Page 127 - Introduction to Continuum Mechanics

112 Kinematics of a Continuum

and

We have already seen in the previous section that W does not contribute to the rate of change
of length of the material vector d\. Thus, Eq. (3.14.3) shows that its effect on dx is simply to
rotate it (without changing its length) with an angular velocity to.
It should be noted however, that the rate of deformation tensor D also contributes to the
rate of change in direction of dxas well so that in general, most material vectors dx rotate with
an angular velocity different from o» (while changing their lengths). Indeed, it can be proven
that in general, only the three material vectors which are in the principal direction of D do
rotate with the angular velocity to, (while changing their length), (see Prob. 3.47)
We also note that in fluid mechanics literature, 2W is called the vorticity tensor.

3.15 Equation of Conservation of Mass

If we follow an infinitesimal volume of material through its motion, its volume dV and
density/) may change, but its total masspdVwill remain unchanged. That is,

i.e.,

Using Eq. (3.13.7), we obtain

Or, in invariant form,

where in spatial description,

Equation (3.15.2) is the equation of conservation of mass, also known as the equation of
continuity.
In Cartesian coordinates, Eq. (3.15.2b) reads: