Nonlinear Finite Elements for Continua and Structures

This updated and expanded edition of the bestselling textbook provides a comprehensive introduction to the methods and theory of nonlinear finite element analysis. New material provides a concise introduction to some of the cutting-edge methods that have evolved in recent years in the field of nonlinear finite element modeling, and includes the eXtended Finite Element Method (XFEM), multiresolution continuum theory for multiscale microstructures, and dislocation- density-based crystalline plasticity.

Nonlinear Finite Elements for Continua and Structures, Second Edition focuses on the formulation and solution of discrete equations for various classes of problems that are of principal interest in applications to solid and structural mechanics. Topics covered include the discretization by finite elements of continua in one dimension and in multi-dimensions; the formulation of constitutive equations for nonlinear materials and large deformations; procedures for the solution of the discrete equations, including considerations of both numerical and multiscale physical instabilities; and the treatment of structural and contact-impact problems.

Key features:

• Presents a detailed and rigorous treatment of nonlinear solid mechanics and how it can be implemented in finite element analysis
• Covers many of the material laws used in today’s software and research
• Introduces advanced topics in nonlinear finite element modelling of continua
• Introduction of multiresolution continuum theory and XFEM
• Accompanied by a website hosting a solution manual and MATLAB^® and FORTRAN code

Nonlinear Finite Elements for Continua and Structures, Second Edition is a must-have textbook for graduate students in mechanical engineering, civil engineering, applied mathematics, engineering mechanics, and materials science, and is also an excellent source of information for researchers and practitioners.

Ted Belytschko, Northwestern University, USA
Ted Belytschko is a Walter P. Murphy Professor and McCormick Distinguished Professor of Computational Mechanics at Northwestern University. His main research interest is in computational methods for modeling the behavior of solids, with particular emphasis on failure and fracture. He is currently the Editor-in-Chief of the International Journal for Numerical Methods in Engineering.

Wing Kam Liu, Northwestern University, USA
Wing Kam Liu is a Walter P. Murphy Professor in the Department of Mechanical Engineering at Northwestern University. His main areas of research include nonlinear finite elements, multiscale methods for materials design and engineering simulation.

Brian Moran, Northwestern University, USA
Brian Moran is a Professor in the Department of Mechanical Engineering at Northwestern University. His research interests include computational methods, continuum and fracture mechanics, micromechanics and composites.

Khalil I. Elkhodary, Northwestern University, USA
Khalil I. Elkhodary is a post-doctoral researcher in the Department of Mechanical Engineering at Northwestern University. His research interests focus on theoretical and applied mechanics.

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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