Hyperelastic material

Abstract: non linear elastic deformations iwsun non linear elastic deformations erpd non linear elastic deformations hneun non-linear elastic deformations (dover civil and non-linear elastic deformations of multi-phase fluid systems non linear elastic deformations dover civil and mechanical ogden nonlinear elastic deformations pdf wordpress non-linear, elastic researchgate chapter 6 non linear material models international journal of nonlinear mechanics nonlinear elastic deformations ogden pdfslibforme international journal of non-linear mechanics 1 rubber elasticity: basic concepts and behavior non linear elastic deformations dover civil and mechanical on a non-linear wave equation in elasticity non linear elastic deformations (pdf) by r. w. ogden (ebook) exact formulations of non-linear planar and spatial euler introduction to nonlinear analysis mit opencourseware manual for the calculation of elastic-plastic materials non linear elastic axisymmetric deformation of membranes types of analysis: linear static, linear dynamic and non fracture mechanics, damage and fatigue non linear fracture chapter 2 linear elasticity freie universität the influence of non-linear elastic systems on the a simple geometric model for elastic deformations

Abstract: Growth and remodeling in tissues may be modulated by mechanical factors such as stress For example, in cardiac hypertrophy, alterations in wall stress arising from changes in mechanical loading lead to cardiac growth and remodeling A general continuum formulation for finite volumetric growth in soft elastic tissues is therefore proposed The shape change of an unloaded tissue during growth is described by a mapping analogous to the deformation gradient tensor This mapping is decomposed into a transformation of the local zero-stress reference state and an accompanying elastic deformation that ensures the compatibility of the total growth deformation Residual stress arises from this elastic deformation Hence, a complete kinematic formulation for growth in general requires a knowledge of the constitutive law for stress in the tissue Since growth may in turn be affected by stress in the tissue, a general form for the stress-dependent growth law is proposed as a relation between the symmetric growth-rate tensor and the stress tensor With a thick-walled hollow cylinder of incompressible, isotropic hyperelastic material as an example, the mechanics of left ventricular hypertrophy are investigated The results show that transmurally uniform pure circumferential growth, which may be similar to eccentric ventricular hypertrophy, changes the state of residual stress in the heart wall A model of axially loaded bone is used to test a simple stress-dependent growth law in which growth rate depends on the difference between the stress due to loading and a predetermined growth equilibrium stress

Abstract: 1 Overview of Solid Mechanics DEFINING A PROBLEM IN SOLID MECHANICS 2 Governing Equations MATHEMATICAL DESCRIPTION OF SHAPE CHANGES IN SOLIDS MATHEMATICAL DESCRIPTION OF INTERNAL FORCES IN SOLIDS EQUATIONS OF MOTION AND EQUILIBRIUM FOR DEFORMABLE SOLIDS WORK DONE BY STRESSES: PRINCIPLE OF VIRTUAL WORK 3 Constitutive Models: Relations between Stress and Strain GENERAL REQUIREMENTS FOR CONSTITUTIVE EQUATIONS LINEAR ELASTIC MATERIAL BEHAVIORSY HYPOELASTICITY: ELASTIC MATERIALS WITH A NONLINEAR STRESS-STRAIN RELATION UNDER SMALL DEFORMATION GENERALIZED HOOKE'S LAW: ELASTIC MATERIALS SUBJECTED TO SMALL STRETCHES BUT LARGE ROTATIONS HYPERELASTICITY: TIME-INDEPENDENT BEHAVIOR OF RUBBERS AND FOAMS SUBJECTED TO LARGE STRAINS LINEAR VISCOELASTIC MATERIALS: TIME-DEPENDENT BEHAVIOR OF POLYMERS AT SMALL STRAINS