Foundations of Elastoplasticity: Subloading Surface Model

Description

This book is the standard text book for elastoplasticity/viscoplasticity which is explained comprehensively covering the rate-independent to -dependent finite deformations of metals, soils, polymers, crystal plasticity, etc. and the friction phenomenon. Concise explanations on vector-tensor analysis and continuum mechanics are provided first, covering the underlying physical concepts, e.g. various time-derivatives, pull-back and push-forward operations, work-conjugacy and multiplicative decomposition of deformation gradient tensor. Then, the rigorous elastoplastic/viscoplastic model, called the subloading surface model, is explained comprehensively, which is based on the subloading surface concept to describe the continuous development of the plastic/viscoplastic strain rate as the stress approaches to the yield surface, while it can never be described by the other plasticity models, e.g. the Chaboche-Ohno and the Dafalias-Yoshida models assuming the purely-elastic domain. The main features of the subloading surface model are as follows: 1)  The subloading surface concept underling the cyclic plasticity is introduced, which insists that the plastic deformation develops as the stress approaches the yield surface. Thus, the smooth elastic-plastic transition leading to the continuous variation of the tangent stiffness modulus is described always. 2) The subloading-overstress model is formulated by which the elastoplastic deformation during the quasi-static loading and the viscoplastic deformation during the dynamic and impact loading can be described by the unified equation. Then, only this model can be used to describe the deformation in the general rate of deformation, disusing the elastoplastic constitutive equation. 3) The hyperelastic-based (visco)plasticity based on the multiplicative decomposition of deformation gradient tensor and the subloading surface model is formulated for the exact descriptions of the finite elastic and (visco)plastic deformations. 4) The subloading-friction model is formulated for the exact description of the dry and the fluid (lubricated) frictions at the general rate of sliding from the static to the impact sliding. Thus, all the elastic and inelastic deformation/sliding phenomena of solids can be described accurately in the unified equation by the subloading-overstress model. The subloading surface model will be engraved as the governing law of irreversible deformation of solids in the history of solid mechanics.   Preface Contents 1 Mathematical Preliminaries: Vector and Tensor Analysis 1.1 Conventions and Symbols 1.1.1 Summation Convention 1.1.2 Kronecker’s Delta and Permutation Symbol 1.1.3 Matrix and Determinant 1.2 Vector 1.2.1 Definition of Vector 1.2.2 Operations of Vectors 1.2.3 Coordinate Transformation of Vector 1.3 Tensor 1.3.1 Definition of Tensor 1.3.2 Quotient Law 1.3.3 Notations of Tensors 1.3.4 Orthogonal Tensor 1.4 Operations of Tensors 1.4.1 Notations in Tensor Operations 1.4.2 Trace 1.4.3 Various Tensors 1.5 Eigenvalues and Eigenvectors 1.6 Calculations of Eigenvalues and Eigenvectors 1.6.1 Eigenvalues 1.6.2 Eigenvectors 1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor 1.8 Cayley-Hamilton Theorem 1.9 Scalar Triple Products with Invariants 1.10 Positive Definite Tensor 1.11 Polar Decomposition 1.12 Isotropic Tensor-Valued Tensor Function 1.13 Representation of Tensor in Principal Space 1.14 Two-Dimensional State 1.15 Tensor Functions 1.16 Partial Differential Calculi 1.17 Differentiation and Integration in Tensor Field 1.18 Representation in General Coordinate System 1.18.1 Primary and Reciprocal Base Vectors 1.18.2 Metric Tensor and Base Vector Algebra 1.18.3 Tensor Representations 2 Description of Motion 2.1 Motion of Material Point 2.2 Time-Derivatives 2.3 Variations and Rates of Geometrical Elements 2.3.1 Deformation Gradient and Variations of Line, Surface and Volume Elements 2.3.2 Velocity Gradient and Rates of Line, Surface and Volume Elements 2.4 Material-Time Derivative of Volume Integration 3 Description of Tensor (Rate) in Convected Coordinate