Foundations of Elastoplasticity: Subloading Surface Model

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