The Combining Effect of Creep Flow and Surface and Grain Boundary

NUMERICAL STUDY OF MICRO-SCALE DAMAGE EVOLUTION IN TIME DEPENDENT FRACTURE MECHANICS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University

Joonyoung Oh, M.S.

*****

The Ohio State University 2005

Dissertation Committee: Approved by Professor Noriko Katsube, Adviser

Professor Daniel A. Mendelsohn ______Adviser Professor Mark E Walter

Professor Ten-hwang steve Lai Graduate Program in Mechanical Engineering ABSTRACT

In part 1, intergranular cavity growth in regimes, where both surface diffusion and deformation enhanced grain boundary diffusion are important, is studied. In order to continuously simulate the cavity shape evolution and cavity growth rate, a fully-coupled numerical method is proposed. Based on the fully-coupled numerical method, a gradual cavity shape change is predicted and this leads to the adverse effect on the cavity growth rate. As the portion of the cavity volume growth due to jacking and viscoplastic deformation in the total cavity volume growth increases, spherical cavity evolves to V- shaped cavity. The obtained numerical results are physically more realistic compared to results in the previous works. The present numerical results suggest that the cavity shape evolution and cavity growth rate based on the assumed cavity shape, spherical or crack- like, simply cannot be used in this regime.

In part 2, intergranular creep failure of high temperature service material under a stress-controlled unbalanced cyclic loading condition is studied. The experimentally verified Murakami-Ohno strain hardening creep law and Norton’s creep law are incorporated into the Tvegaard’s axis-symmetric model for the constrained grain boundary rupture analysis. Based on the physically realistic Murakami-Ohno creep law,

ii it is shown that the cavity growth becomes unconstrained upon the stress reversal from compression to tension. This leads to the prediction that the material life under a cyclic loading condition is shorter than that under a constant loading. Based on the classical

Norton’s law, the predicted material life under a cyclic loading condition remains the same as that under a constant loading. The obtained numerical results qualitatively match with recent experimental results by Arai, where the life under a cyclic loading can be much shorter than that under a constant loading. There are many cases where engineers use a simple Norton’s creep law because of its simplicity. The present work suggests that more physically realistic creep laws should be used when cyclic loading must be considered.

iii

DEDICATED TO

my wife Wonjoo, my parents, and my Angels Andrew, Allison, Emily

iv ACKNOWLEDGMENTS

God has blessed me with an opportunity to study at The Ohio State University and

kept my family together in peace, for which I am most grateful.

I wish to thank my adviser, Professor Noriko Katsube, for intellectual support,

encouragement, and enthusiasm which made this possible, and especially for her patience throughout the study. I sincerely thank Dr. Brust at Battelle Memorial Institute for giving me a chance to work with him, and for his valuable guidance and support. I would also like to thank the members of my committee: Professors Daniel A. Mendelsohn, Mark E

Walter.

I would especially like to thank my lovely wife, Wonjoo Lee, for her pray,

patience and understanding, and my son, Andrew, and my daughters, Allison and Emily,

for their precious smiles. I would also like to express my heartwarming gratitude to my

parents for their love.

v VITA

February 25, 1969…………….... Born-Taegu, Korea

1993……………………………. B.S. Material Science and Engineering, Hanyang University, Seoul, Korea

1994-1995..……………………. M.S. Material Science and Engineering, Hanyang University, Seoul, Korea

1995.2 – 1995.12………………. Researcher, Research Institute of Industrial Science, Hanyang University, Seoul, Korea

1996-1997……………………. M.S. Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO

1998.1-2002.6……………………… Graduate Research Associate, The Ohio State University, Columbus, OH

2002.8-2005.3…………………… Research Scientist Battelle Memorial Institute, Columbus, OH

vi TABLE OF CONTENTS

Page

Dedication ………………………………………………………………………………. iv

Acknowledgments ……………………………………………………………………..... v

Vita ……………………………………………………………………………………… vi

List of Tables …………………………………………………………………………… x

List of Figures …………………………………………………………………………. xi

PART 1 NUMERICAL ANALYSIS OF THE EFFECT OF DIFFUSION AND CREEP

FLOW ON CAVITY GROWTH………………………………………………………… 1

CHAPTERS :

1.1 INTRODUCTION AND LITERATURE REVIEW…………………………………. 1

1.2 METHODOLOGY …...... 5

1.2.1 Finite Element Method…………………...... 10

1.2.2 Finite Difference Form…………………….…………………………………...13

1.3 RESULTS…………………………………………………………………………...13

1.3.1 Modeling of Grain Boundary Diffusion Controlled Cavity Growth …………. 13

1.3.2 Modeling of Surface Diffusion Controlled Cavity Growth…………………… 13

vii 1.3.3 Transition from quasi-equilibrium mode to crack-like mode…………………. 13

1.3.4 Comparison of numerical cavity growth prediction with experiments………... 13

1.4 DISCUSSION…………………………………………………………………….... 38

1.5 CONCLUSION……………………………………………………………………… 41

PART 2 STUDIES ON EFFECT OF CYCLIC LOADING ON GRAIN BOUNDARY

RUPTURE TIME……………………………………………………………………….. 75

CHAPTERS :

2.1 INTRODUCTION AND LITERATURE REVIEW………………………………….75

2.2 CONSTITUTIVE MODELING AND FINITE ELEMENT ALGORITHM FOR

THE MURAKAMI-OHNO MODEL...... 79

2.3 METHODOLOGY………………………………………………………………… 83

2.4 RESULTS…………………………………………………………………………... 90

2.4.1 Verification of FEM code and constitutive law………………………………. 90

2.4.2 Grain boundary rupture analysis……………………………………………… 91

2.5 DISCUSSION…………………………………………………………………….... 97

2.6 CONCLUSION…………………………………………………………………..… 100

APPENDIX A Derivation of the functional, F gb , in Eq. (1.5) (Needleman and Rice,

1980)…………………………………………………………………………………… 118

APPENDIX B Derivation of Eq. (1.9)…………………………...... 123

APPENDIX C Derivation of Eq. (1.10)…………………………...... 125

viii APPENDIX D Derivation of Eq. (1.19)…………………………...... 126

APPENDIX E Derivation of Eq. (1.22)…………………………...... 127

APPENDIX F Derivation of Eq. (1.25)…………………………...... 128

APPENDIX G.….……………………..…………………………...... 129

APPENDIX H ….……………………..…………………………...... 131

APPENDIX I ….……………………..…………………………...... 134

APPENDIX J..….……………………..…………………………...... 138

REFERENCES………………………………………………………………………….143

ix LIST OF TABLES

Table Page

1.1 Experimental conditions reported by Goods and Nix [1978] and a/L value….... 44

1.2 Material properties of silver……………………………………………………. 45

1.3 α and the stress ratio between the initial sintering stress and the remote stress . 46

2.1 The material property of Inconel 617 at 950°C ………………………………. 100

2.2 The material property of 1.25Cr-0.25Mo steel at 538°C (Murakami and

Ohno [1982])…………………………………………………………………. 101

x LIST OF FIGURES Figure Page

1.1 (a) Hull-Rimmer type diffusion flow along cavity surface and grain boundary. The grain boundary separate as rigid bodies. (b) The effect of creep flow on grain boundary diffusion. The deformation of the grain material causes local accomodation at the cavity tip……….………………………………………… 47

1.2 Principal curvatures at cavity tip, κ1 in the plane perpendicular to the grain boundary and κ2 in the plane of the grain boundary, are shown, where arrow shows the tangent direction of each curvature. Intersection of the grain boundary and the cavity surface is shown by Γ (Γ is in the plane of the grain boundary)………………………………………………………………………. 48

1.3 Illustration of numerical calculation structure for single cavity growth model. The unified numerical method, which combines finite element method and finite difference method, starts with the known spherical-cap shape cavity geometry. For the given time step, which is chosen to be sufficiently short, finite element method and finite difference method are employed to simulate cavity shape evolution………………………………………………………… 49

1.4 Equilibrium dihedral angle (αo) satisfying the local tension equilibrium condition, where γs and γgb are cavity surface tension and grain boundary surface tension, respectively………………………………………………….. 50

1.5 (a) The cavity surface and the grain boundary shape at time t+∆t after FEM analysis is done. The additional flow δjs is not considered. The grain boundary and the cavity surface at time t moves up due to the volume of the diffused atoms. The newly created cavity surface at the cavity tip is not at equilibrium angle (b) The cavity surface and the grain boundary shape at time t+∆t before starting FDM analysis. The additional flow δjs is necessary to satisfy the equilibrium angle at the newly created surface …………………. 51

1.6 Sintering stress (σo) and the unit normal vector (mα) at the cavity tip ………... 53

1.7 Unit cell model with a spherical-cap shaped cavity with the major radius ‘a’, the minor radius ‘c’, and the cavity half distance ‘b’. The boundary conditions on the grain boundary satisfy the linear kinetic law for the atomic

xi flux and the matter conservation law. The boundary conditions on the outer surface of the grain material satisfy the axis-symmetric condition……………. 54

1.8 Boundary conditions along the cavity surface for the finite difference method. 55

1.9 Discretized cavity surface is schematically represented, which is used in the finite difference method………………………………………………………... 56

1.10 Finite difference numerical scheme showing how the ith point on the cavity surface moves to the new position (hollow point) after time increment ∆t and the definition of the angle βi…………………………………………………… 57

1.11 Definition of the surface normal vector (n) at the node on the cavity surface… 58

1.12 (a) Nondimensionalized cavity major radius as a function of nondimensionalized time for aI/L=0.316, aI/bI=0.1, and Ds/Dgb=171(fast surface diffusion). When a/b is 0~0.5, the present numerical results (with jacking effect) reproduce the Needleman and Rice results. In this range, the Chen and Argon results also match with the Needleman and Rice results. When jacking effect is not considered, present result deviates with the other three results. That implies jacking effect is significant in this a/L range (b) Nondimensionalized cavity major radius as a function of nondimensionalized time for aI/L=0.316, aI/bI=0.1, and Ds/Dgb=171(fast surface diffusion). When a/b is 0.5~1.0, the present results still match well with the Needleman and Rice results. However, Chen and Argon result starts to deviate from the Needleman Rice results ……………………………………..………………... 59

1.13 Nondimensionalized cavity volume growth rate as a function of nondimensionalized cavity major radius for aI/L=0.316, aI/bI=0.1, and Ds/Dgb=171(fast surface diffusion). Present results match well with the Needleman and Rice results (reported at a/b=0.1, 0.33, and 0.66). Chen and Argon results start to deviates when a/b is larger than 0.5…………………….. 61

1.14 Nondimensionalized grain boundary rupture time (when a/b reaches to 1) vs. nondimensionalized applied stress under different diffusivity ratios -4 (f=Dgb/Ds), for aI/bI=0.1, σ∞/E=10 , and cavity tip angle=70°. Chuang et al. [1979] (C&R) analytically calculated surface diffusion controlled cavity growth rate on the assumption of crack-like cavity. Martinez and Nix [1982] (M&N) used Finite Difference Method to evaluate the cavity shape evolution. When f=1 and 10 (surface diffusion controlled region), the present results match well with both M & N and C & R………………………………………. 62

1.15 The nondimensionalized normal stress of each element above the grain boundary vs. the nondimensionalized grain boundary length is shown for xii -4 σ∞/E=10 and cavity tip angle 70°. When a/b=0.1, stress distribution is the same for both f=1 and f=10 (surface diffusion controlled) cases. The stress along the grain boundary increases as the cavity grows for both cases, f=1 and f=10, since the grain boundary area decreases. For f=10, the sintering stress (σo), which is the normal stress at the cavity tip, increases as the cavity grows. For f=1, the sintering stress decreases when a = 0.3 and 0.5…………………... 63

-4 1.16 Evolution in the cavity shape for σ∞/E≈10 and cavity tip angle 70°, (a) f = 1 (surface diffusion controlled), (b) f = 10 (more surface diffusion controlled case). In all cases aI/bI=0.1 and the cavity changes to crack-like. The node removal-creation procedure is clearly shown for both cases. Cavity shape becomes more crack-like when f=10…………………………………………... 64

1.17 Nondimensionalized cavity growth rate vs. nondimensionalized cavity radius is shown for aI/L=0.1 and α=16.2. An initial spherical cavity evolves to a crack like cavity with much higher da/dt values compared to the prediction by Chen and Argon [1981]; the maximum difference occurs when a/L=0.18, where the current result predicts 1624 and the analysis by Chen and Argon [1981] predicts 951…………………………………………………………….. 65

1.18 Nondimensionalized cavity growth rate vs. nondimensionalized cavity radius is shown for aI/L=0.1 and α=10.0. An initial spherical cavity evolves to a crack like cavity with much higher da/dt values compared to the prediction by Chen and Argon [1981]; the maximum difference occurs when a/L=0.28, where the current result predicts 565 and the analysis by Chen and Argon [1981] predicts 350…………………………………………………………….. 66

1.19 Nondimensionalized cavity growth rate vs. nondimensionalized cavity radius is shown for aI/L=0.316 and α=5.82. Ιnitial nondimensionalized cavity growth rates predicted by the fully-coupled method are slightly higher than those by Chen Argon [1981]. As a/L increases, cavity growth rates do not follow those based on the assumed cavity shape…………………………………………….. 67

1.20 Nondimensionalized cavity growth rate vs. nondimensionalized cavity radius is shown for aI/L=1.0 and α=1.0. Since the cavity volume change due to creep flow (jacking effect and the cavity shape change due to creep flow) is significant in this a/L range, the analysis by Chen and Argon [1981], which assumes that the cavity volume growth rate is all related with the atomic flow rate at cavity tip, can not accurately predict the cavity major radius rate and cavity shape change……………………………………………………………. 68

1.21 Cavity aspect ratio variation during evolution of spherical-cap shape cavity for aI/L=0.1 and α=16.2 is shown. The aspect ratio by the fully-coupled method decrease gradually……………………………………………………………… 69

xiii 1.22 Cavity aspect ratio variation during evolution of spherical-cap shape cavity for aI/L=0.1 and α=10.0 is shown. Cavity maintains the initial aspect ratio until a/L reaches 0.16, since surface diffusivity is higher compared to the case of Fig. 1.21 (lower α value)………………………………………………………. 70

1.23 Cavity aspect ratio variation during evolution of spherical-cap shape cavity for aI/L=0.316 and α=5.82 is shown. The aspect ratio by the fully-coupled method saturates at much higher value (c/a=0.4) than that based on the crack like cavity shape assumption (c/a=0.09)……………………………………….. 71

1.24 Cavity aspect ratio variation during evolution of spherical-cap shape cavity for aI/L=1 and α=1 is shown. The cavity aspect ratio by the fully-coupled method does not follow those based on the assumed cavity shape. It clearly shows that the initial spherical-cap shape cavity does not evolve to become crack like cavity…………………………………………………………………………… 72

1.25 Cavity evolution from spherical-cap shape to V-shape for aI/L=1, α=1 is shown. Initial spherical-cap shape cavity maintains its original shape until a/L=3. When a/L reaches to 5, cavity shape changes to V-shape, since surface diffusivity is slow and material creep flow effect is significant……………….. 73

1.26 Normalized cavity growth rate vs. normalized cavity radius are shown. The experimental results are bounded by the cavity growth rates predicted by the fully coupled method (aI/L=0.1, α=16.2, σo/σ∞=0 (hollow circle) and aI/L=0.12, α=24, σo/σ∞=0.8 (hollow square)). Since different material constants for surface diffusion are reported and the simple assumptions are employed on obtaining the data points for the experimental results, the direct comparison between the each experimental data points and the numerical simulation results is not possible. However, from the distribution of each data points in this figure, it is clear that the cavity evolves from the spherical-cap shape to other shape in this range of a/L and α values………………………… 74

2.1 (a) Isolated cavitating grain boundary in the polycrystal. σ∞ is the remote axial stress and σn is the local normal stress. Since cavity growth causes grain boundary displacement, the local stress can be different from the remote stress depending on the material deformability. (b) cavitites on the grain boundary (c) axis-symmetric finite element model used in this study…………………… 102

2.2 Verification of ABAQUS UMAT code for Murakami-Ohno cyclic creep flow law. The numerical prediction (solid line) by UMAT code is compared with the experimental results (hollow circle) and numerical prediction by traditional strain hardening type constitutive law (dotted line). Murakami- Ohno law predicts experimental creep strain variation more accurately than the traditional strain hardening law……………………………………………. 103

xiv 2.3 (a)Verification of ABAQUS UMAT code based on Murakami-Ohno constitutive equation with the experimental result (displacement controlled cyclic test for inconel 617 at 950 ºC); displacement controlled cyclic condition (b) Verification of ABAQUS UMAT code based on Murakami-Ohno constitutive equation with the experimental result (displacement controlled cyclic test for inconel 617 at 950 ºC); stress prediction based on classical strain hardening creep law + Norton creep law (c) Verification of ABAQUS UMAT code based on Murakami-Ohno constitutive equation with the experimental result (displacement controlled cyclic test for inconel 617 at 950 ºC); stress prediction based on Murakami-Ohno cyclic creep law + Norton creep law. The suggested cyclic creep law more accurately predicts the stress relaxation than the result in Fig. 2.3 (b)…………………………….……….. 104

2.4 Stress-controlled remote loading condition is shown. To simplify the problem, compressive loading time (tc) and transition time (tm) are set to be small compared to the reference time. Norton’s flow law is used to calculate the effective creep strain rate in the reference time (tr)……………………………. 107

2.5 Normalized cavity radius increase with time for cavity at r/d = 0 with initial conditions (a/L∞) I =0.01, a I /b I =0.1, and b I /d I =0.1. When Murakami-Ohno + Norton material property is applied, grain boundary damage accelerates under the cyclic loading condition, tt/tc=100, compared to the constant loading case. When Norton’s type material property is employed, there is not difference in the final grain boundary rupture time. The same trend is observed for the cavity at r/d = 0.5. The overall cavity growth rate is slightly faster at the center of grain boundary………………………………………….. 108

2.6 Normalized cavity radius increase with time for cavity at r/d = 0 with initial conditions (a/L∞) I =0.015, a I /b I =0.1, and b I /d I =0.1. The overall cavity growth results are similar to those in the Fig. 2.5……………………………… 109

2.7 Normalized cavity radius increase with time for cavity at r/d = 0 with initial conditions (a/L∞) I =0.03, a I /b I =0.1, and b I /d I =0.1. The overall cavity growth results are similar to those in the Fig. 2.6……………………………… 110

2.8 Normalized cavity radius increase with time for the cavity at r/d = 0 with initial conditions (a/L∞) I =0.055, a I /b I =0.1, and b I /d I =0.1. Both material flow rule (“Murakami-Ohno+Norton” and “Norton”) predicts similar final grain boundary rupture time under constant loading condition. Grain material constraint is not significant when (a/L∞) I ≥0.06. Therefore, initial fast creep strain rate predicted by Murakami-Ohno law does not result in a significant change in the overall cavity growth rate………………………………………. 111

2.9 Development of the normalized normal stress around the cavity at r/d = 0 with (a/L∞) I =0.01, a I /b I =0.1, and a I /d I =0.1 under constant loading condition.

xv The initial normal stress (σn/σ∞=1 at t/tr=0) is decreased due to material constraint. For “MO+N’ case, this constraint process is slower than “N” case due to the fast creep strain rate by Murakami-Ohno flow rule. After this transition time, local normal stress saturates to the same value for both case…. 112

2.10 Normal stress variation after stress transition from compression to tension (at nd st 2 and 21 cycles) around the cavity at r/d=0 for (a/L∞) I =0.01, a I /b I =0.1, b I /d I =0.1 and tt/tc=100. The normal stress increase after stress reversal explains the fast cavity growth rate under cyclic loading condition for “MO+N” case especially at 2nd cycle. At 21st cycle, stress increase upon stress reversal is similar for both “MO+N” and “N” case……………………………. 113

2.11 Mises type effective creep strain rate variation after stress transition from compression to tension (at 2nd and 21st cycles) around the cavity at r/d=0 for (a/L∞) I =0.01, a I /b I =0.1, b I /d I =0.1 and tt/tc=100. The effective creep strain rate increase upon stress reversal for “MO+N” case at 2nd cycle shows the possibility that creep flow enhanced cavity growth (shorter diffusion length) can happen for constrained cavity growth under cyclic loading condition……. 114

2.12 Development of the normal stress state around the cavity at r/d=0 with (a/L∞)I =0.055, a I /b I =0.1, and b I /d I =0.1 under constant loading condition. Grain material constraint is not significant when (a/L∞) I ≥0.06. Therefore, initial normalized normal stress value decrease gradually compared to the result ((a/L∞)I =0.01) in Fig. 2.9……………………………………………………… 115

2.13 The effect of the number of cycles on the grain boundary rupture time for cavity (a) at r/d =0.and (b) at r/d=0.5 The compressive loading time(tc) and the transition time(tm) are set fixed and the loading time(tt) is changed to give different number of cycle. The grain boundary rupture time under cyclic loading condition decreased up to 12% compared to the constant loading case. The grain boundary rupture time slightly increases when the number of cycles exceeds 300. Increase of total compressive loading time is relevant with this behavior but the reason is not clear at this time. ……………………………… 116

J.1 user element for a micromechanical model……………………………………. 142

xvi PART 1

NUMERICAL ANALYSIS OF THE EFFECT OF DIFFUSION AND CREEP

FLOW ON CAVITY GROWTH

CHAPTER 1.1 INTRODUCTION AND LITERATURE REVIEW

At high temperature, cavity initiation and cavity growth are important phenomena in understanding the failure mechanism, and in predicting the lifetime of various parts in service in the area of a power plant and aero space applications, among others. Such nucleation and growth phenomena are explained by a diffusion of atomic flux (from cavity surface to grain boundary), creep flow, and grain boundary sliding. Because of the complexity of the physical phenomenon, in most of the previous works, one of the two extreme cases, fast grain boundary diffusion or fast surface diffusion, with or without the consideration of grain material creep flow, is assumed.

When grain boundary diffusivity is much faster than surface diffusivity (surface diffusion controlled process), the cavity shape will be similar to a crack because the atomic flow rate along the cavity surface is not fast enough to reduce the surface curvature at the cavity tip. On the other hand, when surface diffusivity is much faster

1 than grain boundary diffusivity (grain boundary diffusion controlled process), the cavity shape will be a spherical cap shape.

The basic model for predicting a grain boundary diffusion dominant cavitation process was first proposed by Hull and Rimmer [1959] (see Fig. (1.1) (a)). Speight and

Harris [1967] and Weertman [1973] included proper boundary conditions in the Hull-

Rimmer [1959] model. Their model predicts the cavity growth rate in the rigid surrounding material with the assumption of a grain boundary diffusion controlled process. Vitek [1978] calculated cavity growth rates taking into consideration the deformation of the surrounding elastic material. Raj [1975] considered the events occurring during the elastic transient time. However, in the above models, plastic material deformation of the grain material is neglected, and the surface diffusivity is assumed to be much faster than the grain boundary diffusivity.

Beere and Speight [1978] and Edward and Ashby [1979] attempted to model the combined effect of creep flow of the surrounding grain material and the grain boundary diffusion process on the cavity growth. However, in these two models, it is assumed that elastic material surrounds the cavity. Needleman and Rice [1980] established a numerical approach based on the variational principle for creep flow and grain boundary diffusion coupled problems and obtained a finite element solution for the cavity growth rate.

Figure (1.1) (b) shows the effect of the creep flow on the grain boundary diffusion and the updated spherical cavity profile based on the assumption of the fast surface diffusion.

The assumption of the grain boundary controlled cavitation may not always be satisfied, and elongated rupture cavities are sometimes observed. Thus, Chuang and Rice

[1973], Chuang et al. [1979], and Pharr and Nix [1979] analyzed the surface diffusion

2 controlled process (no deformation or grain boundary diffusion included). Chuang et al.

[1979] studied crack-like cavity volume growth by solving the Nernst-Einstein surface diffusion equation. Assuming an initial crack-like cavity shape, they obtained the steady state solution for cavity growth. Martinez and Nix [1982] extended the Finite Difference approach by Pharr and Nix [1979] to study cavity evolution from an initial spherical- shaped cavity to a crack-like cavity and verified their results against the experimental data by Goods and Nix [1978].

Since the development of the variational principle approach by Needleman and Rice

[1980], further studies have been performed to analyze the combined phenomena of the grain boundary diffusion and the surface diffusion (Pan and Cocks [1993a, 1993b, 1995],

Cocks and Pan [1993], Suo and Wang [1994], and Sun, Suo and Cocks [1996]). The main problems in coupling the surface and grain boundary diffusions are to satisfy three physical boundary conditions at the cavity tip, i.e., the continuity of the chemical potential, equilibrium dihedral angle, and matter conservation law. While they successfully overcome these problems, the diffusion element used by the above researchers can be used only for a rigid material.

Despite its apparent importance, a fully coupled continuous cavity growth analysis has never been attempted. The numerical studies of these combined effects on the cavity growth will provide a basic understanding of these synergies, and they will be useful in identifying the critical conditions where the combined effects become important.

In this work, the fully-coupled numerical method is proposed to study cavity shape evolution and cavity growth rate, where the surface and grain boundary diffusion and material viscoplastic deformation are considered. The proposed fully-coupled method

3 continuously describes the cavity shape change and cavity growth in the regimes where both surface diffusion and deformation enhanced grain boundary diffusion are important.

It is shown that the cavity growth rate based on the fully-coupled method is faster than that based on the combined cavity shape assumption in the cavity shape transition region.

This is attributed to the gradual cavity shape change from the spherical to the crack-like.

In addition, it is shown that the spherical cavity evolves to the V-shaped cavity under conditions where both surface diffusion and deformation enhanced grain boundary diffusion are important. It is attributed to the jacking and cavity shape change due to viscoplastic deformation in the fully-coupled method. The cavity shape evolution from the spherical cavity to the V-shaped cavity, which was experimentally observed but has never been predicted, is numerically predicted in this work.

It is shown that the cavity shape change is directly connected to the life prediction of the structure. Therefore, the fully-coupled method which can monitor the continuous cavity shape change is crucial for accurately predicting the remaining life of the structure.

4 CHAPTER 1.2 METHODOLOGY

The atomic flow rates are proportional to driving forces, which are chemical potential gradients of the atom. When tensile stress is applied on the grain boundary, atoms diffuse from the cavity surface to the grain boundary due to the chemical potential gradient. As atoms diffuse from the cavity wall to the grain boundary, the grain boundary should accommodate the diffused atoms.

The chemical potential of the atom on the cavity surface ‘µs ’ and that on the grain

boundary ‘µgb ’ respectively, are given by

µs = −γ s (κ1 + κ 2 )Ω

µ gb = −σ N Ω . (1.1)

The atomic volume, surface tension at cavity surface, two principal curvatures of the cavity surface, and normal stress along the grain boundary are respectively represented

by ‘Ω’, ‘γs’, ‘κ1’, ‘κ2’, and ‘ σ N ’. In Fig. (1.2), the definition of the two principal curvatures for the axis-symmetric cavity surface is shown. The driving force of the surface diffusion denoted by ‘Fs’ and that of the grain boundary diffusion denoted by

‘Fgb’, respectively, are

∂µ F = − s s ∂S (1.2) ∂µ F = − gb gb ∂r

where ‘S’ is the curvilinear coordinate along the cavity surface and ‘r’ is the radial coordinate from the center of the cavity. Assuming the linear kinetic law, the surface and grain boundary atomic flow rates, denoted by ‘js’ and ‘jgb’, respectively, are given by

Equation (1.3).

j = M F s s s (1.3) jgb = Mgb Fgb

where ‘Ms’ and ‘Mgb’ are given by

D Q M = s , D = D δ exp(− s ), s kT s so s RT (1.4) D Q M = gb , D = D δ exp(− gb ). gb kT gb gbo gb RT

In the above equation, ‘Ds’, ‘Dgb’, ‘k’, ‘T’, ‘Dsoδs’, ‘Dgboδgb’, ‘Qs’, ‘Qgb’, and ‘R’ respectively, are the surface diffusivity, grain boundary diffusivity, Boltzman constant, absolute temperature, surface diffusion coefficient, grain boundary diffusion coefficient,

6 activation energy for surface diffusion, activation energy for grain boundary diffusion, and gas constant.

In this work, cavity growth rates and cavity shape evolution are calculated by combining finite element and finite difference methods for a given time step. First, an extended version of the FE method by Needleman and Rice [1980] is used to evaluate the cavity growth rate. This consists of the contribution due to the atomic flow rate from the cavity surface to the grain boundary and that due to the cavity shape change caused by deformation and the phenomena called “jacking”. The “jacking” effect refers to the upward movement of the grain boundary and the cavity surface perpendicular to the grain boundary due to the atomic diffusion from the cavity surface to the grain boundary.

Second, the open ended finite difference method by Pharr and Nix [1979] is used to update the cavity shape for a given atomic rate jo (jo = jgb at the cavity tip). After the cavity shape evolves for the current time step, the chemical potential of the atom at the cavity tip is calculated approximately from the value of principal curvatures of the node next to the cavity tip. At the next time step, the cavity tip stress (σo) is calculated from the cavity tip curvatures and will be used as an input for the FEM portion of the analysis.

The same procedure is repeated until cavity coalescence occurs. In Fig. (1.3), the structure of this numerical procedure is shown.

In combining the FEM and FDM method, numerical determination of the cavity tip position is necessary. The following numerical procedure is developed for this purpose.

After modeling the grain boundary diffusion and grain material deformation (Finite

Element analysis), the original cavity tip node moves upward due to the “jacking” effect.

However, before starting the cavity evolution simulation (Finite Difference analysis), this 7 node needs to move back to somewhere on the grain boundary since the cavity tip node must always be on the original grain boundary. The new position is determined so that the local tension equilibrium condition shown in Fig. (1.4) is satisfied.

As shown in Fig. (1.4), the force equilibrium condition between the cavity surface tension (γs) and the grain boundary tension (γgb) determines the equilibrium angle αo.

The position of the cavity tip node is numerically determined to satisfy this equilibrium angle. In order to physically explain this nodal movement and the associated local cavity shape change, additional matter diffusion within the cavity surface needs to be introduced.

In Fig. (1.5), this additional matter diffusion accompanied by the “jacking” phenomena is explained. In Fig. (1.5) (a), the movement of the cavity surface and the grain boundary after FEM analysis at time t+∆t is shown. The cavity surface and the grain boundary move up due to the volume of the diffused atoms and new cavity surface is created. In order to maintain the equilibrium angle at the newly created cavity surface, local rearrangement of atoms (the additional matter flow δjs) becomes necessary. In Fig.

(1.5) (b), the cavity surface satisfying the equilibrium angle is shown, where the additional matter flow δjs is considered. This surface shape is used for the FDM analysis.

1.2.1 Finite Element Method

As the starting point, the finite element formulation by Needleman and Rice [1980] is implemented to consider the effect of viscoplastic material deformation and grain boundary diffusion on cavity growth. With this approach, the diffusive flux along the 8 cavity surface was not considered and the cavity shape could not be changed from the

initial spherical cap shape. Following Needleman and Rice [1980], the functional,‘ Fgb ’, is given by (detailed derivation is explained in Appendix A),

• jα jα F gb = ω(ε kl )dV − T v dS + dA + σ m j dΓ (1.5) ∫ V ∫ S T i i ∫ A gb ∫ Γ o α α 2M gb Ω

• for all kinematically associated fields, ‘ vi ’, ‘ ε kl ’, and ‘ jα ’, which are material velocity, strain rate, and flow rate, respectively. Greek subscripts (α) have the range 1,2 (repeated indices represent the summation) and refer to a local set of cartesian coordinates in the

• grain boundary. In addition, ‘Ti’, ‘ ω(εkl ) ’, and ‘Agb’ represent tractions, stress power rate, and the grain boundary surface. As shown in Fig. (1.6), ‘m’ is the unit normal vector to the arc (‘ Γ ’, see Fig. (1.2)) of the intersection of the grain boundary and the cavity

surface, and ‘ σo ’ is the normal stress at the cavity tip (stress component normal to the grain boundary), respectively. The last term in Eq. (1.5) comes from the condition of continuity of the chemical potential at the cavity tip.

In order to have a bounded flux at the cavity tip, the chemical potential must be

continuous at the cavity tip. Therefore, the normal stress at the cavity tip (‘ σo ’), which is known as the sintering stress of the cavity, is a function of the specific surface tension

(‘ γs ’) and cavity principal curvatures at the cavity tip (‘ κ1 and κ2 ’ shown in Fig. (1.2)) as follows.

9 σo = γ s (κ1 + κ 2 ) . (1.6)

A cylinder containing a cavity at the center as shown in Fig. (1.7) represents the unit cell model for FE analysis. The material is assumed to be incompressible, and this leads to the velocity boundary condition on the outer boundary of the cylinder (r = b). Due to the symmetric geometric condition, only one quarter of the unit cell is sufficient for the analysis. The far-field stress state is assumed to be uniaxial. The remote creep strain rate

• in the z-direction is represented by ‘ ε∞ ’, and the corresponding remote stress in the z-

direction is denoted by ‘ σ∞ ’.

In this analysis, the grain material is assumed to be elastic and non-linear viscous, with the nonlinear creep specifically of the power law form

• 1/ n σ = Λ ε (1.7)

in the uniaxial tension where ‘ Λ ’and ‘n’ are material constants. Following Needleman

and Rice [1980], we consider an incremental form of a functional, ∆Fgb , given by

1 ∆ F gb = σ : ∆ddV − T ⋅ ∆vdS + ∆j ⋅ ∆jdA + σ o m ⋅ ∆jdΓ (1.8) ∫V ∫S ∫A ∫ T gb 2M gb Ω Γ

for all kinematically associated fields, the rate of deformation tensor, ‘d ’, the velocity,

‘ v ’, and the volumetric flux, ‘ j ’.

10 Based on the matter conservation law, the incremental volumetric flux crossing the

unit length in the grain boundary,‘ ∆jgb ’, are related to the incremental grain boundary

velocity, ‘∆vN’. Due to the axis-symmetric nature of problem, the relevant components

of the flux and velocity, respectively, reduce to ‘ ∆jr ’ and ‘∆vz’ (Appendix B).

2 b ∆jgb = ∆jr = ∫ r'∆vz (r',z = 0)dr' . (1.9) r r

Using the above equation, the last two terms of the functional becomes (Appendix C),

2 4π b 1⎛ b ⎞ b ∆Fd = ⎜ r'∆v (r',0)dr'⎟ dr + 4πσ r'∆v (r',0)dr'. (1.10) ∫∫⎜ z ⎟ o ∫ z MgbΩ a r ⎝ r ⎠ a

In this work, linear shape functions are employed for the incremental normal velocities as follows.

N ∆v z = ∑Φ i (r)∆v zi (1.11) i=1 where

r − ri−1 Φ i (r) = for ri−1 ≤ r ≤ ri ri − ri−1

ri+1 − r Φ i (r) = for ri ≤ r ≤ ri+1 (1.12) ri+1 − ri

Φ i (r) = 0 for r < ri−1 or r > ri+1

11 and ‘N’ and ‘ ∆vzi ’ are the total number of nodal points and the incremental nodal velocity at point ‘i’.

After some algebraic manipulations, ∆Fd becomes

N N N ∆Fd = ∑∑Cij∆v zi ∆v zj + ∑ pi ∆v zi (1.13) i==1 j 1 i=1

where the stiffness matrix ‘Cij’ and ‘pi’ are given by,

4π b 1 C = g (r)g (r)dr ij ∫ i j M gbΩ a r (1.14)

pi = 4πσo g i (a) with

b gi ()r = ∫ r'Φi (r')dr' . (1.15) r

Numerical integration is performed to evaluate the components ‘Cij’ and ‘pi’. The stiffness matrix from the functional (1.8) is constructed and then the effect of creep flow along the grain boundary on the grain boundary diffusion is analyzed numerically. The

cavity growth rate is calculated for the given stress (‘ σ∞ ’), grain boundary mobility

(‘Mgb’, see Eq. (1.4)), cavity tip geometry (‘k1 and k2’, see Fig. (1.2)), and material constitutive relation, Eq. (1.7). The solution of this problem gives two results: the diffusive flux (‘jo’) at the cavity tip and the velocity of nodes along cavity surface due to

“jacking” effect and viscoplastic deformation of the surrounding material. 12

1.2.2 Finite Difference Form

For the second step in the solution process, the cavity evolution due to the surface diffusion is solved. As discussed earlier, Needleman and Rice [1980] assumed that a spherical cavity shape was maintained during the cavity growth for the grain boundary diffusion controlled problem. However, experimental results (Goods and Nix [1978] and

Raj [1978]) have shown that the shape of a cavity becomes a crack rather than a sphere under the surface diffusion controlled condition. Therefore, an analysis, which includes the surface diffusion process, will produce physically more general results than the problems analyzed by Needleman and Rice [1980]. In this numerical procedure, the cavity profile will be updated at each time step, satisfying boundary conditions.

Figure (1.8) shows the axis-symmetric spherical-shaped cavity with boundary conditions at the cavity top and the cavity tip. Due to the symmetry, only one quarter of the cavity is sufficient for the cavity evolution simulation. At the cavity top, the atomic flux, ‘js’, and the angle, ‘α’, are zero because of the symmetry condition. At the cavity tip, the angle, ‘αο’ (see Fig. (1.4)) between the cavity surface and grain boundary remains unchanged to maintain the local force equilibrium condition. In addition, the cavity tip flow, ‘js’, is the same as the flow to grain boundary, ‘jo’. Following the numerical simulations by Martinez and Nix [1982], the cavity tip node moves freely to satisfy this boundary condition. The four boundary conditions imposed for the cavity evolution are:

13 ⎧α = 0 ⎨ at the cavity top ⎩js = 0 ⎧ ⎛ γ ⎞ (1.17) −1⎜ gb ⎟ ⎪α = αo = cos ⎜ ⎟ ⎨ ⎝ 2γs ⎠ at the cavity tip ⎪ ⎩js = jo

For a given step time, the cavity shape evolves in a manner satisfying the matter conservation law and the above boundary conditions.

From Eqs. (1.1)1, (1.2)1, and (1.3)1, the atomic flow driven by the chemical potential difference along cavity surface, ‘js’, is

d(κ + κ ) j = M Ωγ 1 2 (1.18) s s s dS

where ‘dS’ is the surface arc element in the direction of the flux. The diffusive flux (‘jo’) at the cavity tip is given from the previously discussed finite element portion of the

analysis. The cavity surface velocity, vN, is determined from the matter conservation law as follows (Appendix D).

1 d(j r) v = s (1.19) N r dS

Equations (1.18) and (1.19) are the kinetic equations which can describe cavity evolution.

The basic numerical scheme of the finite difference method is used to calculate the movement of each point on the cavity surface satisfying the matter conservation law 14 along the cavity surface. Figure (1.9) shows total ‘n’ discrete points along the cavity surface. Point ‘1’ is located at the cavity top and point ‘n’ is at the cavity tip. The direction of positive arc length ‘S’ is also shown.

Figure (1.10) and (1.11) show the definition of the normal direction of each node. It is necessary to calculate the principal curvatures and the atomic flux to obtain the

th velocity at each node. The ‘i ’ point on the cavity surface has coordinates ‘ (ri (t),z i (t)) ’ at time t. The normal direction at a node is constructed geometrically as the line passing the node and center of circle, which passes ‘i-1th’, ‘ith’, and ‘i+1th’ nodes.

The first primary curvature ‘ κ1 ’ is chosen to be in the plane of the figure and it is approximated to be the same as the inverse of the radius of the inscribed circle. The sign of the curvature is positive when the center of circle is inside the cavity and negative when it is outside. The second primary curvature is given by

cosβi ()κ2 i = (1.20) ri

on the assumption of the axis-symmetric geometry, where angle ‘βi ’ is described on Fig.

(1.10).

The atomic flux at the ‘ith’ point is obtained by Eq. (1.18) using the finite difference method

[(κ1 + κ 2 )i − (κ1 + κ 2 )i−1 ] ()js i = M sΩγ s i (1.21) []∆S i−1

i th th where ‘ []∆S i−1 ’ represents the distance between ‘i ’ point and ‘i-1 ’ point.

Differentiation of Eq. (1.19) gives the more detailed form of displacement velocity (see

Appendix E) :

⎛ djs js ⎞ v N = ⎜ + sinβ⎟ . (1.22) ⎝ dS r ⎠

Using the open ended finite difference method, the above equation can be approximated as

⎛ (j ) − (j ) (j ) ⎞ ()v = ⎜ s i+1 s i + s i sinβ ⎟ (1.23) N i ⎜ i+1 i ⎟ ⎝ []∆S i ri ⎠

i+1 th th where ‘ []∆S i ’ is the distance between ‘i ’ point and ‘i+1 ’ point. In the above

equation, ‘ (v N )i ’ is obtained by using ‘ (js )i ’ and ‘ (js )i+1 ’, where ‘ (js )i ’ is obtained from the curvatures at ‘ith’ point and ‘i-1th’ point and the distance between these two points.

Normal velocity representation, Eq. (1.23), is centralized since information from ‘i-2th’,

‘i-1th’, ‘ith’, ‘i+1th’, and ‘i+2th’ feeds into Eq. (1.23). The normal velocity is calculated from Eq. (1.23) and the normal direction is also obtained from the geometric information at each point on the cavity surface.

16 An appropriate time increment ‘∆t ’, which depends on the flux at the cavity tip, is

chosen. The points on the cavity surface moves by an amount ‘ v N ∆t ’ to its new coordinates, which are approximated as

r (t + ∆t) = r (t) + (v ) ∆t cosβ i i N i i (1.24) zi (t + ∆t)) = z i (t) + (v N )i ∆t sinβi .

All points along the cavity surface moves in a manner based on the scheme described above.

The 1st and nth points (the points at the cavity top and at the cavity tip), however, move differently because of the boundary conditions. First, the point at the cavity top

moves to satisfy the two boundary conditions, the angle β = π/2 and the flux (js )1 = 0.

Since β = π/2 from the geometric symmetry boundary condition, it is reasonable to set an

imaginary node by reflecting point 2 across the z-axis to calculate ‘ κ1 ’ at cavity top. The second primary curvature ‘ κ2 ’ is the same as ‘ κ1 ’ due to the axis-symmetric cavity

dj geometry i.e. ()κ = (κ ) . When r = 0, the matter conservation requires that v = 2 s 2 1 1 1 N dS

(see Appendix F) and it can be approximated by

⎛ (j ) − (j ) ⎞ ()v = 2⎜ s 2 s 1 ⎟ . (1.25) N 1 ⎜ 2 ⎟ ⎝ []∆S 1 ⎠

17 The other end point, the ‘nth’ point, should move along the r-axis satisfying the local tension equilibrium. Therefore, the velocity calculation according to Eq. (1.23) is not attempted for this point. Instead, after the ‘n-1th’ point moves to a new position, the ‘nth’ point floats to a new position in order to satisfy the surface tension equilibrium. It is based on the physical phenomenon that the atoms near the cavity tip instantaneously move to satisfy the local surface tension. The distance between the ‘nth’ and ‘n-1th’ point was kept within ‘0.1aI’ where ‘aI’ is the initial void radius. This guarantees that both the equilibrium condition of surface tension at the cavity tip and the cavity growth rate are accurately predicted throughout the numerical simulation. The flux at the ‘nth’ point,

‘(js)n’, is set equal to ‘jo’ obtained from the finite element analysis and that is used to

th calculate velocity of ‘n-1 ’ point, ‘(vN)n-1’ as in Eq. (1.23).

