Page 334 - The Combined Finite-Discrete Element Method

DEFORMABILITY WITH FINITE ROTATIONS IN 3D 317

The matrix of the velocity gradient tensor

 
∂v xc ∂v xc ∂v xc
 ∂x i ∂y i ∂z i 
 
 ∂v yc ∂v yc ∂v yc 
(10.12)
 
 ∂x i ∂y i ∂z i 
 
∂v zc ∂v zc ∂v zc
∂x i ∂y i ∂z i
is stored as the two-dimensional array L[3][3].
In Listing 10.40, strain tensors are evaluated. These are as follows:

• Left Cauchy–Green strain tensor. The matrix of this tensor is stored as two-dimensional
array B[3][3].
• Green–St.Venant strain tensor. The matrix of this tensor is stored as array E[3][3].
• Rate of deformation tensor. The matrix of the rate of deformation tensor is stored using
array D[3][3].

In Listing 10.41, the Cauchy stress tensor is calculated. This is achieved by employing
the appropriate constitutive law, depending on the material. At this point, a MACRO
resolving the stress-strain relationship is called. A detailed description of a set of physical
equations for homogeneous isotropic elastic material is given in Chapter 4. However, any
constitutive law can be employed at this point, including material nonlinearity such as
plastic, plastic hardening or softening material.

.............continued from previous listing..............
for(i=0;i<3;i++)
{ for(j=0;j<3;j++)
{ B[i][j]=R0;
for(k=0;k<3;k++)
{ B[i][j]=B[i][j]+F[i][k]*F[j][k]; /* left Cauchy-Green strain */
}
D[i][j]=RP5*(L[i][j]+L[j][i]); /* rate of deformation */
if(i==j)
{ E[i][j]=RP5*(B[i][j]-R1); /* Green-St.Venant strain */
}
else
{ E[i][j]=RP5*B[i][j];
}}}
Listing 10.40 Strain calculation.

..........continued from previous listing...........
/* Cauchy stress */
{ ConstitutiveLaw(E,T,volc/voli);
}
Listing 10.41 Cauchy stress tensor.