MODELING ANISOTROPIC MATERIAL BEHAVIOR UNDER ELASTIC AND PLASTIC DEFORMATIONS
CONTENTS 1 INTRODUCTION 2 FUNDAMENTALS OF CONTINUUM MECHANICS 2 1 FINITE KINEMATICS 2 1 1 REPRESENTATION, TRANSFORMATION AND ROTATION OF TENSORIAL OBJECTS 2 1 2 MOTION OF A BODY 2 2 NOTION OF STRESSES AND HEAT FLUX 2 2 1 STRESSES 2 2 2 HEAT FLUX 2 3 BALANCE PRINCIPLES 2 3 1 BALANCE OF MASS 2 3 2 BALANCE OF LINEAR MOMENTUM 2 3 3 BALANCE OF ANGULAR MOMENTUM 2 3 4 BALANCE OF TOTAL ENERGY 2 3 5 BALANCE OF ENTROPY AND SECOND LAW OF THERMODYNAMICS 2 4 CONSTITUTIVE EQUATIONS 2 4 1 PRINCIPLE OF MATERIAL OBJECTIVITY 2 4 2 MATERIAL SYMMETRY 3 MATERIAL SYMMETRIES — CLASSI?CATION OF SOLIDS 3 1 CONSTRUCTION OF A SPACE LATTICE 3 2 SYMMETRY TRANSFORMATIONS 3 2 1 ROTATIONS 3 2 2 ROTATION-INVERSIONS 3 2 3 TENSOR REPRESENTATIONS 3 2 4 SYMMETRY GROUPS 3 3 THE 14 BRAVAIS LATTICES AND 7 CRYSTAL SYSTEMS 3 3 1 TRICLINIC SYMMETRY 3 3 2 MONOCLINIC SYMMETRY 3 3 3 ORTHORHOMBIC SYMMETRY 3 3 4 TETRAGONAL SYMMETRY 3 3 5 CUBIC SYMMETRY 3 3 6 TRIGONAL AND HEXAGONAL SYMMETRY 3 3 7 SUMMARY 3 4 THE 32 CRYSTAL CLASSES 3 4 1 THE MOTIF — INNER SYMMETRIES 3 4 2 NOTATION OF SYMMETRY 3 5 ICOSAHEDRAL, CYLINDRICAL AND SPHERICAL SYMMETRY CLASSES 3 6 CLASSI?CATION INTO 14 TYPES OF ANISOTROPY 4 REPRESENTATIONS OF ANISOTROPIC TENSOR FUNCTIONS 4 1 DE?NITIONS AND NOTIONS 4 2 ISOTROPIC EXTENSION OF ANISOTROPIC TENSOR FUNCTIONS 4 3 ISOTROPIC FUNCTIONS OF FIRST- AND SECOND-ORDER TENSORS 4 3 1 WANG’S APPROACH 4 3 2 SMITH’S APPROACH 4 3 3 COMPARISON OF THE RESULTS OBTAINED BY WANG AND SMITH 4 4 ISOTROPIC POLYNOMIALS OF FIRST- AND SECOND-ORDER TENSORS 4 4 1 DE?NITIONS 4 4 2 INTEGRITY BASIS FOR SETS OF FIRST-ORDER TENSORS 4 4 3 ISOTROPIC TENSORS 4 4 4 INTEGRITY BASES FOR SETS OF FIRST-ORDER AND SECOND-ORDER TENSORS 4 5 IRREDUCIBILITY OF INTEGRITY BASES AND FUNCTIONAL BASES 4 6 QUADRATIC FUNCTIONS OF A SYMMETRIC SECOND-ORDER TENSOR 4 6 1 TRICLINIC FUNCTIONS — SYMMETRY GROUP CI 4 6 2 MONOCLINIC FUNCTIONS — SYMMETRY GROUP C2H 4 6 3 ORTHORHOMBIC FUNCTIONS — SYMMETRY GROUP D2H 4 6 4 TETRAGONAL FUNCTIONS — SYMMETRY