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Meshless Method for Modeling Large Deformation With MESHLESS METHOD FOR MODELING LARGE DEFORMATION WITH ELASTOPLASTICITY by JIANFENG MA B.S., Beijing Institute of Light Industry, China, 1995 M.S., Beijing Institute of Technology, China, 1998 AN ABSTRACT OF A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Mechanical Engineering College of Engineering KANSAS STATE UNIVERSITY Manhattan, Kansas 2007 Abstract Over the past two decades meshless methods have attracted much attention owing to their advantages in adaptivity, higher degree of solution field continuity, and capability to handle moving boundary and changing geometry. In this work, a meshless integral method based on the regularized boundary integral equation has been developed and applied to two-dimensional linear elasticity and elastoplasticity with small or large deformation. The development of the meshless integral method and its application to two-dimensional linear elasticity is described first. The governing integral equation is obtained from the weak form of elasticity over a local sub-domain, and the moving least-squares approximation is employed for meshless function approximation. This formulation incorporates: a subtraction method for singularity removal in the boundary integral equation, a special numerical integration for the calculation of integrals with weak singularity which further improves accuracy, a collocation method for the imposition of essential boundary conditions, and a method for incorporation of natural boundary conditions in the system governing equation. Next, elastoplastic material behavior with small deformation is introduced into the meshless integral method. The constitutive law is rate-independent flow theory based on von Mises yielding criterion with isotropic hardening. The method is then extended to large deformation plasticity based on Green-Naghdi’s theory using updated Lagrangian description. The Green-Lagrange strain is decomposed into the elastic and plastic part, and the elastoplastic constitutive law is employed that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. Finally, a pre- and post-processor for the meshless method using node- and pixel-based approach is presented. Numerical results from the meshless integral method agree well with available analytical solutions or finite element results, and the comparisons demonstrate that the meshless integral method is accurate and robust. This research lays the foundation for modeling and simulation of metal cutting processes. MESHLESS METHOD FOR MODELING LARGE DEFORMATION WITH ELASTOPLASTICITY by JIANFENG MA B.S., Beijing Institute of Light Industry, China, 1995 M.S., Beijing Institute of Technology, China, 1998 A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Mechanical Engineering College of Engineering KANSAS STATE UNIVERSITY Manhattan, Kansas 2007 Approved by: Approved by: Co-Major Professor Co-Major Professor Dr. Prakash Krishnaswami Dr. Jack Xin Copyright JIANFENG MA 2007 Abstract Over the past two decades meshless methods have attracted much attention owing to their advantages in adaptivity, higher degree of solution field continuity, and capability to handle moving boundary and changing geometry. In this work, a meshless integral method based on the regularized boundary integral equation has been developed and applied to two-dimensional linear elasticity and elastoplasticity with small or large deformation. The development of the meshless integral method and its application to two-dimensional linear elasticity is described first. The governing integral equation is obtained from the weak form of elasticity over a local sub-domain, and the moving least-squares approximation is employed for meshless function approximation. This formulation incorporates: a subtraction method for singularity removal in the boundary integral equation, a special numerical integration for the calculation of integrals with weak singularity which further improves accuracy, a collocation method for the imposition of essential boundary conditions, and a method for incorporation of natural boundary conditions in the system governing equation. Next, elastoplastic material behavior with small deformation is introduced into the meshless integral method. The constitutive law is rate-independent flow theory based on von Mises yielding criterion with isotropic hardening. The method is then extended to large deformation plasticity based on Green-Naghdi’s theory using updated Lagrangian description. The Green-Lagrange strain is decomposed into the elastic and plastic part, and the elastoplastic constitutive law is employed that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. Finally, a pre- and post-processor for the meshless method using node- and pixel-based approach is presented. Numerical results from the meshless integral method agree well with available analytical solutions or finite element results, and the comparisons demonstrate that the meshless integral method is accurate and robust. This research lays the foundation for modeling and simulation of metal cutting processes. Table of Contents List of Figures..................................................................................................................... x List of Tables .................................................................................................................. xvii Acknowledgements........................................................................................................xviii Dedication........................................................................................................................ xix CHAPTER 1 - Introduction ................................................................................................ 1 Reference ........................................................................................................................ 6 CHAPTER 2 - A Meshless Integral Method Based on Regularized Boundary Integral Equation ............................................................................................................................................. 9 2.1 Abstract..................................................................................................................... 9 2.2 Introduction............................................................................................................. 10 2.3 Local Boundary Integral Equation for Linear Elasticity ........................................ 13 2.4 Methods for Evaluating Strongly Singular Integrals .............................................. 17 2.5 Regularized Local Boundary Integral Equation Using Subtraction Method.......... 18 2.6 The Moving Least-squares Approximation ............................................................ 22 2.7 Meshless Implementation ....................................................................................... 25 2.8 Treatment for Weak Singularity ............................................................................. 27 2.9 Imposition of Essential and Natural Boundary Conditions .................................... 28 2.10 Numerical Examples............................................................................................. 32 2.10.1 Constant Stress Patch Tests ........................................................................... 34 2.10.2 Higher-order Patch Tests ............................................................................... 34 2.10.3 Cantilever Beam............................................................................................. 35 2.10.4 Infinite Plate with a Circular Hole ................................................................. 37 2.11 Concluding Remarks............................................................................................. 40 2.12 Acknowledgement ................................................................................................ 41 2.13 References............................................................................................................. 41 CHAPTER 3 - Elastoplastic Meshless Integral Method................................................... 78 3.1 Abstract................................................................................................................... 78 3.2 Introduction............................................................................................................. 79 vi 3.3 Regularized Local Boundary Integral Equation Using Subtraction Method.......... 81 3.4 Elastoplastic Constitutive Equation........................................................................ 85 3.5 The Moving Least-squares Approximation ............................................................ 87 3.6 Meshless Implementation ....................................................................................... 90 3.7 Treatment for Weak Singularity and Imposition of Boundary Conditions ............ 92 3.8 Solution Algorithm for Elastoplasticity.................................................................. 94 3.8.1 Solution algorithm ........................................................................................... 95 3.8.2 The procedure used to compute the stress at each Gaussian point .................. 97 3.8.3 The computation of plastic term ...................................................................... 99 3.9 Numerical Examples..............................................................................................
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