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e,±



A.5.3 TANGENT MODULUS
The derivation of the deviatoric component of the tangent modulus for this
constitutive model follows the same process employed in Section 5.6, with
the only di¬erence being that the ¬xed reference con¬guration is now the
unloaded state at increment n rather than the initial con¬guration. Similar
algebra thus leads to,
3 3
1 ‚„±±
ˆ
c= n± — n± — nβ — nβ ’ 2σ±± n± — n± — n± — n±
J ‚ ln »trial
e,β ±=1
±,β=1
2 2
3
σ±± »trial ’ σββ »trial
e,±
e,β
+ 2 (A.33)
n± — nβ — n± — nβ
2 trial 2
»trial ’ »e,β
e,±
±,β=1
±=β

In order to implement this equation the derivatives in the ¬rst term must
¬rst be evaluated from Equation (A.29) as,
trial
‚„±± 2µ ∆γ ‚„±± ‚ ∆γ
trial
= 1’ ’ 2µ„±±
‚ ln »trial trial
‚ ln »trial
‚ ln »e,β
„ trial „ trial
e,β e,β
(A.34)
where Equation (A.20b) shows that the derivatives of the trial stresses are
given by the elastic modulus as,
trial
‚„±±
= 2µδ±β ’ 2 µ (A.35)
3
trial
‚ ln »e,β
If the material is in an elastic regime, ∆γ = 0, and the second term in
Equation (A.34) vanishes.
12 LARGE INELASTIC DEFORMATIONS



In the elastoplastic regime, ∆γ = 0, and the second derivative of Equa-
tion (A.34) must be evaluated with the help of Equation (A.27) and the
chain rule as,
‚ „ trial
‚ ∆γ 1 1 ∆γ
= ’
2µ + 2 H
‚ ln »trial ‚ ln »trial
„ trial trial „ trial

e,β e,β
3
(A.36)
where simple algebra shows that,
3
‚ „ trial trial
1 ‚„±±
trial
= „±±
trial
‚ ln »trial
‚ ln »e,β „ trial e,β
±=1
3
1 2
trial
= „±± 2µδ±β ’ 3 µ
trial

±=1

= 2µνβ (A.37)
Finally, combining Equations (A.34“37) gives the derivatives needed to eval-
uate the consistent elastoplastic tangent modulus as,
‚„±± 2µ ∆γ
2µδ±β ’ 2 µ
= 1’ 3
‚ ln »trial „ trial
e,β
2µ 2µ ∆γ
’ 2µ ν± νβ ’ (A.38)
2 „ trial
2µ + 3 H
BIBLIOGRAPHY




1. bathe, k-j., Finite element procedures in engineering analysis, Prentice Hall, 1996.
2. crisfield, m. a., Non-linear ¬nite element analysis of solids and structures, Wiley, Volume
1, 1991.
´
3. eterovic, a. l., and bathe, k-l., A hyperelastic-based large strain elasto-plastic constitutive
formulation with combined isotropic-kinematic hardening using logarithmic stress and strain
measures, Int. J. Num. Meth. Engrg., 30, 1099“1114, 1990.
4. gurtin, m., An introduction to continuum mechanics, Academic Press, 1981.
5. hughes, t. j. r., and pister, k. s., Consistent linearization in mechanics of solids and
structures, Compt. & Struct., 8, 391“397, 1978.
6. hughes, t. j. r., The ¬nite element method, Prentice Hall, 1987.
7. lubliner, j., Plasticity theory, Macmillan, 1990.
8. malvern, l. e., Introduction to the mechanics of continuous medium, Prentice Hall, 1969.
9. marsden, j. e., and hughes, t. j. r., Mathematical foundations of Elasticity, Prentice-Hall,
1983.
10. miehe, c., Aspects of the formulation and ¬nite element implementation of large strain
isotropic elasticity, Int. J. Num. Meth. Engrg., 37, 1981“2004, 1994.
11. ogden, r. w., Non-linear elastic deformations, Ellis Horwood, 1984.
´
12. peric, d., owen, d. r. j., and honnor, m. e., A model for ¬nite strain elasto-plasticity
based on logarithmic strains: Computational issues, Comput. Meths. Appl. Mech. Engrg.,
94, 35“61, 1992.
13. schweizerhof, k., and ramm, e., Displacement dependent pressure loads in non-linear ¬nite
element analysis, Compt. & Struct., 18, 1099“1114, 1984.
14. simo, j., A framework for ¬nite strain elasto-plasticity based on a maximum plastic dissipa-
tion and the multiplicative decomposition: Part 1. Continuum formulation, Comput. Meths.
Appl. Mech. Engrg., 66, 199“219, 1988.
15. simo, j., Algorithms for static and dynamic multiplicative plasticity that preserve the classical
return mapping schemes of the in¬nitesimal theory, Comput. Meths. Appl. Mech. Engrg.,
99, 61“112, 1992.
16. simo, j., taylor, r. l., and pister, k. s., Variational and projection methods for the volume
constraint in ¬nite deformation elasto-plasticity, Comput. Meths. Appl. Mech. Engrg., 51,
177“208, 1985.
17. simo, j. c., and ortiz, m., A uni¬ed approach to ¬nite deformation elastoplastic analysis
based on the use of hyperelastic constitutive equations, Comput. Meths. Appl. Mech. Engrg.,
49, 221“245, 1985.
18. simo, j. c., and taylor, r. l., Quasi-incompressible ¬nite elasticity in principal stretches.
Continuum basis and numerical algorithms, Comput. Meths. Appl. Mech. Engrg., 85, 273“
310, 1991.



