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Principles of Hyperplasticity : An Approach to Plasticity Theory Based on Thermodynamic Principles - Guy T. Houlsby

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Principles of Hyperplasticity

An Approach to Plasticity Theory Based on Thermodynamic Principles

By: Guy T. Houlsby, Alexander M. Puzrin

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Hardcover | 30 November 2006

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Hardcover
376 Pages

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24.13 x 16.51 x 2.54

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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.

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Paperback

Published: 13th October 2010

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Introduction	p. 1
Plasticity and Thermodynamics	p. 1
Purpose of this Book	p. 1
Advantages of Our Approach	p. 2
Generality	p. 2
Ziegler's Orthogonality Condition	p. 3
Constitutive Models	p. 4
Context of this Book	p. 4
Notation	p. 5
Some Basic Continuum Mechanics	p. 6
Small Deformations and Small Strains	p. 6
Sign Convention	p. 8
Equations of Continuum Mechanics	p. 8
Equilibrium	p. 9
Compatibility	p. 9
Initial and Boundary Conditions	p. 9
Work Conjugacy	p. 10
Numbers of Variables and Equations	p. 11
Classical Elasticity and Plasticity	p. 13
Elasticity	p. 13
Basic Concepts of Plasticity Theory	p. 16
Incremental Stiffness in Plasticity Models	p. 19
Perfect Plasticity	p. 20
Hardening Plasticity	p. 22
Isotropic Hardening	p. 25
Kinematic Hardening	p. 27
Discussion of Hardening Laws	p. 28
Frictional Plasticity	p. 28
Restrictions on Plasticity Theories	p. 30
Drucker's Stability Postulate	p. 31
Il'iushin's Postulate of Plasticity	p. 32
Thermodynamics	p. 35
Classical Thermodynamics	p. 35
Introduction	p. 35
The First Law	p. 36
The Second Law	p. 38
Thermodynamics of Fluids	p. 40
Energy Functions	p. 42
An Example of an Internal Energy Function	p. 43
Perfect Gases	p. 44
Thermomechanics of Continua	p. 47
Terminology	p. 47
Thermoelasticity	p. 48
Internal Variables and Dissipation	p. 49
The Hyperplastic Formalism	p. 53
Introduction	p. 53
Internal Variables and Generalised Stress	p. 53
Dissipation and Dissipative Generalised Stress	p. 54
The Laws of Thermodynamics	p. 54
Dissipation Function	p. 55
Dissipative Generalised Stress	p. 56
Yield Surface	p. 56
Definition	p. 56
The Flow Rule	p. 57
Convexity	p. 58
Uniqueness of the Yield Function	p. 58
Transformations from Internal Variable to Generalised Stress	p. 59
A Complete Formulation	p. 59
Incremental Response	p. 62
Isothermal and Adiabatic Conditions	p. 66
Plastic Strains	p. 67
Yield Surface in Stress Space	p. 68
Conversions Between Potentials	p. 69
Entropy and Temperature	p. 69
Stress and Strain	p. 70
Internal Variable and Generalised Stress	p. 70
Dissipation Function to Yield Function	p. 70
Yield Function to Dissipation Function	p. 71
Constraints	p. 71
Constraints on Strains	p. 72
Constraints on Plastic Strain Rates	p. 73
Advantages of Hyperplasticity	p. 74
Summary	p. 74
Elastic and Plastic Models in Hyperplasticity	p. 77
Elasticity and Thermoelasticity	p. 77
One-dimensional Elasticity	p. 77
Isotropic Elasticity	p. 78
Incompressible Elasticity	p. 78
Isotropic Thermoelasticity	p. 79
Hierarchy of Isotropic Elastic Models	p. 80
Perfect Elastoplasticity	p. 81
One-dimensional Elastoplasticity	p. 81
Von Mises Elastoplasticity	p. 83
Rigid-plastic Models	p. 84
Frictional Plasticity and Non-associated Flow	p. 84
A Two-dimensional Model	p. 85
Dilation	p. 86
The Drucker-Prager Model with Non-associated Flow	p. 87
Strain Hardening	p. 88
Theory of Strain-hardening Hyperplasticity	p. 88
Isotropic Hardening	p. 91
Kinematic Hardening	p. 96
Mixed Hardening	p. 101
Hierarchy of Plastic Models	p. 102
