# 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:

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

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:

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:

## Linear EOS¶

The pressure in linear EOS is given by

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

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

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

where \(K_0\), \(K_1\) are material parameters, \(V\) is the volume.

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

Note

Murnaghan EOS is independent to the energy.

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

The pressure can be computing by

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.

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

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:

when \(\mu\)<0 and \(E\geq E_s\):

when \(\mu\)<0 and \(E<E_s\):

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