Table of contents for Computational methods for plasticity : theory and applications / Eduardo de Souza Neto, Djordje Peric, David Owens.

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Part One Basic concepts 1
1 Introduction 3
1.1 Aims andscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Readership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Theuseofboxes . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 General schemeofnotation . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Character fonts.General convention . . . . . . . . . . . . . . . . 8
1.3.2 Some importantcharacters . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Indicialnotation,subscripts andsuperscripts . . . . . . . . . . . 13
1.3.4 Other important symbolsandoperations . . . . . . . . . . . . . . 14
2 ELEMENTS OF TENSOR ANALYSIS 17
2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Second-order tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 The transpose.Symmetricandskewtensors . . . . . . . . . . . . 19
2.2.2 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Cartesian components and matrix representation . . . . . . . . . 21
2.2.4 Trace, inner product and Euclidean norm . . . . . . . . . . . . . 22
2.2.5 Inversetensor.Determinant . . . . . . . . . . . . . . . . . . . . 23
2.2.6 Orthogonal tensors. Rotations . . . . . . . . . . . . . . . . . . . 23
2.2.7 Cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.8 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.9 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Higher-order tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Fourth-order tensors . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Generic-order tensors . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Isotropictensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 Isotropic second-order tensors . . . . . . . . . . . . . . . . . . . 30
2.4.2 Isotropicfourth-order tensors . . . . . . . . . . . . . . . . . . . 30
2.5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.1 Thederivativemap.Directionalderivative . . . . . . . . . . . . . 32
2.5.2 Linearisationof a nonlinear function . . . . . . . . . . . . . . . . 32
2.5.3 Thegradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.4 Derivatives of functions of vector and tensor arguments . . . . . . 33
2.5.5 The chainrule . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.6 The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5.7 Thedivergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5.8 Useful relations involving the gradient and the divergence . . . . 38
2.6 Linearisationofnonlinearproblems . . . . . . . . . . . . . . . . . . . . . 38
2.6.1 Thenonlinearproblemandits linearisedform . . . . . . . . . . . 38
2.6.2 Linearisation in infinite-dimensional functional spaces . . . . . . 39
3 ELEMENTS OF CONTINUUM MECHANICS AND THERMODYNAMICS 41
3.1 Kinematicsofdeformation . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Material andspatialfields . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 Material and spatial gradients, divergences and time derivatives . 46
3.1.3 Thedeformationgradient . . . . . . . . . . . . . . . . . . . . . . 46
3.1.4 Volume changes. The determinant of the deformation gradient . . 47
3.1.5 Isochoric/volumetric split of the deformation gradient . . . . . . 49
3.1.6 Polardecomposition.Stretches androtation . . . . . . . . . . . . 49
3.1.7 Strainmeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.8 Thevelocitygradient.Rate ofdeformationandspin . . . . . . . . 55
3.1.9 Rate ofvolumechange . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Infinitesimaldeformations . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.1 The infinitesimal straintensor . . . . . . . . . . . . . . . . . . . 57
3.2.2 Infinitesimal rigiddeformations . . . . . . . . . . . . . . . . . . 58
3.2.3 Infinitesimal isochoric andvolumetricdeformations . . . . . . . 58
3.3 Forces.StressMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.1 Cauchy?saxiom.TheCauchystress vector . . . . . . . . . . . . 61
3.3.2 The axiomofmomentumbalance . . . . . . . . . . . . . . . . . 61
3.3.3 TheCauchystress tensor . . . . . . . . . . . . . . . . . . . . . . 62
3.3.4 The First Piola?Kirchhoff stress . . . . . . . . . . . . . . . . . . 64
3.3.5 The Second Piola?Kirchhoff stress . . . . . . . . . . . . . . . . . 66
3.3.6 The Kirchhoff stress . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Fundamental laws of thermodynamics . . . . . . . . . . . . . . . . . . . . 67
3.4.1 Conservationofmass . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.2 Momentumbalance . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.3 Thefirst principle . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.4 The secondprinciple . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.5 The Clausius?Duhem inequality . . . . . . . . . . . . . . . . . . 69
3.5 Constitutive theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.1 Constitutive axioms . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.2 Thermodynamics with internal variables . . . . . . . . . . . . . . 71
3.5.3 Phenomenologicalandmicromechanicalapproaches . . . . . . . 74
3.5.4 Thepurelymechanical theory . . . . . . . . . . . . . . . . . . . 75
3.5.5 The constitutive initial value problem . . . . . . . . . . . . . . . 76
3.6 Weak equilibrium. The principle of virtual work . . . . . . . . . . . . . . 77
3.6.1 The spatialversion . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6.2 Thematerialversion . . . . . . . . . . . . . . . . . . . . . . . . 78
3.6.3 The infinitesimal case . . . . . . . . . . . . . . . . . . . . . . . 78
3.7 The quasi-static initial boundary value problem . . . . . . . . . . . . . . . 79
3.7.1 Finite deformations . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.7.2 The infinitesimalproblem . . . . . . . . . . . . . . . . . . . . . 80
4 The finite element method in quasi-static nonlinear solid mechanics 83
4.1 Displacement-based finite elements . . . . . . . . . . . . . . . . . . . . . 84
4.1.1 Finite element interpolation . . . . . . . . . . . . . . . . . . . . 85
4.1.2 Thediscretisedvirtualwork . . . . . . . . . . . . . . . . . . . . 86
4.1.3 Some typical isoparametricelements . . . . . . . . . . . . . . . . 90
4.1.4 Example.Linearelasticity . . . . . . . . . . . . . . . . . . . . . 93
4.2 Path-dependentmaterials.The incrementalfinite elementprocedure . . . . 94
4.2.1 The incremental constitutive function . . . . . . . . . . . . . . . 95
4.2.2 The incremental boundary value problem . . . . . . . . . . . . . 95
4.2.3 Thenonlinear incrementalfinite element equation . . . . . . . . . 96
4.2.4 Nonlinear solution. The Newton?Raphson scheme . . . . . . . . 96
4.2.5 The consistent tangent modulus . . . . . . . . . . . . . . . . . . 98
4.2.6 Alternativenonlinearsolutionschemes . . . . . . . . . . . . . . 99
4.2.7 Non-incrementalproceduresforpath-dependentmaterials . . . . 101
4.3 Largestrainformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.1 The incremental constitutive function . . . . . . . . . . . . . . . 102
4.3.2 The incremental boundary value problem . . . . . . . . . . . . . 103
4.3.3 The finite element equilibrium equation . . . . . . . . . . . . . . 103
4.3.4 Linearisation. The consistent spatial tangent modulus . . . . . . . 103
4.3.5 Material andgeometricstiffnesses . . . . . . . . . . . . . . . . . 106
4.3.6 Configuration-dependentloads.The load-stiffnessmatrix . . . . . 106
4.4 Unstable equilibrium. The arc-length method . . . . . . . . . . . . . . . . 107
4.4.1 The arc-lengthmethod . . . . . . . . . . . . . . . . . . . . . . . 107
4.4.2 The combined Newton?Raphson/arc-length procedure . . . . . . 108
4.4.3 Thepredictorsolution . . . . . . . . . . . . . . . . . . . . . . . 111
5 Overview of the program structure 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.2 Remarksonprogramstructure . . . . . . . . . . . . . . . . . . . 116
5.1.3 Portability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Themainprogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3 Data input and initialisation . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.1 Theglobaldatabase . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.2 Main problem-defining data. Subroutine INDATA . . . . . . . . . 119
