/EOS/TILLOTSON

Block Format Keyword Describes the Tillotson equation of state.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/EOS/TILLOTSON/mat_ID/unit_ID
eos_title
C1 C2 a b  
ER ES VS E0  
α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3792@ 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)

 
C1 C1 coefficient .

(Real)

[ Pa ]
C2 C2 coefficient.

(Real)

[ Pa ]
a A coefficient.

(Real)

 
b b coefficient.

(Real)

 
ER Internal energy per unit reference volume.

(Real)

[ J m 3 ]
ES Sublimation energy per unit reference volume.

(Real)

[ J m 3 ]
VS Sublimation relative volume.

(Real)

[ m 3 ]
E0 Initial energy per unit reference volume.

(Real)

[ J m 3 ]
α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3792@ α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3792@ 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 [ g c m 3 ] g/cm3 C1 [ Mbar ] C2 [ Mbar ] a b ER [ Mbar ] α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3792@ β ES [ Mbar ] VS
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

  1. With μ = ρ ρ 0 1

    V = 1 ρ the specific volume

    η = 1 + μ

    x = 1 ρ 0 ρ

    and E being the internal energy per unit reference volume.

    The pressure is defined by:

    Region 1: μ 0 (1)
    P = C 1 μ + C 2 μ 2 + ( a + b ω ) η E

    with ω=1+ E E R η 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDcqGH9aqpcaaIXaGaey4kaSYaaSaaa8aabaWdbiaadwea a8aabaWdbiaadweapaWaaSbaaSqaa8qacaWGsbaapaqabaGccaaMi8 +dbiabeE7aO9aadaahaaWcbeqaa8qacaaIYaaaaaaaaaa@41F5@

    Region 2: μ < 0 V V 0 < V S and E < E S (2)
    P = C 1 μ + ( a + b ω ) η E
    Region 3: μ < 0 , V V 0 > V S or V V 0 < V S and E E S (3)
    P = C 1 e β x e α x 2 μ + ( a + b e α x 2 ω ) η E
  2. 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)