#### MAT_ELASTIC

###### Material properties
*MAT_ELASTIC
"Optional title"
mid, $\rho$, $E$, $\nu$, did, tid
$a$, $b$, $c$, $c_{dec}$
##### Parameter definition
VariableDescription
mid Unique material identification number
$\rho$ Density
$E$ Young's modulus
$\nu$ Poisson's ratio
did Damage property command ID
tid Thermal property command ID
$a$, $b$ Non-linear elasticity parameters
$c$ Damping coefficient
$c_{dec}$ Damping decay coefficient
##### Description

A non-linear elastic constitutive model with damping. The stress is defined as:

$\boldsymbol{\sigma} = -p \mathbf{I} + 2G \cdot [1 + a \varepsilon_{dev}^{geo} + b (\varepsilon_{dev}^{geo})^2] \cdot \boldsymbol{\varepsilon}_{dev} + \displaystyle{ \frac{c}{c_{dec}} \int_0^t \dot{\boldsymbol{\varepsilon}}(\tau) \cdot \mathrm{e}^{(\tau-t)/c_{dec}} \mathrm{d}\tau }$

$G$ is the shear modulus, $\boldsymbol{\varepsilon}_{dev}$ is the deviatoric strain and $\varepsilon_{dev}^{geo}$ is the effective deviatoric geometric strain.

$\varepsilon_{dev}^{geo} = \displaystyle{ \sqrt{ \frac{2}{3} \boldsymbol{\varepsilon}_{dev} : \boldsymbol{\varepsilon}_{dev} } }$

The hydrostatic pressure $p$ is defined as:

$p = -K \varepsilon_v + 3 K \alpha_T (T-T_{ref})$

where $K$ is the bulk modulus, $\varepsilon_v$ is the volumetric strain. $\alpha_T$ is the thermal expansion coefficient and $T_{ref}$ is the reference temperature (see PROP_THERMAL).