5.3.3 External loading. Subroutine INLOAD . . . . . . . . . . . . . . . 119
5.3.4 Initialisation of variable data. Subroutine INITIA . . . . . . . . . 120
5.4 The load incrementation loop. Overview . . . . . . . . . . . . . . . . . . 120
5.4.1 Fixedincrementsoption . . . . . . . . . . . . . . . . . . . . . . 120
5.4.3 Automatic increment cutting . . . . . . . . . . . . . . . . . . . . 123
5.4.4 The linear solver. Subroutine FRONT . . . . . . . . . . . . . . . . 124
5.4.5 Internal force calculation. Subroutine INTFOR . . . . . . . . . . . 124
5.4.6 Switching data. Subroutine SWITCH . . . . . . . . . . . . . . . . 124
5.4.7 Output of converged results. Subroutines OUTPUT and RSTART . . 125
5.5 Material and element modularity . . . . . . . . . . . . . . . . . . . . . . . 125
5.5.1 Example. Modularisation of internal force computation . . . . . . 125
5.6 Elements. Implementationandmanagement . . . . . . . . . . . . . . . . . 128
5.6.1 Element properties. Element routines for data input . . . . . . . . 128
5.6.2 Element interfaces. Internal force and stiffness computation . . . 129
5.6.3 Implementinga newfinite element . . . . . . . . . . . . . . . . . 129
5.7 Materialmodels: implementationandmanagement . . . . . . . . . . . . . 131
5.7.1 Material properties. Material-specific data input . . . . . . . . . . 131
5.7.2 State variables and other Gauss point quantities.
Material-specific state updating routines . . . . . . . . . . . . . . 132
5.7.3 Material-specific switching/initialising routines . . . . . . . . . . 133
5.7.4 Material-specifictangentcomputationroutines . . . . . . . . . . 134
5.7.5 Material-specificresults output routines . . . . . . . . . . . . . . 134
5.7.6 Implementinga newmaterialmodel . . . . . . . . . . . . . . . . 135
Part Two Small strains 137
6 The mathematical theory of plasticity 139
6.1 Phenomenologicalaspects . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2 One-dimensional constitutive model . . . . . . . . . . . . . . . . . . . . . 141
6.2.1 Elastoplastic decomposition of the axial strain . . . . . . . . . . . 142
6.2.2 The elastic uniaxial constitutive law . . . . . . . . . . . . . . . . 143
6.2.3 Theyieldfunctionandtheyieldcriterion . . . . . . . . . . . . . 143
6.2.4 Theplasticflowrule.Loading/unloadingconditions . . . . . . . 144
6.2.5 Thehardeninglaw . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2.6 Summaryof themodel . . . . . . . . . . . . . . . . . . . . . . . 145
6.2.7 Determination of the plastic multiplier . . . . . . . . . . . . . . . 147
6.2.8 The elastoplastic tangent modulus . . . . . . . . . . . . . . . . . 147
6.3 General elastoplastic constitutive model . . . . . . . . . . . . . . . . . . . 148
6.3.1 Additive decomposition of the strain tensor . . . . . . . . . . . . 148
6.3.2 The free energypotentialandthe elastic law. . . . . . . . . . . . 149
6.3.3 Theyieldcriterionandtheyieldsurface . . . . . . . . . . . . . . 150
6.3.4 Plasticflowrule andhardeninglaw . . . . . . . . . . . . . . . . 150
6.3.5 Flowrulesderivedfromaflowpotential . . . . . . . . . . . . . . 152
6.3.6 The plastic multiplier . . . . . . . . . . . . . . . . . . . . . . . . 152
6.3.7 Relation to the general continuum constitutive theory . . . . . . . 153
6.3.8 Rate formandthe elastoplastic tangentoperator . . . . . . . . . . 153
6.3.9 Non-smooth potentials and the subdifferential . . . . . . . . . . . 154
6.4 Classical yieldcriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.4.1 TheTrescayieldcriterion . . . . . . . . . . . . . . . . . . . . . 157
6.4.2 ThevonMises yieldcriterion . . . . . . . . . . . . . . . . . . . 162
6.4.3 TheMohr?Coulombyieldcriterion . . . . . . . . . . . . . . . . 164
6.4.4 TheDrucker?Prageryieldcriterion . . . . . . . . . . . . . . . . 167
6.5 Plasticflowrules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.5.1 Associative and non-associative plasticity . . . . . . . . . . . . . 169
6.5.2 Associative laws and the principle of maximum plastic dissipation 170
6.5.3 Classical flowrules . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.6 Hardeninglaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.6.1 Perfectplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.6.2 Isotropichardening . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.6.3 Thermodynamical aspects. Associative isotropic hardening . . . . 182
