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. 54
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Finger tensor, 64; inverse of second order tensor, 27
see also left Cauchy“Green tensor isoparametric elements, 165
¬nite deformation analysis, 57, 59 isotropic elasticity in principle directions;
¬nite element method see principle directions
summary, 2 isotropic elasticity tensor, 125, 133, 139
¬nite element analysis isotropic material
3
INDEX



de¬nition, 121 multiplicative decomposition, 233
isotropic tensors, 33, 38, 41
natural strain, 5;
Jacobian, 74 see also logarithmic strain
Jaumann stress rate, 115 nearly incompressible materials, 131
neo-Hookean material
Kelvin e¬ect, 126 compressible, 124
Kirchho¬ stress tensor, 106 incompressible, 129
kinematics Newton“Raphson method, 13“8, 43“6,
de¬nition, 57 48, 184
discretized, 165 convergence, 18
solution algorithm, 18, 184“5
nodes, 2, 166
Lagrangian description, 59, 60
numbering of, 197
Lagrange multipliers (for incompressibility),
nonlinear computational mechanics
154“5
de¬nition, 1
Lagrangian elasticity tensor, 119, 130, 132, 137
nonlinear equations, 16
Lagrangian strain tensor, 65, 72;
general solution of, 44“7
see also Green strain
Lam` constants, 42, 125, 194
e
objectivity, 92, 101
large inelastic deformation;
objective stress rates, 113“4
see inelastic materials
Ogden materials, 145
left Cauchy“Green tensor, 64, 70
Oldroyd stress rate, 114
discretized, 167
orthogonal tensor, 28
Lie derivative, 87, 114, 121
out-of-balance force, 7, 103;
limit points, 7, 187
see also residual force
line search method, 185
linear stability analysis, 4 patch test, 223
linearization, 13“7, 43“51, 146 penalty method
algebraic equations, 16, 48 for incompressibility, 157
of determinant, 16, 50 penalty number, 131, 140
of inverse of tensor, 50 permanent deformation, 233
linearized perturbed Lagrangian functional, 157
deformation gradient, 78 Piola“Kirchho¬ stress tensor
Eulerian virtual work, 150 ¬rst, 107“9
external virtual work, 150“3 second, 10, 109;
Green™s strain, 79 see also stress tensor
Lagrangian virtual work, 149 Piola transformation, 111, 114, 121
right Cauchy“Green tensor, 79 planar deformation, 94 (Q.3)
virtual work, 147 plane strain, 142, 194
volume change, 80 plane stress, 142, 194
load increments, 17, 184 Poisson™s ratio, 42, 139, 143
locking polar decomposition, 28, 68“72
shear 161 (ex. 6.4) pressure, 112“3, 128, 131
volumetric, 158, 182 discretized, 182
logarithmic strain, 5, 73 pressure forces
stretch, 138 enclosed boundary, 153
linearization of, 151
mass conservation, 74 principle directions, 68“73, 88“9, 100
material description, 59“60 isotropic elasticity in, 134“44
material elasticity tensor, 119, 136 principle stresses, 100
material strain rate tensor, 85; principle of virtual work
see also Green strain material, 106“7, 110
material time derivative, 81 spatial, 104
material vector triad, 70 pull back, 63, 67, 79, 86, 110, 149
mean dilatation, 159, 160 pure dilatation;
deformation gradient, 161 see dilatation
discretization of, 182 push forward, 63, 67, 86, 110, 149
minimization of a function
simply supported beam, 48“9 radial return, 239
modi¬ed Newton“Raphson method, 18 rate of deformation, 84, 86
Mooney“Rivlin materials, 130 physical interpretation, 86“7
rate of volume change, 90
residual force, 7, 103, 171, 172
4 INDEX



return mapping, 238 tangent matrix, 17, 48
right Cauchy“Green tensor, 64, 68 assembled, 180“2
discretized, 167 constitutive component
distortional, 75 indicial form, 175
plane stress, 144 matrix form, 177
rigid body motion, 87, 92 dilatation component, 184
rotational equilibrium, 103 external force component, 179
rotation tensor, 68 initial stress component, 178
mean dilatation method, 182“4
scalar product, 22 source of, 147, 174
second order tensor, 26“36 tangent modulus, 241
inverse, 27 Taylor™s series expansion, 46
isotropic, 33 tensor analysis;
linearity of, 26 see gradient, divergence and integration
trace, 34 theorems
transpose, 27 tensor product, 28
second Piola“Kirchho¬ stress tensor, 10, 110; components of, 31
see also stress tensor properties of, 29
shape functions, 166 third order tensors, 37“9
shear, simple, 76, 126, 132 total potential energy, 14, 49, 154
simply supported beam, 48 trace
skew tensor, 27, 31, 37, 41 properties of, 34
small strain tensor, 10, 79 second order tensor, 34
snap back, 187 traction vector, 97, 102, 106
snap through behavior, 7 trial deformation gradient 236
spatial description, 59“60 transformation tensor, 28
spatial elasticity tensor, 120, 130, 137, 138 transpose
spatial vector triad, 70 second order tensor, 27
spatial virtual work equation, 106 Truesdell stress rate, 114
spin tensor, 87 truss member, 6
sti¬ness, 8 two-point tensor, 62, 69, 71, 107
strain energy function, 118;
see elastic potential uniaxial motion, 60
see also volumetric strain energy User instructions for computer program
strain measures FLagSHyP, 192
one dimensional nonlinear, 5;
variational statement
see also Green, Almansi
Hu-Washizu, 158
stress objectivity, 101
total potential energy, 154
stress tensor, 33
vectors, 22“6
Cauchy, 6, 96“100, 123, 124, 129, 139,
vector (cross) product, 26
140, 143
modulus (magnitude) of, 34
deviatoric, 112“3
transformation of, 24“6
¬rst Piola“Kirchho¬, 107, 118
velocity, 81
Kirchho¬, 106
velocity gradient, 83
physical interpretations, 108, 111
virtual work, 15;
second Piola“Kirchho¬, 10, 109, 110, 119,
see principle of
122, 124, 127, 128, 131, 134, 135
volume change, 73
stress rates
linearized, 80
convective, 114
rate of, 90
Green“Naghdi, 115
volumetric strain energy, 131, 140
Jaumann, 115
volumetric locking, 157, 158, 182
Oldroyd, 114
von Mises plasticity, 238
Truesdell, 114
stress vector, 172
work conjugacy, 106
stretch, 70
WWW address, xiv, 192
stretch tensor, 68“70
St. Venant“Kirchho¬ material, 120
Young™s modulus, 7, 42, 139, 143
superimposed rigid body motion, 92;
see also objectivity
surface forces
linearization of, 151;
see also pressure forces
symmetric tensor, 27, 41, 42

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. 54
( 54 .)