Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2015 Coupled Gradient Enhanced Damage - Viscoplasticity Model Using Phase Field Method Navid Mozaffari Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Civil and Environmental Engineering Commons Recommended Citation Mozaffari, Navid, "Coupled Gradient Enhanced Damage - Viscoplasticity Model Using Phase Field Method" (2015). LSU Doctoral Dissertations. 1502. https://digitalcommons.lsu.edu/gradschool_dissertations/1502 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. COUPLED GRADIENT ENHANCED DAMAGE – VISCOPLASTICITY MODEL USING PHASE FIELD METHOD A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Civil and Environmental Engineering by Navid Mozaffari B.S., Isfahan University of Technology, 2005 M.S., University of Tehran, 2008 M.S., Louisiana State University, 2014 August 2015 It is my genuine gratefulness to dedicate this dissertation to my parents, who have always supported me unconditionally and have taught me to work hard for things that I aspire to achieve. Without their patience, help and support, the completion of this work would not have been possible. I would also like to dedicate this work to my beloved sisters, Noushin and Negin, for their emotional support in every single step of my research journey. ii ACKNOWLEDGMENTS The work presented in this dissertation was carried out in the Department of Civil and Environmental Engineering at Louisiana State University during the period of Fall 2009 to Spring 2015. I would like to express my special thanks to Prof. George Voyiadjis, the major professor and committee chairman, for his help, encouragement and support during the course of this research. I would also like to acknowledge the invaluable assistance and support from my minor advisor, Prof. Dorel Moldovan. I would like to thank Prof. Yitshak Ram and Dr. Suresh Moorthy, members of my committee, for their time, help and invaluable comments on this dissertation. I would also like to express my appreciation to Prof. Mehdy Sabbaghian, emeritus professor of Mechanical Engineering at LSU and Dr. Mahmood Sabahi, adjunct faculty of Chemical Engineering for their attendance in my final examination. I would like to express my sincere gratitude to Dr. Blaise Bourdin, Professor of Mathematics at Louisiana State University and the Center of Computation and Technology for his invaluable and constructive advice, patient guidance and enthusiastic encouragement during the course of this research work. I greatly benefitted from discussions with him owing to his insightful remarks on the mathematical analysis of the gradient damage models and phase field method. I also wish to offer my special thanks to my friend, Dr. Ata Mesgarnejad, postdoc fellow at the Center of Computation and Technology of Louisiana State University who helped me a great deal during the time I was doing this dissertation. The accomplishment of this work, specifically computational part would not have been possible without his support and help. I would like to extend my thanks to all my dear friends, Kasra Fattah, Dr. Hessam Babaee, Arash Mirahmadi, Dr. Mohammadreza Ebrahimi, Alireza Eslami, Amir Shoaei, Saman Azadi, Erfan Danaei and Dr. Siavash Mirahmadi for the emotional support they have given to me during my research journey. iii TABLE OF CONTENTS ACKNOWLEDGMENTS ..........................................................................................................iii ABSTRACT ............................................................................................................................. vii 1. ISOTROPIC NONLOCAL DAMAGE MODEL USING PHASE FIELD METHOD ............ 1 1.1. Introduction ..................................................................................................................... 1 1.2. General framework of phase field models ........................................................................ 3 1.2.1. Order parameter .........................................................................................................3 1.2.2. Framework of phase field method ..............................................................................4 1.3. Phase field theory for isotropic continuum damage mechanics theory .............................. 8 1.3.1. Order parameter .........................................................................................................8 1.4. Nonlocal thermodynamic formulation of isotropic damage mechanics ............................10 1.5. Comparing with the variational formulation....................................................................12 1.5.1. Positive Elasticity .................................................................................................... 12 1.5.2. Decreasing Stiffness................................................................................................. 13 1.5.3. Dissipation ............................................................................................................... 13 1.5.4. Irreversibility ........................................................................................................... 14 1.6. New implicit damage variable ........................................................................................14 1.6.1. Strain Energy Equivalence ....................................................................................... 15 1.6.2. Strain relations in the proposed new model .............................................................. 18 1.6.3. Thermodynamic conjugate force due to damage ....................................................... 21 1.6.4. Damage Criterion..................................................................................................... 22 1.6.5. Boundary conditions ................................................................................................ 22 1.7. Numerical Aspects, Algorithm and 1D implementation ..................................................23 1.7.1. Numerical Aspects ................................................................................................... 23 1.7.2. Explicit in Space, Explicit in time ............................................................................ 23 1.7.3. Implicit in Space, Explicit in time ............................................................................ 25 1.7.4. Implicit in Space, Explicit in time (with Crank Nicolson scheme in space)............... 26 1.7.5. Rate independent material ........................................................................................ 28 1.7.6. Rate dependent material ........................................................................................... 28 1.8. Numerical Algorithm ......................................................................................................29 1.9. Numerical Examples ......................................................................................................29 1.10. Conclusions ..................................................................................................................36 1.11. References ....................................................................................................................37 2. ANISOTROPIC NONLOCAL DAMAGE MODEL USING PHASE FIELD METHOD ......44 2.1. Introduction ....................................................................................................................44 2.2. Generalized Phase field method framework ....................................................................46 2.3. Anisotropic damage description ......................................................................................49 2.4. Phase field theory for anisotropic continuum damage mechanics theory .........................53 2.4.1. Order parameters ..................................................................................................... 53 2.5. Nonlocal thermodynamic formulation of anisotropic damage mechanics ........................55 2.6. Matrix representation of Damage effect and Damage rate tensors in 2D .........................60 iv 2.7. Plane Stress problem ......................................................................................................61 2.8. Numerical implementation .............................................................................................66 2.8.1. Geometry and material parameters ........................................................................... 66 2.8.2. Computational algorithm ......................................................................................... 67 2.8.3. Numerical results and discussion ............................................................................. 68 2.9. Conclusions ....................................................................................................................71 2.10. References ....................................................................................................................74