Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented:
First, the Cauchy stress tensor is split into a hydrostatic and a deviatoric part:
$$\sigma^{ij} = -p\delta^{ij} + S^{ij}$$
The pressure is found using an equation of state. Often the following isothermal approach is used: $$p = c_0^2(\rho - \rho_0)$$
$c_0^2$ being the adiabatic sound speed, $\rho$ the density and $\rho_0$ the reference density. Then, it is stated that Hooke's law is assumed and the deviatoric part of the stress tensor evolves as follows:
$$\frac{dS^{ij}}{dt} = 2\mu(\dot{\epsilon}^{ij} - \frac{1}{3}\delta^{ij}\dot{\epsilon}^{kk}) + S^{ij}\Omega^{jk} + \Omega^{ik}S^{kj}$$
where $\mu$ is the shear modulus, $$\Omega^{ij} = \frac{1}{2} (\frac{\delta v^i}{\delta x^j} - \frac{\delta v^j}{\delta x^i})$$ is the spin tensor and $$\dot{\epsilon}^{ij} = \frac{1}{2} (\frac{\delta v^i}{\delta x^j} + \frac{\delta v^j}{\delta x^i})$$ is the rate of deformation tensor. Now, since:
$$\dot{\sigma} = \overset{\Delta J}{\sigma} + \sigma\cdot\Omega + \Omega\cdot\sigma$$
where $\overset{\Delta J}{\sigma}$ is the Jaumann rate it holds that:
$$ \overset{\Delta J}{\sigma}_{ij} = 2\mu(\dot{\epsilon}^{ij} - \frac{1}{3}\delta^{ij}\dot{\epsilon}^{kk})$$
Now on to my questions:
How does one come up with the above equation for the Jaumman Rate? Or, particularly, how does the assumption of Hooke's law yield that equation for the Jaumann rate?
Is that equation for the Jaumann rate also valid for other objective stress rates? For example for the Truesdell rate, giving a stress update as follows:
$$\frac{dS}{dt} = 2\mu(\dot{\epsilon} - \frac{1}{3}\mathbf{1}{\rm Tr}(\dot{\epsilon})) - {\rm Tr}(\dot{\epsilon})S + \dot{\epsilon}\cdot S + S\dot{\epsilon}^T$$
($\rm Tr(.)$ being the trace)