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• SPH

EOS_TILLOTSON

SPH
*EOS_TILLOTSON
eosid, $a$, $b$, $A$, $B$, $\alpha$, $\beta$, $e_0$
$e_{IV}$, $e_{CV}$, $\eta_{min}$, $\eta_{max}$, $p_{spall}$

Parameter definition

VariableDescription
eosid Unique EOS identification number
$a$ Tillotson parameter
$b$ Tillotson parameter
$A$ Bulk modulus
$B$ Tillotson parameter
$\alpha$ Tillotson parameter
$\beta$ Tillotson parameter
$e_0$ Initial specific internal energy
$e_{IV}$ Incipient vaporization specific energy
$e_{CV}$ Complete vaporization specific energy
$\eta_{min}$ Compression threshold
default: 0.4
$\eta_{max}$ Compression threshold
default: 2.5
$p_{spall}$ Spall pressure

Description

This command is only supported by $\gamma SPH$.

State 1 - For compressed states where $\rho \geq \rho_0, e \geq 0$.

$\displaystyle{ p_1(\rho, e) = \left[ a - \frac{b}{\frac{e}{e_0 \eta^2} + 1} \right] \rho e + A \mu + B \mu^2 }$

where $\eta = \rho / \rho_0$ and $\mu = \eta - 1$.

State 2 - For cold expanded states where $\rho_0 \gt \rho, e \leq e_{IV}$.

$\displaystyle{ p_2(\rho, e) = \left[ a - \frac{b}{\frac{e}{e_0 \eta^2} + 1} \right] \rho e + A \mu + B \mu^2 }$

State 3 - For a mixed state $\rho_0 \gt \rho, e_{CV} \gt e \gt e_{IV}$.

$\displaystyle{ p_3(\rho, e) = \frac{(e - e_{IV}) p_4 + (e_{CV} - e) p_2}{e_{CV} - e} }$

State 4 - For hot expanded states where $\rho_0 \gt \rho, e \geq e_{CV}$.

$\displaystyle{ p_4(\rho, e) = a \rho e + \left[ \frac{b \rho e}{\frac{e}{e_0 \eta^2} + 1} + A \mu {\mathrm e}^{ -\beta \left( \frac{1}{\eta} - 1 \right) } \right] {\mathrm e}^{-\alpha \left( \frac{1}{\eta} - 1 \right)^2} }$

For condensed states (state $\leq 2$) and if $p \lt p_{spall}$, particle spalls and $p \lt 0$ is never allowed.