6.6.4 Kinematichardening.TheBauschingereffect . . . . . . . . . . . 185
6.6.5 Mixedisotropic/kinematichardening . . . . . . . . . . . . . . . 189
7 Finite elements in small-strain plasticity problems 191
7.1 Preliminaryimplementationaspects . . . . . . . . . . . . . . . . . . . . . 192
7.2 General numerical integration algorithm for elastoplastic constitutive
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.2.1 The elastoplastic constitutive initial value problem . . . . . . . . 193
7.2.2 Euler discretisation: the incremental constitutive problem . . . . . 194
7.2.3 The elastic predictor/plasticcorrectoralgorithm . . . . . . . . . . 196
7.2.4 Solutionof the return-mappingequations . . . . . . . . . . . . . 198
7.2.5 Closest pointprojectioninterpretation . . . . . . . . . . . . . . . 200
7.2.6 Alternativejustification:operatorsplitmethod . . . . . . . . . . 201
7.2.7 Other elastic predictor/return-mappingschemes . . . . . . . . . . 202
7.2.8 Plasticity anddifferential-algebraicequations . . . . . . . . . . . 210
7.2.9 Alternativemathematicalprogramming-basedalgorithms . . . . . 210
7.2.10 Accuracy and stability considerations . . . . . . . . . . . . . . . 211
7.3 Application: integration algorithm for the isotropically hardening
vonMisesmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.3.1 The implementedmodel . . . . . . . . . . . . . . . . . . . . . . 216
7.3.2 The implicit elastic predictor/return-mappingscheme . . . . . . . 217
7.3.3 The incremental constitutive function for the stress . . . . . . . . 221
7.3.4 Linear isotropic hardening and perfect plasticity: the
closed-formreturnmapping . . . . . . . . . . . . . . . . . . . . 223
7.3.5 Subroutine SUVM . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.4 The consistent tangent modulus . . . . . . . . . . . . . . . . . . . . . . . 229
7.4.1 Consistent tangentoperatorsin elastoplasticity . . . . . . . . . . 229
7.4.2 The elastoplastic consistent tangent for the von Mises model
withisotropichardening . . . . . . . . . . . . . . . . . . . . . . 233
7.4.3 Subroutine CTVM . . . . . . . . . . . . . . . . . . . . . . . . . . 236
7.4.4 The general elastoplastic consistent tangent operator for implicit
returnmappings . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.4.5 Illustration: the vonMisesmodelwithisotropichardening . . . . 241
7.4.6 Tangentoperator symmetry: incrementalpotentials . . . . . . . . 243
7.5 Numericalexampleswith thevonMisesmodel . . . . . . . . . . . . . . . 244
7.5.1 Internallypressurisedcylinder . . . . . . . . . . . . . . . . . . . 245
7.5.2 Internallypressurisedspherical shell . . . . . . . . . . . . . . . . 247
7.5.3 Uniformlyloadedcircularplate . . . . . . . . . . . . . . . . . . 250
7.5.4 Strip-footingcollapse . . . . . . . . . . . . . . . . . . . . . . . . 251
7.5.5 Double-notched tensile specimen . . . . . . . . . . . . . . . . . 253
7.6 Further application: the von Mises model with nonlinear mixed hardening . 256
7.6.1 Themixedhardeningmodel: summary . . . . . . . . . . . . . . 258
7.6.2 The implicit return-mappingscheme . . . . . . . . . . . . . . . . 259
7.6.3 The incremental constitutive function . . . . . . . . . . . . . . . 260
7.6.4 Linearhardening:closed-formreturnmapping . . . . . . . . . . 261
7.6.5 Computational implementationaspects . . . . . . . . . . . . . . 262
7.6.6 The elastoplastic consistent tangent . . . . . . . . . . . . . . . . 262
8 Computations with other basic plasticity models 265
8.1 TheTrescamodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.1.1 The implicit integration algorithm in principal stresses . . . . . . 268
8.1.2 Subroutine SUTR . . . . . . . . . . . . . . . . . . . . . . . . . . 279
8.1.3 Finite step accuracy: iso-error maps . . . . . . . . . . . . . . . . 283
8.1.4 The consistent tangentoperator for theTrescamodel . . . . . . . 286
8.1.5 Subroutine CTTR . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8.2 TheMohr?Coulombmodel . . . . . . . . . . . . . . . . . . . . . . . . . 295
8.2.1 Integrationalgorithmfor theMohr?Coulombmodel . . . . . . . 297
8.2.2 Subroutine SUMC . . . . . . . . . . . . . . . . . . . . . . . . . . 310
8.2.3 Accuracy: iso-errormaps . . . . . . . . . . . . . . . . . . . . . . 315
8.2.4 Consistent tangent operator for the Mohr?Coulomb model . . . . 316
