
positive stress ratio at a given value of AK. This effect is more
significant for the thinnest plate and also increases at lower
testing frequencies.
3. The compressive portion of the load cycle at negative stress ratios
does contribute to crack growth. The contribution was found dependent
on the length of the fatigue crack, measured from the notch-
compressive load causing more fatigue damage in short cracks than
in long ones.
4. For the range of thickness tested, 12 mm, 24 mm and 50 mm, crack
growth rates increase with thickness at a given AK value. The
effect of thickness is more pronounced at low stress ratios (for
example R = 0.08) but less so at high stress ratios (for example
R = 0.7). However, at low stress ratios and at growth rates higher
than 3 x 10-4 mm/cycles, fatigue cracks grow faster in the thinner
plate. The effect of thickness has been adequately explained in
•
10 mm/cycle),
150
terms of crack closure, localised necking around the crack tip
and 'steady state' cracking as Kmax approaches Kc.
5. Crack growth rates increase with decrease in frequency for the
range of frequency tested, 0.02 Hz to 30 Hz. The effect of frequency
is more pronounced at high stress ratios for a given frequency. For
example growth rates are doubled by changing the frequency from
30 Hz to 0.25 Hz at R = 0.7 while no significant effect is observed
at R = 0.08. `The effect of frequency was adequately explained in
terms of environmental effects of moist air.
6. Crack growth rates in air for the three thicknesses tested can be
predicted by the expression
da _ C (1-bR)me (DK )e dN e 1-R eff
where Ce = 7.6 x 10-9' me = 3.07, ~Keff
is the effective stress
intensity factor range corrected for crack closure, R is the stress
ratio and b is an empirical constant dependent on frequency (and
also on environment). For all the air data at 30 Hz b = 1 whereas
for air data at 0.25 Hz, b was evaluated at 0.87 and 0.9 for the
12 mm and 24 mm thick plates. da/dN is in mm/cycle and AKeff is in
MN/m3/2.
7. Salt water solution (here 3.5% NaC1) does enhance fatigue crack
growth rates. Corrosion fatigue crack growth rates increase with
stress ratio and with decrease in frequency. However, this increase
in growth rates seems to 'saturate' at some stress ratio or at some
low frequency. The upper bound of growth rates in salt-water solution
is represented by
da dN =
2.11 x 10-7(AK)2.63
•
151
where dN is in mm/cycle and AK is in MN/m3/2. The crack growth
rates in salt-water can be represented by a superposition model
of the form
da B S
dN = 6.71 x
10-9(K ff + f (Keff) 1
where f is the frequency and Bland Sl are dependent on
Reff' such
that: for Reff < 20.6 MN/m3/2, Bl = 6.11 x 10-16
and 61 = 8.2;
for 20.6 < Keff < 35.3 MN/m3/2, B1 = 1.44 x 10-9 and 61 = 3.40
and for Reff > 35.3 MN/m3/2, Bl = 1.36 x 10-2
and $1 = -1.09.
8. The results of the present investigation are compared with those of
another investigation carried out by Scott and Silvester [110] on
the same material. It is found that results of both investigations
are comparable under similar test conditions (e.g. thickness and
environment). However, Scott and Silvester did not carry out tests
at growth rates lower than 5 x 10-5 mm/cycle where effects of stress
ratio and thickness are found significant in the present investigation.
There is also a large difference between the present results, for the
12 mm thick plate tested at R = 0.08 and the results by Scott and
Silvester for 25 mm thick plate tested at R = 0.1. It is found
that at lower stress intensities, the present results (12 mm thick
plate at R = 0.08) give much lower growth rates than those of the
other authors; for example, at a AK of 12 MN/m3/2, the growth rates
in the present results are 5 times lower than those obtained in
the latter investigation.
• •
I I I I I I i I l I 2Go 20 4o Go So Ioo 't20 14o 1Goo "Go 20o 22o
NuMBER oF cct_ES (.'o) 1
Figure 4.1. Crack length, a, versus number of cycles, N, for a constant stress ratio fatigue test.
•
1-3 UI N
S
153
A R:: 0.06 0 R=0'7
A A
A A
A 0
1
0 004
OE,
`0 I I ! I I ( I I! t l ,
10 20 40 C0 80 loo ZKCMSi/rn3/2)
Figure 4.2. da/dN versus AY; B = 50 mm, 30 Hz, in air.
154 •
A R : O•og ❑ R=0'5 O R. 0.7
A
ā A g
A A
105
143
O A 00 6
o ~
O
5 so 20 40 I I I l l t t l
Figure 4.3. da/dN versus AK; B = 24 mm, 30 Hz, in air.
CcO 80 100
2 -a --d -cs
5
155
6 R=O'OS
o R=O'5 o R= 0'7
20 . 40 00 eo 100 ~K(N1 N/m '0/2.)
Figure 4.4. da/dN versus ~K; B = 12 mm, 30 Hz, in air.
103
1ō5
156
A R =O.0S KME+NI = 20 Mr Im 3/2
v K MEaN = 4o MN /m 312
A V n
VTA
AA
•
A
nA
5 ? 10 20 30 40 000 80 100 dk CMN 0/2)
Figure 4.5. da/dN versus AK, at constant Kmean
; B = 24 mm, 30 Hz, in air.
163
157
A R= O.08
A 14MEst1= 20 MN!m5/2
V KMT = 10MNIm3/2
A A
•
u 104 5) u
E
-a
4
165
1 S
,•k •A
V• 0 i I I k A 1 I I 7 10 20 30 40
4k (MN/ir,3(2 )
I 1 I I I GO 100
Figure 4.6. da/dN versus AK, at constant K mean;
B = 12 mm, 30 Hz, in air.
•
AR=0.08
0 R=O'7
A A
A
A A
AA
A
A
ot 4 O A
o O A
OO A 0
08 A
09 0
' 1OR ~
O A
O 66
E6 0
O A 00 0
0 o A O AA
00 A
0 A -0 o
1-, 31 IA r f r ! t I I i r
25 10 2o . 40 Go 80 OO dK(MN/m3/2)
158
Figure 4.7. da/dN versus AK; B = 24 mm, 0.25 Hz, in air.
I I I I I coo 80 1.00
159 •
0
Q
A A A
o R. o•08 3 0 Q=o•7
0 0
o 0-1 A
S ° ~ 0 ,-, 0 J 0
ū
RO AA
(§ o Q
~ D
o 0 0
-cs o Q d
o Ar CO A 0
A A
~•5
O A
I ,,,. I I I 5 14 20 40
AK (M'4/rn/2 )
Figure 4.8. da/dN versus AK; B = 12 mm, 0.25 Hz, in air.
o R=-2 o R=-o•7 0
0 0 0
160
0 0
EL O Oo
0 0 0
0
Oo 0
R=0-08 AT 30 Hz
O C1') I t• I IIIII
20 • 4o GO 80 100 A le. Tell (MN/rna/2)
Figure 4.9. Effect of negative stress ratio on da/dN versus AKTen• B = 24 mm, at 1-10 Hz, in air.
63
A IN
A
A
A
i 0
0
A o 1 e
• R= 0•O ■ R=--O.17
o R=- 0.7
• 161
5 10 20 40 Coo so 100 (MN/m3k )
Figure 4.10. Effect of negative stress ratio on da/dN versus AK; B = 12 mm, 0.12 Hz, in air.
162
A0 0 E
•
162 AR =O.O ill R --0.17
0 R=-0"7
• i / ii 1
it .41
st• . ii. • ta
Sf • • 9 •~ h • il
Z= O.OB AT 30Hz
4
11111I I I I I I I t 1
5 10
aKT 20
CMN(m3(2) $0 100
en
Figure 4.11. da/dN versus AKTen; B 12 mm, 0.12 Hz, in air.
•ga,eO O A
L A A A
A
00. o O Afill AO A
KT¢n:45k4f1it rz FOR ALL CASES
163
FATLC(UE CRAC.I<, LENGTH FRQlvM I4oTCH, LSA mrr+
Figure 4.12. Stress ratio effect on crack initiation and growth from a notch; B = 24 mm.
0 A
•
0 OO
0c3 0 D O DO ❑❑❑
1 1 2 3 4 Co
120 IGDO Z0O 240 280 320 3C20 400 Nom• or C•fCLes,N X lo 3 ON
Figure 4.13. R versus N at various values of As measured from the notch (from Figure 4.12).
165 •
J 10
E E
Z
a -,
20 40 Coo 8o doo QKCMNfm3f2 )
Figure 4.14. Effect of thickness on da/dN versus AK; R = 0.08, 30 Hz, in air.
166
•
8=5Omm g = 24 m m 5 12mret
1
/
-5 10
/
/
20 AK N/m34 60 10Q
Figure 4.15. Effect of thickness on da/dN versus AK; R = 0.7, 30 Hz, in air.
167
20 40 (0 80 100 AK (Mt.. j m312)
•
Figure 4.16. Effect of thickness on da/dN versus 4K; 0.25 Hz, in air.
0.25 1-Iz 30 Hz
14
R=0.7
da f
dt ,
(mm
/CYC
LE)
=0
20 Ax(MN/m 312) 0 80 100
168
Figure 4.17. Effect of frequency on da/dN versus AK; B = 24 mm, in air.
2.0 40 Eo 80 coI LAK (MN/biz/2)
169
Figure 4.18. Effect of frequency on da/dN versus AK; B = 12 mm, in air.
•
10
do./d4 (
mm
f cY
CL£
)
170 •
105
A . 30 Hz 0.251-1z
Q 0.06 Hz o 0.02.142
1 1 11 1 1 1 1
5 10 20 40 6o 100 AK NW/n-1 5/2
Figure 4.19. Effect of frequency on da/dN versus OK; B = 24 mm, R = 0.08, in air.
1 I 1 1 1 1 I L 20 40 Coo 100.
6.1( (Mi•1 frn3/Z )
15
1 I I
5
171
❑ o•2.4 0/ 0 0•70
❑ /
ā/
ō/ r4/ q/
•
03
A R=o•08 / r
AIR (R:0.05, o•S k 0.7)
Figure 4.20. da/dN versus AK; B = 24 mm, 30 Fz, in 3.5% NaC1 solution.
1
A R : 0.08 0 R = 0'?
A A
A A
Q A
1 0~/ A
A
AAA AA
•
R=0.7 ii LAIR
AA - O
00 / 0 / ,-
A
R=o.o8 IN AIR
172 •
10s I ( ( ? / ! ( I 1 (III 5 - 10 20 40 GO 80 loo
dk (NINI fm3/2
Figure 4.21. da/dN versus AK; B = 24 mm, 0.25 Hz, in 3.5% NaC1 solution.
'1d
da/ d
nl (m
rn /c
YcL
E.)
3.3°/s No-CL SOLUTION v o.oro Mt. A 0.25 Mz ❑ 3 biz, 0 3o Hz v v
A 3OHz N AtQ
AA v ~b
ava
A
0 0
0
ri.•0•251-iz tN AiQ
0
,ss 10 20 4o (O 80 100
AK (MN f rn3/2)
173,
Figure 4.22. Effect of frequency on da/dN versus AK; $ = 24 mm, R = 0.08, in 3.5% NaC1 solution.
O'
l I I II 24 40 Go 100
AK (MINIPY1312 )
106
174
Figure 4.23. Effect of frequency on da/dN versus AK; B = 24 mm, R = 0.7, in 3.5% NaC1 solution.
175 •
•
i •• • ■ OA
■ • li
• ■
■ i tvlEANI Lir 1E
3 4111 daidN= 2.88 Y.16 9Cap)
23
5 { •• OF SCOTT # 51LVES'jER 10 RESULTS U103 FOR
• 854300- 50D STEEL • B= 2.5.4mm
o• LHz.-1OKz .
ō R4 0~1
ōQ
1 I '7AI I I 1 I I 1 1 I 1 5 10' 2o 30 40 60 So 1o0
AK(M N / rnnj2 )
Figure 4.24. da/dN versus AK; B = 24 mm, 30 Hz, in air - compared with Scott & Silvester's [Ref. 110] results.
A R=0.08 R= 0.6
0 R= O.7
10
MEA N1 Li KE: da/dN 2.8S .10"q Cp14)3.23 OF SCOTT* SILVESTER RESULTS FoR 5543GO- sop 5TEEL
2.B.4 rn+~ 1Hz-~toKz R t 0.1
I j I i l t t 20 Bo 40 GO 80 100 OK (MN 1rn3/2 )
00 &
8~ SE pp OP o .
►f'1 r to
Figure 4.25. da/dN versus OK; B = 12 mm, 30 Hz, in air - compared with Scott & Silvester's [Ref. 110] results.
r t 5
176 •
177
• -
da j d
s1 (r
hm / c
YcL
E)
0
A ca
A iZr.o•08 5=24mrn 0 R=0•1 A R.:: 0.0815 _ 12. wirv1 o gr-o.-7
1 I II 1 1 I I 'i I- b2 50 4o CCC 80 100
C .7-k----/ OKePFi (MNfrn3l2 )
Figure 4.26. da/dN versus AK 4c; B 12 mm & 24 mm, 0.25 Hz, in air. r
• - 178
•
102
UPPER BOUND FOR ALL SALT- WATER DATA do i T. 2.11 x10 C1K) "3
UPPER SOUND FOR SALT WATER
103 DATA AT R= O.08
._ UPPER 8OUA4D FOR SALT=WATER
— DATA AT ,_ R= 0'7
1 L_
MEAN LINE īHROUC4H AIR DATAATBOHz:- caidN G 11,00cl aKf° a
I I
I
10 1 1 1 1 1(
1 1 1 l 5 10 20 40 Go 80 loo
dK (MN /m312.)
Figure 4.27. da/dN versus OK; B = 24 mm, showing upper bounds for salt-water data.
10 .10
EqUATIOtJ C4•f8)
-- -- EXPERIMENTAL DATA
1Hz.
o•06H
UPPER BOUND OF — EXPERIMENTAL
DATA
0.25 Hz da
/ d tJ
(m
m /c
ycL
E)
/
179 •
10 '5 I I I I I I I t 10 20 40 60 8O '100
AK (M N / m 312 )
Figure 4.28. Correlation of da/dN versus AK data in 3.5% NaC1 solution using a superposition model.
180
0
CHAPTER FIVE
THRESHOLD AND NEAR-THRESHOLD FATIGUE CRACK
GROWTH IN BS4360-50D STEEL
181
5.1 INTRODUCTION
Fatigue crack growth in steels has been successfully described
by means of a sigmoidal relationship between crack growth rate, da/dN,
and the stress intensity factor range, AK. This relationship predicts
the existence of a threshold, AKth, for non-propagating cracks. Most
of the fatigue crack growth data, however, have been generated for
growth rates above 10-5 mm/cycle. Although this information is vital
for determining safe non-destructive inspection intervals in certain
engineering structures, it has been increasingly recognized that defects
introduced during fabrication or developed in service may become potential
sources of catastrophic failures from subcritical crack growth at
apparently immeasurable rates. The relevance of thresholds and low
fatigue crack growth data, therefore, cannot be under-estimated. Such
data are specially important in design calculations for extended lives,
or for components subjected to high frequency and low amplitude cycling
and also in welded structures where stress fluctuations are superimposed
on a mean stress which result in extremely high stress ratios.
Attempts have been made to define criteria for non-propagating
cracks in terms of other material properties, as discussed in section
2.8.1. However, this approach has not so far accounted for the experi-
mentally observed sensitivity of AKth and near-threshold growth rates
to such factors as stress ratio, environment, stress history and micro-
structure, as was discussed in section 2.8. No specific results have
been reported on thickness effect on thresholds and near-threshold crack `
growth rates. It has usually been assumed that the effects of thickness
are those associated with changes in the state of stress at the crack tip
observed at high stress intensities. Since plane strain conditions are
prevalent in low stress intensities, no distinct thickness effect should
be observed.
•
182
Results presented in chapter four have indicated a significant
influence of stress ratio and thickness on low fatigue crack growth
rates and a continuation of that work to lower growth rates will be
presented in this chapter. The merits of crack closure and environmental
effects as a possible explanation of the stress ratio and thickness
dependence of low crack growth rates will be discussed.
5.2 EXPERIMENTAL PROCEDURE
Using compact specimens, tests were carried out on the Dowty electro-
hydraulic and the Mayes servo-hydraulic fatigue machine. Threshold
fatigue tests were performed in air at stress ratios, R, varying from
-0.7 to 0.9 and at 30 Hz cyclic frequency for the three thicknesses,
i.e. 12 mm, 24 mm and 50 mm. Additional tests on the 24 mm thickness
were carried out in 3.5% NaCl solution to investigate the effect of
environment on low crack growth rates.
Two different methods for shedding the load were used to obtain
low crack growth rates and thresholds. In the first technique, referred
to here as the 'constant R' technique (usually used in the literature for
determining tKth) the value of AK was progressively reduced by steps of
10% or less. During each AK-step the crack was allowed to grow past
the plastic zone (equation (4.1)) created by Kmax of the previous step.
Several readings of crack length against number of cycles were then taken
before making the next AK-step. This procedure was continued until
threshold conditions were attained. The threshold value, AKth, was
considered to have been reached if, after the last AK-step 'no growth'
was recorded in 3 to 4 million cycles. This represented growth rates of
the order of 5 x 10-9 mm/cycle according to the accuracy of the crack
length monitoring equipment (i.e. 0.02 mm).
•
183
The above procedure of obtaining AKth was found time consuming
due to the delays in the crack growth observed to occur following each
AK-step down - arising from load-history effects. Consequently, a
second and much faster technique, to be called 'increasing R' technique,
was devised. This involves the stepping down of AK at increasing stress
ratios, R. Starting the test from some value of max and at a low
stress ratio (preferably R = 0.0) the maximum load, Pmax , was kept
constant, but the minimum load increased in steps. It was found that
the values of AK could be reduced by even larger steps than 30% without
noticeable load history effects. The testing time to obtain threshold
conditions could therefore be reduced by as much as half of that taken
using the 'constant R' load shedding technique. However, it should be
noted that the 'increasing R' technique should normally be used to
obtain AKth
values at relatively high stress ratios, depending upon the
initial value of K max'
Crack length measurements were made using both a travelling microscope
and also the AC potential drop crack microgauge [190], both of which could
detect crack increments of the order of 0.02 mm. Before the fatigue
tests, calibration curves for each specimen thickness were produced using
both specimens with saw cuts and fatigue cracks. The probe terminals
for the crack microgauge were spot-welded onto the specimen at a distance
of 15 mm from the loading line, as discussed in chapter three. This
position was found suitable for the crack measurement sensitivity required
for these tests.
The fatigue crack growth rate, da/dN, was calculated from the experi-
mental data using the secant method. The stress intensity factor, K, was
calculated using the standard formulae recommended for the CS geometry
(see section 4.6).
184
5.3 THE RESULTS
Figure 5.1(a) & (b) shows typical a versus N curves obtained during
the threshold tests using both the 'constant R' and the 'increasing R'
techniques. This figure shows that the number of cycles required to
reach AKth at high stress ratios is considerably reduced by adopting
the 'increasing R' technique.
Figure 5.2 shows a typical variation of digital voltage change with
crack length increase during a threshold test and compared with the
previously determined calibration curve. This figure shows the good
reproducibility of the crack microgauge readings during both the
calibration and the actual tests.
The fatigue crack growth results for the three thicknesses tested
in air and those in salt water solution have been represented in terms
of da/dN versus AK. These results are shown in Figures 5.3, 5.4 and 5.5.
Figure 5.3 shows the data for the 12 mm thick plate tested in air at
stress ratios, R, of -0.7, 0.08, 0.3, 0.5, 0.7 and 0.9. The results
show a considerable effect of stress ratio on low crack growth rates
and thresholds, i.e. growth rates increase with stress ratio and AKth
values decrease with stress ratio. -
Figure 5.4 shows data for the 24 mm thickness, tested at stress
ratios of -0.7, 0.0, 0.08 and 0.7. Again this figure shows a significant
stress ratio effect on thresholds and low crack growth rates. However,
the effect is reduced compared to the data for the 12 mm thickness
The data for the 50 mm thickness, shown in Figure 5.5,seem to show
no significant effect of stress ratio, at least for the two stress ratios
(i.e. 0.08 and 0.7) tested.
In Figures 5.3 and 5.4 the data for the negative stress ratio
(R = 0.7) are plotted using the total range of AK for comparison purposes.
•
185
However, when the compressive part of AK is subtracted from the total
range, the resulting data is found to correspond to those obtained at
zero stress ratio indicating that at low stress intensities, unlike
high stress intensities, as was found in the previous chapter, the
compressive part of the cycle was essentially non-damaging, at least for
the -0.7 stress ratio.
It can be seen from Figures 5.3 and 5.4 that all curves for R are
shifted towards lower AK with increasing R. The amount of shift seems
to increase with decreasing growth rates and also to disappear at high
R-ratios and at high growth rates.
The effect of thickness at a low stress ratio of 0.08 is illustrated
in'Figure 5.6, where mean lines from the data in Figures 5.3, 5,4 and 5.5
are drawn. The trend in Figure 5.6 bears some resemblance to those in
Figures 5.4 and 5.5 suggesting that varying the thickness at low stress
ratios may produce effects similar to those obtained by varying stress
ratios for thinner sections. The effect of thickness on low growth rates
seems to disappear at higher stress ratios.
Figure 5.7 shows the data for the 24 mm thickness tested in salt-
water at R-ratios of 0.08 and 0.7 compared to corresponding data in air.
The results at R = 0.7 show no significant effect of salt water environ-
ment on low crack growth rates and on AKth. However, at R = 0.08 lower
growth rates are obtained in salt-water solution as compared to air
results. A great amount of scatter in the data at R = 0,08 was also
observed. AKth was observed to increase from 5.5 MN/m3/2 in air to about
10 MN/m3/2 in salt water. It was found that an introduction of salt
water into the crack, previously growing in air at AK values below
10 MN/m3/2 and at R = 0.08, caused a rather unexpected crack arrest.
The crack stopped very suddenly or in some instances, after a short growth.
This behaviour needs further investigation under different loading
conditions.
•
186
In Figure 5.8, AKth values are plotted against the respective
stress ratios. This figure shows that the effect of R on AKth is
dependent on thickness. It is also seen that AKth
decreases with
increasing R to a minimum value, AKc th
, and then remains constant.
This minimum value of AKth
seems to be the same for all three thicknesses
and averages about 3 MN/m3/2.
Figure 5.9 shows the variation of AKth
with thickness at low and
high stress ratios. The trend of the curves again bears some resemblance
to those in Figure 5.8. No effect of thickness on AKth is observed at
high stress ratios.
In Figure 5.10 is plotted the data, for the 24 mm thick specimens,
obtained using the 'increasing R' technique and also the best line
through the 'constant R' data at R = 0.7 is drawn for comparison purposes.
The 'increasing R' data were obtained at different values of Kmax by
keeping P max constant and varying Kmin from zero until threshold
conditions were reached. The data show no significant effect of Kmax
on either the thresholds or the near-threshold growth rates. By
comparing these data with those obtained 'constant R' tests at R = 0.7,
it can be seen that at growth rates below 10-6 mm/cycle the 'increasing R'
technique gives higher_ growth rates than the 'constant R' technique.
This phenomenon is due to the load history effects present in the
'constant R' data despite the great care taken during the step-down
procedure. For max.th
values of 8.3 MN/m3/2, 13.6 MN/m3/2 and
24.27 MN/m3/2 and the corresponding Rth values of 0.64, 0.78 and 0.88
respectively values of AKth very close to 3 MN/m3/2 were obtained. The
above data show that at high values of Kmin
(and thus high R-ratios) the
thresholds and the low growth rates are independent of Kmax but only
dependent on the cyclic value of AK.
•
187
5.4 DISCUSSION OF RESULTS
The value of AKth
is dependent on the conditions and test methods
used for it§ determination. Thus, some comment may be appropriate here
before general discussion of the results. To highlight this point it
would be appropriate to cite the results of Bates and Clark [218].
