I have a volume which is deforming (using explicit time-integration scheme) uniformly with velocity gradient $L$ and stress tensor $\sigma$. I would like to determine work done by the volume deformation during one timestep $\Delta t$, knowing both current and previous values of $L$, $\sigma$ and volume $V$.
I've seen somewhere the formula $\Delta W={\mathrm tr}(L\sigma)V\Delta t$, but I don't know if it is correct and how to derive it. $L\sigma$ should be energy density, but why are its non-diagonal terms discarded?
Remark: the integration scheme actually is leap-frog, but I ignored mid-step/on-step business for now and supposed everything are on-step values. The formula above would correctly compute the increment mid-step, reading $\Delta W(t-\Delta t/2)={\mathrm tr}\left(L(t-\Delta t/2)\frac{\sigma(t-\Delta t)+\sigma(t)}{2}\right)\frac{V(t-\Delta t)+V(t)}{2}\Delta t$
