Page 333 - The Combined Finite-Discrete Element Method

316 ALGORITHM IMPLEMENTATION

The array FX[3][3] also represents the matrix of the deformation gradient:
∂x c ∂x c ∂x c
 
 ∂
x i ∂
y i ∂
z i 


 ∂y c ∂y c ∂y c 
(10.8)
F = 

 ∂
x i ∂
y i ∂
z i 
 
∂z c ∂z c ∂z c
∂
x i ∂
y i ∂
z i
This matrix represents the deformation gradient tensor using the base vectors of the initial
frame (undeformed tetrahedron). The matrix of the deformation gradient that uses the base
vectors of the global frame is obtained as follows:
   
∂x c ∂x c ∂x c ∂x c ∂x c ∂x c
 ∂x i ∂y i ∂z i   ∂
x i ∂
y i ∂
z i   
i
x j
x k
x
   
 ∂y c ∂y c ∂y c   ∂y c ∂y c ∂y c 
 i
y  (10.9)
j
y k
y
 =  
F = 
 ∂x i ∂y i ∂z i   ∂
x i ∂
y i ∂
z i 
i
z j
z k
z
   
∂z c ∂z c ∂z c ∂z c ∂z c ∂z c
∂x i ∂y i ∂z i ∂
x i ∂
y i ∂
z i
These calculations are performed in Listing 10.39, where the deformation gradient matrix
 
∂x c ∂x c ∂x c
 ∂x i ∂y i ∂z i 
 
 ∂y c ∂y c ∂y c 
(10.10)
 
 ∂x i ∂y i ∂z i 
 
∂z c ∂z c ∂z c
∂x i ∂y i ∂z i
is represented by the array F[3][3].
In Listing 10.39, the matrix of the velocity gradient tensor is also calculated using the
following formula:
   
∂v xc ∂v xc ∂v xc ∂v xc ∂v xc ∂v xc
 ∂x i ∂y i ∂z i   ∂
x i ∂
y i ∂
z i   
i
x j
x k
x


 ∂v yc ∂v yc ∂v yc   ∂v yc ∂v yc ∂v yc 


i
y j
y k
y
  (10.11)
L = 
 =  
 ∂x i ∂y i ∂z i   ∂
x i ∂
y i ∂
z i 
i
z j
z k
z
∂v zc ∂v zc ∂v zc ∂v zc ∂v zc ∂v zc
   
∂x i ∂y i ∂z i ∂
x i ∂
y i ∂
z i
.............continued from previous listing..............
for(i=0;i<3;i++)
{ for(j=0;j<3;j++)
{ F[i][j]=R0;
L[i][j]=R0;
for(k=0;k<3;k++)
{ F[i][j]=F[i][j]+FX[i][k]*F0inv[k][j];/* deform. gradient */
L[i][j]=L[i][j]+LX[i][k]*FXinv[k][j];/* velocity gradient */
}}}
Listing 10.39 Deformation and velocity gradients.