An expression for stress power

I have seen it written that for a continuum undergoing deformation, if we ignore body forces and heat transfer, the work done is equal to stress power:

$\cfrac{dW}{dt}=\sigma_{ij}D_{ij}$, where $D_{ij}$ is the velocity gradient.

Then we have $\cfrac{dW}{dt}=\sigma_{ij}D_{ij}=\sigma_{ij}v_{i,j}=\sigma_{ij}\cfrac{dF_{iA}}{dt}F^{-1}_{Aj}=\cfrac{1}{J}P_{Bi}x_{j,B}\cfrac{dF_{iA}}{dt}F^{-1}_{Aj}$, where $P_{Bi}$ is the first Piola-Kirchhoff tensor and we have used the fact that $J\sigma_{ij}=P_{Ai}\cfrac{dF_{iA}}{dt}$.

This result simplifies to $\cfrac{dW}{dt}=\cfrac{1}{J}P_{Ai}\cfrac{dF_{iA}}{dt}$.

However, in other sources I see it written that $\cfrac{dW}{dt}=P_{Ai}\cfrac{dF_{iA}}{dt}$, and I see no reason to assume that the Jacobian should be equal to 1. Could you please tell me where I went wrong? Thank you.

• Aren't the "other sources" writing the power in the vicinity of the reference configuration? – Joce Apr 25 '16 at 13:36

The stress power is specified per unit volume, but you have to specify which unit of volume. The stress power integrated over an entire body in the current configuration $\Omega_t$ is given by:
Now, taking $dv_t$ to be a differential volume in the current configuration and $dV$ to be a differential volume in the reference configuration, we use the relation $dv_t = J dV$ to change to integration over the reference body $\Omega_0$: