Foreword xxi
Preface xxiii
List of Boxes xxvii
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 17
2 Lagrangian and Eulerian Finite Elements in One Dimension
19
2.1 Introduction 19
2.2 Governing Equations for Total Lagrangian Formulation 21
2.3 Weak Form for Total Lagrangian Formulation 28
2.4 Finite Element Discretization in Total Lagrangian
Formulation 34
2.5 Element and Global Matrices 40
2.6 Governing Equations for Updated Lagrangian Formulation
51
2.7 Weak Form for Updated Lagrangian Formulation 53
2.8 Element Equations for Updated Lagrangian Formulation 55
2.10 Weak Forms for Eulerian Mesh Equations 68
2.11 Finite Element Equations 69
2.12 Solution Methods 72
2.13 Summary 74
2.14 Exercises 75
3 Continuum Mechanics 77
3.1 Introduction 77
3.2 Deformation and Motion 78
3.3 Strain Measures 95
3.4 Stress Measures 104
3.5 Conservation Equations 111
3.6 Lagrangian Conservation Equations 123
3.7 Polar Decomposition and Frame-Invariance 130
3.8 Exercises 143
4 Lagrangian Meshes 147
4.1 Introduction 147
4.2 Governing Equations 148
4.3 Weak Form: Principle of Virtual Power 152
4.4 Updated Lagrangian Finite Element Discretization 158
4.5 Implementation 168
4.6 Corotational Formulations 194
4.7 Total Lagrangian Formulation 203
4.8 Total Lagrangian Weak Form 206
4.9 Finite Element Semidiscretization 209
4.10 Exercises 225
5 Constitutive Models 227
5.1 Introduction 227
5.2 The Stress?Strain Curve 228
5.3 One-Dimensional Elasticity 233
5.4 Nonlinear Elasticity 237
5.5 One-Dimensional Plasticity 254
5.6 Multiaxial Plasticity 262
5.7 Hyperelastic?Plastic Models 281
5.8 Viscoelasticity 292
5.9 Stress Update Algorithms 294
5.10 Continuum Mechanics and Constitutive Models 314
5.11 Exercises 328
6 Solution Methods and Stability 329
6.1 Introduction 329
6.2 Explicit Methods 330
6.3 Equilibrium Solutions and Implicit Time Integration 337
6.4 Linearization 358
6.5 Stability and Continuation Methods 375
6.6 Numerical Stability 391
6.7 Material Stability 407
6.8 Exercises 415
7 Arbitrary Lagrangian Eulerian Formulations 417
7.1 Introduction 417
7.2 ALE Continuum Mechanics 419
7.3 Conservation Laws in ALE Description 426
7.4 ALE Governing Equations 428
7.5 Weak Forms 429
7.6 Introduction to the Petrov?Galerkin Method 433
7.7 Petrov?Galerkin Formulation of Momentum Equation
442
7.8 Path-Dependent Materials 445
7.9 Linearization of the Discrete Equations 457
7.10 Mesh Update Equations 460
7.11 Numerical Example: An Elastic?Plastic Wave
Propagation Problem 468
7.12 Total ALE Formulations 471
7.13 Exercises 475
8 Element Technology 477
8.1 Introduction 477
8.2 Element Performance 479
8.3 Element Properties and Patch Tests 487
8.4 Q4 and Volumetric Locking 496
8.5 Multi-Field Weak Forms and Elements 501
8.6 Multi-Field Quadrilaterals 514
8.7 One-Point Quadrature Elements 518
8.8 Examples 527
8.9 Stability 531
8.10 Exercises 533
9 Beams and Shells 535
9.1 Introduction 535
9.2 Beam Theories 537
9.3 Continuum-Based Beam 540
9.4 Analysis of the CB Beam 551
9.5 Continuum-Based Shell Implementation 563
9.6 CB Shell Theory 578
9.7 Shear and Membrane Locking 584
9.8 Assumed Strain Elements 589
9.9 One-Point Quadrature Elements 592
9.10 Exercises 595
10 Contact-Impact 597
10.1 Introduction 597
10.2 Contact Interface Equations 598
10.3 Friction Models 609
10.4 Weak Forms 614
10.5 Finite Element Discretization 624
10.6 On Explicit Methods 638
11 EXtended Finite Element Method (XFEM) 643
11.1 Introduction 643
11.2 Partition of Unity and Enrichments 647
11.3 One-Dimensional XFEM 648
11.4 Multi-Dimension XFEM 656
11.5 Weak and Strong Forms 660
11.6 Discrete Equations 662
11.7 Level Set Method 668
11.8 The Phantom Node Method 670
11.9 Integration 673
11.10 An Example of XFEM Simulation 675
11.11 Exercise 678
12 Introduction to Multiresolution Theory 681
12.1 Motivation: Materials are Structured Continua 681
12.2 Bulk Deformation of Microstructured Continua 685
12.3 Generalizing Mechanics to Bulk Microstructured Continua
686
12.4 Multiscale Microstructures and the Multiresolution
Continuum Theory 696
12.5 Governing Equations for MCT 699
12.6 Constructing MCT Constitutive Relationships 701
12.7 Basic Guidelines for RVE Modeling 705
12.8 Finite Element Implementation of MCT 710
12.9 Numerical Example 712
12.10 Future Research Directions of MCT Modeling 718
12.11 Exercises 719
13 Single-Crystal Plasticity 721
13.1 Introduction 721
13.2 Crystallographic Description of Cubic and Non-Cubic
Crystals 723
13.3 Atomic Origins of Plasticity and the Burgers Vector in
Single Crystals 726
13.4 Defining Slip Planes and Directions in General Single
Crystals 729
13.5 Kinematics of Single Crystal Plasticity 735
13.6 Dislocation Density Evolution 740
13.7 Stress Required for Dislocation Motion 742
13.8 Stress Update in Rate-Dependent Single-Crystal Plasticity
743
13.9 Algorithm for Rate-Dependent Dislocation-Density Based
Crystal Plasticity 745
13.10 Numerical Example: Localized Shear and Inhomogeneous
Deformation 747
13.11 Exercises 750
Appendix 1 Voigt Notation 751
Appendix 2 Norms 757
Appendix 3 Element Shape Functions 761
Appendix 4 Euler Angles From Pole Figures 767
Appendix 5 Example of Dislocation-Density Evolutionary Equations
771
Glossary 777
References 781
Index 795
