Elastoplasticity Theory (eBook)

Title Page 2
Preface 6
Contents 11
Tensor Analysis 16
Conventions and Symbols 16
Vector 23
Tensor 29
Operations of Second-Order Tensor 35
Eigenvalues and Eigenvectors 41
Calculations of Eigenvalues and Eigenvectors 46
Eigenvalues and Eigenvectors of Skew-Symmetric Tensor 48
Cayley-Hamilton’s Theorem 50
Positive Definite Tensor 50
Polar Decomposition 51
Isotropic Tensor-Valued Tensor Function 52
Representation of Tensor in Principal Space 55
Two-Dimensional State 59
Partial Differential Calculi 62
Time Derivatives 65
Differentiation and Integration in Field 66
Motion and Strain (Rate) 72
Motion and Deformation 72
Strain Tensor 79
Strain Rate and Spin Tensors 85
Various Simple Deformations 96
Surface Element, Volume Element and Their Rates 112
Conservation Laws and Stress Tensors 115
Conservation Law of Mass 115
Conservation Law of Momentum 115
Conservation Law of Angular Momentum 116
Stress Tensor 116
Equilibrium Equation 119
Equilibrium Equation of Moment 121
Virtual Work Principle 122
Objectivity and Corotational Rate Tensor 124
Objectivity 124
Influence of Rigid-Body Rotation on Various Mechanical Quantities 125
Rate of State Variable and Corotational Rate Tensor 127
Transformation of Material-Time Derivative of Scalar Function to Its Corotational Derivative 132
Various Objective Stress Rate Tensors 135
Work Conjugacy 137
Elastic Constitutive Equations 139
Hyperelasticity 139
Cauchy Elasticity 142
Hypoelasticity 143
Basic Formulations for Elastoplastic Constitutive Equations 146
Multiplicative Decomposition of Deformation Gradient and Additive Decomposition of Strain Rate 146
Conventional Elastoplastic Constitutive Equations 153
Loading Criterion 159
Associated Flow Rule 162
Anisotropy 167
Incorporation of Tangential-Inelastic Strain Rate 170
Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory 176
Unconventional Elastoplasticity Model: Subloading Surface Model 182
Mechanical Requirements 182
Subloading Surface Model 185
Salient Features of Subloading Surface Model 192
On Bounding Surface and Bounding Surface Model 195
Incorporation of Anisotropy 197
Incorporation of Tangential Inelastic Strain Rate 198
Cyclic Plasticity Model: Extended Subloading Surface Model 201
Classification of Cyclic Plasticity Models 201
Translation of Subyield Surface(s): Extension of Kinematic Hardening 201
Extended Subloading Surface Model 206
Modification of Reloading Curve 215
Incorporation of Tangential-Inelastic Strain Rate 218
Viscoplastic Constitutive Equations 220
History of Viscoplastic Constitutive Equations 220
Mechanical Response of Ordinary Overstress Model 223
Modification of Overstress Model: Extension to General Rate of Deformation 224
Incorporation of Subloading Surface Concept: Subloading Overstress Model 226
Constitutive Equations of Metals 230
Isotropic and Kinematic Hardening 230
Cyclic Stagnation of Isotropic Hardening 234
On Calculation of the Normal-Yield Ratio 241
Comparisons of Test Results 241
Orthotropic Anisotropy 247
Representation of Isotropic Mises Yield Condition 253
Plane Stress State 254
{/it Plane Strain State} 257
Constitutive Equations of Soils 258
Isotropic Consolidation Characteristics 258
Yield Conditions 262
Isotropic Hardening Function 268
Rotational Hardening 270
Extended Subloading Surface Model 274
Partial Derivatives of Subloading Surface Function 276
Calculation of Normal-Yield Ratio 280
Simulations of Test Results 284
Simple Subloading Surface Model 290
Super-Yield Surface for Structured Soils in Natural Deposits 300
Numerical Analysis of Footing Settlement Problem 310
Corotational Rate Tensor 317
Hypoelasticity 317
Kinematic Hardening Material 321
Plastic Spin 325
Localization of Deformation 334
Element Test 334
Gradient Theory 335
Shear-Band Embedded Model: Smeared Crack Model 338
Necessary Condition for Shear Band Inception 340
Numerical Calculation 344
Numerical Ability of Subloading Surface Model 344
Return-Mapping Formulation for Subloading Surface Model 347
Constitutive Equation for Friction 355
History of Constitutive Equation for Friction 355
Decomposition of Sliding Velocity 356
Normal Sliding-Yield and Sliding-Subloading Surfaces 360
Evolution Rules of Sliding-Hardening Function and Normal Sliding-Yield Ratio 361
Relations of Contact Traction Rate and Sliding Velocity 363
Loading Criterion 365
Sliding-Yield Surfaces 366
Basic Mechanical Behavior of Subloading-Friction Model 371
Extension to Orthotropic Anisotropy 381
Appendixes 393
Appendix 1: Projection of Area 393
Appendix 2: Proof of $/partial(F_{jA} / J) / /partial{x}_{j}$=0 394
Appendix 3: Euler’s Theorem for Homogeneous Function 394
Appendix 4: Normal Vector of Surface 395
Appendix 5: Convexity of Two-Dimensional Curve 396
Appendix 6: Derivation of Eq. (11.19) 397
Appendix 7: Numerical Experiments for Deformation Behavior Near Yield State 398
References 400
Index 411