From their crack growth test data for a low alloy steel, ASTM A533-
Grade B and Ti-6A1-4V aluminium alloy, these authors suggested that DKth
for these materials were 27.5 MN/m3/2 and 22 MN/m3/2 respectively. The
tests were performed in air at R = 0.0 and 10-30 Hz, and the authors
suggested that threshold could be deduced from crack growth rates of the
order of 2.5 x 10-5 mm/cycle. These authors clearly failed to test for
sufficiently long periods to achieve lower growth rates. Of course,
since then test methods have been regularised and it is now commonplace
to find AKth values quoted for growth rates of the order of 10
-7 mm/cycle
or even lower.
In order to achieve low growth rates and true thresholds unaffected
by load history effects, it is necessary to make sufficiently small load
changes and test for sufficiently long periods preferably at high
frequencies. The crack length monitoring equipment must also be
sufficiently sensitive to detect small crack length increments. It is
not possible to specify the amount of crack length extension that should
•be allowed between load steps. This will not only depend on the
sensitivity of the monitoring equipment but also on the geometry of the
specimen, as regards the rate of change in compliance (and thus K) with
crack increase. To allow for sufficient crack increments between load
steps without too big changes in K values, the present tests were therefore
performed on specimens with relatively short crack lengths (i.e.
0.3 < a/W < 0.45) where the change in specimen compliance with crack length
was small. During the present investigation it was also found that
188
reducing AK, during 'constant R' tests, by more than 10% at higher
growth rates and more than 5% near-threshold caused long crack growth
delays and sometimes arrest resulting in a 'false' threshold value due
to the load history effects.
It was suggested by Pook [5] that the minimum crack growth per
cycle in terms of continuum mechanics, is of the order of one lattice
spacing. For most materials, this would account for growth rates of the
order of 3-5 x 10-7 mm/cycle. While accepting this lower limit as
fairly conservative for some engineering applications, it would prove
dangerously large for long lives of the order of 1010 to 1012 cycles.
Since growth rates lower than the atomic spacing are not unknown
[94,130,144,146] it is likely that cracks will reach critical dimensions
during the design life of a structure with extended life. It is also
known that the crack growth mechanism near the threshold can change from
a continuous crack front advance to a discontinuous ledge nucleation and
growth mechanism [219], which is highly sensitive to changes in OK. The
use of K as a characterising parameter for such discontinuous crack
growth is open to discussion and further investigation of these extremely
slow processes is needed, but the present definition of thresholds at
growth rates of the order of 10-8 mm/cycle, under the present test
conditions, is considered adequate.
The present investigation has shown that crack growth rates
generally increase with stress ratio at low stress intensities. This
behaviour is prominent for the 12 mm thick specimens but almost absent
in the 50 mm thick specimens. The stress ratio sensitivity of low
growth rates obtained in the data in Figures 5.3 and 5.4, for the 12 mm
and 24 mm thick specimens is similar to that reported by Cooke et al.
[131], Sasaki et al. [132] and Paris et al. [142, 143] for other steels.
The present AKth results are compared with some recently reported
investigations on other steels in Figure 5.11. A similar comparison of
I
189
both ferrite-pearlite and martensitic steels has recently been made by
Vosikovsky [134]. From these results no simple relationship between
AKth and R for all the steels seems to exist. However, a linear
correlation has been obtained for most steels with the two exceptions of
the data by Branco et al. [61] and by Davonport and Brook [137].
Moreover, all the results tend to point towards a limiting value of
th' reached at high positive stress ratios. This trend AKth' AKc
contradicts the suggestion by Blacktop and Brook [220] that LKth
can be
extrapolated to zero at a R-ratio of 1. It will be realised that, at
high stress ratios, the maximum stress intensity factor, Kmax
, will also
be large and both plasticity effects and slow stable crack growth by
static modes will be substantial.
Looking again closely at the results in Figure 5.11 and also at
those for other steels presented elsewhere [134] it is observed that
AKc,th varies between 3.0 and 3.5 MN/m3/2 for most ferrite-pearlite steels
and between 2.0 and 2.5 MN/m3/2 for the martensitic steels. On the
other hand, AKO,th (i.e. AKth at R = 0.0) is found to vary widely for
both ferrite-pearlite and martensitic steels. For the former type of
steels Kth varies between 6 and 13 MN/m3/2 while for the latter steels
it varies between 3 and 9 MN/m3/2; in this case decreasing with increasing
strength. The wide variation of AKOth (compared to that of AKc th)
make it a less realistic parameter to use to characterise low growth
rates. This variation of AKOth in ferrite-pearlite steels (with no
relative change in strength) could be due to the procedures used to
determine the thresholds and/or (and more likely) due to crack closure
effects. In the case of martensitic steels the variation of AKOth
is due to the influence of microstructure which, though not yet still
clearly understood, is apparently related to environmental effects, such
as hydrogen embrittlement from moist air. At high stress ratios the
0
190
effect of crack closure is reduced or eliminated altogether (Ka, < Km in)
while that of the environment is 'saturated' and thus a critical AKc th
(which seems a constant for most steels) is reached. In the next
paragraphs, the effects of environment and crack closure, as they
influence the stress ratio effects, will be discussed in detail.
The effect of stress ratio on low growth rates has often been
explained in terms of either environmental interactions or the crack
closure phenomenon [66]. Examination of existing data obtained in various
environments is not conclusive as to the actual contribution of the
environment to the stress ratio effect observed in steels. Nevertheless,
by examining behaviour in high-strength steels, where the primary
mechanism of environmental' attack during fatigue crack growth in moist
air and salt water environments is hydrogen embrittlement [103], some
seemingly consistent explanation has been put forward [127]. This
explanation is supported by some data obtained in inert environments
(such as in vacuum) which show lack of, or markedly reduced, stress ratio
effect at low growth rates [131,145,146]. It is also argued, by the
above authors, that the lack of stress ratio effect at higher stress
intensities, where the crack velocity is too fast for any environmental
factors to have effect, is consistent with the above explanation. However,
such an explanation of the fatigue crack growth behaviour in low strength
steels and at low stress intensities, solely in terms of hydrogen
embrittlement, may be questioned. In these steels, and especially at low
stress intensities, the hydrogen enrichment ahead of the crack tip, as
predicted by Richie [127], would be very small owing to the lower magni-
tude of the hydrostatic tension.
The present results in salt water show no significant environmental
influence compared with the results in air, Figure 5.7. The decrease in
growth rates at low stress ratios observed in this figure at R = 0.08
191
could be explained in terms of possible effects of corrosion fluid and
corrosion products at the crack tip. These may become trapped at the
crack tip during the unloading part of the cycle and their 'wedging
effects' may cause crack tip blunting [61,105] which consequently
reduces the effective stress intensity and thus the growth rates. The
present results in salt water do not show any crack growth rate enhance-
ment due to the environment and therefore any explanation of the stress
ratio effect using environmental arguments (at least for the present
results) seems unsatisfactory.
Although only limited data are available on low fatigue crack growth
rates for specimens tested in a salt water environment [5,61,105,130,
141-143], the present results at R = 0.7 seem to support the general view
that, in low alloy steels, low growth rates and AKth are not strongly
affected by the environment. However, exceptions have been reported
[5,139,140] where salt water environment was found to enhance the growth
rates. More data is required before any general conclusions could be
drawn from these results on the exact influence of the environment.
In view of the trend of the present results both in air and in
salt water environments it can be concluded that environmental arguments
alone cannot adequately explain the observed effects of stress ratio and
thickness. Therefore, an alternative explanation, namely the crack
closure concept, is advanced.
The crack closure concept has previously been used to explain the
stress ratio effect on low fatigue crack growth rates [66,94,132,142].
This concept, first described by Elber [66] for crack growth at high
stress intensities, relies on the fact that, as a result of plastic
deformation left in the wake of a growing fatigue crack, some closure of
the crack surface may occur at positive loads during the loading cycle.
This phenomenon reduces the effective stress intensity factor causing
192
crack growth. It follows from this argument that a threshold for fatigue
crack growth will be reached when the crack remains closed throughout
the entire load cycle. However, this claim has not yet been fully
substantiated [147]. To explain the effect of stress ratio on low growth
rates it is argued that as the stress ratio is raised, the crack will
remain open for a larger portion of the cycle, thereby increasing the
effective stress intensity factor range, dKeff, and hence the growth
rates.
There is now a large body of evidence [147] to suggest that crack
closure, as proposed by Elber [66] is essentially a surface (plane stress)
effect, having a minimal effect on crack growth under plane strain
conditions. At low stress intensities and under plane strain conditions -
conditions prevailing at the low fatigue crack growth rates - this type
of crack closure should be insignificant. It is,- therefore, reasonable
to assume that at low stress intensities crack closure is primarily due
to microstresses already present in the plate. These microstresses might
have been induced in•the plate during manufacturing processes, or during
subsequent machining of the specimens. These microstresses will be used
to explain the effects of stress ratio and thickness presently observed,
and will be assumed to cause a constant crack closure stress at the
crack tip.
The stress intensity factor, K, is a characterising parameter for
stresses and strains in the crack tip region only if tensile stresses and
strains are produced by the applied load. Fatigue crack growth rates,
when characterised by K, can only be looked upon as caused by that part
of AK which, when applied, results in tensile stresses and strains at
the crack tip. Using this strain characterisation analogy to crack
closure, the minimum stress intensity factor corresponding to zero stresses
and strains at the crack tip will be defined as the crack closure stress
193
intensity factor, KCL. This interpretation of crack closure will later
be used to derive a strain-controlled crack growth model (see section 5.4).
Therefore the effective stress intensity factor range, AKeff,
responsible for crack growth is given by
AKeff max - KCL for min < KCL
and (5.1) .
AKeff AK for min > KCL
The crack closure stress intensity factor, KCL, can be determined
from physical measurements of crack closure using various methods [221].
Some of these methods use surface measurements which do not take into
account the crack tip behaviour at the interior of the specimen. It is
also difficult to measure crack closure at very low stress intensities
since a great amount of accuracy is required. Alternatively, it is there-
fore proposed that KCL may be estimated from da/dN versus AK data obtained
at various stress ratios.
Assuming that KCL is constant at low stress intensities, its value
can be estimated from the curves of AK versus R constructed at levels
of constant growth rates. It is hereby further assumed that crack growth
is only caused by cyclic loading and no additional growth occurs by
static modes. The procedure for constructing AK versus R curves is
illustrated in Figures 5.12 and 5.13 for the 12 mm thickness. Figure 5.12
shows the points used to construct AK versus R curves which are in turn
plotted in Figure 5.13. The knee of each curve gives the specific value
of R and K, designated as Rc and AKc respectively, at which the applied
min equals KCL and at which further Increase of R, at a constant AK,
does not cause any increase in growth rates. Thus KCL may be expressed as
R
KCL 1 -cR . AKc
c (5.2)
194
Using the above construction and also using equation (5.2) an average
value of KCL equal to 6.3 MN/m3/2 was obtained for the 12 mm thickness.
This method, however, requires data at a sufficient number of stress
ratios both below and above Rc to enable a reliable AK versus R
construction.
Using equation (5.2), the value of Rc,th
at threshold can be
obtained from
_ KCL Rc,th KCL + AKc th
(5.3)
A value of Rc,th of 0.68 for the 12 mm thick specimens is therefore
obtained. The data for AKth versus R, for R < Rc,th
, can be represented
by a straight line in the form
AKth = A1R + B'
(5.4)
and after solving for the constants Al and B' we obtain
AKth = KCL(1 - R/Rc th) + AKc th (5 .5)
and substituting for Rc,th in equation (5.3), equation (5.5) reduces to
AKth = (1-R)(KCL + AKc,th) (5.6)
The value of KCL may also be estimated from the comparison of data
at a low R (R = 0) and that at a high R(R > Rc) for same growth rates.
This comparison is based on the assumption that constant crack growth
rates may be achieved at varying stress ratios and at constant KCL only
by keeping Kmax
constant and varying Kmin
for min
< KCL' This assumption
was confirmed by the present results where max was found to be virtually
constant for different stress ratios at constant growth rates and for
R < R. For growth rates above threshold equation (5.6) can be written as
195
and therefore
AK = (KCL + c)(1-R)R<R c
KCL = (Kmax)R<R - (AK)R>R c - c
(5.7)
(5.8)
Using equation (5.8) the values of KCL were estimated for both the 12 man
and 24 mm thicknesses at various stress ratios, as plotted in Figure 5.14.
Average values of 6.3 MN/m3/2 and 3.1 MN/m3/2 respectively were obtained.
A Rc,th of 0.51 for the 24 mm thickness was obtained. The results for
the 50 mm thickness obtained at R = 0.08 and R = 0.7 suggest a very low
value of KCL.
Some preliminary crack closure measurements using the compliance
method [221] were made, and the values of KCL obtained for the 24 mm and
50 mm thicknesses are shown in Figure 5.15-. Most of the low growth rates
results were obtained at K < 15 MN/m3/2 and based on this K value max max
values of KCL of about 3 MN/m3/2 and 1.4 MN/m3/2 are estimated for the
24 mm and 50 mm thicknesses respectively. These results seem to confirm
the values of KCL estimated from da/dN versus AK data. However, more
KCL measurement results, especially for the 12 mm thickness, are required
before making any generalised conclusions.
The values of KCL estimated above show that crack closure at low
stress intensities is dependent on the plate thickness; decreasing with
increasing thickness. Previously reported results on some aluminium
alloy [93] also showed thickness dependence of crack closure though
unlike the present results, these showed lower crack closure values in
the thinner plates. Therefore, no simple relationship appears to exist
between crack closure on thickness. More data on other materials will be
needed before any firm conclusions can be drawn.
Using the above estimated values of KCL the low crack growth rate
data was reanalysed in terms of AKeff
using equation (5,1). The da/dN
•
196
versus AKeff
data are shown in Figures 5.16 and 5.17. Again using the
estimated values of KCL, the threshold values, dKth, are calculated
using equation (5.6) and compared with the measured values in Figure 5.18.
The above three figures show a good correlation of the data, suggesting
that the crack closure concept has successfully been used to account for
the observed stress ratio and thickness effects on low growth rates and
thresholds.
The above analysis shows that the use of data obtained at low
stress ratios and from thin sections - conditions maximising the crack
closure effects - could prove too conservative and unrealistic. They
are certainly so when applied to the design applications in welded
structures (like those in the North Sea area) where high mean stresses
are superimposed on the cyclic stresses. To obtain realistic data it is
therefore proposed that low crack growth and threshold data should be
obtained at high stress ratios. Such data are easy to obtain using the
-'increasing R'-technique devised during the present investigation. The
data from the present tests, Figure 5.10, have shown that low crack
growth rates and thresholds are only dependent on the 1Keff and that max
has no effect as long as static modes of cracking and plasticity effects
do not occur.
At this point in the discussion, it is worthwhile to note the
observation made by some authors [128,131,146] on the effect of Kmax
onAKth. These authors plotted the crack growth rates against Kmax
and observed that (in the low-strength ferritic steels they tested) AKth
occurred at a constant value of Kmax irrespective of the stress ratio, R.
This behaviour has not been observed in higher strength steels or other
materials [147]. Although the above authors failed to explain the Kmax
dependence of OKth, this behaviour is consistent with the crack closure
effects discussed above. Constant growth rates are obtained at constant
•
197
Kmax, for various stress ratios, when R < Rc. This also applies to the
threshold condition, which can be viewed as being obtained at some
constant growth rate level. Therefore as long as Rth < Rc th then
1Kth should occur at a constant Kmax.
On the other hand, when Rth > Rc th'
or if additional Kmax
dependent fracture modes occur (as in the case
with high-strength steels) then AKth will not occur at a constant Kmax'
In the next section a simplified model, based on a strain-controlled
fracture criterion and AKeff'
will be formulated for predicting low
crack growth rates in ductile materials and in particular the BS4360-50D
steel used in the present investigation.
5.5 A SIMPLIFIED CRACK GROWTH MODEL
5.5.1 Introduction:
Existing literature contains a number of attempts to model fatigue
crack growth over a wide range of growth rates through manipulation of
the constants and of the stress intensity factors in empirical expressions
(see section 2.4). This line of approach has been adopted mainly because
of the fact that a single mechanism is not responsible for crack growth
over such a wide range of growth rates.
In mechanistic terms, crack growth rates have been characterised
into three growth rate regimes [37]. At low crack growth rates a small
scale of crack tip deformation well below the grain size of the material
is involved and growth rates aria therefore found sensitive to micro-
structures. In the intermediate growth rate regime crack tip deformations
may spread over several grains, consequently a continuum crack growth
mechanism is involved. At high growth rates, crack growth increasingly
involves material inhomogeneities (e.g. microvoids, inclusions, second
phase particles, etc.) ahead of the crack tip and thus growth becomes
•
198
discontinuous, involving coalescence of the resulting microvoids.
Nevertheless several investigators have attempted to model fatigue
crack growth in terms of some mechanical and fatigue properties of the
material [49-53,219]. Most of these models are based on the general
acceptance that the process of fatigue crack growth in engineering
materials is a function of the plastic deformation at the crack tip
[46,89,222]. Each crack growth model must include at least two elements.
Firstly, a description of the elastic-plastic stress and strain fields
ahead of the crack tip taking into account the finiteness of the plastic
strains in the deformed zone. Secondly, a definition of a realistic
failure criterion for crack growth to take place in a material which
has undergone a considerable plastic deformation with the possibility of
subsequent accumulation of microdefects.
In the present analysis a strain distribution ahead of the crack
tip from the work by Rice [223] and recently employed by other authors
[49,51,219] is modified to obtain finite plastic strains at the crack
tip. It is further assumed that failure of material elements at the
crack tip occurs when the accumulated cyclic strain at the crack tip
reaches some large value, which may be of the order of the true fracture
strain, sf, of the material. It is, however, recognized that at high
stress intensities and in materials containing a high proportion of
brittle particles, inclusions or embrittled gain boundaries, accelerated
growth rates may occur by void coalescence and subsequent brittle or
ductile rupture of the material elements.
r
• 199
5.5.2 Mechanical and fatigue properties of the material:
The material is assumed to obey the following cyclic stress-strain
behaviour,
Ao = EAE for 6e < aeyc, Aa < OQyc (elastic)
&=a(e )n for De > LE yc AE yc
yc (plastic) (5.9)
or Ae = De (_&1 )1/n for aa > t.c yc yc
yc
Where E is the elastic modulus, DEyc is the cyclic yield strain,
AQyc
is the cyclic yield stress corresponding to Aeyc which marks the
onset of appreciable plastic deformation and n is the cyclic strain
s hardening exponent.
5.5.3 Stress and strain distribution ahead of crack tip
•
The stress and strain distribution which is frequently used to
characterise stress and strain ranges at the crack tip is that derived
by Rice [223] for a stationary crack in antiplane shear (Mode III) under
small scale yielding. A similar analysis for Mode I is not available but,
McClintock [224] had earlier discussed the analogy between Mode III and
Mode I for the case where displacements parallel to the crack are small
compared to those normal to the crack surface. Therefore for a tensile
crack in a material obeying the cyclic stress-strain behaviour in
equation (5.9) the stress and strain ranges can be approximated by [51].
n
AK2 l+n
Aa(x) - 2oyc
(5.10)
4(1+n)irc 2.x yc -
and
1 AK2 n+1
Ae(x) = 2e and yc 4(1+n)Trary2x
(5.11)
• 200
The product of equations (5.10) and (5.11) reduces to
AaAe = AK2 (l+n) TrEx
To estimate the cyclic plastic zone size, ryc , we set Aa = 2a yc,
Ac = 2ey and x = ryc in equation (5.12) and obtain
r AK2
- yc 4(1+n)rcy2
(5.12)
(5 .13)
Equation (5.13) gives the cyclic plastic zone size under plane stress
conditions. Plastic zone sizes determined using various criteria, such
as von Mises, Tresca, octahedral shear stress and the plastic cohesive
force theory have been compared and discussed by Lal and Carg [225,226].
Values of ryc obtained using the above criteria compare very well with
equation (5.13).
For plane strain conditions equation (5.13) may be written as
__ AK2 ryc 4a(1+n)Trayc
(5.14)
where a = 3 according to Rice [227] and a = 1 2 - 6 according to (1-2v)2
McClintock and Irwin [228].
Therefore equation (5.11) may be modified for a plane strain case as 1
2 l+n Se(x) = 2e AK (5.15)
yc 4a( l+n)Tr6 2 .x yc
1 2 l+n
Ae(x) = 2e AK
yc [;a(l+n)way2(x +rc) (5.16)
201
Equations (5.10), (5.11) and (5.15) are physically unrealistic
since they predict infinite stresses and strains as x =>0, while for
real engineering materials, the crack tip first blunts on application of
a load and therefore Aa and to will not exceed some finite values,
probably of the order of the ultimate tensile strength, auts,
and the
true fracture strain, ef, of the material respectively. To overcome this
anomaly a constant, rc, will be introduced into equation (5.15). re is
assumed to be a critical cyclic crack tip opening displacement which has
to be reached before fatigue crack growth occurs. As will be seen later
in the analysis rc is assumed to be related to the condition for a non-
propagating fatigue crack.
Thus equation (5.15) becomes
•
5.5.4 The effective stress intensity factor range, AKeff:
The stress intensity factor, K, can be looked upon as a characterising
parameter for stresses and strains in the crack tip region only if the
stresses and strains that are produced by the applied stress are tensile.
Therefore when K is used to characterise fatigue crack growth, only
tensile stresses and strains can be assumed to contribute to crack growth.
During the unloading part of the cycle, stresses and strains in the cyclic
plastic zone,_where the fatigue processes are thought to take place,
decrease and may become compressive due to the elastic unloading of the
material in the surrounding relatively large monotonic plastic zone.
This decrease in effect terminates the growth process until the next loading
half cycle. During the next loading cycle the induced compressive stresses
and strains at the crack tip will need to be overcome before any further
202
crack tip deformation takes place.
It is therefore necessary to define an effective stress intensity
factor range, AKeff' producing only tensile stress and strain excursions
at the crack tip. Thus in terms of AKeff
and tensile strain ranges,
AeT(x), equation (5.16) becomes
•
1
2 l+n
AKeff 1
Ae (x) = 2e
(5.17)
yc I4a(l+n)rayc x+rc
5.5.5 Fatigue crack growth process:
The process of fatigue crack growth is assumed to occur when the
level of plastic deformation or tensile strain range ahead of the crack
tip reaches some large but finite value, decf' sufficient to cause
material separation. The crack is assumed to advance cycle by cycle
by an average increment ahead of the crack tip whenever the above conditions
are satisfied.
When AKeff
is applied from the minimum value to the maximum value,
the crack will advance by a distance Aa, Figure 5.19. Over that distance ,
the tensile strain range equals or exceeds the critical value, Aecf, for
fracture. The amount by which the crack tip has advanced at Kmax is
given by substituting aecf for AsT(x) and Aa for x in equation (5.17),
and rearranging,
2
Aa + r = (2Eyc)1+n AKeff
c Aecf 4a(l+n)ira 2
yc
(5.18)
Upon unloading the new crack tip position will be a distance Da from
that at the previous cycle. Therefore, growth per cycle, da/dN, is
given by equating Aa to da/dN in equation (5.18) and rearranging,
r• dN AEcf 4a(1+n)Tra
2 c yc
2 da = (2Eyc)l+n AKeff (5.19)
203
•
The value of rc in equation (5.19) may be estimated by assuming
that crack growth does not occur when EET(0) < Aecf. The condition for
a non-propagating crack is thus assumed to occur when AKeff is less than
or equal to some critical value, AKc th' at which DET(0) < aECf. Under
such a condition then dN
=>0, and therefore
r
2 (2Eyc)l+n AKc,th
• c AEcf 4a(l+n)rra
2 yc
(5.20)
Substituting equation (5.20) in equation (5.19) and rearranging we
obtain
22 da __ (2Eyc)l+n AKeff - AKc,th dN aECf 4a(l+n)'ira 2 yc
(5.21)
Equation (5.21) can be modified by letting Eyc = ayc/E' where E' = E for
plane stress and E' = E/(1-v2) for plane strain and assuming that Aecf
is of the order of the true fracture strain, Ef, which can be
approximated by the plane strain ductility of the material under cyclic
conditions. Thus equation (5.21) reduces to
da 21+n
(1-v2)1+n(AKeff - AKc2th) (5.22)
dN 4a(l+n)7oyn(
E)1+ne l+n yc
204
5.5.6 Determination of LKc,th:
The value of AKc th
for a non-propagating crack can be determined
either experimentally or by theoretical considerations. The simplest and
most accurate method of determining AKc,th
experimentally is the
'increasing R' technique described and discussed in sections 5.2 and 5.3
of this chapter. Other experimental techniques for estimation of AKc th
involve the determination of the crack closure stress,KCL, as discussed
in section 5.3. However, these latter methods are only approximate.
