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Bonet J., Wood R.D. Nonlinear Continuum Mechanics for Finite Element Analysis
Bonet J., Wood R.D.  Nonlinear Continuum Mechanics for Finite Element Analysis







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: Nonlinear Continuum Mechanics for Finite Element Analysis

: Bonet J., Wood R.D.

:

This book provides a look at the theory behind the programs engineers use for the computer simulation of nonlinear structural behaviour. It establishes the mathematical foundations for the development of computer programs that can predict the behaviour of mechanical and structural components. After a thorough but succinct introduction, the book delves into mathematical preliminaries, kinematics, stress and equilibrium. Subsequent sections address hyperelasticity for compressible and incompressible materials, finite element discretisation, equation solution and computer implementation. A short appendix extends the kinematics chapter to cater for elasto-plastic deformation. The book provides user instructions, program description and examples for the FLAGSHYP computer implementation for which the source code is available free on the Internet. Worked examples and exercises complete each chapter, making it an essential resource for engineers and researchers.


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ed2k: ed2k stats

: 1997

: 268

: 20.02.2005

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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)
CauchyGreen tensor      see Right and left CauchyGreen tensor
Column, simple      3
Compressible neo-Hookean material      124
Computer implementation      191230
Computer implementation, constitutive equations      194 201204
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      1920
Conservation of mass      74
Conservation of mass, rate form of      90
Constitutive, equations      117
Constitutive, matrix      176
Constitutive, tangent matrix      174175
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      7477
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      112113
Deviatoric tensor      42
Dilatation, pure      125 132
Directional derivative      13 1417 4351
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      172173
Discretized equilibrium equations, tensor form      171172
Distortional, deformation gradient      7477
Distortional, stretches      140
Divergence      53
Divergence, discretized average      183
Divergence, properties      5354
Dot product      see Scalar product
Double contraction      35 3840
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      140141
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)      120121 133 138 142
Elasticity tensor, Lagrangian (material)      119120 121122 137
Engineering strain      5
Equilibrium equations, differential      103 108
Equilibrium equations, discretized      171173
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      5960
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      3942
Fourth order tensor, identity      4041
Fourth order tensor, isotropic      41
Fourth order tensor, symmetric      42
Ftp address      192
Gauss point numbering      197
Gauss theorem      5455
Generalized strain measures      72
Geometric stiffness      see Initial stress stifness
Gradient      52
Gradient, properties of      5354
Green (or Greens) strain      5 6 12 65
Green Naghdi stress rate      115
Greens strain, linearized      79
Greens strain, physical interpretation      66
Greens 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      134144
Hyperelasticity, definition      118
Hyperelasticity, incompressible and nearly incompressible      126134
Hyperelasticity, isotropic      121126
Identity tensor      27
Incompressibility, Lagrange multiplier approach      154
Incompressibility, mean dilatation approach      160 182
Incompressibility, penalty approach      157
Incompressible materials      126134
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 177178 190
Integration theorems      54
Internal equivalent nodal forces      see Equivalent forces
Internal virtual work      106
Invariants, tensor      33 3436 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)      154155
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      1317 4351 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      1503
Linearized, Greens 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      5960
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      4849
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      1318 4346 48 184
Newton Raphson method, convergence      18
Newton Raphson method, solution algorithm      18 184185
nodes      2 166
Nodes, numbering of      197
Nonlinear computational mechanics, definition      1
Nonlinear equations      16
Nonlinear equations, general solution of      4447
Objective stress rates      113114
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      107109
Piola Kirchhoff stress tensor, second      10 109
Planar deformation      94 (Q.3)
Plane strain      142 194
Plane stress      142 194
Poissons ratio      42 139 143
Polar decomposition      28 6872
Pressure      112113 128 131
Pressure forces, enclosed boundary      153
Pressure forces, linearization of      151
Pressure, discretized      182
Principle directions      6873 8889 100
Principle directions, isotropic elasticity in      134144
Principle of virtual work, material      106107 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      8687
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      2636
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      5960
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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