Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2011 Numerical Implementation of Continuum Dislocation Theory Shengxu Xia Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES NUMERICAL IMPLEMENTATION OF CONTINUUM DISLOCATION THEORY By SHENGXU XIA A Thesis submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Master of Science Degree Awarded: Fall Semester, 2011 Shengxu Xia defended this thesis on November 7, 2011. The members of the supervisory committee were: Anter El-Azab Professor Directing Thesis Tomasz Plewa Committee Member Xiaoqiang Wang Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the thesis has been approved in accordance with the university requirements. ii To my parents and friends iii ACKNOWLEDGEMENTS It has been a great honour to spend two years in Florida State University, where I constantly felt inspired by the intelligence and humanity surrounding me. The completion of my thesis has been a joyful journey and I could not have succeeded without the invaluable assistance of many individuals, so I want to express my deepest appreciation to them. First of all, I would like to give special thanks to my advisor, Professor Anter El-Azab, for his patient, friendly and encouraging support over the past two years, and for his knowledge and enthusiasm which made this thesis possible. I am very grateful to all members of my thesis committee as well as faculty and staff at scientific computing Department for their input, valuable discussions and accessibility. In particular, I would like to thank Professor Max Gunzburger, Professor Xiaoqiang Wang, Professor Tomasz Plewa for their support. The discussion with professionals in this research area is always interesting and cheerful. In particular, the communication with Dr. Thomas Hochrainer, Dr. Jie Deng, Dr. Stefan Sandfeld, Dr. Giacomo Po, and Dr. Mamdouh Mohamed benefits me a lot. I will take this chance to thank them for all the help they offer. In addition to the academic assistance above, I received equally important assistance from my friends and family. I am grateful to my parents for their love, support and trust throughout not only the course of my Master study, but throughout my entire life. iv TABLE OF CONTENTS List of Figures ....................................... vi Abstract ........................................... viii 1 Introduction 1 2 Discrete Dislocation Dynamics Model 4 2.1 Basic Concepts of Dislocations ......................... 4 2.2 Burgers Vector .................................. 6 2.3 Slip System and Dislocation Motion ...................... 6 2.3.1 Slip Systems ............................... 6 2.3.2 Dislocation Motion ............................ 7 3 Continuum Theory of Dislocation 8 3.1 Crystal Plasticity ................................. 8 3.2 Dislocation Plasticity Theory .......................... 10 3.3 Dislocation Evolution Equation ......................... 12 4 Numerical Scheme for Simulation 14 4.1 Numerical Scheme for Eigenstrain Problem . 14 4.2 Numerical Scheme for Curl Equation ...................... 15 4.2.1 Least Square Finite Element Method(LSFEM) . 15 4.2.2 Finite Difference Method ........................ 17 4.3 Coupled Scheme for Stress-Dislocation Simulation . 18 5 Numerical Results 20 5.1 Test Problem for Equilibrium Equation .................... 20 5.2 Test Problem for Curl Equation ......................... 24 5.2.1 Simulation Results Obtained by LSFEM . 24 5.2.2 Comparison between Different Numerical Schemes for Curl Equation 27 5.3 Test Problem for Stress-Dislocation Coupled Scheme . 29 6 Summary and outlook 34 References .......................................... 36 Biographical Sketch .................................... 40 v LIST OF FIGURES 2.1 edge dislocation and screw dislocation ...................... 5 2.2 mixed dislocation pattern formation ....................... 5 2.3 determination of Burger’s vector ......................... 6 3.1 kinematics of crystal plasticity .......................... 9 4.1 coupled scheme for the simulation of dislocation evolution . 19 5.1 formation of a single edge dislocation ...................... 20 5.2 γ field distribution ................................ 21 5.3 distorted shape due to an edge dialocation ................... 22 5.4 σxx component field ................................ 23 5.5 σzz component field ................................ 23 5.6 σxz component field ................................ 24 5.7 the movement of a straight dislocation line ................... 25 5.8 movement of a tilt dislocation line ........................ 26 5.9 loop expansion ................................... 