#### MAT_MOONEY_RIVLIN

###### Material properties
*MAT_MOONEY_RIVLIN
"Optional title"
mid, $\rho$, $K$, ., ., tid
$C_1$, $C_2$, $\alpha_1$, $\beta_1$, $\alpha_2$, $\beta_2$, $\alpha_3$, $\beta_3$
$\alpha_4$, $\beta_4$
##### Parameter definition
VariableDescription
mid Unique material identification number
$\rho$ Density
$K$ Bulk modulus
tid Thermal property ID
$C_1$ Shear stiffness parameter
$C_2$ Shear stiffness parameter
$\alpha_1$ Viscous stiffness parameter
$\beta_1$ Viscous decay parameter
$\alpha_2$ Viscous stiffness parameter
$\beta_2$ Viscous decay parameter
$\alpha_3$ Viscous stiffness parameter
$\beta_3$ Viscous decay parameter
$\alpha_4$ Viscous stiffness parameter
$\beta_4$ Viscous decay parameter
##### Description

This is a visco-elastic model for rubber materials. The total stress $\boldsymbol{\sigma}$ is the sum of a rate independent elastic stress tensor $\boldsymbol{\sigma}_e$ and a viscous deviatoric stress tensor $\boldsymbol{\sigma}_v$.

$\boldsymbol{\sigma} = \boldsymbol{\sigma}_e + \boldsymbol{\sigma}_v$

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 the deviatoric part of Cauchy-Green's right stretch tensor ${\mathrm{dev}\mathbf C} = {\mathrm{dev}\mathbf F}^\mathrm{T} {\mathrm{dev}\mathbf F}$. The corresponding principal directions are $\mathbf{n}_i$ and the rate independent stress tensor $\boldsymbol{\sigma}_e$ can be expressed as:

$\boldsymbol{\sigma}_e = \displaystyle{ \sum_{i=1}^3 \sigma_i \mathbf{n}_i \otimes \mathbf{n}_i }$

The pressure $p$ is a linear function of the volumetric strain $\varepsilon_v$:

$p = -K \varepsilon_v$

The viscous stresses $\boldsymbol{\sigma}_v$ are purely deviatoric and are controlled by parameters $\alpha_k$ and $\beta_k, k=[1,4]$.

$\boldsymbol{\sigma}_v(t) = \displaystyle{ \sum_{k=1}^4 \frac{2\alpha_k}{\beta_k} \int_0^t \dot{\boldsymbol{\varepsilon}}_{dev}(\tau) \mathrm{e}^{(\tau-t)/\beta_k} \mathrm{d}\tau }$

Here $t$ is the current time and $\dot{\boldsymbol{\varepsilon}}_{dev}$ is the deviatoric strain rate. Note that, given a constant deviatoric strain rate $\dot{\boldsymbol{\varepsilon}}_{dev}$, the viscous stress response will asymptotically approach:

$\displaystyle{ \lim_{t \to \infty} \boldsymbol{\sigma}_v = \sum_{k=1}^4 {2 \alpha_k} \dot{\boldsymbol{\varepsilon}}_{dev} }$