/MAT/LAW5 (JWL)

Block Format Keyword This law describes the Jones-Wilkins-Lee EOS for detonation products of high explosives. Optional afterburning modeling is available.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW5/mat_ID/unit_ID or /MAT/JWL/mat_ID/unit_ID
mat_title
ρ i ρ 0            
A B R1 R2 ω
D PCJ E0 Eadd IBFRAC QOPT
P0 Psh Bunreacted      
Insert if Eadd > 0 and QOPT = 0,1,2 (time controlled afterburning)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Tstart Tstop            
Insert if Eadd and QOPT = 3 (Miller's extension)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
a m n      

Definition

Field Contents SI Unit Example
mat_ID Material identifier

(Integer, maximum 10 digits)

 
unit_ID Unit identifier

(Integer, maximum 10 digits)

 
mat_title Material title

(Character, maximum 100 characters)

 
ρ i Initial density

(Real)

[ kg m 3 ]
ρ 0 Reference density used in E.O.S (equation of state)

Default = ρ 0 = ρ i (Real)

[ kg m 3 ]
A A parameter of equation of state

(Real)

[ Pa ]
B B parameter of equation of state

(Real)

[ Pa ]
R1 R1 parameter of equation of state.

(Real)

 
R2 R2 parameter of equation of state.

(Real)

 
ω ω parameter of equation of state.

(Real)

 
D Detonation velocity.

(Real)

[ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
PCJ Chapman Jouguet pressure.

(Real)

[ Pa ]
E0 Detonation energy per unit volume.

(Real)

[ J m 3 ]
Eadd Additional energy per unit volume.
= 0
Afterburning parameters have no effect

(Real)

[ J m 3 ]
IBFRAC Burn fraction calculation flag. 3
= 0
Volumetric Compression + Burning Time.
= 1
Volumetric Compression only.
= 2
Burning Time only.

(Integer)

 
QOPT Optional afterburning model (if Eadd > 0).
= 0
Instantaneous release on Tstart.
= 1
Constant rate from Tstart to Tstop.
= 2
Linear rate from Tstart to Tstop.
= 3
Miller’s Extension.

(Integer)

 
P0 Initial pressure.

(Real)

[ Pa ]
Psh Pressure shift.

(Real)

[ Pa ]
Bunreacted Unreacted explosive bulk modulus. 9 10 11

(Real)

[ Pa ]
Tstart Start time for additional energy (QOPT = 0,1,2).

(Real)

[ s ]
Tstop Stop time for additional energy (QOPT = 0,1,2).

(Real)

[ s ]
a Optional Miller parameter if QOPT = 3.

(Real)

[ s 1 P a n ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGZbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaamiuaiaadggadaah aaWcbeqaaiabgkHiTiaad6gaaaaakiaawUfacaGLDbaaaaa@3E8F@
m Optional Miller parameter if QOPT = 3.

(Real)

 
n Optional Miller parameter if QOPT = 3.

(Real)

 

Example (TNT)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/JWL/2/123
TNT - data from example 46 with unit: (g-cm-mus) - Standard JWL , No Afterburning
#              RHO_I
                1.63
#                  A                   B                  R1                  R2               OMEGA
              3.7121               .0323                4.15                 .95                  .3
#                  D                P_CJ                  E0                Eadd   I_BFRAC     Q_OPT
                .693                 .21                 .07                   0         0         0		
#                 P0                 Psh          Bunreacted
                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/123
Miller’s extension unit system
                   g                  cm                 mus
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

