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The Combining Effect of Creep Flow and Surface and Grain Boundary

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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 BY 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
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