Page 157 - The Combined Finite-Discrete Element Method

140 DEFORMABILITY OF DISCRETE ELEMENTS

˜
j = j x i + j y j + j z k
˜
˜
˜
˜
˜
k = k x i + k y j + k z k
˜
˜
In a similar way, base vectors of the local frame are expressed using base vectors of the
deformed local frame:
˜
i = i ˜x i + i ˜y j + i ˜z k (4.40)
˜
˜
j = j i + j j + j k
˜
˜
˜
˜ z
˜ y
˜ x
˜
k = k ˜x i + k ˜y j + k ˜z k
˜
˜
Any particular vector a can be expressed using either frame:
a = a x i + a y j + a z k (4.41)
= a x (i ˜x i + i ˜y j + i ˜z k)
˜
˜
˜
+ a y (j i + j j + j k)
˜
˜
˜
˜ z
˜ y
˜ x
+ a z (k ˜x i + k ˜y j + k ˜z k)
˜
˜
˜
As explained above, base vectors of the deformed local frame are obtained from the base
vectors of the local frame, thus
˜
˜
a = a ˜x i + a ˜y j + a ˜z k ˜ (4.42)

∂u ∂v ∂w
= a ˜x 1 + i + j + k
∂x ∂x ∂x
∂u ∂v ∂w

+ a ˜y i + 1 + j + k
∂y ∂y ∂y
∂u ∂v ∂w

+ a ˜z i + j + 1 + k
∂z ∂z ∂z
= a x i + a y j + a z k
The components of vector a in the local frame are therefore calculated from the vector
components in the deformed local frame as follows:
˜ ˜ ˜
 
  i x j x k x  
a x a ˜x
 
=  i y j y k y 
 a y  ˜ ˜ ˜  a ˜y  (4.43)
a z ˜ ˜ ˜ a ˜z
i z j z k z