Page 163 - The Combined Finite-Discrete Element Method

146 DEFORMABILITY OF DISCRETE ELEMENTS

material element in the initial configuration is unit volume, then the same material element
after deformation occupies a volume equal to (det F). The second term can be written
as follows:

−1 T
F −T = (F )
∂u ∂u ∂u
 −1 T

1 +
 ∂x ∂y ∂z 
 
 
 
∂v ∂v ∂v
 
 
 
 1 +  (4.71)
=  
 ∂x ∂y ∂z 
 
 
 
 ∂w ∂w ∂w 
 
1 +
∂x ∂y ∂z
−T
 
˜ ˜ ˜
i x j x k x
i j
˜ ˜ ˜ ˜ ]
  −T
=  i y j y k y  = [ ˜ ˜ k
˜ ˜
i z j z k z
Matrix
 
˜ i x j x k x
˜
˜
 ˜ ˜ ˜  (4.72)
 i y j y k y 
˜ ˜ ˜
i z j z k z
is not an orthogonal matrix, thus the inverse matrix of this matrix is a non-
orthogonal matrix:
 −1  T
∂u ∂u ∂u

1 +   T
 ∂x ∂y ∂z  ˆ  ˆ ˆ ˆ 
  ˜ ˜ ˜ ˜
  i i x j k x
  x
∂v ∂v ∂v
 
     
   ˆ ˆ ˆ ˆ ˆ
˜
 1 +   =  ˆ  = i ˜ j ˜ ˜ ˆ  = [˜ ˜ ˜ ]
 j  y y i j k
∂x ∂y ∂z  
  k y 
 
  ˆ ˆ ˆ ˆ
  ˜ ˜ ˜ ˜
 ∂w ∂w ∂w  k i z j k z
  z
1 +
∂x ∂y ∂z
(4.73)
This inverse matrix represents the global components of a new triad of vectors
ˆ ˆ ˆ
(i, j, k) (4.74)
˜ ˜ ˜
This triad of vectors is associated with a deformed configuration. Vectors of this triad
have the following property:
ˆ ˆ ˆ
˜ ˜
˜ ˜
i · i = 1; j · i = 0; k · i = 0
˜ ˜
ˆ ˆ ˆ
˜ ˜ ˜ ˜ k · j = 0 (4.75)
i · j = 0;
j · j = 0;
˜ ˜
ˆ ˆ ˆ
i · k = 0;
˜ ˜ j · k = 0; k · k = 1
˜ ˜
˜ ˜