Nonlinear Finite Elements for Continua and Structures

Ted Belytschko, Northwestern University, USAWing Kam Liu, Northwestern University, USABrian Moran, King Abdullah University of Science and Technology, The Kingdom of Saudi ArabiaKhalil I. Elkhodary, The American University in Cairo, Egypt

• Foreword xxiPreface xxiiiList of Boxes xxvii1 Introduction 11.1 Nonlinear Finite Elements in Design 11.2 Related Books and a Brief History of Nonlinear Finite Elements 41.3 Notation 71.4 Mesh Descriptions 91.5 Classification of Partial Differential Equations 131.6 Exercises 172 Lagrangian and Eulerian Finite Elements in One Dimension 192.1 Introduction 192.2 Governing Equations for Total Lagrangian Formulation 212.3 Weak Form for Total Lagrangian Formulation 282.4 Finite Element Discretization in Total Lagrangian Formulation 342.5 Element and Global Matrices 402.6 Governing Equations for Updated Lagrangian Formulation 512.7 Weak Form for Updated Lagrangian Formulation 532.8 Element Equations for Updated Lagrangian Formulation 552.10 Weak Forms for Eulerian Mesh Equations 682.11 Finite Element Equations 692.12 Solution Methods 722.13 Summary 742.14 Exercises 753 Continuum Mechanics 773.1 Introduction 773.2 Deformation and Motion 783.3 Strain Measures 953.4 Stress Measures 1043.5 Conservation Equations 1113.6 Lagrangian Conservation Equations 1233.7 Polar Decomposition and Frame-Invariance 1303.8 Exercises 1434 Lagrangian Meshes 1474.1 Introduction 1474.2 Governing Equations 1484.3 Weak Form: Principle of Virtual Power 1524.4 Updated Lagrangian Finite Element Discretization 1584.5 Implementation 1684.6 Corotational Formulations 1944.7 Total Lagrangian Formulation 2034.8 Total Lagrangian Weak Form 2064.9 Finite Element Semidiscretization 2094.10 Exercises 2255 Constitutive Models 2275.1 Introduction 2275.2 The Stress–Strain Curve 2285.3 One-Dimensional Elasticity 2335.4 Nonlinear Elasticity 2375.5 One-Dimensional Plasticity 2545.6 Multiaxial Plasticity 2625.7 Hyperelastic–Plastic Models 2815.8 Viscoelasticity 2925.9 Stress Update Algorithms 2945.10 Continuum Mechanics and Constitutive Models 3145.11 Exercises 3286 Solution Methods and Stability 3296.1 Introduction 3296.2 Explicit Methods 3306.3 Equilibrium Solutions and Implicit Time Integration 3376.4 Linearization 3586.5 Stability and Continuation Methods 3756.6 Numerical Stability 3916.7 Material Stability 4076.8 Exercises 4157 Arbitrary Lagrangian Eulerian Formulations 4177.1 Introduction 4177.2 ALE Continuum Mechanics 4197.3 Conservation Laws in ALE Description 4267.4 ALE Governing Equations 4287.5 Weak Forms 4297.6 Introduction to the Petrov–Galerkin Method 4337.7 Petrov–Galerkin Formulation of Momentum Equation 4427.8 Path-Dependent Materials 4457.9 Linearization of the Discrete Equations 4577.10 Mesh Update Equations 4607.11 Numerical Example: An Elastic–Plastic Wave Propagation Problem 4687.12 Total ALE Formulations 4717.13 Exercises 4758 Element Technology 4778.1 Introduction 4778.2 Element Performance 4798.3 Element Properties and Patch Tests 4878.4 Q4 and Volumetric Locking 4968.5 Multi-Field Weak Forms and Elements 5018.6 Multi-Field Quadrilaterals 5148.7 One-Point Quadrature Elements 5188.8 Examples 5278.9 Stability 5318.10 Exercises 5339 Beams and Shells 5359.1 Introduction 5359.2 Beam Theories 5379.3 Continuum-Based Beam 5409.4 Analysis of the CB Beam 5519.5 Continuum-Based Shell Implementation 5639.6 CB Shell Theory 5789.7 Shear and Membrane Locking 5849.8 Assumed Strain Elements 5899.9 One-Point Quadrature Elements 5929.10 Exercises 59510 Contact-Impact 59710.1 Introduction 59710.2 Contact Interface Equations 59810.3 Friction Models 60910.4 Weak Forms 61410.5 Finite Element Discretization 62410.6 On Explicit Methods 63811 EXtended Finite Element Method (XFEM) 64311.1 Introduction 64311.2 Partition of Unity and Enrichments 64711.3 One-Dimensional XFEM 64811.4 Multi-Dimension XFEM 65611.5 Weak and Strong Forms 66011.6 Discrete Equations 66211.7 Level Set Method 66811.8 The Phantom Node Method 67011.9 Integration 67311.10 An Example of XFEM Simulation 67511.11 Exercise 67812 Introduction to Multiresolution Theory 68112.1 Motivation: Materials are Structured Continua 68112.2 Bulk Deformation of Microstructured Continua 68512.3 Generalizing Mechanics to Bulk Microstructured Continua 68612.4 Multiscale Microstructures and the Multiresolution Continuum Theory 69612.5 Governing Equations for MCT 69912.6 Constructing MCT Constitutive Relationships 70112.7 Basic Guidelines for RVE Modeling 70512.8 Finite Element Implementation of MCT 71012.9 Numerical Example 71212.10 Future Research Directions of MCT Modeling 71812.11 Exercises 71913 Single-Crystal Plasticity 72113.1 Introduction 72113.2 Crystallographic Description of Cubic and Non-Cubic Crystals 72313.3 Atomic Origins of Plasticity and the Burgers Vector in Single Crystals 72613.4 Defining Slip Planes and Directions in General Single Crystals 72913.5 Kinematics of Single Crystal Plasticity 73513.6 Dislocation Density Evolution 74013.7 Stress Required for Dislocation Motion 74213.8 Stress Update in Rate-Dependent Single-Crystal Plasticity 74313.9 Algorithm for Rate-Dependent Dislocation-Density Based Crystal Plasticity 74513.10 Numerical Example: Localized Shear and Inhomogeneous Deformation 74713.11 Exercises 750Appendix 1 Voigt Notation 751Appendix 2 Norms 757Appendix 3 Element Shape Functions 761Appendix 4 Euler Angles From Pole Figures 767Appendix 5 Example of Dislocation-Density Evolutionary Equations 771Glossary 777References 781Index 795