System 3.1 Reference and Current Primary and Reciprocal Base Vectors 3.2 Description of Deformation Gradient Tensor by Embedded Base Vectors 3.3 Pull-Back and Push-Forward Operations 3.4 Convected Time-Derivative 3.4.1 General Convected Derivative 3.4.2 Corotational Rate 3.4.3 On Adoption of Convected Rate Tensor in Hypoelastic Constitutive Equation 3.4.4 Time-Integration of Convected Rate Tensor 4 Deformation/Rotation Tensors 4.1 Deformation Tensors 4.2 Strain Tensors 4.3 Volumetric and Isochoric Parts of Deformation Gradient Tensor 4.4 Strain Rate and Spin Tensors 4.5 Logarithmic (True) and Infinitesimal (Nominal) Strains 5 Stress Tensors and Conservation Laws 5.1 Stress Tensor 5.2 Conservation Law of Mass 5.3 Conservation Law of Linear Momentum 5.4 Conservation Law of Angular Momentum 5.5 Equilibrium Equation 5.6 Equilibrium Equation of Angular Moment 5.7 Virtual Work Principle 5.8 Conservation Law of Energy 5.9 Work Conjugacy 5.10 Various Simple Deformations 5.10.1 Uniaxial Loading 5.10.2 Simple Shear 5.10.3 Combination of Tension and Distortion 6 Objectivity and Objective (Rate) Tensors 6.1 Objectivity 6.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities 6.3 Material-Time Derivative of Tensor 6.4 Objectivity of Convective Time-Derivative and Corotational Rate 6.5 Various Objective Stress Rate Tensors 6.6 Jaumann Rate with Plastic Spin 6.7 Time-Derivative of Scalar-Valued Tensor Function 7 Elastic Constitutive Equations 7.1 Definition of Hyperelasticity 7.2 Hyperelastic Equations 7.3 Explicit Hyperelastic Models 7.3.1 St.Venant-Kirchhoff Model 7.3.2 Neo-Hookean Model 7.3.3 Mooney Model 7.3.4 Ogden Model 7.4 Rate Forms of Hyperelastic Equation 7.5 Infinitesimal Strain-Based Elastic Equation 7.6 Cauchy Elasticity 7.7 Hypoelasticity 8 Elastoplastic Constitutive Equations 8.1 Fundamental Requirements for Elastoplastic Constitutive Equations 8.2 Classification of Elastoplastic Constitutive Equations 8.2.1 Infinitesimal Hyperelastic-Based Plasticity 8.2.2 Hypoelastic-Based Plasticity 8.2.3 Multiplicative Hyperelastic-Based Plasticity 8.3 Conventional Plastic Constitutive Equation 8.4 Constitutive Equation of Metals 8.5 Formulation of General Loading Criterion 8.6 Physical Backgrounds of Associated Flow Rule 8.6.1 Positiveness of Second-Order Plastic Work Rate: Prager’s Interpretation 8.6.2 Principle of Maximum Plastic Work 8.6.3 Positiveness of Work Done During Stress Cycle: Drucker’s Interpretation 8.6.4 Positiveness of Second-Order Plastic Relaxation Work Rate 8.6.5 Comparison of Interpretations for Associated Flow Rule 8.7 Anisotropy 8.7.1 Definition of Isotropy 8.7.2 Elastoplastic Constitutive Equation with Kinematic Hardening 8.7.3 Kinematic Hardening Rules 8.8 Plastic Spin 8.9 Physical Interpretation of Nonlinear Kinematic Hardening Rule 8.10 Limitations of Conventional Elastoplasticity 9 Unconventional Elastoplasticity Model: Subloading Surface Model 9.1 Mechanical Requirements 9.1.1 Continuity Condition in the Small 9.1.2 Continuity Condition in the Large: Smoothness Condition 9.2 Subloading Surface (Hashiguchi) Model 9.3 Distinguished Advantages of Subloading Surface Model 9.4 Numerical Performance of Subloading Surface Model 9.5 On Bounding Surface Model with Radial-Mapping: Misuse of Subloading Surface Concept 9.6 Incorporation of Kinematic Hardening 9.7 Incorporation of Tangential-Inelastic Strain Rate 9.8 Limitation of Initial Subloading Surface Model 10 Classification of Plasticity Models: Critical Reviews and Assessments 10.1 Cyclic Loading Behavior 10.2 Classification and Assessment of Plasticity Models 10.3 Plasticity Models with Elastic Domain 10.3.1 Common Drawbacks in Models with Elastic-Domain 10.3.2 Cylindrical Yield Surface (Chaboche) Model: Ad Hoc. Primitive Conventional Model Limited to Simple Metal Behavior 10.3.3 Multi-surface (Mroz) Model: Incapable of Describing Mechanical Ratchetting 10.3.4 Two Surface (Dafalias) Model: Incapable of Describing Plastic Strain Rate in Unloading Process 10.4 Extended Subloading Surface (Hashiguchi) Model: Capable of Describing General Loading Behavior 10.5 Overall Assessment of Plasticity Models 11 Extended