Existing criteria for non-propagating cracks (123,125-127] do not
predict a critical value of AKth as observed in the present results for
the BS4360-50D steel and those of other steels. It is therefore proposed
to use energy considerations of a growing crack to predict AKc th'
Consider the total energy change when a crack advances by an amount
Aa into the elements of a ductile material. The energy change, AU,
is given by:
AU = AUs + AUes + AUPS (5.23)
where AUs is the energy change required to create new fracture surfaces,
AUeS is the decrease in elastic strain energy or potential energy
in the system due to an increase in system compliance as a
result of crack growth, and
AUFS is the plastic strain energy dissipated by producing plastic
deformation at the crack tip.
Assuming that the propagating crack front advances continuously by
an average amount of åa across material of unit thickness after a given
number of cycles then AUs, AUeS and AUFS can be approximated as follows;
•
205
AUs = [d(Asys)/da] Aa (5.24)
Where As is the area of the fractured cross-section of unit thickness
and is equal to 2 x 1 x Aa and ys is the specific surface free energy
[228]
. . AUs = 2ys.Aa (5.25)
Ues is estimated by considering the elastic energy change of the system
at constant AKeff when the crack advances by an amount Aa. This is shown
to be given by [229]
AU = es
2 AKeff . Aa (5.26)
TrE
AUps can be expressed as
AUps = {d(Vpyp)/da}Aa (5.27)
where y is the plastic strain energy per unit volume of plastically
deformed material and V is the critical volume near the crack front in P
which plastic flow occurs. y can be estimated from the stress-strain
curve by auts.c
• where auts is the ultimate tensile strength and of is
the true fracture strain under the particular stress-state conditions.
Vp can be viewed as ZA5.COD, where COD is the crack-tip opening displacement
given by [51]
2
COD - AKeff
(5.28)
2 (1+n) TrEc yc
for a strain hardening material. Therefore
2 AKeff.ef.auts AU ps - 2Tr(l+n)oyc E Aa (5.29)
206
Substituting equations (5.25), (5.27) and (5.29) into equation
(5.23) we obtain,
2 2
DU _ AKeff + 2 + AKeffEf.auts
t,a E ys 27(1+n)a E yc (5.30)
conditions for non-propagating cracks is satisfied by letting ~a 4> 0
th and therefore equation (5.30) reduces to as LKeff = AK
c
i
1/2
AKc,th
4TTEy s
(5.31) •
outsEf 2
(1+n)a
yc
For an elastic perfect-plastic material outs = o
yc and n = 0 and therefore
equation (5.31) reduces to
4TrEys 1/2
AKc,th 2 -Ef
(5.32)
5.5.7 Evaluation of crack growth rates and AKc th.
In order to evaluate the crack growth rates and AKc th'
predicted
using equations (5.22) and (5.31) respectively, it was necessary to
determine the respective material constants. Of these ayc' outs E' Ef
and n were determined experimentally and they have been given in Table 3.1.
ys was estimated using theoretical approximations as described below:
Cottrell [230] developed a theory which assumes that the critical
stage of the fracture process is the growth of microcracks within each
microstructural unit of size d (taken as the grain size). This assumption
leads to the following relationship between the critical fracture stress,
acf' and d
207
4G Ys -1/2 acf k
d y
(5.33)
where G' is the shear modulus, ys is the specific surface free energy
and k is the Fetch's constant [230]. Since G' - 2(l+v) then it
follows that
•
Ys 2(l+v)acf.ky 1/2
4E d (5.34)
The value of k has recently been evaluated for different steels [231]
and found to average about 0.13 MN/m3/2.
Green and Hundy [232] have shown that acf can be expressed by
acf = 2Ts(l+2- 2 (5.35)
where Ts is yield stress in pure shear and g is the notch angle. For a
sharp notch (e = 0) and using von Mises yield criterion (T = ay/✓3), we
obtain
acf = 2.97 ayc
and therefore for BS4360.50D steel, acf = 1130 MN/m2.
(5.36)
Ritchie and Knott [233] have obtained a plot of acf and d-1/2 for
steels and from this plot, knowing either acf or d-1/2 the other can be
estimated. Using acf = 1130 MN/m2 for BS4360-50D steel a value of
d = 27 pn was estimated. Thence a value of ys = 2.39 J/m2 was calculated
using equation (5.34).
The above value of ys compares well with the value of 2 J/m2 for
pure iron and also with the average value of 1.5 J/m2 given in a recent
report [234] for steels.
Therefore, from equation (5.31) and using ys = 2.39 J/m2 we obtain
AKc th = 3.51 MN/m3/2. If ys = 1.5 J/m2 is used a value of AKc th
2.80 MN/m3/2 is obtained.
208
AKc th
of other materials were estimated (using data reported in
the literature) using the above procedure and these are presented in
Table 5.1. These values are compared with the experimental values of
AKc th
obtained at high stress ratios and, in view of the approximations
involved, the comparison is good.
The fatigue crack growth rates were estimated using equation (5.22)
and a value of a = 4.5 (average of 3 and 1/(1-2v)2). The predicted crack
growth rates are compared with the growth rates, at R = 0.7 and for the
three thicknesses, in Figure 5.20. The correlation is excellent. The
prediction was also applied to two other materials, HT80 steel in
Figure 5.21(R = 0.8) and a C-Mn steel in Figure 5.22 (R = 0.72), and
again the correlation is good. The mechanical and fatigue properties
for the above steels, and others in Table 5.1, were estimated from values
of similar steels given by Majumdar and Morrow [51].
5.5.8 Discussion of the crack growth model:
The strain distribution assumed in the present analysis is only
applicable for small-scale yielding. This condition is well satisfied
at low stress intensities and therefore the above assumption is justified.
At low stress intensities, plane strain conditions prevail and
therefore the use of a plane strain plastic zone size in the present
analysis is justified. The influence of the state of stress on the plastic
zone size and on crack growth has been discussed (135,136]. However, at
present no expression exists for the plastic zone sizes under mixed mode
conditions. The ratio of the plane stress to the plane strain plastic
zone sizes, a, is usually given at 3(227] or 1/(1-2v)2(228] which differ
by a factor of two. Kang and Liu (89] made measurements of plastic zone
sizes under both plane strain and plane stress conditions and obtained
•
209
values of a which varied from 4 to 8. Therefore, the use of an average
value of a = 4.5 in the present analysis is justified.
It will be realised that the strain-controlled failure criterion
used in the present analysis has some limitations. It assumes a homo-
geneous, ductile material undergoing plastic deformations. However, if
the material contains brittle inclusions, particles or embrittled grain
boundaries, the fracture process will be accelerated by superimposed
cleavage. These secondary fracture processes are insignificant at low
stress intensities and in high toughness structural steels. It should
also be noted tnac-at the crack tip intense deformations take place, and
since the fracture process is a cycle by cycle process it would be right
to assume that strains of the order of the fracture strain, cf, are
reached at the crack tip.
The mechanism of crack growth near threshold is found to be micro-
structurally sensitive. In using a critical crack tip radius, rc, in
the present analysis to characterise crack growth rates at threshold it
is hoped that this dimension is characteristic of the microstructural
unit. Calculation of rc from equation (5.20) gives a value of
rc = 6.7 x 10-7 mm which is close to the atomic spacing of 3-5 x 10-7 mm
for steels [5].
The prediction of the crack growth rates and of AKc th using the
present model seems to give good results, at least for the materials
considered. It is therefore proposed that this simple model could be
applied to other ductile materials. However, for brittle or for ultra-
high strength-low toughness material, where static modes of failure are
important even at low stress intensities, the above model should be used
with some caution.
The above analysis will be extended to elastic-plastic crack growth
in the next chapter using the J-integral.
•
210
5.6 CONCLUSIONS
The fatigue crack growth behaviour of BS4360-50D steel has been
investigated at very low stress intensities and the following conclusions
drawn:
1. Low crack growth rates generally increase with stress ratio, R.
However, the effect of stress ratio on low growth rates is found
to be thickness dependent, especially at low stress ratios. It is
most pronounced for 12 mm, but almost insignificant for the 50 mm
thick plates. Growth rates are highest in the thickest plate at
low stress ratios (eq. at R = 0.08). At high stress ratios no
significant effect of thickness is observed.
2. AKth decreases with increasing stress ratio; the value of AKth at
low stress ratios is thickness dependent. For example, values of
AKth of 9, 5.5 and 3.2 MN/m3/2 for the 12 mm, 24 mm and 50 mm thick
plates respectively, at R = 0.08, are obtained. However, at high
stress ratios AKth, for all thicknesses, converge to a minimum
th' of about 3 MN/m3/2. This behaviour is found common value, AKc
to most steels where values of AKc th of 3-3.5 MN/m3/2 for ferrite
and pearlitic steels and of 2-2.5 MN/m3/2 for martensitic steels are
observed from results in the literature.
3. 3.5% NaC1 solution does not enhance low crack growth rates and
threshold as compared with the data in air. However, a significant
decrease in growth rates at low stress ratios is observed and crack
tip blunting could probably explain this behaviour.
4. The stress ratio and thickness effects can be adequately explained
in terms of the crack closure concept using an effective stress
intensity factor range, .Keff'
given by
•
211
AKeff Kmax - KCL for
min < KCL
and AKeff - AK for
Km. > KCL'
where KCL is the closure stress intensity factor.
5. Kth can also be related to R by the following expressions using
the crack closure concept:
AKth (KCL + AKc th)(1-R) for R < Rc
th'
and AKth
AKc > th >
for R > Rc th'
6. KCL can be estimated from da/dN versus AK data by constructing AK
versus R curves at constant growth rates. Values of KCL of
6.3 MN/m3/2 and 3.1 MN/m3/2 and values of Rc th
of 0.68 and 0.51
were estimated for the 12 mm and 24 mm thick plates respectively.
Data for the 50 mm thickness.predict low values of KCL. The
above values of KCL are found consistent with preliminary crack
closure measurements.
7. The low crack growth rates can be predicted using the AKeff
instead
of the total applied AK. Crack growth data at high stress ratios,
where AK = AKeff'
are more relevant to design application at high
mean stresses, such as occur in welded structures in the North Sea
area. A fast method of obtaining such data, even at near-threshold
conditions, called the 'increasing R' technique is devised and
successfully used in the present investigation.
8. Crack growth rates can be predicted using the expression
da a
dN C (aKeff - ~Kc,th)
Z
•
212
where C, m and a1 are material constants. An attempt is made to
determine theoretically the above material constants in terms of
mechanical and fatigue properties and this leads to the following
expression
da 21+n(1-v2)1+n(AKeff - AKc2th)
dN 4a(l+n)fra1 nEl+nefl+n y
eymue where n is the,1strain hardening exponent,v the Poisson's ratio,
a a constant of state of stress (a = 1 for plane stress and a = 4.5
for plane strain), ayc is the cyclic yield strength, E is the
Young's Modulus and sf is the true fracture strain. The prediction
of growth rates using the above expression is found excellent for
BS4360-50D steel and two other steels, i.e. HT80 steel and a C-Mn
steel.
9. The critical threshold value, AKc th can be estimated using energetic
considerations. An expression of the form
1/2
AKc,th
4rEys
autsef (l+n)a
yc
is obtained where nuts is the ultimate tensile strength and ys is
the specific surface free energy of the material. Estimated values
of AKc th
using the above expression compare well with the experi-
mentally observed values for BS4360-50D steel and for other steels.
213
TABLE 5.1
Comparison between experimental and theoretical values
of AKc th for various steels
Material AKc,th (Experimental)
(MN/m3/2)
AK theoretical
apprx. (MN/m3/2)
BS4360-50D [Present results]
HY 130 [133]
HT 80 [68]
SM50A [132]
C-Mnsteel [128]
SM 58Q [132]
3.00
3.07
2.80
3.52
3.40
2.60
3.51
2.70
2.78
3.84
4.0
2.78
goo
,t_ - __-b-----0
214
$_ 12 mm RI: 0.7mm .p= BO 14z
4 QK-STEP-DOWN
t A<-STP'
1
2.7
4 6 S N°• OF cYCLES Y-106
B = 24 rnrn C_ BO Hz
i LAK STEP DOS N
I 2 3
N°• OF CY N cLES~ x1O6 4
(a) 'Constant R' test.
(b) 'Increasing R' test.
Figure 5.1. Typical crack growth curves, a versus N, during threshold tests.
•
Calibration Curve
R = 0.5 B= 12 mm
iil
w 300 o V Z
U C5 Q H J
200
400
600
500
100
I I I I IFI I
• 215
I 2 3 4 5 6 7 8 CRACK LENGTH INCREASE Aa , mm
Figure 5.2. Digital voltage change versus crack length increment during a threshold test, compared with the prior calibration curve.
216
da /
dN
(m
m/c
ycle
)
1 I
R=-O.7
R=O.O8
R = 0.3
R = O.5
R = 0.7
R = O.9
Figure 5.3. da/dN versus AK; B = 12 mm, 30 Hz, in air.
10-5
I0 -6
10-7
-8 10
217 •
Figure 5.4. da/dN versus AK; B = 24 mm, 30 Hz, in air.
I
A R= 0.08 0 R=0•7
OLP
~ 00
O 107 0
A
A
106
a
218 •
10-5
-,
10-8 -8 l I_ 11 1 1 11 1
1 2 5 10 20 30 LAK (MN/m312)
Figure 5.5. da/dN versus AK; B = 50 mm, 30 Hz, in air.
1 1 L
/ I i
-
B=5Omm B=24mm I B=12mm
I I I I I 1 1 II 1 I
I0
d ū A tJ 1 E E
Z v 0 IQ
107
—6 I0
-8 I0
219 •
-5
I 2 5 3 IO AK (MN /m/2)
Figure 5.6. Effect of thickness on da/dN versus AK; R = 0.08, in air.
50
10-5 I I I I. I I I I I ~a
A — R = 0.08
O— R=0.7 NaCt Solution
A
AA
A _O. Air —
220 •
G1 ū 6" E E
pA
A /O °a
01 I 0 S
/o I ° 1 ' g
, o
r I
01 1 °
I
I
I O I
I
I I
I
i ii 1 IIIII 2 AK (MN/m3/2 )
Figure 5.7. da/dN versus AK; B = 24 mm, in 3.5% NaCl solution, compared to air data.
R =0.7: in Air
10 7 I R=0.08=in Air
10-8 1 30
• •
• B=12mm
B=24 mm
• B =50 m m
v ū IO E E
m 10 8
I6
14
12
~I
z v
0 6
4J a
_N 4 E
2 2 6 Kc, th
- O.8 -0.6 - 0.4 -0-2 0 0.2 0.4 0.6 0.8 1.0 STRESS RATIO , R
O- - 1.0 N N 1-4
Figure 5.8. Variation of AKth with R; in air.
IO 20
30
40
50 THICKNESS B, mm
E10 E in d ū >' 8 u
co 10
II z ▪ 6 v 73
• a a
Figure 5.9. Variation of AKth
with thickness, in air.
N N N
n— 17.8<Kmax< 2427 MN/m312
Roh=0.88
— ° — 11.4 <Km <13.6
R =0.78 •
~— 722<Kmax< 8.3 ■❑
Rc,th 0.64 0
0
"Constar-it R" test data at R=0-7
L
7.1 I 11111
1 2 4 6 8 10 AK, (MN/m312 )
I I
20 30
0 0 0
223 •
10-5
0
CLI
E
1C8
Figure 5.10. da/dN versus AK data from 'increasing R' test.
I I I ! I
1,I, ■ BS436O -5OD [Present results) O BS15 MILD STEEL [Ref.'61 l ` O 0.15°/oC 1.5°/oMn STEEL -
[Ref.1371 A 13 Cr CAST STEEL [Ref.1301 ✓ ASTM A533B STEEL [Ref.1421
I I I 1
A
I I I I I i I I I I I 10 0.8 0.2 0.4 0.6
• • a •
16
14
12
I0
(
M E 8
Z 2
Y d
6
4
2
0 -1.4 -1.2 -1.0 -0.8 - 0.6 --0.4 -0.2 0
STRESS RATIO R
N N
Figure 5.11. Variation of AK with R for different steels.
225
J
1 1 I
I 1 I
0-POSITIONS FOR K ESTIMATION
a 10
-6
I
0
R=0.9
R =07
--~—~•- n . . 1 2 3 4 5 6 10 20 30
AK(MN /m3/2 )
Figure 5.12. Positions for estimating KCL from da/dN versus AK data; B = 12 mm.
-7 10
•
V x
d a/N (mm /cycle )
-- 2x1Ō6 — 1x10-6
o — 4x10-7 o — 1x 167 A — 1 x 108
KcL(MN/ m 2)
6.18
6.48
6.20
6.21
6.38
40 - -0.8 -0.6 -0.4 -0.2 0.2
Stress ratio R '
Y d
Figure 5.13. AK versus R curves for estimation of KCL; B = 12 mm.
u A
* 0
, 1
0
0
Q (6.31 MN/m /2)
e0 0
0 0
0
O—B=12mm
A--B:24mm
3.08 MNIm3I2)
-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 STRESS RATIO R
Figure 5.14. Estimated values of KCL; B = 12 mm and B = 24 mm.
I
N N V
6
228
5i
O — B= 24mm
A B= 50 mm
4
1
0
NNW
I
0 10 20 30 40 50
Kmax (MN/m3/2)
Figure 5.15. Measured values of KCL plotted against Kmax.
-5 IO
• R=-0.7 o R=0d08
• R=0.3 0 R=0.5 O R=0.7 • R=0.9
-6 I0
0 .10
-8 I0
0
229
2 AKeff(MN /m3/2)
,o
Figure 5.16. da/dN versus iKeff;
B = 12 mm, in air.
•
16-
14-
12-
6-
4
2-
o-B=12mm
o-B=24mm
P-B=50mm
0
N
' -T
c2 ci
A
A Kc th.
0 -10 -0.8 -06 -04 -02 0 02 04 06 07 08 10 `,'
Stress ratio R
Figure 5.18. Correlation of AKth with R using crack closure concept.
IE.r(o) =O arc
STRAIN! otsTRteu-rtotJ ~ AT AKQFF = dKo , th
VIVAtt.t DISTR18UTtotJ AT KmQ%.
aET
a6T(o)= aecp A=`..1 E--- ---1I----ro AET(o) _Aecp
isepo
0..) 5t-1A.RP CRACK TIP. b). CRACK TIP 'FULLY •BWNTED C) CRA[.K HA5 ADVANCED BUT NO C{ROWTH.
Figure 5.19. Crack tip deformation and crack growth during a fatigue cycle (schematic).
Aa. AT K.Inckx SUT THE IS STILL BLuMTEO.
BY TIP
N w N
233 •
AK (MN fm3/2 ) 10
Figure 5.20. da/dN versus OK at R = 0.7, in air; compared with theoretical crack growth rates: BS4360-50D steel.
EQuAMON (5'22)
00 0
0
0
1081 I .I 1 11 1 1 I
LAK (Mnt/m3/2) 10
da►./d
t l (m
m j
cc l
e)
234
Figure 5.21 da/dN versus AK at R = 0.8, in air; compared with theoretical crack growth rates: HT80 steel [Ref. 68].
•
dc.%
(AK
! (m
m/ c
c 1e
)
EQUATION (5.22)
IV
s I i o l I I I IlI
LAK (Nisi /in 34) 10
235
Figure 5.22. da/dN versus AK at R = 0.72 in air; compared with theoretical crack growth rates: C-Mn steel [Ref. 128].
•
236 •
CHAPTER SIX
ELASTIC-PLASTIC CRACK GROWTH BEHAVIOUR IN
BS4360.-50D STEEL
237
6.1 INTRODUCTION
In the previous two chapters linear elastic fracture mechanics
(lefm) was used successfully to correlate fatigue crack growth at
intermediate and low growth rate regimes. However, substantial limita-
tions on the use of LEFM arise when materials capable of large plastic
deformations, such as tough structural steels, are considered. If the
plastic deformation around the crack tip is large, the plastic effects
are also large and the stress intensity factor, K, loses its original
significance. Therefore, under conditions of large plasticity at the
crack tip an alternative characterising parameter to K has to be
employed.
Recent research has shown that the J-integral is a satisfactory
failure criterion for those materials which fail only after substantial
plastic deformation followed by slow stable crack growth. The concept
has sometimes been interpreted as a measure of the intensity of the
characteristic crack tip plastic strain field [202]. This interpretation,
similar in philosophy to the LEFM stress intensity factor for elastic
stresses and strains may account for the success of the J-integral as a
static and fatigue fracture criterion under elastic-plastic conditions
[150-156].
The amount of plastic deformation and of slow stable crack growth,
although primarily dependent on the material's properties and temperature,
is influenced by the state of stress at the crack tip, and consequently
by the size and geometry of the specimen or component. Recently, a
significant research effort, using the J-integral, has been directed
towards the investigation of the effect of size and geometry on both
initiation and subsequent slow stable crack growth. This investigation
has led to the definition of the plane strain initiation value of J,
considered equivalent to Glc, and hence given the notation Jlc [202].
•
238
This value is said to be size-independent [199,203,237], however, some
recent results have indicated variation of Jlc with geometry and size
[238] .
The deformation theory of plasticity on which the J-integral is
based does not allow for the unloading that Occurs during cyclic loading.
However, this is an area with important practical consequences in many
application problems. It is for this reason that recent attempts have
been made, with some success, to correlate fatigue crack growth under
elastic-plastic conditions with the cyclic J-integral. The results from
these investigations have shown that even with cyclic unloading the
J-analysis must still be applicable.
In the present chapter the influence of thickness on the initiation
of slow stable crack growth and on the elastic-plastic fatigue crack
growth is investigated. An attempt is also made to predict elastic-
plastic fatigue crack growth rates in terms of the material's fatigue
properties.
6.2 EXPERIMENTAL PROCEDURE
Both cyclic and monotonic crack growth tests were performed on compact
specimens of three plate thicknesses, 12 mm, 24 mm and 50 mm. The
specimens were provided with suitable knife edges at the loading line
for load-line displacement measurements and also with deep notches, so
that a/W > 0.48, All the tests were performed in laboratory air at 21 °C
in ambient conditions, using two machines: the Dowty and the Mayes fatigue
machines described in section 3.2.
The experimental procedure and data reduction techniques used have
been described in chapter 3 in a more detailed form. However, it is
worthy mentioning here that deflections were measured using a standard
clip gauge. J and MJ values were determined from the area under the
•
239
load-displacement curves with the help of a planimeter. The load-
displacement curves were recorded using a Bryans X-Y plotter.
6.2.1 Compliance calibration curve:
A compliance calibration curve for the specimen geometry (i.e. CS)
used was generated prior to the cyclic and monotonic tests. A fatigue
crack was grown in a 24 mm thick specimen from a/W = 0.3 to a/W = 0.8,
and the load limits were continuously adjusted to keep the Kmax value
betrre-n 15 MN/m3/2 and 16 MN/m3/2 throughout the test. The crack length
was monitored on the polished surface of the specimen using a travelling
microscope, while at the same time compliance plots were made for selected
crack lengths. At the end of the test, the specimen was opened up and
the crack front profiles recorded. The average crack length for each
profile was then estimated. Crack length surface measurements were then
corrected by reference to the crack front profiles and normalised
compliance values, (BESLL)/P, where SLL is the load-line displacement and
P is the applied load, plotted against the corresponding normalised crack
length, (a/W). Additional compliance measurements were made on the 12 mm
and 50 mm thick specimens and the whole compliance calibration curve is
shown in Figure 6.1. The experimental points were compared with the
compliance curve recently derived (Saxena and Hudak (201]) in the form
of a compliance polynomial foi compact specimens, for a/W > 0.2, i.e.
