# Equations of State (EOS)¶

MatEditor allows you to define the EOS material properties. The supported properties are listed below.

• Compaction
• Gruneisen
• Ideal Gas
• Linear
• LSZK (Landau-Stanyukovich-Zeldovich-Kompaneets)
• Murnaghan
• NASG (Noble-Abel Stiffened Gas)
• Noble-Abel
• Osborne
• Polynomial
• Puff
• Stiff Gas
• Tillotson

## Compaction EOS¶

Plastic compaction is along path defined by equation:

$p=C_0 + C_1 \mu +C_2 \mu^2 + C_3 \mu^3$

where $$P$$ is the hydrodynamic pressure in material. $$\mu$$ is the volumetric strain that can be obtained by $$\mu=\dfrac{\rho}{\rho_0}-1$$.

Unloading bulk modulus $$B$$ is the bulk modules for the unloading process.

Pressure Shift $$P_{sh}$$ is used to model the relative pressure formulation.

## Gruneisen EOS¶

In the Gruneisen EOS model, the hydrodynamic pressure is described by the following equations:

For the compressed material, $$\mu$$>0

$p = \dfrac{\rho_0C^2\mu[1+(1-\dfrac{\gamma_0}{2})\mu-\dfrac{\alpha}{2}\mu^2]}{[1-(S_1-1)\mu-S_2\dfrac{\mu^2}{\mu+1}-S_3\dfrac{\mu^3}{(\mu+1)^2}]^2} + (\gamma_0+\alpha\mu)E$

For the expanding material, $$\mu$$<0 $$p = \rho_0C^2\mu + (\gamma_0+\alpha\mu)E$$

where the $$\mu=\dfrac{\rho}{\rho_0}-1$$.

## Ideal Gas EOS¶

The pressure in the Ideal Gas model can be represented by the function:

$p = (\gamma-1)(1+\mu)E$

where unitless parameter $$\gamma$$ is determined by the heat capacity $$C_v$$ and $$C_p$$, $$\gamma=\dfrac{C_p}{C_v}$$. The initial heat capacity $$C_v$$ is calculated from the initial conditions:

$C_v=\dfrac{E_0}{\rho_0T_0}$

## Linear EOS¶

The pressure in linear EOS is given by

$p = p_0 + B\mu$

where $$p_0$$ i initial pressure and $$B$$ is the initial bulk modulus. Linear EOS is a simplified form of polynomial EOS:

$p=C_0+C_1\mu + C_2\mu + C_3\mu + (C_4+C_5)E_0$

where, $$C_0=p_0$$, $$C_1=B$$, $$C_2=C_3 = C_4 = C_5 = 0$$.

Bulk modulus is usually treated as $$B=\rho_0c_0^2$$, where $$c_0$$ is the initial sound speed.

## LSZK (Landau-Stanyukovich-Zeldovich-Kompaneets) EOS¶

This EOS model is the short for the Landau-Stanyukovich-Zeldovich-Kompaneets EOS, used for the detonation modeling. The pressure is given by

$p = (\gamma-1)\rho e + a \rho^b$

where $$\rho$$ is the mass density, $$e$$ is the internal energy density by mass, $$b$$ is the material parameter.

## Murnaghan EOS¶

This EOS is also known as Tait EOS. The pressure is defined by

$p = \dfrac{K_0}{K_1}[(\dfrac{V}{V_0})^{-K_1}-1]$

where $$K_0$$, $$K_1$$ are material parameters, $$V$$ is the volume.

This model is also expressed in terms of the compressibility $$\mu$$:

$p = p_0 + \dfrac{K_0}{K_1}[(1+\mu)^{K_1}-1]$

Note

Murnaghan EOS is independent to the energy.

## NASG (Noble-Abel Stiffened Gas) EOS¶

The pressure can be computing by

$p = \dfrac{(\gamma-1)(1+\mu)(E-\rho_0q)}{1-b\rho_0(1+\mu)} - \gamma p_{\infty}$

where $$p_{\infty}$$ is the stiffness parameter.

## Noble-Abel EOS¶

This EOS can apply to dense gases at high pressure, as the volume occupied by the moledules is no longer negligible.

$p = \dfrac{(\gamma-1)(1+\mu)E}{1-b\rho_0(1+\mu)}$

where $$\gamma=\dfrac{C_p}{C_v}$$

Note

Covolume parameter b is usually in the range between [0.9e-3, 1.1e-3] $$m^3/kg$$.

## Osborne EOS¶

This EOS is also called quadratic EOS.

$$p = \dfrac{A_1\mu+A_2\mu |\mu| + (B_0+B_1\mu+B_2\mu^2)E + (C_0 + C_1\mu)E^2 }{E+D_0}$$ where $$E$$ is the internal energy by initial volume.

## Polynomial EOS¶

The pressure for the linear polynomial EOS can be calculated by

$p=C_0+C_1\mu + C_2\mu + C_3\mu + (C_4+C_5)E$

where $$E$$ is the internal energy density by volume.

Note

For the expanding status ($$\mu$$<0), the term $$C_2\mu^2$$=0.

## Puff EOS¶

This EOS model describes pressure accroding to the compressibility $$\mu$$ and sublimation energy density by volume $$E_s$$.

When $$\mu\geq$$ 0:

$p = (C_1\mu+C_2\mu^2+C_3\mu^3)(1-\dfrac{\gamma\mu}{2})+\gamma(1+\mu)E$

when $$\mu$$<0 and $$E\geq E_s$$:

$p = (T_1\mu+T_2\mu^2)(1-\dfrac{\gamma\mu}{2})+\gamma(1+\mu)E$

when $$\mu$$<0 and $$E<E_s$$:

$p = \eta[H+(\gamma_0-H)\sqrt{\eta}][E-E_s(1-exp(\dfrac{N(\eta-1)}{\eta^2}))]$

with $$N=\dfrac{C_1\eta}{\gamma_0E_s}$$.

## Stiffened Gas EOS¶

This EOS was originally designed to simulate water for underwater explosions.

The pressure can be calculated by $$p = (\gamma-1)(1+\mu)E - \gamma p_{\star}$$

where $$E=\dfrac{E_{int}}{V_0}$$, $$\mu=\dfrac{\rho}{\rho_0}-1$$. The additional pressure term $$p^{\star}$$ is introduced here.

This EOS can be derived from the Polynomial EOS: $$p=C_0+C_1\mu + C_2\mu + C_3\mu + (C_4+C_5)E$$ when $$C_0 = -\gamma p^{\star}$$, $$C_1=C_2=C3=0$$, $$C_4=C_5=\gamma-1$$, $$E_0=\dfrac{P_0-C_0}{C_4}$$.

## Tillotson EOS¶

The pressure is defined by

$$p = C_1\mu + C_2\mu^2 +(a+\dfrac{b}{\omega})\eta E$$ with $$\omega=1+\dfrac{E}{E_r}\eta^2$$ for the region $$\mu \geq$$ 0.

$$p = C_1\mu+(a+\dfrac{b}{\omega})\eta E$$ for the region $$\mu<0$$, $$\dfrac{V}{V_0}<V_s$$, and $$E<E_s$$.

and $$p = C_1 e^{\beta x} e^{-\alpha x^2}\mu + (a + \dfrac{be^{-\alpha x^2}}{\omega}) \eta E$$