Page 152 - The Combined Finite-Discrete Element Method

DEFORMATION GRADIENT 135

Since the triad vectors of the deformed frame are fixed to the material points in the
vicinity of material point p, it follows that the magnitude and orientation of this vectors
changes with deformation gradient in the vicinity of the point p, and is given by

∂u ∂u ∂u  ∂u 
 
1 + 1 +
∂x ∂y ∂z
     ∂x 
  1
∂v ∂v ∂v  
  ∂v
˜ i = Fi =  1 +      (4.16)
0
 ∂x ∂y ∂z  =  ∂x 
0  
   
∂w ∂w ∂w
  ∂w
1 +
∂x ∂y ∂z ∂x
∂u ∂v ∂w

= 1 + i + j + k
∂x ∂x ∂x
In the same way,
∂u ∂u ∂u ∂u
   
1 +
∂x ∂y ∂z
     ∂y 
  0  
∂v ∂v ∂v
   ∂v 
˜ j = Fj =  1 +    =   (4.17)
1
 ∂x ∂y ∂z   ∂y 
0
   
∂w ∂w ∂w ∂w
   
1 +
∂x ∂y ∂z ∂y
∂u
∂v ∂w
= i + 1 + j + k
∂y ∂y ∂y
and
∂u ∂u ∂u ∂u
   
1 +
∂x ∂y ∂z
     ∂z 
∂v ∂v ∂v
  0  
   ∂v 
˜
k = Fk =  1 +   0  =   (4.18)
∂x ∂y ∂z
   
  1  ∂z 
 ∂w ∂w ∂w   ∂w 
1 +
∂x ∂y ∂z ∂z
∂u ∂v
∂w
= i + j + 1 + k
∂z ∂z ∂z
In other words, through the deformation process, vectors of the local triad
(i, j, k) (4.19)
are mapped onto the vectors of the deformed local triad:
(i, j, k) (4.20)
˜ ˜ ˜