1
2 BIBLIOGRAPHY



19. simmonds, j. g., A brief on tensor analysis, Springer-Verlag, 2nd edition, 1994.
20. spencer, a. j. m., Continuum mechanics, Longman, 1980.
21. weber, g., and anand, l., Finite deformation constitutive equations and a time integration
procedure for isotropic, hyperelastic-viscoplastic solids, Comput. Meths. Appl. Mech. Engrg.,
79, 173“202, 1990.
22. zienkiewicz, o. c., and taylor, r. l., The ¬nite element method, McGraw-Hill, 4th edition,
Volumes 1 and 2, 1994.
INDEX




Almansi strain, 5 tangent matrix, 174“5
physical interpretation of, 67; tangent modulus, 240;
see also Eulerian strain see also elasticity tensor
tensor, 65, 72 cross product;
alternating tensor, 37 see vector product
angular velocity vector, 88 computer program, xiv
arc-length method, 187 simple, 19“20
area change, 77 computer implementation, 191“230
area ratio, 106 constitutive equations, 194, 201“4
assembly process, 171, 181 dictionary of main variables, 227;
augmented lagrangian method, 226 see also www and ftp addresses
element types, 193, 197
bifurcation point, 3, 4 solution algorithm, 206
Biot stress tensor, 112 solver, 200
body forces, 97, 102, 106 structure of, 205
linearization of, 151 user instructions, 192
buckling, 4
bulk modulus, 131, 140, 194 deformation gradient, 62
average, 162
cantilever discretized, 167
simple, 2 distortional, 74“7
Cartesian coordinates, 22, 58 incremental, 94 (Q.2), 236
Cauchy stress tensor, 6, 99 linearized, 78
objectivity of, 101 in principle directions, 70
in principle directions, 100 time derivative of, 83;
symmetry of, 104; see also polar decomposition
see also stress tensor density, 59
Cauchy“Green tensor; determinant of a tensor, 35
see right and left Cauchy“Green tensor linearization of, 16, 50
column deviatoric stress tensor, 112“3
simple, 3 deviatoric tensor, 42
continuity equation, 74, 91 dilatation
convective pure, 125, 132
derivative, 82 directional derivative, 13, 14“7, 43“51
stress rate, 114 of a determinant, 16, 50
compressible neo-Hookean material, 124 of inverse of a tensor, 50
conservation of mass, 74 linearity of, 47
rate form of, 90 properties of, 47
constitutive and time rates, 82
equations, 117 of volume element, 80
matrix, 176 discretized equilibrium equations



1
2 INDEX



matrix-vector form, 172“3 plane strain strip, 225
tensor form, 171“2 simple patch test, 223
distortional strip with a hole, 225
deformation gradient, 74“7 truss member, 224
stretches, 140 ¬rst Piola“Kirchho¬ stress tensor, 107;
divergence, 53 see also stress tensor
discretized average, 183 fourth order tensor, 39“42
properties, 53“4 identity, 40“1
dot product; isotropic, 41
see scalar product symmetric, 42
double contraction, 35, 38“40 ftp address, 192
properties of, 35, 39, 40
dyadic product; Gauss point numbering, 197
see tensor product Gauss theorem, 54“5
generalized strain measures, 72
e¬ective Lam` coe¬cients, 42, 125, 142
e geometric sti¬ness;
Einstein summation convention, 22 see initial stress stifness
eigenvectors gradient, 52
of second order tensors, 36 properties of, 53“4
eigenvalues Green (or Green™s) strain, 5, 6, 12, 65
of second order tensors, 36 linearized, 79
elastic potential, 118 physical interpretation, 66
compressible neo-Hookean, 124 time derivative of, 85, 89
incompressible neo-Hookean, 129 Green“Naghdi stress rate, 115
Mooney“Rivlin, 130
nearly incompressible, 131 homogeneous potential, 128
St. Venant“Kirchho¬, 120 Hu-Washizu variational principle
elastic potential in principle directions, 134 six ¬eld, 164
nearly incompressible, 140“1 three ¬eld, 158
plane strain, 142 hydrostatic pressure;
plane stress, 142 see pressure
simple stretch based, 138 hyperelasticity
uniaxial rod, 143 de¬nition, 118
elasticity tensor, 40, 42 incompressible and nearly incompressible,
Eulerian (spatial), 120“1, 133, 138, 142; 126“34
see also isotropic elasticity tensor isotropic, 121“6
Lagrangian (material), 119“20, 121“2, 137 in principle directions, 134“44
engineering strain, 5
equilibrium equations identity tensor, 27
di¬erential, 103, 108 incompressible materials, 126“34
discretized, 171“3 incompressibility
linearized, 146; Lagrange multiplier approach, 154
see also principal of virtual work mean dilatation approach, 160, 182
rotational, 103 penalty approach, 157
translational, 101 inelastic materials, 231
Euler buckling load, 4 incremental kinematics for, 236
Eulerian description, 59“60 radial return, 239,
Eulerian elasticity tensor, 120, 138 stress evaluation, 238
Eulerian strain tensor, 65, 72 tangent modulus, 241
physical interpretation of, 67; initial stress sti¬ness, 10, 177“8, 190
see Almansi strain integration theorems, 54
equivalent nodal forces internal equivalent nodal forces;
external, 170, 171, 190 see equivalent forces
internal, 170, 171, 173 internal virtual work, 106
equivalent strain, 239 invariants
e-mail addresses, 192 tensor, 33, 34“36, 121, 123, 134
vector, 34

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