SMALL STRAIN, RATE-INDEPENDENT PLASTICITY: METALS LOADED BEYOND YIELD SMALL-STRAIN VISCOPLASTICITY: CREEP AND HIGH STRAIN RATE DEFORMATION OF CRYSTALLINE SOLIDS LARGE STRAIN, RATE-DEPENDENT PLASTICITY LARGE STRAIN VISCOELASTICITY CRITICAL STATE MODELS FOR SOILS CONSTITUTIVE MODELS FOR METAL SINGLE CRYSTALS CONSTITUTIVE MODELS FOR CONTACTING SURFACES AND INTERFACES IN SOLIDS 4 Solutions to Simple Boundary and Initial Value Problems AXIALLY AND SPHERICALLY SYMMETRIC SOLUTIONS TO QUASI-STATIC LINEAR ELASTIC PROBLEMS AXIALLY AND SPHERICALLY SYMMETRIC SOLUTIONS TO QUASI-STATIC ELASTIC-PLASTIC PROBLEMS SPHERICALLY SYMMETRIC SOLUTION TO QUASI-STATIC LARGE STRAIN ELASTICITY PROBLEMS SIMPLE DYNAMIC SOLUTIONS FOR LINEAR ELASTIC MATERIALS 5 Solutions for Linear Elastic Solids GENERAL PRINCIPLES AIRY FUNCTION SOLUTION TO PLANE STRESS AND STRAIN STATIC LINEAR ELASTIC PROBLEMS COMPLEX VARIABLE SOLUTION TO PLANE STRAIN STATIC LINEAR ELASTIC PROBLEMS SOLUTIONS TO 3D STATIC PROBLEMS IN LINEAR ELASTICITY SOLUTIONS TO GENERALIZED PLANE PROBLEMS FOR ANISOTROPIC LINEAR ELASTIC SOLIDS SOLUTIONS TO DYNAMIC PROBLEMS FOR ISOTROPIC LINEAR ELASTIC SOLIDS ENERGY METHODS FOR SOLVING STATIC LINEAR ELASTICITY PROBLEMS THE RECIPROCAL THEOREM AND APPLICATIONS ENERGETICS OF DISLOCATIONS IN ELASTIC SOLIDS RAYLEIGH-RITZ METHOD FOR ESTIMATING NATURAL FREQUENCY OF AN ELASTIC SOLID 6 Solutions for Plastic Solids SLIP-LINE FIELD THEORY BOUNDING THEOREMS IN PLASTICITY AND THEIR APPLICATIONS 7 Finite Element Analysis: An Introduction A GUIDE TO USING FINITE ELEMENT SOFTWARE A SIMPLE FINITE ELEMENT PROGRAM 8 Finite Element Analysis: Theory and Implementation GENERALIZED FEM FOR STATIC LINEAR ELASTICITY THE FEM FOR DYNAMIC LINEAR ELASTICITY FEM FOR NONLINEAR (HYPOELASTIC) MATERIALS FEM FOR LARGE DEFORMATIONS: HYPERELASTIC MATERIALS THE FEM FOR VISCOPLASTICITY ADVANCED ELEMENT FORMULATIONS: INCOMPATIBLE MODES, REDUCED INTEGRATION, AND HYBRID ELEMENTS LIST OF EXAMPLE FEA PROGRAMS AND INPUT FILES 9 Modeling Material Failure SUMMARY OF MECHANISMS OF FRACTURE AND FATIGUE UNDER STATIC AND CYCLIC LOADING STRESS- AND STRAIN-BASED FRACTURE AND FATIGUE CRITERIA MODELING FAILURE BY CRACK GROWTH: LINEAR ELASTIC FRACTURE MECHANICS ENERGY METHODS IN FRACTURE MECHANICS PLASTIC FRACTURE MECHANICS LINEAR ELASTIC FRACTURE MECHANICS OF INTERFACES 10 Solutions for Rods, Beams, Membranes, Plates, and Shells PRELIMINARIES: DYADIC NOTATION FOR VECTORS AND TENSORS MOTION AND DEFORMATION OF SLENDER RODS SIMPLIFIED VERSIONS OF THE GENERAL THEORY OF DEFORMABLE ROD EXACT SOLUTIONS TO SIMPLE PROBLEMS INVOLVING ELASTIC RODS MOTION AND DEFORMATION OF THIN SHELLS: GENERAL THEORY SIMPLIFIED VERSIONS OF GENERAL SHELL THEORY: FLAT PLATES AND MEMBRANES SOLUTIONS TO SIMPLE PROBLEMS INVOLVING MEMBRANES, PLATES, AND SHELLS Appendix A: Review of Vectors and Matrices A.1. VECTORS A.2. VECTOR FIELDS AND VECTOR CALCULUS A.3. MATRICES Appendix B: Introduction to Tensors and Their Properties B.1. BASIC PROPERTIES OF TENSORS B.2. OPERATIONS ON SECOND-ORDER TENSORS B.3. SPECIAL TENSORS Appendix C: Index Notation for Vector and Tensor Operations C.1. VECTOR AND TENSOR COMPONENTS C.2. CONVENTIONS AND SPECIAL SYMBOLS FOR INDEX NOTATION C.3. RULES OF INDEX NOTATION C.4. VECTOR OPERATIONS EXPRESSED USING INDEX NOTATION C.5. TENSOR OPERATIONS EXPRESSED USING INDEX NOTATION C.6. CALCULUS USING INDEX NOTATION C.7. EXAMPLES OF ALGEBRAIC MANIPULATIONS USING INDEX NOTATION Appendix D: Vectors and Tensor Operations in Polar Coordinates D.1. SPHERICAL-POLAR COORDINATES D.2. CYLINDRICAL-POLAR COORDINATES Appendix E: Miscellaneous Derivations E.1. RELATION BETWEEN THE AREAS OF THE FACES OF A TETRAHEDRON E.2. RELATION BETWEEN AREA ELEMENTS BEFORE AND AFTER DEFORMATION E.3. TIME DERIVATIVES OF INTEGRALS OVER VOLUMES WITHIN A DEFORMING SOLID E.4. TIME DERIVATIVES OF THE CURVATURE VECTOR FOR A DEFORMING ROD References