8.2.5 Subroutine CTMC . . . . . . . . . . . . . . . . . . . . . . . . . . 319
8.3 TheDrucker?Pragermodel . . . . . . . . . . . . . . . . . . . . . . . . . . 324
8.3.1 Integrationalgorithmfor theDrucker?Pragermodel . . . . . . . 325
8.3.2 Subroutine SUDP . . . . . . . . . . . . . . . . . . . . . . . . . . 334
8.3.3 Iso-errormap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
8.3.4 Consistent tangent operator for the Drucker?Prager model . . . . 337
8.3.5 Subroutine CTDP . . . . . . . . . . . . . . . . . . . . . . . . . . 340
8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
8.4.1 Bendingof aV-notchedTresca bar . . . . . . . . . . . . . . . . . 343
8.4.2 End-loaded tapered cantilever . . . . . . . . . . . . . . . . . . . 344
8.4.3 Strip-footingcollapse . . . . . . . . . . . . . . . . . . . . . . . . 346
8.4.4 Circular-footingcollapse . . . . . . . . . . . . . . . . . . . . . . 350
8.4.5 Slope stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
9 Plane stress plasticity 357
9.1 Thebasicplanestress plasticityproblem . . . . . . . . . . . . . . . . . . 357
9.1.1 Plane stress linear elasticity . . . . . . . . . . . . . . . . . . . . 358
9.1.2 The constrained elastoplastic initial value problem . . . . . . . . 359
9.1.3 Procedures forplane stress plasticity . . . . . . . . . . . . . . . . 360
9.2 Plane stress constraintat theGauss point level . . . . . . . . . . . . . . . 361
9.2.1 Implementationaspects . . . . . . . . . . . . . . . . . . . . . . . 362
9.2.2 Plane stress enforcementwith nestediterations . . . . . . . . . . 362
9.2.3 Plane stress vonMiseswith nestediterations . . . . . . . . . . . 364
9.2.4 The consistent tangent for thenestediterationprocedure . . . . . 366
9.2.5 Consistent tangent computationfor thevonMisesmodel . . . . . 366
9.3 Plane stress constraint at the structural level . . . . . . . . . . . . . . . . . 367
9.3.1 Themethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
9.3.2 The implementation . . . . . . . . . . . . . . . . . . . . . . . . 368
9.4 Plane stress-projectedplasticitymodels . . . . . . . . . . . . . . . . . . . 370
9.4.1 Theplane stress-projectedvonMisesmodel . . . . . . . . . . . . 371
9.4.2 Theplane stress-projectedintegrationalgorithm. . . . . . . . . . 373
9.4.3 Subroutine SUVMPS . . . . . . . . . . . . . . . . . . . . . . . . . 378
9.4.4 The elastoplastic consistent tangentoperator . . . . . . . . . . . 382
9.4.5 Subroutine CTVMPS . . . . . . . . . . . . . . . . . . . . . . . . . 383
9.5 Numericalexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.5.1 Collapse of an end-loaded cantilever . . . . . . . . . . . . . . . . 387
9.5.2 Infiniteplatewitha circularhole . . . . . . . . . . . . . . . . . . 387
9.5.3 Stretching of a perforated rectangular plate . . . . . . . . . . . . 390
9.5.4 Uniformloadingof a concreteshearwall . . . . . . . . . . . . . 391
9.6 Other stress-constrainedstates . . . . . . . . . . . . . . . . . . . . . . . . 396
9.6.1 Athree-dimensionalvonMisesTimoshenkobeam . . . . . . . . 396
9.6.2 Thebeamstate-projectedintegrationalgorithm . . . . . . . . . . 400
10 Advanced plasticity models 403
10.1 AmodifiedCam-Claymodel for soils . . . . . . . . . . . . . . . . . . . . 403
10.1.1 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
10.1.2 Computational implementation . . . . . . . . . . . . . . . . . . . 406
10.2 AcappedDrucker?Pragermodel forgeomaterials . . . . . . . . . . . . . . 409
10.2.1 CappedDrucker?Pragermodel . . . . . . . . . . . . . . . . . . . 410
10.2.2 The implicit integrationalgorithm . . . . . . . . . . . . . . . . . 412
10.2.3 The elastoplastic consistent tangentoperator . . . . . . . . . . . 413
10.3 Anisotropic plasticity: the Hill, Hoffman and Barlat?Lian models . . . . . 414
10.3.1 TheHill orthotropicmodel . . . . . . . . . . . . . . . . . . . . . 414
10.3.2 Tension?compression distinction: the Hoffman model . . . . . . 420
10.3.3 Implementationof theHoffmanmodel . . . . . . . . . . . . . . . 423
10.3.4 TheBarlat?Lianmodel for sheetmetals . . . . . . . . . . . . . . 427
10.3.5 Implementationof theBarlat?Lianmodel . . . . . . . . . . . . . 431
11 Viscoplasticity 435
11.1 Viscoplasticity:phenomenologicalaspects . . . . . . . . . . . . . . . . . 436
11.2 One-dimensionalviscoplasticitymodel . . . . . . . . . . . . . . . . . . . 437
11.2.1 Elastoplastic decomposition of the axial strain . . . . . . . . . . . 437
