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Preface xi List of Boxes xv 1 Introduction 1 1.1 Nonlinear finite elements in design 1 1.2 Related books and a brief history of nonlinear finite elements 4 1.3 Notation 7 1.4 Mesh descriptions 9 1.5 Classification of partial differential equations 13 1.6 Exercises 18 2 Lagrangian and Eulerian finite elements in one dimension 19 2.1 Introduction 19 2.2 Governing equations for total Lagrangian formulation 20 2.3 Weak form for total Lagrangian formulation 27 2.4 Finite element discretization in total Lagrangian formulation 33 2.5 Element and global matrices 38 2.6 Governing equations for updated Lagrangian formulation 48 2.7 Weak form for updated Lagrangian formulation 51 2.8 Element equations for updated Lagrangian formulation 52 2.9 Governing equations for Eulerian formulation 64 2.10 Weak forms for Eulerian mesh equations 65 2.11 Finite element equations 66 2.12 Solution methods 70 2.13 Summary 72 2.14 Exercises 72 3 Continuum mechanics 75 3.1 Introduction 75 3.2 Deformation and motion 76 3.3 Strain measures 92 3.4 Stress measures 101 3.5 Conservation equations 108 3.6 Lagrangian conservation equations 119 3.7 Polar decomposition and frame-invariance 125 3.8 Exercises 137 4 Lagrangian meshes 141 4.1 Introduction 141 4.2 Governing equations 142 4.3 Weak form: principle of virtual power 145 4.4 Updated Lagrangian finite element discretization 152 4.5 Implementation 162 4.6 Corotational formulations 185 4.7 Total Lagrangian formulation 193 4.8 Total Lagrangian weak form 196 4.9 Finite element semidiscretization 198 4.10 Exercise 213 5 Constitutive models 215 5.1 Introduction 215 5.2 The stress-strain curve 216 5.3 One-dimensional elasticity 221 5.4 Nonlinear elasticity 225 5.5 One-dimensional plasticity 240 5.6 Multiaxial plasticity 247 5.7 Hyperelastic-plastic models 264 5.8 Viscoelasticity 274 5.9 Stress update algorithms 277 5.10 Continuum mechanics and constitutive models 294 5.11 Exercises 308 6 Solution methods and stability 309 6.1 Introduction 309 6.2 Explicit methods 310 6.3 Equilibrium solutions and implicit time integration 317 6.4 Linearization 337 6.5 Stability and continuation methods 353 6.6 Numerical stability 369 6.7 Material stability 384 6.8 Exercises 392 7 Arbitrary Lagrangian Eulerian formulations 393 7.1 Introduction 393 7.2 ALE continuum mechanics 395 7.3 Conservation laws in ALE description 402 7.4 ALE governing equations 403 7.5 Weak forms 404 7.6 Introduction to the Petrov-Galerkin method 408 7.7 Petrov-Galerkin formulation of momentum equation 417 7.8 Path-dependent materials 420 7.9 Linearization of the discrete equations 432 7.10 Mesh update equations 435 7.11 Numerical example: an elastic-plastic wave propagation problem 442 7.12 Total ALE formulations 443 8 Element technology 451 8.1 Introduction 451 8.2 Element performance 453 8.3 Element properties and patch tests 461 8.4 Q4 and volumetric locking 469 8.5 Multi-field weak forms and elements 474 8.6 Multi-field quadrilaterals 487 8.7 One-point quadrature elements 491 8.8 Examples 500 8.9 Stability 504 8.10 Exercises 507 9 Beams and shells 509 9.1 Introduction 509 9.2 Beam theories 511 9.3 Continuum-based beam 514 9.4 Analysis of CB beam 524 9.5 Continuum-based shell implementation 536 9.6 CB shell theory 550 9.7 Shear and membrane locking 555 9.8 Assumed strain elements 560 9.9 One-point quadrature elements 563 9.10 Exercises 566 10 Contact-impact 569 10.1 Introduction 569 10.2 Contact interface equations 570 10.3 Friction models 580 10.4 Weak forms 585 10.5 Finite element discretization 595 10.6 On explicit methods 609 11 XFEM  11.1. INTRODUCTION  11.2. PARTITION OF UNITY AND ENRICHMENTS  11.3. ONE DIMENSIONAL XFEM  11.4. MULTI-DIMENSION XFEM  11.5. WEAK AND STRONG FORMS  11.6. DISCRETE EQUATIONS  11.7. LEVEL SET METHOD  11.8. XFEM IMPLEMENTATION STRATEGY  11.9. INTEGRATION  11.10. AN EXAMPLE OF XFEM SIMULATION  11.11. EXERCISE 12  Introduction to multiresolution theory 12.1 MOTIVATION: MATERIALS ARE STRUCTURED CONTINUA 12.2 BULK DEFORMATION OF MICROSTRUCTURED CONTINUA 12.3 GENERALIZING MECHANICS TO BULK MICROSTRUCTURED CONTINUA 12.4 MULTISCALE MICROSTRUCTURES AND THE MULTIRESOLUTION CONTINUUM THEORY 12.5 GOVERNING EQUATIONS FOR MCT 12.6 CONSTRUCTING MCT CONSTITUTIVE RELATIONSHIPS 12.7 BASIC GUIDELINES FOR RVE MODELS 12.8 FINITE ELEMENT IMPLEMENTATION OF MCT 12.9 NUMERICAL EXAMPLE 12.10 FUTURE RESEARCH DIRECTION OF MCT MODELING 12.11 EXERCISES 13 Single-crystal plasticity 13.1 Introduction 13.2 Crystallographic description of cubic and non-cubic crystals 13.3 Atomic origins of plasticity and the burgers vector in single crystals 13.4 Defining slip planes and directions in general single crystals 13.5 Kinematics of single crystal plasticity 13.6 Dislocation density evolution 13.7 Stress required for dislocation motion. 13.8 Stress update in rate-dependent single-crystal plasticity 13.9 Algorithm for rate-dependent dislocation-density based crystal plasticity 13.10 Numerical example 13.11 Exercises Appendix 1 Voigt notation 615 Appendix 2 Norms 619 Appendix 3 Element shape functions 622 Appendix 4 Euler angles from pole figures Appendix 5 Example of dislocation density evolutionary equations Glossary 627 References 631 Index 641

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