Nonlinear Finite Element Analysis of Solids and Structures

Mike Crisfield (deceased), Imperial College, London; René de Borst, Joris Remmers & Clemens Verhoosel, TU Eindhoven, Netherlands Professor Mike Crisfield (deceased) joined the Transport & Road Research Laboratory (TRRL) in 1971, where he rose to the rank of Deputy Chief Scientific Officer. In 1989 he was appointed as first holder of the FEA Chair in Computational Mechanics in the aeronautics department at Imperial College, London, the department that had pioneered FEA in the 1950s and 1960s. Shortly before he died, a list of the most cited engineering researchers in the UK was published included Mike in the top 20, and he received an IACM Research Achievement Award in recognition of his extraordinary achievements in the field of non-linear computational mechanics. An eminent researcher and a scholar, he was reputed as an innovative thinker who adopted a 'hands-on' approach. René de Borst was appointed Dean and Distinguished University Professor of the Faculty of Mechanical Engineering of TU Eindhoven in May 2007 after a long tenure as Professor and deputy Dean at TU Delft. He is Editor for the International Journal for Numerical Methods in Engineering and International Journal for Numerical and Analytical Methods in Geomechanics and Editor for the Encyclopedia of Computational Mechanics. His many awards and the outstanding assessment of his work by the scientific community attest to his reputation as a world leading scientist and researcher within the field of computational mechanics. Joris Remmers is an assistant professor within René de Borst's group at TU Eindhoven. Clemens Verhoosel is an assistant professor within René de Borst's group at TU Eindhoven.

• Preface xiSeries Preface xiiiNotation xvAbout the Code xxiPART I BASIC CONCEPTS AND SOLUTION TECHNIQUES1 Preliminaries 31.1 A Simple Example of Non-linear Behaviour 31.2 A Review of Concepts from Linear Algebra 51.3 Vectors and Tensors 121.4 Stress and Strain Tensors 171.5 Elasticity 231.6 The PyFEM Finite Element Library 25References 292 Non-linear Finite Element Analysis 312.1 Equilibrium and Virtual Work 312.2 Spatial Discretisation by Finite Elements 332.3 PyFEM: Shape Function Utilities 382.4 Incremental-iterative Analysis 412.5 Load versus Displacement Control 502.6 PyFEM: A Linear Finite Element Code with Displacement Control 53References 623 Geometrically Non-linear Analysis 633.1 Truss Elements 643.2 PyFEM: The Shallow Truss Problem 763.3 Stress and Deformation Measures in Continua 853.4 Geometrically Non-linear Formulation of Continuum Elements 913.5 Linear Buckling Analysis 1003.6 PyFEM: A Geometrically Non-linear Continuum Element 103References 1104 Solution Techniques in Quasi-static Analysis 1134.1 Line Searches 1134.2 Path-following or Arc-length Methods 1164.3 PyFEM: Implementation of Riks’ Arc-length Solver 1244.4 Stability and Uniqueness in Discretised Systems 1294.5 Load Stepping and Convergence Criteria 1344.6 Quasi-Newton Methods 138References 1415 Solution Techniques for Non-linear Dynamics 1435.1 The Semi-discrete Equations 1435.2 Explicit Time Integration 1445.3 PyFEM: Implementation of an Explicit Solver 1495.4 Implicit Time Integration 1525.5 Stability and Accuracy in the Presence of Non-linearities 1565.6 Energy-conserving Algorithms 1615.7 Time Step Size Control and Element Technology 164References 165PART II MATERIAL NON-LINEARITIES6 Damage Mechanics 1696.1 The Concept of Damage 1696.2 Isotropic Elasticity-based Damage 1716.3 PyFEM: A Plane-strain Damage Model 1756.4 Stability, Ellipticity and Mesh Sensitivity 1796.5 Cohesive-zone Models 1856.6 Element Technology: Embedded Discontinuities 1906.7 Complex Damage Models 1986.8 Crack Models for Concrete and Other Quasi-brittle Materials 2016.8.1 Elasticity-based Smeared Crack Models 2016.8.2 Reinforcement and Tension Stiffening 2066.9 Regularised Damage Models 210References 2157 Plasticity 2197.1 A Simple Slip Model 2197.2 Flow Theory of Plasticity 2237.3 Integration of the Stress–strain Relation 2397.4 Tangent Stiffness Operators 2497.5 Multi-surface Plasticity 2527.6 Soil Plasticity: Cam-clay Model 2677.7 Coupled Damage–Plasticity Models 2707.8 Element Technology: Volumetric Locking 271References 2778 Time-dependent Material Models 2818.1 Linear Visco-elasticity 2818.2 Creep Models 2878.3 Visco-plasticity 289References 303PART III STRUCTURAL ELEMENTS9 Beams and Arches 3079.1 A Shallow Arch 3079.2 PyFEM: A Kirchhoff Beam Element 3179.3 Corotational Elements 3219.4 A Two-dimensional Isoparametric Degenerate Continuum Beam Element 3289.5 A Three-dimensional Isoparametric Degenerate Continuum Beam Element 333References 34110 Plates and Shells 34310.1 Shallow-shell Formulations 34410.2 An Isoparametric Degenerate Continuum Shell Element 35110.3 Solid-like Shell Elements 35610.4 Shell Plasticity: Ilyushin’s Criterion 357References 361PART IV LARGE STRAINS11 Hyperelasticity 36511.1 More Continuum Mechanics 36511.2 Strain Energy Functions 37411.3 Element Technology 389References 39812 Large-strain Elasto-plasticity 40112.1 Eulerian Formulations 40212.2 Multiplicative Elasto-plasticity 40712.3 Multiplicative Elasto-plasticity versus Rate Formulations 41112.4 Integration of the Rate Equations 41412.5 Exponential Return-mapping Algorithms 418References 422PART V ADVANCED DISCRETISATION CONCEPTS13 Interfaces and Discontinuities 42713.1 Interface Elements 42813.2 Discontinuous Galerkin Methods 436References 43914 Meshless and Partition-of-unity Methods 44114.1 Meshless Methods 44214.2 Partition-of-unity Approaches 451References 47015 Isogeometric Finite Element Analysis 47315.1 Basis Functions in Computer Aided Geometric Design 47315.2 Isogeometric Finite Elements 48315.3 PyFEM: Shape Functions for Isogeometric Analysis 48715.4 Isogeometric Analysis in Non-linear Solid Mechanics 490References 506Index 509