11.2.2 The elastic law . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
11.2.3 Theyieldfunctionandthe elastic domain . . . . . . . . . . . . . 438
11.2.4 Viscoplasticflowrule . . . . . . . . . . . . . . . . . . . . . . . . 438
11.2.5 Hardeninglaw . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
11.2.6 Summaryof themodel . . . . . . . . . . . . . . . . . . . . . . . 439
11.2.7 Some simple analytical solutions . . . . . . . . . . . . . . . . . . 439
11.3 A von Mises-based multidimensional model . . . . . . . . . . . . . . . . 445
11.3.1 A von Mises-type viscoplastic model with isotropic strain
hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
11.3.2 Alternativeplastic strain ratedefinitions . . . . . . . . . . . . . . 446
11.3.3 Other isotropicandkinematichardeninglaws . . . . . . . . . . . 448
11.3.4 Viscoplastic models without a yield surface . . . . . . . . . . . . 449
11.4 General viscoplastic constitutive model . . . . . . . . . . . . . . . . . . . 450
11.4.1 Relation to the general continuum constitutive theory . . . . . . . 451
11.4.2 Potential structure and dissipation inequality . . . . . . . . . . . 452
11.4.3 Rate-independentplasticityas a limit case . . . . . . . . . . . . . 452
11.5 Generalnumerical framework . . . . . . . . . . . . . . . . . . . . . . . . 455
11.5.1 Ageneral implicit integrationalgorithm . . . . . . . . . . . . . . 455
11.5.2 AlternativeEuler-basedalgorithms . . . . . . . . . . . . . . . . . 457
11.5.3 Generalconsistent tangentoperator . . . . . . . . . . . . . . . . 459
11.6 Application: computational implementation of a von Mises-based model . 460
11.6.1 Integrationalgorithm . . . . . . . . . . . . . . . . . . . . . . . . 460
11.6.2 Iso-errormaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
11.6.3 Consistent tangentoperator . . . . . . . . . . . . . . . . . . . . . 464
11.6.4 Perzyna-typemodel implementation . . . . . . . . . . . . . . . . 467
11.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
11.7.1 Double-notched tensile specimen . . . . . . . . . . . . . . . . . 468
11.7.2 Plane stress: stretchingof aperforatedplate . . . . . . . . . . . . 470
12 Damage mechanics 471
12.1 Physicalaspects of internaldamagein solids . . . . . . . . . . . . . . . . 472
12.1.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
12.1.2 Rubbery polymers . . . . . . . . . . . . . . . . . . . . . . . . . 473
12.2 Continuum damage mechanics . . . . . . . . . . . . . . . . . . . . . . . . 473
12.2.1 Originaldevelopment:creep-damage . . . . . . . . . . . . . . . 474
12.2.2 Other theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
12.2.3 Remarksonthenatureof thedamagevariable . . . . . . . . . . . 476
12.3 Lemaitre?s elastoplasticdamagetheory . . . . . . . . . . . . . . . . . . . 478
12.3.1 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
12.3.2 Integrationalgorithm . . . . . . . . . . . . . . . . . . . . . . . . 482
12.3.3 The tangentoperators . . . . . . . . . . . . . . . . . . . . . . . . 485
12.4 AsimplifiedversionofLemaitre?smodel . . . . . . . . . . . . . . . . . . 486
12.4.1 The single-equationintegrationalgorithm . . . . . . . . . . . . . 486
12.4.2 The tangentoperator . . . . . . . . . . . . . . . . . . . . . . . . 491
12.4.3 Example. Fracturing of a cylindrical notched specimen . . . . . . 493
12.5 Gurson?svoidgrowthmodel . . . . . . . . . . . . . . . . . . . . . . . . . 496
12.5.1 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
12.5.2 Integrationalgorithm . . . . . . . . . . . . . . . . . . . . . . . . 500
12.5.3 The tangentoperator . . . . . . . . . . . . . . . . . . . . . . . . 502
12.6 Further issues in damage modelling . . . . . . . . . . . . . . . . . . . . . 504
12.6.1 Crackclosureeffects in damagedelasticmaterials . . . . . . . . 504
12.6.2 Crackclosure effects indamageevolution . . . . . . . . . . . . . 511
12.6.3 Anisotropic ductile damage . . . . . . . . . . . . . . . . . . . . 512
Part Three Large strains 517
13 Finite strain hyperelasticity 519
13.1 Hyperelasticity:basic concepts . . . . . . . . . . . . . . . . . . . . . . . . 520
13.1.1 Material objectivity: reduced form of the free-energy function . . 520
13.1.2 Isotropic hyperelasticity . . . . . . . . . . . . . . . . . . . . . . 521
13.1.3 Incompressible hyperelasticity . . . . . . . . . . . . . . . . . . . 524
