/EOS/STIFFGAS
Block Format Keyword Describes the ideal gas equation of state for stiffed gas $P\left(\mu ,E\right)=\left(\gamma 1\right)\left(1+\mu \right)E\gamma {P}^{*}$ .
This equation of state was originally designed to model water for underwater explosions.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/EOS/STIFFGAS/mat_ID/unit_ID  
eos_title  
$\gamma $  P_{0}  P_{sh}  ${P}^{*}$  ${\rho}_{0}$ 
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) 

$\gamma $  Heat capacity ratio
$\gamma =\frac{{C}_{p}}{{C}_{v}}$
. (Real) 

P_{0}  Initial pressure. (Real) 
$\left[\text{Pa}\right]$ 
P_{sh}  Pressure shift. (Real) 
$\left[\text{Pa}\right]$ 
${P}^{*}$  EOS additional pressure
term. (Real) 
$\left[\text{Pa}\right]$ 
${\rho}_{0}$  Reference density. Default = material density (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
Example
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
/MAT/LAW06/7/1
water
# RHO_I
0.001
# NU PMIN
/EOS/STIFFGAS/7/1
STIFF_GAS_WATER
# GAMMA P0 PSH P_STAR RHO0
6.1 0.10 0 368.85
/ALE/MAT/7
#12345678910
#enddata
Comments

(1) $$P\left(\mu ,E\right)=\left(\gamma 1\right)\left(1+\mu \right)E\gamma {P}^{*}$$Where, $E=\frac{{E}_{int}}{{V}_{0}}$
 $\mu =\frac{\rho}{{\rho}_{0}}1$
 The stiffened gas EOS introduces an
additional pressure term (
${P}^{*}$
) with respect to the ideal gas equation of state. The EOS is
stated in this form because the speed of sound in water is:
(2) $${c}_{s}=\sqrt{\frac{\gamma (P+{P}^{*})}{\rho}}$$This way, by considering ${P}^{*}\gg P$ , the speed of sound will not be affected by $P$ fluctuations and ${c}_{s}\approx \sqrt{\frac{\gamma {P}^{*}}{\rho}}$ .

${P}^{*}$
can be calculated with:
(3) $${P}^{*}=\frac{\rho {c}_{s}^{2}}{\gamma}P$$As ${P}^{*}\gg P$ , ${P}^{*}\approx \frac{\rho {c}_{s}^{2}}{\gamma}$ leads to a good estimate of ${P}^{*}$ .
 The stiffened gas EOS ^{1} can be derived from a polynomial EOS:
(4) $$P={C}_{0}+{C}_{1}\mu +{C}_{2}{\mu}^{2}+{C}_{3}{\mu}^{3}+\left({C}_{4}+{C}_{5}\mu \right){E}_{0}$$Where, ${C}_{0}=\gamma {P}^{*}$
 ${C}_{1}={C}_{2}={C}_{3}=0$
 ${C}_{4}={C}_{5}=\gamma 1$
 ${E}_{0}=\frac{{P}_{0}{C}_{0}}{{C}_{4}}$
 For example, for water, $\rho =0.001g.m{m}^{3}$ , ${c}_{s}=1500mm.m{s}^{1}$ and $\gamma =6.1$ can be found in literature. ^{1} This leads to ${P}^{*}=368.852MPa$ . Thus, the stiffened gas EOS will give the following behavior at atmospheric pressure:
 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 (HENSELSPITTEL)