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108 Kinematics of a Continuum
3.13 The Rate of Deformation Tensor
The velocity gradient (Vv) can be decomposed into a symmetric part and an antisymmetric
part as follows:
where D is the symmetric part, i.e.,
and W is the antisymmetric part, i.e.,
The symmetric tensor D is known as the rate of deformation tensor and the antisymmetric
tensor W is known as the spin tensor. The reason for these names will be apparent soon.
With respect to rectangular Cartesian coordinates, the components of D and W are given
by:
\ / \
With respect to cylindrical and spherical coordinates the matrices take the form given in
Eq. (3.7.11) and Eq. (3.7.12).
We now show that the rate of change of length of dx is described by the tensor D whereas
the rate of rotation of dx is described by the tensor W.
Let dx — dsn, where n is a unit vector, then
Taking the material derivatives of the above equation gives