GROUP C4H 4 6 5 TETRAGONAL FUNCTIONS — SYMMETRY GROUP D4H 4 6 6 TRIGONAL FUNCTIONS — SYMMETRY GROUP S6 4 6 7 TRIGONAL FUNCTIONS — SYMMETRY GROUP D3D 4 6 8 HEXAGONAL FUNCTIONS — SYMMETRY GROUP C6H 4 6 9 HEXAGONAL FUNCTIONS — SYMMETRY GROUP D6H 4 6 10 CUBIC FUNCTIONS — SYMMETRY GROUP OH 4 6 11 CUBIC FUNCTIONS — SYMMETRY GROUP TH 4 6 12 TRANSVERSELY ISOTROPIC FUNCTIONS — SYMMETRY GROUP C?H 4 6 13 TRANSVERSELY ISOTROPIC FUNCTIONS — SYMMETRY GROUP D?H 4 6 14 CONCLUSION 5 ANISOTROPIC ELASTICITY 5 1 GENERAL FRAMEWORK 5 2 MODEL PROBLEM 5 2 1 FIBER-REINFORCED TECHNICAL RUBBER 5 3 NUMERICAL EXAMPLE: TENSION TEST OF A FIBER-REINFORCED BAR 5 4 NUMERICAL EXAMPLE: IN?ATION OF A FIBER-REINFORCED SHEET 6 APPROACHES TO ANISOTROPIC PLASTICITY 6 1 KINEMATIC APPROACH IN TERMS OF A PLASTIC MAP 6 1 1 GEOMETRY OF MULTIPLICATIVE PLASTICITY, STRESS TENSORS 6 2 CONSTITUTIVE MODEL FOR PLASTIC-MAP PLASTICITY 6 2 1 ENERGY STORAGE AND ELASTIC STRESS RESPONSE 6 2 2 DISSIPATION AND PLASTIC FLOW RESPONSE 6 2 3 DECOUPLING OF THE CONSTITUTIVE FUNCTIONS 6 2 4 CONTINUOUS TANGENT MODULI 6 3 ALGORITHMIC FORMULATION 6 3 1 OUTLINE OF THE STANDARD STRESS UPDATE ALGORITHMS 6 3 2 IMPLICIT STRESS UPDATE ALGORITHM (U1) 6 3 3 EXPLICIT STRESS UPDATE ALGORITHM (U2) 6 3 4 ALGORITHMIC TANGENT MODULI 6 4 VARIATIONAL FORMULATION 6 4 1 STANDARD FORMULATION OF RATE-INDEPENDENT PLASTICITY 6 4 2 INCREMENTAL VARIATIONAL FORMULATION 6 4 3 SPECI?CATION TO MULTI-SURFACE PLASTICITY 6 4 4 IMPLICIT DISCRETE VARIATIONAL FORMULATION (V1) 6 4 5 ALGORITHMIC SOLUTION OF THE DISCRETE VARIATIONAL FORMULATION (V1) 6 4 6 STRESSES AND ALGORITHMIC TANGENT MODULI (V1) 6 4 7 APPLICATION OF THE ALGORITHM (V1) TO THE MODEL PROBLEM 6 4 8 EXPLICIT DISCRETE VARIATIONAL FORMULATION (V2) 6 4 9 ALGORITHMIC SOLUTION OF THE DISCRETE VARIATIONAL FORMULATION (V2) 6 4 10 STRESSES AND ALGORITHMIC TANGENT MODULI (V2) 6 4 11 APPLICATION TO MODEL PROBLEM 6 5 MODEL PROBLEM: DOUBLE SLIP PLASTICITY 6 5 1 NUMERICAL EXAMPLE: ROTATION OF THE SLIP SYSTEMS 6 5 2 NUMERICAL EXAMPLE: DRAWING OF A FLANGE 6 6 MODEL PROBLEM: PLASTICITY