13.1.4 Compressibleregularisation . . . . . . . . . . . . . . . . . . . . 525
13.2 Someparticularmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
13.2.1 The Mooney?Rivlin and the neo-Hookean models . . . . . . . . 525
13.2.2 TheOgdenmaterialmodel . . . . . . . . . . . . . . . . . . . . . 527
13.2.3 TheHenckymaterial . . . . . . . . . . . . . . . . . . . . . . . . 528
13.2.4 TheBlatz?Komaterial . . . . . . . . . . . . . . . . . . . . . . . 530
13.3 Isotropic finite hyperelasticity in plane stress . . . . . . . . . . . . . . . . 530
13.3.1 Theplane stress incompressibleOgdenmodel . . . . . . . . . . . 531
13.3.2 Theplane stressHenckymodel . . . . . . . . . . . . . . . . . . 532
13.3.3 Plane stresswithnestediterations . . . . . . . . . . . . . . . . . 533
13.4 Tangentmoduli: the elasticity tensors . . . . . . . . . . . . . . . . . . . . 534
13.4.1 Regularisedneo-Hookeanmodel . . . . . . . . . . . . . . . . . . 535
13.4.2 Principal stretches representation:Ogdenmodel . . . . . . . . . . 535
13.4.3 Henckymodel . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
13.4.4 Blatz?Komaterial . . . . . . . . . . . . . . . . . . . . . . . . . 537
13.5 Application:Ogdenmaterial implementation . . . . . . . . . . . . . . . . 538
13.5.1 Subroutine SUOGD . . . . . . . . . . . . . . . . . . . . . . . . . 538
13.5.2 Subroutine CSTOGD . . . . . . . . . . . . . . . . . . . . . . . . . 542
13.6 Numericalexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
13.6.1 Axisymmetric extension of an annular plate . . . . . . . . . . . . 547
13.6.2 Stretching of a square perforated rubber sheet . . . . . . . . . . . 547
13.6.3 Inflation of a spherical rubber balloon . . . . . . . . . . . . . . . 548
13.6.4 Rugby ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
13.6.5 Inflation of initially flat membranes . . . . . . . . . . . . . . . . 552
13.6.6 Rubber cylinder pressed between two plates . . . . . . . . . . . . 554
13.6.7 Elastomericbeadcompression . . . . . . . . . . . . . . . . . . . 555
13.7 Hyperelasticity with damage: the Mullins effect . . . . . . . . . . . . . . . 556
13.7.1 TheGurtin?Francisuniaxialmodel . . . . . . . . . . . . . . . . 560
13.7.2 Three-dimensionalmodelling.Abrief review . . . . . . . . . . . 562
13.7.3 Asimple rate-independent three-dimensionalmodel . . . . . . . 562
13.7.4 Example: themodelproblem . . . . . . . . . . . . . . . . . . . . 565
13.7.5 Computational implementation . . . . . . . . . . . . . . . . . . . 565
13.7.6 Example: inflation/deflation of a damageable rubber balloon . . . 569
14 Finite strain elastoplasticity 573
14.1 Finite strainelastoplasticity: a brief review . . . . . . . . . . . . . . . . . 574
14.2 One-dimensionalfinite plasticitymodel . . . . . . . . . . . . . . . . . . . 574
14.2.1 The multiplicative split of the axial stretch . . . . . . . . . . . . . 575
14.2.2 Logarithmic stretches and the Hencky hyperelastic law . . . . . . 575
14.2.3 Theyieldfunction . . . . . . . . . . . . . . . . . . . . . . . . . 576
14.2.4 Theplasticflowrule . . . . . . . . . . . . . . . . . . . . . . . . 576
14.2.5 Thehardeninglaw . . . . . . . . . . . . . . . . . . . . . . . . . 577
14.2.6 The plastic multiplier . . . . . . . . . . . . . . . . . . . . . . . . 577
14.3 General hyperelastic-based multiplicative plasticity model . . . . . . . . . 578
14.3.1 Multiplicative elastoplasticity kinematics . . . . . . . . . . . . . 578
14.3.2 The logarithmicelastic strainmeasure . . . . . . . . . . . . . . . 582
14.3.3 Ageneral isotropiclarge-strainplasticitymodel . . . . . . . . . . 583
14.3.4 The dissipation inequality . . . . . . . . . . . . . . . . . . . . . 586
14.3.5 Finite strain extensionto infinitesimal theories . . . . . . . . . . 588
14.4 Thegeneralelastic predictor/return-mappingalgorithm . . . . . . . . . . . 590
14.4.1 The basic constitutive initial value problem . . . . . . . . . . . . 590
14.4.2 Exponential map backward discretisation . . . . . . . . . . . . . 591
14.4.3 Computational implementation of the general algorithm . . . . . 595
14.5 The consistent spatial tangent modulus . . . . . . . . . . . . . . . . . . . 597
14.5.1 Derivation of the spatial tangent modulus . . . . . . . . . . . . . 598
14.5.2 Computational implementation. . . . . . . . . . . . . . . . . . . 599
