ABAQUS user subroutines for the simulation of viscoplastic - loicz
ABAQUS user subroutines for the simulation of viscoplastic - loicz
ABAQUS user subroutines for the simulation of viscoplastic - loicz
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W internal variable, eq. (20b)<br />
fourth order tensors<br />
< 4><br />
Ai < 4><br />
I<br />
< 4> <br />
P , ε Pεe < 4><br />
R<br />
< 4><br />
S<br />
< 4><br />
T<br />
operations<br />
ab = a i b j e i e j<br />
a ⋅ b = a ib i<br />
8<br />
material tensors (i = 1, 2, 3, 4, 5), eqs. (21a-e)<br />
identity tensor<br />
positive spectral projection operator <strong>for</strong> total and elastic strain tensor, eqs. (10a,b)<br />
lattice tensor, eq. (5)<br />
damage characteristic tensor, eq. (3a)<br />
positive projection operator, eq. (12)<br />
AB = A ij B kl e ie ke je l<br />
A ⋅ B= A ijB jk e ie k<br />
A : B= A ij B ji<br />
tensor product <strong>of</strong> two vectors<br />
scalar product <strong>of</strong> two vectors<br />
tensor product <strong>of</strong> two (second order) tensors<br />
scalar product <strong>of</strong> two (second order) tensors<br />
double scalar product <strong>of</strong> two (second order) tensors<br />
A = A 2 = A ij A ji EUKLIDean norm <strong>of</strong> second order tensor<br />
< 4><br />
C : A = C A e e ijkl kl i j<br />
double scalar product <strong>of</strong> a fourth and a second order tensor