By trial and error, it was found that 15 points, which are equally spaced along the cavity surface, are sufficient for an accurate analysis. If ‘∆β’ exceeds 18°, a new node is inserted to reduce ‘∆β ’, while the point is removed if ‘ ∆β ’ becomes less than 2°. This procedure reduces the number of points at low curvature regions and increases the

i+1 number of points at high curvature regions. In addition, ‘[∆S]i ’ is restricted not to become too large in the region of high curvature. The time increment ‘ ∆t ’ is limited so that the cavity radius increment ‘ ∆a ’ does not exceed ‘0.005aI’.

The proposed numerical method provides a physically based cavity growth rate where two important aspects of the cavity growth mechanism are considered. First, it includes the effect of the viscoplastic material deformation on the single cavity growth. Second, the cavity profile is updated so that the physical boundary conditions such as the matter

18 conservation law, equilibrium dihedral angle, and continuous chemical potential condition are satisfied. In section 1.3.1 and 1.3.2, the proposed numerical method is verified for two extreme cases; grain boundary diffusion controlled cavity growth and surface diffusion controlled cavity growth. In section 1.3.3 and 1.3.4, cavity shape evolution of an initial spherical cavity is numerically analyzed and cavity growth predictions using the current numerical method are compared with experimental results.

19 CHAPTER 1.3 RESULTS

1.3.1 Modeling of Grain Boundary Diffusion Controlled Cavity Growth

When the surface diffusion is very fast, the grain boundary diffusion controls cavity growth and the spherical-shaped cavity maintains its original shape while it grows.

Based on these assumptions, Needleman and Rice [1980] calculated cavity growth rates and final rupture time of a spherical-shaped cavity. Chen and Argon [1981] proposed an approximate cavity growth equation, which reproduces the Needleman and Rice [1980] numerical results within an error of 30%. In the following, grain boundary diffusion controlled cavity growth is calculated using the current numerical methods, and we verify the proposed numerical method by comparing the obtained result with the work by

Needleman and Rice [1980] and that by Chen and Argon [1981].

In this study, numerical analysis of an initial spherical-shaped cavity was carried out for aI/b = 0.1 and aI/L=0.316, where ‘L’ is defined by

1 ⎛ M σ Ω ⎞ 3 L = ⎜ gb ∞ ⎟ . (1.26) ⎜ • ⎟ ⎝ ε∞ ⎠

20 The parameter ‘L’ is introduced by Rice [1979] and it physically represents “effective diffusion length”. When the effective diffusion length ‘L’ is smaller than the cavity half- distance ‘b’, it is approximately assumed that atoms diffuse from the cavity tip to ‘L’ instead of diffusing from the cavity tip to ‘b’ Chen and Argon [1981] considered the

“effective diffusion length” effect in the cavity growth rate equation (see Appendix G).

The equilibrium dihedral angle was chosen to be 70° and the creep exponent was taken to be 4.5. These conditions are the same as those used by Needleman and Rice [1980]. In order to reproduce fast surface diffusion assumption employed by Needleman and Rice

[1980], a large value of diffusivity ratio, Ds/Dgb = 171, is chosen in this numerical simulation.

In Fig. (1.12) (a) and (b), the nondimensionalized cavity radius, a/b, is plotted as a

• c r • c r function of nondimensionalized time ‘ t ε e ’, where ‘ ε e ’ is the equivalent creep strain rate. Proposed numerical results with or without jacking effect are compared with the work by Needleman and Rice [1980] and those by Chen and Argon [1981]. Cavity growth rates by Needleman and Rice [1980] are only available for a/b = 0.1, 0.33, and

0.67. Therefore, the cavity growth rate for other ‘a/b’ values are obtained by the interpolation method suggested by Needleman and Rice [1980]. Although they proposed an extrapolation method to calculate cavity growth rates for a/b>0.67, it was not used here.

From Fig. (1.12), it is clear that the proposed numerical method exactly reproduces

Needleman and Rice [1980] result by choosing a large surface diffusivity. In addition, it also demonstrates that the assumption of the maintenance of the spherical cavity shape

21 during the growth (which is employed by Needleman and Rice [1980]) is valid under the fast-surface diffusion condition. The proposed unified numerical method can simulate

cavity growth by considering only atomic diffusion contribution (denoted as V& 1 ) to the

total cavity volume ( V& tot = V& 1 + V& 2 ) increase. When the cavity volume increase rate due

to ‘jacking’ and deformation (denoted as V& 2 ) is not included, the present model overestimates cavity coalescence time. Needleman and Rice [1980] pointed out the importance of cavity shape change due to ‘jacking’ and deformation. The analysis with

and without this effect quantitatively demonstrates the importance of V& 2 contribution.

The result by Chen and Argon [1981] shows significant deviation from those by

• c r Needleman and Rice [1980] and by the proposed model in t εe > 0.2. This deviation may be attributed to their assumption that the grain boundary displacement (‘jacking’ effect) due to matter flow is constant over diffusion distance ‘L’.

• cr ⎛ 3 ⎞ Figure (1.13) shows nondimensionalized cavity volume growth rates, ‘ V /⎜ e a ⎟ ’, & ⎜ε I ⎟ ⎝ ⎠ as a function of nondimensionalized cavity radius ‘a/b’. Cavity volume growth rates predicted by the current numerical method match well with the Needleman and Rice

[1980] numerical result at a/b = 0.1, 0.33, and 0.67. Chen and Argons [1981] equation underestimates cavity volume growth rate when a/b > 0.5 under the assumption of fast surface diffusivity.

As opposed to the work by Needleman and Rice [1980], the continuous cavity growth is numerically predicted. Only the initial cavity radius is required in the present work.

22 One to one match between our simulation and the results by Needleman and Rice [1980] demonstrates the validity of the approach provided here.

1.3.2 Modeling of Surface Diffusion Controlled Cavity Growth

When surface diffusion controls cavity growth, the cavity tends to elongate in the direction normal to the applied stress and the cavity shape also tends to become crack- like. In the following, surface diffusion controlled cavity growth is calculated using the fully-coupled numerical method and we verify our approach against the results by

Martinez and Nix [1982] and Chuang et al. [1979].

For this purpose, we use the same parameters as in Martinez and Nix [1982]. The diffusivity ratio is expressed as

D f = gb . (1.27) Ds

Nondimensionalized time and nondimensionalized stress are given by

t T = 4 (1.28) a I M s Ωγ s

σa Σ = I . (1.29) γ s

23 In order to compare the results from the current fully-coupled method to those from the previous works (Martinez and Nix [1982], Chuang et al. [1979]), where the material deformation is not considered, the remote normal stress is set to be small so that the resulting viscoplastic deformation is negligible. The ratio of the applied normal stress to

σ −4 Young’s modulus, ∞ , is chosen to be 10 and initial ‘a /b ’ is 0.1. The capillarity E I I angle at cavity tip is assumed to be 70° and the creep exponent is taken to be 4.5.

In Fig. (1.14), a log-log plot of the nondimensionalized rupture time vs. nondimensionalized applied stress is shown. For f = 1 and f = 10 (surface diffusion controlled region), the current numerical results agree well with the results of Martinez and Nix [1982](denoted as M & N) and Chuang et al. [1979] (denoted as C & R). In reality, the f values of most metals do not exceed 10.

Martinez and Nix [1982] reported stress exponents for fracture, which is the slope of the Fig. (1.14), as -1.7 and -2.2 for f=1 and f=10, respectively. The corresponding values based on the proposed methods are determined to be -1.69 and -2.19 for f=1 and f=10, respectively. These analyses are performed here in the region where the creep deformation is of second order inportance. However, the small differences between these values can be attributed to the creep effect, which is automatically included in the proposed method.

Since the proposed method is fully coupled, the sintering stress (the normal stress at the cavity tip) and the stress along the grain boundary can be calculated including the viscoplastic deformation. In addition, this detailed stress distribution can be continuously monitored as the cavity changes its shape. Figure (1.15) shows the stress distribution

24 along the grain boundary when a/b = 0.1, 0.3, 0.5, 0.7, and 0.9 for f = 1 and f =10, respectively. f=10 implies more surface diffusion controlled than f=1 as in Eq. (1.28).

The normal stress along the grain boundary increases as the cavity grows for both cases

(f=1 and f=10) since the grain boundary area decreases. For f=10, the sintering stress

(σo), increases as the cavity grows. For f=1, the sintering stress decreases when a = 0.3 and 0.5. As shown in Eq. (1.6), the sintering stress is proportional to the summation of the two curvatures (κ1 and κ2). Therefore, the difference between the sintering stress for f=1 and f=10 reflects the difference in the cavity surface geometry near the tip, which can be further analyzed as follows.

Since a cavity becomes crack-like as it grows for both cases (f=1 and f=10), the first primary curvature (κ1) increases at the cavity tip. The second primary curvature (κ2), which is calculated according to Eq. (1.20), decreases as the radius increases. For f=10, the sintering stress increases as the cavity grows because the increase of the first primary curvature is dominant compared to the decrease of the second primary curvature.

However, for f = 1, the decrease of the second primary curvature (κ2) leads to sintering stress decrease when a = 0.3, 0.5. These analyses imply that the cavity shape for f=10 becomes sharper than that for f=1. This point is clearly demonstrated by continuously monitoring the cavity shape change.

Fig. (1.16) (a) and (b) show the cavity shape when a = 0.1, 0.3, 0.5, and 0.7 for f = 1 and f =10, respectively. Figure (1.16) clearly demonstrates that the cavity shape for f=10 becomes sharper than that for f=1. The nodal points on the cavity surface were marked along the cavity surface. Initially, 15 nodes are used along the cavity surface with the same separation distance. As the cavity shape becomes more crack-like, additional nodal 25 points are automatically created along the high curvature area based on the nodal

‘removal-creation’ algorithm employed in this work.

As opposed to the work by Martinez and Nix [1982], the cavity coalescence time is numerically obtained with the deformation consideration. One to one match between our simulation and the results by Martinez and Nix [1982] and Chuang et al. [1979] demonstrates that our approach is valid for the simulation of the surface diffusion dominant cavity growth.

Due to fully coupled nature of our method, the detailed stress distribution including the deformation can be analyzed as shown in Fig. (1.15). In practical applications, an applied remote load does not remain constant, which can change grain boundary stability.

The detailed stress analysis as in Fig. (1.15) will be crucial in predicting grain boundary rupture without cavity coalescence.

1.3.3 Transition from quasi-equilibrium mode to crack-like mode

Based on the results from section 1.3.1 and 1.3.2, the proposed numerical model is validated against two extreme cases; grain boundary and surface diffusion controlled cavity growth cases. In section 1.3.2, the continuous cavity shape change is monitored, but the viscoplastic deformation effect on the cavity growth is not included. Since the viscoplatic deformation effects on the cavity volume growth rate is important in the deformation enhanced diffusion regime as shown in section 1.3.1, it is necessary to study the cavity shape change while considering creep deformation. In this section, we perform

26 analyses in regimes where both surface diffusion and deformation enhanced grain boundary diffusion are important.

In order to include the effect of the material viscoplastic deformation on the cavity growth, the remote normal stress is set to be large so that the ratio between the initial cavity radius, aI, and the effective grain boundary diffusion length, L, is larger than 0.1.

Three different (aI/L) ratios, 0.1, 0.316, and 1.0, are chosen, and the grain boundary and surface diffusivity are chosen so that the cavity is subjected to shape change. Previously

Chen and Argon [1981] successfully calculated the cavity radius growth rate in this regime based on the assumed (spherical or crack-like) cavity shape (see Appendix H). As opposed to their work (Chen and Argon [1981]), in the proposed fully-coupled method, the cavity shape is determined as a result of the numerical simulation. In order to examine the effect of continuous cavity shape change on the cavity growth rate, we organize our numerical results using the same nondimensional parameter α, defined by

⎛ ⎞ ⎜ ⎟ 1 ⎜ ⎟ 4πh(ψ) ⎛⎛ D ⎞⎛ σ L ⎞⎞ 2 α = ⎜ ⎟ × ⎜⎜ gb ⎟⎜ ∞ ⎟⎟ ⎜ 3 ⎟ ⎜⎜ D ⎟⎜ γ ⎟⎟ , (1.30) 2 ⎝⎝ s ⎠⎝ s ⎠⎠ ⎜ ⎛ ⎛ ψ ⎞⎞ ⎟ ⎜ ⎜4sin⎜ ⎟⎟ ⎟ ⎝ ⎝ ⎝ 2 ⎠⎠ ⎠

as in the work by Chen and Argon [1981] where h(ψ) = V /(4πa 3 / 3) =

()1/(1+ cosψ) − cosψ / 2 / sin ψ .

27 Figure (1.17)~(1.20) show the nondimensionalized cavity growth rate vs. the nondimensionalized cavity radius for various cases. In order to properly model the cavity shape transition from the spherical to the crack-like, the smaller α, which means higher surface diffusivity, is chosen for the higher a/L values. The cavity growth rate predictions based on the spherical cavity shape assumption, the crack-like cavity shape assumption, and the sphere and crack-like combined assumption by Chen and Argon

[1981] are included for the comparison purpose.

In Fig. (1.17) (aI/L =0.1 and α = 16.2), the overall cavity growth rate predicted by the fully-coupled method is larger than that by Chen and Argon [1981] prediction. The maximum difference occurs at a/L=0.18, where the proposed method predicts 1624 and

the result by Chen and Argon [1981] predicts 951, and the ratio, a& current / a& C&A , between

the cavity growth rate from the proposed method, a& current , and that by Chen and Argon

[1981], a& C&A , is 1.71. The result matches only at the initial point (a/L=0.1) and at the final point (a/L=0.5). Since the initial cavity shape is spherical in the fully-coupled method, it matches with the prediction based on the spherical shape assumption by Chen and Argon [1981]. The fact that the two predictions match at the final point implies the crack-like cavity shape at a/L=0.5. This result implies that, as opposed to an abrupt change from a sphere to a crack, gradual shape change occurs between these two points.

This demonstrates the adverse effect of gradual cavity shape change on the cavity growth rate, and the cavity growth rate based on the assumed cavity shape simply cannot be used in this transition range.

28 In Fig. (1.18) (aI/L =0.1 and α = 10), the proposed numerical analysis predicts faster cavity growth rates in the transition range and the gradual transition is again observed.

The maximum difference occurs at a/L=0.28, where the proposed result predicts 565 and the result by Chen and Argon [1981] predicts 350. The cavity growth rate ratio,

a& current / a& C&A , at a/L = 0.2, 0.25, 0.28, 0.3, 0.35, and 0.4 is 1.5, 1.6, 1.61, 1.6, 1.4, and 1.3, respectively. The cavity growth rate predicted by the fully-coupled method is larger than the prediction based on the assumed cavity shape (sphere) for a/L=0.14~0.7. The smaller

α value means larger surface diffusion as in Eq. (1.30). Therefore, the cavity shape tends to remain spherical at least in the initial stage of growth. This is clearly demonstrated in the initial region, a//L=0.1~0.14, where the prediction by the fully-coupled method matches with the prediction based on the spherical cavity shape assumption. The fact that the two predictions closely match for a/L=0.5~0.7 implies the crack-like cavity shape in this range. This result again demonstrates the adverse effect of gradual cavity shape change on the cavity growth rate.

In Fig. (1.19) ((aI/L) =0.316 and α = 5.82), the cavity growth rate predictions are compared for a/L=0.316~2.5. In this case, the larger a/L range compared to the previous two cases is considered. The larger a/L value means the smaller effective diffusion length ‘L’ for the same cavity radius ‘a’. As ‘L’ decreases, the effect of the deformation on the cavity growth becomes more important. The cavity growth rate predicted by the fully-coupled method is faster than that by Chen and Argon [1981] for a/L=0.316~0.79.

This implies the transition from spherical to crack-like occurs in this range. For a/L=0.79~2.5, the cavity growth rate predicted by the fully-coupled method is completely

29 different from that (Chen and Argon [1981]) based on the assumed cavity shape. The cavity growth rate predicted by the fully-coupled method is slower than that by Chen and

Argon [1981] for a/L=0.79~2.22 and is faster than that by Chen and Argon [1981] for a/L=2.22~2.5. This implies that the cavity shape does not converge to crack-like shape.

In the cavity growth rate calculation based on the assumed cavity shape (Chen and

Argon [1981]), it is assumed that the cavity volume growth rate ( V& 1 and V& 2 ) is all related with the atomic flux at cavity tip, ‘js(tip)’. Therefore, Chen and Argon [1981] assumed two types of final cavity shape: spherical cavity shape when the surface diffusion is fast or crack-like cavity shape when the surface diffusion is slow. However, the volume

growth rate V& 2 , which is the cavity volume growth rate due to deformation, contributes to the elongation of the cavity in the loading direction. Therefore, as a/L increases and the surface diffusion controls the cavity growth, the final cavity shape cannot be simply assumed to be spherical or crack-like. The cavity shape evolution will be further investigated later in this section.

In Fig. (1.20) ((aI/L) =1.0 and α = 1.0), the cavity growth rate predicted by the fully- coupled method is completely different from that by Chen and Argon [1981] for a/L=1.0~9.0. In this a/L range, the volume growth rate due to deformation and jacking,

V& 2 , becomes more important compared to the previous three cases. The cavity growth rate predicted by the fully-coupled method is slower than the prediction (Chen and Argon

[1981]) based on the assumed cavity shape for the entire a/L range. This implies that the cavity growth rate based on the assumed cavity shape simply cannot be used for this case.

30 Figure (1.21) ~ (1.24) show the cavity aspect ratio variation (c/a) versus a/L for the cases shown in Fig. (1.17) ~ (1.20), where ‘c’ is the cavity minor radius (as shown in Fig.

(1.7)). The aspect ratio variation based on the spherical cavity shape assumption, the crack-like cavity shape assumption, and the sphere and crack-like combined assumption

(Chen and Argon [1981]) are included for the comparison purpose. The initial aspect ratio is 0.615 for all cases.

In Fig. (1.21) (aI/L =0.1 and α = 16.2), the initial aspect ratio predicted by the fully- coupled method is 0.615 and the final aspect ratio is 0.32. The initial aspect ratio based on the crack-like shape assumption is 0.96 and the final aspect ratio is 0.23 while the aspect ratio based on the spherical shape assumption remains 0.615. The aspect ratio change based on the combined cavity shape assumption shows the abrupt change at a/L=0.18. The aspect ratio predicted by the fully-coupled method decreases gradually in the transition range. This demonstrates that the initial spherical shape cavity gradually changes to crack-like cavity. In addition, the gradual decrease of the aspect ratio predicted by the fully-coupled method explains the faster cavity growth rate compared to the cavity growth rate based on the combined cavity shape assumption as shown in Fig.

(1.17). The final aspect ratio predicted by the fully-coupled method does not match with the aspect ratio based on the crack-like assumption at a/L=0.5 even if the cavity growth rate converges as shown in Fig. (1.17).

In Fig. (1.22) (aI/L =0.1 and α = 10), the final aspect ratio predicted by the fully- coupled method is 0.37. The initial aspect ratio based on the crack-like shape assumption is 1.56 and the final aspect ratio is 0.25. The aspect ratio predicted by the fully-coupled method is similar as the initial value for a/L=0.1~0.16. This demonstrates that the 31 surface diffusion is fast enough to maintain the spherical shape for this range. It explains the reason why the cavity growth rate match for the initial transition range as shown in

Fig. (1.18). The aspect ratio based on the combined cavity shape assumption shows the abrupt change at a/L=0.31. The aspect ratio predicted by the fully-coupled method decrease gradually for a/L=0.16~0.7. It implies again that the initial spherical shape cavity gradually changes to crack-like cavity for this case. The final aspect ratio predicted by the fully-coupled method, 0.37, is higher than that for the previous case shown in Fig. (1.21), 0.32, since the lower α value is employed in this case.

In Fig. (1.23) (aI/L =0.316 and α = 5.82), the final aspect ratio predicted by the fully- coupled method is 0.4. The initial aspect ratio based on the crack-like shape assumption is 1.46 and the final aspect ratio is 0.09. The aspect ratio change based on the assumed cavity shape shows the abrupt change at a/L=0.8. The aspect ratio predicted by the fully- coupled method decrease gradually and then saturate to 0.4. The aspect ratio from the fully-coupled analysis does not follow that based on the crack-like shape assumption.

This demonstrates that the cavity does not change to crack-like shape. In addition, it explains the difference in the cavity growth rate as shown in Fig. (1.19).

In Fig. (1.24) ((aI/L) =1.0 and α = 1.0), the final aspect ratio predicted by the fully- coupled method is 1.24. The initial aspect ratio based on the crack-like shape assumption is 1.61 and the final aspect ratio is 0.047. The aspect ratio change based on the combined cavity shape assumption shows the abrupt change at a/L=2. The aspect ratio predicted by the fully-coupled method is completely different from that based on the assumed cavity shape. This demonstrates that the cavity shape evolves to the different shape. The cavity shape evolution for this case is further investigated as follows. 32 Figure (1.25) shows cavity shapes for the case of (aI/L) =1.0 and α = 1.0 when a/L =

1, 3, 5, 7, and 9. The cavity shape is completed by using the axis-symmetric condition.

When a/L = 3, the initial spherical-shaped cavity maintains its original shape with slightly increased aspect ratio. When a/L = 5 and 7, some nodes have negative curvature value because of cavity elongation at the cavity tip in the r-direction due to material diffusion and cavity elongation at the cavity top in z-direction due to the ‘jacking’ and creep deformation. When a/L = 9, cavity shape becomes V-shaped.

1.3.4 Comparison of numerical cavity growth prediction with experiments

Based on the results from section 1.3.3, it is demonstrated that the result by the fully- coupled method is physically more reasonable than that based on the combined cavity shape assumption in regimes where both surface diffusion and deformation enhanced grain boundary diffusion are important. In this section, the fully-coupled method is employed to predict the experimental result by Goods and Nix [1978].

