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Constitutive equations for fully bonded metal matrix composites

Herath, Kulothdeepthi Ravindra Bandara, Ph.D.

University of California, Santa Barbara, 1992

Copyright ©1992 by Herath, Kulothdeepthi Ravindra Bandara. All rights reserved.

U M I 300 N. ZeebRd. Ann Arbor, MI 48106

UNIVERSITY OF CALIFORNIA

Santa Barbara

Constitutive equations for fully bonded metal matrix composites A dissertation submitted in partial satisfaction of the

requirement for the degree of

Doctor of Philosophy

in

Mechanical Engineering

by

Kulothdeepthi Ravindra Bandara Herath

Committee:

Professor Frederick A. Leckie, Chairperson

Professor Robert M. McMeeking

Professor Zhigang Suo

Professor G. Lucus

July 1992

The dissertation of Kulothdeepthi Ravindra Bandara Herath is approved

f l . WW

Committee Chairperson

July 1992

ii

copyright by


1992

iii

Acknowledgments

First of all it is my great honor and pleasure to thank my advisor Prof. F. A.

Leckie for advising, assisting and teaching me throughout this study at UCSB. I

would also specially like to thank Dr. S. Jansson for having very useful

discussions and suggestions throughout this study and shearing his previous

knowledge on this material system. I will also appreciate their steady concern for

my professional development and personal welfare. I am also grateful for the

continuous financial support they provided through the Air Force Grant number

AFOSR-90-0132 during this study. In addition to them, I am also indebted to Prof.

M. P. Ranaweera of the University of Peradeniya for his continuous advice and

concern during the past period. I also have to thank him for arranging me to work

with the research group at UCSB. I would also like to thank the Professors who

were in my dissertation committee. In addition to them, I have been very lucky to

be surrounded by a number of very supportive and intellectual people in Mechanical

Engineering and Materials departments of UCSB.

In addition, I would like to thank my office mates Dov Sherman, Shrwai Ho,

Shobha (my wife), Francois Hild and Dominic Dal Bello for their helpful

discussions and sharing ideas.

In addition, I thank my family, relatives and friends for their constant

encouragement through all of my endeavors.

Finally I would like to thank Prof. Keith Kedward for lending his new

Macintosh computer for my dissertation writing, and Mrs. Leah Pollard for many

clerical assistance during the last few years.

K. R. B. Herath

July, 1992

iv

VITA

December 13,1962~Born~Kandy, Sri Lanka.

1985--B.Sc. in Civil Engineering, University of Peradeniya, Sri Lanka.

1986-Instructor, Civil Engineering, University of Peradeniya, Sri Lanka

1987--Assistant Lecturer, Civil Engineering, University of Peradeniya, Sri Lanka

1987-1989—Teaching Assistant, Theoretical and Applied Mechanics, University of

Illinois at Urbana-Champaign, USA.

1989~M.Sc. in Theoretical and Applied Mechanics, University of Illinois at

Urbana-Champaign, USA.

1990-1992—Teaching/ Research assistant, Mechanical Engineering, University of

California at Santa Barbara, USA.

1992-PhD in Mechanical Engineering, University of California at Santa Barbara,

USA.

1992—Lecturer, Civil Engineering, University of Peradeniya, Sri Lanka.

v

ABSTRACT

Constitutive equations for fully bonded metal matrix composites

by


Unidirectionally reinforced metal-matrix composites (MMCs) have excellent

strength and stiffness properties when they are loaded in the fiber direction. An

advantage of these MMCs is that they can also carry relatively high transverse and

shear loads. Therefore these materials can be utilized effectively to make multiaxial

load carrying structural components. To use composites to design such components

it is necessary to have anisotropic constitutive equations developed in general form.

In this study elasto-plastic constitutive equations are developed to describe the

overall mechanical behavior of fully bonded metal-matrix composites when

subjected to cyclic loads.

The constitutive equations developed in this study are implemented in ABAQUS

finite element code as a new user defined material subroutine (UMAT). With the

use of these constitutive laws some structural analysis were performed to predict the

stress/ strain behavior and the failure mechanisms of representative components

such as a plate with a hole, a plate with notches etc. The ability of this model to

describe cyclic loading behavior is also demonstrated. The predictions from these

analyses are useful in design of structural components made of metal matrix

composites.

vi

TABLE OF CONTENTS

INTRODUCTION 1

CHAPTER 1: Proposed Outline of Study 3

1.1) Research Plan 3

1.2) The Composite Material 4

1.3) General Structure of the Yield Surface for a Transversely

Isotropic Material 5

CHAPTER 2: Axisymmetric Component of the Flow Potential 9

2.1) Yield Surfaces and Hardening Rule 9

2.1.1) Initial Yield Surface 12

2.1.2) Subsequent Yield Surface 13

2.1.3) Hardening Rule 15

2.2) Comparison of the Axisymmetric model with Finite Element

Computing Predictions

2.3) Modified Current Yield Surface

2.4) Compressibility

2.5) Summary of the Constitutive Equations Developed for

Axisymmetric Behavior of the Composite Material

Figure Captions and Figures

CHAPTER 3: Shear Component of the Flow Potential

3.1) Yield Surfaces and Hardening Rule

3.1.1) Initial Yield Surface

3.1.2) Limit Surface

3.1.3) Subsequent Yield Surface and Hardening Rules

3.2) Compressibility

vii

19

20

22

23

26

31

31

31

32

33

36

3.3) "Comparison of the shear Model with Experimental Results 37

3.4) Summary of the Constitutive Equations Developed for Shear

Behavior of the Composite Material 37

Figure Captions and Figures 39

CHAPTER 4: Constitutive Equations 43

4.1) Combination of Axisymmetric and Shear Models 43

4.1.1) Elasticity 44

4.1.2) Plasticity 45

4.2) Determination of Parameters in Constitutive Equations 47

4.2.1) Longitudinal Tension Test, K'L 48

4.2.2) Shear Test, K's, K*s and A 48

4.2.3) Transverse Tension Test, K'T 49

4.2.3) Hydrostatic Tension Test, K'LT 50

4.3) Verification of the Constitutive Model from other Experiments 50

4.4) Remarks 52


CHAPTER 5: Implementation of Constitutive Law in ABAQUS 60

5.1) User Material Subroutine 60

5.2) Iterative Scheme 61

5.3) Input Parameters to the ABAQUS 63

5.4) Output Variables from ABAQUS 66

CHAPTER 6: Applications to Structures: Part 1 69

6.1) Plate with a Circular Hole 70

6.1.1) Monotonic Loading 70

6.1.2) Cyclic Loading 73

6.2) Plate with a Center Notch 74

6.3) Plate with Double Edge Notches 77

viii

6.4)'Summary

Figure Captions and Figures

80

83

CHAPTER 7: Applications to Structures: Part 2 116

7.1) Uniformly Loaded Ring 116

7.2) Partially Loaded Ring 119

7.3) Summary 120


CHAPTER 8: Conclusions 140

8.1) Summary 140

8.2) Implications for Future Work 141

REFERENCES 143

APPENDIX 1 147

ix

INTRODUCTION

Unidirectionally reinforced metal-matrix composites (MMCs) have excellent

strength and stiffness properties when they are loaded in the fiber direction. An

advantage of these MMCs is that they can also carry relatively high transverse and

shear loads. Due to this reason MMCs can be used to make structural components

which can carry multiaxial stresses. Therefore we need to have the constitutive

equations and failure criteria to do the necessary design calculations for MMCs

when they are subjected to multiaxial variable loading. The main purpose of the

present study is to develop elastic-plastic constitutive equations for a fully bonded

metal-matrix composite under variable loads and to demonstrate some applications

of these constitutive equations.

When a metal-matrix composite is subjected to proportional loads, it has been

shown that isotropic hardening is sufficient to describe the material behavior, but

when it is subjected to non-proportional and cyclic loading both isotropic and

kinematic hardening are required for an adequate prediction. One of the simplest

constitutive equations which describe both isotropic and kinematic hardening was

that suggested by Krieg [1975]. The constitutive equations for Aluminum based

metal-matrix composites (B/Al and Gr/Al) have been developed by Dvorak and co

workers [1973,1976,1987,1988,1991]. Their theory uses a hardening law

1

suggested by Phillips and co-workers [1976,1979]. In their formulation of

constitutive equations the assumptions which are valid for the hardening of the

Aluminum based metals have been extended to the composite material. It has been

observed that these assumptions are not directly valid for MMC's. It is noticed that

the subsequent yield surfaces obtained using Dvorak's hardening rules are not in

good agreement with the calculated yield surfaces for FP/Al composite system.

Later Aboudi and co-workers [1989,1990] developed a micromechanical model to

predict initial and subsequent yield surfaces for MMCs using the method of cells

and the unified viscoplasticity theory. Although it is based on simple assumptions,

Aboudi's constitutive equations are too complicated for practical use. Jansson

(1992) has developed constitutive equations for MMCs based on a detailed

numerical study. These constitutive equations are developed only for proportional

loading and they describe the mechanical behavior of the FP/Al composite material

very well.

In the present study new constitutive equations are developed for MMCs when

they are subjected to multiaxial variable loading. Therefore this constitutive model

can be used to predict the cyclic loading behavior of components made of the

composite material. The constitutive equations developed in this study are

implemented into the ABAQUS finite element code as a new user material model.

Then using these constitutive equations some numerical calculations are performed

for structural components made of FP/Al to understand their stress-strain behavior

and the failure mechanisms.

2

CHAPTER 1

Proposed Outline of Study

1.1 Research Plan

Because of the limited availability of composite materials in quantity and shape

the information available from standard mechanical tests is small. Using the least

possible amount of information (part experiment and part finite element

calculations) we shall attempt to describe the behavior of metat-matrix composite in

the following way,

1) Develop the theoretical frame work for constitutive equations which

describes the multiaxial behavior of a given MMC system under variable

loads.

2) Determine the necessary parameters in the constitutive equations from

available experimental data.

3) Implement the constitutive equations as a UMAT routine in the ABAQUS

[Hibbit et al 1988] finite element code.

3

4) Evaluate the ability of the constitutive equations to predict the behavior of

structural components with representative stress concentrations, when

subjected to cyclic loading.

1.2 The Composite Material

The composite material studied herein is the Du Pont's A1 matrix [Champion et

al, 1978] with continuous FP fibers in a unidirectional lay up. It is fabricated by

preparing the FP fibers into tapes by using a fugitive binder and the tapes are

subsequendy laid up in a metal mold in the desired orientation. The binder is burned

away and the mold is vacuum-infiltrated with the molten matrix. The composite

was available in the form of a plate 150x150x12.5 mm. The fiber volume fraction

was determined to be 55%.

The matrix material is 2 wt% Li-Al binary alloy, which exhibits kinematic and

isotropic hardening behavior. The modulus of the matrix Em 68.9 GPa, Poisson's

ratio vm 0.32, the yield strength Ym 94 MPa, the ultimate strength 130 MPa, and

the failure strain is 30% [Jansson 1991].

The FP fiber consists of 99% pure crystalline a-alumina (AI2O3) coated with

Silica that improves the strength of the fiber and aids the wetting by the molten

4

metal. The Lithium also promotes the wetting of the Alumina fibers that forms a

strong matrix-fiber interface [Jansson 1991]. Fibers are considered to be isotropic.

The modulus of fiber Ef 344.5 GPa, the Poisson's ratio VF 0.26, approximate fiber

diameter 20 |im, tensile strength Cf for a 6.4 mm gage length specimen is 1.9-2.1

GPa, and the fracture strain of the fiber is 0.3-0.4 % [Jansson 1991].

Based on the experimental and computational results [Jansson 1991] the

interface of this composite material is assumed to be fully bonded.

1.3 General Structure of the Yield Surface for a Transversely

Isotropic Material

For modeling purposes the fibers are assumed to be long parallel circular

cylinders; in reality they have slightly different diameters which are distributed

randomly. However the fiber volume fraction is taken to be nearly uniform in the

transverse plane. The fully bonded metal-matrix composite system is taken to be

homogeneous and transversely isotropic.