Abstract: Continuum elastoplastic damage models employing irreversible thermodynamics and internal state variables are developed within two alternative dual frameworks. In a strain [stress] -based formulation, damage is characterized through the effective stress [strain] concept together with the hypothesis of strain [stress] equivalence , and plastic flow is introduced by means of an additive split of the stress [strain] tensor . In a strain -based formulation we redefine the equivalent strain , usually defined as the J 2 -norm of the strain tensor, as the (undamaged) energy norm of the strain tensor. In a stress -based approach we employ the complementary energy norm of the stress tensor. These thermodynamically motivated definitions result, for ductile damage, in symmetric elastic-damage moduli. For brittle damage, a simple strain -based anisotropic characterization of damage is proposed that can predict crack development parallel to the axis of loading (splitting mode). The strain- and stress-based frameworks lead to dual but not equivalent formulations, neither physically nor computationally. A viscous regularization of strain-based, rate-independent damage models is also developed, with a structure analogous to viscoplasticity of the Perzyna type, which produces retardation of microcrack growth at higher strain rates. This regularization leads to well-posed initial value problems. Application is made to the cap model with an isotropic strain-based damage mechanism. Comparisons with experimental results and numerical simulations are undertaken in Part II of this work.

Abstract: Part One Basic concepts 1 Introduction 1.1 Aims and scope 1.2 Layout 1.3 General scheme of notation 2 ELEMENTS OF TENSOR ANALYSIS 2.1 Vectors 2.2 Second-order tensors 2.3 Higher-order tensors 2.4 Isotropic tensors 2.5 Differentiation 2.6 Linearisation of nonlinear problems 3 THERMODYNAMICS 3.1 Kinematics of deformation 3.2 Infinitesimal deformations 3.3 Forces. Stress Measures 3.4 Fundamental laws of thermodynamics 3.5 Constitutive theory 3.6 Weak equilibrium. The principle of virtual work 3.7 The quasi-static initial boundary value problem 4 The finite element method in quasi-static nonlinear solid mechanics 4.1 Displacement-based finite elements 4.2 Path-dependent materials. The incremental finite element procedure 4.3 Large strain formulation 4.4 Unstable equilibrium. The arc-length method 5 Overview of the program structure 5.1 Introduction 5.2 The main program 5.3 Data input and initialisation 5.4 The load incrementation loop. Overview 5.5 Material and element modularity 5.6 Elements. Implementation and management 5.7 Material models: implementation and management Part Two Small strains 6 The mathematical theory of plasticity 6.1 Phenomenological aspects 6.2 One-dimensional constitutive model 6.3 General elastoplastic constitutive model 6.4 Classical yield criteria 6.5 Plastic flow rules 6.6 Hardening laws 7 Finite elements in small-strain plasticity problems 7.1 Preliminary implementation aspects 7.2 General numerical integration algorithm for elastoplastic constitutive equations 7.3 Application: integration algorithm for the isotropically hardening von Mises model 7.4 The consistent tangent modulus 7.5 Numerical examples with the von Mises model 7.6 Further application: the von Mises model with nonlinear mixed hardening 8 Computations with other basic plasticity models 8.1 The Tresca model 8.2 The Mohr-Coulomb model 8.3 The Drucker-Prager model 8.4 Examples 9 Plane stress plasticity 9.1 