Work-conjugate stress and strain ($\mathcal{P}=T:\dot{E}$)

A stress measure $T$ is work-conjugate to a strain measure $E$ when their contraction gives the mechanical power per specified volume for every admissible deformation rate. The reference-volume power is $P_{Ii}{\dot{F}}_{iI}=\tau :D$, with Kirchhoff stress tensor $\tau$ and rate-of-strain tensor $D$. A strain constraint can leave the conjugate stress undetermined by a reaction component that does no admissible work; uniqueness should therefore not be assumed on a restricted set of rates.

Use the material-first nominal stress $P_{Ii}$ and $F=RU$. The symmetric Biot stress conjugate to the Biot strain tensor is $T^{(1)}=\mathrm{sym}(PR)$. To prove it, set $M=P^T$ and differentiate $F=RU$. Conservation of angular momentum makes $M{F}^T$ symmetric, so the contraction of $R^{T}M$ with $R^T\dot{R}U$ is zero: it is a contraction of a symmetric tensor with a skew tensor after cyclically moving $U$. Thus $M:\dot{F}=\mathrm{sym}(R^{T}M):\dot{U}=T^{(1)}:{\dot{E}}^{(1)}$. No assumption that $PR$ itself is symmetric is needed.