•
BES LL - (i+ā W)2 PLL + 12.219(a/W) - 20.065(a/W)2
- 0.9925(a/W)3 + 20.609(a/W)4 - 9.9314(a/W)5} (6.1)
The correlation between the experimental compliance and equation (6.1)
was excellent and thus this equation was used for crack length determina-
tion for the three thicknesses used in the present investigation. For
240
the few specimens where the clip gauge was placed at the front face of
the specimens the compliance was determined using another of Saxena
and Hudak expressions, i.e.
BEV0 = (1 0./W 125)(1+a/W)2
W{1.61369 + 12.6778(a/W) - 14.2311(a/W)2
a-a/
- 16.6102(a/W)3 + 35.0499(a/W)4 - 14.4943(a/W)5} ( 6.2)
for 0.2 < a/W < 0.975, which gave results in agreement with the
experimentally determined values.
6.2.2 The monotonic J-tests:
These tests were performed on specimens which were precracked by
fatigue loading to a/W < 0.48 at low Kmax values of between 15 MN/m
3/2
and 16 MN/m3/2. Specimens for each of the three thicknesses were tested
using the single-specimen method [2001. The specimens were loaded to
failure while generating load-displacement curves using intermittent
10% unloading. Compliance values were determined from the slopes of the
unloading lines (Figure 6.2) and thus crack growth increment, da,
obtained from Figure 6.1. In addition, some 24 mm thick specimens were
loaded to pre-determined displacements (using Figure 6.2) and then
unloaded completed. The extent of ductile tearing was marked by
subsequent fatigue cracking at much lower'Kmax values. The specimen
was then broken into two for the examination of the fracture surface.
Some of the specimens' fracture surfaces were photographed to show the
configuration of the ductile tearing.
The areas under the load-displacement curves were then measured
with the help of a planimeter and J values then calculated using
equation (3.12) with values of X as given in Table 3.2. Resistance
curves, J versus Da, were then plotted for the three thicknesses.
•
241
6.2.3 The cyclic J-tests:
Elastic-plastic fatigue crack growth tests were conducted using
load cyclic control at a load ratio, R, of 0.08 to facilitate comparison
with the previous elastic crack growth results (chapter four). Tests
were performed on precracked specimens with crack lengths such that
a/W > 0.48 and at AK values of 60 MN/m3/2 and higher. In order to apply
a constant loading rate, the cyclic frequency, initially set at 0.25 Hz,
had to be continuously reduced. Hysteresis loops, Figure 6.3, were
recorded at suitable crack length increments. Shorter crack lengths
were determined by the surface measurements and the compliance measure-
ments using load-displacement unloading lines. However, at the higher
growth rates, the surface measurements could not be made accurately
because of the great amount of crack tip blunting and also due to the
tunnelling effects at the centre of the specimen, hence only the
compliance measurements were made. For consistency of the results only
crack lengths determined by the compliance method were used in the data
analysis.
The crack growth rates were established from the graphs of a versus
number of cycles, N, using the secant method (see chapter 3). Experi-
mental values of AJ were determined from the hysteresis loops using
Dowling's method [150]. These AJ values were then plotted against the
corresponding crack lengths and the best curve fitted through the data
points. The AJ values corresponding to the mean crack lengths at the
calculated growth rates, i,e. (a. 1 4- a.)/2, were then read off from the
a versus MJ curve, and these were plotted against the respective da/dN
on a log-log scale.
The values of AK from the linear elastic cyclic tests were used to
compute elastic cyclic MJ values from the linear elastic strain energy
release rate, C, using the basic formula:
•
242
__ AK2 = AJelastic OG E
(6.3)
These results were plotted together with the elastic-plastic cyclic
crack growth results.
6.3 RESULTS
6.3.1 Monotonic J versus to resistance curves:
The J versus da resistance curve for the 24 mm thick specimens is
shown in Figure 6.4. The curve consists of two parts; one having a steep
slope for small crack extensions and the other of a shallow slope corres-
ponding to large crack extensions. The intersection of the resistance
curve with the theoretical blunting line (i.e. La = J/2aflow) indicates
a value of J, Jc, of 150 N/mm. The experimental data below the knee of
the resistance curve are generally above the theoretical blunting line.
If the lower part of J versus Da curve is extrapolated to as = 0, a much
lower value of J (to be called J-initiation, Ji) of just over 50 N/m is
obtained.
Figure 6.5 shows the best lines drawn through the experimental data
for all the three thicknesses. These resistance curves indicate Jc
values of about 200 N/mm, 150 N/mm and 120 N/mm for the 12 mm, 24 mm
and 50 mm thick specimens respectively. The corresponding J. values are
about 65 N/mm, 50 N/mm and 48 N/m respectively.
If the crack growth lines, i.e. the second parts of the J versus
a curves, are directly extrapolated to Aa = 0 (without considering the
initial part of the curve), values of J (here called Jī2) of 85 N/mm,
90 N/mm and 85 N/mm for the 12 mm, 24 mm and 50 mm thicknesses, respectively,
are obtained. The variation of .1 , Ji and 3. with thickness is illustrated12 c
in Figure 6.6. From this figure it is seen that Jc is definitely thickness
dependent while Ji and Ji2 may be considered thickness independent.
•
243
Photographs of some of the 24 mm thick specimens broken open after
various amounts of ductile tearing are shown in Figure 6.7. These
photographs show that almost all the crack growth occurred in the
interior of the specimens where the plane strain conditions were
predominant. The same crack front tunnelling was also observed in the
elastic-plastic fatigue test specimens.
It is interesting to note that compliance measurements of two of
the monotonic test specimens, with complete unloading, showed Da values
of 0.08 mm and 0.96 mm, while in the centre of the specimen maximum crack
growth values of 1.25 mm and 2.15 mm respectively, were measured. In
both cases, no measurable growth had taken place on either extreme sides
of the crack front or on the specimen surfaces. Average values of La
measured on the fracture surfaces (average of nine positions equally
spaced across the thickness) showed comparable values with the compliance
determined Aa values.
6.3.2 Cyclic crack growth results:
The results obtained in the elastic-plastic crack growth tests are
shown in Figure 6.8. It is observed from this figure that crack growth
rates are generally higher in the thicker specimens, a pattern already
noted in the linear elastic test results in chapter four. The growth
rates for the 12 mm and 24 mm thick specimens differ by substantial
amounts although at most by a factor of two. However, the difference in
growth rates between 24 mm and 50 mm is negligible for most of the growth
rate range tested. It will be noted that at growth rates above
3 x 10-3
mm/cycle, the stable crack growth by ductile tearing strongly
interferes with the cyclic crack growth.
Figure 6.9 shows the cyclic and monotonic crack growth data plotted
together. This figure shows that at high growth rates, the cyclic crack
•
244
growth data correspond to crack growth by ductile tearing.
From the hysteresis loops recorded just before final fracture,
the maximum values of J, (Jmax = MJ/(1-R)) were 85 N/mm and 96 N/mm for
the 50 mm and 24 mm thicknesses respectively, while for the 12 mm
thickness values higher than 110 N/m were calculated, and this is
certainly a large increase. However, this maximum value of J was
found to depend on the crack length.
6.4 DISCUSSION
The compliance method used for crack length determination was
found satisfactory. Examination of the fracture surface have confirmed
that ductile tearing cannot be measured using visual observations on the
specimen surface since most of the ductile tearing occurs in the interior
of the specimen. It is difficult to assess the overall effect of this
non-uniform crack growth on the observed change in the compliance. It
can only be assumed that the change in the compliance indicates the
equivalent crack growth across the entire thickness and this has been
shown to hold true in previous tests using both compliance and fracture
surface measurements for t.a [239,240]. Results from the few specimens
broken open for measurement of to in the present tests also justifies the
use of the compliance method.
It was also found that drawing lines through the short 10% unloading
lines, for compliance determination, caused a certain amount of uncertainty
and consequently inaccuracies in the determination of 1a. Recent results
by Castro et al. [200] have suggested that a larger percentage of
unloading would increase the accuracy of the evaluation of the crack
length. These authors discussed the effects of cyclic and partial unloading
on the J-da resistance curve using 30% unloading. 10% and 30% unloading
•
245
correspond to cycling at high mean roads using the stress ratio, R, of
0.9 and 0.7 respectively. As expected, under identical test conditions,
fatigue crack growth has been found to take place (chapter four). This
would indicate that more crack growth could occur in the tests with
unloading than without unloading for the same load and displacement.
This argument is contrary to Castro's observations. However, other
material properties, such as strain hardening and cyclic creep not
considered here, may influence the shape of the resistance curves.
The present results show that the initiation of stable crack growth
can be thickness dependent, or independent, depending on the definition
of the critical value of J. Clarke et al. [119] found no specimen size
or geometry effects on the onset of stable crack growth, Jc, in the
materials they tested. However, these authors noted that the effect of
size was dependent on the position of the measurement of Jc on the load-
displacement curve. Hence, the Jc value obtained from large crack
extensions would be more size dependent.
Griffis and Yoder [237] extrapolated the resistance curve back to
zero crack growth and achieved a J. value independent of thickness for is
2024-Al alloy. Landes and Begley [203] included the stretch zone in
their measurements of crack extension and therefore defined Jc as the
intersection of the approximate resistance curve with the line describing
the growth of the stretch zone, i.e. Da = J/2aflow. They noted that this
Jc value was size independent. This procedure, forms the basis of the
standard procedures for determination of a size-independent J.
Keller and Munz [238] performed J-tests on aluminium alloy 7475
and Ni-Cr-Mo steel using compact specimens of varying thicknesses. Their
results showed that for the alloy steel 13 mm < B < 90 mm, Ji was
independent of the specimen thickness, whereas between B = 5 mm and 13 mm,
J. increased with decrease in thickness. The Jc values determined at the
246
intersection of the resistance with the Aa = J/2 flow
line were found
substantially size-dependent and somewhat over-estimated. These Jc
values increased with increasing thickness, a result contrary to the
present results.
It would be noted from Figure 6.6 that there is minimal size
dependence of Ji and Jic when using the extrapolation of the two parts
of the resistance curve to to = 0. However, the value of Jc, determined
at the intersection of the resistance curve with the theoretical
blunting line as originally suggested by Landes and Begley [203] is
highly size dependent and increases with decreasing thickness. Also,
these values seem to be too high to define the onset of slow stable
crack growth, since finite growth is measurable at lower J values. Tests
on the 24 mm specimens which were broken open after small amounts of crack
growth support the claim that Jc is too high to define initiation of crack
growth. It may therefore be suggested that for a size independent
J-integral criterion, the Ji value obtained at the intersection of the
resistance curve with the Aa = 0 axis would be more realistic than the Jc
value defined by Landes and Begley method, at least for this material
and specimen geometry used.
The present results show that crack growth rate is controlled by the
J parameter for the linear elastic range and a part of the elastic-
plastic crack growth rate region. It can be noted that above
3 x l0-3 mm/cycle the crack growth rate accelerates and the fast tear
interferes with cyclic crack growth. This process is probably associated
with superposition of cyclic creep and ductile tearing on the fatigue
crack growth. At higher growth rates, of the order of 10-1 mm/cycle,
instability seems to set in and specimen fracture occurs on further
cycling. The cyclic growth rates at the high values correspond to growth
rates under monotonic loading indicating that the mechanism of crack
•
247
growth under both types of loading is the same. The elastic and the
elastic-plastic fatigue crack growth results seem to be correlated
fairly well using J and fit on expression of the form obtained by
Dowling [150] and Branco [152] (equation (2.45)) for other steels.
In the next section a simplified model, which combines crack growth
by crack tip blunting mechanism and that by other static modes, such as
void coalescence, will be derived. This model is an. extension of that
derived in chapter 5 for the linear elastic case.
6.5 THEORETICAL CONSIDERATIONS
6.5.1 Introduction
In chapter five a model for crack growth at low stress intensities
was derived in terms of linear elastic fracture mechanics (LEFM). This
concept assumes that the crack tip plastic zone size is small compared
to planar distances to other boundaries (or load points, etc.), that is,
to say for small-scale yielding. For cases of general yielding, as
found in ductile materials, the elastic-plastic fracture mechanics must
be adopted. The general applicability of the J-integral (as with K in
LEFM) comes from viewing the stress and strain fields surrounding the
crack tip with an appropriate rationale.
Consider three distinct levels of viewing the surrounding field as
noted in Figure 6.10. Region (1) can be viewed as an elastic field
surrounding the crack tip where LEFM is applicable. Region (2) is an
elastic-plastic field surrounding crack tip where general yielding
fracture mechanics (YFM) is applicable. Region (3) is a zone of intense
deformation which is currently incapable of full analysis. If zone (3)
is comparatively small, then region (2) can be analysed using plasticity
theories. To extend fatigue crack growth to the elastic-plastic regime
•
, 248
the J-integral will be used. The same procedure adopted for the elastic
case, dealt with in chapter five, is followed.
6.5.2 Stress and strain fields:
Using an earlier analysis by Rice and Rosengren [241] and
Hutchinson [242], McClintock [243], derived the stress and strain distri-
bution in terms of J-integral (Appendix) for power hardening material,
Figure 6.10. Applying a method suggested by Lamba [153], the strain
distribution at the crack tip, under cyclic loading, can be expressed as:
•
Ae(x) = 2eyc 4Ion eJ o ' yc yc
1. l+n
(6.5)
cycue where
On is a function of theA strain hardening exponent, n, the state of
stress at the crack tip and of 8 in the direction of crack growth
AK2 (i.e. 8 = 0). For linear elastic case AJ =
and equation (6.5) reduces
to the strain distribution used in chapter five. Equation (6.5) is
corrected for the finite strains at the crack tip experienced in ductile
materials -by introducing a finite crack opening 2rc and can then be
rewritten as
1 l+n
Ae(x) = 2eyc
MJ 1 (6.6)
41One Y c6 Yc ' x+r
6.5.3 Fatigue crack growth process:
The process. of crack growth is assumed to occur when the plastic
strains at the crack tip become large, i.e. of the order of the true
fracture strain, ef. The crack tip of the originally sharp crack
becomes blunted and its flanks, initially closed, will separate by a
finite distance yet remaining almost parallel. Further damage occurs
only when this separation reaches a critical value, 2rc, Figure 6.11.
249
The crack then advances by Da, as a result of a crack-tip blunting
mechanism, on further increase of the load. On still further increase
of the load, voids form and coalesce by ductile tearing and/or cleavage
cracking and the crack extends by a further amount of Aadt. The total
crack growth will then be given by the sum of crack increments due to
the two mechanisms. It is proposed that the two mechanisms be modelled
separately and then combined by addition.
To account for crack closure effects it is necessary to define an
effective AJ, LJeff' as in the elastic case. It is assumed that crack
growth does not take place when the stresses and strains are compressive.
To predict crack growth by a crack tip blunting mechanism for a
homogeneous fully ductile material undergoing large plastic deformations,
it is assumed that crack growth is a cycle by cycle process occurring at
the crack tip wherever Ac(x) -} ef. The crack advance per cycle,
Aab (E da/dN), is given by substituting Aab for x and for Ac(x) in
equation (6.3);
sf = 2syc
Jeff 1
(6.7)
41Onc y ca yc Aab+rc
Rearranging, we obtain
da __ 1 n+l (2eyc)nAJeff dN (sf)2oycl0n
rc (6.8)
Value of rc can be arrived at by allowing for da/dN -} 0 at a critical
value of Jeff'
bJc th which is the critical threshold for a non-
propagating crack (c.f. AKc,th).
Thus, the crack growth rate is given by
2 n+1 _ da _ (J eff ~Jc,th) (6.9) dN 41 Enc 1-n(s l+n
On yc f
250
Equation (6.9) is similar to equation (5.22) for the linear elastic case.
If the intense zone of deformation is large, additional crack
growth will occur at fairly low strain ranges due to the void formation
and coalescence ahead of the crack tip. Therefore application of
equation (6.9) is limited to those regions of LEFM and YFM where the
intense zone of deformation, w, (Figure 6.10) is small compared with the
planar dimensions [244]. However, when w is large the expression (6.6)
would underestimate the actual growth rate. At present, no simple analysis
exists except that by Rice and Johnson [244] who approximated deformation'
behaviour in this region using slip-line theory. The stress 1= ,...at
which voids start forming at the crack tip will depend on the material
ductility and the amount of second phase particles and inclusions present
in the material. For an ideal homogeneous ductile material voids may
not form until after large plastic deformations. In this case
equation (6.9) would be applicable under both elastic and elastic-plastic
conditions.
The lack of analytical solutions for zones of intense deformation
may explain why fatigue crack growth models are frequently modified. An
empirical factor in the form of Kc/(Kc-Kmax)
is often included in order
to account for the increased growth rates that occur at high stress
intensities due to void coalescence, creep and other processes. At high
stress intensities cyclic crack growth rates are found to correspond to
the monotonic slow stable crack extensions (e.g. Figure 6.9). This
observation suggests that the mechanism of crack growth is the same for
both types of loading. To account for the static mode of crack growth
superimposed in the 'pure' fatigue process, it is suggested that two
approximations may be made to equation (6.9). The first of these approxi-
mations is achieved by multiplying the above equation by an empirically
derived function of aJeff to fit the experimental data. Then equation
(6.9) becomes
•
251
1+n 1+82 da 2 A2(AJeff - AJc,th)
(dN)total 41 E1+no E
1-n l+n On yc f
(6.10)
where A2 and $2 are constants and are a measure of the amount of static
crack growth superimposed in the cyclic fatigue process by the crack
tip blunting mechanism. For an ideal, homogeneous, ductile material
A2 =1 and $2 =O.
Alternatively, it is suggested that a double mechanism fatigue
crack growth process may be assumed to take place.
At high stress intensities the process is Jmax dependent and the
crack growth may be estimated using crack growth resistance curves
(R-curves) [245,246]. The resistance curve may be approximated by [246]
DaR = C(JR)m+ (6.11)
where C' and m' are material constants determined experimentally from
the R-curve. It could be reasonably assumed that the crack tip blunting
mechanism predominates in crack growth at low stress intensities whereas
void coalescence predominates at high stress intensities. Although
there is present no evidence available to show that these processes are
additive, it is suggested that at least at intermediate stress intensities
both mechanisms may occur simultaneously. Therefore, the total crack
growth can be expressed by direct summation of equations (6.9) and (6.11),
i.e.
l+n ( da '
2 Jeff - dJc~th) + C'(J
)m ( ) dNtotal -
4I En o 1-n l+n max
On yc f
where JR in equation (6.11) is now replaced, max the maximum
value of the applied AJ and given by Jmax = MJ/(1-R).
(6.12)
•
252
It may be noted here that the transition region between true
cyclic crack growth and stable slow crack growth was investigated by
Leevers et al. [247] using polymers at room temperature and a wide range
of frequencies. In that study the high stress intensity cyclic growth
was successfully predicted from data obtained under static loads. It
could be speculated therefore that a similar approach could be applied
in ductile metals.
6.5.4 Crack growth evaluation:
In order to evaluate equations (6.10) and (6.12) for the total
crack growth under elastic-plastic crack growth, the empirical constants
and material properties for BS4360-50D steel were determined.
The results for the stable crack growth under monotonic loading
are plotted in Figure 6.12. The results show a linear relationship for
low crack extensions and these can be fitted by a mean R-curve in the
form:
LiaR = 7.5 x 10-18 JR2.60 (6.13)
At high crack extensions, greater scatter and also deviation from
linearity sets in and this seems to be most prominent in the thinner
specimens. It is therefore assumed that crack extensions predicted by
equation (6.13) can be extrapolated to lower values of J, with a threshold
value below J.. i
The value of I0n
in equation (6.12) can be approximated by assuming
plane strain conditions and also that IOn is constant for both small-scale
yielding and general yielding conditions. For n = 0.19, IOn = 4.57r(l+n) =
16.81. This value is comparable to those tabulated by McClintock [241]
from which a value of IOn = 15.08 can be approximated. Therefore using
M
253
the material constants presented in Table 3.1 the experimental data in
Figure 6.8 was fitted to equation (6.10) and values of A2 and S2 of
24.26 and 0.59 respectively were obtained. Thus under elastic-plastic
conditions crack growth rates could be approximated by
da dN =
4.18 x 10-10 (AJeff)1.59
(6.14)
Using the double mechanism model of equation (6.12) the predicted
growth rates are compared with the experimental growth rates in Figure 6.13.
It will be noted that although a very simplified crack growth model
involving superposition of two growth mechanisms has been used, the
prediction is good. Figure 6.13 indicates that for BS4360-50D steel the
blunting mechanism is dominant at low growth rates <10-4
mm/cycle,
whereas at high stress intensities void coalescence involving dimple
formation and ductile tearing becomes the principal fracture mechanism.
At intermediate growth rates the two mechanisms seem to act simultan-
eously.
In Figure 6.14 the data for the whole range of growth rates from
threshold to fast fracture are presented. Equation (6.12) is seen to
fit the data fairly well.
6.5.5 Discussion of the double-mechanism model:
The crack tip blunting mechanism described by equation (6.9)
above has certain limitations. It assumes a homogeneous, ductile
material undergoing large plastic deformations and is therefore ideal
for high toughness, ductile materials. However, if the material contains
brittle inclusions, particles or embrittled grain boundaries, the
fracture process will be accelerated by superimposed cleavage and void
coalescence occurring at high stress intensities. It may be recalled
•
254
that, in the LEFM, equation (6.9) implies a da/dN versus J relationship
with the value of the exponent, m16, (equation (2.45)) of 1. This
relationship would imply that when m16 = 1, the crack growth mechanism
is predominantly that of blunting. This process has actually been
observed in high toughness steels and aluminium alloys. In these
materials the striation spacing distance equals the macroscopic crack
growth rate.
On the other hand, in low toughness materials with high values of
m16, fractographic analyses have revealed striation spacing with super-
imposed cleavage and intergranular separation as well as void coales-
cence [219]. Such fracture modes cannot be analysed accurately using'
current plasticity theories [244] and the use of R-curves to predict
crack growth rates by such modes would serve as a first approximation.
The stress and strain distribution assumed in the analysis can
only be evaluated for either plane strain or plane stress conditions.
However, in most practical cases, and especially under elastic-plastic
conditions, the state of stress is of mixed mode. Consequently, the
function IOn in equation (6.9) varies with J for it is most likely that
stress state will change from plane strain at low stress intensities to
plane stress at high stress intensities.
Despite the limitations discussed in the previous paragraphs, the
results for BS4360-50D steel appear satisfactory. The correlation of
the growth rates with AJ in Figure 6.14 is similar to that provided by
Dowling [150] and Branco et al. [152], confirming the applicability of
the J-integral analysis for cyclic loading. The apparent crack
acceleration and instability at growth rates above 3 x 10-3 mm/cycle in
the present results was also observed by Dowling at growth rates above
10-2 mm/cycle and more recently by Proctor [151] at growth rates also
above 10-2
mm/cycle. However, Branco's results do not show the above
r
255
acceleration behaviour and, in fact, these results conform very closely
to crack growth predictions using the crack tip blunting mechanism
(equation 6.9) as shown in Figure 6.15.
The present results were obtained at low stress ratios, R < 0.1.
It was noted that although crack closure effects were significant at
low stress intensities (chapter five), where crack closure was found
to occur at positive applied loads, at the high stress intensities
discussed here, the crack needed compressive loads to cause closure.
The data in the elastic region was corrected for crack closure effects,
as discussed in chapters four and five.
6.6 CONCLUSIONS
The elastic-plastic fatigue crack growth and slow stable crack
growth behaviour in BS4360-50D steel have been investigated using the
J-integral. From the investigation it can be concluded that:
1. The J-integral can be used to characterise crack growth under
both monotonic and cyclic loading.
2. The original JC criterion, based on the intersection of the J versus
Aa curve with the Da = J/2oflow line seems to over-estimate the
onset of slow stable crack growth. The Jc value appears to depend
on the thickness of the specimen - increasing with decreasing
thickness, at least for this material and for the CS geometry used.
3. The slow stable crack growth begins at the centre of the specimen
where plane strain conditions are fully developed.
4. The extrapolation of both linear parts of the J versus as curve
yields thickness independent values of Ji and 3i2 amounting to
•
256
55 N/mm and 86 N/mm (average values), respectively. The evidence
available at present indicates that the lower J value (Ji) clearly
represents the initiation of the slow stable crack growth.