The basic plane stress plasticity problem 9.2 Plane stress constraint at the Gauss point level 9.3 Plane stress constraint at the structural level 9.4 Plane stress-projected plasticity models 9.5 Numerical examples 9.6 Other stress-constrained states 10 Advanced plasticity models 10.1 A modified Cam-Clay model for soils 10.2 A capped Drucker-Prager model for geomaterials 10.3 Anisotropic plasticity: the Hill, Hoffman and Barlat-Lian models 11 Viscoplasticity 11.1 Viscoplasticity: phenomenological aspects 11.2 One-dimensional viscoplasticity model 11.3 A von Mises-based multidimensional model 11.4 General viscoplastic constitutive model 11.5 General numerical framework 11.6 Application: computational implementation of a von Mises-based model 11.7 Examples 12 Damage mechanics 12.1 Physical aspects of internal damage in solids 12.2 Continuum damage mechanics 12.3 Lemaitre's elastoplastic damage theory 12.4 A simplified version of Lemaitre's model 12.5 Gurson's void growth model 12.6 Further issues in damage modelling Part Three Large strains 13 Finite strain hyperelasticity 13.1 Hyperelasticity: basic concepts 13.2 Some particular models 13.3 Isotropic finite hyperelasticity in plane stress 13.4 Tangent moduli: the elasticity tensors 13.5 Application: Ogden material implementation 13.6 Numerical examples 13.7 Hyperelasticity with damage: the Mullins effect 14 Finite strain elastoplasticity 14.1 Finite strain elastoplasticity: a brief review 14.2 One-dimensional finite plasticity model 14.3 General hyperelastic-based multiplicative plasticity model 14.4 The general elastic predictor/return-mapping algorithm 14.5 The consistent spatial tangent modulus 14.6 Principal stress space-based implementation 14.7 Finite plasticity in plane stress 14.8 Finite viscoplasticity 14.9 Examples 14.10 Rate forms: hypoelastic-based plasticity models 14.11 Finite plasticity with kinematic hardening 15 Finite elements for large-strain incompressibility 15.1 The F-bar methodology 15.2 Enhanced assumed strain methods 15.3 Mixed u/p formulations 16 Anisotropic finite plasticity: Single crystals 16.1 Physical aspects 16.2 Plastic slip and the Schmid resolved shear stress 16.3 Single crystal simulation: a brief review 16.4 A general continuum model of single crystals 16.5 A general integration algorithm 16.6 An algorithm for a planar double-slip model 16.7 The consistent spatial tangent modulus 16.8 Numerical examples 16.9 Viscoplastic single crystals Appendices A Isotropic functions of a symmetric tensor A.1 Isotropic scalar-valued functions A.1.1 Representation A.1.2 The derivative of anisotropic scalar function A.2 Isotropic tensor-valued functions A.2.1 Representation A.2.2 The derivative of anisotropic tensor function A.3 The two-dimensional case A.3.1 Tensor function derivative A.3.2 Plane strain and axisymmetric problems A.4 The three-dimensional case A.4.1 Function computation A.4.2 Computation of the function derivative A.5 A particular class of isotropic tensor functions A.5.1 Two dimensions A.5.2 Three dimensions A.6 Alternative procedures B The tensor exponential B.1 The tensor exponential function B.1.1 Some properties of the tensor exponential function B.1.2 Computation of the tensor exponential function B.2 The tensor exponential derivative B.2.1 Computer implementation B.3 Exponential map integrators B.3.1 The generalised exponential map midpoint rule C Linearisation of the virtual work C.1 Infinitesimal deformations C.2 Finite strains and deformations C.2.1 Material description C.2.2 Spatial description D Array notation for computations with tensors D.1 Second-order tensors D.2 Fourth-order tensors D.2.1 Operations with non-symmetric tensors References Index