Description
This is a visco-elastic model for rubber materials. The total stress $\mathbf{\sigma}$ is the sum of a rate independent elastic stress
tensor $\mathbf{\sigma}_e$ and a viscous deviatoric stress tensor $\mathbf{\sigma}_v$.
$\mathbf{\sigma} = \mathbf{\sigma}_e + \mathbf{\sigma}_v$
It is to be noted that the viscous stresses are not part of the original Mooney-Rivlin material model.
The rate independent deviatoric response is non-linear and it is controlled by the two parameters $C_1$ and $C_2$.
The principal stresses are defined as:
$\begin{array}{l}
\sigma_1 = \displaystyle{\frac{2C_1}{3} ( 2 \lambda_1 - \lambda_2 - \lambda_3 ) - \frac{2 C_2}{3} (2/\lambda_1 - 1/\lambda_2 - 1/\lambda_3) - p} \\ \\
\sigma_2 = \displaystyle{\frac{2C_1}{3} ( 2 \lambda_2 - \lambda_3 - \lambda_1 ) - \frac{2 C_2}{3} (2/\lambda_2 - 1/\lambda_3 - 1/\lambda_1) - p} \\ \\
\sigma_3 = \displaystyle{\frac{2C_1}{3} ( 2 \lambda_3 - \lambda_1 - \lambda_2 ) - \frac{2 C_2}{3} (2/\lambda_3 - 1/\lambda_1 - 1/\lambda_2) - p}
\end{array}$
where $\lambda_i, i=[1,3]$ are eigenvalues of Cauchy-Green's right stretch tensor ${\mathbf C} = {\mathbf F}^\mathrm{T} {\mathbf F}$.
The corresponding principal directions are $\mathbf{n}_i$ and the rate independent stress tensor $\mathbf{\sigma}_e$ can be expressed as:
$\mathbf{\sigma}_e = \displaystyle{ \sum_{i=1}^3 \sigma_i \mathbf{n}_i \otimes \mathbf{n}_i }$
The pressure $p$ is a linear function of the volumetric strain $\epsilon_v$:
$p = -K \epsilon_v$
The viscous stresses $\mathbf{\sigma}_v$ are purely deviatoric and are controlled by parameters $\alpha_k$ and $\beta_k, k=[1,4]$.
$\mathbf{\sigma}_v(t) = \displaystyle{ \sum_{k=1}^4 \frac{2\alpha_k}{\beta_k} \int_0^t \dot{\mathbf{\epsilon}}_{dev}(\tau) \mathrm{e}^{(\tau-t)/\beta_k} \mathrm{d}\tau }$
Here $t$ is the current time and $\dot{\mathbf{\epsilon}}_{dev}$ is the deviatoric strain rate. Note that, given a constant
deviatoric strain rate $\dot{\mathbf{\epsilon}}_{dev}$, the viscous stress response will asymptotically approach:
$\displaystyle{ \lim_{t \to \infty} \mathbf{\sigma}_v = \sum_{k=1}^4 {2 \alpha_k} \dot{\mathbf{\epsilon}}_{dev} }$
