/EOS/MURNAGHAN

Block Format Keyword Describes the Murnaghan equation of state.

Format

(1)	(3)	(5)	(7)	(9)
/EOS/MURNAGHAN/`mat_ID`/`unit_ID`
`eos_title`
`K`₀	`K`₁	`P`₀	`P`_sh	$ρ_{0}$ $ρ_{0}$

Definitions

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	Unit Identifier. (Integer, maximum 10 digits)
`eos_title`	EOS title. (Character, maximum 100 characters)
`K`₀	Material parameter. (Real)	$[Pa]$ $[Pa]$
`K`₁	Material parameter. (Real)
`P`₀	Initial pressure. (Real)	$[Pa]$ $[Pa]$
`P`_sh	Pressure shift. (Real)	$[Pa]$ $[Pa]$
$ρ_{0}$ $ρ_{0}$	Reference density. Default = material density (Real)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$

Example

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                   g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/HYDPLA/7
Articficial Linear Material Law
#              RHO_I               RHO_0
             1.22e-3             1.22e-3
#                  E                  nu
                   0                   0
#                  a                   b                   n             eps_max           sigma_max
                1E30                   0                   0                   0                   0
#               Pmin 
                   0
/EOS/MURNAGHAN/7
EoS for NaCl  at atmospheric pressure
#                 K0                  K1                  P0                 PSH                RHO0      
               24000               5.390                  .1                   0            2.165e-3
/ALE/MAT/7

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

This equation of state is also known as Tait Equation of State.(1)
$P (V) = \frac{K_{0}}{K_{1}} [{(\frac{V}{V_{0}})}^{- K_{1}} - 1]$ $P (V) = \frac{K_{0}}{K_{1}} [{(\frac{V}{V_{0}})}^{- K_{1}} - 1]$

Where $K_{0}$ $K_{0}$ and $K_{1}$ $K_{1}$ are material parameters.
This equation can also be found with the following form:(2)
$\frac{Δ v}{V_{0}} = 1 - {[1 + \frac{K_{1}}{K_{0}} p]}^{\frac{- 1}{K_{1}}}$ $\frac{Δ v}{V_{0}} = 1 - {[1 + \frac{K_{1}}{K_{0}} p]}^{\frac{- 1}{K_{1}}}$

With, $Δ v = V_{0} - V$ $Δ v = V_{0} - V$ and $p = P - P_{0}$ $p = P - P_{0}$
In some publications, the material parameters $K_{0}$ $K_{0}$ and $K_{1}$ $K_{1}$ are replaced by $c$ $c$ and $k$ $k$ with, $K_{0} = c$ $K_{0} = c$ and $K_{1} = c \times k$ $K_{1} = c \times k$ .
Another way to express this equation is with the compressibility $μ$ $μ$ .(3)
$P (μ) = P_{0} + \frac{K_{0}}{K_{1}} [{(1 + μ)}^{K_{1}} - 1]$ $P (μ) = P_{0} + \frac{K_{0}}{K_{1}} [{(1 + μ)}^{K_{1}} - 1]$

with $μ = \frac{ρ}{ρ_{0}} - 1 = \frac{V_{0}}{V} - 1$ $μ = \frac{ρ}{ρ_{0}} - 1 = \frac{V_{0}}{V} - 1$
Murnaghan EOS does not depend on energy.
Equations of state are used by Radioss to compute the hydrodynamic pressure and are compatible with the material laws:
- /MAT/LAW3 (HYDPLA)
- /MAT/LAW4 (HYD_JCOOK)
- /MAT/LAW6 (HYDRO or HYD_VISC)
- /MAT/LAW10 (DPRAG1)
- /MAT/LAW12 (3D_COMP)
- /MAT/LAW49 (STEINB)
- /MAT/LAW102 (DPRAG2)
- /MAT/LAW103 (HENSEL-SPITTEL)

Murnaghan, F. D. "The compressibility of media under extreme pressures." Proceedings of the National Academy of Sciences 30, no. 9 (1944): 244-247