Subloading Surface Model 11.1 Normal-Yield and Subloading Surfaces 11.2 Evolution Rule of Elastic-Core 11.3 Plastic Strain Rate 11.4 Stain Rate Versus Stress Rate Relations 11.5 Calculation of Normal-Yield Ratio in Unloading Process 11.6 Improvement of Inverse and Reloading Responses 11.7 Loading Criterion for Large Loading Increment 11.7.1 Exact Judgment of Loading 11.7.2 Initial Value of Normal-Yield Ratio in Plastic Corrector Step 11.8 Plastic Spin 11.9 Incorporation of Tangential-Inelastic Strain Rate 12 Constitutive Equations of Metals 12.1 Yield Surface, Isotropic, Kinematic Hardening and Elastic-Core 12.2 Cyclic Stagnation of Isotropic Hardening 12.3 Calculation of Normal-Yield Ratio in Unloading Process 12.4 Implicit Stress-Integration 12.5 Material Parameters and Comparisons with Test Data 12.5.1 Material Parameters 12.5.2 Comparisons with Test Data 12.6 Analyses of Engineering Phenomena 12.7 Orthotropic Anisotropy 12.7.1 Representation of Isotropic Mises Yield Condition 12.7.1.1 Plane Stress State 12.7.2 Plane Strain State 12.8 Subloading Surface Model with Orthotropic Anisotropy 12.8.1 Subloading Surface with Orthotropic Anisotropy 12.8.2 Plastic Strain Rate 12.8.3 Normal-Yield Ratio 12.8.4 Elastic-Core Yield Ratio 12.8.5 Cyclic Stagnation of Isotropic Hardening 13 Constitutive Equations of Soils 13.1 Isotropic Consolidation Characteristics 13.2 Yield Conditions 13.2.1 Yield Functions 13.2.2 Critical State Surface Taken Account of Third Deviatoric Invariant 13.3 Subloading Surface Model for Soils 13.4 Extension of Material Functions 13.4.1 Yield Surface with Tensile Strength 13.4.2 Rotational Hardening 13.5 Extended Subloading Surface Model 13.5.1 Superyield, Normal-Yield and Subloading Surfaces 13.5.2 Evolution Rules of Internal Variables 13.5.3 Plastic Strain Rate 13.5.4 Yield Stress Function 13.5.5 Partial Derivatives of Subloading Surface Function 13.5.6 Calculation of Normal-Yield Ratio 13.6 Simulations of Test Results 13.7 Numerical Analysis of Footing Settlement Problem 13.8 Hyperelastic Equation of Soils 14 Viscoplastic Constitutive Equations with Subloading Surface Concept 14.1 Rate-Dependent Deformation of Solids 14.2 History of Viscoplastic Constitutive Equations 14.3 Irrationality of Creep Model 14.4 Mechanical Response of Past Overstress Model 14.5 Subloading Overstress Model: Extension to Description of General Rate of Deformation 14.5.1 Static and Limit Subloading Surfaces 14.5.2 Viscoplastic Strain Rate 14.5.3 Strain Rate Versus Stress Rate Relation 14.6 Comparison with Test Data 14.6.1 Dynamic Loading Process Inducing Elastic-Viscoplastic Deformation 14.6.2 Quasi-static Loading Process Inducing Elastoplastic Deformation Behaviors 14.7 Temperature Dependence of Elasto-Viscoplastic Deformation Behavior 15 Continuum Damage Model with Subloading Surface Concept 15.1 Basic Hypothesis of Strain Equivalence in Constitutive Equation with Brittle Damage 15.2 Hyperelastic Equation in Undamaged Variables 15.3 Hyperelastic Equations with Damage 15.3.1 Bilateral Damage 15.3.2 Unilateral Damage 15.4 Evolution of Damage Variable 15.4.1 Bilateral Damage 15.4.2 Unilateral Damage 15.5 Elastoplastic-Damage Model with Subloading Surface Model 15.5.1 Plastic Strain Rate 15.5.2 Stress(Rate) Versus Strain(Rate) Relation and Stress Integration 15.5.2.1 Bilateral Damage 15.5.2.2 Unilateral Damage 15.6 Anisotropic (Orthotropic) Damage Tensor 15.7 Subloading-Overstress Damage Model 15.7.1 Bilateral Damage 15.7.2 Unilateral Damage 15.8 Subloading-Gurson Model for Ductile Damage 15.9 High Cycle Fatigue: Redundancy of Two-Scale Damage Model and Necessity of Subloading-Damage Surface Model 16 Subloading Phase-Transformation Model 16.1 Constitutive Equation 16.1.1 Elastic Strain Increment 16.1.2 Plastic Strain Increment Based on Subloading Surface Model 16.2 Thermal and Transformation Strain Increments 16.2.1 Heat-Transformation Strain Increment 16.2.2 Transformation-Plastic Strain Increment 16.3 Stress Rate Versus Strain Rate Relation 17 Multiplicative Hyperelastic-Based Plasticity with Subloading Surface Concept 17.1 Exact Elastic–Plastic Decomposition of Deformation Measure 