Advanced Plasticity Theories	p. 105
Developments of Classical Plasticity Theory	p. 105
Bounding Surface Plasticity	p. 105
Nested Surface Plasticity	p. 107
Multiple Surface Plasticity	p. 110
Remarks on the Intersection of Yield Surfaces	p. 112
The Non-intersection Condition	p. 112
Example of Intersecting Surfaces	p. 112
What Occurs when the Surfaces Intersect?	p. 115
Alternative Approaches to Material Non-linearity	p. 117
Comparison of Advanced Plasticity Models	p. 118
Multisurface Hyperplasticity	p. 119
Motivation	p. 119
Multiple Internal Variables	p. 120
Kinematic Hardening with Multiple Yield Surfaces	p. 121
Potential Functions	p. 121
Link to Conventional Plasticity	p. 121
Incremental Response	p. 123
One-dimensional Example (the Iwan Model)	p. 125
Multidimensional Example (von Mises Yield Surfaces)	p. 128
Summary	p. 131
Continuous Hyperplasticity	p. 133
Generalised Thermodynamics and Rational Mechanics	p. 133
Internal Functions	p. 134
Energy and Dissipation Functionals	p. 134
Energy Functional	p. 134
Generalised Stress Function	p. 135
Dissipation Functional	p. 136
Dissipative Generalised Stress Function	p. 136
Legendre Transformations of the Functionals	p. 137
Legendre Transformations of the Energy Functional	p. 137
Legendre Transformation of the Dissipation Functional	p. 138
Incremental Response	p. 138
Kinematic Hardening with Infinitely Many Yield Surfaces	p. 142
Potential Functionals	p. 142
Link to Conventional Plasticity	p. 143
Incremental Response	p. 145
Example: One-dimensional Continuous Hyperplastic Model	p. 146
Calibration of Continuous Kinematic Hardening Models	p. 148
Example: Calibration of the Weighting Function	p. 148
Formulation of the One-dimensional Model	p. 148
Analogy with the Extended Iwan's Model	p. 149
Model Calibration Using the Initial Loading Curve	p. 150
Unloading Behaviour	p. 151
Example: Calibration of the Plastic Modulus Function	p. 151
Formulation of the Multidimensional von Mises Model	p. 151
Model Calibration Using the Initial Loading Curve	p. 154
Analogy with an Advanced Plasticity Model	p. 155
Hierarchy of Multisurface and Continuous Models	p. 155
Applications in Geomechanics: Elasticity and Small Strains	p. 159
Special Features of Mechanical Behaviour of Soils	p. 159
Sign Convention and Triaxial Variables	p. 159
Effective Stresses	p. 160
Dependence of Stiffness on Pressure	p. 162
Linear and Non-linear Isotropic Hyperelasticity	p. 163
Proposed Hyperelastic Potential	p. 167
Elastic-plastic Coupling in Clays	p. 172
Effects of Elasticity on Plastic Behaviour	p. 175
Small Strain Plasticity, Non-linearity, and Anisotropy	p. 176
Continuous Hyperplastic Form of a Small Strain Model	p. 177
Derivation of the Model from Potential Functions	p. 178
Behaviour of the Model During Initial Proportional Loading	p. 180
Behaviour of the Model During Proportional Cyclic Loading	p. 184
Concluding Remarks	p. 186
Applications in Geomechanics: Plasticity and Friction	p. 187
Critical State Models	p. 187
Hyperplastic Formulation of Modified Cam-Clay	p. 187
Non-uniqueness of the Energy Functions	p. 190
Towards Unified Soil Models	p. 191
Small Strain Non-linearity: Hyperbolic Stress-strain Law	p. 191
Modified Forms of the Energy Functionals	p. 193
Combining Small-strain and Critical State Behaviour	p. 195
Examples	p. 198
Continuous Hyperplastic Modified Cam-Clay	p. 203
Frictional Behaviour and Non-associated Flow	p. 204
The Dissipation to Yield Surface Transformation	p. 205
The Yield Surface to Dissipation Transformation	p. 207
Tensorial Form	p. 209
Further Applications of Hyperplasticity in Geomechanics	p. 209
Rate Effects	p. 211
Theoretical Background	p. 211
Preliminaries	p. 211
The Force Potential and the Flow Potential	p. 213
Incremental Response	p. 215
Examples	p. 216
One-dimensional Model with Additive Viscous Term	p. 216
A Non-linear Viscosity Model	p. 219