5. For ductile materials, the elastic-plastic cyclic crack growth
can be described either by an empirical expression of the form
da dN = C13(&J)
m16
where C13 and m16 are constants, or by a double-mechanism crack
growth model of the form
n+1(J
- OJ ) da _ 2 eff c,th +
CIO )- dN 4I Ena 1-nE 1+n max On yc f
where C' and m' are constants derived from the monotonic resistance
curve (R-curve); IOn
is a function of the strain hardening exponent,
n, and state of stress (IOn = 4.57r(l+n) for plane strain) and
ayc, n and of are cyclic fatigue properties of material. The
constants C16 and m16 were evaluated at 4.18 x 10 10 and 1.59
respectively, and values of C' and m' were evaluated at 7.5 x 10-18
and 2.6 respectively for BS4360-50D steel (with da/dN in mm/cycle
and J in N/m).
6. Further evidence of the additive character of the mechanisms of
crack tip blunting and of void coalescence is needed and a more
rigorous analysis will be required.
•
200
REF. (201)
180
C10 EXPERIMENTAL POINTS FOR DIFFERENT
SPECIMENS 160
J 140 J
w a
co—
120
w U Z
100 -J a 2 0 U 80
NO
RM
ALIS
ED
60
40
20
257
•
! I I I I 00 2 0.3 0.4 0.5 0-6 0.7 0.8
NORMALISED CRACK LENGTH (o/w)
Figure 6.1. Compliance calibration curve for the CS geometry used.
50 FRACTURE
40
Z
- 30 Q 0
20
10
I I I I I I I I
• • •
2 3 4 5 6 7 8
LOAD LINE DISPLACEMENT (mm)
Figure 6.2. J-test using 10% unloading technique; B = 24 mm.
• 4
0 0 0 o Ln n • - - l^ ^ ° m r m ,t -4
▪ -
-4.-4- i
▪
f z
Fracture
30
20
10
Z
0 J
1 2
Clip gauge displacement, e6, mm
Figure 6.3. Typical hysteresis loops during cyclic loading.
—800 E E
Z 1 600
400 10 0/0 UNLOADING
• COMPLETE UNLOADING
. 1.200
1000
200
I • •
1.0 2.0
CRACK GROWTH A a , mm
3.0 4-0 ' N 0
Figure 6.4. J versus to (resistance curve) for B = 24 mm.
• •
1.0 2.0 3.0
CRACK GROWTH , A a, imm)
Figure 6.5. J versus to for B = 12 mm, 24 mm and 50 mm.
4.0 N ON I-.
150
100
50
CRIT
ICA
L J
, (N
/mm
)
262
•
IMEN
0 10 20 30 40 50 60
SPECIMEN THICKNESS B. (mrn)
Figure 6.6. Critical values of J versus specimen thickness.
Fatigue
Ductile tearing
Fatigue
Figure 6.7. Fracture surfaces showing extent of ductile tearing during J-tests.
0 A 0 101
102
• 103 D
264 •
100
o- B = 12 mm - A- B=25mm -. a- B=50 mm
A A
0
0 0
A A A 0 ❑ 0 0 o o
AA A
A A o'0 0 A 0
A o ō 0
0
1 ī~0 3 1 i 1 10 A J (N/m)
I 1 11 10̀ '
1 1 1 F I
105
• Figure 6.8. Elastic-plastic fatigue crack growth data; R = 0.08.
a
laO j •
0
.265
E
0 4
a u V ° fa
E A
■ e
10 1OI t O•
0 0
b ❑ 0 9 A I O I- °
0 °d0 .
t9 °
4 go
CYCLIC MONOTONIC ° 0
ū 10 A 00 0 B=12mm • O 0 0 A 8=24mm A
°° 00 p 0 B= 50mm ❑ 0
-3 I0 10 105 106
°J , N/m, J N /m
Figure 6.9. Cyclic and monotonic crack growth at high stress intensities.
cQ $0
084 °Cb
A. c
266
w<r« planar dimensions
REGION 1 - AN ELASTIC FIELD SURROUNDING TEE CRACK TIP
REGION 2 - AN ELASTIC-PLASTIC FIELD SURROUNDING THE CRACC TIP
REGION 3 - AN IPIENSE ZONE OF DEFORMATION ■
Figure 6.10. Crack tip stress and strain fields (schematic).
CRACK TIP CLOSED AT ZERO LOAD
CRACK TIP OPENS BUT STILL SEARP:
.APPLIED LOAD BALANCING CLOSURE STRESS.
CRACK TIP FULLY BLUNTED BUT NO CRACK GROWTH
increasing loan
2 rc
• CRACK ADVANCES BY AA WHEREVER TEE CRACK TIP
STRAINS REACH TEE FRACTURE cf .
)0000 DIMPLES FORM =FRONT OF THE CRACK TIP
AND GROW
DL`PLES COALESCE AND CRACK EXTENDS
CIO BY A FURTHER daft DUE TO DLYPLE
FORMATION AND DUCTILE TEARING
Lay
j LadtH Total crack growth = Lay +Ladt
Figure 6.11. Crack tip deformations and crack growth under increasing load.
MU
A
101
267
• - B-12mm
A- B =24mm
■ - B=50mm
•
0=
./A Y .
/11 ' .:7:.
ba~ 7.5x1018JR•60
/ /
/
I i I 1 1 1 1 I i 1 I I I [ I 1 I
JR(N/m) . 105
106
3 10104 10
0- - / A. /V
i /
C 0 .I C W X W
Y u 0 c- v
_2
Figure 6.12. Crack growth by monotonic loading.
o— B:12mm A— B- 25mm o— B=50mm
A A
A 0
0 ~0 0
O O
o0
A
■Is•
. • / . • ,.
•• ', /
0, / . / ~~ /
1 1•
• A / / 5. 0
•
Q • ///
O O
AJ (N/m)
268
100_
Crack - tip blunting crack growth, equ (6,9)
Monotonic crack growth ❑ equ (6,13)
Double mechanism crack growth , equ (6,12) AA
A A
A ~0
A
A
102
103
104 3 10
w
CU
E E
d
Figure 6.13. Modelling of elastic-plastic crack growth in BS4360-50D steel.
I 1 [ I Il r4 t
10
o' 8
10 fi i 0Q4 " i 101 102
3 Jeff(N/mi
11 0
o -B= 12 mm-B=24mm
a- B =50mm
1Ō3
Equation(6.12)
0.1
10]
1Ō=
Figure 6.14. Experimental and predicted da/dN versus
Jeff data: BS4360-50D steel.
10E
102
• 269
I I I
LINEAR ELAS lC ELASTIC- P LAST( G
EQUATION( 6.9)
_co 10
l06 10
103 104 105
1ō
270
- a 3, K2/E (Ki r")
Figure 6.15. Experimental and predicted da/dN versus AJ data: BS15 steel.
271 •
CHAPTER SEVEN
LOAD HISTORY EFFECTS DURING TWO-LEVEL BLOCK
LOADING IN BS 4360-50D STEEL
272
7.1 INTRODUCTION
The importance of retardation in the rate of fatigue crack growth,
produced by load interaction in variable-amplitude loading, on the
accurate prediction of fatigue lives of engineering structures has
been recognized for some time [163,164,170-175]. However, relatively
less attention has been focussed on the load sequences which result in
crack acceleration [163,164]. Crack growth acceleration occurs within
a small crack growth region following the load sequence change.
Therefore this phenomenon is not easily detectable from macroscopic
crack growth observations. More accurate measurements of the crack
increment and fractographic analysis, however, do reveal locally acceler-
ated crack growth.
During a complex load sequence, it is difficult to observe the
local interaction effects separately. The overall effect, however, can
easily be deduced from macroscopic measurements of the crack growth.
These observations and others obtained from fractographic techniques
can reveal the nature of interactions during a complex load sequence.
Generally, crack growth rates derived from crack growth measurements
are found to be significantly lower than those predicted from constant
amplitude cumulative damage approach. Apparently, retardation effects
dominate over the acceleration effects in complex load sequences.
However, the effects of crack acceleration on the fatigue life
of a structure cannot be completely overlooked. In certain situations
the crack retardation effects may be considerably reduced during the
load sequence. This may be the case when large compressive loads are
frequently occurring in the load sequence. These large compressive
loads have been found to reduce or even eliminate the retardation effects
if they are applied immediately after overloads or blocks of overloads
[163]. In such cases then crack acceleration effects would be significant.
273
Crack growth retardation and acceleration effects have been
explained in terms of mechanisms occurring at the crack tip (163-178].
These mechanisms have been reviewed in detail in section 2.10.2.
In the present study the use of an A.C. crack microgauge for
measurement of crack retardation and acceleration during simple two-
level block loading was investigated. A description of crack retardation
and acceleration in terms of crack tip deformation is offered to explain
the present results.
7.2 EXPERIMENTAL PROCEDURE
The tests for this programme were carried out on the 24 mm thick
compact specimens prepared as discussed in Chapter Three. All the tests
were performed on the Dowty fatigue machine.
All the tests were performed at constant AK for the individual load
blocks. To maintain a constant AK and crack growth rates during the
test, the load limits had to be manually reduced as the crack extended.
The load limits were adjusted after every 0.25 mm of crack increment.
Typical reductions in load limits at two consecutive crack length
readings, to keep AK constant, amounted to no more than 2% of the total
load range. The tests were also performed at crack lengths in the
range 0.3 < a/W < 0.60. In this range the compliance,change with crack
length increase was not so high as to affect the AK values between the
load reductions. A computer print-out listing values of crack length
(at intervals of 0.25 mm) against the corresponding values of load
limits, for required AK values, was prepared and this was used for the
load adjustments.
The cyclic frequency was chosen for each particular block of loading
such that the velocity of crack extension (da/dt) was maintained constant
at about 1 x 10-3 mm/sec. At this slow velocity, it was convenient to
274
adjust the load limits after every four to five minutes. The constant
crack velocity was also chosen to.give similar environmental conditions
at the crack tip during the tests.
To change from one block to another, the machine had to be stopped
(at a mean load). The mean load was adjusted to the required value,
cycling was begun and then the amplitude adjusted. This was done fast
and carefully so that the transient effects could be recorded. Crack
length measurements were taken at very short intervals immediately after
load change. The test was then run until constant growth rates were
attained.
To measure the crack length both the A.C. crack microgauge and
a travelling microscope were, used. The current terminals for the
crack microgauge were fixed symmetrically onto the front face of the
specimen, a distance 20 mm apart. The probe terminals were spot-
welded onto the face of one side of the specimen, 10 mm apart, equi-
distant from the crack line and at a distance 25 mm from the loading
line. Details of the specimen set-up were discussed in section 3.3.1
and are illustrated in Figure 3.5. The probe output voltage was
connected through the crack microgauge to a chart recorder. Use of
the time base of the recorder enables a continuous record of crack
length as the test proceeds. The whole apparatus, showing the crack
microgauge and the chart recorder, is shown in Figure 7.1. During all
the tests the microgauge was used with a gain setting at 2-4. The
chart recorder was set at 200 mV and a 10 min/cm chart speed was
used. These settings were found to give the best results within the
accuracy of crack length measurement required for the present tests.
Before the block-loading tests were performed, both the crack
microgauge and the chart recorder were calibrated. A single specimen
was used for this purpose. The specimen, with the original notch length,
was set in the machine. Readings of both the microgauge meter and the
•
275
chart recorder were taken. A fatigue crack was then initiated and
propagated, at a AK value of about 16 MN/m3/2 and R = 0.1, to length
of about 3 mm from the notch. Several readings of crack length,
using the microscope, the microgauge meter and chart recorder were
taken as the crack propagated over another 3 mm. The specimen was
then removed from the machine and the crack extended further using a
fine saw cut. It was then put back to the machine and readings of
the microscope, the microgauge meter and the chart recorder taken.
A fatigue crack was again initiated and propagated taking the respective
readings as was done previously. This process of alternating a saw cut
with a fatigue crack was continued until a total crack length of about
50 mm was reached. This value of crack length corresponded to the
maximum readings on both the microgauge and the chart recorder for the
chosen settings. The results of the above calibration are shown in
Figures 7.2 and 7.3 for the microgauge and the chart recorder
respectively. Additional points were added on Figure 7.2 during the
fatigue tests to check on the repeatability of the crack microgauge
readings. As seen from the above two figures the crack length could
be measured to about 0.02 mm and 0.08 mm using the microgauge and the
chart recorder respectively.
When the block-loading tests were started it was found that the
microgauge readings (and thus the readings on the chart recorder) were
affected by the load changes. It was therefore decided to investigate
the effect of crack tip stress on the microgauge meter reading. A
specimen was fatigued at different stress ratios and after each period
of cycling a record of the microgauge readings at increasing load was
made. A graph of digital voltage change, AVD, against applied K
(in MN/m3/2) was drawn. This is shown in Figure 7.4. The linear
portions of these curves were found to have slopes varying between
-1.05 and -1.44 with an average value of -1.31. It was also observed
276
that the dynamic reading (during cycling) of the microgauge was very
close to that obtained at the mean load of the fatigue cycle. It was
then concluded that the change in the microgauge reading during load
changes resulted from a corresponding mean load change. Further
examination of the load change effects revealed that the change in
microgauge reading was abrupt when following a mean load change during
fatigue. The subsequent variation of the microgauge reading and the
crack length was again linear-and could accurately be predicted using
the calibration curve. However, when the load changes were very large,
such as to cause large plastic deformation at the crack tip, which
resulted in significant changes in crack direction, the subsequent
microgauge readings deviated from the calibration curve. It was there-
fore decided to limit the load changes to moderate values (i.e.
Kmax
< 60 MN/m3/2).
To investigate crack growth acceleration and retardation during
block loading tests were performed using three types of loading
histories, namely: constant amplitude loading with an increase in
Kmax at R = 0.0; with an increase in mean stress at constant
AK = 15 MN/m3/2; and with a constant Kmax of 15 MN/m3/2 and a decrease
in Kmin
from zero to -30 MN/m3/2. In these latter tests the crack
lengths could not be monitored continuously using the chart recorder
when the test was still running (i.e. during cycling). This was
because the contact between the fracture surfaces, during the compressive
part of the cycle, produced erratic readings on the microgauge meter.
Therefore, the test had to be stopped in order to take the crack length
readings. These readings were taken at a static load corresponding to
about 7.5 MN/m3/2 (to avbid crack closure effects).
The above load histories were chosen in order to investigate the
crack growth interaction effects produced by changes in Kmax, AK and
also in K involving large compressive load cycling. min
r
277
The crack growth rates were calculated using the secant method
as discussed earlier in chapter three. The stress intensity factor,
K, was calculated using the standard expressions for the CS geometry
(i.e. equations (3.6) and (3.8)).
7.3 THE RESULTS
Figures 7.5 to 7.8 show experimental results obtained using both
the A.C. microgauge and the microscope for crack length measurement.
Figure 7.5 shows the Aa versus AN data for Hi-low block loading at
R = 0 where Kmax was reduced from 20 MN/m3/2 and 30 MN/m3/2 to
15 MN/m3/2 and 25 MN/m3/2 respectively. These results show that
(especially for the block loading with the lower Kmax2
value) the A.C.
microgauge records a higher value of crack length compared to the
microscope surface measurement. It is also observed that the delay
period deduced from the microscope crack length measurements is slightly
longer than that deduced from the microgauge measurements. The above
observation is illustrated further in Figure 7.6 where the same results
are plotted in terms of da/dN versus La. In this figure it can be seen
that during the delay period higher growth rates are recorded using the
microgauge technique. However, after the crack has advanced through
the region affected by the retardation, the growth rates deduced from
both crack length measurement techniques are similar.
Figure 7.7 shows to versus AN results for the Lo-Hi block loading
at R = 0 where Kmax
was increased from 15 MN/m3/2 to 20 MN/m3/2 and
also from 15 MN/m3/2 to 30 MN/m3/2. These results show that the
microscope records higher values of crack length than the microgauge.
In Figure 7.8 the same data is plotted in terms of da/dN versus 1a.
This figure shows that while the growth rates recorded using the
microscope decrease at a constant rate to a constant value, those
•
278
recorded using the microgauge first increase and then decrease to the
same constant value. The difference in growth rates is confined to a
short crack increment of the order of 0.2 mm.
Figures 7.9 and 7.10 show the Hi-Lo data plotted in terms of
da/dN versus Aa for results calculated from crack lengths measured
using the microgauge. Figure, 7.9 shows the data where Kmax was reduced
from 30 MN/m3/2 to 25 MN/m3/2, from 30 MN/m3/2 to 20 MN/m3/2 and from
30 MN/m3/2 to 15 MN/m3/2. In these tests the delay periods were
25,000 cycles and 160,000 cycles for the 30-25 MN/m3/2
and 30-20 MN/m3/2
block loading, respectively, while no crack growth was recorded after
1,500,000 cycles for the 30-15 MN/m3/2 block loading. Figure 7.10
shows da/dN versus Aa data where Kmax
was reduced from 60 MN/m3/2,
45 MN/m3/2, 30 MN/m3/2 and 20 MN/m3/2 to 45 MN/m3f2, 30 MN/m3/2,
25 MN/m3/2 and 15 MN/m3/2 respectively. Figures 7.9 and 7.10 show that
immediately after reduction in Kmax
the growth rates decrease to a
minimum value and then increase to a constant value after the crack
has advanced across the affected crack length increment. The crack
increment corresponding to the minimum growth rates, DaDM, is found to
increase with increasing Kmaxl but seems independent of Kmax2. AaDT
is also found to increase with Kmaxi'
Figures 7.11 and 7.12 show the data obtained in the Lo-Hi load
sequence at R = 0. In Figure 7.11 are plotted the results obtained
where Kmax
was increased from a value of 15 MN/m3/2 to higher values of
20 MN/m3/2, 30 MN/m3/2 and 60 MN/m3/2 in different tests. In Figure 7.12
are also plotted the results where max was increased from 25 MN/m3/2 and 30 MN/m3/2 to 30 MN/m3/2 and 45 MN/m3/2 respectively. In both
figures the growth rates are found to increase first over a short crack
length increment of up to 0.3 mm and then to decrease at a constant
rate to a constant value for the particular AK2. The increment of
crack length, AaA, over which acceleration is observed to occur is found
•
279
to vary between 0.65 mm and 1.35 mm. The values of aaA do not appear
to depend on either K or K maxi max2'
In Figure 7.13 are plotted data obtained with an increase of mean
stress at a constant AK of 15 MN/m3/2. The results do not show a
decrease in growth rates immediately after the load change. No growth
was observed following the load change until after a certain number of
cycles. For the 30-20 MN/m3/2 and 45-30 MN/m3/2 block loading the
periods during which no growth was observed were 70,000 cycles and
25,000 cycles respectively. The affected crack length increments,
"DT' a found to be 1.40 mm and 1.90 mm respectively.
Figure 7.14 shows crack acceleration results obtained by changing
the mean stress at a constant AK of 15 MN/m3/2. Although the crack
lengths measured on the surface using a microscope showed an immediate
crack acceleration following the load change, the microgauge readings
showed a delayed acceleration. The affect crack length increment, AaA,
is observed to vary between 0.85 mm and 1.15 mm for the three changes
in mean stress. Again there is no simple dependence of taA on either
maxi or max2'
In Figure 7.15 are shown the results obtained at a constant Kmax
of 15 MN/m3/2 and with a decrease in Kin from zero to -30MN/m3/2
(i.e. R = -2). The results show higher growth rates at R = -2 than
at R = 0. However, no transient effects were observed immediately
following the change in Kmin
from zero to -30MN/m3/2 or from -30 MN/m3/2
to zero. In both cases constant growth rates were obtained immediately
after the load changes.
Figure 7.16 shows the accelerated growth rates obtained in the
Lo-Hi load sequence tests plotted together with the constant amplitude
growth rates. The Lo-Hi growth rates immediately after load change are
higher than those under constant amplitude loading by a factor of two.
The upper bound for the accelerated growth rates can be approximately
•
280
described by
da dN =
7.48 x 10-9 (AK)3.24
where da/dN is in mm/cycle and K is in MN/m3/2.
(7.2)
7.4 DISCUSSION OF RESULTS
The calibration results for both the crack microgauge and the
chart recorder showed a linear variation of the crack length, a, with
the digital voltage, VD. The readings obtained from a saw cut and
from a fatigue crack were found consistent after they were corrected for
the saw cut width and the crack front curvature respectively. It should
be noted that to obtain such consistent results the probe terminals
had to be spot-welded accurately at identical positions. Some •
specimens which did not meet this condition were found to give incon-
sistent results and the probe terminals had to be removed and re-welded
accurately. The chart recorder could be read to the nearest 0.5 mm
and this meant a crack length increment of about 0.08 mm. Therefore
the chart recorder could not detect such small changes of crack length
as the microscope or the microgauge meter reading. For observation of
small crack length increments immediately after the load changes during
the block loading tests the readings of the microgauge meter were taken
directly and used for crack length determination. The chart recorder
readings were used for crack length evaluation during initiation and
propagation at constant growth rates. However, in some cases transient
effects were also deduced from the chart record but only at crack
increments of more than 0.25 mm.
The use of the A.C. microgauge with the chart recorder was found to
produce best results during constant amplitude loading where small
increments in crack length were not required, The continuous recording
•
281
of the crack length as the test proceeds is an added advantage in
that one does not have to sit and take readings every few minutes.
It is recommended that the A.C. microgauge should be connected directly
to a control-processing unit (e.g. a microprocessor) which controls the
load signal and processes the data directly. Such a system would
considerably reduce the testing time and also make the fatigue tests
less tiring to the operator.
The effect of crack-tip stress on the crack microgauge reading was
investigated during the present study. It was shown that (Figure 7.4)
the microgauge reading should be interpreted with care when investigating
transient effects. The changes in the microgauge reading with changes
in crack-tip stress could be wrongly interpreted as corresponding to
. an increase or a decrease in crack length. The present results have
shown that an increase in K of 1 MN/m3/2 produced a corresponding
decrease in the microgauge reading of 1.31 DVolts. This meant that
changing Kmean
from 7.5 MN/m3/2 to 15 MN/m3/2 (as during the Lo-Hi
sequence when Kmax was changed from 15 MN/m3/2 to 30 MN/m3/2) could
produce an apparent 'decrease' in crack length of the order of 0.2 mm.
However, it was further observed that the change in the microgauge
reading occurred simultaneously as the crack-tip stress changes.
Therefore, it was assumed that the change in crack-tip stress did not
affect the subsequent microgauge reading as the crack increased in
length. In addition to the stress effects other factors, such as
crack closure and crack tip geometry changes may influence the micro-
gauge readings. For the present investigation such effects are difficult
to quantify and were neglected.
The procedure adopted in the present tests for keeping AK constant,
by reducing the load limits every 0.25 mm of crack increment, was found
to give satisfactory results for crack lengths in the range
282
0.3 < a/W < 0.6. However, for much longer crack lengths (i.e. a/W > 0.6)
higher growth rates were found than those obtained for shorter crack
lengths under the same AK values. This feature is illustrated in
Figure 7.17 where growth rates obtained at crack length of 31.14 mm,
34.69 mm and 62.05 mm using a constant AK of 30 MN/m3/2 are compared.
This effect of crack length could be partly associated with the fast
change in specimen compliance (and thus K) with crack length increase
occurring at long crack lengths for the CS geometry. This change in
compliance may cause significant increases in AK between load adjust-
ments, resulting in increased growth rates. Other factors such as
crack closure effects, which may vary with crack length, could also
explain the above observed effect of crack length on the growth rates.
To overcome this difficulty the present tests were performed at crack
lengths where the effect was small, i.e. a/W < 0.6.
The present results (Figures 7.5 to 7..8) have shown that the values
of crack length measured using the microgauge were slightly different
from those measured on the specimen surface. During the Hi-Lo sequence
the microgauge measured larger values of crack length than the micro-
scope. This behaviour could be explained in terms of crack-tip changes.
When the stress is reduced from a high to a low value the crack closes
due to the residual stresses. The crack closure is more significant
on the specimen surface because of the shear lips formation. Therefore
the crack may grow at a faster rate in the interior of the specimen,
where crack closure effects are less significant, than on the surface.
