The approach to plasticity theory developed here is firmly rooted in thermodynamics. Emphasis is placed on the use of potentials and the derivation of incremental response, necessary for numerical analysis. The derivation of constitutive models for irreversible behaviour entirely from two scalar potentials is shown.
The use of potentials allows models to be very simply defined, classified and, if necessary, developed and it permits dependent and independent variables to be interchanged, making possible different forms of a model for different applications.
The theory is extended to include treatment of rate-dependent materials and a powerful concept, in which a single plastic strain is replaced by a plastic strain function, allowing smooth transitions between elastic and plastic behaviour is introduced.
This monograph will benefit academic researchers in mechanics, civil engineering and geomechanics and practising geotechnical engineers; it will also interest numerical analysts in engineering mechanics.
Author(s): Guy T. Houlsby
Edition: 1
Publisher: Springer
Year: 2006
Language: English
Pages: 363
Tags: Механика;Механика деформируемого твердого тела;Теория пластичности;
Contents......Page 8
1.1.1 Purpose of this Book......Page 23
1.1.3 Generality......Page 24
1.1.4 Ziegler's Orthogonality Condition......Page 25
1.2 Context of this Book......Page 26
1.3 Notation......Page 27
1.4.1 Small Deformations and Small Strains......Page 28
1.5 Equations of Continuum Mechanics......Page 30
1.5.3 Initial and Boundary Conditions......Page 31
1.5.4 Work Conjugacy......Page 32
1.5.5 Numbers of Variables and Equations......Page 33
2.1 Elasticity......Page 34
2.2 Basic Concepts of Plasticity Theory......Page 37
2.3 Incremental Stiffness in Plasticity Models......Page 40
2.3.1 Perfect Plasticity......Page 41
2.3.2 Hardening Plasticity......Page 43
2.3.3 Isotropic Hardening......Page 46
2.3.4 Kinematic Hardening......Page 48
2.4 Frictional Plasticity......Page 49
2.5 Restrictions on Plasticity Theories......Page 51
2.5.1 Drucker's Stability Postulate......Page 52
2.5.2 Il'iushin's Postulate of Plasticity......Page 53
3.1.1 Introduction......Page 55
3.1.2 The First Law......Page 56
3.1.3 The Second Law......Page 58
3.2 Thermodynamics of Fluids......Page 60
3.2.1 Energy Functions......Page 62
3.2.2 An Example of an Internal Energy Function......Page 63
3.2.3 Perfect Gases......Page 64
3.3.1 Terminology......Page 67
3.3.2 Thermoelasticity......Page 68
3.3.3 Internal Variables and Dissipation......Page 69
4.2 Internal Variables and Generalised Stress......Page 72
4.3.1 The Laws of Thermodynamics......Page 73
4.3.2 Dissipation Function......Page 74
4.4.1 Definition......Page 75
4.4.2 The Flow Rule......Page 76
4.4.4 Uniqueness of the Yield Function......Page 77
4.6 A Complete Formulation......Page 78
4.7 Incremental Response......Page 81
4.8 Isothermal and Adiabatic Conditions......Page 85
4.9 Plastic Strains......Page 86
4.10 Yield Surface in Stress Space......Page 87
4.11.1 Entropy and Temperature......Page 88
4.11.4 Dissipation Function to Yield Function......Page 89
4.12 Constraints......Page 90
4.12.1 Constraints on Strains......Page 91
4.12.2 Constraints on Plastic Strain Rates......Page 92
4.14 Summary......Page 93
5.1.1 One-dimensional Elasticity......Page 95
5.1.3 Incompressible Elasticity......Page 96
5.1.4 Isotropic Thermoelasticity......Page 97
5.1.5 Hierarchy of Isotropic Elastic Models......Page 98
5.2.1 One-dimensional Elastoplasticity......Page 99
5.2.2 Von Mises Elastoplasticity......Page 101
5.3 Frictional Plasticity and Non-associated Flow......Page 102
5.3.1 A Two-dimensional Model......Page 103
