2.9 Rate of Deformation and Rate of Spin Formulation
Rate of deformation tensor is also called the stretching tensor or velocity of strain
(strain rate) tensor. The spin tensor is also called the vorticity tensor.
We realize that the rate of deformation tensor is defined differently by different
authors. We will subscribe to Malvern’s (1969) definition and use his formulation
and derivation with few modifications (Fig. 2.29).
The relative velocity components of dv i of point A’ relative to point B’ can be
given by
de v i ¼
∂e v i
∂x m
dx m
or in matrix form
de v i
½ Š ¼ e v i,m
½ Š dx m
½
Š
ð3:14Þ
or in tensorial notation
de v ¼ L Á dx ¼ dx Á L
T
where L is
L im ¼ e v i,m , L
T
ð Þ im ¼ e v m,i
The components of tensor L are spatial gradients of the velocity. L im ¼ e v i,m can be
written as the sum of a symmetric tensor D, which we will call the rate-of-deformation tensor (also called stretching tensor) and a skew-symmetric tensor W called the
spin tensor (also called vorticity tensor).
Hence, we can write
Figure 2.29 Relative
velocity of de v i of point A’
relative to particle B’
2.9 Rate of Deformation and Rate of Spin Formulation
43