Since the a.c. microgauge measures the average crack length across the
whole crack front (see chapter 3) larger crack increments are recorded
using this technique than when using the surface measurement. Examination
of the fracture surfaces showed that there was more crack front tunnelling
during the Hi-Lo sequence than in the Lo-Hi sequence except in the
latter sequence, when high stress intensities (causing static tearing)
were involved.
•
283
During the Lo-Hi sequence the crack measurement on the surface
produced larger values of crack length increment than those obtained
using the microgauge. This behaviour also can be explained using crack-
tip geometry changes. At the higher Kmax value the crack front tends to
straighten out. Thus more growth takes place near the surface of the
specimen than in the specimen interior, resulting in higher observed
values of crack length on the surface than the average crack length
increase measured by the a.c. microgauge.
On the basis of the above observations it seems that the crack
microgauge would be the best technique for monitoring crack length
during variable amplitude loading. The main advantage with this
instrument is that it measures the average crack length taking into
account the overall changes occurring at the crack tip. Crack length
measurements on the surface may give misleading results since this
method does not consider the crack front profile changes which have
been observed to occur during block loading.
Several explanations of the fatigue crack retardation and acceler-
ation phenomena have been proposed but the most often discussed theories
in this regard are: (i) crack tip blunting [169,170], (ii) crack front
incompatibility [171,172], (iii) residual stresses at the crack tip
[163,166,173-177] and (iv) crack closure [66,163,178]. At present,
no single mechanism appears to be capable of satisfactorily explaining
all the load interaction effects observed in variable amplitude loading.
In reality, the load interactions effects are more likely caused by a
combination of the above mechanisms.
The recently proposed mechanism [163,184] based on the overall
deformations caused by the propagating crack seems to offer the best
explanation of the load interaction effects. It is argued that sequence
effects mainly occur due to interaction of the plastic zones of the
284
respective loads in the subsequent load cycles. Of predominant
importance in this case are the Kmax -controlled plastic zones. The
residual deformations build up within the area of the KmaX controlled
plastic zones hinder the formation of a stable condition corresponding
to constant-amplitude loading. It is observed that crack closure,
cyclic strain hardening or softening of the material at the crack tip
and along the plastic zone, blunting of the crack tip, incompatible
crack front, etc., all become simultaneously active. The above
'combined' mechanism will be used to explain the interaction effects
observed in the present tests.
7.4.1 Crack growth retardation behaviour in Hi-Lo block loading:
To explain the crack retardation behaviour observed in the Hi-Lo
block loading an attempt will be made to rationalise the crack tip
deformations in terms of the cyclic and monotonic plastic zones and
crack front changes. At the end of the high loading period, plastic
zones due to Kmaxl
and AK1 are present. The deformations along the
path of monotonic plastic zone are in an equilibrium condition corres-
ponding to the high load level. When Knax is reduced to the lower level,
the crack growth rates fall below those under constant amplitude Km2,
loading because the residual deformations generated in the previous
loading period reduce the crack tip stress intensity factor range. The
delayed decrease in da/dN following the load change could be associated
with the incompatibility of the crack front changes as well as with
the changes in the cyclic plastic zone at the crack tip where the crack
growth process takes place. This proposition seems to be supported by
the results in Figure 7.13 where AK1 is small. In these results the
delay occurred immediately after load change and AaDM 0. The values
of AaDM were plotted against the corresponding values of al on a
•
285
log-log basis in Figure 7.18. A linear relationship of the form
AaDM = 3.34 x 10-4(AK1)
2.12 (7.3)
seems to exist, where AaDM
is in mm and 4K1 in MN/m3/2. In the same
figure the curve for the cyclic plastic zone sizes predicted using
equation (4.1) (i.e. ryc = 3~(OK1/2ayc)2) is plotted. It is found that
AaDM is about 2.8 times larger than the cyclic plastic zone size of
AK1. However, it is not possible at the present moment to explain
quantitatively, in terms of crack tip deformations, the simple relation-
ship obtained between ta.M and ryc.
When the crack tip has passed through the point of minimum growth
rates, the growth rates increase rapidly until they become constant after
the crack length has increased by AaDT. Ideally the value of AaDT
should equal the difference between the maximum plastic zone sizes
immediately prior to and following the load reduction [175] i.e.
K 2 - K 2 _ 1 ( maxi max2)
(7.4) aDT 37 2
ay
This observation infers that the retardation affected crack length
increment is equal to the amount of crack growth required to extend the
new maximum zone corresponding to Kmax2' into material not previously
plastically strained by the higher loading conditions. However, in
reality aaDT
is often found much larger than the value predicted using
equation (7.4). This is because the influence of the residual stresses
due to the Kmaxl
plastic zone may persist even when the crack has
physically passed through the plastic zone. Crack closure effects due
to these stresses, reduce the effective stress intensity factor range.
Plotting the values of naDT
against (Kmaxl - Kmax2
) did not indicate a
simple relationship. However, when AaDT
was plotted against Kmax1
(Figure 7.19) a simple relationship was found to exist:
286
= 4.76 x 10-2 AaDT Kmaxl (7.5)
where AaDT is in mm and maxi
in (MN/m3/2
). In the same figure is
plotted the values of the monotonic plastic zone size predicted using
equation (4.1). It is seen that only at high values of Kmaxl
does the
value of AaDT approach the plastic zone size created byKmaxl'
At
low Kmaxl values, LaDT values are much larger than the Kmaxl plastic
zone sizes.
Another interesting result obtained by comparing craDM
and AaDT
values is that the difference between the two values (i.e. taDT AaDM)
is almost a constant (see Table 7.1(a)) for the tests at R = 0. This
constant was found to average about 0.92 mm. It is not possible to
explain the significance of this phenomenon which was not observed in
the Hi-Lo block loading tests at a constant OK of 15 MN/m3/2.
7.4.2 Crack acceleration behaviour in Lo-Hi block loading:
The crack growth results obtained during the Lo-Hi block loading
tests showed that crack growth rates increased above the constant
amplitude loading rates immediately after the load change. This increase
in growth rates is explained as follows:
At the end of the low mail constant amplitude cyclic loading, the
plastic zones due to maxi and AK1 exist and these control the growth
rates. Residual deformations are in equilibrium and within the cyclic
plastic zone a stable strain hardening profile corresponding to the low
mean stress loading level exists. During the first high load cycle the
material at the very tip of the crack is subjected to high tensile
strains. Since this material is already cyclic hardened from the preceding
loading period, its deformation capacity is limited. Its brittle
characteristics increase the crack growth rates. Furthermore, a
•
287
considerable amount of crack tip blunting occurs at the instant when
the applied load is stepped up; this blunting precludes the possibility
of crack closure behind the blunted crack in the subsequent load
cycling. Thus immediately after the load step-up has taken place,
crack closure occurs only in the small area adjacent to the current
crack tip. Consequently, the total residual stresses acting on the
closed surfaces is also small immediately after the load step up. This
behaviour has been observed by Nakagaki and Atluri [248] during their
recent elastic-plastic finite element analysis of crack closure effects
under spectrum loading.
The load step up may also cause the shape of the crack front to
change in order to become compatible with the new loading conditions.
This may take the form of tunnelling if the maximum applied load (i.e.
Kmax
) is sufficient to cause static tearing at the crack tip. This
behaviour would result in more crack growth at the interior of the
specimen than on the surface. However, at moderate Kmax values, the
crack front, which is usually more curved at low stress intensities,
may begin to straighten out, resulting in more crack growth near the
specimen surface than in the interior. The observed behaviour due to
the crack front changes will therefore depend on the method of crack
length measurement. The latter crack front changes (i.e. crack front
straightening out) were found to occur during the present tests.
As the crack tip rapidly enters material regions which are less
cyclic strain hardened and also because of the fact that higher tensile
deformations develop in the large Kmax2 plastic zone, which reduce the
effective stress intensity factor range, the growth rates decrease.
This decrease in growth rates proceeds until new stable plastic zones
at the crack tip are built up and growth rates then stabilise to the
constant amplitude loading level.
288
Therefore, cyclic strain hardened crack tip material, crack
closure and crack tip incompatibility are reasons for the crack
acceleration in Lo-Hi block loading sequence. To quantify the contri-
bution of each of these mechanisms to crack acceleration would be
a difficult task, however, the overall effect could be derived from
experimental crack tip strain measurements and fractographic analysis.
This exercise was recently performed by Nowack et al. [184] whose results
seem to confirm the abode explanation.
The present results show that the crack length increment affected
by acceleration, AaA, varies between 0.65 mm to 1.35 mm. These limits
are consistent with those reported in the literature for other materials
[163,164]. However, it is worth citing the results by Scott and
Silvester [110] who observed crack growth acceleration over crack
increments of 5 mm to 7 mm during their crack growth tests in sea water
at different electrochemical potentials. Similar tests in air showed
only a slight load history effect.
The values of AaA obtained in the present Lo-Hi tests are tabulated
in Table 7.1(b). They are also plotted in Figure 7.20 against
Kmax2/Kmaxl. This figure shows that LaA has a maximum at K
max2/Kmaxl
value of about 1.3 and decreases almost linearly with increase in
Kmax2/Kmaxl. AaA can be estimated using the expression
LaA = 1.47 - 0.22 max2/Kmaxl (7.6)
for Kmax2/Kmaxl > 1.3. For Kmax2/ maxl < 1.3 the variation of iaA is
not clear from the present data.
0 289
7.4.3 Modelling Crack Growth Retardation and Acceleration
An estimation of crack growth retardation and acceleration within
the affected crack length increment may be obtained by employing an
effective stress intensity concept obtained by comparing the measured
crack growth rates with those obtained under constant amplitude
conditions. This concept is similar to that proposed by Wheeler [173]
(equation (2.51)). However, since the present results do not fit the
Wheeler model in terms of the plastic zone sizes an empirical effective
stress intensity factor will be defined. Starting from the crack
growth rate expression for constant amplitude loading as the reference
growth rates, i.e.
da dN
CR(AK) ( 7 .7)
(where CR and mR are determined from constant amplitude loading),
instantaneous values of effective stress intensity factor can be defined
as
(AKeff)ins da
1/mR
dN)ins / CR] (7.8)
where (da/dN). are the instantaneous values of crack growth rates in ins
the affected crack length region. The instantaneous growth rates can
be expressed as a function of the experimental variables, i.e.
(da) = F(K ,K ,AK ,AK ,Aa,A ,da ,Da etc.) (7.9)
ins ns maxi max2 1 2 aDM DT A
Using regression analysis based on empirical trends in the experimental
data equation (7.9) can be used for both acceleration and retardation,
including the condition for crack arrest in the latter case.
The present results from the Hi-Lo block loading tests indicate
that (da/dN).
is a function of most of the variables in equation (7.9)
• and therefore this expression would need computational methods to solve.
290
This exercise was not possible to perform for the present results due
to the time factor. However, the crack acceleration results seem to
indicate that a simplified relationship can be arrived at by making
certain approximations.
Using the da/dN versus a data in Figures 7.11 and 7.12 And also
the data in Figure 7.16, values of (AKeff)ins were estimated at various
values of Aa. The ratio Aa/AaA was plotted against AK2/(AKeff)ins'
An empirical relationship of the form
4.5AaA- (AKeff )ins 3.6AaA ♦ Aa .AK2 (7.10)
is suggested for Aa < AaA. Therefore the instantaneous crack growth
rates after load change in Lo-Hi block loading can be approximated by
using the expression
da 4.5AaA mRTIQR
(dN)acc CR(3.6AaA ♦ Aa) (~K2) (7.11)
for Aa < AaA' CR = 3.13 x 10-9
, mR = 3.24 and AaA is given by
expression (7.6) (da/dN in mm/cycle, Aa in mm and AK2 in MN/m3/2).
7.5 CONCLUSIONS
The use of an a.c. crack microgauge to measure crack length during
two-level block loading fatigue crack growth in BS4360-50D steel was
investigated. Crack growth retardation and acceleration were studied
using linear elastic fracture mechanics. The following conclusions
were drawn:
1. The A.C. crack microgauge can be used to measure crack length
during ,two level block loading. Although the microgauge meter
reading is affected by load changes during block loading these
291
effects occur at the moment the load is changed and do not
significantly affect the crack length measurements during subsequent
cycling. The a.c. crack microgauge measures the average crack
length taking into account the crack front configurations.
2. Use of surface crack length measurement techniques during block
loading may sometimes produce misleading results. This is
especially the case when crack front changes occur (e.g. tunnelling,
'straightening', etc.) following load changes.
The a.c. crack microgauge can be used, with the help of a suitable
chart recorder, to continuously monitor the crack length during
a fatigue test.
4. During Hi-Lo block-loading tests crack growth retardation, and
even complete arrest of the crack, is observed to occur in
BS4360-50D steel. The crack growth rates first decrease to a
minimum value and then increase to the constant amplitude level
as the crack propagates through the retardation-affected crack
length increment, DaDT. The retarded crack growth rates are
dependent on both the previous and the current loading conditions.
5. The value of crack length increment at which minimum growth rates
occur (i.e. AaDM) is found to be related to a1 using the
expression
paDM = 3.34 x 10-4(al)2.12
where AaDM
is in mm and AK1 in MN/m3/2. This expression suggests
that daDM
is about 2.8 times larger than the AKl cyclic plastic
zone under plane strain conditions.
292
6. The total crack length increment affected by the crack growth
retardation (aaDT) is found to be unrelated to the monotonic
plastic zone sizes of maxi
and Kmax2.
Values of AaDT are found
to increase linearly with Kmaxi
, i.e.
AaDT = 4.76 x 10-2 Kmaxl
where iaDT
is in mm and Kmaxl
in MN/m3/2. The above variation
of~aDT with
maxi cannot, at present, by explained.
7. The crack length increment affected by crack growth acceleration,
AaA, is found to vary between 0.65 and 1.35 mm for the Lo-Hi
tests performed. For max2/Kmaxl > 1.
3 approximate values of AaA
can be obtained from the expression
AaA = 1.47 - 0.22 Kmax2/Kmaxl
8. Crack growth retardation and acceleration can be explained in
terms of crack closure, strain hardening of crack tip material,
crack tip blunting, as well as crack front incompatibility. The
effects of these mechanisms do not seem additive in nature;
interaction between them seems to occur.
9. Crack growth retardation and acceleration can be modelled using
an empirically derived effective stress intensity factor range,
AKeff. This AKeff can be derived by comparing the instantaneous
crack growth rates during load interactions with those under
constant amplitude loading. To evaluate AKeff
for the crack growth
retardation results numerical methods would be required. For the
crack acceleration results AKeff
is given, approximately, by
293
4.5AaA
~Keff 3.60aA + da . AK2
for to < AaA. Therefore accelerated growth rates can be obtained
using the expression
da _9 4.5~aA 3.24 3.24
(dN
)acc
= 3.13 x 10 ( 3.6AaA+oa)
)3.24( AK
where AaA is given by the expression in conclusion 7; da/dN is in
mm/cycle, Ea and AaA in mm and AK2 in MN/m3/2.
•
294
TABLE 7.1
Crack length increments affected by load interaction effects
(a)
Kmaxl
Values of AaDM and AaDT
Kmax2 AK1
AK2 AaDM AaDT ~aDT AaDM
60 45 60 45 2.1 3.0 0.9
45 30 45 30 1.0 2.1 1.1
30 20 30 20 0.4 1.3 0.9
20 15 20 15 0.20 1.0 0.8
30 25 30 25 0.35 1.25 0.9
45 30 15 15 =0 1.9 1.9
30 20 15 15 =0 1.4 1.4
(b) Values of AaA
Kmaxl Kmax2 4K1
AK2 AaA
15 20 15 20 0.65
15 30 15 30 0.96
15 30 15 30 1.05
15 60 15 60 0.65
25 30 25 30 1.15
30 45 30 45 1.35
15 45 15 15 0.80
15 30 15 15 1.03
30 45 15 15 1.15
•
• I •
N VD
Figure 7.1. Apparatus used for the block-loading testing programme.
2000
Taco
PROBE -rM W.ALS
SPECI).4E 4
25 rnrn
A sow cu - crz.0 o,
4 aRowtWq F-1CoE O CRACK
SPECtMEr NOTCH iposmoN
1 1000
:to
I I I 20 4o
C.PAC$< LENGTH) rrwil
• 296
Figure 7.2. Calibration curve for the crack microgauge.
A ►W CUT CRACK
A GcovtNC4 FATtGuE C.RACk,
/ / I
O le4 +oner, % r.,".3
3D
297 •
•
GRAC.K LEtJC(T1•{ l cx, rr►m
Figure 7.3. Calibration curve for the chart recorder.
r
X-.•"....X.".""......-X R= O•O
0 R=o• o
R= CI- 1
R=o.o I
5 10 45 20 25 30
APPLIED K (MN /m3/2)
298
AVERAC.IE d d K o) - -1. 31 DNOL•rg/ MN 1 m3/2
LINEAR PoRT1ONS
Figure 7.4. The effect of stress on the digital voltage
reading of the crack microgauge.
r
I meuci
kMin1 ir 14min2' O TIME
n 2 _ MNJm512 K
1 0
2 0 Kr„ 2 Z5MNIrn512
I ;1 0/ J 1 / ill
_/ Y O AC MI GROG giJC4E
A MICROSCOPE_ GORF,ACE
40
ry0~ A w
• •
go 100 12o 44o 4 0 48o
AFTER LOAD . CHANCtE, dN )00 3 OF CYCLES
Figure 7.5. Aa versus AN; crack length measurements using both a microscope and the a.c. microgauge in Hi-Lo block loading.
Kranz, 15 mph. -o-11— - — --6—• —o.
0 A.G. MtGROC A UC4E
• 300
K flGC 2
Kmiri Kmin2` O "TIME
10
L1 MIGQOSCOPE– SURAE
1012̀ -05 I 1 1 1 I I .
0 40 20 30 CRACK 1NCREMEWT, Ao-ICnI+m)
Figure 7.6. da/dN versus Aa; retarded growth rates detected using both microscope and the a.c. microgauge.
k max 2= 2OMNIn3/2
0 AC WNCROC4At C.E.
A MICRp5C0PE- SURFACE ■
O/ I
—10 -tSMNIr~x K,nnxt
I 1 I 1 ____II 10 20 30 40
N°' OF CKCLES AFTER LOAL) CHANGE, LN, x 1O 5
K1.-K mox 2 Kl
TIME Kmm h
KmxZ= 30 MNjrn t
Figure 7.7. Aa versus AN; crack length measurements using both a microscope and the a.c. microgauge in, Lo-Hi block loading.
•
J
W 0
302
K m 2
••. Kmin= krn;n2 =0 Time
I 1 I I 1 I .1 1•o 2.O 3.0
CRACK IIJCREi 1EN-r, Ao.) (rnv i)
- 0.5 0
•
Figure 7.8. da/dN versus Aa; accelerated growth rates detected using both the a.c. microgauge and a microscope.
rnox K max Z
f r «. IC tnit~1 Time
KninZ -°•°
e°e°
• 4,0
e•"". -- •— e— —,
• K mox 2 = 25 Mt■1l rn3/2
♦ Kmox2 = 20 MN1rn3/2
l(marz = 15 MN Jm312' NO CRACK GROWTH AFTER tSooj»o' crcc.z✓s
i 1 2.O
106 0.5 0 1.0
Kr1 = 30 MN jm3/2
303
CRACK LENGTH INCREMENT to., (rfly )
Figure 7.9. da/dN versus da data for Hi-Lo block loading at R = 0.
•
— K mous -f )jK mcac 2 _
— . K mtn' Kruin 2' TiME
X
0
103
K12" 45 MNf-d
A
ma„f 30 MN' m31z 0-0_"0
®~ rnoa 2=25MNfm312 p _.. O — p 0 _ ..—
Kmax2 = I5MN1m3'2 p p
45
304
1.O 2.0 3.0 CRACK LENCaTH tI4CREMENT, A°̀ ,(""”)
Figure 7.10. da/dN versus Ea data for Hi-Lo block loading R = 0.
0
X
K t 15 Mt4Irri8f2 maxi
I .I 1
- 14 rno.% 2 - 14rna,c1 = 15NiNiir I
' 4'►1ir11 TIME
/III 4 K rr~o 2" G0MNim3/2
U U ® ® ®—
.b3
N
I K rn".2' 2 rf tylal2
1.0 2'0 0
305
CRACK IZCREMENIT, dal( rnrn)
Figure 7.11. da/dN versus to for Lo-Hi block loading, at R = 0 and for
Kmaxl 15 MN/m3/2.
a
KA
1
306
r K rnosc 2 K encac 'ti — KtT ni' % mtn2" O.O
TIME
10 KracgcZ=45MNJrr 2. c 0 0 0
E ~
2 K= 30Miai `, ~A ` K ~oac2 =30 MN ! 3/z
0 0 A-- — -~- -- o - 4 26Natqcns±j io
-5 I 1 I is -0-5 0 1.0 2.0
CRACK LENGTH INCREMENT)Az,., (rnm)
Figure 7.12. da/dN versus Aa for Lo-Hi block loading at R = 0 and at various Kmaxl values.
•
W J
in1
Tim( hlits Z
--.6—&• .A
A
Q K maxl - 45 ION Ī rn312
K max 2 = 30 MN f r 3/2
r,, ti = 30 MN f ni 12 K fro* _ 20 MN/1'n 3/2
1•o 2.0 3•0 CRACK LENC,TH M1CREMEIJT Aa, (troll)
307
--A
•
Figure 7.13. da/dN versus Qa for Hi-Lo block loading at same AK but different K
mean '
308
•
A
—K maxi Kmaxi =15 MN/m3/2
Kmin2 ---0--0-K
max2=45MN/m /2
_A aJ =15MN/m312 n~
T{M Kmax2maxi
=30MN/m3/2 E Kmax1=30MNIm312
~~ Kmax2 = 45MN/m3/2
MCI = AK2 =15MN/m312
1
1Ō3
Kmaxt Kmirn
105
10 3,0 0, 1,0 2,0
CRACK INCREMENT, Aa, (mm)
Figure 7.14. da/dN versus Aa for Lo-Hi block loading at same AK but different K
mean'
309
K 15/4N/m3/2,
-30M4m312 ti-
T1Z1E
•
L0_ ai,— o®— -o* —o _0 ..... AAA Q—PA A_ i
40 o 1 Kmin 2 = 0, • J Kn,ox2 _15 MN /m312'
Krnimi =-30 MN J nW2 k1= 16 Mt Im'2
106 -10 I
0 I I I I
1.0 20
CRACK INCREMEN1J ct, (mm)
Figure 7.15. da/dN versus as data; at constant Kmax and with changing K from zero to -30 MN/m3/2 and bac21Po zero.
10
da % d
N (
mm
Ic%f
cLE
)
104
310
UPPER BOUND FOR CRACK ACCELERATION ill
7 48xt09(46K))•24 dN
• ACCELERATED GROWTH RATES it.f Lo—H s S4-0CK LOADtN4 .
O CONSTANT AMPUTUDE CROWN RATE.
100 DIC (MN/ m312)
Figure 7.16. Upper bound for the crack growth acceleration.
AK MN/m/2 , R=0,0 B =24mm
• a = 31,14 mm . a = 34,69 mm • a = 62,05 mm
da/dN= 2,63 x10m / m CYCLE
da/dN=1,83 x 10 mm/ CYCLE
5 6 7 NO. OF CYCLES, AN, X103
10
s
Figure 7.17. The effect of crack length on growth rates during block loading test.
5,0
2,0
1,0
0
/ PREDICTED CYCLIC /..4------PLASTIC ZONE SIZE
_ 1 Ki 2 -~
Crc 311 (Tyc
7
0,2
0,1
0 ,05_ °1t 20 50
1 I I 100
200
~K1 (MN /m3/2 )
312
Figure 7.18. Variation of GaDM with GK1.
f
313
20 50 100
Kmaxi (MN/m3/2 )
Figure 7.19. Variation of daDT with Kmaxi.
200
3
•
2,0
I
I • /
,o 2P - 3,0
'A Kmax2/ Kmaxi
Figure 7.20. Variation of AaA with Kmax2~Kmaxl
during Lo-Hi block loading.