14.6 Principal stress space-based implementation . . . . . . . . . . . . . . . . . 600
14.6.1 Stress-updating algorithm . . . . . . . . . . . . . . . . . . . . . 600
14.6.2 Tangent modulus computation . . . . . . . . . . . . . . . . . . . 601
14.7 Finiteplasticityin planestress . . . . . . . . . . . . . . . . . . . . . . . . 601
14.7.1 Theplanestress-projectedfinite vonMisesmodel . . . . . . . . . 601
14.7.2 Nestediterationforplane stress enforcement . . . . . . . . . . . 604
14.8 Finiteviscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
14.8.1 Numerical treatment . . . . . . . . . . . . . . . . . . . . . . . . 606
14.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
14.9.1 Finite strain bendingof aV-notchedTrescabar . . . . . . . . . . 606
14.9.2 Neckingof a cylindricalbar . . . . . . . . . . . . . . . . . . . . 607
14.9.3 Plane strainlocalisation . . . . . . . . . . . . . . . . . . . . . . 611
14.9.4 Stretchingof aperforatedplate . . . . . . . . . . . . . . . . . . . 613
14.9.5 Thinsheetmetal-formingapplication . . . . . . . . . . . . . . . 614
14.10 Rate forms: hypoelastic-based plasticity models . . . . . . . . . . . . . . . 615
14.10.1 Objectivestress rates . . . . . . . . . . . . . . . . . . . . . . . . 619
14.10.2 Hypoelastic-based plasticity models . . . . . . . . . . . . . . . . 621
14.10.3 The Jaumannrate-basedmodel . . . . . . . . . . . . . . . . . . . 622
14.10.4 Hyperelastic-basedmodels andequivalent rate forms . . . . . . . 624
14.10.5 Integrationalgorithmsandincrementalobjectivity . . . . . . . . 625
14.10.6 Objectivealgorithmfor Jaumannrate-basedmodels . . . . . . . . 628
14.10.7 Integration of Green?Naghdi rate-based models . . . . . . . . . . 632
14.11 Finiteplasticitywith kinematichardening . . . . . . . . . . . . . . . . . . 633
14.11.1 Amodeloffinite strainkinematichardening . . . . . . . . . . . 633
14.11.2 Integrationalgorithm . . . . . . . . . . . . . . . . . . . . . . . . 638
14.11.3 Spatial tangentoperator . . . . . . . . . . . . . . . . . . . . . . 642
14.11.4 Remarks on predictive capability . . . . . . . . . . . . . . . . . . 644
14.11.5 Alternativedescriptions . . . . . . . . . . . . . . . . . . . . . . 644
15 Finite elements for large-strain incompressibility 647
15.1 The F-bar methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
15.1.1 Stress computation: the F-bar deformationgradient . . . . . . . . 649
15.1.2 The internal forcevector . . . . . . . . . . . . . . . . . . . . . . 651
15.1.3 Consistent linearisation: the tangent stiffness . . . . . . . . . . . 652
15.1.4 Plane strain implementation . . . . . . . . . . . . . . . . . . . . 655
15.1.5 Computational implementationaspects . . . . . . . . . . . . . . 656
15.1.6 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 656
15.1.7 Other centroid sampling-based F-bar elements . . . . . . . . . . 662
15.1.8 A more general F-bar methodology . . . . . . . . . . . . . . . . 662
15.1.9 The F-bar-PatchMethodfor simplexelements . . . . . . . . . . 665
15.2 Enhanced assumed strain methods . . . . . . . . . . . . . . . . . . . . . . 669
15.2.1 Enhancedthree-fieldvariationalprinciple . . . . . . . . . . . . . 669
15.2.2 EASfinite elements . . . . . . . . . . . . . . . . . . . . . . . . 671
15.2.3 Finite element equations: static condensation . . . . . . . . . . . 676
15.2.4 Implementationaspects . . . . . . . . . . . . . . . . . . . . . . . 680
15.2.5 The stability of EAS elements . . . . . . . . . . . . . . . . . . . 680
15.3 Mixed u/p formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 683
15.3.1 The two-fieldvariationalprinciple . . . . . . . . . . . . . . . . . 684
15.3.2 Finite elementequations . . . . . . . . . . . . . . . . . . . . . . 686
15.3.3 Solution: static condensation . . . . . . . . . . . . . . . . . . . . 688
15.3.4 Implementationaspects . . . . . . . . . . . . . . . . . . . . . . . 690
16 Anisotropic finite plasticity: Single crystals 691
16.1 Physical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
16.1.1 Plastic deformationbyslip: slip-systems . . . . . . . . . . . . . . 692
16.2 Plastic slip andtheSchmidresolvedshear stress . . . . . . . . . . . . . . 693
16.3 Single crystal simulation:a brief review . . . . . . . . . . . . . . . . . . . 694
16.4 A general continuum model of single crystals . . . . . . . . . . . . . . . . 694
