Page 158 - The Combined Finite-Discrete Element Method

HOMOGENEOUS DEFORMATION 141

where transformation matrix is given by
∂u ∂u ∂u
 
1 +
  ∂x ∂y ∂z
˜ ˜ ˜  
i x j x k x
∂v ∂v ∂v
 
 
˜ ˜ ˜
 
 i y j y k y  =  1 +  (4.44)
 ∂x ∂y ∂z 
˜ ˜ ˜  
i z j z k z  ∂w ∂w ∂w 
1 +
∂x ∂y ∂z
By analogy, the components of vector a in the local deformed frame are calculated from
the vector components in the local frame using the following transformation:
 
  j  
i ˜x ˜ x k ˜x
a ˜x a x
 
a ˜y =  i ˜y a y
˜ y k ˜y 
  j   (4.45)
a ˜z a z
i ˜z j k ˜z
˜ z
where the transformation matrix is given by
∂u ∂u ∂u
 −1
1 +
−1
   ˜ ˜ ˜ ∂x ∂y ∂z
i ˜x j ˜ x k ˜x i x j x k x  

∂v ∂v ∂v

 
 j   ˜ ˜ ˜  (4.46)
 i ˜y ˜ y k ˜y  =  i y j y =  1 + 
k y 
 ∂x ∂y ∂z 
˜
˜
i ˜z j ˜ z k ˜z ˜ i z j z k z  ∂w ∂w ∂w 


1 +
∂x ∂y ∂z
4.3 HOMOGENEOUS DEFORMATION
Homogeneous deformation can be expressed as a composition of rotation g and stretch s:
f(p) = g◦s 1 = s 2 g (4.47)
◦
The deformation gradient for homogeneous deformation is therefore given by
F = RU = VR (4.48)
where
R =∇g
(4.49)
U =∇s 1
V =∇s 2
It is worth mentioning that by definition of homogeneous deformation tensors F, R and U
are constant tensors, i.e. they do not change from point to point (over the spatial domain).