1,0
315
•
CHAPTER EIGHT
GENERAL CONCLUSIONS AND PROPOSALS FOR
FUTURE WORK
316
8.1 INTRODUCTION
A fracture mechanics approach was successfully applied in the
investigation of fatigue crack growth characteristics in BS4360-50D
low-alloy structural steel. The results of an extensive programme
have been presented and discussed in the previous five chapters. Also
presented in the individual chapters are the conclusions drawn from
each of the topics studied. This chapter draws together the main
conclusions of the investigation and final proposals of areas in which
further work may be needed.
8.2 GENERAL CONCLUSIONS
The main conclusions drawn from the present investigation are as
follows:
1. The a.c. crack microgauge is suitable for crack length measurement
under both constant amplitude and two-level block loading. The
main advantages of the microgauge are that: (i) for the CS
geometry tested, it measures the average crack length over the
whole crack front; (ii) its accuracy can be varied for different
situations by varying the gain value; and (iii) the instrument
can, by the use of a chart recorder,- be made to record the crack
length continuously during the fatigue test.
2. Crack growth rates are generally found to increase with both
stress ratio and specimen thickness. The effect of stress ratio
is most significant at low growth rates especially in the thinnest
material. The effect of thickness is most prominent at low stress
ratios and at low growth rates. At high growth rates no appreciable
effect of thickness is observed. The effect of stress ratio and
thickness at low growth rates can be adequately explained in terms
317
of the crack closure concept using an effective stress intensity
factor.
3. The compressive portion of the tension-compression load cycle
was found to be damaging. This was especially so in the case of
short cracks growing from a notch.
4. Crack growth rates increase with decrease in frequency. This
effect of frequency is more pronounced at high stress ratios for
a given frequency. The effect of frequency was adequately
explained in terms of environmental effects.
S. Salt-water solution environment was found to enhance crack growth
rates at low frequencies and at high stress intensities. Near-
threshold growth rates were found not to be affected by the salt-
water solution.
6. The original Jc criterion, based on the intersection of the J
versus da curve with the da = J/2aflow
line seems to over-
estimate the onset of slow stable crack growth. This value is
thickness dependent. The Ji value seems to represent the
initiation of slow stable crack growth.
7. Both crack growth retardation and acceleration occur in two-level
block loading. Retardation was recorded under Hi-Lo block loading
while acceleration was recorded under Lo-Hi block loading. These
phenomena can adequately be explained in terms of crack closure,
crack tip blunting, cyclic strain hardening of the crack tip
material, as well as crack front incompatibility; all the mechanisms
acting simultaneously.
318
8. The following expressions can be used to express crack growth
rates for BS4360-50D steel under various loading and environmental
conditions:
(i) In air and at high frequency:
da = 7.6 x 10-9 (AK )3.07
where AKeff max - KCL' Values of KCL vary with thickness
and approximate values of KCL are 6.3 MN/m3/2, 3.1 MN/m3/2
and negligible value (by experimental trends) for the 12 mm,
24 mm and 50 mm thick specimens respectively.
(ii) In air and at low frequency
da - 7.6 x 1 9 (1 - bR)3.07 (AK )3.07 dN 1 -R eff
where values of b are given 0.87 and 0.9 for the 12 mm and
24 mm thicknesses respectively.
(iii) Upper bound for growth rates in salt-water solution:
da dN = 2.11 x 10
-7 (AK)2.63.
and growth rates in salt-water:
da_ B s
dN - 6.71 x 10 9(OKeff)3 + f (Keff) 1
wh ere B1 and 81 have values given in chapter four.
(iv) Low crack growth rates can be modelled in terms of material
fatigue properties:
•
319
2 da _ 21+n(1-112)1+n(aKe2f
- LKc2th)
dN 4a(l+n)TrQy~-n E1
+nefl+n
where the critical value of AKth, AKc th, is given by
AKc,th
4TrEy s
2 autsEf (l+n)a
yc.._
derived by using energy balance approach.
(v) Under elastic plastic conditions growth rates are predicted
using a double-mechanism model in terms of the J-integral
1+n da _ 2 (LU
eff - ~Jc~th) +
C'(J )m,
dN 41 En 1-n 1+n max On aye
where first term in the r.h.s. predicts crack growth by
crack tip blunting mechanism and the second term predicts
crack growth by ductile tearing mechanism. Values of the
respective constants are given in Chapter Six.
(vi) Accelerated growth rates can be estimated from
da _ 4.5GaAt mR
dN ̂ CR(3. 6AaA ♦ Aa) (dK)
for La < LaA, where CR and mR are 3.13 x 10-9 and 3.24,
respectively, and MaA is given by:
MA = 1.47 - 0.22 max2/Kmax1, for max21 maxi '— 1.3.
9. A fast method of obtaining AKth and low growth rates, called the
"increasing R" technique, is devised and successfully used in the
present investigation.
320
8.3. PROPOSALS FOR FUTURE WORK
The present investigation has indicated areas which require
further work. These include the following:
1. The possibility of using a microprocessor for control of the
machine and analysis of the fatigue data during fatigue tests
should be explored. It would be advantageous to use the a.c.
microgauge in such an investigation.
2. The use of the 'increasing R' technique for determining thresholds
and low growth rates, devised during the present investigation,
should be applied to other materials.
3. The magnitude and effects of the microstresses in the plate
material should be evaluated. Further work on other materials
should be carried out to investigate whether these microstresses
are the main cause of the thickness effects observed in the
present investigation.
4. The present results indicate that the compressive portion of a
tension-compression load cycle is very damaging for short cracks
growing from a notch. Further investigation is required to
assess the effect of compressive cyclic loading on cracks growing
from areas of high stress concentration in welded joints.
5. Further work is required in testing under variable amplitude
loading. The present results show that a computational method
is required to derive an empirical expression, based on present
data, to accurately predict crack growth acceleration and
retardation, especially the latter.
•
321
•
M
APPENDIX
FUNDAMENTAL CONCEPTS OF FRACTURE
MECHANICS
322
A.1. INTRODUCTION
The review given below is intended only as a brief introduction
to the fundamental concepts of fracture mechanics. It is included as
an explanation of the fracture mechanics terms used elsewhere in the
text. Comprehensive reviews and analyses of fracture mechanics can
be found in standard text books [13,14].
Fracture mechanics analysis of a cracked body is based on the
theory that for the crack to grow two conditions must be fulfilled.
Firstly the stress at the crack tip should be sufficiently high for a
decohesion to occur and hence for some mechanism of crack growth to
operate. Secondly, sufficient energy must be available in the region
of the crack tip to supply the work necessary for the creation of the
new fracture surfaces. The second criterion is of fundamental importance,
as the very high crack tip stresses predicted by elastic stress analysis
suggest that a sharply cracked body could sustain no load, which is
obviously not the case in practice.
A.2. LINEAR ELASTIC FRACTURE MECHANICS
A.2.1. The energy balance approach to fracture
The energy balance approach to the study of fracture phenomenon
in cracked bodies was originally proposed by Griffith [249] in 1921.
His basic premise was that the unstable propagation of a crack takes
place if an increment of crack growth results in more stored energy
being released than is absorbed by the creation of the new crack surface.
Thus the energy condition for crack growth is
dU dU
da + da < 0 (A.1)
r
323
where Ue is the elastic strain energy and Us is the surface energy.
Using Inglis' [250] stress field calculations for an elliptical flaw
in an infinite plate Griffith produced an energy balance such that
unstable growth would occur when
d (_ a2ira(a-1) + 4a ) = 0 da 4G' ys (A.2)
where a is the remote direct stress,
ys is the specific surface energy,
G' is the shear modulus and
a is (3-4v) for plane strain and (i+-) for plane stress.
The instability condition of equation (A.2) shows that fracture
should occur at a stress af' defined-by
2Ey af/(ara) = ( 2 )
1-v - plane strain
and (A.3)
of ( = 2 - plane stress
Griffith found, from tests on glass that the above condition for fracture
was satisfied. However, the value of of predicted by equation (A.3) was
not satisfactory.
The inability to predict the fracture stress, af, using the
Griffith approach arises because the analysis is limited to the fully
brittle situation. In nearly all fractures some plasticity occurs and
this increases the energy required to create the new surfaces. In this
case the strain energy is largely dissipated by producing plastic flow
at the crack tip.
Irwin [251] and Orowan [2521 later suggested an extension of the
Griffith energy balance to localised yielding situations. They suggested
324
that ys in equation (A.3) be replaced by ys + y , where y represents P P
the localised plastic energy dissipation unit area at the crack tip.
Orowan showed that y , the plastic work term, varies widely from
material to material, and was also temperature sensitive. In some
cases y was up to 105 times greater than ys and hence the energy balance
was basically dependent upon y rather than ys. The problem is that yp
cannot be measured directly. This difficulty was overcome by Irwin and
Kies [253] who suggested an experimental method for calculating energy
release rate in an arbitrary body. The energy release rate or change in
strain energy dU/da was shown to be
dU __ P2 dC da 2 ' da
(A.4)
where P is the applied load and C is the compliance. dC/da can be
found from a series of load displacement curves for different crack
lengths in the same geometry. The critical value of energy release rate
and hence 2ys can be found by substituting the fracture load and the
appropriate value of dC/da in equation (A.4).
Irwin [254] termed the strain energy release rate (dU/da) G, which
he defined as the crack extension force, and the critical value at which
unstable crack growth would occur, Gc. If the fracture process is
controlled by the energy balance then the fracture should occur at the
same energy release rate for different loadings and geometries. Thus
Gc should be a material property and as such should be measurable in
fracture tests.
The most complete mathematical treatment of the energy balance for
crack growth in an elastic body was provided by Rice [255] and
Cherapanov [2561. They found that G can be expressed as
G = _ dU da
(A.5)
•
325
where U is the potential energy of an elastic body. G is then more
generally defined as the rate at which potential energy decreases with
crack length. A convenient way of visualising the potential energy
decrease is as the area between the overall load-deflection curves
before and after crack extension. This definition holds true for non-
linear elastic as well as for linear elastic behaviour.
A.2.2 The stress intensity factor (K) approach
A.2.2.1 Elastic stress distribution at the crack tip
Inglis [250] in 1913 derived the first expression for the maximum
stress at the crack tip of an elliptical notch with major axis a and
minor axis b' under a stress a, i.e.
(ayy)max = a(1 + (ba)) (A.6)
As the ellipse becomes long and narrow a becomes very large and YY
is approximated by
(a ) = 2a(a)z Yymax p
where p = b'2/a is the radius of curvature.
(A.7)
Many years later, using the Westergaard [257] solutions for a
number of configurations involving cracks in tensile stress fields, a
general solution for the stress system at the crack tip of an infinitely
sharp crack was derived by Irwin [254] and Williams [258]. The near
crack tip stresses and displacements could be written in the form
(2a-l)cos 2 - cos 32
(2a+1)sin 2 - sin 32
(A.9) u
= 8a( ?r K
v
KI P-~0 2 lim KtoP
'
326
oxx
a yy
K I
cos 2(1 - sin 2 sin 32)
cos 2(1 + sin 2 sin 32
cos 2 sin 2 cos 32
(A.8)
y X
•
where a = 3-4v in plane strain and a = (2-v)/(l+v) in plane stress, r and
8 are the radial coordinates centred at the crack tip.
The parameter KI, the stress intensity factor, is independent of r
and 8. It dictates the magnitude but not the distribution of the stress
fields in different crack configurations. For an infinite plate
containing a crack of length 2a, subjected to a remote tensile stress
field, a, the Westergaard solution gives
K1 a (A.10)
For more complex cases K1 is given by
KI = Ya (A.11)
where Y is a dimensionless parameter accounting for such variables as
proximity of a free surface, crack shape and distribution of load. KI
can be related to the elastic stress concentration factor Kt by
(A.12)
The above equation, derived by Irwin [251], implies that KI is
valid only when a sharp crack is considered.
Thus far attention has been centred on the case of a crack in a
tensile stress field. Results analogous to equations (A.8) and (A.9)
E G = K2 1-v2 (A.16)
327
can be obtained for the case of a crack subjected to in-plane shear, i.e.
--sin 2 (2 + cos 0 cos 32 )
KII sin 2 cos 32 cos 32
cos 2 (1 - sin 2 sin 26)
(A.13)
and out of plane shear
e -sin
cos 2
T xz
T yz
KIII
(A.14)
KI, KII and KIII
are usually referred to as the mode I, mode II and
mode III stress intensity factors (see Figure A.1). The present analysis
is concerned with mode I only and the stress intensity factor will
simply be referred to as K.
It will be recalled that the Griffiths energy balance gave
dU + dUs
da • da - 0
where
dU _ a2ira _ da E
Thus for plane stress
2 G = Ē or K = AT (A.15)
Using a virtual work approach Irwin has confirmed this and showed that
for plane strain
G
where v is Poisson's ratio.
328
A.2.2.2 Fracture analysis using K-linear elastic fracture mechanics (LEFM):
The stress fields at the crack tip in a purely linear elastic body
are completely specified once the stress intensity factor K is known.
On the assumption that fracture is governed by conditions at the crack
tip, it must be expected that fracture will occur when K reaches a
critical value Kc. Kc may be obtained from a laboratory test in which
failure stress, af, crack length, a, and geometry factor, Y, are
determined. The difficulty in applying LEFM lies in ensuring that K0
is obtained under crack tip conditions which are reproduced in the
structure to be analysed. For purely elastic behaviour this is no
problem, since the stress field at the crack tip is completely specified
by K for all configurations. However, for a real material perfect elasti-
city is never completely realised.
Equation (A.8) indicates that as r -} 0 crack tip stresses would
become infinite for any value of the applied stress. Hence plastic
deformation will occur at the crack tip. The order of magnitude of the
plastic zone, r , can be estimated from equation (A.8) by setting
ayy = ay, the yield stress of the material, leading to
ry = 2r
(6 )2 y
for plane stress conditions.
(A.17)
An elastic stress distribution with a singularity at the crack tip
describes the stress field ahead of the crack where plastic yielding
occurs.- If the extent of the plastic zone is small in comparison with
the dimensions of the elastically stressed region of the structure, the
elastic model will closely represent the real conditions. Since LEFM
provides a method of measuring the fracture strength of a brittle
material, one must ensure that errors introduced by plastic yielding
are minor or can be corrected. For plasticity effects to be negligible,
•
329
the plastic zone must be small compared with the crack length and the
remaining ligament. At the onset of fracture, at which K = Kc, the
error introduced by plastic yielding may be estimated from the ratio
-1' _- a 2zra
(6c)2 _ 2 (Qf)2
Y Y
(A.18)
where of is the gross fracture stress. Thus the LEFM approach is
acceptable as long as the gross fracture stress is small compared to
the yield stress of the material. To correct the small scale plastic
yielding effect it was suggested by Irwin [254] that the physical crack
length be increased by an amount ry to obtain an effective crack length,
aeff, i.e.
aeff a+ry,. (A.19)
The fracture toughness Kc is dependent on, among other factors, the
thickness. The role of thickness is to provide a triaxial stress field
ahead of the crack, which increases the hydrostatic stress component
and thus reduces the plastic zone size. Experimentally it has been
verified that Kc decreases as thickness increases (Figure Ā.2) attaining
a minimum value, Klc, corresponding to plane strain conditions. The
major effect explaining the above trend is that the tensile stress rises
as thickness is increased. This favours the cleavage mode of fracture
over the shear mode, which consequently reduces Kc.
330
. A.3. NON-LINEAR (YIELDING) FRACTURE MECHANICS (YFM)
The stress-strain field at the'tip of a crack in an elastic
medium is completely described by the stress intensity factor K. With
an elastic-plastic material this parameter (K) can only be applied if
deformation is localised in the crack tip region (small-scale yielding).
However, if extensive plastic deformation occurs the linear elastic
approach is no longer valid and an elastic-plastic (non-linear) fracture
parameter must be sought.
A large number of possible fracture criteria have been suggested,
but the two parameters which have received most attention in recent
years are the crack opening displacement (COD) and the J-integral.
A.3.1 The Dugdale model and the crack opening displacement:
Dugdale (259] considered an isolated straight crack of length 2a
in an infinite sheet with remotely applied stress a normal to the crack
plane. Yielding is confined to a narrow band along the line of the
crack and extending to a distance x from the crack tip. He assumed
that a was constant over this distance, and that all stresses in the y
plate were to remain finite.
Although the Dugdale model was originally suggested to correlate
experimentally observed plastic zone sizes in mild steel, it can also
be used to obtain the crack opening displacement (COD) at the crack
tip, as
8a a v = ~Ē (ln sec 2a )
y
(A.20)
where v is the crack opening displacement. Expanding equation (A.20)
as a series we obtain
•
331
• ] (A.21)
(A.22)
86Ya 1 1 Q 2 1 ir a 4 1 /r a 6 - ETr [2(2 a ) + 12(2 a ) + ) 45 (2 a +
y y y
Considering the first term only, then
v = a2Tra - Ca Ea y
•
y
The COD may be used as a failure criterion in materials which
exhibit extensive plastic deformation at the crack tip, i.e. failure
occurs when COD reaches a critical value. This was initially proposed
by Wells [260].
Early experimental COD measurement was performed using a rotating
paddle meter. This technique was developed by Burdekin and Stone [261].
Due to the large experimental errors involved, this technique was
abandoned and measurement of the notch opening using a clip gauge was
substituted. The notch opening can be related to the COD by assuming
that the crack opens about a centre of rotation located at a certain
distance beyond the crack tip. Wells [262] suggested an analytical
expression relating v and the notch opening, and more recently
Sumpter [263] has computed COD for a large variety of geometries, using
the finite element method.
A3.2 The J-integral:
Rice [149] defined the J contour integral for a crack aligned in
the x direction in a two dimensional body of unit thickness, Figure A.3,
as
du.
dx ds r (A.23)
332
where w is the strain energy density, s is the contour circumference,
Ti are surface tractions, ui are surface displacements in the direction
of Ti, is the path of the integral and x,y are the normal rectangular
coordinates.
Rice [264] showed that this integral is independent of the choice
of path r around the crack taken in an anti-clockwise direction from
the lower face of the crack to the upper.
The change in potential energy of the body in growing a crack an
amount da is defined as the sum of the change of the internal strain
energy and the change of potential energy of external forces, i.e.
-AU = f wdA - f Tiuids (A.24) A sT
where A is the area of the body and sT is that part of the circumference
on which the forces are fixed.
Rice [2641 stated that the potential energy is expressible in
terms of the difference in overall load deflection curves for two
bodies with crack length differing by da. When the loading is entirely
by prescribed surface tractions proportional to load P, and ō is the
corresponding generalised displacement, then -AU is the shaded area in
Figure A.4(a). Similarly when loading is by displacements proportional
to ō, -AU is the shaded area in Figure A.4(a).
Rice [264] showed that the J integral is equal to the potential energy
release rate, i.e.
•
J = - 8U Tit
x (A.25)
To simplify the derivations consider steady state conditions (not
a necessary assumption). Then
= - āa a
(A.26)
333
From equation (A.24), and since the second integral can be taken
over s since it is zero where displacements are fixed, equation (A.24)
becomes
au. J
f dx 8x dy - f Ti 8x~ ds
By Green's theorem,
f āx dx dy = f dy A
then equation (A.27) becomes
3u. J = f wdy-Ti ax
3
(A.27) (A.27)
(A.28)
(A.29)
Rice showed that the J integral (defined by equation (A.23) to be
zero around a closed path, so that on any contours such as t1 and r2
in Figure A.5, the integrals must be equal, i.e. independent of path
since the two paths differ only by terms along the crack face for which
dy and T. are both zero. Therefore the contour s need not be the
boundary of the body.
For linear elasticity,
J = - 8a = G (A.30)
and also
K2 J = G = Ē' (A.31)
334
A3.3 Crack tip stress and strain fields under elastic-plastic conditions
For the linear elastic stress and strain fields around a crack
there is a singularity at the tip described by the stress intensity
factor K. Fracture occurs at a critical value of this parameter which
can therefore be used as a representative measure of its toughness or
resistance to fracture.
It is desirable to find a similar parameter to characterise the
elastic-plastic situation and it could be postulated that a parameter
exists which describes all crack tip situations from linear elastic
behaviour to beyond general yield. Rice and Rosengren [242] and
Hutchinson [243] showed that a singularity still exists at the tip of
the crack in the presence of local plasticity. In these analyses
plasticity was incorporated by introducing non-linearity into the stress-
strain relationship only.
Analysis of the crack tip region proved to be difficult; however,
the development of the J integral provided a means of describing crack
tip conditions from far field behaviour. McClintock [241] used the
earlier analyses by Rice and Rosengren [242] and Hutchinson [243] to
express the stress and strain singularities in terms of the J-integral
for a strain hardening material, i.e. n
13
a..(r,e) = ayI J 1 T;;
ra e E y y ij
(8,n) (A.32)
1
c..(r,8) = Ey
[J _1' ra e E (e,n)
13Y y ij
(A.33)
where E..(6,n) and E..(6,n) are stress and strain functions, respectively,13
dependent on a and the strain hardening exponent n, ay and a are the
yield stress and strain respectively, r and 0 are the radial coordinates.
335
The derivation of J is only strictly valid for linear and non-
linear elasticity and its application to the elastic-plastic situation
is uncertain.
Hayes [265] showed that using the Prandt-Reuss laws of plasticity
(which are more representative of the elastic plastic situation) J is
approximately path independent for monotonic loading. So it would
seem that J can be applied to the elastic-plastic regime provided the
loading curve is assumed to follow a non-linear law, i.e.
2U J = aa
where Up is the pseudo potential energy since the plastic portion of the
potential energy is not recoverable.
Begley and Landes [202] by analogy with the elastic problem
indicated that J can be measured from the loading curves of adjacently
cracked elastic-plastic bodies. For a body of thickness B J can be
expressed as
_ 1 dU _ - B da
0
J
where U is the potential energy and a the crack length.
Figure A.I. Loading modes.
Kc
KM
THICKNESS
336
Figure A.2. Kc versus specimen thickness (schematic).
Figure A.3. The J-integral.
I
AU
337
(a) Constant load (b) Constant displacement
Figure A.4. Evaluation of potential energy balance.
Figure A.5. Contours around a narrow crack.
338
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LIST OF PUBLICATIONS
1. Musuva, J.R. and Radon, J.C. "The Effects of Stress Ratio,
Thickness and Frequency on Fatigue Crack Growth in Structural
Steel used in Marine Technology", Proc. 2nd European Conference
on Fracture, ICF 2, Darmstadt, W. Germany (1978). Published by
VDI, Dusseldorf, pp.286-310 (1979).
2. Musuva, J.K. and Radon, J.C. "The Effects of Stress Ratio and
Frequency on Fatigue Crack Growth", Fatigue of Engineering Materials
and Structures, Vol. 1, No. 4, pp.457-470 (1979). (Enclosed).
3. Musuva, J.K. and Radon, J.C. "Size Effects and the J-integral
Approach to Low Cycle Fatigue Crack Growth", DVM, Stuttgart,
pp.479-494 (1979).
4. Musuva, J.K. and Radon, J.C. "Influence of Thickness on the
Fatigue Crack Propagation in a Low Alloy Steel". Proc. 2nd Int.
Conf. on Behaviour of Off-shore Structures","BOSS '79", paper 47,
pp.619-624 (1979).
5. Musuva, J.K. and Radon, J.C. "An Elastic-Plastic Crack Growth
Analysis Using the J-integral Concept". Proc. 3rd European Conf.
on Fracture," "Fracture and Fatigue", ed. Radon, J.C., London,
pp.129-141 (1980).
6. Musuva, J.K. and Radon, J.C. "Threshold of Fatigue Crack Growth
in a Low Alloy Steel", to be presented at the 5th Int. Conf. on
Fract., ICF5, Cannes, France. (29th March - 3rd April, 1981).
7. Musuva, J.K. and Radon, J.C. "Fatigue Crack Growth at Low Stress
Intensities", to be presented at the Int. Conf. on Fatigue
"Fatigue '81", Warwick, U.K. (24th - 27th March, 1981).
Fatigue of Engineering Materials and Structures Vol. 1, pp 457-470 Perpamon Press Printed in Great Britain. Fatigue of Engineering Materials Ltd. 1979
THE EFFECT OF STRESS RATIO AND FREQUENCY ON FATIGUE CRACK GROWTH-
J. K. MUSUVA and J. C. RADON Department of Mechanical Engineering, Imperial College of Science and Technology,
London SW7 2BX, U.K.