16.4.1 Theplasticflowequation . . . . . . . . . . . . . . . . . . . . . . 695
16.4.2 The resolvedSchmidshear stress . . . . . . . . . . . . . . . . . 696
16.4.3 Multisurface formulation of the flow rule . . . . . . . . . . . . . 696
16.4.4 IsotropicTaylorhardening . . . . . . . . . . . . . . . . . . . . . 698
16.4.5 The hyperelastic law . . . . . . . . . . . . . . . . . . . . . . . . 698
16.5 Ageneral integrationalgorithm . . . . . . . . . . . . . . . . . . . . . . . 699
16.5.1 The searchforanactive set of slip systems . . . . . . . . . . . . 703
16.6 An algorithm for a planar double-slip model . . . . . . . . . . . . . . . . 705
16.6.1 A planar double-slip model . . . . . . . . . . . . . . . . . . . . . 705
16.6.2 The integrationalgorithm . . . . . . . . . . . . . . . . . . . . . 707
16.6.3 Example: themodelproblem . . . . . . . . . . . . . . . . . . . . 709
16.7 The consistent spatial tangent modulus . . . . . . . . . . . . . . . . . . . 714
16.7.1 The elastic modulus: compressible neo-Hookean model . . . . . . 714
16.7.2 The elastoplastic consistent tangent modulus . . . . . . . . . . . 714
16.8 Numericalexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
16.8.1 Symmetric strain localisation on a rectangular strip . . . . . . . . 717
16.8.2 Unsymmetriclocalisation . . . . . . . . . . . . . . . . . . . . . 720
16.9 Viscoplastic single crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 723
16.9.1 Rate-dependent formulation . . . . . . . . . . . . . . . . . . . . 723
16.9.2 The exponential map-based integration algorithm . . . . . . . . . 725
16.9.3 The spatial tangent modulus: neo-Hookean-based model . . . . . 727
16.9.4 Rate-dependentcrystal:modelproblem . . . . . . . . . . . . . . 727
Appendices 728
A Isotropic functions of a symmetric tensor 731
A.1 Isotropicscalar-valuedfunctions . . . . . . . . . . . . . . . . . . . . . . . 731
A.1.1 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
A.1.2 Thederivativeof anisotropicscalar function . . . . . . . . . . . 732
A.2 Isotropictensor-valuedfunctions . . . . . . . . . . . . . . . . . . . . . . . 733
A.2.1 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 733
A.2.2 Thederivativeof anisotropictensor function . . . . . . . . . . . 734
A.3 The two-dimensionalcase . . . . . . . . . . . . . . . . . . . . . . . . . . 735
A.3.1 Tensor functionderivative . . . . . . . . . . . . . . . . . . . . . 735
A.3.2 Plane strainandaxisymmetricproblems . . . . . . . . . . . . . . 737
A.4 The three-dimensionalcase . . . . . . . . . . . . . . . . . . . . . . . . . 739
A.4.1 Functioncomputation . . . . . . . . . . . . . . . . . . . . . . . 739
A.4.2 Computationof the functionderivative . . . . . . . . . . . . . . 740
A.5 Aparticularclass of isotropic tensor functions . . . . . . . . . . . . . . . 741
A.5.1 Twodimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 743
A.5.2 Threedimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 744
A.6 Alternativeprocedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
B The tensor exponential 747
B.1 The tensor exponential function . . . . . . . . . . . . . . . . . . . . . . . 747
B.1.1 Some properties of the tensor exponential function . . . . . . . . 748
B.1.2 Computation of the tensor exponential function . . . . . . . . . . 749
B.2 The tensor exponential derivative . . . . . . . . . . . . . . . . . . . . . . 749
B.2.1 Computer implementation . . . . . . . . . . . . . . . . . . . . . 751
B.3 Exponential map integrators . . . . . . . . . . . . . . . . . . . . . . . . . 751
B.3.1 The generalised exponential map midpoint rule . . . . . . . . . . 751
C Linearisation of the virtual work 753
C.1 Infinitesimaldeformations . . . . . . . . . . . . . . . . . . . . . . . . . . 753
C.2 Finite strains anddeformations . . . . . . . . . . . . . . . . . . . . . . . . 755
C.2.1 Materialdescription . . . . . . . . . . . . . . . . . . . . . . . . 755
C.2.2 Spatialdescription . . . . . . . . . . . . . . . . . . . . . . . . . 756
D Array notation for computations with tensors 759
D.1 Second-order tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
D.2 Fourth-order tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
D.2.1 Operations with non-symmetric tensors . . . . . . . . . . . . . . 763
References 765
Index 783

Library of Congress Subject Headings for this publication:

Plasticity -- Mathematical models.