Plasticity of Pressure-Sensitive Materials (eBook)

Preface 5
Contents 7
Contributors 8
1 Basic Equations of Continuum Mechanics 10
1 Kinematics and Deformations 11
1.1 Lagrangian and Eulerian Description 11
1.2 Time Derivatives and Nabla Operator 12
1.3 Strains and Deformation Gradient 13
1.4 Velocities, Velocity Gradients 14
1.5 Strains and Strain Measures 15
1.6 Displacements, Displacement Gradient, Linearizations 17
2 Stress State 19
2.1 Classification of External Actions 19
2.2 Cauchy's Stress Vector and Tensor 21
2.3 Equilibrium Equations, Equations of Motion 23
2.4 Stress Vectors and Tensors After Piola-Kirchhoff 24
3 Balance Equations 25
3.1 General Formulation of Balance Equations 25
3.2 Mass Balance and Mass Conservation 28
3.3 Balance of Moment of Momentum 31
3.4 Balance of Energy 32
3.5 Balance of Entropy 35
4 Constitutive Modeling 37
4.1 Basics of Material Theory 38
4.2 General Constitutive Equations of Thermo-Mechanical Materials 39
4.3 Elastic Simple Material 40
4.4 Models with Internal Variables 43
5 Governing Equations of Mechanics of Hyperelastic Materials 45
6 Constitutive Equations of Incompressible Materials 49
6.1 Polynomial Approximation 49
6.2 Treloar (Neo-Hookean) Material 49
6.3 Mooney Material 49
6.4 Rivlin-Saunders Material 50
6.5 Biderman Model 50
6.6 Non-Polynomial Approximation 50
6.7 Incompressible Ogden Material 51
6.8 Chernykh-Shubina Material 51
6.9 Bartenev-Khazanovich Material 51
7 Constitutive Equations of Compressible Materials 52
7.1 Compressible Neo-Hookean Material 52
7.2 Mooney-Rivlin Material 52
7.3 Compressible Ogden Material 53
7.4 Arruda-Boyce Material 53
7.5 Gent Material 54
7.6 Blatz-Ko Material 54
7.7 Other Material Models 54
References 55
2 Phenomenological Yield and Failure Criteria 57
1 Need of Criteria 58
2 Classical Strength Theories 59
2.1 Normal Stress Hypothesis 60
2.2 Tresca Hypothesis 61
2.3 Huber-vonMises-Hencky Hypothesis 62
2.4 Schmidt-Ishlinsky Hypothesis 62
3 Basic Stress States 63
4 Inelastic Poisson's Ratio 65
5 Ratios for a Torsion Bar 72
6 Standard Criteria 76
6.1 Strain Criterion 76
6.2 Mohr-Coulomb Criterion 77
6.3 Pisarenko-Lebedev Criterion 77
6.4 Burzy?ski-Yagn Criterion 78
7 Mathematical Formulations 80
7.1 Criterion of Altenbach-ZolochevskyI 80
7.2 Model in Terms of the Integrity Basis 82
7.3 Models Based on the Stress Deviator 83
8 Visualization Methods 84
8.1 Spatial Representation of the Limit Surface 84
8.2 Burzy?ski-Plane 86
8.3 ?-Plane 88
9 Pressure-Insensitive Criteria 88
9.1 Yield Surfaces with Trigonal Symmetry 90
9.2 Yield Surfaces with Hexagonal Symmetry 97
10 Pressure-Sensitive Criteria 102
10.1 Compressible Generalization 102
10.2 Unified Strength Theory of Yu 104
10.3 Models for Applications 108
11 Combined Criteria 114
11.1 Criteria with C0-Transition 115
11.2 Criteria with C1-Transition 118
12 Fitting 124
12.1 Mathematical Criteria 124
12.2 Physical Criteria 126
12.3 Geometrical Criteria 128
13 Applications 129
13.1 Measurements of Coffin for Gray Cast Iron 129
13.2 Measurements by Pae for Poly(oxymethylene) (POM) 136
13.3 Measurements of Cristensen for PVC Hard Foam 137
14 Summary and Outlook 146
15 Invariants 147
15.1 Principal Invariants 147
15.2 Irreducible Invariants 148
15.3 Axiatoric-Deviatoric Invariants 149
15.4 Cylindrical Invariants 149
16 Criteria of this Chapter 2 151
References 151
3 Plasticity of Cellular Metals (Foams) 161
1 Introduction 162
2 Constitutive Modeling: Basics 163
2.1 Introduction 163
2.2 Mathematical Notation 163
2.3 The Stress State 165
2.4 Deformation and Strain 169
2.5 Formal Introduction to Elasto-Plasticity 171
3 Deformation Mechanisms and Yielding in Cellular Metals 173
3.1 Onset of Failure 173