5.3.2 Dilation......Page 104
5.3.3 The Drucker-Prager Model with Non-associated Flow......Page 105
5.4.1 Theory of Strain-hardening Hyperplasticity......Page 106
5.4.2 Isotropic Hardening......Page 109
5.4.3 Kinematic Hardening......Page 114
5.4.4 Mixed Hardening......Page 119
5.5 Hierarchy of Plastic Models......Page 120
6.2 Bounding Surface Plasticity......Page 122
6.3 Nested Surface Plasticity......Page 124
6.4 Multiple Surface Plasticity......Page 127
6.5.2 Example of Intersecting Surfaces......Page 129
6.5.3 What Occurs when the Surfaces Intersect?......Page 132
6.6 Alternative Approaches to Material Non-linearity......Page 134
6.7 Comparison of Advanced Plasticity Models......Page 135
7.1 Motivation......Page 136
7.2 Multiple Internal Variables......Page 137
7.3.2 Link to Conventional Plasticity......Page 138
7.3.3 Incremental Response......Page 140
7.4 One-dimensional Example (the Iwan Model)......Page 142
7.5 Multidimensional Example (von Mises Yield Surfaces)......Page 145
7.6 Summary......Page 148
8.1 Generalised Thermodynamics and Rational Mechanics......Page 149
8.3.1 Energy Functional......Page 150
8.3.2 Generalised Stress Function......Page 151
8.3.4 Dissipative Generalised Stress Function......Page 152
8.4.1 Legendre Transformations of the Energy Functional......Page 153
8.5 Incremental Response......Page 154
8.6.1 Potential Functionals......Page 158
8.6.2 Link to Conventional Plasticity......Page 159
8.6.3 Incremental Response......Page 161
8.7 Example: One-dimensional Continuous Hyperplastic Model......Page 162
8.9.1 Formulation of the One-dimensional Model......Page 164
8.9.2 Analogy with the Extended Iwan's Model......Page 165
8.9.3 Model Calibration Using the Initial Loading Curve......Page 166
8.10.1 Formulation of the Multidimensional von Mises Model......Page 167
8.10.2 Model Calibration Using the Initial Loading Curve......Page 170
8.11 Hierarchy of Multisurface and Continuous Models......Page 171
9.2 Sign Convention and Triaxial Variables......Page 175
9.3 Effective Stresses......Page 176
9.4 Dependence of Stiffness on Pressure......Page 178
9.4.1 Linear and Non-linear Isotropic Hyperelasticity......Page 179
9.4.2 Proposed Hyperelastic Potential......Page 183
9.4.3 Elastic-plastic Coupling in Clays......Page 188
9.4.4 Effects of Elasticity on Plastic Behaviour......Page 191
9.5 Small Strain Plasticity, Non-linearity, and Anisotropy......Page 192
9.5.1 Continuous Hyperplastic Form of a Small Strain Model......Page 193
9.5.2 Derivation of the Model from Potential Functions......Page 194
9.5.3 Behaviour of the Model During Initial Proportional Loading......Page 196
9.5.4 Behaviour of the Model During Proportional Cyclic Loading......Page 200
9.5.5 Concluding Remarks......Page 202
10.1.1 Hyperplastic Formulation of Modified Cam-Clay......Page 203
10.1.2 Non-uniqueness of the Energy Functions......Page 206
10.2.1 Small Strain Non-linearity: Hyperbolic Stress-strain Law......Page 207
10.2.2 Modified Forms of the Energy Functionals......Page 209
10.2.3 Combining Small-strain and Critical State Behaviour......Page 211
10.2.4 Examples......Page 214
10.2.5 Continuous Hyperplastic Modified Cam-Clay......Page 219
10.3 Frictional Behaviour and Non-associated Flow......Page 220
10.3.1 The Dissipation to Yield Surface Transformation......Page 221
10.3.2 The Yield Surface to Dissipation Transformation......Page 223
10.4 Further Applications of Hyperplasticity in Geomechanics......Page 225
11.1.1 Preliminaries......Page 227
11.1.2 The Force Potential and the Flow Potential......Page 229
11.1.3 Incremental Response......Page 231
11.2.1 One-dimensional Model with Additive Viscous Term......Page 232
