NON-LINEAR FINITE ELEMENT ANALYSIS OF SOLIDS AND STRUCTURES

By ANTHONY JOHNSON

CONTENTS PART I BASIC CONCEPTS AND SOLUTION TECHNIQUES PRELIMINARIES A SIMPLE EXAMPLE OF NON-LINEAR BEHAVIOUR A REVIEW OF CONCEPTS FROM LINEAR ALGEBRA VECTORS AND TENSORS STRESS AND STRAIN TENSORS ELASTICITY THE PYFEM FINITE ELEMENT LIBRARY REFERENCES NON-LINEAR FINITE ELEMENT ANALYSIS EQUILIBRIUM AND VIRTUAL WORK SPATIAL DISCRETISATION BY FINITE ELEMENTS PYFEM: SHAPE FUNCTION UTILITIES INCREMENTAL-ITERATIVE ANALYSIS LOAD VERSUS DISPLACEMENT CONTROL PYFEM: A LINEAR FINITE ELEMENT CODE WITH DISPLACEMENT CONTROL GEOMETRICALLY NON-LINEAR ANALYSIS TRUSS ELEMENTS TOTAL LAGRANGE FORMULATION UPDATED LAGRANGE FORMULATION COROTATIONAL FORMULATION PYFEM: THE SHALLOW TRUSS PROBLEM STRESS AND DEFORMATION MEASURES IN CONTINUA GEOMETRICALLY NON-LINEAR FORMULATION OF CONTINUUM ELEMENTS TOTAL AND UPDATED LAGRANGE FORMULATIONS LINEAR BUCKLING ANALYSIS PYFEM: A GEOMETRICALLY NON-LINEAR CONTINUUM ELEMENT SOLUTION TECHNIQUES IN QUASI-STATIC ANALYSIS LINE SEARCHES PATH-FOLLOWING OR ARC-LENGTH METHODS PYFEM: IMPLEMENTATION OF RIKS’ ARC-LENGTH SOLVER STABILITY AND UNIQUENESS IN DISCRETISED SYSTEMS STABILITY OF A DISCRETE SYSTEM UNIQUENESS AND BIFURCATION IN A DISCRETE SYSTEM BRANCH SWITCHING LOAD STEPPING AND CONVERGENCE CRITERIA QUASI-NEWTON METHODS SOLUTION TECHNIQUES FOR NON-LINEAR DYNAMICS THE SEMI-DISCRETE EQUATIONS EXPLICIT TIME INTEGRATION PYFEM: IMPLEMENTATION OF AN EXPLICIT SOLVER IMPLICIT TIME INTEGRATION THE NEWMARK FAMILY THE HHT ?-METHOD ALTERNATIVE IMPLICIT METHODS FOR TIME INTEGRATION STABILITY AND ACCURACY IN THE PRESENCE OF NON-LINEARITIES ENERGY-CONSERVING ALGORITHMS TIME STEP SIZE CONTROL AND ELEMENT TECHNOLOGY PART II MATERIAL NON-LINEARITIES DAMAGE MECHANICS THE CONCEPT OF DAMAGE ISOTROPIC ELASTICITY-BASED DAMAGE PYFEM: A PLANE-STRAIN DAMAGE MODEL STABILITY, ELLIPTICITY AND MESH SENSITIVITY STABILITY AND ELLIPTICITY MESH SENSITIVITY COHESIVE-ZONE MODELS ELEMENT TECHNOLOGY: EMBEDDED DISCONTINUITIES COMPLEX DAMAGE MODELS ANISOTROPIC DAMAGE MODELS MICROPLANE MODELS CRACK MODELS FOR CONCRETE AND OTHER QUASI-BRITTLE MATERIALS ELASTICITY-BASED SMEARED CRACK MODELS REINFORCEMENT AND TENSION STIFFENING REGULARISED DAMAGE MODELS NON-LOCAL DAMAGE MODELS GRADIENT DAMAGE MODELS PLASTICITY A SIMPLE SLIP MODEL FLOW THEORY OF PLASTICITY YIELD FUNCTION FLOW RULE HARDENING BEHAVIOUR INTEGRATION OF THE STRESS–STRAIN RELATION TANGENT STIFFNESS OPERATORS MULTI-SURFACE PLASTICITY KOITER’S GENERALISATION RANKINE PLASTICITY FOR CONCRETE TRESCA AND MOHR–COULOMB PLASTICITY SOIL PLASTICITY: CAM-CLAY MODEL COUPLED DAMAGE–PLASTICITY MODELS ELEMENT TECHNOLOGY: VOLUMETRIC LOCKING TIME-DEPENDENT MATERIAL MODELS LINEAR