Page 154 - The Combined Finite-Discrete Element Method

DEFORMATION GRADIENT 137

∂u
∂v ∂w
˜
j = i + 1 + j + k
∂y ∂y ∂y
∂u ∂v
∂w
˜
k = i + j + 1 + k
∂z ∂z ∂z
As can be seen from Figure 4.3, these vectors are in general non-orthogonal to each other.
In addition, these vectors are not unit vectors. Thus, a cube shaped material element of
unit volume changes both its volume and it original cubic shape.
Special types of deformation include the deformation with constant displacement and
deformation with constant deformation gradient.
The deformation with constant displacement

u(p) = const (4.24)

is referred to as translation (Figure 4.4). As can be seen from the figure, the initial material
element is identical in shape, size and orientation to the deformed initial volume, except
that it is translated. Translation therefore does not produce any straining of the material.
The deformation with constant deformation gradient

F(p) = const (4.25)

is referred to as homogeneous. It can be expressed as

f(p) = f(q) + F(p − q) (4.26)

Two important examples of homogeneous deformation are stretch from q and rotation
about q. Stretch from q can be written as follows:

f(p) = q + U(p − q) (4.27)

where U is a symmetric and positive definite tensor. Spectral decomposition of U in
the form
Ue = se (4.28)

y

x

z

Figure 4.4 Translation.