Other Rates of Change (Volume and Area)

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

Volume:
If dV=ds_1ds_2ds_3, where ds_1, ds_2, ds_3 are the lengths of the sides of a “box” that is oriented in the principal directions of \mathbf{D}, then:
\frac{d}{dt}(dV)=\underbrace{\frac{d(ds_1)}{dt}}_{D_{11}ds_1}ds_2ds_3+\underbrace{\frac{d(ds_2)}{dt}}_{D_{22}ds_2}ds_1ds_3+\underbrace{\frac{d(ds_3)}{dt}}_{D_{33}ds_3}ds_1ds_2

(1)   \begin{equation*} =(D_{11}+D_{22}+D_{33})\underbrace{ds_1ds_2ds_3}_{dV}=D_{kk}dV \text{     note:  } tr(\mathbf{D})=D_{kk} \end{equation*}

Area:
Consider the following time derivative:
\frac{d}{dt}(dS\mathbf{n}) = \frac{dS}{dt}\mathbf{n}+dS\frac{d\mathbf{n}}{dt}
where dS is an area (as opposed to ds_1, ds_2, ds_3, which are lengths).

dS\mathbf{n} is known from eq. 2 in Section 2: Volume and Area Change. So, we can take the time derivative. We can also re-write \frac{d\mathbf{n}}{dt} in terms of D_{nn}.

(2)   \begin{equation*} \longrightarrow \frac{d}{dt}(dS)=(\underbrace{D_{kk}}_{\text{sum}}-\underbrace{D_{nn}}_{\text{no sum}})dS \end{equation*}

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}
    }