Goods and Nix [1978] used silver in their study of cavity growth and recorded fracture time ‘tf’, and fracture strain ‘εf’, for each constant stress creep test at several temperatures between 200°C and 550°C. These experimental data and test conditions

(test temperature, T, and the remote stress, σ∞) for each test are shown in Table. 1.1, and the physical constants for silver are shown in Table. 1.2. In this study we assumed that the initial cavity radius, ‘aI’, is to be 0.785 µm and ‘af’, is to be 1.1 µm, which was first reported in Goods and Nix’s [1978] experiment and was later modified due to geometric consideration by other authors (Chen and Argon [1981], Pharr and Nix [1979]).

33 In order to use the experimental results in verifying the fully-coupled method, the

4h(ψ) da normalized cavity growth rate, • , versus the normalized cavity radius, a/L, are ε a dt obtained from the experimental results by Goods and Nix [1978]. Following the

4h(ψ) da procedure by Chen and Argon [1981], the normalized cavity growth rate • is ε a dt assumed to be constant due to short testing time and can be approximated (Chen and

Argon [1981]) by

4πh(ψ) da 4πh(ψ) a ≅ ln f , (1.31) • dt ε a εa f I

where ψ is assumed to be 70 degree (Table 1.2). In calculating L from Eq. (1.26) and

ε Eq. (1.4), creep strain rate is approximated by f and the material properties of silver, tf such as Dgbo, δgb,, Qgb, and Ω in Table 1.2, are used. The a/L value is approximated by aI/L and values for each test are shown in Table. 1.1.

In order to employ the fully-coupled method in predicting the experimental result, it is necessary to calculate two parameters: diffusivity ratio, α (Eq. (1.30)), and the ratio between the initial sintering stress and the remote stress, (σo)I/σ∞. The initial sintering stress, (σo)I, is obtained using the following relation,

σo = γs (κ1 + κ2 ) = γs (sin ψ / a I + sin ψ / a I ) = 2γs sin ψ / a I (1.32)

where the relation shown in Eq. (1.6) and the initial spherical cavity shape assumption are used. The values of these two parameters, α and (σo)I/σ∞, depend on test condition. In calculating these parameters, physical constants for silver are required. As shown in

Table. 1.2, different values for surface diffusion constants for silver, including the values from Goods and Nix experiment [1978], are reported. Goods and Nix [1978] calculated the surface pre-exponent value, Dso, and the surface activation energy, Qs, from the rupture time vs. stress plot and the reciprocal of the time to rupture vs. the reciprocal of the test temperature plot, respectively. In order to obtain these plots, they (Goods and

Nix [1978]) used the constant stress creep test results. Therefore physical constants reported by Goods and Nix [1978] are used in this study.

Ds and Dgb are obtained from Eq. (1.4) and Table 1.2. The diffusivity ratio, Ds/Dgb is used in obtaining the α value as in Eq. (1.30). These values for each test are listed in

Table. 1.3. As shown in Table. 1.3, the largest α value is 24 and the smallest α value is

16.2. (σo)I/σ∞ should be less than 1 under a cavity growth condition. These values for test 1 or 2 are larger than 1 or very close to 1, which implies no cavity growth. Therefore they are omitted from further analysis. This leads to the largest (σo)I/σ∞ value, 0.8, and the smallest (σo)I/σ∞ value, 0. In applying the fully-coupled method to this experimental condition, initial spherical cavity is assumed to have one of the following two conditions: aI/L = 0.1, α=16.2, (σo)I/σ∞ =0.0 and aI/L = 0.12, α=24, (σo)I/σ∞ =0.8. The reason for choosing the specific aI/L, α, and (σo)I/σ∞ values for the two conditions is as follows.

35 The aI/L value is assumed to be around 0.1 since the smallest aI/L value reported in

Table. 1.1 is 0.13. According to the reported results by Needleman and Rice [1980] and the current study, an increase of the (σo)I/σ∞ value significantly decreases the cavity growth rate. Therefore, the cavity growth rate with (σo)I/σ∞ =0.0 is the fastest and the cavity growth rate with (σo)I/σ∞ =0.8 is the slowest for a given constant α value.

Compared to the effect of the stress ratio, (σo)I/σ∞, on the cavity growth rate, the effect of

α value on the cavity growth rate is not significant. Therefore, any combination of α and

(σo)I/σ∞ from α=16.2~24, and (σo)I/σ∞ =0.0 represent the upper bound (fast normalized cavity growth rate) and any combination of α and (σo)I/σ∞ from α=16.2~24, and (σo)I/σ∞

=0.8 represent the lower bound (slow normalized cavity growth). In this study, the condition, aI/L = 0.1, α=16.2, and (σo)I/σ∞ =0.0, is chosen to represent the upper bound and the condition, aI/L = 0.12, α=24, and (σo)I/σ∞ =0.8, is chosen to represent the lower bound. This condition represents the narrowest bound of any possible combination of α and (σo)I/σ∞.

Figure (1.26) shows the normalized cavity growth rate versus the normalized cavity radius for Goods and Nix [1978] experiment, the prediction from the fully-coupled method, and the prediction based on the spherical cavity shape assumption with (σo)I/σ∞

=0.0 and 0.8. As shown in Fig. (1.26), some of the experimental data points (denoted by

●) are bounded by the prediction based on spherical cavity shape assumption with

(σo)I/σ∞ =0.0 and 0.8. However, some of the experimental data points (denoted by ▲) are not bounded by the prediction based on the spherical cavity shape assumption. This implies that some of initial cavities considered in Goods and Nix [1978] are already

36 crack-like at the start of the analysis and the cavity shape evolution from the spherical to the crack-like needs to be considered in predicting the experimental data points. As shown in Fig. (1.26), all the experimental data points excluding test 1 and test 2 are bounded by the fully-coupled method prediction for the two extreme cases. This demonstrates validity of the fully-coupled method developed in this work.

In this study, the one-to-one match between the experimental result and prediction by the fully-coupled method is not possible, since the initial cavity shape is not known and precise value of the initial and final cavity radius is not reported for each test. If these information are available, individual test data can be predicted based on the fully coupled method.

Chen and Argon [1981] successfully predicted the experimental results by Goods and

Nix [1978] using the cavity growth rate prediction based on the combined cavity shape assumption. However, specific values of surface diffusion material properties employed for the analysis (Chen and Argon [1981]) are not mentioned in their work and it is not clear how two extreme values of α, 13 and 4, are chosen. In this study, the α and

(σo)I/σ∞ values are based on the material data obtained by Goods and Nix [1978], and they are more carefully chosen compared to those by Chen and Argon [1981]. In addition, Chen and Argon [1981] assumed that the sintering stress is zero. In this study, no such assumption is employed in obtaining the results.

37 CHAPTER 1.4 DISCUSSION

Arai et al. [1996] reported cavity shape evolution from the spherical-shape to the V- shape under load controlled cyclic test conditions. Since the fully-coupled method developed in this work is developed for the cases of constant loading, the quantitative comparison between the experimental result (Arai et al. [1996]) and the prediction based on the fully-coupled method is not readily available. However, it is possible to make a qualitative comparison.

They (Arai et al. [1996]) argued that the sharp creep strain rate increases upon stress reversal or accumulated plastic strain can be possible reasons for this cavity shape evolution. According to our recent numerical study (Oh et al., will appear on J. of Eng.

Mat. Tech.) on the constrained cavity growth, it was shown that, under load controlled cyclic test conditions, the grain material experiences sharp creep strain rate increases at stress reversals, which increases creep flow effects on the cavity growth. In this study, it was shown that the initial spherical cavity evolves to the V-shaped cavity when the creep strain rate increases (a/L > 1.0). While the cyclic loading condition used in the experiments was not simulated, the cavity shape change from a sphere to a V-shape qualitatively matches the experimental observation. This qualitative match can be attributed to the fully coupled nature of our developed method combining the surface diffusion and the deformation enhanced grain boundary diffusion.

38 The material viscoplastic deformation, which causes cavity shape change, ‘jacking’ at the cavity tip, and the decrease of the effective diffusion length along the grain boundary, have never been fully coupled with the surface/grain boundary diffusion in the cavity growth study. In this work, possible cavity shape evolution from the spherical to the V- shaped is numerically predicted for the first time and the adverse effect of gradual cavity shape change on the cavity growth rate is demonstrated.

In order to achieve the quantitative agreement between the prediction from the fully- coupled method and the experimental result (Arai et al. [1996]), the following features need to be considered in the future. In this study, numerical simulations under uniaxial tensile loading condition were studied. According to our recent numerical study (Oh et al., will appear on J. of Eng. Mat. Tech.) on the grain boundary cavitation, it was shown that under uniaxial tensile loading conditions, the multiaxial stress state develops around the cavities on the cavitating grain boundary. The effect of the multiaxial stress on the cavity growth rate is numerically studied by Sham and Needleman (Sham and Needleman

[1982], Sham and Needleman [1983]). They reported that cavity growth rate depends on the stress ratio between the Mises type effective stress and the hydrostatic stress state based on the assumption of fast surface diffusion. In order to accurately predict the cavity growth rate under multiaxial stress condition, the fully coupled method proposed in this study needs to be expanded to include the multiaxial stress state in the boundary condition.

In this study, a steady state creep flow rule is employed to represent the material inelastic constitutive relation. The steady state creep flow rule, which is implemented in the proposed fully-coupled method, is accurate enough when a constant loading is

39 applied or cavity is nucleated when the strain hardening of the grain material is saturated.

However, in all practical loading conditions, a cyclic loading condition is more realistic than a constant loading condition. The recent numerical study (Oh et al., will appear on

J. of Eng. Mat. Tech.) on the grain boundary cavitation shows that strain hardening type creep flow rule is more accurate than the steady state creep flow rule in representing the material inelastic constitutive relation under cyclic loading condition. Under this condition, it is necessary to employ the strain hardening type creep flow rule in the fully- coupled method.

40 CHAPTER 1.5 CONCLUSION

In this study, a unified numerical method is proposed, where the surface and grain boundary diffusion and material viscoplastic deformation are considered. The proposed unified method is verified by comparing the obtained results for extreme cases with the previous works (Needleman and Rice [1980], Martinez and Nix [1982]).

In the first extreme case, the loading condition and the material constants are chosen so that the deformation is not important and matter diffusion is the dominant cavity growth mechanism. When the surface diffusion controls the overall diffusion process, it is shown that the cavity shape changes from spherical-cap shape to crack-like shape as shown by Martinez and Nix [1982]. In addition, it is shown that the cavity coalescence time matches with the previous works (Chuang et al. [1979], Martinez and Nix [1982]).

In the second extreme case, the loading condition and the material constants are chosen so that matter diffusion is enhanced by the material viscoplastic deformation.

When the grain boundary diffusion controls the overall diffusion process, the cavity is shown to maintain initial spherical shape as it grows. As opposed to the work by

Needleman and Rice [1980], the continuous cavity growth is numerically predicted based on the given initial cavity geometry. Therefore, this result confirms the previous work

(Needleman and Rice [1980]) where the spherical cavity is assumed to maintain its spherical shape as it grows under the fast surface diffusion. In the previous numerical

41 work (Needleman and Rice [1980]), the cavity growth rate is discretely calculated for chosen cavity radius. The cavity volume growth rates are calculated at the same cavity radius as in Needleman and Rice [1980]. It is shown that one to one match between our results and the previous results (Needleman and Rice [1980]) is achieved. This quantitative verification demonstrates the validity of our approach.

Upon verification of the proposed method by considering extreme cases, a fully coupled case of the surface diffusion controlled cavity growth in the deformation enhanced diffusion region is examined. When the diffusion is more important than the phenomena called “jacking” (upward movement of cavity surface) and material viscoplastic deformation, the transitional shape change from the initial spherical-cap shaped cavity to the crack-like cavity is predicted. The cavity growth rate in this transition range is found to be much faster than that obtained by combining the cavity growth rates with two extreme cavity shape (Chen and Argon [1981]) (spherical and crack-like). This demonstrates that the transitional cavity shape change, which has never been examined in the previous work, significantly accelerates the cavity growth rates.

Therefore, a fully coupled model needs to be employed for accurate life prediction.

When the “jacking” and material viscoplastic deformation is more important than diffusion, the transitional shape change from the initial spherical cavity to the V-shaped cavity is predicted. While V-shaped cavity has never been predicted in the previous work, Arai et. al [1996] experimentally observed the V-shaped cavity under cyclic loading condition. Unfortunately, quantitative comparison between the experimental results and the proposed method could not be made due to unknown material parameters and different loading condition. However, the proposed fully coupled approach provides

42 the research community a possible mechanism of experimentally observed transitional cavity shape change from a spherical cavity to V-shaped cavity.

Upon obtaining physically reasonable result for a fully coupled case of the surface diffusion controlled cavity growth in the deformation enhanced diffusion region, the fully-coupled method is employed to predict the experimental result (Goods and Nix

[1978]). The one-to-one match between the experimental result and prediction based on the developed method is not possible due to lack of parameters such as initial cavity shape and dimension. However, it is shown that all the cavity growth rates reported in the experiment are bounded by the fully-coupled method prediction for the two extreme cases. This demonstrates the validity of the fully-coupled method in regimes where both surface diffusion and deformation enhanced grain boundary diffusion are important. .

As a future work, it is desirable to include features, such as the multiaxial stress state and the strain hardening type creep flow rule, in the fully coupled method so that it can be readily applied to life prediction under cyclic loading condition.

Test T (K) σ∞ (Pa) tf (Sec.) εf • L (m) a/L ε(≅ εf / tf ) (/Sec.) no.

1 823 1.38E+06 3.07E+05 0.006 1.95E-08 1.07E-05 0.07 2 823 2.76E+06 5.58E+04 0.008 1.34E-07 7.10E-06 0.11 3 823 3.45E+06 2.52E+04 0.006 2.50E-07 6.22E-06 0.13 4 823 4.83E+06 1.10E+04 0.006 5.82E-07 5.25E-06 0.15 5 823 6.89E+06 2.20E+03 0.006 2.91E-06 3.46E-06 0.23 6 823 8.69E+06 1.29E+03 0.009 6.59E-06 2.85E-06 0.28 7 823 1.09E+07 3.20E+02 0.010 3.09E-05 1.83E-06 0.43 8 823 1.38E+07 1.35E+02 0.019 1.44E-04 1.19E-06 0.66 9 673 6.89E+06 7.70E+04 0.006 8.18E-08 4.58E-06 0.17 10 673 9.74E+06 1.75E+04 0.012 6.57E-07 2.56E-06 0.31 11 673 1.38E+07 4.99E+03 0.020 4.07E-06 1.57E-06 0.50 12 673 2.07E+07 9.24E+02 0.022 2.42E-05 9.90E-07 0.79 13 673 2.76E+07 3.90E+02 0.033 8.51E-05 7.17E-07 1.09 14 673 3.45E+07 1.80E+02 0.025 1.36E-04 6.61E-07 1.19 15 673 4.48E+07 5.62E+01 0.027 4.82E-04 4.73E-07 1.66 16 573 1.09E+07 3.42E+05 0.012 3.36E-08 2.97E-06 0.26 17 573 1.72E+07 2.70E+04 0.016 6.07E-07 1.32E-06 0.60 18 573 2.76E+07 4.50E+03 0.017 3.87E-06 8.32E-07 0.94 19 573 3.45E+07 2.14E+03 0.024 1.12E-05 6.28E-07 1.25 20 573 4.48E+07 6.00E+02 0.033 5.45E-05 4.05E-07 1.94 21 573 5.52E+07 8.10E+02 0.030 3.70E-05 4.94E-07 1.59 22 573 6.89E+07 3.15E+02 0.025 7.94E-05 4.12E-07 1.90 23 573 8.68E+07 1.85E+01 0.007 3.89E-04 2.62E-07 3.00 24 473 2.17E+07 3.96E+05 0.013 3.18E-08 1.07E-06 0.73 25 473 3.45E+07 5.04E+04 0.032 6.29E-07 4.62E-07 1.70 26 473 4.83E+07 1.76E+04 0.028 1.59E-06 3.79E-07 2.07 27 473 6.89E+07 3.31E+03 0.014 4.23E-06 3.08E-07 2.55

Table 1.1 Experimental conditions reported by Goods and Nix [1978] and a/L value

γs ψ Dgbo δgb Qgb Ω Dso δs Qs

J/m2 Deg. m2/sec m J/mole m3 m2/sec m J/mole

2.5E-061 845001 1.71E- 2.57E- 1.14* 70* 1.2E-5* 5E-10* 90016.2* 4.5E-062 491002 29* 10* 1.4E-063 674003

* S.H.GOODS & W.D.NIX, Acta metall, 26, 739, 1978, T.Z.CHUANG et.al. Acta metall, 27, 265, 1979 J.T. Robinson and N.J. Peterson, Surf. Sci. 31, 586, 1972 D. Turnbull, J. appl. Phys. 22, 634, 1951 1 S.H.GOODS & W.D.NIX, Acta metall, 26, 739, 1978 2 T.Z.CHUANG et.al. Acta metall, 27, 265, 1979 3 N.A.Gjostein, Diffusion Seminar, ASM, Cleveland 1974

Table 1.2 Material properties of silver

Test no. Temp (K) Ds/Dgb α (σo)I/σ∞

1 823 2.40E-01 1.62E+01 1.98E+00 2 823 2.40E-01 1.87E+01 9.89E-01 3 823 2.40E-01 1.95E+01 7.91E-01 4 823 2.40E-01 2.12E+01 5.65E-01 5 823 2.40E-01 2.06E+01 3.96E-01 6 823 2.40E-01 2.10E+01 3.14E-01 7 823 2.40E-01 1.89E+01 2.50E-01 8 823 2.40E-01 1.71E+01 1.98E-01 9 673 2.35E-01 2.39E+01 3.96E-01 10 673 2.35E-01 2.13E+01 2.80E-01 11 673 2.35E-01 1.98E+01 1.98E-01 12 673 2.35E-01 1.93E+01 1.32E-01 13 673 2.35E-01 1.90E+01 9.90E-02 14 673 2.35E-01 2.03E+01 7.92E-02 15 673 2.35E-01 1.96E+01 6.09E-02 16 573 2.38E-01 2.41E+01 2.50E-01 17 573 2.38E-01 2.02E+01 1.58E-01 18 573 2.38E-01 2.03E+01 9.90E-02 19 573 2.38E-01 1.97E+01 7.92E-02 20 573 2.38E-01 1.81E+01 6.09E-02 21 573 2.38E-01 2.21E+01 4.95E-02 22 573 2.38E-01 2.26E+01 3.96E-02 23 573 2.38E-01 2.02E+01 3.14E-02 24 473 2.51E-01 1.99E+01 1.26E-01 25 473 2.51E-01 1.65E+01 7.92E-02 26 473 2.51E-01 1.77E+01 5.65E-02 27 473 2.51E-01 1.90E+01 3.96E-02

Table 1.3 α and the stress ratio between the initial sintering stress

and the remote stress

σ∞

2b Cavity surface Js ψ

2a Jgb Grain boundary

(a)

Original shape Creep flow effect Final cavity shape due to fast surface diffusion

Grain boundary

(b)

Fig. 1.1 (a) Hull-Rimmer type diffusion flow along cavity surface and grain boundary. The grain boundary separate as rigid bodies. (b) The effect of creep flow on grain boundary diffusion. The deformation of the grain material causes local accomodation at the cavity tip.

κ2 κ1

Fig. 1.2 Principal curvatures at cavity tip, κ1 in the plane perpendicular to the grain boundary and κ2 in the plane of the grain boundary, are shown, where arrow shows the tangent direction of each curvature. Intersection of the grain boundary and the cavity surface is shown by Γ (Γ is in the plane of the grain boundary).

48 Initial spherical-cap shape cavity

Cavity growth rate calculation Finite Element Method Cavity shape evolution due to creep flow and jacking effect

Calculate atomic flow rate(jo)

jo σo

Cavity surface profile update Finite Difference Method Cavity shape evolution due to atomic flow

Calculate approximate chemical potential at the cavity tip(σo)

Fig. 1.3 Illustration of numerical calculation structure for single cavity growth model. The unified numerical method, which combines finite element method and finite difference method, starts with the known spherical-cap shape cavity geometry. For the given time step, which is chosen to be sufficiently short, finite element method and finite difference method are employed to simulate cavity shape evolution.

49 γs 2γs cosαο = γgb αο γgb

γs

Fig. 1.4 Equilibrium dihedral angle (αo) satisfying the local tension equilibrium condition, where γs and γgb are cavity surface tension and grain boundary surface tension, respectively.

50 Cavity surface after FEM analysis

Before δjs is considered Original cavity surface Diffused atoms from

Newly created cavity surface (jo) surface

(a)

Fig.1.5 (a) The cavity surface and the grain boundary shape at time t+∆t after FEM analysis is done. The additional flow δjs is not considered. The grain boundary and the cavity surface at time t moves up due to the volume of the diffused atoms. The newly created cavity surface at the cavity tip is not at equilibrium angle. (b) The cavity surface and the grain boundary shape at time t+∆t before starting FDM analysis. The additional flow δjs is necessary to satisfy the equilibrium angle at the newly created surface.

(continued)

51 (figure 1.5 continued)

Cavity surface after FEM analysis

Before δjs is considered

δjs αo

Cavity surface after FEM analysis

After δjs is considered

(b)

σo (normal stress at the cavity tip)

mα

Grain boundary Cavity surface

Fig. 1.6 Sintering stress (σo) and the unit normal vector (mα) at the cavity tip

53 σzz= σ∞,σrz= 0 z

σrz= 0 Grain material

1 • v = − b ε ∞ h b r 2 • n (ε ∞ = Bσ ) Cavity surface ∞ c r a 1 ∂ (rj ) + 2v = 0 r ∂r gb z Grain boundary ∂ j = M Ω (σ ) j = 0 gb gb ∂r zz gb Boundary conditions on the grain boundary 1 ∂ (rj ) + 2v = 0:matterconservation law r ∂r gb z ∂ j = M Ω (σ ) : linear kinetic law for atomic flux gb gb ∂r zz

σo is specified at r = a : the continuous chemical potential condition

jgb = 0 at r = b : axis - symmetric condition

Boundary conditions on the outer surface

On z = h: σ z = σ∞ , σ rz = 0 1 • On r = b: σ = 0, vr = − ε ∞ b zr 2 On r = 0: ur = 0. Fig. 1.7 Unit cell model with a spherical-cap shaped cavity with the major radius ‘a’, the minor radius ‘c’, and the cavity half distance ‘b’. The boundary conditions on the grain boundary satisfy the linear kinetic law for the atomic flux and the matter conservation law. The boundary conditions on the outer surface of the grain material satisfy the axis-symmetric condition. 54 Boundary conditions at the cavity top ⎛α = 0 ⎜ ⎝ js = 0

Grain

α Boundary conditions at the cavity tip z ⎛α = α = cos−1 γ / 2γ ⎜ o ()gb s cavity ⎜ ⎝ js = jo r Grain boundary

Fig. 1.8 Boundary conditions along the cavity surface for the finite difference method

55 +S

Node 1 n-1

Node n

Fig. 1.9 Discretized cavity surface is schematically represented, which is used in the finite difference method.

