For completeness, we’ll quickly derive the rate of volume change and the rate of area change:
Consider the following time derivative:
where is an area (as opposed to , , , which are lengths).
is known from eq. 2 in Section 2: Volume and Area Change. So, we can take the time derivative. We can also re-write in terms of .
The derivation was skipped here, but can be found in [Holzapfel].
We’ll come back to rates again when we get to rate-form constitutive relationships.