VISCO-ELASTICITY ONE-DIMENSIONAL LINEAR VISCO-ELASTICITY THREE-DIMENSIONAL VISCO-ELASTICITY ALGORITHMIC ASPECTS CREEP MODELS VISCO-PLASTICITY ONE-DIMENSIONAL VISCO-PLASTICITY INTEGRATION OF THE RATE EQUATIONS PERZYNA VISCO-PLASTICITY DUVAUT–LIONS VISCO-PLASTICITY CONSISTENCY MODEL PROPAGATIVE OR DYNAMIC INSTABILITIES PART III STRUCTURAL ELEMENTS BEAMS AND ARCHES A SHALLOW ARCH KIRCHHOFF FORMULATION INCLUDING SHEAR DEFORMATION: TIMOSHENKO BEAM PYFEM: A KIRCHHOFF BEAM ELEMENT COROTATIONAL ELEMENTS KIRCHHOFF THEORY TIMOSHENKO BEAM THEORY A TWO-DIMENSIONAL ISOPARAMETRIC DEGENERATE CONTINUUM BEAM ELEMENT A THREE-DIMENSIONAL ISOPARAMETRIC DEGENERATE CONTINUUM BEAM ELEMENT 10 PLATES AND SHELLS 10 1 SHALLOW-SHELL FORMULATIONS 10 2 AN ISOPARAMETRIC DEGENERATE CONTINUUM SHELL ELEMENT 10 3 SOLID-LIKE SHELL ELEMENTS 10 4 SHELL PLASTICITY: ILYUSHIN’S CRITERION PART IV LARGE STRAINS 11 HYPERELASTICITY 11 1 MORE CONTINUUM MECHANICS 11 1 1 MOMENTUM BALANCE AND STRESS TENSORS 11 1 2 OBJECTIVE STRESS RATES 11 1 3 PRINCIPAL STRETCHES AND INVARIANTS 11 2 STRAIN ENERGY FUNCTIONS 11 2 1 INCOMPRESSIBILITY AND NEAR-INCOMPRESSIBILITY 11 2 2 STRAIN ENERGY AS A FUNCTION OF STRETCH INVARIANTS 11 2 3 STRAIN ENERGY AS A FUNCTION OF PRINCIPAL STRETCHES 11 2 4 LOGARITHMIC EXTENSION OF LINEAR ELASTICITY: HENCKY MODEL 11 3 ELEMENT TECHNOLOGY 11 3 1 U/P FORMULATION 11 3 2 ENHANCED ASSUMED STRAIN ELEMENTS 11 3 3 F-BAR APPROACH 11 3 4 COROTATIONAL APPROACH 12 LARGE-STRAIN ELASTO-PLASTICITY 12 1 EULERIAN FORMULATIONS 12 2 MULTIPLICATIVE ELASTO-PLASTICITY 12 3 MULTIPLICATIVE ELASTO-PLASTICITY VERSUS RATE FORMULATIONS 12 4 INTEGRATION OF THE RATE EQUATIONS 12 5 EXPONENTIAL RETURN-MAPPING ALGORITHMS PART V ADVANCED DISCRETISATION CONCEPTS 13 INTERFACES AND DISCONTINUITIES 13 1 INTERFACE ELEMENTS 13 2 DISCONTINUOUS GALERKIN METHODS 14 MESHLESS AND PARTITION-OF-UNITY METHODS 14 1 MESHLESS METHODS 14 1 1 THE ELEMENT-FREE GALERKIN METHOD 14 1 2 APPLICATION TO FRACTURE 14 1 3 HIGHER-ORDER DAMAGE MECHANICS 14 1 4 VOLUMETRIC LOCKING 14 2 PARTITION-OF-UNITY APPROACHES 14 2 1 APPLICATION TO FRACTURE 14 2 2 EXTENSION TO LARGE DEFORMATIONS 14 2 3 DYNAMIC FRACTURE 14 2 4 WEAK DISCONTINUITIES 15 ISOGEOMETRIC FINITE ELEMENT ANALYSIS 15 1 BASIS FUNCTIONS IN COMPUTER AIDED GEOMETRIC DESIGN 15 1 1 UNIVARIATE B-SPLINES 15 1 2 UNIVARIATE NURBS 15 1 3 MULTIVARIATE B-SPLINES AND NURBS PATCHES 15 1 4 T-SPLINES 15 2 ISOGEOMETRIC FINITE ELEMENTS 15 2 1 B´EZIER ELEMENT REPRESENTATION 15 2 2 B´EZIER EXTRACTION 15 3 PYFEM: SHAPE FUNCTIONS FOR ISOGEOMETRIC ANALYSIS 15 4 ISOGEOMETRIC ANALYSIS IN NON-LINEAR SOLID MECHANICS 15 4 1 DESIGN-THROUGH-ANALYSIS OF SHELL STRUCTURES 15 4 2 HIGHER-ORDER DAMAGE MODELS 15 4 3 COHESIVE ZONE MODELS