The Cartesian coordinate axes system xi is defined as indicated in figure 2.1

where the axis xi is parallel to the fiber direction and axes X2, X3 are in the

transverse plane. For a transversely isotropic material the invariants of the stress

tensor a are,

Il-Oll . 12=2^022+033) . I3=^(ct22-033)2 +C$3 , l4=0?2+0?3 , I5HCT| (1.1)

II is directly related to the stresses in longitudinal direction. I2 and I3 can be

related to the Mohr's circle in the transverse plane, where I2 is the center of the

circle which is the mean normal stress on transverse plane and VI3 is the radius of

the circle which relates to the shear stresses acting on the transverse plane. I4 is

related to the anti plane shear stresses acting on the composite. I5 is the only term

which is cubic in stress.

In general the initial yield function can be written in terms of the stress

invariants

F=F(I1II2I I3) I4,15) (1.2)

This is still a very general result and simplifications are required if it is to be

useful. For proportional loading Jansson [1992a] proposed the following initial

yield function for FP/A1,

6

r„ C?1 (q22+g33)2 gll(CT22+g33) fo22'^)^^ Q?2 + q?3

K' 2 K" ' 2 " K" ' 2 K" ' 2 K" ' 2 IV L JV T 1S. L T FT>TS K AS

where the K's are material constants.

This is the simplest form of a quadratic yield function. The first three terms in

the right hand side of equation 1.3 are the contribution from axisymmetric stresses

with respect to the fiber direction and the other terms are the contribution from shear

stresses acting on the matrix. Longitudinal and transverse biaxial loading with

stress components of same sign will load the fiber in the longitudinal direction so

the initial yield stress for those loadings are large. For transverse

tension/compression, transverse shear and anti-plane shear loading the fiber will

induce a stress concentration which will result in an initial yield stress which is

smaller than that for the matrix. Therefore K'L , K'T and K'LJ are expected to be

larger than K'TS and K'AS. Experiments (as explained in section 4.2) indicated this

is to be the case where the constants are found to be K'L=300 MPa, K'T=360 MPa,

K'lt=247 MPa and K'As=K,Ts=42 MPa.

The composite behavior will be divided into two parts, one for the fiber

dominated behavior for axisymmetric loading and the other for the matrix

dominated behavior for shear loading. An axisymmetric unit cylinder model is

studied to describe the fiber dominated behavior and is developed in Chapter 2. In

7

this model the matrix is assumed to be an elastic-perfectly plastic material which

obeys von Mises yield condition. The matrix dominated behavior studied in Chapter

3 characterises the behavior by using the two surface plasticity model (Krieg 1975)

in which the kinematic and isotropic hardening of the matrix is used to describe the

hardening behavior of the composite. By combining the fiber dominated and matrix

dominated models we develop a rather complete set of constitutive equations which

describe the elastic-plastic behavior of the metal-matrix composite. These combined

equations are developed in Chapter 4 and some experiments are also proposed to

determine the necessary parameters in the constitutive equations.

The constitutive equations are then implemented into ABAQUS finite element

code as a new user defined material subroutine. The procedure is explained in

Chapter 5. Using this new user material subroutine some structural components

(with stress concentrations) are analyzed under different loading conditions to

determine the stress and strain distributions. The results obtained from these studies

are discussed in Chapters 6 and 7. These stress and strain distributions will provide

an insight into the local failure conditions in the composite.

8

CHAPTER 2

Axisymmetric Component of the Flow Potential

2.1 Yield Surfaces and Hardening Rule

In this Chapter we shall study the properties when the composite material is

subjected to axisymmetric loading. A detailed micro-mechanical study is performed

to obtain the current yield surface and the hardening rule by using a unit cylinder

model with a single fiber surrounded by the matrix material. The fiber volume

fraction is designated by c which is 0.55 for FP/A1. The matrix can be taken as a

thin layer around the fiber due to this high volume fraction of fibers in the FP/A1

system. Therefore the radial stress is assumed to be constant throughout the matrix

layer.

The axisymmetric applied stress vector to the unit cylinder (figure 2.2) is

defined with the use of the global coordinate system (figure 2.1) as follows,

Q I = (2.1)

P

9

where Q=CTn and P=^-<022+033) . 2

To obtain a general solution to the elasticity problem we have to satisfy both

equilibrium and compatibility equations. This is complicated and numerical

procedures are necessary to get the solution. One of the simplest ways to solve this

problem approximately is to assume uniform stresses in the entire matrix phase

which will satisfy the equilibrium equations in both phases. When this assumption

is made it is quickly discovered that the yield surface in the P, Q space consists of

two parallel straight lines given by

-^-+-^± 1=0. KL KT

Computational studies, referred to later, suggest that for equal biaxial stress states

this condition is an overestimate of the yield condition. Therefore to get a better

yield surface the equilibrium equation in the hoop direction is relaxed and the

compatibility of the displacements across the interface is satisfied in this study.

Since we assume a perfect bond condition for FP/A1 composite system, the

displacements ui are continuous across the interface. Therefore

u? = u£ no sliding of fibers

up = uf no debonding of matrix

10

where superscripts m and f denote the matrix and the fiber, Subscripts r and z

denote the direction of the displacement in the r and z direction according to the axes

system shown in figure 2.2.

Now by considering the equilibrium in the fiber direction and the compatibility

of displacements at the interface, the elasticity problem for the unit cylinder model is

solved approximately. The elastic solution gives the matrix (local) stresses at the

interface (figure 2.2) in terms of global stresses £ as,

By solving the cylinder model Ly are found to be functions of the elastic properties

of the constituents and the fiber volume fraction. General expressions for Ly are

given in Appendix 1. For the FP/A1 metal-matrix composite this tensor is

[A]=[L][Z] (2.2)

CJZ

where [o]= G0

.°r

Lu L12

and [L]= LJJ L22

0 1

0.318 0.349

[L]= 0.02 0.59

0

11

2.1.1 Initial Yield Surface

Because the behavior under the current loading is dominated by the elastic

properties of the fiber the analysis is simplified by assuming that the matrix is an

elastic-perfectly plastic material which obeys the von Mises yield condition.

Therefore the matrix stresses must satisfy the following condition at yield,

where a'ij are the deviatoric stresses at the interface of the matrix and ^eff is the

effective stress. Ym is the yield stress of the matrix which is 94 MPa.

The matrix stresses a'ij in equation 2.3 are substituted by applied stress vector

£ (equation 2.2) to obtain the yield condition in terms of applied stresses as,

f=S£+ El--PQ . i = o (2.4) KI K$ Klj

where KL, KJ and KLT are constants which are functions of the elastic properties

of the constituents and the fiber volume fraction. General expressions for KL, KT

arid Klt are obtained using the elastic solution of the cylinder model and are given

in Appendix 1. From the unit cylinder model, using equations in Appendix 1 it

is found that KL=305 MPa, KT=165 MPa and KLT=175 MPa for FP/A1. They are

12

relatively large compared to K's (which is explained in Chapter 3) because they

strongly depend on the fiber behavior.

The initial yield surface defined by equation 2.4 is a general equation of a conic

section. For a stable material yield surface has to be convex everywhere. Therefore

using the properties of conic sections [Nichols, Kalin 1972] it is found that the

initial yield surface given in equation 2.4 represents an ellipse when it satisfies the

following condition

K j T >^l (2 .5)

As expected the above inequality is satisfied by the initial yield surface given in

equation 2.4 for FP/A1. The equality condition in the equation 2.5 holds only when

the initial yield surface is given by two parallel straight lines.

2.1.2 Subsequent Yield Surface

After the initial yielding, the matrix has both elastic and plastic deformations

while the fiber deforms elastically. The plastic deformations result in a self

equilibrating stress distribution so that the matrix stresses aNi can be written in the

form,

13

O? — ®z " Pz

°e = CTe - Pe (2-6)

O_f = 0r

where a; are the current elastic stresses and pj are the components of the current

residual stresses in the matrix. Since the matrix is assumed to be a thin layer around

the fiber the radial stress is always equal to the applied stress P and there is no

residual stress in the matrix in the radial direction.

In order to achieve a simple mathematical format for the current yield surface it

is useful to define the residual stresses by quantities ai, where

[p]=[K][M][a]. (2.7)

In this equation

[p] = pz Pe

, [K] = Ln L12

L21 L22 . , [M] =

1 rm n

0 i n

and [a] = az

-Or-

Here the constants m and n are functions of the elastic properties of the constituents

and defined in Appendix 1. For FP/A1 these constants are found to be m= -2.14

and n= -1.59.

14

The current yield surface is obtained by substituting the current matrix stresses

given in equation 2.6 into the yield condition (equation 2.3). Then using applied

stresses P, Q (equation 2.1) and the residual stresses ai (equation 2.7) the current

yield surface can be written in the following mathematical form.

f _ (Q - az)2 [ (P - cxr)2 (Q - ocz)(P - ar) 1=0

p2 KL p2 K$ p2 KLT

It is found that the P depends on the current values of residual stress ar. Analytical

expression for p is given in Appendix 1. Therefore the size of the current yield

surface changes with current loading. The current yield surface given in equation

2.8 is also an ellipse and it does not rotate w.r.t. the initial yield surface.

2.1.3 Hardening Rule

Applying equilibrium in the axial direction and compatibility conditions at the

interface for the current stress state the relation between local plastic strains and

local residual stresses can be found as follows,

[epm] = [A][p] * (2.9)

15

where [epm] =

,pm

pm

,pm

[A] = An A12

A21 A22

A31 A32 .

and

The constants Ay depend on the elastic properties of the constituents and the

fiber volume fraction. By solving the elasto-plastic unit cylinder model, we can get

analytical expressions for Ajj which are given in Appendix 1. For FP/A1 we

obtain,

[A] = 10 -11 1.689 -0.464

-0.526 1.451

-1.163 -0.987 J

Pa"

Using the virtual work principle [Cocks and Leckie 1987] for the same unit

cylinder model, the matrix (local) plastic strains are related to the composite (global)

plastic strains through the tensor Ljj (defined in equation 2.2) as follows,

[de1*] =(l-c)[L]T[depm] (2.10)

16

where [deP°] = deT

de?c are the composite plastic strains.

Using equations 2.7, 2.9 and 2.10 the relation between composite plastic

strains and the residual stresses are determined as follows,

[da]=lH][d£pc]. (2.11)

where [H] = H i i H 1 2

H21 H22 = {(1 -c)[L]T[ A] [K] [M]}_1

The constants Hy depends on the elastic properties of the constituents and the fiber

volume fraction. For FP/A1 we obtain,

[H] = 10- 52.39 22.28

21.84 12.49 J MPa

The relations given in equation 2.11 define the hardening rule.

Although the current yield surface and the hardening rules for this model are

obtained in the cylindrical coordinate system, we need to have them in the global

Cartesian coordinate system for future use (see Chapters 4 and 5). Therefore the

17

residual Stresses and the plastic strains in cylindrical coordinate system can be

transformed into Cartesian coordinate system through following transformations,

<Xii = Oz and Ot2+3 = 20r (2.12)

£l! = ef and 8^ =2erpc (2.13)

Here an, (X2+3> epli and ePa are in Cartesian coordinate system. The last

definition given in equation 2.13 is the relation between area strain and radial strain.

Where £Pa = ep22 + ep33 •

Now with the use of equations 2.12 and 2.13, the hardening rule obtained from

the model (equation 2.11) can be rewritten in the Cartesian coordinate system as

follows,

da i i " z„ Z I 2 " de f i

. da2+3 . . Z2i Z22 . X.

where Zy depend on the elastic properties of the constituents and the fiber volume

fraction. For FP/A1 we obtain,

18

[Z]= 1(T 52.39 11.14

43.68 12.49 J MPa

The general expressions for Zij are given in Appendix 1. We can see from this

model the hardening rule obtained for axisymmetric loading is linear. For the

completeness of the hardening rules we need to know the increments of the plastic

strains. They are obtained from the normality condition as explained in section 2.3

of this Chapter.

2.2 Comparison of the Axisymmetric Model with Finite Element

Computing Predictions

Using the unit cylinder model the motion of the yield surface can be found for

both proportional and non-proportional axisymmetric loading. The predictions of

the model are compared with those obtained using finite element calculations.

