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. |