BASED ON QUADRATIC FUNCTIONS 6 6 1 ELASTIC RESPONSE 6 6 2 PLASTIC RESPONSE 6 6 3 D2H-, OH-, D?H AND O(3)-INVARIANT FOURTH-ORDER TENSORS 6 6 4 COMPARISON OF THE STRESS UPDATE ALGORITHMS 6 6 5 KELVIN-MODE DECOMPOSITION OF FOURTH-ORDER TENSORS 6 7 CONSTITUTIVE MODEL FOR PLASTIC-METRIC PLASTICITY 6 7 1 ENERGY STORAGE AND ELASTIC STRESS RESPONSE 6 7 2 DISSIPATION AND PLASTIC FLOW RESPONSE 6 7 3 DECOUPLING OF THE CONSTITUTIVE FUNCTIONS 6 7 4 CONTINUOUS TANGENT MODULI 6 8 ALGORITHMIC FORMULATION 6 8 1 IMPLICIT STRESS UPDATE ALGORITHM 6 8 2 ALGORITHMIC TANGENT MODULI 7 1 KINEMATIC APPROACH IN TERMS OF A PLASTIC METRIC 7 1 1 CURRENT METRIC, PLASTIC METRIC AND STRESSES 7 2 CONSTITUTIVE MODEL IN THE LOGARITHMIC STRAIN SPACE 7 2 1 ENERGY STORAGE AND ELASTIC STRESS RESPONSE 7 2 2 DISSIPATION AND PLASTIC FLOW RESPONSE 7 2 3 CONSIDERED MODEL PROBLEM 7 2 4 CONTINUOUS ELASTIC-PLASTIC TANGENT MODULI 7 3 ALGORITHMIC FORMULATION 7 3 1 STRESS UPDATE ALGORITHM 7 3 2 ALGORITHMIC TANGENT MODULI 7 4 VARIATIONAL FORMULATION IN THE LOGARITHMIC STRAIN SPACE 7 4 1 STANDARD FORMULATION OF INELASTICITY 7 4 2 INCREMENTAL VARIATIONAL FORMULATION 7 4 3 SPECI?CATION TO MULTI-SURFACE MODELS OF ELASTO-PLASTICITY 7 4 4 ALGORITHMIC SOLUTION 7 4 5 STRESSES AND MODULI 7 4 6 APPLICATION TO MODEL PROBLEM 8 1 PARAMETERIZATION OF THE SHELL-LIKE CONTINUUM 8 2 FINITE ELEMENT APPROXIMATION 8 2 1 COMPATIBLE DISPLACEMENT APPROACH 8 2 2 ASSUMED STRAIN MODI?CATIONS 8 2 3 ENHANCED STRAIN MODI?CATIONS 8 3 VARIATIONAL FORMULATION 8 4 GRADIENT-TYPE INTERFACE TO CONSTITUTIVE MODELS 9 1 NECKING OF AN ISOTROPIC ROD 9 2 NECKING OF AN ISOTROPIC RECTANGULAR STRIP 9 3 DRAWING OF A CIRCULAR FLANGE 9 3 1 COMPARISON OF ADDITIVE AND MULTIPLICATIVE PLASTICITY 9 3 2 COMPARISON OF THE STRESS UPDATE ALGORITHMS 9 3 3 COMPARISON OF ADDITIVE AND MULTIPLICATIVE SHELL ELEMENT DESIGN 9 4 DEEP DRAWING OF CUBIC AND ORTHOTROPIC SHEETS 9 4 1 COMPARISON OF ADDITIVE AND MULTIPLICATIVE PLASTICITY 9 4 2 COMPARISON OF THE STRESS UPDATE ALGORITHMS 9 4 3 COMPARISON OF MULTIPLICATIVE AND ADDITIVE SHELL ELEMENT DESIGN