In the following, I use the standard notation for (solid) mechanics and conservation laws, i.e. $F$ the formation gradient, $H$ the cofactor, $v$ the velocity field and $J$ the Jacobian. Moreover, $X$ is for the initial configuration where functions and fields are expressed using a bar and $x$ is for the mapped {current) configuration.
The rate of deformation gradient can be expressed as: $$ \frac{\partial F}{\partial t} = (\nabla \bar{v}) F. $$ Now I know how to derive the total (TLF) $$\frac{d}{dt}\int_V F \; dV = \int_{\partial V} \bar{v}\otimes N \; dA$$ and updated (ULF) $$\int_{v(t)} J^{-1} \left.\frac{\partial F}{\partial t}\right\vert_{X} \; dv = \int_{v(t)} \nabla\cdot (v\otimes H) dv$$ lagrangian formulations.
I would like to get, if it exists, the Eulerian formulation of the above, as well as the ALE and Total ALE formulation of it.
To do so, I tried to start from TLF, I switch to the current configuration $v(t)$ using $J$ and I use the Reynold's transport theorem to get rid of the total time derivative on the integral and to get the convective term. I end up with:
$$ \int_{v(t)} \frac{\partial J^{-1}F}{\partial t} + \nabla\cdot(J^{-1}F\otimes v) \; dv = \int_{v(t)} \nabla\cdot (v\otimes H^{-1}) \; dv $$
Are the mathematics correct? And can I reach these formulations? May I say that I prefer integral forms..