Figure 2.3 shows the motion of the composite yield surface for various

axisymmetric proportional loading paths. These loads are chosen to be much higher

than the operating loading range for FP/A1 system. Then as the next step we want to

investigate the behavior for different non-proportional loading paths. The motion of

the composite yield surface for axisymmetric non-proportional loading paths is

shown in figure 2.4. Maximum and minimum applied load levels for both P and

Q were 900 MPa and -900 MPa. Notations shown in figure 2.4 are explained

here. In figure 2.4 Q=P corresponds to a proportional loading path where the

19

ratio Q/P was kept as 1 throughout the loading path. Then Q-->P means load Q was

applied first from 0 to maximum (900 MPa) or minimum (-900 MPa) level of Q

while keeping P at zero. After Q has reached to 900 MPa or -900 MPa (depending

on the loading direction) the load Q was kept constant at that final load level and

load P was applied from 0 to maximum (900 MPa) or minimum (-900 MPa) level

of P (again depending on the loading direction). Similarly P~>Q means vice versa

of Q-->P. These two variable loading procedures will produce non-proportional

loading paths. The current yield surfaces obtained from the model agree very well

with the finite element predictions for both proportional and non-proportional

loading as shown in figures 2.3 and 2.4.

2.3 Modified Current Yield Surface

The model was tested for small and very large (out of the operating loading

range for FP/A1 material system) proportional and non-proportional loading as

explained in section 2.2. It is found that the change in the size of the current yield

surface given in equation 2.8 is negligible compared to the initial yield surface

(equation 2.4) for all of those possible loading. Therefore using these observations,

we can simplify the form of the current yield surface given in equation 2.8 as a

kinematic hardening behavior by taking p=l. Now the modified current yield

surface can be written as,

20

„ (Q-OQ2 (P-OR)2 (Q-OGCP-O,.) f = ^ +- — - ^ — - 1=0 (2.15)

K2 Kf KlT

in which KL, KT and KLT are the same constants as defined for the initial yield

surface. Therefore the current yield surface can be represented by pure translation

of the initial yield surface.

Using equations 2.1 and 2.12 with equation 2.15 the current yield surface can

be written in the Cartesian coordinate system as follows,

f_(gn-an)~ ^ (q22+CT33-a2+3)2 (gii'a11)(g22+<*33-^2+3) 1=q (2 16)

K'l2 K'2 K'2t

where K'L=KL, K'T= 2KJ and K'LT= ^ KLT- For FP/A1 these constants are

found to be K'L=305 MPa, K'T=330 MPa and K'LT=247.5 from the equations

given in Appendix 1.

The composite plastic strains are obtained by assuming that the plastic strain

rates are normal to the current yield surface, ie.

< = (2.17)

21

where f is the composite flow potential given in equation 2.16. The plastic

multiplier dA, is positive and depends on the stress and deformation history. During

the plastic loading process dA. can be found by the consistency condition, df=0, as

follows,

d f =f d o ' + 5s i r d a ' > + 5 i^=° (2.18)

dan and da2+3 in equation 2.18 are given in terms of plastic strain increments

in equation 2.14. Substituting the normality condition of equation 2.17 for plastic

strains, equation 2.18 can be simplified to give dA. as,

dA =

df_ dOi

dq

(2.19)

where T = 9f da li

df d022

Zn

L Z21

2Z12

2 Z22 J

_3f_ 9CTH

3f .da 22.

(2.20)

2 . 4 C o m p r e s s i b i l i t y

For isotropic materials plastic deformation is incompressible, ie.

dePn+deP22+deP33=0. However it can be deduced from the current anisotropic

22

model that the volumetric strain during the plastic deformation is non zero and is

given by,

dV = dA, |2(oi i-aii) 1 1 [K'L2 K',2

tJ + (CT22+a33-a2+3) 4

K'f 1

K'2 LT

(2.21)

where dV=dePn+deP22+deP33.

For a uniaxial tension test in longitudinal direction the quantity dV/dePn is

calculated from the model and found that it varies from -2.07 to -2.30 (see Table

2.1). Since the plastic flow is constrained in the fiber direction, the plastic strains

in the transverse directions are found to be larger than the plastic strains in the fiber

direction. For an isotropic material, the quantity dV/dePn is zero during the plastic

deformation. When the total strains are considered for the above test the ratio of the

volumetric strain to longitudinal strain, e'kk/e'll. is 0.44 when the material is elastic

and drops to 0.28 during the plastic deformation (see Table 2.1).

2.5 Summary of the Constitutive Equations Developed for

Axisymmetric Behavior of the Composite Material

The flow potential and hardening rules developed in this study for the

axisymmetric behavior of the metal-matrix composite can be summarized as

follows,

23

Flow potential: f= (giran)2 + (cf22+o'33-a2+3)2 _ (Pi _ j _ q

K'L K'lt

Flow rule def = dX 1 0C?i

Hardening rule: doci = Z± dePk

and from the consistency condition dX, can be found as,

dX =

3f ^ — da, da

where T = 8f 3f 3<Jll d(S22

Zn

L Z21

2 Z12

2 Z22 J

8f 8CTH

8f _d(J22.

24

ai i (MPa) dV/dePn e'kk/e'll

304 -0.00 0.44

348 -2.07 0.42

416 -2.16 0.38

487 -2.23 0.33

527 -2.26 0.32

561 -2.28 0.29

590 -2.29 0.29

611 -2.30 0.28

Table 2.1 Amount of compressibility in a longitudinal tension test

25

FIGURE CAPTIONS: CHAPTER 2

Figure 2.1 Cartesian coordinate axes system for fiber reinforced metal matrix

composite

Figure 2.2 Axisymmetric unit cylinder model with applied loads P and Q

Figure 2.3 Kinematic motion of composite yield surface for various

axisymmetric proportional loading

Figure 2.4 Comparison of kinematic motion of composite yield surfaces for

axisymmetric proportional and non-proportional loading

26

a,.

X3

Figure 2.1

27

Matrix

Fiber.

Stresses in the matrix at the interface

Figure 2.2

28

Q=991 MPa P=0 *• MODEL

• FEM

P=-1052 MPa Q=0

litial

P=1052 MPa Q=0

-10

Q=-991 MPa P=0

-20 -20 - 1 5 - 1 0 5 0 5 1 0 15 20

P/Ym

Figure 2.3

29

CHAPTER 3

Shear Component of the Flow Potential

3.1 Yield Surfaces and Hardening Rule

In this Chapter we shall study the properties when the composite material is

subjected to shear loading. The behavior of the composite material under shear

loading is dominated by the matrix behavior. When attempts have been made to

describe the behavior of composite material when subjected to cyclic shear loading

it was discovered that both isotropic and kinematic hardening are required for an

adequate prediction of the behavior. For this kind of behavior one of the simplest

constitutive equations developed was that suggested by Krieg [1975]. His theory

involved a memory of a second order tensor and some scalars, where the tensor is

related to back stress which comes from kinematic hardening and the scalars are

related to isotropic hardening.

3 . 1 . 1 I n i t i a l Y i e l d S u r f a c e

Figure 3.1 shows the shear stress-strain diagrams for metal-matrix composite

for transverse and anti plane shear tests. Both shear experiments followed almost

31

the same stress-strain path as shown in figure 3.1. Also from the numerical

calculations for FP/A1 it has been observed that the calculated initial yield surface is

a circle in a 12, <*13 plane [Jansson 1991]. These observations allow a great

simplification to the formulation of constitutive equations for shear and enable us to

assume that the transverse and anti plane shear responses are the same for the

composite material. Therefore the initial yield stress in shear can be characterized by

a single constant K's for both transverse and anti plane shear loading (ie.

K'S=K'TS=K'AS)- The value of initial yield stress K's is found to be 42 MPa from

the experiments shown in figure 3.1. From the results obtained in Chapter 2 it

can be immediately seen that K'L, K'T and K'LT » K'S. This is due to the

influence of the strong elastic fibers in the composite system.

Based on the above assumptions when multiaxial shear stresses are acting on

the composite the initial yield surface can be represented by a sphere of radius K's

in the shear stress space and can be written as follows,

-ka22-C33)2+C232+Cn22+<Tl 32

F = 4 1 = 0 ( 3 .

K's2

3 . 1 . 2 L i m i t S u r f a c e

When the composite is subjected to transverse and anti plane shear loading the

material exhibits a hardening behavior after the initial yielding and gradually

32

approaches a limit stress level as shown in figure 3.1. This is expected because

the behavior of the composite material under the shear loading is dominated by the

behavior of the matrix material. Since the composite material behavior is assumed to

be the same for transverse and anti plane shear loading the limit shear stress has

been taken to be 105 MPa from the experimental observations (figure 3.1).

Therefore when the composite is subjected to multiaxial shear stresses the limit

surface can be simply written as,

1 / * * x 2 * 2 * 2 * 2 7(^22-^33) +023 +<*12 +Oi3

F = 4 1=0 (3.2)

Ks2

This limit surface represents a sphere in the shear stress space, where the radius

K*s is the limit shear strength and it is 105 MPa for the considered material system.

3 . 1 . 3 S u b s e q u e n t Y i e l d S u r f a c e a n d H a r d e n i n g R u l e s

When the composite material was subjected to cyclic shear loading the

experimental stress-strain behavior observed is shown in figure 3.2 (Jansson

1991). The experimental observations show that both kinematic and isotropic

hardening are necessary to describe the material behavior. To capture these features

the two surface plasticity model [Krieg 1975, McDowell 1985, 1989 and Ohno

1986], as briefly explained below, is used to obtain the current yield surface and

33

hardening rule for the shear part of the flow potential. Using a simple monotonic

shear loading experiment we can obtain the necessary constants to describe the two

surface model by curve fitting. Then this model can be used to describe the

behavior of the composite material when it is subjected to cyclic shear loading.

In order to describe the transient elastoplastic behavior of materials observed

just after initial yielding (or reverse yielding), a limiting surface (F* = 0) is

introduced inside which a yield surface (F=0) is allowed to translate (in kinematic

hardening), as shown in figure 3.3, where a denotes the center of the current

yield surface, while K*s and Ks are the radii of limit and yield surfaces. After the

onset of yielding, the yield surface is assumed to translate so that stress a

approaches the stress point a* on F* = 0 at which the outward normal is

codirectional with the outward normal to the yield surface (F=0) at stress point <T.

The translation of the yield surface within the limit surface describes the transient

elastoplastic behavior after yielding. The plastic tangent modulus is taken to be

larger when the stress state is remote from the limit surface. Therefore the model

varies smoothly from a high to a low stiffness during the plastic loading process.

Using this model with notations shown in Figure 3.3 we obtain the following

current yield surface for the shear component of the flow potential,

34

-1 = 0 (3.3)

Where a defines the back stress tensor which comes from the kinematic hardening.

The hardening rule is directly obtained from the two surface model and is given

where A is a constant which defines the hardening rule for the shear model. It is

calculated by curve fitting (see section 4.2.2) and is 11.014 for this material, and

T|=CT*-0, where T| is the distance between possible contact points in two surfaces.

We can easily see equation 3.4 gives a non linear hardening behavior for the shear

part of the flow potential.

The composite plastic strains are obtained from the normality condition as

follows,

by,

da = Atari (3.4))

d i f -A i t (3.5)

35

where F is the composite flow potential given in equation 3.3. The plastic multiplier

dA. is positive and depends on the stress and deformation history. During the plastic

loading process dA, can be found by the consistency condition as follows,

3F 3 F dF = — dCi + —dttj = 0 (3.6)

aai a«i

dtXj in equation 3.6 are given in terms of dA, in equation 3.4. Therefore equation

3.6 can be simplified to give dA, as follows,

3F . —do;

dA, = — (3.7) aF A — Arh doj

where r\\ = a*j - aj.

3 . 2 C o m p r e s s i b i l i t y

From the model developed for the shear behavior of the composite material, it

can be seen that there is no volumetric strain during the plastic deformation, ie

dePi i+deP22+dep33=0.

36

3 . 3 C o m p a r i s o n o f t h e S h e a r M o d e l w i t h E x p e r i m e n t a l R e s u l t s

Figure 3.1 shows the shear stress-strain diagrams for transverse and anti

plane shear tests. After curve fitting with these monotonically loaded shear

experimental results, the necessary parameters (K's, K*s and A) to describe the

constitutive model were obtained as explained in section 4.2.2. Then this model

was used to generate the stress-strain behavior for a cyclic shear test. The model is

found to be in a good agreement with the experimental result (Jansson 1991) as

shown in figure 3.2.

3 . 4 S u m m a r y o f t h e C o n s t i t u t i v e E q u a t i o n s D e v e l o p e d f o r S h e a r

Behavior of the Composite Material

The flow potential, limit surface and hardening rules developed in this study for

the shear behavior of the metal-matrix composite can be summarized as follows,

1 0 0 0 2 J-{022-CJ33-CX2-3)"+(023-(X23)"+(012-CX 12)~+(C 13"(* 13)

Flow potential: F = 4 1 = 0 K's2

i * * \ 2 * 2 * 2 # 2 + -(a22-033) +a23 +c12 +<*13

Limit surface: F = 1=0

Ks2

37

3F Flow rule: def = dA, —

00i

Hardening rule: da, = A dA, T|i

and from the consistency condition dA. can be found as,

r—Arli 8aj

38


Figure 3.1 Shear stress - strain diagrams for transverse and anti plane shear

loading experiments [Jansson 1991]

Figure 3.2 Cyclic anti plane shear test

Figure 3.3 Limit and yield surfaces in two surface plasticity model

39

Transverse - experiment

j-oqoc Anti plane - experiment « • • • • Transverse - computed

Model

0.5 '.0

r •*) I 5 2.0

Figure 3.1

40

15

10

5

0

•5

10 expenment model

15 •0.6 •0.4 •0.2 0 0.2 0.4 0.6 0.8

Y12 <%>

Figure 3.2

41

F=0 , Yield surface

F*=0 , Limit surface

Figure 3.3

42

CHAPTER 4

Constitutive Equations

4.1 Combination of Axisymmetric and Shear Models

Two constitutive models have been developed in this study to describe the

behavior of composite material under different loading conditions; one for the fiber

dominated axisymmetric behavior (Chapter 2) and the other for the matrix

dominated shear behavior (Chapter 3). In this Chapter these two models are

combined to describe the multiaxial behavior of the composite under general loading

and a systematic procedure for determining the material parameters is developed.

This combined model will give the complete set of constitutive equations for a

metal-matrix composite.

When the composite is loaded beyond the initial yielding, the increments of the

total strains in the composite are given by

dej= def+deP (4.1)

43

where eej' are the elastic strains and ePj are the plastic strains in the composite. Each

of these vector consists of six components; den, de22, d£33, dei2, dei3 and d£23.

4 . 1 . 1 E l a s t i c i t y

For a transversely isotropic material, such as FP/A1 system, increments of the

elastic strains are given by

def

de°

de 33

dYf2

1 -VL -VL E, E, E,

-V|. 1 ~VT E, ET Et

-v,. -v1

E,.

0 0 0

0 0 0

0 0 0

0 0 0

0 0

1 0 0 0 0 — 0

CL

0 0 0 0 0 1

'da, '

da 22

da 33

dx,

dx,

dx 23

(4.2)

where El, Ej, Gl, Vl and Vt are the five independent elastic constants for a

transversely isotropic material. These constants can be determined experimentally

44

[Jansson' 1991] and for the studied composite material system EL=225 GPa,

ET=150 GPa, GT=55 GPa, GL=58 GPa, VL=0.28 and VX=0.31. The transverse

shear modulus Gr is given by the relation

GT—^L-2(1+vr)

4 . 1 . 2 P l a s t i c i t y

When the composite material is subjected to multiaxial stresses the current yield

surface can be obtained by combining the axisymmetric model developed in Chapter

2 and the shear model developed in Chapter 3 as follows,

P_(qn-<*n)2 (q2Z+q33-^2+3)2 (qil-a|lXq22+<*33-ttZt3)

(4.3) Kt 2 T-r ' 2 V I 2

L K- T LT

4i022-q33-«2-3)2+(023-023)2+(012-ctI2)2+(013-Cl3)2 4 [ = 0

K'<?

where K'L, K'T, K'LT and K's are constants and calculated from the initial yield

stresses of the anisotropic composite material and aj are the back stresses which are

zero at the initial yield. The first three terms in the equation 4.3 are obtained from

the fiber dominated axisymmetric model and the last four terms are obtained from

the matrix dominated shear model.

45

The function F is such that

dey = 0 for F< 0

dey = 0 for F= 0 and dF < 0

d£y * 0 for F= 0 and dF = 0

(elasticity)

(elastic unloading) (4.4)

(plastic loading)

The plastic strains are obtained by assuming the normality condition, ie.

de»P-^ 55T (4,5)

where F is the flow potential given in equation 4.3 and dA. is the plastic multiplier.

The consistency condition, dF=0, is used to find the current value of dA. as follows.

3F . K—ctoi

dX = - "55 *5 (4"6)

^LZ- — + — A n -doti 1J So] 3 a, 1

where Zjj are constants related to axisymmetric hardening and A is the constant

related to shear hardening, rji is the distance between possible contact points in limit

and yield surfaces in shear as shown in figure 3.3. ie rji = a*i - Gj.

For the fiber dominated axisymmetric behavior the hardening rules are given

by,

46

da; = Zy dePj (4.7)

and for the matrix dominated shear behavior the hardening rules are given by,

doti = A dA, rii (4.8)

The limit surface for the matrix dominated shear behavior is required in the

constitutive equations and is given by

1 / * * \ 2 * 2 * 2 * 2 . 7(022-033) +°23 +Ol2 +CJ13

F = * 1 = 0 ( 4 . 9 ) Ks*2

Equations 4.1 - 4.9 are the complete set of constitutive equations to describe the

composite material behavior under general loading.

4.2 Determination of Parameters in Constitutive Equations

A systematic procedure to find the parameters in constitutive equations is

explained in this section. Some of the parameters are determined by calculations and

others by experiments. Since the experimental and computed (finite element) stress-

strain graphs for longitudinal tension and transverse tension tests were in good

agreement the finite element procedures are convenient and used to find K'L and

47

K'T. A finite element calculation for a hydrostatic tension test is used to find K'LT

and the monotonically loaded shear experiments are used to obtain K's, K*s and

A. The constants Zjj in the constitutive equations are obtained from the cylinder

model as explained in Chapter 2.

4.2.1 Longitudinal Tension Test, K'l

The stress-strain diagram obtained from a longitudinal tension test is shown in

figure 4.L and is used to find K'L- The stress corresponding to the initial yield is

K'L and found to be 300 MPa (as indicated in the figure 4.1) from the test.

4.2.2 Shear Test, K's, K*s and A

The stress-strain diagrams for transverse and anti plane shear tests are shown in

figure 3.1 of Chapter 3 and are used to obtain K's, K*s and A. The stress-strain

diagrams (figure 3.1) exhibit a hardening behavior after the initial yielding and

gradually approaches a limit stress level. From those experimental observations

initial yield stress, K's, and limit stress, K*s, are selected to be 42 MPa and 105

MPa. The only other necessary parameter to describe the shear behavior is the

constant A in equation 4.8 which defines the slope of the hardening curve. It is

determined by curve fitting the model with shear experimental results as explained

below. During the plastic deformation the flow potential, hardening rule and

consistent condition can be used to obtain the following relation

48

di. dy to

(Ks--tb)K's(l-1- di ) G dy TO

where io is a shear stress selected in the plastic region and Idx/d'ylxo is the

corresponding slope at the stress-strain diagram for that stress To as indicated in

figure 4.6. The constant G is the shear modulus and K's and K*s are as defined

above. Using the experimental shear stress-strain diagrams in figure 3.1, the

constant A can be calculated by choosing a stress level in plastic region and using

equation 4.10. Then using this calculated A, the entire shear stress-strain behavior

is predicted from the model and compared with experimental results. This

procedure has been repeated until the best fit of the model is obtained. The best fit

of the model is obtained when A is 11.014.

4.2.3 Transverse Tension Test, K'T

The stress-strain diagram obtained from a transverse tension test is shown in

figure 4.3 and is used to calculate K'T. The only non zero stress for this test is

033. When the initial yield stress, a, in the transverse tension is obtained from the

test, K'T can be calculated using the equation 4.3 as follows,

49

1 _ 1 l_ K't Cf2 4 K's

(4.10)

Knowing K's from the shear test, K'T is calculated to be 360 MPa from the

transverse tension test.

4 . 2 . 4 H y d r o s t a t i c T e n s i o n T e s t , K'LT

From finite element calculations the stress-strain diagram can be obtained for the

composite material when it is subjected to hydrostatic tension (ie <Ji 1=022=^33)

and is used to calculate K'lt- IF the initial yield stress for the hydrostatic tension

test is <Jh then K'lt can be calculated using the equation 4.3 as follows,

Knowing K'l from the longitudinal tension test and K't from the transverse

tension test, K'lt is calculated to be 247 MPa from the hydrostatic tension test.

4.3 Verification of the Constitutive Model from Other Experiments

Initial yield stresses from both longitudinal tension test and transverse tension

test are used to find the parameters in the constitutive equations as explained in

section 4.2. Then the other experimental and numerical results available from both

2 , 1 , 4 I (4.11) rrt 2 tt( 2 TT »2 IV LT K-L ^ T °H

50

longitudinal and transverse tension tests are used to verify the predictions from the

constitutive model as explained in this section.

Figures 4.1 and 4.2 are from a longitudinal tension test, where figure 4.1

is the stress-strain diagram and figure 4.2 is the relation between transverse

contraction and longitudinal strain for this test. The initial yield stress from this test

has already been used to calculate K'L as explained in section 4.2.1. Then using the

model longitudinal stress-strain diagram and the transverse contraction are predicted

and shown in figures 4.1 and 4.2. It can be seen that the predictions from the

model are in very good agreement with experimental and numerical results for this

test.

Figures 4.3, 4.4, and 4.5 are from a transverse tension test, where figure

4.3 is the stress-strain diagram, figure 4.4 is the relation between out of plane

and transverse strain and figure 4.5 is the relation between longitudinal and

transverse strain for this test. The initial yield stress from this test is used to

calculate K'x as explained in section 4.2.3. Then the model was used to predict the

stress-strain behavior of the material under transverse tension loading and compared

with experimental and computed results as shown in figures 4.3, 4.4, and 4.5.

It can be seen that the predictions obtained from the model for this test are also in

good agreement with the experimental and numerical results.

51

4.4 Remarks

The constitutive laws developed in this study represent the mechanical behavior

of the fully bonded composite material system (FP/A1) very well. This model has

the ability to predict the material behavior of the composite under complex

multiaxial loading. Remainder of this study is devoted to demonstrate how to use

these constitutive equations in design of structural components.

52


Figure 4.1 Stress - strain curve for longitudinal tension test [Jansson 1991]

Figure 4.2 The relation between transverse contraction and longitudinal strain

for specimen loaded in longitudinal tension [Jansson 1991]

Figure 4.3 Stress - strain curve for transverse tension test [Jansson 1991]

Figure 4.4 The relation between out of plane and transverse strain for specimen

loaded in transverse tension [Jansson 1991]

Figure 4.5 The relation between longitudinal and transverse strain for specimen

loaded in transverse tension [Jansson 1991]

Figure 4.6 Typical shear stress-strain diagram

53

Experiment Computed

Model

K'L

0

0. 3 0 . 0 0. 4 0. I 0 . 2

fin (S)

Figure 4.1

54

0.00

0*11

Experiment Computed

Mode L

-0.02

-0.C6

-0.08

-0 .10

-0.12 0.3 0.2 0.0

C i i ( * )

Figure 4.2

55

Experiment Computed Wotrix

••••••r Model

0 2 0.4 0.6

£33 (*)

Figure 4.3

56

-0 2

*

: o 0-2 0-4 C-o oa £i3 (X)

Figure 4.4

57

0.00

CTlS

- Experiment Computed

ModeL

-o.ot

0.02

0.04 o.t 0.1 0.0 0.4 0.2

*u (»)

Figure 4.5

58

K's

y %

Figure 4.6

59

CHAPTER 5

Implementation of Constitutive Law in ABAQUS

For the design purpose it is important to understand the stress and strain

distributions of structural components made of MMCs. But due to the limited

availability of these materials, experiments will be expensive to perform on

components. However the constitutive models developed for these materials can be

used to analyze structural problems numerically. This has been done by

implementing constitutive law in ABAQUS finite element code as a user defined

material subroutine and that procedure is explained in this Chapter.

5 . 1 U s e r M a t e r i a l S u b r o u t i n e

The user material subroutine called UMAT in ABAQUS provides an extremely

powerful and flexible tool for analysis. During the structural calculations this

subroutine will be used to define the mechanical constitutive behavior of the

material at each material calculation point. This is very flexible to use in analysis

where the user defines the number of material parameters necessary for the UMAT

and their magnitudes and the number of solution dependent state variables with

60

input data for the structure. These parameters with other variables such as stresses,

strains etc. are passed to the subroutine at each Gauss integration point of the

element. The subroutine calculates the Jacobian matrix of the constitutive model,

3(A<Ji)/3(Aej), which is the change in the i1*1 component of stress at the end of the

increment caused by a change in the j1*1 component of the strain increment array.

The stress tensor and the state variables are also updated to the current values at the

end of the increment. The distributions and magnitudes of these state variables can

be printed out as additional information for the analysis.

5 . 2 I t e r a t i v e S c h e m e

When the deformation during the increment is elastic the stresses satisfy the

condition F < 0 where F is the yield function. Iterations are not necessary in this

case and only the elasticity relations are used to define the Jacobian matrix for the

next increment as

J"j+1 = Ey (5.1)

where Ey is the 6x6 elasticity tensor (which is the inverse of the compliance tensor

given in equation 4.2). The stress increment Aoi and the current stress of"1 are

given by,

61

Actj = E;; A& (5.2)

o?+1 = of + Aq

If of+1 gives the condition F > 0 then the plastic deformation occurs within the

increment. Therefore corrections are needed for the stresses Aai and corresponding p

AX, Aoii and Ae;. To find these corrections an implicit iterative scheme is used

within the subroutine which will ensure the stability of the global convergence. The

method used by Doghri et al, 1989 and Hild et al 1990 is implemented for this

purpose. In that scheme the following non linear equations are simultaneously

solved by the Newton's method.

Yield condition : F"+1 = 0 (5.3)

t)F Flow rule : gjf+1 = Ae? - AX = 0 (5.4)

1

Hardening rule: h"+1 = Aoc, - AX D; =0 (5.5)

Elastic law: of"1"1 - Ejj( ej+1 - e? - Ae? )= 0 (5.6)

p The variables au A8;, Aaj and dX at the end of the time increment must satisfy

the yield condition, flow rule, hardening rule and the elastic law. Since the yield

62

condition F=0 is satisfied throughout the plastic deformation the consistency

condition, dF=0, is automatically satisfied.

The iterative scheme is considered to be converged when the corrections are

smaller than the specified tolerances. After the convergence is achieved the stresses,

strains and the state variables are known at the end of the increment. The Jacobian

matrix, Jij+1, is then calculated as follows,

j?+1 = E - -Jlj Mj

3F 3F (Eik — )(- Emj)

dOjc OCTm

3F ^ 3F 3F _ Ek, Dfe

d<Jk 3ci dak

n+l

(5.7)

where 3Aej

5.3 Input Parameters to the ABAQUS

The following input parameters must be defined in the input file for ABAQUS

with the *USER MATERIAL key word.

Ell (or En): Elastic modulus of the composite in longitudinal direction. For the

FP/AL system Ell is 225 GPa.

63

vtl (or v31): Poisson's ratio of the composite for transverse strain when stressed in

the longitudinal direction. For the FP/Al system vjl is 0.18.

ETT (or E33): Elastic modulus of the composite in transverse direction. For the

FP/Al system ETT is 150 GPa.

vtt (or v33): Poisson's ratio of the composite for transverse strain when stressed in

the other transverse direction. For the FP/Al system Vtt is 0.31

K'L, K'T, K'LT: These constants are related to the yield stresses of the composite

for different experiments. For the FP/Al system K'L=304.86 MPa,

K't=330.22 MPa and K'LT=247.47 MPa.

K's: This is the yield stress of the composite in shear. For the FP/Al system K's is

42 MPa.

Zn, Z12, Z21, Z22: These constants will describe the hardening rule for the

axisymmetric part of the flow potential. For the FP/Al system these

constants are found to be Zn=5239 GPa, Zi2=1114 GPa, Z2i=4368 GPa

and Z22=1250 GPa.

64

A: This constant will define the hardening rule for the shear part of the flow

potential. For the FP/A1 system A is 11.014.

K*s: This is the limit strength of the composite when it is subjected to shear stress.

For the FP/A1 system K*s is 105 MPa.

Ln, L12, L21, L22: These constants will define the relation between the elastic

stresses in the matrix and the applied stresses when the composite is

subjected to axisymmetric loading. For the FP/A1 system Ln=0.318,

Li2=0.349, L21=0.02 and L22=0.59.

m, n: For the FP/A1 system m= -2.14 and n= -1.59.

c: Fiber volume fraction. For the FP/A1 system c=0.55.

Ym: Yield strength of the matrix in uniaxial tension. For A1 matrix Ym=94 MPa.

Efaii: Amount of ductility (or the failure strain) of the matrix material. For A1 matrix

£fail=0.3.

An example of special key words necessary when a user material is used is

given below.

"•MATERIAL, NAME = FP/AL

65

•USER MATERIAL, CONSTANTS = 23

*DEPVAR

* SOLID SECTION, ELSET = ALL, MATERIAL = FP/AL

5.4 Output Variables from ABAQUS

The user material subroutine has the capability to define any number of solution

dependent state variables in addition to the standard output variables such as

stresses and total strains. Using this state variable approach additional macro level

composite variables such as plastic strains, residual stresses and micro level

constituent variables such as fiber and matrix stresses can be determined. These

state variables are calculated at every Gauss integration point and available as

tabular print out form or as contour form in ABAQUS. A list of available solution

dependent state variables are given below.

SDV1-SDV6: Residual stresses developed in the composite. They can be written

according to the order of output as AN, OC2+3, 012-3, A12> «13> and OC23..

66

SDV7-SDV12: Plastic strains developed in the composite. They can be written

according to the order of output as ePn, EP22, eP33>7P12. ̂ 13. and 7P23.

SDV13: Accumulated value of plastic multiplier, ie.

t

X = (5.14) t=o

SDV14: Increment in the plastic multiplier (Ak) for that time increment

SDV15: This is the indicator for the convergence of the local iteration scheme for

constitutive equations. The possible range for this variable is between 0 and

50.

SDV16: This variable gives the amount of axisymmetric hardening. It is defined as

2 2 an +

a2+3 + ail <*2+3 (525

Kt 2 v 2 |/i 2 L ̂T K LT

SDV17: This variable gives the amount of shear hardening. It is defined as

^-ai.3 + ah + a?2 + a?3

K'l (5.16)

67

SDV18: This variable is the equivalent plastic strain for composite. This is defined

as

This is the total accumulation of plastic strain.

SDV19: This variable gives the fiber stress. Knowing of the fiber stresses are

important for the prediction of fiber failure in the composite.

SDV20: This variable gives the void growth ratio (G]± / aeff) in the matrix during its

plastic deformation. Where a^k is the sum of principal stresses in the matrix

and aeff is the effective stress in the matrix.

(5.17)

68

CHAPTER 6

Applications to Structures: Part 1

Some structural components with representative stress concentrations were

analyzed using this new user material subroutine to demonstrate the ability of

constitutive equations to predict the material behavior. It is important to understand

the distribution of stresses and strains in these structural components to get an

insight to the final failure of the structure for design puiposes. An attempt has been

made to establish the governing failure mechanisms for the structures analyzed in

this study. Anisotropic strength properties and failure strains for FP/A1 are

summarized in Table 6.1 and will be used to establish the failure criteria.

Three structural components, a plate with a hole, a plate with a center notch,

and a plate with double edge notches, were analyzed under monotonic loading

using finite element method and the results are discussed in this Chapter. Then the

plate with a hole is analyzed when subjected to cyclic loading and the results are

also discussed in this Chapter. An analysis of a ring reinforced in hoop direction is

performed under two different loading conditions, one with uniform radial loads

69

and the other with partial radial loads, and the results are discussed in the next

Chapter.

6.1 Plate with a Circular Hole

6.1.1 Monotonic Loading

A plate with a hole which is loaded in the fiber direction is analyzed and

discussed in this section. Dimensions and boundary conditions of the problem are

shown in figure 6.1. The length to width ratio is taken to be 5/3 while the hole

diameter is one half of the width of the plate. Due to the symmetries of the structure

only one fourth of the plate was considered for the analysis. A plane stress

calculation was done using displacement controlled boundary conditions as shown

in figure 6.1(b). The ultimate longitudinal tensile stress of the unnotched

composite (CUTS), which is 600 MPa for considered FP/A1 system, is used to

normalize the average stress at the load carrying ligament (C[ig) of the plate and the

load level is defined as % = Oiig / ours-

The stress-strain diagram obtained from this analysis is shown in figure 6.2.

Yielding first occurs at the load level ^,=0.1457 and is indicated in figure 6.2. It

should be noted that the linear behavior of the stress-strain diagram is not

significantly altered due to yielding. When the elastic stresses in the composite are

considered the stress concentration factor at point B (figure 6.1(b)) of the hole

surface was found to be 4.6. The distribution of longitudinal stress in the composite

along the ligament AB is shown in figure 6.3 for three load levels, one before

yielding (A.=0.004) and other two after yielding (?i=0.293 and X=0.566). An

important feature which can be noticed from the figure 6.3 is that the stress

concentration factor has not changed significantly due to yielding in the matrix.

This is probably due to the fact that the elastic fibers carry most of the longitudinal

stress since the matrix yield stress is relatively small in this material. For the load

level of A.=0.57, the highest longitudinal stress and the strain in the composite (at

point B) were found to be 780 MPa and 0.37% respectively, which are larger than

the reported longitudinal failure stress (600 MPa) and strain (0.3%) for the FP/A1

system. The distribution of fiber stress which is computed from the axisymmetric

model (Chapter 2) is plotted along the ligament AB (figure 6.4) for the same

load levels as above. It can be observed that the stress concentration in the fiber is

7.0, which is higher than the stress concentration of 4.6 for the composite. Also

from figure 6.4 it can be seen that the fiber stress concentration has increased

from 7.0 to 8.0 after the yielding in the matrix occurred. The horizontal fiber along

the critical section BC (figure 6.1(b)) is the one which is highly stressed and the

distribution of fiber stress along BC is shown in figure 6.5 for the same load

levels as above. The highest stress in the fiber is found at point B, where it is 1350

MPa for A.=0.57 and it drops to a lower value within a half of the hole radius

distance and remains uniform at a stress of 280 MPa as shown in figure 6.5.

71

The Variation of the shear stress, %\2, along the section BC is shown in figure

6.6 for load levels X=0.293 and ^,=0.566. The limit strength in shear, K*s (105

MPa), is used to normalize t\2 in figure 6.6. It can be seen that the high shear

stresses are limited to a distance of one hole radius from point B. Since the initial

yield stress in shear, K's, is 42 MPa for this material, yielding can be observed in

regions where Ti2/K*s is greater than 0.4.

Distributions of the effective plastic strains are shown in figure 6.7 for three

different load levels X=0.15, A,=0.22, ^.=0.35 and X=0.57. Yielding was first

observed at point D (which is found at 20° from the Y-axis) when X=0.15 and

indicated in figure 6.7(a). As the load increases yield zone has extended in the

fiber direction as illustrated in figures 6.7(b), (c) and (d). Although a diffuse

yield zone is observed, the high plastic strain gradient is confined to the region near

the hole surface as shown in figure 6.7(d). This yielding is mainly due to the

high shear stresses along the fiber direction. Maximum strain component of Yl2 was

found at point D where it is 0.8% at the end of the loading and is much less than the

failure strain of the composite in shear. It can be seen that the high plastic strains in

this structure are not within the critical load carrying ligament of AB. Therefore

during the monotonic loading the expected failure of this structure is due to the fiber

failure at the critical load carrying ligament.

72

6 . 1 . 2 C y c l i c L o a d i n g

Another analysis was performed for a plate with a hole with dimensions a=4.75

mm, b=19.05 mm, c=76.2 mm and thickness = 2.6 mm to understand the behavior

when subjected to cyclic loading. A load control analysis was performed for this

test with the applied stress varying between 9 MPa and 180 MPa. The above

geometry and load levels were selected to be the same as the experimental data

available for cyclic loading (Tsangarakis et al 1985). The global stress-strain

diagram obtained for the first cycle of this analysis is shown in figure 6.8 and for

all other cycles the same stress-strain behavior is observed. The loading and

unloading paths followed the same stress-strain curve and the linearity in the stress-

strain curve has not changed due to yielding near the hole surface. In this cycle

yielding was first observed at 53 MPa during loading and at 77 MPa during

unloading, but residual strains were negligibly small at the end of the cycle.

First yielding was observed on the hole surface at the same place as in the

previous problem (point D of figure 6.7(a)) and the plastic strains are also found

to be confined to a small region near the hole surface (same as in the previous

problem). The highest shear stress in the plate is found at point D and the cyclic

shear stress-strain diagram at this point is shown in figure 6.9(a). As the number

of cycles increases the mean stress level of the cyclic shear behavior has dropped to

zero as shown in figure 6.9(a) and in ten cycles local plastic shake down

behavior is observed with a reversal strain, 2yr, of 0.33% (figure 6.9(a)). Due

to this behavior, the Aluminum matrix will have cyclic plastic deformation of

0.33% at point D and fatigue will be expected to initiate there. The variation of At

(difference between maximum and minimum shear stress within a cycle) at point D

is shown in figure 6.9(b) for the first ten cycles. The highest difference in shear

stress is At/K*s=0.99 and it is observed in the first cycle. Then Ax gradually drops

as the number of cycles increases and remains almost constant after eight cycles.

The variation of the shear stress, T12, along the horizontal section BC (in

figure 6.1(b)) for the maximum and minimum levels of the applied stresses (ie.

for 180 MPa and 9 MPa) in the tenth cycle is shown in figure 6.9(c). It can be

observed from figure 6.9(c) that the maximum difference in the shear stress in

the tenth cycle is Ax/K*s=0.91 and it occurs near the hole surface. Shear stresses

are found to be zero away from the hole surface.

6.2 Plate with a Center Notch

A plate with a center notch which is loaded in the fiber direction is analyzed and


shown in figure 6.10. The length to width ratio is taken to be 5/3 (same as in the

hole problem) while the notch length is one half of the width of the plate. The notch

tip was blunted by introducing a root radius of 0.1 mm. Due to the symmetries of

the structure only one fourth of the plate was considered for the analysis. A plane

stress calculation was done using displacement controlled boundary conditions as

74

shown in figure 6.10(b). The ultimate longitudinal tensile stress of the

unnotched composite (otjTS). which is 600 MPa for the considered FP/A1 system,

is used to normalize the average stress at the load carrying ligament (ang) of the

plate and the load level is defined as X = AUG / (JUTS-

The stress-strain diagram obtained from this analysis is shown in figure

6.11. Yielding first occurs at load level X=0.051 and is indicated in figure 6.11.

It should be noted that the linear behavior of the stress-strain diagram is only

slightly changed due to yielding. The distribution of the longitudinal stress in the

composite along the ligament AB is shown in figure 6.12(a) for three load

levels, one before yielding (^=0.0008) and other two after yielding (^=0.3287 and

X=0.5257). It is noticed from the figure 6.12(a) that there is a high stress

gradient near the notch tip. Figure 6.12(b) shows the variation of composite

stress just ahead of the notch tip for both AB and BC directions. The distance in

this figure is normalized using the root radius of the notch. It can be seen that the

high stresses near the tip die down within a distance of five times the root radius in

both AB and BC directions. Outside this region the longitudinal stress in the

composite remained uniform and equal to applied stress. When the elastic stresses

in the composite are considered the stress concentration factor at the notch tip (point

B) was found to be 15. The stress concentration factor did not change significantly

after yielding illustrating a behavior similar to the plate with a hole problem.

Figure 6.12(c) shows the distribution of composite stress just ahead of the notch

tip along Y-direction in a log-log plot. Using the gradient of that graph it is found

that the stresses near the notch tip has r -°-6 singularity. From the analysis it is

found that the longitudinal stress in the composite at point B exceeds the

longitudinal failure stress (600 MPa) at a low load level of A,=0.16. The distribution

of the fiber stress along the ligament AB is shown in figure 6.13 for three load

levels, one before yielding (X=0.0008) and other two after yielding (^=0.3287 and

A.=0.5257) and it is observed that elastic stress concentration of the fiber is 22,

which is higher than the stress concentration of 15 for the composite. The

horizontal fiber along the critical section BC is the one which is highly stressed and

the distribution of fiber stress along BC is shown in figure 6.14 for the same load

levels as above. The highest stress in the fiber is at the notch tip (point B) and that

high stress drops to a lower value within the distance of five times the root radius

and then gradually approaches to a uniform stress level as shown in figure 6.14.

This indicates that only a very short length of fiber is highly stressed near the tip.

The variation of the normalized shear stress T12 along the section BC is shown

in figure 6.15 for two load levels X,=0.328 and X=0.526. Since the initial yield

stress in shear, K's, is 42 MPa for the composite material, yielding is observed in

regions where Ti2/K*s is greater than 0.4. Also it is found that the shear stresses

are less than the limit shear strength of 105 MPa.

Distribution of the effective plastic strains are shown in figure 6.16 for two

different load levels A,=0.15 and X=0.53. Yielding was first observed at the notch

tip as shown in figure 6.16(a). As the load increases yield zone has extended in

the fiber .direction as illustrated in figure 6.16(b) and high plastic strains were

confined to a very small region near the notch tip. Maximum strain component of

yl2 was found to be 3.1% at the end of the loading and it is still less than the

reported failure strain of the composite in shear. Yielding in the fiber direction is

mainly due to the high shear stresses near the tip. As the load increases yielding on

the free surface of the notch was observed (figure 6.16(b)) due to compressive

transverse stresses. With all of these observations, it can be concluded that the

expected failure of the plate with a center notch during the monotonic loading will

be due to the fiber failure at the critical load carrying ligament. A statistical study

will be necessary to obtain the failure strength of this structure.

6.3 Plate with Double Edge Notches

A plate with double edge notches loaded in the fiber direction is analyzed and


shown in figure 6.17. The size of the plate and the total lengths of the notches are

selected to be same as in the center notch problem. Both notch tips are blunted by

introducing a root radius of 0.1 mm. Due to the symmetries of the structure only

one fourth of the plate was considered for the analysis. A plane stress calculation

was done using displacement controlled boundary conditions as shown in figure

6.17(b). The load level, X, is defined in the same way as in the center notch

problem.

77

The siress-strain diagram obtained from this analysis is shown in figure 6.18.

Yielding first occurs at load level A.=0.051 and is indicated in figure 6.18. It is

observed that the load level corresponds to initial yield was the same as for the

center notch problem. In this problem also it is observed that the linear behavior of

the stress-strain diagram is only slightly changed due to yielding. The distribution

of longitudinal stress in the composite along the ligament AB is shown in figure

6.19(a) for three load levels, one before yielding (A,=0.0016) and other two after

yielding ( X=0.3247 and A,=0.4492). The longitudinal stress distribution was very

similar to the results obtained from the center notch problem and the high stress

gradient was near the notch tip. Figure 6.19(b) shows the variation of composite

stress just ahead of the notch tip for both AB and BC directions. The distance in

this figure is normalized using the root radius of the notch. It can be seen that the

high stresses near the tip die down within a distance of five times the root radius in

both AB and BC directions. Outside this region the longitudinal stress in the

composite remained uniform and equal to applied stress. The stress concentration

factor at the notch tip (point B) was found to be 15 and it was same as in the center

notch problem. The stress concentration factor did not change significantly after

yielding. Figure 6.19(c) shows the distribution of composite stress just ahead of

the notch tip along Y-direction in a log-log plot. Using the gradient of that graph it

is found that the stresses near the notch tip has r -°-6 singularity. From the analysis

it is obtained that the longitudinal stress in the composite at point B exceeds the

longitudinal failure stress (600 MPa) at a low load level of A.=0,16. The distribution

of the fiber stress along the ligament AB is shown in figure 6.20 for three load

78

levels, one before yielding (A.=0.0016) and other two after yielding (?i=0.3247 and

^=0.4492) and the distribution was very similar to the results obtained from the

center notch problem. The horizontal fiber along the critical section BC is the one

which is highly stressed and the distribution of fiber stress along BC is shown in

figure 6.21 for the same load levels as above. Again the distribution of fiber

stress along BC is very similar to the results obtained from the center notch

problem. The highest stress in the fiber is near the notch tip (point B) and that high

stress drops to a lower value within the distance of five times the root radius and

then gradually approaches to a uniform stress level as shown in figure 6.21. This

indicates that only a very short length of fiber is highly stressed near the tip.

The variation of the normalized shear stress T12 along the section BC is shown

in figure 6.22 for two load levels X=0.325 and ?i=0.449. Since the initial yield

stress in shear, K's, is 42 MPa for the composite, yielding is observed in regions

where Ti2/K*s is greater than 0.4. Also it is found that the shear stresses are less

than the limit shear strength of 105 MPa.

Distribution of the effective plastic strains are shown in figure 6.23 for

different load levels X.=0.16 and A,=0.45. Yielding was first observed at the notch

tip as shown in figure 6.23(a). The yield zone has extended in the fiber direction

(see figure 6.23(b)) as the load increases showing similar behavior to center

notch and hole problems. High plastic strains were also confined to a very small

79

region near the notch tip. Maximum strain component of yl2 was found to be 2.2%

at the end of the loading and it is still less than the reported failure strain of the

composite in shear. Yielding in the fiber direction is mainly due to the high shear

stresses near the tip. With all of these observations, it can be concluded that the

expected failure of the plate with double edge notches during the monotonic loading

will also be due to the fiber failure at the critical load carrying ligament.

6 . 4 S u m m a r y

The three problems studied in this Chapter had a very similar overall behavior

during longitudinal monotonic loading. From observing the stress-strain diagrams

for longitudinal monotonic loading shown in figure 6.24 for three problems, it

can be seen that the both notch problems had almost the same stress-strain behavior

and the plate with a hole was compliant compared to the notches because of the area

reduction due to the hole. During the monotonic loading the high plastic strains

were not within the critical load carrying ligament and therefore the final failure of

these structures can be predicted from the failure of fibers in the load carrying

ligament due to high stress concentrations.

It is found that the results obtained from both load and displacement control

tests were similar for above three structural components. But the displacement

control calculations were easy to converge than the load control calculations during

the plastic deformations of the structures.

80

When the plate with a hole is subjected to cyclic loading it is found that the

fatigue is expected to initiate at point D (20° away from the vertical axis of the hole)

where the plastic strains are found to be high during the cycles.

81

Property Definition Magnitude

(FP/Al)

Longitudinal Ultimate Tensile Strength, (Tuts (MPa) 600

Transverse Limit Strength, Otl (MPa) 210

Initial Yield Strength in Shear, K's (MPa) 42

Limit Strength in Shear, K*s (MPa) 105

Longitudinal Failure Strain, efl (%) 0.3

Transverse Failure Strain, efr (%) 0.8

Anti plane Shear Failure Strain, yfs (%) 20

Average Strength of Uncoated Fibers (MPa)

(Gauge length 6.25 mm, Weibull Modulus 6.5)

1480

Table 6.1 Strengths and Failure Strains of FP/Al

82


Figure 6.1 Plate with a circular hole - Longitudinal loading

a) Notations and dimensions

b) Boundary conditions

Figure 6.2 Stress-strain diagram

Figure 6.3 Normalized longitudinal composite stress distribution along AB

Figure 6.4 Normalized fiber stress distribution along AB

Figure 6.5 Normalized fiber stress distribution along BC

Figure 6.6 Normalized shear stress distribution along BC

Figure 6.7 Distribution of effective plastic strains in the plate

Figure 6.8 Applied stress-strain diagram for cyclic loading

83

Figure 63 (a) Shear stress-strain diagram at point D for cyclic loading

(b) Variation of Ax at point D with number of cycles

(c) Variation shear stress along BC at tenth cycle

Figure 6.10 Plate with a center notch - Longitudinal loading




Figure 6.12 Normalized longitudinal composite stress distribution

a) Along the entire ligament AB

b) Just ahead of the notch tip along AB and BC

c) Same as b) in a log-log plot





84

Figure 6.17 Plate with double edge notches - Longitudinal loading




Figure 6.19 Normalized longitudinal composite stress distribution

a) Along the entire ligament AB

b) Just ahead of the notch tip along AB and BC

c) Same as b) in a log-log plot





Figure 6.24 Comparison of Stress-Strain behaviors of Hole, Center notch and

Double edge notches

85

fiber direction

2c

2a = 30 mm , 2b = 60 mm , 2c = 100 mm

Figure 6.1(a)

G. lig

'app

/ t 9 / ? i f / Prrf

°iig " O b/(b-a) app

Figure 6.1(b)

86

Yielding

B. a. eg

2~

0.1 0 0.2 0.3 0.4 0.5

y / b

Figure 6.3

88

Ti »-*• era § Os

oo VO

g /a

o NJ 00 o ©

o

o to

o ji.

o l/l

X = 0.0035

- X = 0.2928 5~

X = 0.5659

0.4 0.6 0 0.2 0.8 1

x / c

Figure 6.5

90

-0.2

-0.4-

-0.6" \ = 0.2928

\ = 0.5659 -0.8-

0 0.2 0.4 0.6 0.8 1 x / c

Figure 6.6

91

1 = 0.0e-0 2 = 5.0e-7 1= 0.0e-0

2 = 1.0e-4 3 = 2.0e-4 4 = 3.0c-4

a) X = 0.1457 b) X. = 0.2163

l=0.0e-0 2 = 3.0e-4 3 =1.0e-3 4 = 2.0c-3

1 = 0.0e-0 2 = 8.0e-4 3 = 2.5e-3 4 = 4.0e-3

c)X = 0.3498 d) X = 0.5659

Figure 6.7

92

200

150-

100-

50-

0

Loading -

Yielding Unloading

Yielding

T I I I

0 0.04 0.06 0.08

e (%) app

0.1

Figure 6.8

93

0.5 & 0.2

a = 180 MPa max

a . = 9 MPa min

0 H

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 "

YI2(%)

Figure 6.9(a)

94

0.99-

0.98-e <

0.97

0.96

1 2 3 4 5 6 7 8 9 1 0

No. of Cycles

Figure 6.9(b)

95

A x / K

<S e

y a = 9 M P a

jfJl

a = 180 MPa app

X / C

Figure 6.9(c)

96

fiber direction t Y

V 1 1

2a ^

X

' J f—— 2r

2c

2a = 30mm , 2b = 60mm , 2c = 100mm , 2r = 0.2mm

Figure 6.10(a)

G lig /o

a

°iig = a b / (b -a) app

Figure 6.10(b)

97

200

app app

150-

100-o. B.

50-

Yielding

0.02 0.04 0.06 0.08 0 0.1 0.12

e (%) app

Figure 6.11

98

1 5 "

a. Q.

0.2 0.4 0.5 0.1 0.3 0.0 y / b

Figure 6.12(a)

99

15-CU a. 03

10-Along AB

5-

Along BC

0 5 10 15 20 25

distance / r

Figure 6.12(b)

100

y/r

Figure 6.12(c)

101

= 10-

0.0 0.1 0.2 0.3 0.4 0.5 y / b

Figure 6.13

102

— X = 0.0008

— X = 0.3287

— A, = 0.5257

20-

15-

10-

5-

0.02 0.04 0.06 0.08 0.1 0

X / c

Figure 6.14

103

-0.1

-0.2 -j1

«« -0.3

-0.4

- X = 0.3287 -0.5

-0.6-; X = 0.5257

-0.8-

0 0.2 0.4 0.6 0.8 1

X / c

Figure 6.15

104

l=0.0e-3 2=3.0e-3

»1

a) X, = 0.1527

l=0.0e-3 2=5.0e-3

^ 3=3.0e-2

32 1

\l

b) X = 0.5257

Figure 6.16

105

X 2r-Hk-

2a = 30mm , 2b = 60mm , 2c = 100mm , 2r = 0.2mm

Figure 6.17(a)

lig

app

0Ug - b / (b -a)

app

Figure 6.17(b)

106

140

120- app app

100-

80-

60-

40-

Yielding 20-

0.1 0.04 0.06 0.08 0.02 0

e (%) app

Figure 6.18

107

0.0 0.1 0.3 0.2 0.4 0.5 y / b

Figure 6.19(a)

108

15-

10-

Along AB

5-

Along BC

0 15 5 10 20 25

distance / r

Figure 6.19(b)

109

20

10

a. a. at

1 0.5 1 10 20

y / r

Figure 6.19(c)

I I I I I I I 1 1 1 1 I I |

110

o. o.

0.4 0.2 0.3 0.5 0.0 0.1

y / b

Figure 6.20

111

— X = 0.0016 — X = 0.3247

— X = 0.4492

20-

B. P. a

15-

10"

5-

0.04 0 0.02 0.06 0.08 0.1

X / C

Figure 6.21

112

0.8 -i

X = 0.3247

0.6 X = 0.4492

0.4-

0.2

0.2 0.4 0.6 0.8 0 1 X / c

Figure 6.22

113

l=O.Oe-3 2=3.0e-3

»1

a) X = 0.1629

l=0.0e-3 2=3.0e-3 3=2.0e-2

3

b) X = 0.4492

Figure 6.23

114

200

150

100 a, eu CO

Center Notch

Double Edge Notch

Hole

0 0.05 0.1 0.15

e (%) app

Figure 6.24

115

CHAPTER 7

Applications to Structures: Part 2

An analysis of a ring reinforced in hoop direction is performed under two

different applied loading conditions. The first analysis is done when the outer

boundary of the ring is subjected to a uniformly distributed radial tensile stresses

and the second when the ring is subjected to discontinuous radial tensile stresses.

Information which is important in design are discussed for these problems.

Anisotropic strength properties and failure strains for FP/A1 which are summarized

in Table 6.1 will be used to establish the dominant failure mechanisms.

7 . 1 U n i f o r m l y L o a d e d R i n g

A composite ring reinforced in its hoop direction and loaded uniformly at the

outer surface is studied in this section. The dimensions and boundary conditions

used are shown in figure 7.1(a) and (b). The outer diameter is taken to be twice

the inner diameter. A plane stress displacement control analysis was performed in r-

0 plane using the symmetry of the problem as indicated in figure 7.1(b).

116

The stress-displacement curve is shown in figure 7.2(a). The non linearity of

this curve is due to yielding which was first observed at 81 MPa as shown in the

figure. The corresponding stress-strain diagram is shown in figure 7.2(b) and it

shows a behavior similar to the transverse tensile behavior of the material (figure

4.2). The strain in figure 7.2(b) is obtained from the finite element calculation as

the radial strain at the outer boundary. Calculations were performed until the loads

reached a level when numerical convergence was impossible. The radial strain

corresponding to this displacement was about 0.2% and the applied stress was 143

MPa. The load level X is redefined for this problem as X = 0appA*TL> where aapp is

the applied stress and cjtl is the transverse limit strength of the composite (210

MPa). Yielding was first observed at the outer surface and progressed inwards as

the applied load increases. The variation of effective plastic strain along the radial

direction is shown in figure 7.3 for the final load level of ^,=0.68. It can be

observed that the plastic strains are high near the outer boundary and almost zero at

the inner boundary. The maximum plastic strain observed for the above load level is

0.16% at the outer surface.

The variation of composite hoop stress, <199, along the radial direction is shown

in figures 7.4(a) and (b) for two load levels, one before yielding (A, =0.3) and

the other after yielding (X.=0.68) which is the final load level. In figure 7.4(a) the

composite hoop stress is normalized w.r.t. (JUTS and in figure 7.4(b) the same

stress is normalized w.r.t. applied stress. For both load levels the maximum

composite hoop stress is observed at the inner surface of the ring. The distribution

117

of the matrix hoop stress along the radial direction is shown in figure 7.4(c). It

can be observed from figure 7.4(b) that after matrix yielding the composite hoop

stress has increased near the outer surface. This is because in the yielding regions

matrix can not carry any higher stresses and forces the fibers to carry that extra

stress in the hoop direction. But still at the outer surface composite hoop stress

occurred is only 48% of the ultimate strength of the composite and at the inner

surface it is 60% of the ultimate strength as shown in figure 7.4(a). Therefore

fiber failure is not expected at the inner surface and better design of reinforcement

pattern may be needed to take the maximum advantage of fibers in this structure.

The distribution of normalized composite radial stress, an-, is shown in figure

7.5 for the same load levels as above and it is observed that the stress distribution

does not change significantly after yielding.

The variation of strains Eqq and En- are shown in figures 7.6 and 7.7 for three

load levels, one before yielding (>.=0.3) and other two after yielding (X=0.39 and

?l=0.68). There is a rapid increase in the radial strain, En-, near outer boundary due

to the plasticity as the applied load increases. This high radial strain will cause

instability in the ring at the outer boundary at that load level before any fiber failure

begins to occur at the inner boundary.

118

7 . 2 P a r t i a l l y L o a d e d R i n g

A ring which has the same dimensions as in the previous section is analyzed

when discontinuous loads are applied on its outer surface as shown in figure

7.8(a). When the symmetry of the problem is considered, only the portion ABCD

is analyzed with boundary conditions shown in figure 7.8(b). This analysis was

done using load control test.

The load level X is defined as X = CTapp/aTL. where aapp is the applied stress

and CTjl is the transverse limit strength of the composite. The applied load was

increased until the numerical convergence problems occurred when the final load

level was X=0.74. Yielding was first observed at the outer boundary near the load

discontinuity point as shown in figure 7.9(a). As the applied load increases

plastic zone spreaded inwards from the outer boundary and the high plastic strains

were observed near the loaded regions as shown in figures 7.9(b). Distribution

of normalized transverse tension, On-, is shown in figure 7.10 for the final load

level of X=0.74. It is observed that the highest transverse tension is at the loaded

surface and confined to a small region near the loaded ligament of the structure. The

distribution of the normalized shear stress, xr0, is shown in figure 7.11 for the

load level X=0.74. It can be seen from figure 7.11 that there is an infinitely large

shear stress at the load discontinuity point of the structure. The high transverse

119

tension and shear stresses around the load discontinuity region will cause the matrix

to fail in that region.

The normalized hoop stress, (Tee, distribution in the structure is shown in

figure 7.12 for the final load level of A,=0.74. The highest hoop stress is

observed at the loaded region of the outer surface and also the high stresses are

found at the inner surface (see figure 7.12). However these stresses are found to

be less than 50% of the composite ultimate strength in longitudinal tension (OUTS)-

The fiber stress, CTfib, distribution in the structure is shown in figure 7.13 for the

final load level of X=0.74. The highest fiber stress of 400 MPa is observed at the

loaded region of the outer surface and at the inner surface as shown in figure

7.13. These fiber stresses are much below the reported failure strength (Table

6.1) of the fiber. Therefore the fiber failure would not expect to occur at this load

level for this structure. From these observations it can be concluded that the

maximum advantage of fibers are not taken in this structure and a better distribution

of reinforcements would be necessary to make the structure more effective.

7 . 3 S u m m a r y

A ring reinforced in hoop direction is studied in this Chapter under two different

applied loading conditions. When uniform radial stresses are applied to the outer

boundary of the structure it is found that the plastic strains at the outer boundary are

120

high and-there is a rapid increase in the radial strain near the outer boundary. This

high radial strain will expect to cause instability in the ring at the outer boundary.