17.1.1 Necessity of Multiplicative Decomposition of Deformation Gradient Tensor 17.1.2 Embedded Base Vectors in Intermediate Configuration 17.2 Deformation Tensors 17.2.1 Elastic and Plastic Right Cauchy-Green Deformation Tensor 17.2.2 Strain Rate and Spin Tensors 17.3 On Limitation of Hypoelastic-Based Plasticity 17.4 Further Multiplicative Decomposition of Plastic Deformation Gradient Tensor 17.5 Formulation and Calculation in Intermediate Configuration: Isoclinic Concept 17.6 Stress Measures 17.7 Internal Variables 17.8 Normal-Yield, Subloading and Elastic-Core Surfaces 17.9 Plastic Flow Rules 17.10 Plastic Strain Rate 17.11 Material Functions for Metals and Soils 17.11.1 Metals 17.11.2 Soils 17.12 Calculation Procedures 17.13 Isotropic Hardening Stagnation 17.14 Subloading-Overstress Model 17.14.1 Constitutive Equation 17.14.2 Calculation Procedure 18 Viscoelastic-Viscoplastic Model of Polymers 18.1 Viscoelastic Rheological Model 18.2 Viscoelastic Deformation with Elastic Strain Energy Function 18.2.1 Elastic Strain Free-Energy Function 18.2.2 Second Piola–Kirchhoff Stress Tensor 18.3 Viscoelastic-Damage Model: Subloading-Mullins Effect 18.4 Viscoplastic Constitutive Equation in Glassy State 19 Corotational Rate Tensors 19.1 Hypoelasticity 19.1.1 Zaremba-Jaumann Rate 19.1.2 Green-Naghdi Rate 19.2 Kinematic Hardening Material 19.2.1 Zaremba-Jaumann Rate 19.2.2 Green-Naghdi Rate 19.3 Plastic Spin 20 Localization of Deformation 20.1 Element Test 20.2 Gradient Theory 20.3 Shear-Band Embedded Model: Smeared Crack Model 20.4 Necessary Condition for Shear Band Inception 21 Hypoelastic- and Multiplicative Hyperelastic-Based Crystal Plasticity 21.1 Description of Strain Rate and Spin by Crystal Lattice Vectors 21.2 Resolved Shear Stress (Rate) 21.3 Stress Rate Versus Plastic Shear Strain Rate Relation 21.4 Conventional Crystal Plasticity Model 21.4.1 Yield Condition and Flow Rule 21.4.2 Evolution of Isotropic Hardening 21.4.3 Evolution of Kinematic Hardening 21.4.4 Stress Rate Versus Strain Rate Relation 21.5 Subloading Crystal Plasticity Model 21.6 Subloading-Overstress Crystal Plasticity Model 21.7 Extension to Description of Cyclic Loading Behavior 21.8 Uniqueness of Slip Rate Mode 21.9 Various Schemes for Calculation of Shear Strain Rates 21.9.1 Singular Value Decomposition 21.9.2 Regularized Schmid Law 21.9.3 On Creep-Type Crystal Plasticity Model 22 Constitutive Equation for Friction: Subloading-Friction Model 22.1 History of Constitutive Equation for Friction 22.2 Sliding Displacement and Contact Traction 22.3 Hyperelastic Sliding Behavior 22.4 Elastoplastic Sliding Velocity 22.4.1 Sliding Normal-Yield and Subloading Surfaces 22.4.2 Evolution Rule of Sliding Hardening Function 22.4.3 Evolution Rule of Sliding Normal-Yield Ratio 22.4.4 Elastoplastic Sliding Velocity 22.5 Loading Criterion 22.6 Calculation of Normal Sliding-Yield Ratio 22.7 Fundamental Mechanical Behavior of Subloading-Friction Model 22.7.1 Relation of Tangential Contact Stress Rate and Sliding Velocity 22.7.2 Numerical Experiments and Comparisons with Test Data 22.8 Stick–Slip Phenomenon 22.9 Friction Condition with Saturation of Tangential Contact Stress 22.10 Subloading-Overstress Friction Model 22.10.1 Subloading-Overstress (Viscoelastic) Friction Model 22.10.2 Numerical Experiments 22.10.3 Comparison with Test Data 22.11 Extension to Rotational and Orthotropic Anisotropy Final Remarks Appendix A: Projection of Area Appendix B: Logarithmic Spin Appendix C: Matrix Representation of Tensor Relations Appendix D: Euler’s Theorem for Homogeneous Function Appendix E: Outward-Normal Tensor of Surface Appendix F: Relationships of Material Constants in \ln v - \ln p and e - \ln p Linear Relations Appendix G: Derivative in Critical State Appendix H: Convexity of Two-Dimensional Curve Appendix I: Normal Tensor to Subloading Surface with Anisotropic Damage Appendix J: Tensor Exponential Map for Time-Integration of First-Order Linear Differential Equation Appendix K: Eyring Equation Appendix L: Computer Programs of Subloading Surface Models References Books on Solid Mechanics and Tensor Analysis Research Articles Index