11.2.2 A Non-linear Viscosity Model......Page 235
11.2.3 Rate Process Theory......Page 237
11.2.4 A Continuum Model......Page 239
11.3 Models with Multiple Internal Variables......Page 240
11.3.2 Incremental Response......Page 241
11.3.3 Example......Page 242
11.4.1 Energy Potential Functional......Page 244
11.4.2 Force Potential Functional......Page 245
11.4.4 Incremental Response......Page 246
11.4.5 Example......Page 247
11.5.1 Formulation......Page 249
11.5.2 Incremental Response......Page 250
11.5.3 Comparison with Experimental Results......Page 251
11.5.4 Extension of the Model to Three Dimensions......Page 254
11.6 Advantages of the Rate-dependent Formulation......Page 255
12.1 Introduction......Page 256
12.2 Thermomechanical Framework......Page 257
12.2.1 Density Definitions, Velocities, and Balance Laws......Page 258
12.2.2 Tractions, Stresses, Work, and Energy......Page 260
12.2.3 The First Law......Page 261
12.2.5 The Second Law......Page 263
12.2.6 Combining the First and Second Laws......Page 264
12.2.8 The Dissipation Function and Force Potential......Page 266
12.2.9 Constitutive Equations......Page 267
12.2.10 Discussion......Page 269
12.3.1 Modifications to Account for Tortuosity......Page 270
12.4 Legendre-Fenchel Transforms......Page 271
12.5 Small Strain Formulation......Page 272
12.6 Example......Page 273
12.7 Conclusions......Page 276
13.1 Introduction......Page 278
13.2 Hyperplasticity Re-expressed in Convex Analytical Terms......Page 279
13.3 Examples from Elasticity......Page 280
13.4 The Yield Surface Revisited......Page 283
13.5 Examples from Plasticity......Page 285
14.1 Introduction......Page 287
14.2 Damage Mechanics......Page 288
14.3.1 Pin-jointed Structures......Page 291
14.3.2 More General Structures......Page 293
14.3.3 Assemblies of Rigid Elements......Page 295
14.4 Bending of Prismatic Beams......Page 298
14.5 Large Deformation Rubber Elasticity......Page 300
14.6 Fibre-reinforced Material......Page 302
14.7.1 Rigid Pile under Vertical Loading......Page 304
14.7.2 Flexible Pile under Vertical Loading......Page 308
14.7.3 Rigid Pile under Lateral Loading......Page 311
14.7.4 Flexible Pile under Lateral Loading......Page 312
15.1 Summary of the Complete Formalism......Page 315
15.3 Some Future Directions......Page 317
15.4 Concluding Remarks......Page 318
A.1 Functions and Functionals......Page 319
A.2 Some Special Functions......Page 320
A.3 Derivatives and Differentials......Page 321
A.4.1 Frechet Derivatives of Integrals......Page 323
A.4.2 Frechet Derivatives of Integrals Containing Differential Terms......Page 324
B.1 Tensor Definitions and Identities......Page 325
B.2.1 Differentials of Invariants of Tensors......Page 327
C.2 Geometrical Representation in (n + 1)-dimensional Space......Page 329
C.3 Geometrical Representation in n-dimensional Space......Page 331
C.4 Homogeneous Functions......Page 332
C.5 Partial Legendre Transformations......Page 333
C.6 The Singular Transformation......Page 334
C.7.1 Integral Functional of a Single Function......Page 335
C.7.2 Integral Functional of Multiple Functions......Page 336
C.7.3 The Singular Transformation......Page 337
D.2 Some Terminology of Sets......Page 339
D.3 Convex Sets and Functions......Page 341
D.4 Subdifferentials and Subgradients......Page 342
D.5 Functions Defined for Convex Sets......Page 343
D.6 Legendre-Fenchel Transformation......Page 345
D.7 The Support Function......Page 346
D.9 Summary of Results for Plasticity Theory......Page 348
D.10 Some Special Functions......Page 350
References......Page 352
C......Page 357
F......Page 358
I......Page 359
P......Page 360
S......Page 361
U......Page 362
Z......Page 363
