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| Bonet J., Wood R.D. — Nonlinear Continuum Mechanics for Finite Element Analysis |
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| Ïðåäìåòíûé óêàçàòåëü |
Almansi strain 5
Almansi strain, physical interpretation of 67 (see also “Eulerian strain”)
Almansi strain, tensor 65 72
Alternating tensor 37
Angular velocity vector 88
Arc-length method 187
Area change 77
Area ratio 106
Assembly process 171 181
Augmented Lagrangian method 226
Bifurcation point 3 4
Biot stress tensor 112
Body forces 97 102 106
Body forces, linearization of 151
Buckling 4
bulk modulus 131 140 194
Cantilever, simple 2
Cartesian coordinates 22 58
Cauchy stress tensor 6 99
Cauchy stress tensor in principle directions 100
Cauchy stress tensor, objectivity of 101
Cauchy stress tensor, symmetry of 104 (see also “Stress tensor”)
Cauchy–Green tensor see “Right and left Cauchy–Green tensor”
Column, simple 3
Compressible neo-Hookean material 124
Computer implementation 191—230
Computer implementation, constitutive equations 194 201—204
Computer implementation, dictionary of main variables 227 (see also “www and ftp addresses”)
Computer implementation, element types 193 197
Computer implementation, solution algorithm 206
Computer implementation, solver 200
Computer implementation, structure of 205
Computer implementation, user instructions 192
Computer program xiv
Computer program, simple 19—20
Conservation of mass 74
Conservation of mass, rate form of 90
Constitutive, equations 117
Constitutive, matrix 176
Constitutive, tangent matrix 174—175
Constitutive, tangent modulus 240 (see also “Elasticity tensor”)
Continuity equation 74 91
Convective, derivative 82
Convective, stress rate 114
Cross product see “Vector product”
Deformation gradient 62
Deformation gradient in principle directions 70
Deformation gradient, average 162
Deformation gradient, discretized 167
Deformation gradient, distortional 74—77
Deformation gradient, incremental 94 (Q.2) 236
Deformation gradient, linearized 78
Deformation gradient, time derivative of 83 (see also “Polar decomposition”)
density 59
Determinant of a tensor 35
Determinant of a tensor, linearization of 16 50
Deviatoric stress tensor 112—113
Deviatoric tensor 42
Dilatation, pure 125 132
Directional derivative 13 14—17 43—51
Directional derivative and time rates 82
Directional derivative of a determinant 16 50
Directional derivative of inverse of a tensor 50
Directional derivative of volume element 80
Directional derivative, linearity of 47
Directional derivative, properties of 47
Discretized equilibrium equations, matrix-vector form 172—173
Discretized equilibrium equations, tensor form 171—172
Distortional, deformation gradient 74—77
Distortional, stretches 140
Divergence 53
Divergence, discretized average 183
Divergence, properties 53—54
Dot product see “Scalar product”
Double contraction 35 38—40
Double contraction, properties of 35 39 40
Dyadic product see “Tensor product”
E-mail addresses 192
Effective Lame coefficients 42 125 142
Eigenvalues of second order tensors 36
Eigenvectors of second order tensors 36
Einstein summation convention 22
Elastic potential 118
Elastic potential in principle directions 134
Elastic potential in principle directions, nearly incompressible 140—141
Elastic potential in principle directions, plane strain 142
Elastic potential in principle directions, plane stress 142
Elastic potential in principle directions, simple stretch based 138
Elastic potential in principle directions, uniaxial rod 143
Elastic potential, compressible neo-Hookean 124
Elastic potential, incompressible neo-Hookean 129
Elastic potential, Mooney — Rivlin 130
Elastic potential, nearly incompressible 131
Elastic potential, St. Venant — Kirchhoff 120
Elasticity tensor 40 42
Elasticity tensor, Eulerian (spatial) 120—121 133 138 142
Elasticity tensor, Lagrangian (material) 119—120 121—122 137
Engineering strain 5
Equilibrium equations, differential 103 108
Equilibrium equations, discretized 171—173
Equilibrium equations, linearized 146 (see also “Principal of virtual work”)
Equilibrium equations, rotational 103
Equilibrium equations, translational 101
Equivalent nodal forces, external 170 171 190
Equivalent nodal forces, internal 170 171 173
Equivalent strain 239
Euler buckling load 4
Eulerian description 59—60
Eulerian elasticity tensor 120 138
Eulerian strain tensor 65 72
Eulerian strain tensor, physical interpretation of 67 (see also “Almansi strain”)
Finger tensor 64 (see also “Left Cauchy — Green tensor”)
Finite deformation analysis 57 59
Finite element analysis, plane strain strip 225
Finite element analysis, simple patch test 223
Finite element analysis, strip with a hole 225
Finite element analysis, truss member 224
Finite element method summary 2
First Piola — Kirchhoff stress tensor 107 (see also “Stress tensor”)
Fourth order tensor 39—42
Fourth order tensor, identity 40—41
Fourth order tensor, isotropic 41
Fourth order tensor, symmetric 42
Ftp address 192
Gauss point numbering 197
Gauss theorem 54—55
Generalized strain measures 72
Geometric stiffness see “Initial stress stifness”
Gradient 52
Gradient, properties of 53—54
Green (or Green’s) strain 5 6 12 65
Green — Naghdi stress rate 115
Green’s strain, linearized 79
Green’s strain, physical interpretation 66
Green’s strain, time derivative of 85 89
Homogeneous potential 128
Hu-Washizu variational principle, six field 164
Hu-Washizu variational principle, three field 158
