/EOS/TILLOTSON

Block Format Keyword Describes the Tillotson equation of state.

Format


(1)	(3)	(5)	(7)
/EOS/TILLOTSON/`mat_ID`/`unit_ID`
`eos_title`
`C`₁	`C`₂	`a`	`b`
`E`_R	`E`_S	`V`_S	`E`₀
$α$	$β$

Definition


Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
Unit Identifier	Unit Identifier. (Integer, maximum 10 digits)
`eos_title`	EOS title. (Character, maximum 100 characters)
`C`₁	`C`₁ coefficient . (Real)	$[Pa]$
`C`₂	`C`₂ coefficient. (Real)	$[Pa]$
`a`	A coefficient. (Real)
`b`	b coefficient. (Real)
`E`_R	Internal energy per unit reference volume. (Real)	$[\frac{J}{m^{3}}]$
`E`_S	Sublimation energy per unit reference volume. (Real)	$[\frac{J}{m^{3}}]$
`V`_S	Sublimation relative volume. (Real)	$[m^{3}]$
`E`₀	Initial energy per unit reference volume. (Real)	$[\frac{J}{m^{3}}]$
$α$	$α$ coefficient. (Real)
$β$	$β$ coefficient. (Real)

Example (Aluminum)

#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----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/HYD_JCOOK/1/1
Aluminum
#              RHO_I               RHO_0              
                 2.8                   0                   
#                 E                   nu
                .734                 .33
#                  A                   B                   n              epsmax              sigmax
               .0024               .0042                  .8                   0               .0068
#               Pmin
              -.0223
#                  C           EPS_DOT_0                   M               Tmelt               Tmax
                .062                1E-6                   1                1220                   0
#              RHOCP                                                         T_r
             2.59E-5                                                           0
/EOS/TILLOTSON/1/1
Aluminum
#                 C1                  C2                   A                   B
                .752                 .65                  .5                1.63
#                 ER                  ES                  VS                  E0               
                .135                .081                 1.1                   0                   
#              ALPHA                BETA
                   5                   5
/FAIL/JOHNSON/3
#                 D1                  D2                  D3                  D4                  D5
                .112                .123                -1.5                .007                   0
#              EPS_0  Ifail_sh  Ifail_so                                    Dadv               Ixfem
                1E-6         0         1                                       0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Input Example

"Metallic Equation of State for Hypervelocity Impact" J.H. Tillotson, General Dynamics, 1962:


Material	$ρ_{0} [\frac{g}{c m^{3}}]$ g/cm³	`C`₁ $[Mbar]$	`C`₂ $[Mbar]$	`a`	`b`	`E`_R $[Mbar]$	$α$	$β$	`E`_S $[Mbar]$	`V`_S
Cu	8.9	1.390	1.10	0.5	1.50	2.892	5	5	0.123	1.18
Fe	7.8	1.279	1.05	0.5	1.50	0.741	5	5	0.190	1.21
Al	2.7	0.752	0.65	0.5	1.63	0.135	5	5	0.081	1.10

Comments

With $μ = \frac{ρ}{ρ_{0}} - 1$
$V = \frac{1}{ρ}$ the specific volume
$η = 1 + μ$
$x = 1 - \frac{ρ_{0}}{ρ}$
and $E$ being the internal energy per unit reference volume.
The pressure is defined by:
Region 1: $μ \geq 0$
$P = C_{1} μ + C_{2} μ^{2} + (a + \frac{b}{ω}) η E$
with $ω = 1 + \frac{E}{E_{R} η^{2}}$
Region 2: $μ < 0$ $\frac{V}{V_{0}} < V_{S}$ and $E < E_{S}$
$P = C_{1} μ + (a + \frac{b}{ω}) η E$
Region 3: $μ < 0$ , $\frac{V}{V_{0}} > V_{S}$ or $\frac{V}{V_{0}} < V_{S}$ and $E \geq E_{S}$
$P = C_{1} e^{β x} e^{- α x^{2}} μ + (a + \frac{b e^{- α x^{2}}}{ω}) η E$
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)
- /MAT/LAW109