#### EOS_GRUNEISEN

###### Material properties
*EOS_GRUNEISEN
eosid, $S_1$, $\Gamma$, $L$, $p_{cut}$
$S_2$, $S_3$, $a$
##### Parameter definition
VariableDescription
eosid Unique EOS identification number
$S_1$ Linear Hugoniot slope coefficient
$\Gamma$ Gruneisen gamma
$L$ Characteristic element size in shock viscosity calculation
default: automatically calculated
$p_{cut}$ Pressure cut-off
$S_2$ Quadratic coefficient ($\gamma$SPH only)
$S_3$ Cubic coefficient ($\gamma$SPH only)
$a$ Gruneisen coefficient ($\gamma$SPH only)
##### Description

This is the Mie-Gruneisen equation-of-state. Note that the linear bulk modulus, $K$, is determined from the elastic properties in the material command.

$p = \displaystyle{ \frac{K \eta}{(1-S_1 \eta)^2} \cdot \left( 1 - \frac{\Gamma \eta}{2} \right) + \Gamma \rho_0 e }$

where $\rho$ is the current density, $\rho_0$ is the initial density and $e$ is the specific internal energy. $\eta$ is a measure of the volumetric compression.

$\displaystyle{ \eta = 1 - \frac{\rho_0}{\rho}}$

Only $\gamma$SPH:

Pressure for compressed material:

$\displaystyle{ p = \frac{\rho_0 c^2 \mu \left[ 1 + (1 - \frac{\Gamma}{2})\mu - \frac{a}{2}\mu^2 \right]} {1 - (S_1 - 1)\mu - \frac{S_2 \mu^2}{\mu + 1} - \frac{S_3 \mu^3}{(\mu + 1)^2}} + (\Gamma + a\mu)E}$

Pressure for expanded material:

$\displaystyle{ p = \rho_0 c^2 \mu + (\Gamma + a\mu) E}$

where $\rho$ is the current density, $\rho_0$ is the initial density and $E$ is the internal energy. $\mu$ is a measure of the volumetric compression.

$\displaystyle{ \mu = \frac{\rho}{\rho_0} - 1}$