Page 188 - The Combined Finite-Discrete Element Method

CONSTANT STRAIN TETRAHEDRON FINITE ELEMENT 171

A deformation gradient in the form

∂x c ∂x c ∂x c
 
 ∂x i ∂y i ∂z i 
 
 ∂y c ∂y c ∂y c 
F =   (4.209)
 
∂x i ∂y i
 ∂z i 
∂z c ∂z c ∂z c
 
∂x i ∂y i ∂z i
is obtained using the initial base. This base is given by the matrix

   

i x
j x
k x x 1i − x 0i x 2i − x 0i x 3i − x 0i


=  y 1i − y 0i (4.210)
i y j y
 y 2i − y 0i y 3i − y 0i 
k y

i z
j z
k z z 1i − z 0i z 2i − z 0i z 3i − z 0i
It is worth noting that node 0 in the initial configuration coincides with the origin of the
initial frame.
Using coordinate transformations explained above, the following expression for the
deformation gradient is obtained:

   
∂x c ∂x c ∂x c ∂x c ∂x c ∂x c
 ∂x i ∂y i ∂z i   ∂
x i ∂
y i ∂
z i   −1

   i x j x k x
 ∂y c ∂y c ∂y c   ∂y c ∂y c ∂y c 
F =   =   
i y
j y
 (4.211)
k y
∂x i ∂y i ∂
x i ∂
y i ∂
z 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
In a similar way, the velocity gradient is given by
   
∂v xc ∂v xc ∂v xc ∂v xc ∂v xc ∂v xc
 ∂x i ∂y i ∂z i   ∂
x i ∂
y i ∂
z i   −1

   i x j x k x
 ∂v yc ∂v yc ∂v yc   ∂v yc ∂v yc ∂v yc 
L =   =   
i y
j y
 (4.212)
k y
∂x i ∂y i ∂
x i ∂
y i ∂
z 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
As explained before, the deformation gradient is comprised of either rotation followed
by stretch
F = VR (4.213)
or stretch followed by rotation
F = RU (4.214)