/EOS/GRUNEISEN

Block Format Keyword Describes the Gruneisen equation of state.

Format


(1)	(3)	(5)	(7)
/EOS/GRUNEISEN/`mat_ID`/`unit_ID`
`eos_title`
`C`	`S`₁	`S`₂	`S`₃
$γ_{0}$	`a`	`E`₀
`P0`	`PSH`

Definition


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)
`C`	C sound speed. 1 (Real)	$[\frac{m}{s}]$
`S`₁	`S`₁ material constant. 1 (Real)
`S`₂	`S`₂ material constant. (Real)
`S`₃	`S`₃ material constant. (Real)
$γ_{0}$	$γ_{0}$ coefficient. (Real)
`a`	`a` coefficient (see equation below). Default = $γ_{0}$ (Real)
`E`₀	Initial energy per unit reference volume. (Real)	$[\frac{J}{m^{3}}]$
`P0`	Initial pressure. 2 (Real)	$[\frac{N}{m^{2}}]$
`PSH`	Pressure Shift. It allows to model relative pressure by shifting the pressure. 3 (Real)	$[\frac{N}{m^{2}}]$

Example (Copper)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                   g                  cm                 mus
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/HYD_JCOOK/1/1
Copper (data from Example 46 - TNT Cylinder Expansion Test)
#              RHO_I               RHO_0
                8.96                   0
#                 E                   nu
                1.24                 .35
#                  A                   B                   n              epsmax              sigmax
                9E-4              .00292                 .31                   0               .0066
#               Pmin
               -1E30
#                  C           EPS_DOT_0                   M               Tmelt               Tmax
                .025                1E-6                1.09                1656                1E30
#              RHOCP                                                         T_r
            3.461E-5                                                           0
/EOS/GRUNEISEN/1/1
Copper
#                  C                  S1                  S2                  S3
                .394               1.489                   0                   0
#             GAMMA0               ALPHA                  E0               
                1.97                 .47                   0               
#                 P0                 PSH
                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA

Comments

C, S₁, S₂ and S₃ are the coefficients of the cubic equation relating the shock velocity to the particle velocity.
When P0 is provided, E₀ is automatically computed such as $P (μ_{0}, E_{0}) = P 0$ .
With the volumetric strain $μ = \frac{ρ}{ρ_{0}} - 1$ , if $μ > 0$ , the pressure is given by:
$P = P S H + \frac{ρ_{0} C^{2} μ [1 + (1 - \frac{γ_{0}}{2}) μ - \frac{a}{2} μ^{2}]}{{[1 - (S_{1} - 1) μ - S_{2} \frac{μ^{2}}{μ + 1} - S_{3} \frac{μ^{3}}{{(μ + 1)}^{2}}]}^{2}} + (γ_{0} + a μ) E$
If $μ < 0$ , the pressure is given by:
$P = P S H + ρ_{0} C^{2} μ + (γ_{0} + a μ) E$
Equations of state are used by Radioss to compute the hydrodynamic pressure and are compatible with the material laws:

Input example with units: grams, cm,

μ s

(micro-seconds):


Material	$ρ_{0} [\frac{g}{c m^{3}}]$	$C [\frac{c m}{μ s}]$	S₁	$γ_{0}$	a
Cu	8.9	0.394	1.489	1.97	0.47
Stainless Steel ^¹	7.9	0.457	1.490	2.00	0.50
Al	2.7	0.533	1.338	2.18	0.48
Al Alloy ^²	2.7855	0.533	1.338	2.18	0.48
Be	1.8519	0.8	1.124	1.16	0.16
Mg Alloy ^³	1.7794	0.452	1.242	1.63	0.33
Ti	4.5249	0.47	1.146	1.3	0.20
Ni	8.8968	0.465	1.445	2	0.50
Pb	11.3379	0.201	1.54	2.84	0.54

¹ Alloy 304, 72% Fe, 19% Cr, 9% Ni

² Alloy 2024, 93,5% Al, 4.5% Cu, 1.5% Mg

³ Alloy AZ31B, 96%Mg, 3% Al, 1% Zn