#### MAT_YEOH

###### Material properties
Attention: This command is in the beta stage and the format may change over time.
*MAT_YEOH
"Optional title"
mid, $\rho$, $K$, ., ., tid, eosid
$C_{10}$, $C_{20}$, $C_{30}$, $r$, $m$, $q$, $\Psi_f$
$\alpha_1$, $\beta_1$, $\alpha_2$, $\beta_2$, $\alpha_3$, $\beta_3$, $\alpha_4$, $\beta_4$
##### Parameter definition
VariableDescription
mid Unique material identification number
$\rho$ Density
$K$ Bulk modulus
tid Thermal property ID
eosid Equation-of-state ID
$C_{10}$ Shear stiffness parameter 1
$C_{20}$ Shear stiffness parameter 2
$C_{30}$ Shear stiffness parameter 3
$r$ Mullins damage parameter 1
default: not used
$m$ Mullins damage parameter 2
default: not used
$q$ Mullins damage parameter 3
default: not used
$\Psi_f$ Failure strain energy
default: not used
$\alpha_1$ Viscous stiffness parameter 1
$\beta_1$ Viscous decay parameter 1
$\alpha_2$ Viscous stiffness parameter 2
$\beta_2$ Viscous decay parameter 2
$\alpha_3$ Viscous stiffness parameter 3
$\beta_3$ Viscous decay parameter 3
$\alpha_4$ Viscous stiffness parameter 4
$\beta_4$ Viscous decay parameter 4
##### Description

This is a visco-elastic model for rubber materials. The total co-rotational Cauchy 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 stress is not part of the original Yeoh model. The hyper-elastic response is non-linear and it is controlled by the parameters $C_{10}$, $C_{20}$ and $C_{30}$:

$\displaystyle{ \boldsymbol{\sigma}_e = \frac{2 \eta}{\mathrm{det} \mathbf{F}} \left[ C_{10} + 2 C_{20} \left( I_1 - 3 \right) + 3 C_{30} \left( I_1 - 3 \right)^2 \right] \mathrm{dev} \mathbf{C} } - p \mathbf{I}$

$\mathbf{F}$ is the deformation gradient, $\mathbf{C} = \mathbf{F}^t \mathbf{F}$ is the right Cauchy-Green deformation tensor and $I_1 = \mathrm{tr} \left( \mathrm{dev} \mathbf{C} \right)$ is the first invariant of the deviatoric part of $\mathbf{C}$. $\eta$ is an optional softening factor (Mullins). It is defined as:

$\displaystyle{ \eta = 1 - \frac{1}{r} \mathrm{erf} \left( \frac{\Psi_{max} - \Psi}{m + q \Psi_{max}} \right) }$

where $\Psi_{max}$ is the maximum specific deviatoric strain energy the material has experienced and $\Psi$ is the current specific deviatoric strain energy. Note that $r$ and $q$ are dimensionless and $m$ has the units of stress (= energy per unit volume).

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:

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

$\Psi_f$ is an optional failure parameter. The material will fail (element erosion) if the specific elastic deviatoric strain energy $\Psi \geq \Psi_f$.