For completeness, we’ll quickly derive the rate of volume change and the rate of area change:

Volume:

If , where , , are the lengths of the sides of a “box” that is oriented in the principal directions of , then:

(1)

Area:

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 .

(2)

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.

- G. Holzapfel, Nonlinear Solid Mechanics, John Wiley & Sons Ltd., England, 2000.

[Bibtex]`@book{Holzapfel, title={Nonlinear Solid Mechanics}, author={Holzapfel, GA}, year={2000}, publisher={John Wiley \& Sons Ltd., England} }`