56 (ri(t+∆t),zi(t+∆t))

vN·∆t β i-1 i

αi (ri(t),zi(t)) i+1

Fig. 1.10 Finite difference numerical scheme showing how the ith point on the cavity surface moves to the new position (hollow point) after time increment ∆t and the definition of the angle βi

Fig. 1.11 Definition of the surface normal vector (n) at the node on the cavity surface

58 1.01.0 a/ba/b ffuully-lly-coupcoupledled analysis analysis ( (wwitithh j ajackckiningg e effefecct)t) 0.80.8 ChenChen && ArgArgoonn NeedlemNeedlemaann & & Rice Rice ffuully-lly-coupcoupledled analysis analysis ( (wwitithouthout j ajacckking)ing) 0.60.6

0.40.4

0.20.2

0.00.0 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 ••crcr0.20 ttεεee

(a)

Fig. 1.12 (a) Nondimensionalized cavity major radius as a function of nondimensionalized time for aI/L=0.316, aI/bI=0.1, and Ds/Dgb=171(fast surface diffusion). When a/b is 0~0.5, the present numerical results (with jacking effect) reproduce the Needleman and Rice results. In this range, the Chen and Argon results also match with the Needleman and Rice results. When jacking effect is not considered, present result deviates with the other three results. That implies jacking effect is significant in this a/L range. (b) Nondimensionalized cavity major radius as a function of nondimensionalized time for aI/L=0.316, aI/bI=0.1, and Ds/Dgb=171(fast surface diffusion). When a/b is 0.5~1.0, the present results still match well with the Needleman and Rice results. However, Chen and Argon result starts to deviate from the Needleman Rice results.

(continued)

59 (figure 1.12 continued)

1.01.0 a/ba/b 0.80.8

0.60.6

0.40.4 fulfullyly--cocoupledupled a annaalylysissis (w (withith ja jackingcking effect) effect) ChenChen & & Arg Argoonn 0.2 0.2 NeeNeeddlemlemaann & & Rice Rice ffuully-lly-coucouppledled an analysisalysis ( w(witithouthout j ajackckining)g) 0.00.0 0.20.2 00.3.3 0.0.44 00.5.5 0.60.6 00.7.7 0.80.8 ••crcr ttεεee

(b)

• V •cr 3 εe aI

250

200 Present results Chen & Argon Needleman & Rice 150

100

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 a/b

Fig. 1.13 Nondimensionalized cavity volume growth rate as a function of nondimensionalized cavity major radius for aI/L=0.316, aI/bI=0.1, and Ds/Dgb=171(fast surface diffusion). Present results match well with the Needleman and Rice results (reported at a/b=0.1, 0.33, and 0.66). Chen and Argon results start to deviates when a/b is larger than 0.5.

Nondimensionalzed Rupture time(T) 102 Present f=1 Present f=10 M&N f=1 101 M&N f=10 C&R f=1 C&R f=10 100

10-1

10-2 110100 Nondimensionalized Applied stress(Σ)

Fig. 1.14 Nondimensionalized grain boundary rupture time (when a/b reaches to 1) vs. nondimensionalized applied stress under different diffusivity ratios -4 (f=Dgb/Ds), for aI/bI=0.1, σ∞/E=10 , and cavity tip angle=70°. Chuang et al. [1979] (C&R) analytically calculated surface diffusion controlled cavity growth rate on the assumption of crack-like cavity. Martinez and Nix [1982] (M&N) used Finite Difference Method to evaluate the cavity shape evolution. When f=1 and 10 (surface diffusion controlled region), the present results match well with both M & N and C & R.

Nondimensionalized normal stress (σ / σ ) N ∞ f=1, a=0.1 3.0 f=1, a=0.3 f=1, a=0.5 2.5 f=1, a=0.7 f=1, a=0.9 f=10, a=0.1 2.0 f=10, a=0.3 f=10, a=0.5 1.5 f=10, a=0.7 f=10, a=0.9 1.0

0.5

0.0 0.0 0.2 0.4 0.6 0.8 1.0

Nondimensionalized radius (r/b)

Fig. 1.15 The nondimensionalized normal stress of each element above the grain boundary vs. the nondimensionalized grain boundary length is shown for -4 σ∞/E=10 and cavity tip angle 70°. When a/b=0.1, stress distribution is the same for both f=1 and f=10 (surface diffusion controlled) cases. The stress along the grain boundary increases as the cavity grows for both cases, f=1 and f=10, since the grain boundary area decreases. For f=10, the sintering stress (σo), which is the normal stress at the cavity tip, increases as the cavity grows. For f=1, the sintering stress decreases when a = 0.3 and 0.5.

63 (a) (b)

-4 Fig. 1.16 Evolution in the cavity shape for σ∞/E≈10 and cavity tip angle 70°, (a) f = 1 (surface diffusion controlled), (b) f = 10 (more surface diffusion controlled case). In all cases aI/bI=0.1 and the cavity changes to crack- like. The node removal-creation procedure is clearly shown for both cases. Cavity shape becomes more crack-like when f=10.

Nondimensionalized cavity growth rate da • 4πh()ψ / ε∞ a dt fully-coupled analysis overall C & A prediction C & A prediction with spherical cavity shape C & A prediction with crack-like cavity shape 5x103 4x103 3x103

2x103 1624

3 9x10102 8x102 7x102 6x102 951 5x102 4x102 3x102

2x102

102 0.18 0.0 0.1 0.2 0.3 0.4 0.5

Nondimensionalized cavity radius (a/L)

Fig. 1.17 Nondimensionalized cavity growth rate vs. nondimensionalized cavity radius is shown for aI/L=0.1 and α=16.2. An initial spherical cavity evolves to a crack like cavity with much higher da/dt values compared to the prediction by Chen and Argon [1981]; the maximum difference occurs when a/L=0.18, where the current result predicts 1624 and the analysis by Chen and Argon [1981] predicts 951.

Nondimensionalized cavity growth rate • da fully-coupled analysis 4πh()ψ / ε∞ a overall C & A prediction dt C & A prediction with spherical cavity shape C & A prediction with crack-like cavity shape 5x103 4x103 3x103

2x103

3 9x10102 8x102 7x102 6x102 565 5x102 4x102 3x102 350 2x102

2 10 0.28 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Nondimensionalized cavity radius (a/L)

Fig. 1.18 Nondimensionalized cavity growth rate vs. nondimensionalized cavity radius is shown for aI/L=0.1 and α=10.0. An initial spherical cavity evolves to a crack like cavity with much higher da/dt values compared to the prediction by Chen and Argon [1981]; the maximum difference occurs when a/L=0.28, where the current result predicts 565 and the analysis by Chen and Argon [1981] predicts 350.

66 Nondimensionalizezd cavity growth rate da • 4πh()ψ / ε∞ a dt fully-coupled analysis 103 overall C & A prediction C & A prediction with spherical cavity shape C & A prediction with crack-like cavity shape

102

101 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 Nondimensionalized cavity radius (a/L)

Fig. 1.19 Nondimensionalized cavity growth rate vs. nondimensionalized cavity radius is shown for aI/L=0.316 and α=5.82. Ιnitial nondimensionalized cavity growth rates predicted by the fully-coupled method are slightly higher than those by Chen Argon [1981]. As a/L increases, cavity growth rates do not follow those based on the assumed cavity shape.

67 Nondimensionalized cavity growth rate da • 4πh()ψ / ε∞ a dt fully-coupled analysis 103 overall C & A prediction C & A prediction with spherical cavity shape C & A prediction with crack-like cavity shape

102

101

100 0123456789 Nondimensionalized cavity radius (a/L)

Fig. 1.20 Nondimensionalized cavity growth rate vs. nondimensionalized cavity radius is shown for aI/L=1.0 and α=1.0. Since the cavity volume change due to creep flow (jacking effect and the cavity shape change due to creep flow) is significant in this a/L range, the analysis by Chen and Argon [1981], which assumes that the cavity volume growth rate is all related with the atomic flow rate at cavity tip, can not accurately predict the cavity major radius rate and cavity shape change.

68 fully-coupled analysis C & A prediction with spherical cavity shape c/a C & A prediction with crack-like cavity shape overall C&A prediction 1.0 aspect ratio of the initial spherical-cap shape cavity (0.615)

0.5 Final aspect ratio (0.32)

0.0 0.0 0.1 0.2 0.3 0.4 0.5 a/L

Fig. 1.21 Cavity aspect ratio variation during evolution of spherical-cap shape cavity for aI/L=0.1 and α=16.2 is shown. The aspect ratio by the fully- coupled method decrease gradually.

69 fully-coupled analysis c/a C & A prediction with spherical cavity shape C & A prediction with crack-like cavity shape overall C&A prediction

1.5 aspect ratio of the initial spherical-cap shape cavity (0.615) 1.0 Final aspect ratio (0.37)

0.5

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 a/L

Fig. 1.22 Cavity aspect ratio variation during evolution of spherical-cap shape cavity for aI/L=0.1 and α=10.0 is shown. Cavity maintains the initial aspect ratio until a/L reaches 0.16, since surface diffusivity is higher compared to the case of Fig. 1.21 (lower α value).

70 c/a fully-coupled analysis 1.5 C & A prediction with spherical cavity shape C & A prediction with crack-like cavity shape overall C&A prediction

1.0 Aspect ratio starts to decrease gradually

Aspect ratio saturate at 0.4

0.5

0.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 a/L

Fig. 1.23 Cavity aspect ratio variation during evolution of spherical-cap shape cavity for aI/L=0.316 and α=5.82 is shown. The aspect ratio by the fully-coupled method saturates at much higher value (c/a=0.4) than that based on the crack like cavity shape assumption (c/a=0.09)

c/a fully-coupled analysis C & A prediction with spherical cavity shape C & A prediction with crack-like cavity shape 1.5 overall C&A prediction

1.0 Final aspect ratio (1.24)

0.5 Aspect ratio starts to increase gradually due to material creep flow effect

0.0 0123456789 a/L

Fig. 1.24 Cavity aspect ratio variation during evolution of spherical-cap shape cavity for aI/L=1 and α=1 is shown. The cavity aspect ratio by the fully-coupled method does not follow those based on the assumed cavity shape. It clearly shows that the initial spherical-cap shape cavity does not evolve to become crack like cavity.

z/L 15.0 a/L=1 a/L=3 7.5

0.0 a/L=9

a/L=7 -7.5 a/L=5

-15.0 -11-9-7-5-3-1 1 3 5 7 911 r/L

Fig. 1.25 Cavity evolution from spherical-cap shape to V-shape for aI/L=1, α=1 is shown. Initial spherical-cap shape cavity maintains its original shape until a/L=3. When a/L reaches to 5, cavity shape changes to V-shape, since surface diffusivity is slow and material creep flow effect is significant.

fully-coupled analysis (α=16.2,(σo)I/σ∞ =0.0) fully-coupled analysis (α=24, (σo)I/σ∞ =0.8) C & A prediction with spherical cavity shape ((σo)I/σ∞ =0.0) • C & A prediction with spherical cavity shape ((σ ) /σ =0.8) da o I ∞ 4πh()ψ / ε∞ a Experimental results by Goods and Nix, which falls within the C&A prediction based on spherical cavity dt Experimental results by Goods and Nix, 10000 which falls outside the C&A prediction based on spherical cavity

1000

100

1 0.01 0.1 1 10 a/L

Fig. 1.26 Normalized cavity growth rate vs. normalized cavity radius are shown. The experimental results are bounded by the cavity growth rates predicted by the fully coupled method (aI/L=0.1, α=16.2, σo/σ∞=0 (hollow circle) and aI/L=0.12, α=24, σo/σ∞=0.8 (hollow square)). Since different material constants for surface diffusion are reported and the simple assumptions are employed on obtaining the data points for the experimental results, the direct comparison between the each experimental data points and the numerical simulation results is not possible. However, from the distribution of each data points in this figure, it is clear that the cavity evolves from the spherical-cap shape to other shape in this range of a/L and α values.

74 PART 2

STUDIES ON EFFECT OF CYCLIC LOADING ON

GRAIN BOUNDARY RUPTURE TIME

CHAPTER 2.1 INTRODUCTION AND LITERATURE REVIEW

Many structural components, such as nuclear power plants, chemical processing plants, and aero space vehicles, are subject to variable, rather than constant, loading conditions at high temperature. When the structure is under a constant stress at high temperature, which is called the creep condition, cavity initiation and cavity growth are important failure mechanisms that must be understood in order to predict the lifetime of components.

Such nucleation and growth phenomena are explained by atomic diffusion, creep flow of surrounding material, and grain boundary sliding. Cavity growth leads to cavity coalescence, and then grain boundary rupture occurs. The material damage by cyclic loading, which is called fatigue, can accelerate such grain boundary rupture phenomena.

Prediction of the cavity growth rates and final grain boundary rupture time is important in estimating the rupture lifetime of high temperature service components in creep-fatigue conditions since cavity growth examination of the structure in service is difficult.

75 In many studies performed before Dyson’s work [1976], the axial stress applied to single cavity growth models was assumed to be the same as the remotely applied stress.

However, Dyson [1976] pointed out that when cavitation occurs at an isolated grain boundary, as shown in Fig. 2.1 (a) and (b), the cavity growth is constrained by the creep strain rate of the surrounding material.

Riedel [1987] explains the mechanisms behind this constrained cavity growth as follows. Grain boundary rupture occurs on an isolated grain boundary accounting for a substantial portion of the total fracture time since the susceptibility of the grain boundary to cavitation is related to its orientation to the stress axis and on the orientations of surrounding grains. If the surrounding material were rigid, the material flow from cavity to grain boundary cannot be accommodated by the surrounding material and the cavity growth stops. Thus cavity growth is controlled by the deformability of the surrounding material and it is called constrained cavity growth. On the other hand, if the surrounding material deforms fast enough to accommodate this excess volume of material, cavity growth is controlled by material diffusion process. This is called unconstrained cavity growth.

The constraint of the surrounding material exerts stress on the cavitating grain boundary to satisfy the compatibility condition between the cavity volume increase rate and the deformation rate of the surrounding material. Therefore, in the constrained limit, the resulting stress reduces to the order of the sintering stress where sintering stress refers to the stress at the cavity tips. In the unconstrained limit, the resulting stress on the grain boundary is the same as the remote stress.

76 Rice [1981] successfully illustrated the constrained cavity growth phenomena, and

Riedel [1983] and Tvergaard [1984] extended Rice’s work. A recent study by Arai et al.

[1996] has shown that the grain boundary rupture time can be better predicted by the constrained cavity growth rate equation than by the Hull-Rimmer [1959] model.

Under a constant loading condition, the resulting stress on the grain boundary can be obtained from the compatibility condition and it can be readily employed on the single cavity growth rate equation to calculate the final grain boundary rupture time. However, under a stress controlled cyclic loading condition, the creep strain rate is expected to change depending on the material constitutive relation, and the constrained cavity growth process will change to the unconstrained cavity growth process depending on the material deformability.

In this work, a stress controlled unbalanced cyclic loading is used and qualitative studies on the cavity growth, which switches between the constrained and the unconstrained at each cycle, is performed. The stress and the inelastic strain rate change around the grain boundary, and the final rupture time is analyzed using a form of the grain boundary rupture model first proposed by Tvergaard [1984].

Tvergaard [1984] proposed the grain size level model shown in Fig. 2.1 (c) to simulate the constrained/unconstrained cavitation process at an isolated grain boundary.

As shown in Fig. 2.1 (c) a cavitating grain boundary facet is represented by a disk without any void geometry. The description of the cavity growth is provided to the model as the boundary condition of the disk through an approximate equation where the coupled influence of the grain boundary diffusion and the grain material deformation is considered.

77 Tvergaard [1984] used Norton’s power law equation as the material constitutive equation. However, cavities can grow in the primary creep region, and cavities may already exist before the polycrystalline metal is placed in service. In addition, a significant creep strain rate can be introduced around the crack tip or cavity upon reversing the stress state (F.W. Brust [1995]). Therefore, under cyclic loading conditions, constitutive equations which properly describe the cyclic creep region should be used to examine the cavity growth behavior properly.

Murakami-Ohno [1982] proposed a continuum mechanics-based constitutive law to describe cyclic creep response. The notion of a creep hardening surface is used to model the sharp creep strain rate response upon stress reversal resulting from the remobilization of immobilized dislocations. This behavior is clearly observed in experimental test results and cannot be explained by classical constitutive laws. We note that classical creep constitutive laws have extensively been used to study creep damage in prior works.

Motivated by this phenomenon, the effect of a physically-based constitutive model and that of the cyclic loading history on the grain boundary rupture time was simulated numerically based on a grain size level FEM model. The results obtained are compared to numerical predictions based on Norton’s power law material model, and the importance of using a realistic material behavior model will be demonstrated.

78 CHAPTER 2.2 CONSTITUTIVE MODELING AND FINITE ELEMENT

ALGORITHM FOR THE MURAKAMI-OHNO MODEL

Conventional time dependent stress-strain behavior under a constant loading at high temperature is defined as three phases; primary or transient creep, where strain hardening occurs due to dislocation density increase, secondary or steady state creep, where dislocation density saturates, and tertiary creep, where micro cracks link and final creep fracture occurs. Under cyclic loading conditions, when local stresses are reversed, dislocations, which are piled up on obstacles, can have a longer free path which leads to fast creep strain rates after the stress reversal. Hence traditional creep laws (Gibbons

[1989]), such as the Norton law for steady state creep or classical primary creep law, are not adequate for this kind of material behavior.

A number of constitutive equations have been proposed to handle the material softening phenomena. However, it is often difficult to determine material constants for these constitutive laws. Also, the mathematical framework for these material models is often complicated to use in practice since numerical stability is an issue requiring solution times to be excessively long. Murakami and Ohno [1982] proposed a simple model which can successfully describe material behavior under non-steady state of stress.

This model defines ‘Creep Hardening Surface (CHS)’, g = 0, with a center, α ij , and radius ρ, analogous to the elastic boundary in the conventional plasticity theory. When

79 the creep strain is on CHS, g = 0, the creep strain pertains to irreversible deformations.

When the creep strain is inside of CHS, g < 0, the creep strain rate increases due to material softening.

• The total strain rate tensor, εij , is first separated into a time-dependent elastic

• e • cr component, εij , and a time-dependent creep component, εij , given by

• • e • cr εij = εij + εij . (2.1)

Short time plastic strains can be accounted for by including them over short time period.

The creep strain rate by Murakami-Ohno [1982] for the general multi-axial case is given by

• cr 1 m−1 n −m 3 m m εij = mA q σ m s (2.2) 2 e ij

where sij and σe respectively are deviatoric stress component and Mises type effective stress; A, m, and n are material constants, and

⎛ (ε cr − α )⎞ q = ρ + ⎜ ij ij ⎟ s . (2.3) ⎜ ⎟ ij ⎝ σe ⎠

80 Equation (2.2) reduces to the time hardening law for primary creep for a uniaxial specimen under a constant load.

The evolution equation for the center of the yield surface αij and radius ρ are given by

• 1 ⎛ • cr ⎞ • 1 • cr ∂g • cr ij ⎜ kl ⎟ ; ij if g 0 and ij 0 α = ⎜ε ηkl ⎟ηij ρ = ε ηij = ε > 2 ⎝ ⎠ 6 ∂ε

• • ∂g • cr α ij = ρ = 0 if g < 0 or cr εij ≤ 0 (2.4) ∂εij

where ηij is the outward normal vector to CHS defined as

cr εij − α ij ηij = 1 . (2.5) cr cr 2 {}()ε kl − α kl (ε kl − α kl )

The boundary of CHS is described by

2 g = ()ε cr − α ()εcr − α − ρ2 = 0 on CHS and < 0 inside (2.6) 3 ij ij ij ij

Therefore, the radius and the center of the CHS change only when the creep strain state is on the CHS and remain the same when the state of creep strain is inside the CHS.

In addition to its simple constitutive form and accurate description of stress reversal, the

81 practical advantage of this theory is that it coincides with the classical strain-hardening theory for the case of uniaxial constant stress. Therefore, all material constants for this theory can be obtained from uniaxial creep test data, which exists for many materials.

The performance of more than ten creep constitutive models is compared in references

(Gittis [1975], Inoue et al. [1989], Inoue [1985], Chaboche [1977]). It clearly shows that the Murakami-Ohno constitutive laws can model the behavior of complex multi-axial loads cases. This further justifies the choice of Murakami-Ohno constitutive model for this study. The Norton’s creep flow rule is also used in the analysis to study the effect of the constitutive law on the cavity growth.

The third order Runge-Kutta method is used to calculate the creep strain increment.

(explained in Appendix I) The forward time marching scheme is employed in the

ABAQUS analysis with a user subroutine used for the Murakami-Ohno law. Although this scheme does not guarantee a stable convergent solution for large time step sizes, sufficiently accurate solutions can be obtained for small step sizes. Reference

(Krishnaswamy et al. [1995]) provides details of the model.

82 CHAPTER 2.3 METHODOLOGY

In this study, Tvergaard’s [1984] numerical model for grain boundary rupture was used. In this numerical model, the grain boundary rupture process is simulated using the analytical cavity growth rate equation and material conservation law. There are four basic assumptions in this simplified numerical model.

First, cavitating grain boundary is surrounded by non-cavitating grains. That means other cavitating grain boundaries are too far away to affect the cavitating grain boundary.

This assumption assures the fact that only the deformability of surrounding grain material determines either the constrained cavity growth or the unconstrained cavity growth.

Second, the geometry of each cavity on the cavitating grain boundary is not modeled. It is assumed that cavities occupy imaginary positions on the grain boundary. The analytical cavity growth equation, which requires the stress and inelastic strain rate values around the position of the imaginary cavity, is employed to calculate the cavity growth. This assumption simplifies the numerical model and reduces computational time. Third, the grain boundary diffusion controls the cavity growth. The surface diffusivity is assumed to be fast enough to maintain initial cavity shape, which are spherical caps in this study, until the cavities coalescence. Fourth, the grain boundary displacement is constant over the atomic diffusion distance. This constant displacement

83 assumption is not valid when the inelastic grain material deformation occurs. However, in this model, detailed grain boundary displacement is not considered.

In Fig. 2.1 (a), a polycrystalline material with an isolated cavitating grain boundary is

shown. It is also shown that the remote stress, ‘σ∞ ’, may be different from the resulting

stress, ‘ σ n ’, on grain boundary depending on the accommodation rate of the surrounding material. In Fig. 2.1 (b), periodically arrayed cavities on the grain boundary are shown.

Although the model is not limited to the case with evenly spaced cavities, it is assumed

• here to simplify the problem. The cavity growth rate ( V ) of each cavity is related to the

• grain boundary displacement rate ( δ ) according to the matter conservation law.

In Fig. 2.1 (c), the grain size FEM model is shown. The actual cavities on the grain boundary are not modeled in this approach, and the cavitating grain boundary has the displacement boundary condition, ‘ δ/2 ’ due to symmetry. The non-cavitating grain boundary is not modeled in this study, and the continuum-based constitutive material property is assumed for the entire volume. The grain boundary sliding and the continuous cavity initiation are also not included here. The radius of the cavitating grain boundary is denoted by ‘d’, and the height and width of the axis-symmetric model are denoted by ‘H’ and ‘W’, respectively, where ‘W’ represents half of the length between two adjacent cavitating facets.

• The approximate cavity growth rate, ‘ V ’, is based on the work of Chen and Argon

[1981], Sham and Needleman [1982], and Needleman and Rice [1980]. Budiansky,

Hutchinson, and Slutsky [1982] and Tvergaard [1984] proposed cavity growth rate equations due to creep flow. Budiansky, et. al [1982] proposed the cavity growth rate 84 equation in the region where the cavity volume change due to creep flow and atomic diffusion does not play a significant role in the total cavity volume growth. Here

Budiansky’s [1982] work is not included since the cavity growth due to diffusion/creep flow assisted diffusion is simulated in this work.

The analytical form of the cavity growth rate, in which the combined effect of creep flow and grain boundary diffusion is included, is

• σ − (1− h)σ V = 4πM Ω n o (2.7) gb ln(1/ h) − (3 − h)(1− h) / 2

where ‘ σ o ’is the sintering stress at the cavity tip and

D M = gb , (2.8) gb kT

Q D = D δ exp(− gb ), (2.9) gb gbo gb RT

⎛ 2 2 ⎞ ⎜⎛ a ⎞ ⎛ a ⎞ ⎟ h = max ⎜ ⎟ ,⎜ ⎟ , (2.10) ⎜⎝ b ⎠ ⎜ a +1.5L ⎟ ⎟ ⎝ ⎝ gb ⎠ ⎠

1/ 3 ⎛ ⎞ ⎜ M gbσeΩ ⎟ Lgb = . (2.11) ⎜ • cr ⎟ ⎝ ε e ⎠

85 In the above equations, “Dgb”, “k”, “T”, “Dgboδgb”, “Qgb”, “R”, ‘a’, ‘b’, ‘ Lgb ’, ‘Ω ’, ‘ σe ’,

• cr and ‘ ε e ’ are the grain boundary diffusivity, Boltzman constant, absolute temperature, grain boundary diffusion coefficient, activation energy for grain boundary diffusion, gas constant, cavity radius, half distance between the cavities, diffusive length-scale parameter above the grain boundary, atomic volume, Mises type effective stress, and effective creep strain rate above the grain boundary, respectively. When the creep flow of the grain material accelerates grain boundary diffusion, diffusive length for the atomic

flow along the grain boundary shortens from ‘b’ to ‘a+ L gb ’ and the cavity volumetric growth rate increases. The parameter ‘h’ is included to account for this effect and the constant ‘1.5’ is a multiplier used to consider the hydrostatic stress state condition (Sham and Needleman [1982]). The Mises type effective stress and effective creep strain rate,

• cr (‘σe ’ and ‘ ε e ’, respectively), above the grain boundary are used to calculate ‘ Lgb ’. In

this study, the sintering stress, ‘ σ o ’, which is the function of the surface free energy and cavity tip geometry is assumed to be zero.

At each time step, the cavity growth rate (Eq. (2.7)) is obtained using the normal stress and the creep strain rate value at integration points of the elements around each imaginary cavity location along the grain boundary. This cavity growth rate is related to the grain boundary displacement rate by the mass conservation law, as suggested by Rice

[1981]:

86 • • V 2V • δ = − b . (2.12) πb 2 πb3

This grain boundary displacement is applied as the boundary condition along the grain boundary at the next time increment. Also, at every time increment, the radius of the

• imaginary cavity is increased based on the radius increase rate, ‘a ’. The relationship between the cavity volumetric growth rate and the cavity radius increase rate is

• • V = 4πa 2h()ψ a (2.13)

⎛ 1 cosψ ⎞ h()ψ = ⎜ − ⎟ / sin ψ;cosψ = γ gb / 2γ s (2.14) ⎝1+ cosψ 2 ⎠

where ‘ h(ψ)’ is a function of the angle ‘ψ ’ which the void surface makes with the plane

of the grain boundary at its tip, ‘ γ gb ’ and ‘ γ s ’ are the surface energy of the grain boundary and the cavity surface, respectively.

The above equation is based on the assumption of quasi-equilibrium cavity growth.

• Chen and Argon [1981] proposed the cavity radius increase rate, ‘ a ’, for both the spherical-shaped cavity and the crack-like cavity. However, recent studies by these same authors (Oh et al., submitted in J. of Mech. and Phys. Of Solids) show that the cavity growth rate in the transition range from the spherical-shaped cavity to the crack-like cavity is faster than the previously reported values (Chen and Argon [1981]). Also, as

87 the combined effect of the creep flow and grain boundary diffusion on the cavity growth increases, a ‘jacking’ effect, which is caused by the local accommodation of the atomic movement at cavity tip, plays a significant role in the cavity shape change and cavity radius increase rate. Therefore, the application of Eq. (2.7) ~ (2.14) in the calculation of the cavity growth is a valid assumption only for the grain boundary diffusion controlled cavity growth without the consideration of the cavity shape change.

For the analysis of axis-symmetric FEM model, the boundary conditions are specified in the form

σrr = T, σrz = 0, ∆Ur = constant, at r = W

σzz = S, σrz = 0, ∆Uz = constant, at z = H

∆Uz = 0, σrz = 0, at z = 0 and d < r < W (2.15)

∆Uz = δ / 2, σrz = 0, at z = 0 and 0 < r < d

∆Ur = 0, σrz = 0, at r = 0 and 0 < z < H

• • where δ = δ ∆t and δ is given by Eq. (2.12). Remote uniaxial stress, ‘S’, which is perpendicular to the cavitating grain boundary, is applied and the radial stress, ‘T’, is applied to control the hydrostatic stress state.

The ABAQUS commercial package is used, and the UEL subroutine (Smelser and

Becker [1989]) is coded to control the hydrostatic stress condition and also to give the displacement boundary condition on the unit cell outer surface and is summarized in appendix J. The nonlinear geometric effect is considered. The user subroutine DISP is used to give the proper displacement boundary conditions, Eq. (2.12), of the cavitating

88 grain boundary. The Murakami-Ohno constitutive equations are implemented through a specially developed UMAT subroutine in ABAQUS.

In this analysis, two types of material creep flow rules are applied. First, Murakami-

Ohno cyclic creep flow rule and Norton’s steady state creep flow rule are simply added to calculate total creep strain rate (denoted as ‘MO + N’ case). The creep strain rate for

‘MO + N’ case is

• cr • cr • cr ε = ε M−O + ε N (2.16)

• cr • cr where ε M−O is the creep strain rate by Murakami-Ohno flow rule and ε N is the creep strain rate by Norton flow rule. The above equation is based on the assumption that the overall creep strain rate is dominated by either the Murakami-Ohno flow rule or the

Norton flow rule and the transition time, in which both flow rules predict the creep strain rate of the same order, is relatively short compared to the overall grain boundary rupture time or tensile holding time. Second, Norton’s steady state creep flow rule is used to calculate total creep strain rate (denoted as ‘N’ case).

89 CHAPTER 2.4 RESULTS

2.4.1 Verification of FEM code and constitutive law

Figure 2.2 shows the load spectrum and the corresponding creep strain response under a combined tension and torsion at 650°C on type 304 stainless steel tubes. The material constants are given as follows (Ohashi [1982]):

A = 3.1×10−19 hr −0.54 MPa −7.2 ,m = 0.54,n = 7.2.

Figure 2.2 compares the creep strain predictions by the present user subroutine UMAT with the experimental data by Murakami-Ohno [1982]. When the load (and hence stress) is completely reversed, the experimental results show a sharp increase in creep strain due to remobilization of immobilized dislocations (Brust [1995], Murakami and Ohno [1982],

Gibbons [1989]). As shown in Fig. 2.2, the Murakami-Ohno constitutive law can predict the experimentally obtained sharp creep strain rate increase upon stress reversal. As mentioned before, conventional creep constitutive laws, such as the classical strain hardening type law (Gittis [1975]) and Norton’s law, cannot account for this material softening phenomena as shown in Fig. 2.2.

90 In Fig. 2.3, the displacement controlled cyclic loading is applied on the tensile specimen of the Inconel 617 at 950°C and the resulting stress variation is shown. The material properties of the Inconel 617 at 950°C employed for this verification are shown in Table 2.1. Two types of material constitutive laws are considered. First, the combination of the classical strain hardening type creep flow rule and Norton’s flow rule is used (denoted as ‘ST+N’ case). Second, the combination of the Murakami-Ohno and

Norton’s flow rule is used (denoted as ‘MO+N’ case). Figure 2.3 (a) shows the displacement boundary condition. It is clearly shown in Fig. 2.3 (b) and (c) that ‘MO+N’ flow rule predicts stress relaxation phenomenon better than ‘ST+N’ flow rule.

2.4.2 Grain boundary rupture analysis

In the present numerical analysis of the grain boundary rupture process, initial cavities are assumed to be distributed along the grain boundary with the initial conditions

(aI /bI) = 0.1 and bI/ dI = 0.1 where ‘aI’, ‘bI’, and ‘dI’ are the initial cavity radius, the initial cavity half spacing, and the initial half radius of cavitating grain boundary, respectively. Remote normal stress applied is specified by σ∞/E ≅ 1~4E-4 depending on

‘(a/L∞) I’ values and (a/L∞) I= 0.01, 0.015, 0.03, and 0.055. The radial stress is maintained as zero.

When ‘(a/L∞) I’ is larger than 0.06 and the constant loading is applied, the cavity growth is not constrained. In this study, ‘(a/L∞) I’ is confined not to exceed 0.06.

According to the numerical result by Needleman and Rice [1980], the creep flow of the

91 grain material accelerates grain boundary diffusion when ‘aI/Lgb’ is over 0.1. Since the cavity growth is constrained in this study, ‘a/Lgb’ is much less than ‘a/L∞’ and the dominant cavity growth mechanism is atomic diffusion along the grain boundary.

The material properties of the 1.25Cr-0.5Mo steel at 538°C employed in the grain boundary rupture analysis are summarized in Table 2.2.

The reference time used here is defined as (Tvergaard [1984])

(σe )∞ t r = (2.17) ⎛ • cr ⎞ E⎜ε e ⎟ ⎝ ⎠∞

where ‘ (σe )∞ ’ is the Mises type remote effective stress ( (σe )∞ = S − T for axis-

⎛ • cr ⎞ symmetric case) and ‘ ⎜ε e ⎟ ’ is the remote effective creep strain rate based on the ⎝ ⎠∞

Norton’s flow rule. It is assumed that the grain boundary rupture is attained when ‘a/b’ reaches 0.6. The initial cavity tip angle is assumed to be 70° and the sintering stress is taken to be zero, which is a reasonable assumption as long as the normal stress is well above the sintering limit. Most calculations are carried out with the mesh type as shown in Fig. 2.1 (c). This mesh does not present the near tip field of the penny shaped crack accurately, but is considered to be sufficient for the grain boundary rupture analysis

(Tvergaard [1984]).

Figure 2.4 shows the stress controlled loading condition applied for this analysis. The compressive loading time, ‘tc’, is about 0.001tr so that ‘tc’ is negligible compared to the

92 total rupture time and the ratio of the tensile loading time to the compressive loading time

(tt /tc) is set to be 10, 50, and 100, respectively. Since the cavity volumetric growth rate equation, Eq. (2.7), is not experimentally verified for the compressive loading condition, the compressive loading time is set to be a very small value such that the cavity shrinkage during the compressive loading time can be ignored. The focus of this analysis is to examine the effect of number of cycles on the void growth. Therefore the parameters unrelated to the number of cycles, such as ‘tc’ and ‘tm’, are chosen to be very small.

Figures 2.5, 2.6, 2.7, respectively, show normalized cavity radius increase for a cavity at r/d = 0 as a function of normalized time for initial value (a/L∞) I = 0.01, 0.015, and

0.03. A combination of the two types of material flow rule and two types of loading conditions, constant loading and cyclic loading, are considered. The slope in these figures represents the cavity growth rate.

As shown in Fig. 2.5, for all four cases, the initial slope is steep and the slope (cavity growth rate) starts to decrease. The normalized cavity radius, a/b, of ‘MO+N’ case is larger than that of ‘N’ case when the normalized time is small. This reflects the faster creep strain rate of Murakami-Ohno law. In other words, the material constraint process for ‘MO+N’ case progresses more slowly than ‘N’ case. For ‘N’ case, the cavity growth rate and the final rupture time under the cyclic loading are almost the same as those under the constant loading case. For ‘MO+N’ case, however, the cavity growth rate is increased and the final rupture time, which is the time at a/b=0.6, is shortened under the cyclic loading condition compared to that under the constant loading. Figure 2.6 and 2.7 show the similar damage evolution pattern with the result shown in Fig. 2.5.

93 Figures 2.8 shows normalized cavity radius increase for a cavity at r/d = 0 as a function of normalized time for initial value (a/L∞) I = 0.055. A combination of the two types of material flow rule and constant loading is considered. As shown in Fig. 2.8, the initial slope does not change much for both types of material flow rule until the grain boundary failures. This demonstrates that, when (a/ L∞) I = 0.055, the material inelastic deformation rate predicted by either Norton’s law or Murakami-Ohno’s law is fast enough to accommodate the volume increase due to the cavity growth. In addition, the final rupture time is similar for these two material cases. This means that the fast creep strain rate predicted by ‘MO+N’ material flow rule during the primary creep region does not make a significant difference in the final rupture time since the current analysis is stress controlled problem.

Figure 2.9 illustrates the normalized normal stress development as a function of normalized time at r/d = 0 under the constant loading condition for aI/L∞ = 0.01. The

normalized normal stress is defined as σ n / σ∞ , where σn is the local normal stress above the grain boundary. The normalized normal stress, which is 1 initially, abruptly decreases due to the material constraint. For ‘MO+N’ case, the normal stress decreases more slowly than ‘N’ case due to the fast creep strain rate by Murakami-Ohno flow rule.

After the transition time, the normal stress saturates at the small value for both cases.

Figure 2.10 shows the normalized normal stress variation after the load reversal from

nd st the compression to the tension at t/tr ≅ 0.1 (2 cycle) and 1.6 (21 cycle) for aI/L∞ = 0.01 and tt/tc = 100. The normalized normal stress variations under the constant loading

nd condition are also shown for the comparison purpose. During the 2 cycle (t/tr ≅ 0.1),

94 the normalized normal stresses under the constant loading condition do not saturate to the same value for ‘MO+N’ and ‘N’ cases. As explained in Fig. 2.9, ‘MO+N’ predicts higher normal stress in this region. The normal stresses after the stress reversal are higher than those under the constant loading condition for both ‘MO+N’ and ‘N’ cases.

These high stresses under the cyclic loading relax and become the same as those under

st the constant loading condition. During the 21 cycle (t/tr ≅ 1.6), the normalized normal stresses under constant loading condition already saturate to the same value for both

‘MO+N’ and ‘N’ cases and these remain constant. Under the cyclic loading condition, for both ‘MO+N’ and ‘N’ cases, the high stresses caused by stress reversal relax and they saturate to the same value as those under the constant loading. The saturation time is defined as the time in which the peak normal stress after the stress reversal saturates to the normal stress under the constant loading condition. The saturation time at t/tr ≅ 1.6 is much shorter than that at t/tr ≅ 0.1.

The corresponding effective creep strain rate above the grain boundary is shown in

Fig. 2.11. Significant creep strain rate increase after the load reversal alleviates material constraint, which results in the normal stress increase above the grain boundary as shown in Fig. 2.10. For ‘MO+N’ case, the significant increase of the effective creep strain rate caused by the stress reversal decreases ‘Lgb’ in Eq. (2.11). However, the calculated ‘Lgb’ is still longer than ‘b’. That means the effective diffusion length is ‘b’ for the case studied here. Therefore, the overall effect of the cyclic loading on the cavity growth is the change of the local normal stress through the grain material deformation rate change.

Figure 2.12 shows the normalized normal stress variation as a function of the normalized time at r/d = 0 under the constant loading condition for (a/ L∞) I = 0.055. 95 Grain material constraint is not significant when (a/ L∞) I ≥ 0.06. Therefore, the initial normalized normal stress value decreases gradually compared to the result in Fig. 2.9 ((a/

L∞) I = 0.01). Figure 2.13 shows the relation between the number of cycles and normalized cavity coalescence time. The number of cycles is obtained as follows;

Number of cycles = tf, constant loading /(tt + tc) (2.18)

where tf,constant loading is the cavity coalescence time under the constant loading condition and (tt+tc) is the total time for one cycle. The normalized cavity coalescence time is expressed as tf/tf,constant loading, where tf is the cavity coalescence time under given cyclic loading condition. Three different cyclic loading conditions, (tt /tc) = 10, 50, and 100, and three different (a/L∞) I values, (a/L∞) I = 0.01, 0.015, 0.03, are applied. The total cavity coalescence time decreases up to 12% compared to the constant loading case. The grain boundary rupture time slightly increases when the number of cycles exceeds 300.

Increase of total compressive loading time is relevant with this behavior but the reason is not clear at this time.

96 CHAPTER 2.5 DISCUSSION

Arai et. al. [1996] continuously observed the constrained cavity growth under creep and creep-fatigue conditions. The stress waveform for the creep-fatigue experiment is similar to the present cyclic pattern. However, constants for Murakami-Ohno constitutive law are not available for the material used in their work [1996]. In addition,

Arai et. al. [1996] observed the cavities on the specimen surface, while the cavities inside the material are analyzed in our FEM model. Therefore, it is not possible to quantitatively compare the numerical results with the experimental results (Arai et al.

[1996]). However, it is possible to make a qualitative comparison.

In the current result, the cavity coalescence time under a creep-fatigue condition is

10% shorter than that under a pure creep condition. Arai et. al. [1996] reported that the cavity growth rate under a creep-fatigue condition is about 10~100 times faster than that under a creep condition. While the orders of magnitude are different, the current result qualitatively matches the experimental result. This qualitative match can be attributed to the physically realistic constitutive model employed in this analysis.

The numerical simulation for the constrained cavity growth under a cyclic loading condition has never been attempted in the previous numerical works. In all practical loading conditions, a cyclic loading condition is more realistic than a constant loading condition. This work is the first numerical prediction of the possible detrimental effect of

97 a cyclic loading on the service life of the material compared to the service life under a constant loading.

In order to achieve the quantitative agreement between the numerical result and the experimental result, the following assumptions used in the model need to be modified in the future. In this work, the combined effect of the creep flow and the grain boundary diffusion on the cavity growth is considered through the parameter ‘h’, in Eq. (2.10), and

‘Lgb’, in Eq. (2.11). In the cases considered in this paper, the effective atomic diffusion distance ‘a+Lgb’ is longer than ‘b’ for most of the rupture life time even in the cyclic loading case. However, ‘Lgb’ value depends on the variables such as compressive loading time, load transition time, the Murakami-Ohno cyclic creep law constants (A, m, and n), ‘aI/L∞’, ‘aI/bI’, and ‘aI/dI’. When ‘a+Lgb’ is shorter than ‘b’, the assumption of the straight grain boundary movement, which is assumed in the model, should be modified to include the concentration of atomic diffusion around the cavity tip.

Arai et. al. [1996] reported that an initial spherical-shaped cavity deforms to a V- shaped cavity under a creep-fatigue condition. The recent numerical study on the single cavity growth behavior by the current authors (Oh et al. submitted in J. of Mech. and

Phys. of Solids) shows that the cavity shape changes from the spherical-shaped to the V- shaped when ‘a/Lgb’ value is large (fast creep strain rate) and the surface diffusion is also considered. Therefore, the cavity shape change observed in the experiment by Arai et. al.

[1996] may partly be attributed to the significant creep strain rate increase at the moment of stress reversal from compression to tension. In order to make the numerical model more complete, cavity shape changes discussed in Part 1 should be included in the future.