3.2 Progressive Collapse and Densification 179
4 Constitutive Modeling of Cellular Metals 181
4.1 Introduction 181
4.2 Linear Elastic Behavior 183
4.3 The Gibson-Ashby-Zhang-Triantafillou (GAZT) Model 185
4.4 The Miller Model 188
4.5 The Deshpande-Fleck Foam Models 192
4.6 Chen and Lu Metallic Foam Material Model 202
4.7 The Model by Zhang et al. 204
4.8 The Abaqus Crushable Foam Model 205
4.9 The Ehlers Model for Cellular Metals 209
5 Discussion and Conclusions 210
References 211
4 Transmission Conditions for Thin Elasto-Plastic Pressure-Dependent Interphases 213
1 Introduction 214
2 Transmission Conditions for Thin Soft Interphases 215
2.1 Problem Formulation 215
2.2 Evaluation of the Transmission Conditions for Inhomogeneous Linear Interphases 217
2.3 Transmission Conditions for Nonlinear Interphases 220
3 Pressure-Dependent Deformation Theory 223
3.1 Main Assumptions 224
3.2 Example: Uniaxial Stress Test 228
3.3 Example: Uniaxial Deformation Test 237
3.4 Plane Stress State 243
3.5 Plane Strain State 244
4 Validation of the Transmission Conditions: Plane Strain Case 245
4.1 Description of the FEM Model 245
4.2 Evaluation of the Mechanical Values: Displacement, Stress and Strain 246
4.3 Validation of the First and Second Transmission Condition 249
5 Discussions and Conclusions 255
References 258
5 Effect of Pressure-Dependency of the Yield Criterion on the Strain Rate Intensity Factor 260
1 Introduction 261
2 Strain Rate Intensity Factor 261
3 Plane Strain Solutions for Pressure-Independent Material 269
3.1 Basic Equations 269
3.2 Compression of a Plastic Layer Between Parallel Plates 271
3.3 Flow of Plastic Material Through an Infinite Wedge-Shaped Channel 272
3.4 Compression of a Plastic Layer Between Cylindrical Surfaces 275
3.5 Compression of a Plastic Layer Between Rotating Plates I 279
3.6 Compression of a Plastic Layer Between Rotating Plates II 282
3.7 Simultaneous Shearing and Expansion of a Hollow Cylinder 286
4 Axisymmetric Solutions for Pressure-Independent Material 288
4.1 Basic Equations 288
4.2 Compression of a Hollow Cylinder on a Rigid Fibre 292
4.3 Flow of Plastic Material Through an Converging Conical Channel 295
4.4 Radial Flow Between Two Conical Surfaces 299
5 Plane Strain Solutions for the Double-Shearing Model 305
5.1 Basic Equations 305
5.2 Compression of a Plastic Layer Between Parallel Plates 307
5.3 Flow of Plastic Material Through an Infinite Wedge-Shaped Channel 309
5.4 Compression of a Plastic Layer Between Cylindrical Surfaces 311
5.5 Compression of a Plastic Layer Between Rotating Plates I 321
5.6 Compression of a Plastic Layer Between Rotating Plates II 331
5.7 Simultaneous Shearing and Expansion of a Hollow Cylinder 339
6 Axisymmetric Solutions for the Double-Shearing Model 344
6.1 Basic Equations 344
6.2 Flow of Plastic Material Through an Converging Conical Channel 346
6.3 Radial Flow Between Two Conical Surfaces 351
7 Concluding Remarks 354
References 354
6 Mechanical Response of Porous Materials: The Gurson Model 356
1 Introduction 356
2 Gurson Damage Model 358
2.1 Yield Criterion 358
2.2 Evolution Law for the Porosity 361
2.3 Elastic Constants for the Damaged Material 363
2.4 Assessment of Gurson Model 364
3 Influence of the Parameter Values on the Behavior of the Damage Model 365
4 Further Developments and New Trends for Gurson Model 368
4.1 Non-Spherical Voids 369
4.2 Shear Effects 370
4.3 Hardening 371
5 Computational Details 372
5.1 Numerical Implementation 372
5.2 Mesh-Size Dependence 373
5.3 Arbitrary Lagrangian-Eulerian Alternative 374
6 Numerical Examples 375
6.1 Indentation of a Block by a Sphere 375
6.2 Analysis of Metallic Foams 376
References 380