Page 189 - The Combined Finite-Discrete Element Method

172 DEFORMABILITY OF DISCRETE ELEMENTS

Deformed configuration
(first rotated then stretched)
2
Initial (undeformed)
configuration
j 2
y
3 1
k 3
j 0
1
k i x i
0 Rotated configuration
(undeformed)
z

Figure 4.21 Left stretch: rotation of tetrahedron followed by stretch.

Either can be employed. However, employing the left stretch V is computationally more
efficient. The physical meaning of this is that the tetrahedron is first rotated and then
stretched (Figure 4.21).
Using left stretch tensor, the left Cauchy–Green strain tensor is calculated as follows:

   
∂x c ∂x c ∂x c ∂x c ∂y c ∂z c
 ∂x i ∂y i ∂z i   ∂x i ∂x i ∂x i 
   
 ∂y c ∂y c ∂y c   ∂x c ∂y c ∂z c 
T T
B = FF = VV =     (4.215)
∂x i ∂y i ∂y i ∂y i
   
 ∂z i   ∂y i 
∂z c ∂z c ∂z c ∂x c ∂y c ∂z c
   
∂x i ∂y i ∂z i ∂z i ∂z i ∂z i
In a similar way, the rate of deformation is obtained from velocity gradient
∂v xc ∂v xc ∂v xc ∂v xc ∂v yc ∂v zc
   
 ∂x i ∂y i ∂z i   ∂x i ∂x i ∂x i 

1 T 1  ∂v yc ∂v yc ∂v yc   ∂v yc ∂v zc 
 ∂v xc


D = (L + L ) =   +   (4.216)
2 2  ∂x i ∂y i   ∂y i ∂y i 
 ∂z i   ∂y i 
   
∂v zc ∂v zc ∂v zc ∂v xc ∂v yc ∂v zc
∂x i ∂y i ∂z i ∂z i ∂z i ∂z i
From the left Cauchy–Green strain tensor, for small strains a Green–St. Venant strain
tensor is obtained as follows:
1 2 1
˘
E = V − I = B − I (4.217)
2 2
The Green–St. Venant strain tensor due to the shape changing stretch is therefore given by
1 T 1
FF T
˜
E d = (V d V − I) = − I (4.218)
d
2 2 (| det F|) 2/3