Rate Process Theory	p. 221
A Continuum Model	p. 223
Models with Multiple Internal Variables	p. 224
Multiple Internal Variables	p. 225
Incremental Response	p. 225
Example	p. 226
Continuous Models with Internal Functions	p. 228
Energy Potential Functional	p. 228
Force Potential Functional	p. 229
Legendre Transformation of the Force Potential Functional	p. 230
Incremental Response	p. 230
Example	p. 231
Visco-hyperplastic Model for Undrained Behaviour of Clay	p. 233
Formulation	p. 233
Incremental Response	p. 234
Comparison with Experimental Results	p. 235
Extension of the Model to Three Dimensions	p. 238
Advantages of the Rate-dependent Formulation	p. 239
Behaviour of Porous Continua	p. 241
Introduction	p. 241
Thermomechanical Framework	p. 242
Density Definitions, Velocities, and Balance Laws	p. 243
Tractions, Stresses, Work, and Energy	p. 245
The First Law	p. 246
Equations of Motion	p. 248
The Second Law	p. 248
Combining the First and Second Laws	p. 249
The Internal Energy Function	p. 251
The Dissipation Function and Force Potential	p. 251
Constitutive Equations	p. 252
Discussion	p. 254
The Complete Formulation	p. 255
Modifications to Account for Tortuosity	p. 255
Legendre-Fenchel Transforms	p. 256
Small Strain Formulation	p. 257
Example	p. 258
Conclusions	p. 261
Convex Analysis and Hyperplasticity	p. 263
Introduction	p. 263
Hyperplasticity Re-expressed in Convex Analytical Terms	p. 264
Examples from Elasticity	p. 265
The Yield Surface Revisited	p. 268
Examples from Plasticity	p. 270
Further Topics in Hyperplasticity	p. 273
Introduction	p. 273
Damage Mechanics	p. 274
Elementary Structural Analysis	p. 277
Pin-jointed Structures	p. 277
More General Structures	p. 279
Assemblies of Rigid Elements	p. 281
Bending of Prismatic Beams	p. 284
Large Deformation Rubber Elasticity	p. 286
Fibre-reinforced Material	p. 288
Analysis of Axial and Lateral Pile Capacity	p. 290
Rigid Pile under Vertical Loading	p. 290
Flexible Pile under Vertical Loading	p. 294
Rigid Pile under Lateral Loading	p. 297
Flexible Pile under Lateral Loading	p. 298
Concluding Remarks	p. 301
Summary of the Complete Formalism	p. 301
Legendre-Fenchel Transforms	p. 303
Some Future Directions	p. 303
Concluding Remarks	p. 304
Functions, Functionals and their Derivatives	p. 305
Functions and Functionals	p. 305
Some Special Functions	p. 306
Derivatives and Differentials	p. 307
Selected Results	p. 309
Frechet Derivatives of Integrals	p. 309
Frechet Derivatives of Integrals Containing Differential Terms	p. 310
Tensors	p. 311
Tensor Definitions and Identities	p. 311
Mixed Invariants	p. 313
Differentials of Invariants of Tensors	p. 313
Legendre Transformations	p. 315
Introduction	p. 315
Geometrical Representation in (n + 1)-dimensional Space	p. 315
Geometrical Representation in n-dimensional Space	p. 317
Homogeneous Functions	p. 318
Partial Legendre Transformations	p. 319
The Singular Transformation	p. 320
Legendre Transformations of Functionals	p. 321
Integral Functional of a Single Function	p. 321
Integral Functional of Multiple Functions	p. 322
The Singular Transformation	p. 323
Convex Analysis	p. 325
Introduction	p. 325
Some Terminology of Sets	p. 325
Convex Sets and Functions	p. 327
Subdifferentials and Subgradients	p. 328
Functions Defined for Convex Sets	p. 329
Legendre-Fenchel Transformation	p. 331
The Support Function	p. 332
Further Results in Convex Analysis	p. 334
Summary of Results for Plasticity Theory	p. 334
Some Special Functions	p. 336
References	p. 339
Index	p. 345
Table of Contents provided by Ingram. All Rights Reserved.

Principles of Hyperplasticity

An Approach to Plasticity Theory Based on Thermodynamic Principles

At a Glance

Hardcover

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How to return your order

Defective items

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