EOS_MURNAGHAN
γSPH
*EOS_MURNAGHAN
eosid, $e_0$, $\Gamma_0$, $C_v$, $B_0$, $n$, $b$, $T_0$
$p_{spall}$
eosid, $e_0$, $\Gamma_0$, $C_v$, $B_0$, $n$, $b$, $T_0$
$p_{spall}$
Parameter definition
Variable
Description
eosid
Unique EOS identification number
$e_0$
Initial specific internal energy
$\Gamma_0$
Reference Gruneisen $\Gamma$
$C_v$
Specific heat capacity at constant volume
$B_0$
EOS coefficient
$n$
EOS coefficient
$b$
Reference Gruneisen compression dependency coefficient
$T_0$
Initial temperature
$p_{spall}$
Spall pressure
Description
This command is only supported by $\gamma SPH$.
The pressure is defied as:
$\displaystyle{ p(x, e) = p_r(x) + \Gamma(x) \rho (e - e_r(x)) }$
where $x = \rho_0/\rho$ and Gruneisen gamma is defined as:
$\displaystyle{ \Gamma(x) = (\Gamma_0 - b)x + b }$
Further:
$\displaystyle{ p_r(x) = p_0 + \frac{B_0}{n} (x^{-n} - 1) }$
$\displaystyle{ e_r(x) = e_0 + \frac{1}{\rho_0} \int_1^x p_r(y) \mathrm{d}y + C_v T_0 \int_1^x \frac{\Gamma(y)}{y} \mathrm{d}y }$
This gives:
$\displaystyle{ e_r(x) = e_0 + \frac{p_0}{\rho_0}(1-x) + \frac{B_0}{n p_0} \left( \frac{1}{n-1}(x^{1-n} - 1) + x - 1 \right) + C_v T_0 (\Gamma_0 - b)(x-1) + b \mathrm{ln} x }$
Also:
$\displaystyle{ T(x,e) = T_0 + \frac{e - e_r(x)}{C_v} }$
Spalling occurs if $p \lt p_{spall}$ and $p \lt 0$ is never allowed.
