Page 153 - The Combined Finite-Discrete Element Method

136 DEFORMABILITY OF DISCRETE ELEMENTS

with
∂u
   ∂u   ∂u 
1 +
 ∂y   ∂z 
 ∂x 
   
∂v  ∂v   ∂v 
 
˜
˜   ; ˜ j =  1 +  ; k =   (4.21)

i = 
∂x  ∂y   ∂z 
 
   
∂w  ∂w   ∂w 
 
1 +
∂x ∂y ∂z
The physical meaning of the deformation gradient can be explained by taking an infinites-
imal material element in the vicinity of point p, as shown in Figure 4.3. It is assumed
that this material element coincides with a cube of edge of unit length. By choosing a
very small unit for the length, the edge of the cube is made infinitesimally small.
With such an assumption, the local triad at point p is given by:
     
1 0 0
0
1
0
i =   ; j =   ; k =   (4.22)
0 0 1
Because the unit for length is conveniently chosen to be infinitesimally small, the base
vectors of the local triad coincide with the edges of the material element. As the material
in the vicinity of point p deforms, these base vectors are mapped through deformation
into corresponding base vectors of the deformed local triad:
∂u ∂v ∂w

˜ i = 1 + i + j + k (4.23)
∂x ∂x ∂x

~
j

~
i
x = f(p)
~ Displacement
k
u(p)

y

j
j
i
x
p
k i
k
z
Figure 4.3 Physical meaning of deformation gradient.