98 Arai et al. [1996] reported that under creep-fatigue loading conditions, grain boundary rupture finally occurs by the unstable grain boundary debonding. Van Der

Giessen and Tvergaard [1995] also showed that the cavities at grain boundary edge can grow faster than the cavity at grain boundary center when free grain boundary sliding is included in the model. Inclusion of grain boundary sliding and grain boundary element into this model should be considered in the future work.

99 CHAPTER 2.6 CONCLUSION

A physically realistic Murakami-Ohno cyclic creep law is employed in Tvergaard’s approximate numerical method for simulating cavity growth on an isolated cavitating grain boundary. The proposed numerical model properly describes the constrained grain boundary rupture phenomena under a stress controlled cyclic loading condition. It is shown that the damage acceleration under cyclic loading condition is faster than that under constant loading condition. This is attributed to the significant creep strain rate increase upon stress reversal, which causes fast material accommodation rate and high normal stress around the cavity. Final grain boundary rupture time under cyclic loading condition is shown to be more than 10% shorter than under constant loading condition.

Theses results qualitatively agree with the experimental result (Arai et al. [1996]). The

Norton’s law under cyclic loading should be used with caution for all creep conditions.

Norton’s constants Murakami-Ohno’s constants Young’s modulus

B (MPa-nbhr-1) nb A(MPa-nhr-m) m n E (MPa)

1.0637E-14 6.7 1.009E-12 0.375 3.73 143705.6

Table 2.1 The material property of Inconel 617 at 950°C.

100

Norton’s constants Murakami-Ohno’s constants Young’s modulus Yield stress

-nb -1 -n -m B (MPa hr ) nb A(MPa hr ) M n E (MPa) σy (MPa)

4.49E-20 8.0 7.9E-15 0.55 5.66 140.6E+03 131.0

Table 2.2 The material property of 1.25Cr-0.25Mo steel at 538°C (Murakami and Ohno

[1982]).

101

σ∞ δ/2 σn

σn δ/2 σn

• 2d • V 2V • δ = − b σ∞ πb2 πb3

(a) (b) W

FEM MODEL H H z

r W d

(c)

Fig. 2.1 (a) Isolated cavitating grain boundary in the polycrystal. σ∞ is the remote axial stress and σn is the local normal stress. Since cavity growth causes grain boundary displacement, the local stress can be different from the remote stress depending on the material deformability. (b) cavitites on the grain boundary (c) axis-symmetric finite element model used in this study. 102 √3σ (MPa) 12 137.3

Time(hrs)

-137.3

(2/ √3)ε12

0.004

Murakami-Ohno theory Experimental result 0.003 Strain-hardening theory

0.002

0.001

-0.001 01632486480

t (hr)

Fig. 2.2 Verification of ABAQUS UMAT code for Murakami-Ohno cyclic creep flow law. The numerical prediction (solid line) by UMAT code is compared with the experimental results (hollow circle) and numerical prediction by traditional strain hardening type constitutive law (dotted line). Murakami-Ohno law predicts experimental creep strain variation more accurately than the traditional strain hardening law.

103 Displacement(mm)

300x10-6

200x10-6

100x10-6

0 1E-3 hr. -100x10-6

-200x10-6

-300x10-6 0.00.51.01.52.0 Time(hr.)

(a)

Fig. 2.3 (a)Verification of ABAQUS UMAT code based on Murakami-Ohno constitutive equation with the experimental result (displacement controlled cyclic test for inconel 617 at 950 ºC); displacement controlled cyclic condition. (b) Verification of ABAQUS UMAT code based on Murakami-Ohno constitutive equation with the experimental result (displacement controlled cyclic test for inconel 617 at 950 ºC); stress prediction based on classical strain hardening creep law + Norton creep law. (c)Verification of ABAQUS UMAT code based on Murakami-Ohno constitutive equation with the experimental result (displacement controlled cyclic test for inconel 617 at 950 ºC); stress prediction based on Murakami-Ohno cyclic creep law + Norton creep law. The suggested cyclic creep law more accurately predicts the stress relaxation than the result in Fig. 2.3 (b). (continued)

104 (figure 2.3 continued)

StrStreessss(Mpa(Mpa)) Experiment 160160 Experiment MuMurakamrakami-Ohi-Ohnnoo + + No Nortonrton 120120 8080 4040 00 -4-400 -8-800 --120120 --160160 0.0.00 00.5.5 1.01.0 11.5.5 2.02.0 TiTimmee(h(hr.r.))

(b)

(continued)

105 (figure 2.3 continued)

StrStreessss(Mpa(Mpa))

ExperExperiimentment 160160 ClassicalClassical strainstrain hardenihardeningng ty typepe + + Norton Norton 120120 8080 4040 00 -4-400 -8-800 --120120 --160160 0.00.0 00.5.5 1.1.00 11.5.5 2.02.0 TiTimmee(h(hr.r.))

(c)

106 Stress (σ) tm tt

time tc

(σe )∞ tr (reference time) = ⎛ • cr ⎞ E⎜ε ⎟ ⎜ e ⎟ ⎝ ⎠∞

t t : tensile loading time

tc : compressive loading time (tc < 0.002tr )

tm : transition time (tm ≈ 0)

Fig. 2.4 Stress-controlled remote loading condition is shown. To simplify the problem, compressive loading time (tc) and transition time (tm) are set to be small compared to the reference time. Norton’s flow law is used to calculate the effective creep strain rate in the reference time (tr).

107 a/b

0.6

0.5

0.4

0.3 Norton, constant loading

0.2 Norton,tt/tc=100 Murakami-Ohno + Norton, constant loading

0.1 Murakami-Ohno + Norton, tt/tc=100

0.0 0123456

Normalized time(t/tr)

Fig. 2.5 Normalized cavity radius increase with time for cavity at r/d = 0 with initial conditions (a/L∞) I =0.01, a I /b I =0.1, and b I /d I =0.1. When Murakami-Ohno + Norton material property is applied, grain boundary damage accelerates under the cyclic loading condition, tt/tc=100, compared to the constant loading case. When Norton’s type material property is employed, there is not difference in the final grain boundary rupture time. The same trend is observed for the cavity at r/d = 0.5. The overall cavity growth rate is slightly faster at the center of grain boundary.

108 a/b

0.6

0.5

0.4

0.3 Norton, constant loading

0.2 Norton,tt/tc=100 Murakami-Ohno + Norton, constant loading

0.1 Murakami-Ohno + Norton, tt/tc=100

0.0 012345

Normalized time(t/tr)

Fig. 2.6 Normalized cavity radius increase with time for cavity at r/d = 0 with initial conditions (a/L∞) I =0.015, a I /b I =0.1, and b I /d I =0.1. The overall cavity growth results are similar to those in the Fig. 2.5.

109 a/b

0.6

0.5

0.4

0.3 Norton, constant loading 0.2 Norton,tt/tc=100 Murakami-Ohno + Norton, constant loading 0.1 Murakami-Ohno + Norton, tt/tc=100 0.0 0123456

Normalized time(t/tr)

Fig. 2.7 Normalized cavity radius increase with time for cavity at r/d = 0 with initial conditions (a/L∞) I =0.03, a I /b I =0.1, and b I /d I =0.1. The overall cavity growth results are similar to those in the Fig. 2.6.

110

a/b

0.6

0.5

0.4

0.3

0.2

0.1 Norton, constant loading Murakami-Ohno + Norton, constant loading 0.0 024681012

Normalized time(t/tr)

Fig. 2.8 Normalized cavity radius increase with time for the cavity at r/d = 0 with initial conditions (a/L∞) I =0.055, a I /b I =0.1, and b I /d I =0.1. Both material flow rule (“Murakami-Ohno+Norton” and “Norton”) predicts similar final grain boundary rupture time under constant loading condition. Grain material constraint is not significant when (a/L∞) I ≥0.06. Therefore, initial fast creep strain rate predicted by Murakami-Ohno law does not result in a significant change in the overall cavity growth rate.

111 1.0 σn / σ∞ Normal stress, Norton 0.9 Normal stress, Murakami-Ohno + Norton

0.8

0.2

0.1

0.0 0.0 0.5 1.04.0 4.5 5.0

t/tr

Fig. 2.9 Development of the normalized normal stress around the cavity at r/d = 0 with (a/L∞) I =0.01, a I /b I =0.1, and a I /d I =0.1 under constant loading condition. The initial normal stress (σn/σ∞=1 at t/tr=0) is decreased due to material constraint. For “MO+N’ case, this constraint process is slower than “N” case due to the fast creep strain rate by Murakami-Ohno flow rule. After this transition time, local normal stress saturates to the same value for both case.

112 σn

σ∞ Norton, constant loading 0.40 Norton, tt/tc=100 Murakami-Ohno + Norton, constant loading Murakami-Ohno + Norton, t /t =100 0.30 t c

0.20

0.10

0.00 0.07 0.09 0.11 0.13 0.15 1.54 1.56 1.58 1.60 1.62

t/tr

Fig. 2.10 Normal stress variation after stress transition from compression to tension nd st (at 2 and 21 cycles) around the cavity at r/d=0 for (a/L∞) I =0.01, a I /b I =0.1, b I /d I =0.1 and tt/tc=100. The normal stress increase after stress reversal explains the fast cavity growth rate under cyclic loading condition for “MO+N” case especially at 2nd cycle. At 21st cycle, stress increase upon stress reversal is similar for both “MO+N” and “N” case.

113 • cr εe Norton, constant loading 10-5 Norton, tt/tc=100 Murakami-Ohno + Norton, constant loading -6 10 Murakami-Ohno + Norton, tt/tc=100

10-7

10-8

10-9

10-10 0.07 0.09 0.11 0.13 0.15 1.54 1.56 1.58 1.60 1.62

t/tr

Fig. 2.11 Mises type effective creep strain rate variation after stress transition from compression to tension (at 2nd and 21st cycles) around the cavity at r/d=0 for (a/L∞) I =0.01, a I /b I =0.1, b I /d I =0.1 and tt/tc=100. The effective creep strain rate increase upon stress reversal for “MO+N” case at 2nd cycle shows the possibility that creep flow enhanced cavity growth (shorter diffusion length) can happen for constrained cavity growth under cyclic loading condition.

114 1.0 σn

σ∞

0.5 Norton Murakami-Ohno + Norton

0.0 0246810

t/tr

Fig. 2.12 Development of the normal stress state around the cavity at r/d=0 with (a/L∞)I =0.055, a I /b I =0.1, and b I /d I =0.1 under constant loading condition. Grain material constraint is not significant when (a/L∞) I ≥0.06. Therefore, initial normalized normal stress value decrease gradually compared to the result ((a/L∞)I =0.01) in Fig. 2.9.

115 tf/tf,constant loading

1.0

0.9

0.8 Murakami-Ohno + Norton, a/L=0.01 Murakami-Ohno + Norton, a/L=0.015 Murakami-Ohno + Norton, a/L=0.03 0.7

0.6 0 100 200 300 400 500 600 Number of cycles

(a)

Fig. 2.13 The effect of the number of cycles on the grain boundary rupture time for cavity (a) at r/d =0 and (b) at r/d=0.5. The compressive loading time(tc) and the transition time(tm) are set fixed and the loading time(tt) is changed to give different number of cycle. The grain boundary rupture time under cyclic loading condition decreased up to 12% compared to the constant loading case. The grain boundary rupture time slightly increases when the number of cycles exceeds 300. Increase of total compressive loading time is relevant with this behavior but the reason is not clear at this time. (continued)

116 (figure 2.13 continued)

tf/tf,constant loading

1.0

0.9

0.8 Murakami-Ohno + Norton, a/L=0.01 Murakami-Ohno + Norton, a/L=0.015 0.7 Murakami-Ohno + Norton, a/L=0.03

0.6 0 100 200 300 400 500 600 Number of cycles

(b)

117 APPENDIX A

Derivation of the functional, F gb , in Eq. (1.5) (Needleman and Rice, 1980)

Needleman and Rice [1980] obtained the functional, F gb , by considering inelastic material deformation and atomic diffusion along grain boundary. They (Needleman and

Rice [1980]) assumed that grain material is incompressible and non-linear viscous, specifically of the power law form

• 1/ n σ = Λ ε (A.1)

in uniaxial tension where ‘ Λ ’ and ‘n’ are material constants. In order to include the grain boundary diffusion phenomena in the functional, they (Needleman and Rice [1980]) used matter conservation law. According to the matter conservation law, atomic flow rate, ‘j’, is related to the normal direction (denoted as ‘z’ since axis-symmetric coordinate is used) velocity of grain boundary, vz, as shown in the following relation.

∂jα + ∆v z = 0 on A gb (A.2) ∂x α

118 where ‘x’ represents the Cartesian coordinate, ‘α’ have the range 1,2 (repeated indices represent the summation) and refer to a local set of Cartesian coordinates in the grain

+ - + boundary, ‘∆vz’ represent the addition, ‘∆vz = (vz )+(vz )’, of the upper movement, ‘vz ’,

- and downward movement, ‘vz ’, of the grain boundary, and ‘Agb’ represent the grain boundary surface.

Needleman and Rice [1980] did not consider surface diffusion along the cavity surface in their numerical method. In order to simulate grain boundary diffusion without surface diffusion considered, proper boundary condition was assumed at the cavity tip, where cavity surface meets grain boundary. As the proper boundary condition at the cavity tip, they (Needleman and Rice [1980]) used the continuity condition of the chemical potential at the cavity tip according to the Eq. (1.1) as follows.

σo = γ s (κ1 + κ 2 ) (A.3)

The continuity condition of the chemical potential along the collection of arcs ‘Γ’ (see

Fig. (1.2)), where grain boundaries meet free surfaces, was employed in the functional by giving the prescribed values σn

σ n = σo along Γ . (A.4)

In order to establish a variational principle satisfied by the solution to the coupled equations of creep flow (in volume ‘V’) and diffusion (on grain boundary area, ‘Agb’),

119 they (Needleman and Rice [1980]) begin by observing that the true stress field satisfies the principle of virtual work (or of virtual velocities) in the form

∗ * ∗ σ kl ε kl dV = Ti v i dS − σ n ∆v z dA (A.5) ∫V & ∫S ∫A T gb

* ∗ where ‘ v i ’ and ‘ε& kl ’ are arbitrary velocity and arbitrary strain fields, respectively. In

* Eq. (A.5), ‘ vi ’ vanishes on any portion of S where ‘vi’ is prescribed.

The normal velocity, ‘vz’, in the last term in Eq. (A.5), is related to the flow rate according to Eq. (A.2), and expressed as follows.

∗ ∗ ∂jα σ n ∆v z dA = − σ n dA ∫A ∫A gb gb ∂x α

∗ ∂σ n ∗ ∂(σ n jα ) = jα dA − dA (A.6) ∫A ∫A gb ∂x α gb ∂x α

Using the divergence theorem, Eq. (A.6) becomes

∗ ∂σ n ∗ ∂(σ n jα ) ∂σ n ∗ ∗ jα dA − dA = jα dA + σn mα jα dΓ (A.7) ∫A ∫A ∫A ∫Γ gb ∂x α gb ∂x α gb ∂x α

Using Eq. (1.1), (1.2), and (1.3), Eq. (A.7) is expressed in terms of atomic diffusion along grain boundary as follows.

120 ∂σ n ∗ ∗ 1 ∗ ∗ jα dA + σ n m α jα dΓ = jα jα dA + σ n m α jα dΓ (A.8) ∫A ∫Γ ∫A ∫Γ gb gb ∂x α M gbΩ

Substituting Eq. (A.8) into the principle of virtual work (last term in Eq. (A.5)), and making the identifications

* ∗ ∗ v i = δv i , ε& ij = δε& ij , jα = δjα , (A.9)

they (Needleman and Rice [1980]) found that the solution to the coupled creep-diffusion problem satisfies the following variational principle,

δF gb = 0 (A.10)

where F gb is the functional defined by

jα jα F gb = ω(ε kl )dV − Ti v i dS + dA + σ o m α jα dΓ (A.11) ∫V & ∫S ∫A ∫Γ T gb 2M gb Ω

and

ε & kl n (1+ n ) / n ω(ε& kl ) = σ ij dε& ij = Λε& . (A.12) ∫0 1 + n

121 for all kinematically associated fields ‘ v i ’, ‘ ε& kl ’, and ‘ jα ’.

122 APPENDIX B

Derivation of Eq. (1.9)

Axially symmetric version of Eq. (A.2) (in Appendix A) is given as

∂ (2πrj )+ 2πr∆v = 0 (B.1) ∂r gb z

where ‘ ∆v z ’ represents ‘ 2v z ’ as shown in Appendix A. Deviding Eq. (B.1) by 2π and

using the relation of ∆v z = 2v z , Eq. (B.1) becomes

1 ∂ (rj )+ 2v = 0. (B.2) r ∂r gb z

Integration of the above equation from r=r to r=b with the condition of j=0 at r=b leads to

b bjgb |r=b −rjgb |r=r = −2 r'v z ()r',0 dr' ∫r (B.3) 2 b jgb |r=r = jr = r'v z ()r',0 dr' r ∫r

123 and the incremental form the above equation leads to the following relation (Eq. (1.9)).

2 b ∆jgb = ∆jr = r'∆v z (r',z = 0)dr' (B.4) r ∫r

where ‘∆jgb’, ‘∆jz’ , and ‘∆vz’ represent the incremental quantity of ‘jgb’, ‘jz’ and ‘vz’, respectively.

124 APPENDIX C

Derivation of Eq. (1.10)

The third term of Eq. (1.8) is derived as follows.

1 ∆j⋅ ∆jdA ∫A gb 2M gb Ω 2 (using the relation A gb = πr → dA gb = 2πrdr ) b 1 ⎛ 2 b ⎞⎛ 2 b ⎞ (C.1) = ⎜ r'∆v z (r',0)dr '⎟⎜ r'∆v z (r',0)dr '⎟2πrdr ∫∫a ∫r r 2M gb Ω ⎝ r ⎠⎝ r ⎠ 2 4π b1 ⎡ b ⎤ = r'∆v z (r',0)dr ' dr ∫∫a ⎢ r ⎥ M gb Ω r ⎣ ⎦

The fourth term of Eq. (1.8) is derived as follows.

σ m ⋅ ∆jdΓ ∫ o Γ

(using the relation Γ = 2 πa and m ⋅ ∆j = ∆jr=a )

= 2πaσ o ∆jr=a (using the relation shown in Appendix B) (C.2) 2 b = 2πaσ o r'∆v z (r',0)dr ' a ∫a b = 4πσ o r'∆v z (r',0)dr ' ∫a

125 APPENDIX D

Derivation of Eq. (1.19)

From the matter conservation law, the cavity surface velocity ‘vN’ is related with surface flux ‘js’ as follows.

2πrjs − 2πrv N ∆S = 0 1 d()j r (D.1) v = s N r dS

126 APPENDIX E

Derivation of Eq. (1.22)

The differentiation of Eq. (1.19) gives the following equation.

1 d()j r v = s N r dS

1⎛ djs dr ⎞ = ⎜ r + js ⎟ r ⎝ dS dS ⎠ (E.1) ⎛ dr ⎞ ⎜ = sin β from the definition of β in Fig.10⎟ ⎝ dS ⎠ dj j = s + s sin β dS r

127 APPENDIX F

Derivation of Eq. (1.25)

From matter conservation law, surface velocity at cavity top is related with the atomic flux as follows.

v N dS = 2js (F.1)

where atomic flux from both sides contributes to the velocity (multiplication of 2 in the right hand side of the above equation) due to the axis-symmetric condition.

128 APPENDIX G

Rice [1979] first introduced the effective diffusion length ‘L’. Needleman and Rice

[1980] showed that when ‘L’ is large compared to the cavity radius, ‘a’, or the cavity half spacing, ‘b’, the effect of creep flow on the diffusion based cavity growth is negligible.

On the other hand, when ‘L’ is small compared to ‘a’ or ‘b’, the coupling of the creep and the diffusion is significant and creep flow accelerates cavity growth.

As pointed out by Needleman and Rice [1980] and several other authors (Beere and

Speight [1978], Edward and Ashby [1979]), creep flow of the surrounding grain material shortens matter diffusion distance along the grain boundary. Chen and Argon [1981] found that the cavity growth rate equation matches well with the numerical result by

Needleman and Rice [1980] if the cavity half spacing, ‘b’, is replaced with the length scale parameter ‘a+L’ in the cavity growth rate equation by Hull-Rimmer [1959]. The replacement is only valid when a+L is less than b. The equation proposed by Chen and

Argon [1981] is

−1 ⎛ ⎛ 2 2 ⎞ ⎜ 3 ⎛ b ⎞ ⎛ a ⎞ ⎛ 1 ⎛ a ⎞ ⎞ 3 2π()L / a ⎜ln⎜ ⎟ + ⎜ ⎟ × ⎜1− ⎜ ⎟ ⎟ − ⎟ if a + L ≥ b • ⎜ ⎜ ⎜ ⎟ ⎟ V ⎝ ⎝ a ⎠ ⎝ b ⎠ ⎝ 4 ⎝ b ⎠ ⎠ 4 ⎠ = ⎜ . (G1) • ⎜ −1 3 ⎛ 2 2 ⎞ a ε cr ⎜ 3 ⎛ a + L ⎞ ⎛ a ⎞ ⎛ 1 ⎛ a ⎞ ⎞ 3 2π()L / a ⎜ln⎜ ⎟ + ⎜ ⎟ × ⎜1− ⎜ ⎟ ⎟ − ⎟ if a + L < b ⎜ ⎜ a a + L ⎜ 4 a + L ⎟ 4 ⎟ ⎝ ⎝ ⎝ ⎠ ⎝ ⎠ ⎝ ⎝ ⎠ ⎠ ⎠

129 Sham and Needleman [1982] pointed out that the cavity growth rate calculated from the above equation matches well with the Needleman and Rice [1980] numerical result within an error of about 30 % for all values of ‘a/L’ less than 10 and ‘a/b’ less than 0.67.

