Nonlinear Finite Elements for Continua and Structures

Ted Belytschko, Northwestern University, USA Wing Kam Liu, Northwestern University, USA Brian Moran, King Abdullah University of Science and Technology, The Kingdom of Saudi Arabia Khalil I. Elkhodary, The American University in Cairo, Egypt

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