26 5.10 pileup process of a loop .............................. 27 5.11 curl equation solved by four distinct numerical methods . 28 5.12 stress field σ13 in 3D crystal and 2D slip plane . 29 5.13 dislocations absorbed into the boundary ..................... 30 5.14 dislocations absorbed into the boundary(given by upwind scheme) . 31 5.15 Plastic distortion component β31 recedes. .................... 32 vi 5.16 A distorted crystal partly recovers to its original state where the cloud denotes the magnitude of displacement. .......................... 32 vii ABSTRACT This thesis aims at theoretical and computational modelling of the continuum dislocation theory coupled with its internal elastic field. In this continuum description, the space-time evolution of the dislocation density is governed by a set of hyperbolic partial differential equations. These PDEs must be complemented by elastic equilibrium equations in order to obtain the velocity field that drives dislocation motion on slip planes. Simultaneously, the plastic eigenstrain tensor that serves as a known field in equilibrium equations should be updated by the motion of dislocations according to Orowan’s law. Therefore, a stress- dislocation coupled process is involved when a crystal undergoes elastoplastic deformation. The solutions of equilibrium equation and dislocation density evolution equation are tested by a few examples in order to make sure appropriate computational schemes are selected for each. A coupled numerical scheme is proposed, where resolved shear stress and Orowan’s law are two passages that connect these two sets of PDEs. The numerical implementation of this scheme is illustrated by an example that simulates the recovery process of a dislocated cubic crystal. The simulated result demonstrates the possibility to couple macroscopic(stress) and microscopic(dislocation density tensor) physical quantity to obtain crystal mechanical response. viii CHAPTER 1 INTRODUCTION Foundation of various crystal plasticity theories is mainly motivated by the technological need to understand and improve the strength of metallurgic crystals. Although research efforts that first approached this problem dates back to early nineteens, our understanding of the various aspects of crystal plasticity is still far from completion. There are two approaches to investigate into crystal plastic behaviour: phenomenological level and physical level. On the phenomenological level and within the continuum framework, Tresca[54], de St. Venant[44] et al. extended the elasticity theory by introducing phenomenological laws to account for the observed plastic behaviours of metals, such as yielding and work hardening. These phenomenological continuum plasticity theories have been very successful in a wide rang of engineering applications. They operate on length scales where the properties of materials and systems are scale invariant. However, experimental results showed that this scale-invariance breaks down at very small dimensions, say e.g. a few microns. In torsion tests of copper wire[16], strong scale-dependent response was observed in the range of 170µm to 12µm of wire diameter. In corresponding uni-axial tension tests, where deformation is macroscopically uniform, no comparable scale-dependence was recognized for the same range of copper wire diameter. Clear evidence for scale-dependent plasticity also appears in micro-indentation tests[53, 50, 39] where the hardness of metal crystals increases as the indenter decreases in size. In a word, a deeper understanding of crystal plasticity is required to interpret such phenomena. On the physical level, attempts to explain the microscopic mechanism underlying the phenomenological laws lead to the concept of the ’dislocation’ - a line-like defect inside a perfect crystal - which was firstly created as a mathematical concept by Volterra[56] and developed independently by Orowan[37], Polanyi[38] and Taylor[52]. While successfully explaining most of the puzzling phenomenology of crystal plasticity, dislocations continued as a beautiful hypothesis until the late 1950s when first sightings of them were reported in transmission electron microscopy (TEM) experiments[23]. Since then, it was recognized that modeling of the strain hardening behaviour of metallic crystals should be based on the collective behaviour of the underlying dislocation system. Kondo[26], Nye[36], Bilby et al.[4] and Kroner[29] formulated the classical continuum theory of dislocations independently from the viewpoint of plastic distortion. The common characteristic residing in their theories is a second-rank dislocation density tensor(also referred to as Kroner-Nye’s