Nonlinear Finite Element Analysis of Solids and Structures

Preface xi

Series Preface xiii

Notation xv

About the Code xxi

PART I BASIC CONCEPTS AND SOLUTION TECHNIQUES

1 Preliminaries 3

1.1 A Simple Example of Non-linear Behaviour 3

1.2 A Review of Concepts from Linear Algebra 5

1.3 Vectors and Tensors 12

1.4 Stress and Strain Tensors 17

1.5 Elasticity 23

1.6 The PyFEM Finite Element Library 25

References 29

2 Non-linear Finite Element Analysis 31

2.1 Equilibrium and Virtual Work 31

2.2 Spatial Discretisation by Finite Elements 33

2.3 PyFEM: Shape Function Utilities 38

2.4 Incremental-iterative Analysis 41

2.5 Load versus Displacement Control 50

2.6 PyFEM: A Linear Finite Element Code with Displacement Control 53

References 62

3 Geometrically Non-linear Analysis 63

3.1 Truss Elements 64

3.2 PyFEM: The Shallow Truss Problem 76

3.3 Stress and Deformation Measures in Continua 85

3.4 Geometrically Non-linear Formulation of Continuum Elements 91

3.5 Linear Buckling Analysis 100

3.6 PyFEM: A Geometrically Non-linear Continuum Element 103

References 110

4 Solution Techniques in Quasi-static Analysis 113

4.1 Line Searches 113

4.2 Path-following or Arc-length Methods 116

4.3 PyFEM: Implementation of Riks? Arc-length Solver 124

4.4 Stability and Uniqueness in Discretised Systems 129

4.5 Load Stepping and Convergence Criteria 134

4.6 Quasi-Newton Methods 138

References 141

5 Solution Techniques for Non-linear Dynamics 143

5.1 The Semi-discrete Equations 143

5.2 Explicit Time Integration 144

5.3 PyFEM: Implementation of an Explicit Solver 149

5.4 Implicit Time Integration 152

5.5 Stability and Accuracy in the Presence of Non-linearities 156

5.6 Energy-conserving Algorithms 161

5.7 Time Step Size Control and Element Technology 164

References 165

PART II MATERIAL NON-LINEARITIES

6 Damage Mechanics 169

6.1 The Concept of Damage 169

6.2 Isotropic Elasticity-based Damage 171

6.3 PyFEM: A Plane-strain Damage Model 175

6.4 Stability, Ellipticity and Mesh Sensitivity 179

6.5 Cohesive-zone Models 185

6.6 Element Technology: Embedded Discontinuities 190

6.7 Complex Damage Models 198

6.8 Crack Models for Concrete and Other Quasi-brittle Materials 201

6.8.1 Elasticity-based Smeared Crack Models 201

6.8.2 Reinforcement and Tension Stiffening 206

6.9 Regularised Damage Models 210

References 215

7 Plasticity 219

7.1 A Simple Slip Model 219

7.2 Flow Theory of Plasticity 223

7.3 Integration of the Stress?strain Relation 239

7.4 Tangent Stiffness Operators 249

7.5 Multi-surface Plasticity 252

7.6 Soil Plasticity: Cam-clay Model 267

7.7 Coupled Damage?Plasticity Models 270

7.8 Element Technology: Volumetric Locking 271

References 277

8 Time-dependent Material Models 281

8.1 Linear Visco-elasticity 281

8.2 Creep Models 287

8.3 Visco-plasticity 289

References 303

PART III STRUCTURAL ELEMENTS

9 Beams and Arches 307

9.1 A Shallow Arch 307

9.2 PyFEM: A Kirchhoff Beam Element 317

9.3 Corotational Elements 321

9.4 A Two-dimensional Isoparametric Degenerate Continuum Beam Element 328

9.5 A Three-dimensional Isoparametric Degenerate Continuum Beam Element 333

References 341

10 Plates and Shells 343

10.1 Shallow-shell Formulations 344

10.2 An Isoparametric Degenerate Continuum Shell Element 351

10.3 Solid-like Shell Elements 356

10.4 Shell Plasticity: Ilyushin?s Criterion 357

References 361

PART IV LARGE STRAINS

11 Hyperelasticity 365

11.1 More Continuum Mechanics 365

11.2 Strain Energy Functions 374

11.3 Element Technology 389

References 398

12 Large-strain Elasto-plasticity 401

12.1 Eulerian Formulations 402

12.2 Multiplicative Elasto-plasticity 407

12.3 Multiplicative Elasto-plasticity versus Rate Formulations 411

12.4 Integration of the Rate Equations 414

12.5 Exponential Return-mapping Algorithms 418

References 422

PART V ADVANCED DISCRETISATION CONCEPTS

13 Interfaces and Discontinuities 427

13.1 Interface Elements 428

13.2 Discontinuous Galerkin Methods 436

References 439

14 Meshless and Partition-of-unity Methods 441

14.1 Meshless Methods 442

14.2 Partition-of-unity Approaches 451

References 470

15 Isogeometric Finite Element Analysis 473

15.1 Basis Functions in Computer Aided Geometric Design 473

15.2 Isogeometric Finite Elements 483

15.3 PyFEM: Shape Functions for Isogeometric Analysis 487

15.4 Isogeometric Analysis in Non-linear Solid Mechanics 490

References 506

Index 509