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