130 APPENDIX H

Chen and Argon [1981] calculated critical cavity size at the time of transition from quasi-equilibrium mode to crack–like mode as a function of material properties and boundary conditions. The following equation, Eq. (H1), shows the relation between cavity volume growth rate and matter diffusive flux at the cavity tip and major radius of the cavity, ‘a’ which is driven from the mass conservation law.

dV = 4πaj . (H1) dt s(tip)

The left-hand side of the above equation can be related to Eq. (G1), which describes cavity volume growth rates resulting from coupled grain boundary diffusion and grain material creep flow. From Eq. (G1) and Eq. (H1), Chen and Argon [1981] obtained the following equation.

−1 3 2 2 • ⎛ ⎛ ⎞ ⎞ 3 ⎛ L ⎞ ⎜ ⎛ a + L ⎞ ⎛ a ⎞ 1 ⎛ a ⎞ 3 ⎟ 4πaj = ε∞ a ⎜ ⎟ 2π ln⎜ ⎟ + ⎜ ⎟ × ⎜1− ⎜ ⎟ ⎟ − . (H2) s(tip) ⎜ ⎜ ⎟ ⎟ ⎝ a ⎠ ⎝ ⎝ a ⎠ ⎝ a + L ⎠ ⎝ 4 ⎝ a + L ⎠ ⎠ 4 ⎠

131 Chuang and Rice [1973] and Chuang, Kagawa, and Rice [1979] showed that the surface flux is only relevant with the cavity radius, ‘a’, and cavity radius growth rate,

‘da/dt’, provided the cavity geometry is given and the cavity growth is at a ‘quasi-steady’ state. Chuang, Kagawa, and Rice [1979] related surface flux and ‘da/dt’ based on the above condition for spherical and crack-like modes respectively. The suggested equations are:

da j = h(ψ)a for spherical mode s(tip) dt where

V ⎛ 1 cosψ ⎞ h(ψ) = = ⎜ − ⎟ / sin ψ 4 3 (1+ cosψ) 2 πa ⎝ ⎠ 3 and

2 / 3 ψ ⎛ D γ ⎞ ⎛ kTΩ1/ 2 (da / dt) ⎞ ⎛ ⎞ s s ⎜ ⎟ js(tip) = 2sin⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟ for crack-like mode. (H3) ⎝ 2 ⎠ ⎝ kT ⎠ ⎝ Ds γ s ⎠

Substituting ‘js(tip)’ from Eq. (H3) into Eq. (H2), the growth rate of a quasi- equilibrium cavity and crack-like cavity are, respectively, obtained as follows:

−1 3 2 2 ⎛ 4πh(ψ) da ⎞ ⎛ L ⎞ ⎛ ⎛ a + L ⎞ ⎛ a ⎞ ⎛ 1 ⎛ a ⎞ ⎞ 3 ⎞ ⎜ ⎟ = 2π⎜ ⎟ ⎜ln⎜ ⎟ + ⎜ ⎟ × ⎜1− ⎜ ⎟ ⎟ − ⎟ ⎜ • dt ⎟ ⎝ a ⎠ ⎜ ⎝ a ⎠ ⎝ a + L ⎠ ⎜ 4 ⎝ a + L ⎠ ⎟ 4 ⎟ ⎝ ε∞ a ⎠quasi−eqilibrium ⎝ ⎝ ⎠ ⎠ and

132 3 5 − 2 2 2 ⎛ 4πh(ψ) da ⎞ ⎛ L ⎞ 2 ⎛ ⎛ a + L ⎞ ⎛ a ⎞ ⎛ 1 ⎛ a ⎞ ⎞ 3 ⎞ ⎜ ⎟ = α⎜ ⎟ ⎜ln⎜ ⎟ + ⎜ ⎟ × ⎜1− ⎜ ⎟ ⎟ − ⎟ , ⎜ • dt ⎟ ⎝ a ⎠ ⎜ ⎝ a ⎠ ⎝ a + L ⎠ ⎜ 4 ⎝ a + L ⎠ ⎟ 4 ⎟ ⎝ ε∞ a ⎠crack−like ⎝ ⎝ ⎠ ⎠

where ‘α’ represents a coupling parameter (the ratio of effective grain boundary diffusional conductance to surface diffusional conductance) as follows,

⎛ ⎞ ⎜ ⎟ 1 ⎜ ⎟ 4πh(ψ) ⎛⎛ D ⎞⎛ σ L ⎞⎞ 2 α = ⎜ ⎟ × ⎜⎜ gb ⎟⎜ ∞ ⎟⎟ . (H4) ⎜ 3 ⎟ ⎜⎜ D ⎟⎜ γ ⎟⎟ 2 ⎝⎝ s ⎠⎝ s ⎠⎠ ⎜ ⎛ ⎛ ψ ⎞⎞ ⎟ ⎜ ⎜4sin⎜ ⎟⎟ ⎟ ⎝ ⎝ ⎝ 2 ⎠⎠ ⎠

From Eq. (H4), Chen and Argon [1981] calculated cavity growth rates in the transition range and critical cavity size at the transition time.

133 APPENDIX I

The third order Runge-Kutta method was used to calculate the creep strain rate

• cr cr predicted by Murakami-Ohno law. First, vector and state variables, such as ‘ εij ’, ‘ εij ’,

n ‘ α ij ’, ‘ρ ’, and ‘ g max ’ are given from the previous time increment, ‘ t ’. At the start of time increment, ‘ t n+1 ’, creep hardening bound and normal vector are calculated as follows.

cr cr g max,1 = (εij − α ij )(εij − αij )

cr εij − α ij

ηij = 1 . (I.1) cr cr 2 {}()ε kl − α kl ()ε kl − α kl

User subroutine calculates first trial values such as,

⎛ (εcr − α )⎞ q = ρ + ⎜ ij ij ⎟s 1 1 ⎜ ⎟ ij ⎝ σe ⎠

• cr 1 m−1 n−m 3 m m ε ij,1 = mA q σ m s 2 1 e ij

• 1 • cr • 1 • cr α ij,1 = (ε ij,1 ηkl )ηkl ; ρ1 = ε ij,1 ηij if g max,1 ≥ g max 2 6 134 • • αij,1 = ρ1 = 0 if gmax,1 < gmax

• cr cr ∆ε ij,1 = ε ij,1⋅ ∆t

• ∆αij,1 = α ij,1⋅ ∆t

• ∆ρ1 = ρ1⋅ ∆t (I.2)

at trial step 1. At trial step 2 and 3, the corresponding quantities are calculated.

At step 2:

⎛ ⎛ 1 1 ⎞ ⎞ ⎜ ⎛ cr cr ⎞ ⎛ ⎞ ⎟ ⎜⎜εij + ∆εij,1 ⎟ − ⎜αij + ∆αij,1 ⎟⎟ ⎛ 1 ⎞ ⎜ ⎝⎝ 2 ⎠ ⎝ 2 ⎠⎠ ⎟ q = ⎜ρ + ∆ρ ⎟ + s 2 1 2 1 ⎜ σ ⎟ ij ⎝ ⎠ ⎜ e ⎟ ⎜ ⎟ ⎝ ⎠

• cr 1 m−1 n−m 3 m m ε ij,2 = mA q σ m s 2 2 e ij

• 1 • cr • 1 • cr α ij,2 = (ε ij,2 ηkl )ηkl ; ρ = ε ij,2 ηij if g max,1 ≥ g max 2 2 6

• • αij,2 = ρ2 = 0 if gmax,1 < gmax

• cr cr ∆ε ij,2 = ε ij,2 ⋅ ∆t

• ∆αij,2 = α ij,2 ⋅ ∆t

• ∆ρ 2 = ρ 2 ⋅ ∆t (I.3)

135 At step 3:

⎛ ((εcr − ∆εcr + 2∆εcr )− (α − ∆α + 2∆α ))⎞ q = ρ − ∆ρ + 2∆ρ + ⎜ ij ij,1 ij,2 ij ij,1 ij,2 ⎟s 3 ()1 1 2 ⎜ ⎟ ij ⎝ σe ⎠

• cr 1 m−1 n−m 3 m m ε ij,3 = mA q σ m s 2 3 e ij

• 1 • cr • 1 • cr α ij,3 = (ε ij,3 ηkl )ηkl ; ρ3 = ε ij,3 ηij if g max,1 ≥ g max 2 6

• • αij,3 = ρ3 = 0 if gmax,1 < gmax

• cr cr ∆ε ij,3 = ε ij,3 ⋅ ∆t

• ∆αij,3 = αij,3 ⋅ ∆t

• ∆ρ3 = ρ3 ⋅ ∆t (I.4)

From the above trial values, the real step values are obtained as follows:

• cr 1 ⎛ • cr • cr • cr ⎞ ⎜ ⎟ εij = ⎜εij,1 + 4εij,2 + εij,3 ⎟ 6 ⎝ ⎠

• 1 ⎛ • • • ⎞ αij = ⎜αij,1 + 4αij,2 + αij,3 ⎟ 6 ⎝ ⎠

• 1 ⎛ • • • ⎞ ρ = ⎜ρ + 4ρ + ρ ⎟ 6 ⎝ 1 2 3 ⎠

• cr cr ∆ε ij = ε ij ⋅ ∆t 136 • ∆αij = αij ⋅ ∆t

• ∆ρ = ρ⋅ ∆t. (I.5)

At the end of time step ‘ t n+1 ’, stress increment is

∆σ = D(∆ε − ∆ε cr ) (I.6)

where ‘D’ and ‘ ∆ε ’ are elastic stiffness matrix and total strain increment for time increment ‘∆t ’.

137 APPENDIX J

In this user element subroutine, constraint equations are implemented so that all nodes on the face z = H (Fig. (2.1) (C)) have the same displacement in z-direction as the first node of the element, N1 and all nodes on the face r=W have the same displacement in r-direction as the first node of the element, N1. Since the constraints are imposed on the face and on the node, the total force on the face, z=H, in the direction in z-direction have the same force in that direction on node N1. Similar condition is applied on the force in r-direction.

Stiff linear spring connect nodes N1 and N2 of the user element. The displacement is applied to the node N2 and the forces on N2 (force in r-direction, Fr, and force in z- direction, Fz) are related to the displacement by

F = K(u − u ) r r,2 r,1 (J.1) Fz = K(u z,2 − u z,1 )

where the ur,1 and uz,1 refer r-direction and z-direction D.O.F. on node N1 and the ur,2 and uz,2 refer r-direction and z-direction D.O.F. on node N2, respectively.

The relations between the force and stress for these unit cell geometries are:

138 F = 2πHWλσ r (J.2) 2 Fz = πW σ

for axisymmetry, where H and W are the current unit cell dimensions. Here, σ is the normal stress acting on the face z=H and λ is the ratio between σ and the average stress acting on the face r=W.

Equating the force equations from (J.1) and (J.2) gives

f λσ + Ku − Ku = 0 r r,1 r,2 (J.3) f z σ + Ku z,1 − Ku z,2 = 0

where fr and fz are

f = 2πHW r (J.4) 2 f z = πW

for axisymmetry. Taking the variation of (J.3) and using the relation of H=Ho+uz,1 and

W=Wo+ur,1 gives

0 = (a + K)δu + bδu − Kδu + cδσ r,1 z,1 r,2 , (J.5) 0 = dδu r,1 + Kδu z,1 − Kδu z,2 + eδσ

where the coefficients a to e are

139 ⎛ ⎞ ⎜ ∂λ ⎟ a = 2πH⎜λ + W ⎟σ ⎝ ∂u r,1 ⎠ ⎛ ⎞ ⎜ ∂λ ⎟ b = 2πW⎜λ + H ⎟σ ⎝ ∂u z,1 ⎠ c = 2πWHλ (J.6) d = 2πWσ e = πW 2

Here, stress σ is arbitrarily introduced as a ‘displacement type’ second D.O.F. on node

N3. The first D.O.F. of node N3 is not needed and is constrained. Therefore, the D.O.F. of the user element are ur,1, uz,1, ur,2, uz,2, ur,3, σ .

In the construction of element stiffness matrix, the negative of (J.5a) is used as the third equation in the stiffness matrix. The negative of the equation is used to avoid negative eigenvalues. For the displacement driven problem, (J.5b) is the 6th equation and boundary conditions are applied to uz,2. The resulting stiffness matrix and residuals are

⎡ K 0 − K 0 0 0 ⎤ ⎢ ⎥ ⎢ 0 K 0 − K 0 0 ⎥ ⎢− K − a − b K 0 0 − c⎥ ⎢ ⎥ ⎢ 0 − K 0 K 0 0 ⎥ ⎢ 0 0 0 0 1 0 ⎥ ⎢ ⎥ ⎣⎢ d K 0 − K 0 e ⎦⎥

R1 = −K(u r,1 − u r,2 )

R 2 = −K(u z,1 − u z,2 )

R 3 = K(u r,1 − u r,2 ) + f r λσ (J.7)

R 4 = K(u z,1 − u z,2 )(specified DOF)

R 5 = −u r,3 (constrained DOF)

R 6 = −K(u z,1 − u z,2 ) − f z σ

140

Although displacement is applied to uz,2, uz,1 is almost same as uz,2 since very large K value is assigned. The equations for the stress driven problem are obtained by using the negative of (J.5b) as the 4th equation of stiffness matrix, zeroing the nondiagonal terms in the 6th equation and applying the stress as a ‘displacement’ boundary condition in the

ABAQUS input. User can give desired ratio of stress by specifying λ value.

141

Wo • N3 Uz = constant σ N2 K N1

• Ur = constant

Ho λσ

Fig. J.1 user element for a micromechanical model

142 REFERENCES

ABAQUS, Hibbit, Karlsson and Sorensen, Inc. (2001)

Arai, M., Ogata, T. and Nitta, JSME Int. Journal A, 39, 3, 382 (1996)

Ashby, M. F., Acta Metall., 20, 887 (1972)

Beere, W., and Speight, M.V., Metal Sci., 4, 172 (1978)

Bower, A. F., and Chuang, T.-J., Proceedings of CICC-1, Beijing, China (1998)

Brust, F.W., Int. J. Solids Structures, 15, 2191 (1995)

Budiansky, B., Hutchinson, J.W., and Slutsky, S., Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, 13, Pergamon Press, London, (1982)

Budiansky, B., and O’connell, R. J., Int. J. Solids Struct., 12, 81 (1976)

Chaboche J.L., Bull. Acad. Polon. Ser. Sci. Tech., 25, 33, (1977)

Chen, I.-W., and Argon, A.S., Acta Metallurgica, 29, 1759, (1981)

Chuang, T.-J., and Rice, T.R., Acta Metallurgica, 21, 1625, December, (1973)

Chuang, T.-J., Kagawa, K. I., Rice, J. R., and Sills, L. B., Acta Metall. Mater., 27, 265, (1979)

Cocks, A.C.F., and Pan, J, Int. J. of Fracture, 60,249 (1993)

Cormeu, I., Int. J. for Num. Meth. In Eng. 9, 109, (1975)

Cyr, N.A., and Teter, R.D., Computers and Structures, 3, 849, (1973)Dobes, F., Scripta Met., 7, 1231, (1973)

Dyson, B.F., Metal Science, 10, 349, (1976)

143 Dyson, B.F., and Rodgers, M.J., In Fracture 1977, Proceedings of ICF4, 2, 621, University of Waterloo Press, Waterloo, Canada (1977)

Edward, G.H., and Ashby, M.F., Acta Metall., 27, 1505 (1979)

Frost, H.J., and Ashby, M.F., Deformation Mechanism Maps: The Plasticity and Creep of Metals and Ceramics, Pergamon Press, Oxford, (1982)

Gibbons, T.B., “Creep Crack Growth, A State of the Art Report”, VAMAS(Versatile Project on Advanced Materials and Standards) task group report, (1989)

Gittis J., Creep, Viscoelasticity and Creep Fracture in Solids, Appl. Sci. Publishers, London, (1975)

Goods, S.H., and Nix, W.D., Acta Metallurgica, 26, 739, (1978)

Hancock, J.W., Metal Sci., 10, 319 (1976)

Herring, C., Structure and Properties of Solid Surfaces (edited by Gomer, R. and Smith, C. S.), Univ. of Chicago Press (1952)

Herring, C., The Physics of Power Metallurgy, pp. 143 (edited by Kingston, W. E.), McGrow-Hill (1951)

Hughes, T. J. R., and Taylor, R. L., Computers and Structures, 8, 169, (1978)

Hull, D., and Rimmer, D.E., Phil. Mag., 4, 673 (1959)

Inoue T., Nucl. Eng. Design, 90, pp.287-297, (1985)

Inoue et al., Nucl. Eng. Design, 114, pp.295-309, (1989)

Kanchi, M.B., Zienkiewicz, O.C., and Owen, D.R. J., Int. J. Num. Meth. Engng., 12, 169,(1978)

Martinez, L., and Nix, W.D., Metall. Trans. A,12A, 23 (1981)

Miller, D.A., Scripta Metall., 13, 595 (1979)

Mohan, R., Zhang, J., and Brust, F.W., J. of Eng. Mat. Tech., 122, 294, Jul (2000)

Monkman, F. C., and Grant, N. J., Proc. ASTM 56,593 (1956)

Murakami, S., and Ohno, N., Int. J. Solids Structures, 18, No.7, 597 (1982)

144 Nayak, G.C., and Zienkiewicz, O.C., Int. Journ. For Numerical Meth. In Eng., 5, 13 (1972)

Needleman, A., and Rice, J.R., Acta Metallurgica 28,1315 (1980)

Needleman, A., and Shih, C.F., Comp. Meth. Appl. Mech. Engng., 15, 223 (1978)

Nishioka, T., and Atluri, S.N., Engineering Fracture Mechanics, 13, #3, 303, (1983)

Ohashi Y., Ohno N., and Kawai M., ASME J. Engng Mat. Tech., 104 (3), 159, (1982)

Owen, D.R.J., and Hinton, E., Finite Elements in Plasticity : Theory and Practice, Pineridge Press, (1980)

Oh J., Brust F. W., and Katsube N., submitted in J. of Eng. Mat. Tech.

Oh J., Katsube N., and Brust F. W., submitted in J. Mech. Phys. Solids.

Pan, J., and Cocks, A.C.F., Int. J. of Fracture 60,121 (1993a)

Pan, J., and Cocks, A.C.F., Computational Materials Science 1, 95 (1993b)

Pan, J., and Cocks, A.C.F., Acta Metall. Mater. 43, No. 4, 1395 (1995)

Pan, J., Cocks, A.C.F., and Kucherenko, S., Proc. R. Soc. Lond. A, 453, 2161 (1997)

Pandey, M.C., Mukherjee, A.K., and Taplin, D.M.R., Metallurgical Transactions, 15A, 1437 (1984)

Perzyna, P., Advances in Appl. Mech., 9, 243, (1966)

Pharr, G.M., and Nix, W.D., Acta Metallurgica, 27, 1615, (1979)

Raj, R., Metallurgical Transactions A, 6A, 1499, (1975)

Raj, R., Acta Metallurgica, 26, 341, (1978)

Raj, R., and Ashby, M.F., Acta Metall. 23, 653 (1975)

Raj, R., and Ghosh, A.K., Metall. Trans. A, 12A, 1291 (1981)

Raj, R., Shih, H.M., and Johnson, H.H., Scripta Metall., 11, 839 (1977)

145 Rice, J.R., Time Dependent Fracture of Materials at Elevated Temperature (edited by S. Wolf), p. 130, U.S. Department of Energy Report CONF 790236 UC-25, Germantown, Md. (June 1979)

Rice, J.R., Acta Metallurgica, 29, 675 (1981a)

Rice, J.R., and Chuang,T-J, Journal of American ceramic society, 64, 1, 46, (1981b)

Rice, J.R., and Drucker, D.C., Int. J. rac. Mech., 3, [1], 19, (1967)

Riedel, H., In “Mechanical Behavior of Materials-IV”, Vol.1, J. Carlsson and N.G. Ohlson, Eds., Pergamon Press, Oxford, 67 (1983)

Riedel, H., Fracture at High Temperature, (Materials Research and Engineering) Springer-Verlag (1987)

Sham, T.-L., and Needleman, A., Effects of Triaxial Stressing on Creep Cavitation of Grain Boundaries. Division of Engineering, Brown University (1982)

Sham, T.-L., and Needleman, A., Acta Metall. 31, No.6, 919 (1983)

Shewmon, P., and Anderson, P., Acta Metall. Mater., 46, 1861 (1998)

Smelser, R.E., and Becker, R., ABAQUS user’s conference proceedings, 207 May (1989)

Snyder, M.D., and Bathe, K.J., Massachusetts Inst. Of Tech. Report No. 82448-3, June (1977)

Speight, M.V., and Harris, J.E., Metal. Sci. J. 1, 83 (1967)

Stonesifer, R.B., Ph. D. Dissertation presented to the Georgia Institute of Technology, Dec. (1981)

Sun, B., Suo. Z., and Cocks, A.C.F., J. Mech. Phys. 44, No. 4, 559 (1996)

Suo, Z., and Wang, W., J. Appl. Phys. 76(6), 15, Sep., (1994)

Svensson, L.E., and Dunlop, G.L., International Metals Reviews, 26, No.2, 109 (1981)

Svensson, L.E., and Dunlop, G.L., Metal Science, 16, 57 (1982)

Takasugi, T., and Vitek, V., Metall. Trans. A, 12A, 659 (1981)

Trinkaus, H., Phys. Status Solidi., 93, 293 (1979)

146

Tvergaard, V., J. Mech. Phys. Solids, 32, No.5, 373 (1984)

Van Der Giessen, E., and Tvergaard, V., Acta Mater., 7, 2697 (1995)

Vitek, V., Acta Metall. 26, (1978)

Weertman, J., Scripta Metall. 7, 1129 (1973)

Weertman, J., Metall. Trans. 5, 1743 (1974)

Weertman, J.R., Can. Metall. Quart., 18, 73 (1979)

Yamasaki, M., J. Jap. Inst. Met., 32, 403 (1968)

Zienkiewicz, O.C., and Cormeu, I.C., Int. Journ. for Numerical Meth. In Eng., 8, 821 (1974)

147