(Received 6 April 1979)
Abstract—The influence of stress ratio and the loading frequency on fatigue crack growth rates in BS 4360-50p steel was investigated in laboratory air.
Fatigue crack growth tests were performed on compact tension specimens (CTS) made in two thicknesses 12 and 24 mm. Tests were conducted at two frequencies of 0.25 and 30 Hz, applying a stress ratio R varying from —0.7 to 0 7. The results were analysed using the linear elastic fracture mechanics approach. They showed that the increase in both positive and negative R caused increased fatigue crack growth rates. Also an empirical effective stress intensity factor range, AKen, was found more appropriate to correlate the fatigue crack growth data than the AK factor frequently used in crack growth studies.
The loading frequency had only a little influence on crack growth rates at low R. However, at high R, growth rates were significantly higher at lower frequencies. It is suggested that this frequency influence may be associated with environmental effects, due to the embrittlement caused by hydrogen from the moist air, while the crack was fully open.
INTRODUCTION
MANY studies have confirmed that the fatigue crack growth rate, da/dN, is primarily controlled by the stress intensity factor range, AK, through the well-known Paris expression [1]. It has also been established [2, 3] that certain mechanical and metallurgical, as well as environmental factors affect fatigue crack growth rate, the role of stress ratio, frequency and environment being most prominent. To account for the observed effect of stress ratio on crack growth rate, growth models have been proposed and applied by different authors [1-12]. A recent review of these models is found in the paper by Wade and Lee [13]. Evidence at present available suggests that frequency has some effect on fatigue crack growth [14-16].
The effect of the stress ratio on the crack growth rate is influenced by at least three factors : cyclic frequency, the size of the specimen, and the structure of the material. The test conditions in the present investigation where therefore varied to ensure that conservative results were obtained.
EXPERIMENTAL PROCEDURE
The steel investigated was a low alloy structural steel, BS4360-500, available in two plate thicknesses of 12 and 24 mm.
Fatigue crack growth tests were performed on compact tension specimens (CTS) of overall dimensions 100 x 95 mm (W = 79-5 mm), machined from the plates supplied. In all the specimens, the fatigue crack was grown perpendicular to the rolling direction of the plate.
All tests were performed in the laboratory air, ambient conditions and at room temperature, using three different machines. For tests at 30 Hz, a Dowty fatigue testing
457
S
458 J. K. MUSUVA and 3. C. RADON
machine of 60 kN capacity was used. Tests at 0.25 Hz were performed on an Avery fatigue machine of 50 kN capacity, while tests at negative stress ratios and 0.12 Hz were performed on an Instron machine, capacity of 55 kN, provided with special grips enabling reversed cycling. Some check results were made to ensure that all three machines gave similar results under identical testing conditions [17].
To investigate the effect of positive stress ratio-and frequency on both thicknesses, tests were performed at R equal to 0.08 and 0.7, while cycling frequencies were 30 and 425 Hz. To investigate the effect of negative stress ratios, a second series of tests on CT specimens of the same geometry as above was arranged. All the specimens were provided with identical starter crack lengths and a constant maximum positive load limit, P„,ax, was chosen. The minimum load limit, P,,,;,,, was then adjusted for each specimen in order to obtain the required stress ratio value, i.e. R = 0.0, —0.17 and —47. Thus, in these tests, the only variable was the negative part of the Ioad cycle.
In all the tests, the specimen was provided with a sharp notch from which the crack propagated. The surface of one side of the specimen, along the path of the crack, was polished to facilitate viewing of the crack tip. Measurements of the crack length were made optically using microscopes mounted on travelling micrometric stages.
The experimental readings of crack length, a, and the correspondipg number of cycles, N, were recorded periodically after suitable crack increments. To calculate the crack growth rate, da/dN, from the experimental data, a combination of graphical curve-fitting and finite difference (secant) methods were used. It may be mentioned here that an attempt was made to fit the data into a polynomial using the least-squares method [18] from which da/dN was then determined by differentiating the fitted curve at various values of a and N.
The stress intensity factor, K, was calculated using the standard formula recommended for CTS geometry [19] :
K BW# .f (w),
where 0.3 < a/W < 0.7. The crack lengh, a, is the average of the two successive readings used to calculate the corresponding da/dN.
RESULTS
The results of crack growth rate, da/dN, plotted against the corresponding AK values for both thicknesses investigated, are shown in Figs. 1-8 for a range of stress ratios and frequencies. These data are replotted in terms of da/dN vs AKeff in the subsequent Figs. 9 and 10.
Figure 1 shows the results for the 24 mm thick specimens tested at 30 Hz using stress ratios, R, equal to 0.08, 0.5 and 0.7. These fall into a fairly narrow scatter band, indicating only a very small effect of the stress ratio on the crack growth rates. The data correspond well with the Paris expression and the best-line fit through the data points has the form:
da = 6.71 x 10-9 AK3 (2)
dN
where da/dN is in mm/cycle, and AK in MN/m1. The results for the 12 mm thick specimens tested at the same frequency of 30 Hz, and
applying stress ratios, R, of 408 and 0.7, are shown in Fig. 2. They suggest a stronger effect
(1)
I
459 The effect of stress ratio and frequency on fatigue crack growth
a 10 (— A R=008
o P.05 0 R=07
a
1 6,,f 5 7 10 20 30 40 60' 13'0'1C0
LK (MN/m3'2)
Fig. 1. da/dN vs OK at 30 Hz, B = 24 mm. of R than in the previous tests performed on 24 mm thick plate. They also show that the values of constants C and m in the Paris expression are dependent upon the stress ratio R and thus AK is not the best parameter to use for correlation the data available at present.
The results recorded at 0.25 Hz on the 24 and 12 mm thick specimens are shown in Figs. 3 and 4, respectively. These results show an increased effect of stress ratio at the lower frequency.
An effective stress intensity factor, originally proposed by Sullivan and Crooker [11], was found suitable for the correlation of the data shown in Figs. 2-4. This growth model is of the form :
da =C{( )1
R, }m
)
(3)
where the values of the constants for the present tests are shown in Table 1. Table 1
Thickness Frequency C b m (mm) (H)
12 30 4 x 10' 0.89 3.74 24 0-25 1.1 x 10' 0.9 2.91 12 0 25 4.70 x 10-10 0.86 3.73
e a
I03 e R:008 A
CP 0 a=07 a
105
s
460 J K. MUSUVA and J. C. RADON
5 7 10 20 30 40 60 67 100
QK (MNb,3/2)
Fig. 2. da/dN vs AK at 30 Hz, B = 12 mm.
The data for the two thicknesses investigated at 025 Hz are presented in Figs. 9 and 10, respectively, in terms of da/dN vs AKKrr.
Figures 5 and 6 show the results at negative stress ratio for the 12 mm thick specimens. Figure 5 is plotted, taking into account a complete compression—tension load cycle, while in Fig. 6 only the tensile portion of the cycle was used. These results confirm that a part of the compressive cycle does contribute to the cyclic crack growth, but the contribution seems to decrease at higher values of AK. Preliminary results for the 24 mm thick specimens available-until now, indicate that the contribution of compressive cycle to growth is larger for the 24 mm specimens than for the thinner ones:
Finally, in order to evaluate the effect of frequency on the crack growth rate, the best-fit lines through the results in Figs. 1 and 3 are shown in Fig. 7. Similarly, the results from Figs. 2 and 4 are plotted in Fig. 8. Apart from the region of very low growth rates, Fig. 7 shows only a small effect of frequency at R = 0.08. However, at R = 0.7, the growth rates are higher at lower frequencies, and here the influence may be considered significant. A similar, but distinctly larger, effect of frequency may be observed in tests performed on the 12 mm thick specimens (Fig. ,8). These results suggest that the influence of the stress ratio will increase at lower frequencies.
The effect of stress ratio and frequency on fatigue crack growth 461
l03
104
E glā
10
I
Ē n
A 0
i 1
R.O0B R-07
S O n
O 1 0 A
A cP an
P S
J a 6
l . 1
A
A5
A
A
A a
A 0
m l
5 7 10 20 30 40 60 80 100
AK (MN /m3/2)
Fig. 3. da/dN vs AK at 0 25 Hz, B = 24 mm.
DISCUSSION
Stress ratio, R, has frequently been used [4, 9-11] to investigate the effect of mean stress on fatigue crack propagation. Tests at constant mean stress intensity factor, Keaan, have been carried out [5, 13, 20] for the same purpose. The results of one form (constant R ratio) can easily be transformed into the other (constant Kmearl ), provided an adequate number of experimental results is available. Preliminary Km can tests on this material were reported earlier [17]. It will be realised that constant R ratio tests are easier to perform on the specimen geometry used in this study; they are also more appropriate in life calculations for non-redundant structures where the stress ratio, R, remains constant with increasing crack length.
In analysing the results, attempts were also made to evaluate the growth rate data using a range of models found in the literature for comparison. The Paris model [1] was found to be satisfactory when applied to the results from 24 mm thick specimens tested at 30 Hz. Unfortunately, these tests showed only a small effect of stress ratio. For other tests, where the effect of stress ratio was larger, the Paris model was not suitable and the correlation of the data was unsatisfactory. On the other hand, the growth model proposed by Forman [4] was found to over-estimate the stress ratio effects substantially, particularly for
0 p
O' ° 8 °o 0 °
Q0 ~
ō 0 AA 1
0 O 0 Q °
OD
CO °
O 8 ° °
8 $ 0 0
A
° R.008 0 R=0 7
°
462 J. K. MUSUVA and J. C. RADON
103
16 4
10
5 7 10 20 30 40 60 80 100
.6 Ī MN/m3/2
Fig. 4. da/dN vs MC at 025 Hz, B = 12 mm.
tests at higher frequencies, where the influence of R is small. Another difficulty encountered with the Forman model was the choice of a suitable K, value. An attempt was made to use the value of Kmax recorded at the moment of sudden fracture of the specimen during the fatigue test. However, this value is not constant, in that it varies not only with thickness, frequency and stress ratio, but also with the crack length. Consequently, the choice of a realistic value of K, is difficult. The same problem occurs when using other models, such as that proposed by Branco et al. [5].
The effective stress intensity factor range, AKea, based on crack closure is not easily determinable. This is because the closure stress, ad., is a function of a number of mechanical and metallurgical factors [8] ; it is also dependent on the test piece [21]. Consequently, act° would have to be determined at virtually every experimental test point, a highly complex exercise, not warranted in the present analysis. In addition, results of the tests performed at negative stress ratio seem to indicate that fatigue damage may occur, even when the crack is closed [9], thus casting doubt on the use of an effective stress intensity factor range based exclusively on the crack closure concept.
An empirically derived effective stress intensity factor range, normalising data of a series of R values to the crack growth rates at R = 0, seems to be a more convenient and
10 2
• • •
■ • • R=OO
• R=-0 17
• R=-0 7 • •• • ■ •
• ■
f
•
t ■ f
•ii 1 • 1 • • •
• • ■ i
• ■ • • •
1 03
E
SIE
10 4
. I I I I I I 1
The effect of stress ratio and frequency on fatigue crack growth 463
5 7 10 20 30 40 60 60 100
AK (MN / m a/2)
Fig. 5. da/dN vs AK at 0.12 Hz, B= 12 mm.
somewhat simpler method. The models by Sullivan and Crooker [11] and Walker [10] appear to be particularly suitable. The empirical constants in the respective models are easily determinable from experimental data at the specific test conditions and can be evaluated for both positive and negative values of R. The-S -":•'1n and Crooker model was found more suitable for correlating the present results and, as seen in Figs. 9 and 10, the results are satisfactory. The determination of the effective stress intensity factor range is easy and the model is good enough for life prediction in the mid-range of growth rates frequently used in design.
Crack closure has often been used to explain the observed effect of stress ratio on fatigue crack growth rates. According to this concept, at low stress ratios, the crack closure stress, '7700, is above the minimum applied stress and thus the effective range of stress is substantially reduced. At high stress ratios, the crack remains open for a larger part of the load cycle and thus closure effects become less pronounced. It should be pointed out that this argument does not explain why there is no effect of stress ratio in results on the 24 mm thick specimens tested at 30 Hz (Fig. 1). Considering now the results obtained on the 12 mm thick specimens at 30 Hz (Fig. 2), it appears that the effect of stress ratio is diminishing with increasing growth rates. Crack closure alone does not explain this
10
From
/cy
cle
1
1
464 J. K. MUSUVA and J. C RADON
Y
I 102 A R.00
■ R•-0 17
• R=-0 7
-4 10
1 l ! 1 1 1 1 1 1 1 1 1 1
20 30 40 60 80 100
AK,„ (MN/m3/2)
Fig. 6. da/dN vs A/Ca...iiti at 0.12 Hz, B = 12 mm.
behaviour, since the closure is likely to occur at high, as well as low, growth rates. It is also known that there is no influence of stress ratio on growth rates for steels tested in vacuum [22], and this observation further confirms that the crack closure is not the only factor influencing the stress ratio effect.
It is possible that the effect of stress ratio on the growth rate described above could be connected' with environmental effects. It is known that hydrogen in moist air can cause embrittlement at the crack tip, thus leading to increased growth rates. Hydrogen atoms in steel can diffuse to the regions of maximum hydrostatic tension ahead of the crack tip, thereby lowering the cohesive strength of the lattice [23]. This process causing faster growth rates is more significant in high strength steels than in low alloy steels.
Higher fatigue crack growth rates could thus be caused by the increased stress ratio, since the higher value of Kmax creates a large stress gradient to promote hydrogen diffusion. Higher growth rates at high testing frequencies would not provide favourable conditions for hydrogen to diffuse to the crack tip in time to cause embrittlement, since the crack is advancing at a high velocity. However, at low growth rates (i:e. near the threshold) and at low testing frequencies, the environmental effects are maximised. This result is supported by comparing Fig. 1 with Fig. 3 and Fig. 2 with Fig. 4. It also seems possible that the plate
5 7 IO
„I
0
The effect of stress ratio and frequency on fatigue crack growth 465
5 7 10
20 30 40 60 BO 100
QK ( MN /m3/2)
Fig. 7. da/dN vs AK at 0 25 Hz and 30 Hz, B = 24 mm.
thickness could influence the transport of hydrogen from most air to the crack tip. The thicker the material, the less the effect of environment at a certain crack velocity, since it takes more time for hydrogen to permeate along the whole width of the crack front. Comparison of Figs. 3 and 4 would support this proposition.
It could also be argued that, at low stress ratios, the crack is closed during the greater part of the load cycle. Since closure is primarily a surface effect, it may reduce the rate of hydrogen diffusion into the crack front. This conclusion is supported by the very small effect of frequency observed for both thicknesses at a stress ratio of 0.08 (Figs. 7 and 8). However, at high stress ratios, the crack is fully open for a larger part of the cycle and the environmental effects are also maximised. Again, this process is confirmed by the results shown in Figs. 9 and 10. Thus, with the absence of stress ratio influence on growth rates for steels tested in vacuum [22] and a reduced stress ratio effect observed in the present tests at higher growth rates and high testing frequency, the environmental conditions would seem to provide good evidence and an explanation of the stress ratio effects on growth rates; the crack closure process would then be only of secondary influence.
Further, the effect of negative stress ratio could not adequately be explained in terms of crack closure alone. The present results show that the fatigue damage does occur, even
466 J K. MUSUVA and J. C. RADON
1 I 1 1 I L I I
20 30 40 60 B0 100
MN / m3/2
1 1 1 1 I
5 7 ‘10
AK
Fig 8. da/dN vs OK at 0.25 Hz and 30 Hz, B = 12 mm.
when the crack is fully closed. It would appear that, during the application of the compressive load, the crack surfaces are highly compressed, especially in the region close to the crack tip. This would result in a sharpened crack. This process would cause increased stress intensity at the crack tip leading to accelerated growth. However, not all the compressive load is used for collapsing the crack surfaces; some of it is dissipated in reducing and even reversing the residual stresses in the vicinity of the crack tip remaining from the preceding tensile cycle. This means that the residual stresses at the crack tip may substantially reduce or even eliminate the compressive portion of the cycle, thus directly influencing the growth rate. It should be remembered that residual stresses are higher in plane stress (thus in thinner sheets) than in plane strain. The above proposition supports the present results. It is concluded that the crack growth models should include the effect of compressive—tensile cyclic loading, since the compressive portion of the cycle is found to cause damage.
CONCLUSIONS
The linear elastic fracture mechanics method was used to characterise the effect of stress ratio and frequency upon fatigue crack growth rates in BS4360-50C steel tested in air and at 21°C. It was concluded that:
G R-008 0 R = 0 7
A
a
I I I I I 1 I 1 I t till
163
104
10 5
The effect of stress ratio and frequency on fatigue crack growth 467
5 7 10 20 30 40 60 BO 1OO
D K = J (1÷f),.22) OK1 (MN'm 3'2 ) EFF `
Fig. 9. da/dN vs OKcR at 0 25 Hz, B = 24 mm.
1. Stress ratio and frequency may substantially affect crack growth rates in both compressive—tensile and purely tensile fatigue cycling. However, these effects are not of a cumulative nature.
2. In general, the fatigue crack growth rate increases with increasing positive stress ratio at a given value of AK. It is more significant for the thinner plate and also increases at lower testing frequencies. The effect of positive stress ratio can be explained in terms of environmental processes with crack closure being only a secondary influence.
3. The compressive portion of the load cycle at negative stress ratio does contribute to crack growth. Thus, in structures which are subjected to compressive—tensile (or reversed) cyclic loading in service, such as oil platforms in the North Sea, crack growth investigation should include tests at negative stress ratios.
4. Crack closure alone cannot adequately be used to explain the effect of stress ratio, but additional environmental effects due to hydrogen embrittlement offer a more realistic explanation of the changes in crack growth.
5. The stress intensity factor range, AK, may not be the best parameter to correlate the fatigue crack growth data, especially at low frequencies when stress ratio effects are significant. The effective stress intensity factor range, AKe ff, of an empirical nature, such as
AKeff = ((1— bR)/(1—R))AK,
468
l0
77
E E
~o 0I
10
0 !G 0
O
A R=0OB 0 R..07
I I I1 I I 1 111i 1.
J. K. MUSUVA and J C. RADON
5 7 10 20 30 40 60 80 100 Ir I-o•86R
6K = j I _R ) AX ( MNim3'2 EFF `
Fig. 10. da/dN vs LKcR at 0-25 Hz, B = 12 mm.
could be more appropriate. The constant b is a variable of the loading conditions and environment. _
6. Frequency has little influence on crack growth rates at low stress ratios. However, at high stress ratios, the lower frequency may cause higher growth rates. This effect of frequency could again be associated with environmental effects due to hydrogen embrittlement. Tests at low frequencies (0.25 Hz) and high stress ratios are more conservative for life predictions and thus are more appropriate for the design of North Sea structures where such conditions are experienced.
7. Life predictions for cyclic tests on BS4360-50C steel in air and at low frequency could be made using the expression:
da ( _ 2.91
=110x10-s {(11 0R )AK} (4) dN l
for the 24 mm thick material, and:
da IC —1 4,70x 10'10 ī0.8R6R) dN
0
(5)
The effect of stress ratio and frequency on fatigue crack growth 469
for the 12 mm thick material, for R > 0. Values of b [equation (3)] equal to 1.4 and 1.95 for the 12 and 24 mm thick specimens, respectively, could be used to normalise the results, for the negative stress ratio, for the present results. However, more results are being performed to confirm the trend of these results at lower growth rates.
REFERENCES
[1] Paris, P. C. and Erdogan, F. (1963) A critical analysis of crack propagation laws, J. bas. Engng, Trans. Am. Soc. mech. Engrs, Series D 85, 528-534.
[2] Lindley, T. C., Richards, C. E. and Richie, R. O. (1975) The mechanics and mechanisms of fatigue crack growth in metals, Conf on Mechanics and Physics of Fracture, Churchill College, Cambridge.
[3] Roberts, R. and Erdogan, F. (1967) The effect of mean stress on fatigue crack propagation in plates under extension and bending, J. bas. Engng, Trans. Am. Soc. mech. Engrs 89, 885-892.
[4] Forman, R. G., Kearney, V. E. and Engle, R. M. (1967) Numerical analysis of crack propagation in cyclic loaded structures, J. bas Engng, Trans. Am. Soc. mech. Engrs 89, 459-464.
[5] Branco, C. M., Radon, J. C. and Culver, L. E. (1976) Growth of fatigue cracks in steel, Metal Sci. 10, 149-155.
[6] Elber, W. (1971) The significance of fatigue crack closure, A.S.T.M. STP 485, 230-242. [7] Maddox, S. J., Gurney, T. R., Mummery, A. M. and Booth, G. S. (1978) An investigation of the
influence of applied stress ratio on fatigue crack propagation in structural steels, Welding Institute Research Report 72/1978/E.
[8] Ogura, K., Ohji, K. and Honda, K. (1977) Influence of mechanical factors on the fatigue crack closure, 4th Int. Conf. on Fracture, Waterloo, Canada, Vol. 2, pp. 1035-1047.
[9] Ohta, A., Kosuge, M. and Sasaki, E. (1978) Fatigue crack closure over the range of stress ratios from -1 to 0.8 down to stress intensity threshold level in HT80 steel and SUS304 stainless steel, Int. .1. Fract. 14, 251-264.
[10] Walker, E. K. (1970) An effective strain concept for crack propagation and fatigue life with specific application to biaxial stress fatigue, Report AFFD-TR-70-144, U.S. Air Force Flight Dynamics Laboratory, pp. 225-233.
[11] Sullivan, A. M. and Crooker, T. W. (1976) Analysis of fatigue crack growth in high strength steels. Part I: Stress level and stress ratio effects at constant amplitude, J. Press. Vess. Tech., Trans. Am. Soc. mech. Engrs 98, 179-184.
[12] Illg, W. and McEvily, A. J. (1959) The rate of fatigue crack propagation for two aluminium alloys under completely reversed loading, NASA TND-52.
[13] Wade, E. H. R. and Lee, G. M. (1977) The influence of mean stress on fatigue crack propagation in a quenched and tempered alloy steel, J. Strain Analysis 12, 81-88.
[14] Schmidt, R A. and Paris, P. C. (1973) Threshold for fatigue crack propagation and effects of load ratio and frequency, A.S.T.M. STP 536, 79-94.
[15] Schijve, J. (1960) Fatigue crack propagation in light alloy sheet materials and structures, Nat. Aer. & Astr. Res. Inst., Amsterdam, Report No. MP195.
[16] Yokobori, T. and Sato, K. (1976) The effect of frequency on fatigue crack propagation rate and striation spacing in 2024-T3 aluminium alloy and SM-50 steel, Engng Fract. Mech. 18, 81-88.
[17] Musuva, J. K. and Radon, J. C. (1978) The effects of stress ratio, thickness and frequency on fatigue crack growth in structural steel used in marine technology, 2nd European Conf. on Fracture, West Germany. Publ. Proc. ECF2, Dusseldorf, 286-310, (1979).
[18] McCartney, L. N. and Cooper, P. M. (1977) A numerical method of processing fatigue crack propagation data, Engng Fract. Mech. 9, 265-272.
[19] Wessel, E. T. (1968) State of the art of the WOL specimen for K1c fracture toughness testing, Engng Fract. Mech. 1, 77-103.
[20] Dover, W. D. and Hibberd, R. D. (1977) The influence of mean stress and amplitude distribution on random load fatigue crack growth, Engng Fract. Mech. 9, 251-263.
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[21] Dover, W. D. and Boutle, N. F. (1978) The influence of mean stress and thickness on fatigue crack growth behaviour in the aluminium alloy BS2L71, J. Strain Analysis 13, 129-139.
[22] Cooke, R. J., Irving, P. E., Booth, G. S. and Beevers, C. J. (1975) The slow fatigue crack growth and threshold behaviour of a medium carbon alloy steel in air and vacuum, Engng Fract. Mech. 7, 69-77.
[23] Oriani, R. A. and Josephic, P. H. (1974) Equilibrium aspects of hydrogen-induced cracking in steels, Acta metall. 22, 1065-1074. f.