Hydrostatic pressure see “Pressure”
Hyperelasticity in principle directions 134—144
Hyperelasticity, definition 118
Hyperelasticity, incompressible and nearly incompressible 126—134
Hyperelasticity, isotropic 121—126
Identity tensor 27
Incompressibility, Lagrange multiplier approach 154
Incompressibility, mean dilatation approach 160 182
Incompressibility, penalty approach 157
Incompressible materials 126—134
Inelastic materials 231
Inelastic materials, incremental kinematics for 236
| Inelastic materials, radial return 239
Inelastic materials, stress evaluation 238
Inelastic materials, tangent modulus 241
Initial stress stiffness 10 177—178 190
Integration theorems 54
Internal equivalent nodal forces see “Equivalent forces”
Internal virtual work 106
Invariants, tensor 33 34—36 121 123 134
Invariants, vector 34
Inverse of second order tensor 27
Isoparametric elements 165
Isotropic elasticity in principle directions see “Principle directions”
Isotropic elasticity tensor 125 133 139
Isotropic material, definition 121
Isotropic tensors 33 38 41
Jacobian 74
Jaumann stress rate 115
Kelvin effect 126
Kinematics, definition 57
Kinematics, discretized 165
Kirchhoff stress tensor 106
Lagrange multipliers (for incompressibility) 154—155
Lagrangian description 59 60
Lagrangian elasticity tensor 119 130 132 137
Lagrangian strain tensor 65 72
Lame constants 42 125 194
Large inelastic deformation see “Inelastic materials”
Left Cauchy — Green tensor 64 70
Left Cauchy — Green tensor, discretized 167
Lie derivative 87 114 121
Limit points 7 187
Line search method 185
Linear stability analysis 4
Linearization 13—17 43—51 146
Linearization of determinant 16 50
Linearization of inverse of tensor 50
Linearization, algebraic equations 16 48
Linearized, deformation gradient 78
Linearized, Eulerian virtual work 150
Linearized, external virtual work 150–3
Linearized, Green’s strain 79
Linearized, Lagrangian virtual work 149
Linearized, right Cauchy — Green tensor 79
Linearized, virtual work 147
Linearized, volume change 80
Load increments 17 184
Locking shear 161 (ex. 6.4)
Locking volumetric 158 182
Logarithmic strain 5 73
Logarithmic strain, stretch 138
Mass conservation 74
Material description 59—60
Material elasticity tensor 119 136
Material strain rate tensor 85 (see also “Green strain”)
Material time derivative 81
Material vector triad 70
Mean dilatation 159 160
Mean dilatation, deformation gradient 161
Mean dilatation, discretization of 182
Minimization of a function, simply supported beam 48—49
Modified Newton — Raphson method 18
Mooney — Rivlin materials 130
Multiplicative decomposition 233
Natural strain 5 (see also “Logarithmic strain”)
Nearly incompressible materials 131
Neo-Hookean material, compressible 124
Neo-Hookean material, incompressible 129
Newton — Raphson method 13—18 43—46 48 184
Newton — Raphson method, convergence 18
Newton — Raphson method, solution algorithm 18 184—185
nodes 2 166
Nodes, numbering of 197
Nonlinear computational mechanics, definition 1
Nonlinear equations 16
Nonlinear equations, general solution of 44—47
Objective stress rates 113—114
Objectivity 92 101
Ogden materials 145
Oldroyd stress rate 114
Orthogonal tensor 28
Out-of-balance force 7 103
Patch test 223
Penalty method for incompressibility 157
Penalty number 131 140
Permanent deformation 233
Perturbed Lagrangian functional 157
Piola transformation 111 114 121
Piola — Kirchhoff stress tensor, first 107—109
Piola — Kirchhoff stress tensor, second 10 109
Planar deformation 94 (Q.3)
Plane strain 142 194
Plane stress 142 194
Poisson’s ratio 42 139 143
Polar decomposition 28 68—72
Pressure 112—113 128 131
Pressure forces, enclosed boundary 153
Pressure forces, linearization of 151
Pressure, discretized 182
Principle directions 68—73 88—89 100
Principle directions, isotropic elasticity in 134—144
Principle of virtual work, material 106—107 110
Principle of virtual work, spatial 104
Principle stresses 100
Pull back 63 67 79 86 110 149
Pure dilatation see “Dilatation”
Push forward 63 67 86 110 149
Radial return 239
Rate of deformation 84 86
Rate of deformation, physical interpretation 86—87
Rate of volume change 90
Residual force 7 103 171 172
Return mapping 238
Right Cauchy — Green tensor 64 68
Right Cauchy — Green tensor, discretized 167
Right Cauchy — Green tensor, distortional 75
Right Cauchy — Green tensor, plane stress 144
Rigid body motion 87 92
Rotation tensor 68
Rotational equilibrium 103
Scalar product 22
Second order tensor 26—36
Second order tensor, inverse 27
Second order tensor, isotropic 33
Second order tensor, linearity of 26
Second order tensor, trace 34
Second order tensor, transpose 27
Second Piola — Kirchhoff stress tensor 10 110
Shape functions 166
Shear, simple 76 126 132
Simply supported beam 48
Skew tensor 27 31 37 41
Small strain tensor 10 79
Snap back 187
Snap through behavior 7
Spatial description 59—60
Spatial elasticity tensor 120 130 137 138
Spatial vector triad 70
Spatial virtual work equation 106
Spin tensor 87
St. Venant — Kirchhoff material 120
Stiffness 8
Strain energy function 118 (see also “Elastic potential and “Volumetric strain energy”)
Strain measures, one dimensional nonlinear 5 (see also “Green Almansi”)
Stress objectivity 101
Stress rates, convective 114
Stress rates, Green — Naghdi 115
Stress rates, Jaumann 115
Stress rates, Oldroyd 114
Stress rates, Truesdell 114
Stress tensor 33
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