When discontinuous radial stresses are applied to the outer boundary of the

structure it is found that the transverse and shear stresses are high around the load

discontinuity region and will cause the matrix to fail in that region. For both loading

conditions matrix failure is expected before the fiber failure. Therefore a better

design of reinforcement pattern may be needed to take the maximum advantage of

fibers in this structure.

121


Figure 7.1 Uniformly loaded ring

(a) Configuration of the problem

(b) Boundary conditions

Figure 7.2 (a) Stress-displacement diagram

(b) Stress-strain diagram

Figure 7.3 Variation of effective plastic strains along radial direction

Figure 7.4 (a) Variation of composite hoop stress, aee, along radial direction

normalized by ultimate longitudinal tensile strength

(b) Variation of composite hoop stress, Gee. along radial direction

normalized by applied stress

(c) Variation of matrix hoop stress along radial direction

Figure 7.5 Variation of composite radial stress, Gn, along radial direction

Figure 7.6 Variation of composite hoop strain, Eee, along radial direction

122

Figure 7.7 Variation of composite radial strain, En, along radial direction

Figure 7.8 Partially loaded ring

(a) Configuration of the problem

(b) Boundary conditions

Figure 7.9 Distribution of effective plastic strains

Figure 7.10 Distribution of normalized radial stress

Figure 7.11 Distribution of normalized shear stress

Figure 7.12 Distribution of normalized hoop stress

Figure 7.13 Distribution of fiber stresses

123

a = 30 cm , b = 15 cm , thickness = 1 cm

Figure 7.1(a)

Figure 7.1(b)

124

Yielding

Figure 7.2(a)

125

Yielding

Figure 7.2(b)

126

0.2

X = 0.68 0.15-

0.1" 4>

0.05-

0.5 0.6 0.7 0.8 0.9 1 r / a

Figure 7.3

127

0.6

X = 0.68 0.5-

0.4-

0.3-

0.2- & a a 6 AA

0.1 0.6 0.7 0.8 0.5 0.9 1

r / a

Figure 7.4(a)

128

2.6

2.5

2.4

o. a. 05

2.2": ® CD 2.1 "

2.0-i

1.9"

1.8-

1.7-I

1.6 0.6 0.7 0.8 0.9 1.0 0.5

r / a

Figure 7.4(b)

129

0.9

X = 0.3

A, = 0.68 e. 0.8

e>

0.7 o

0.6

0.5 0.5 0.6 0.8 0.7 0.9 1.0

Figure 7.4(c)

130

0.2

$ 0.1 "

co

0.0-K" 0.5

• tl O

OJ

A. = 0.39 31=0.68

0.6 0.7 0.8 0.9 1.0 r / a

Figure 7.6

132

0.2

A. = 0.3 \ = 0.39 X=0.68 /—N

&

U Urn

CO o.i -

o.o 0.6 0.5 0.7 0.8 0.9 1.0

r / a

Figure 7.7

133

a = 30 cm , b = 15 cm , thickness =

Figure 7.8(a)

1 cm

ft ft u o

Figure 7.8(b)

134

1 = 3e-6 2= le-5 3 = 2e-5

a) A, = 0.38

Oe-6 5e-4 le-3 1.5e-3 2e-3

b) X = 0.74

Figure 7.9

135

0

X = 0.74

(Japp= 155 MPa

Distribution of g

TL

Figure 7.10

136

X = 0.74

Oapp= 155 MPa

Distribution of T & K* s

Figure 7.11

137

X = 0.74

0.3

0.3 Oapp= 155 MPa

0.4

Distribution of Z®§ CTUTS

Figure 7.12

138

>00

400. X = 0.74

300

r~\ 300 \

\\200 |

Oapp= 155 MPa

Distribution of (MPa)

Figure 7.13

139

CHAPTER 8

Conclusions

8.1 Summary

In this study constitutive equations were developed for a fully bonded metal-

matrix composite when it is subjected to multiaxial loading. These constitutive

equations can describe the elasto-plastic behavior of the composite material under

variable loading conditions.

The composite material behavior is divided in to two parts, one for the fiber

dominated axisymmetric behavior and the other for the matrix dominated shear

behavior. The combination of these two behaviors describe the complete set of

constitutive equations for the fully bonded metal-matrix composites. Some

experimental and numerical methods are used to determine the necessary parameters

in the constitutive equations.

The constitutive laws are then implemented in the finite element code ABAQUS

as a user defined material subroutine. This subroutine is implemented in a very

general way such that it can be used for any fully bonded MMC system when the

1 4 0

corresponding material parameters are provided as input data. This subroutine can

be used with a variety of element types such as three dimensional, plane stress,

plane strain and axisymmetric elements.

Using the finite element code ABAQUS and the new material subroutine UMAT

some numerical calculations are performed on different structural components made

of FP/A1 system to understand the behavior of these components. Using local stress

and strain distributions, it was possible to propose failure mechanisms for these

different structures. New reinforcement designs can be suggested by evaluating the

finite element results to get the maximum advantage of the fibers.

8.2 Implications for Future Work

In this study the composite is assumed to be a continuous and homogeneous

material. Therefore this material model can only predict the initiation of damage. A

further investigation must be done using damage mechanics or fracture mechanics

concepts to evaluate the evolution of damage. As a breakthrough a few number of

structural components were examined during this study, but the subroutine is

available with ABAQUS to use with any other component design.

In general some mesh refinements are necessary in the places where the stress

and strain gradients are high. The mesh dependence of this material model must be

1 4 1

investigated with a detailed study. The necessary equilibrium tolerances must also

be determined using a similar investigation.

1 4 2

REFERENCES

1972 Nicholes, E.D. and Kalin, R., Analytic Geometry, 174-191, Holt,

Reinehart and Winston inc.

1973 Dvorak, GJ., Rao, M.S.M., and Tarn, J.Q., J. Composite materials, 7(2),

194

1975 Phillips, A. and Weng, G.J., J.appLMech., 42, 375

1975 Krieg, R.D., J.appl.Mech., 42, 641

1976 Dvorak, GJ. and Rao, M.S.M., IntJ.Engng.Sci., 14, 361

1977 Phillips, A., and Moon, H., Acta Mechanica, 27, 91

1977 Phillips, A. and Lee, C.W., IntJ.Solids structures, 15, 715

1978 Champion, A.R., Krueger, W.H., Hartman, H.S. and Dhingra, A.K.,Proc

2nd International conference on composite materials, Toronto, Metallurgical

Soc. AIME, 883-904

1 4 3

1984 Spencer, A.J.M., Continuum theory of the mechanics of fiber-reinforced

composites, Springer-Verlag Wein-New York

1985 Tsangarakis, N., Gruber, J.J., and Nunes, J., J. of Composite materials,

19, 250-268

1985 McDowell, D.L., J.appl.Mech., 52, 298-308

1986 Ohno, N., and Kachi, Y., J.appl.Mech., 53, 395

1987 Cocks, A.C.F., and Leckie, F.A., Advances in appl.Mech., 25, 239

1987 Dvorak, G J. and Bahei-El-Din, Y.A., Acta Mechanica, 69,219

1988 Dvorak, G.J., Bahei-El-Din, Y.A, Macheret, Y. and Liu, C.H., J.Mech.

Phys.Solids, 36 (6) 655

1988 Hibbitt, Karlson, and Sorensen, Inc., ABAQUS Finite element program,

version 4.7

1989 McDowel, D.L., International Journal of Plasticity, 5,29

1 4 4

1989 Karr, D.G., Law, F.P., Fatt, M.H., and Cox, G.F.N., International

Journal of Plasticity, 5, 303

1989 Aboudi, J., Appl.Mech.Rev., 42 (7), 193

1989 Dvorak, G J., Bahei-El-Din, Y.A., Bank, L.C., Engineering Fracture

Mechanics, 34 (1), 87-104

1989 Bahei-El-Din, Y.A., Dvorak, G.J., Wu, Jer-Fang, Engineering Fracture

Mechanics, 34 (1), 105-123

1989 Doghri, I., and Billardon, R., Plastendo, Laboratoire de Mecanique et

Technologie, E.N.S. de Cachan / C.N.R.S./ Universite Paris 6

1990 Hild, F., and Billardon, R., Elastendo, Laboratoire de Mecanique et

Technologie, E.N.S. de Cachan / C.N.R.S./ Universite Paris 6

1990 Aboudi, J., Pindera, M-J., Herakovich, C.T. and Becker, W.,J.Composite

Materials, 24, 2

1990 Aboudi, J., International Journal of Plasticity, 6, 471

1991 Jansson, S., Mechanics of materials, 12, 47-62

1 4 5

1991 Fares, N. and Dvorak, G.J., J.Mech.Phys.Solids, 39 (6), 725

1991 Everett, R.K. and Arsenault, R.J., Metal matrix composites, Academic

Press, inc.

1991 Gunawardena, S.R., Jansson, S., and Leckie, F.A., Proc. of ASME

winter annual meeting

1991 Meletis, E.I., and Chaudhury, S., Composite Structures, 19, 89-103

1992 Bao, G., Ho, S., Suo, Z., and Fan, B., Int. J. Solids structures, 29 (9),

1105-1116

1992a Jansson, S., On the structure of non linear constitutive equations for fiber

reinforced composites, Department of Mechanical and Enviromental

Engineering, University of California at Santa Barbara

1992b Jansson, S., and Leckie, F.A., J. Mech.Phys. Solids, to be published

1992c Jansson, S., and Leckie, F.A., Effect of cyclic thermal loading on the in-

plane shear strength of fiber reinforced MMC's, Department of Mechanical

and Enviromental Engineering, University of California at Santa Barbara

1 4 6

APPENDIX 1

From the elastic solution obtained for unit cylinder model (Chapter 2) we obtain the

transformation tensor Ly for stresses:

T _M(l-vmVf) T (M(Mv2+Mvm+vm-2vrvmvf) T M(vm-vf) Lll~ . l12 . *-21= .

Ai Ai Aj (Al.l)

and L _[cM(MVm+Mym+1 - Vf-2ymVf)+(l- c)(1 +Vf)(Mym+1 -2Vf)]

A,

where Ai=M[cM(l-v&)+(l-c)(l-vmVf)] and . Em

The subscripts denote matrix (m) and fiber (f). E and v are the Young's modulus

and Poisson's ratio of each phase respectively.

For the initial yield surface, the following relations are obtained from the elastic

solution of the cylinder model,

147

tKr Kn Y I 2 2 i 2 2 1 m' L11+L21 -L11L21 <rm' L]2+L22-L12L22 " Ll2 " ^22+1

(A1.2)

Kixf = . _L Ym/ L11+L21 -2(LnLi2+L2iL22) + LnL22 + L2iLi2

For continuous yielding (beyond the initial yielding) of the matrix we also obtain

the following relations from the cylinder model

K2 _ 1

Ym / L12+L22 " L12L22

1 2( Li 1L12+L21L22) - ( Li 1L22 + L21L12)

(K7 \Ym 2( L12+L22 " L12L22) - ( L12+L22)

(A1.3)

_L = —4 L_ t m = A2 A2 K[t

1 v2y 2 y2 T/"2

*LTK7 and n = A2

V"2 v-2 ir2if 2

p2 = 1 - — n2

-L . - t t - .A2 tv-2 T^-2 K2 K7

1 1 TT2it4

ATK3 KLK7.

Plastic strains in the matrix are related to the residual stresses in the matrix through

an elastic tensor Ajj (equation 2.9), where

148

au = cm+1-c , a12 = ̂ m. , a2j = ~tcmvm+(1 ~ c)vf] t

cEf E„, cEf (A1.4)

A22 = ~— > A31 = - (Aj 1 + A21) and A32 = -(Ai2 + A22) •

Using Hy which is defined in Chapter 2 (equations 2.11), we can obtain a relation

between residual stresses and plastic strains. This relation also defines the tensor

Zij, which describes the hardening rule for the axisymmetric part of the flow

potential in Cartesian coordinate system. Therefore we get the following results,

Z11=H„+mH12 , Z12 = Hi2+2mH22 f z21=2nH21 and Z22 = nH22 .

149

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