  1. JWL pressure is:(1)
    P j w l = A ( 1 ω R 1 V ) e R 1 V + B ( 1 ω R 2 V ) e R 2 V + ω ( E + Q ) V
    Radioss then outputs:(2)
    P = B f r a c P j w l + ( 1 B f r a c ) ( P 0 + B u n r e a c t e d . μ ) P s h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaamiuaiabg2da9iaadkeadaWgaaWcbaGaamOzaiaadkhacaWGHbGa am4yaaqabaGccqGHflY1caWGqbWaaSbaaSqaaiaadQgacaWG3bGaam iBaaqabaGccqGHRaWkcaGGOaGaaGymaiabgkHiTiaadkeadaWgaaWc baGaamOzaiaadkhacaWGHbGaam4yaaqabaGccaGGPaGaaiikaiaadc fadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGcbWaaSbaaSqaaiaa dwhacaWGUbGaamOCaiaadwgacaWGHbGaam4yaiaadshacaWGLbGaam izaaqabaGccaGGUaGaeqiVd0MaaiykaiabgkHiTiaadcfadaWgaaWc baGaam4CaiaadIgaaeqaaaaa@6155@
    Where,
    V = V V 0
    Relative volume
    E = E int V 0
    Internal energy per unit initial volume
    ω = γ 1 with γ = C p C V
    Adiabatic constant
    As usual (3)
    μ = ρ ρ 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaeqiVd0Maeyypa0ZaaSaaaeaacqaHbpGCaeaacqaHbpGCdaWgaaWc baGaaGimaaqabaaaaOGaeyOeI0IaaGymaaaa@4150@
    Bfrac
    Burn fraction of explosives 3
    Q = λ E a d d
    Optional afterburning energy
    λ
    Reaction ratio 7
  2. The Jones-Wilkins-Lee Material Law (LAW5) may be used as a boundary for Hydrodynamic Viscous Fluid Material (/MAT/LAW6 (HYDRO or HYD_VISC)) provided the flow direction is from LAW5 to LAW6 (simulation of an explosion), and the gas properties ( γ ) are similar. Nevertheless, this method is not the most accurate one and multi-material law (/MAT/LAW51 (MULTIMAT)) is recommended instead.
  3. Detonation Velocity (D) and Chapman Jouget Pressure (PCJ) are used in the burn fraction calculation ( B frac [ 0 , 1 ] ) . It controls the release of detonation energy and corresponds to a factor which multiplies JWL pressure.

    For a given time: P ( V , E ) = B frac P j wl ( V , E )

    A lighting time, T det , is computed by the Starter from the detonation velocity. During the simulation the burn fraction is computed as: (4)
    B frac = min ( 1 , max ( B f 1 , B f 2 ) )

    Where, B f1 = 1V 1 V CJ = ρ 0 D 2 P CJ ( 1V ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOqamaaBaaaleaacaWGMbGaaGymaaqabaGccqGH9aqpdaWcaaqa aiaaigdacqGHsislcaWGwbaabaGaaGymaiabgkHiTiaadAfadaWgaa WcbaGaam4qaiaadQeaaeqaaaaakiabg2da9maalaaabaGaeqyWdi3a aSbaaSqaaiaaicdaaeqaaOGaamiramaaCaaaleqabaGaaGOmaaaaaO qaaiaadcfadaWgaaWcbaGaam4qaiaadQeaaeqaaaaakmaabmGabaGa aGymaiabgkHiTiaadAfaaiaawIcacaGLPaaaaaa@4F2A@

    B f2 ={ T T det 1.5Δx T T det 0 T< T det MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOqamaaBaaaleaacaWGMbGaaGOmaaqabaGccqGH9aqpdaGabaqa auaabeqaciaaaeaadaWcaaqaaiaadsfacqGHsislcaWGubWaaSbaaS qaaiGacsgacaGGLbGaaiiDaaqabaaakeaacaaIXaGaaiOlaiaaiwda cqqHuoarcaWG4baaaaqaaiaadsfacqGHLjYScaWGubWaaSbaaSqaai GacsgacaGGLbGaaiiDaaqabaaakeaacaaIWaaabaGaamivaiabgYda 8iaadsfadaWgaaWcbaGaciizaiaacwgacaGG0baabeaaaaaakiaawU haaaaa@5469@

    It can take several cycles for the burn fraction to reach its maximum value of 1.00.

    Burn fraction calculation can be changed defining IBFRAC flag:

    IBFRAC = 0: B f r a c = min ( B f 1 , B f 2 ) is the default value

    IBFRAC = 1: B f r a c = min ( 1 , B f 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOqamaaBaaaleaacaWGMbGaamOCaiaadggacaWGJbaabeaakiab g2da9iGac2gacaGGPbGaaiOBamaabmaabaGaaGymaiaacYcacaWGcb WaaSbaaSqaaiaadAgacaaIXaaabeaaaOGaayjkaiaawMcaaaaa@4688@

    IBFRAC= 2: B f r a c = min ( 1 , B f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOqamaaBaaaleaacaWGMbGaamOCaiaadggacaWGJbaabeaakiab g2da9iGac2gacaGGPbGaaiOBamaabmaabaGaaGymaiaacYcacaWGcb WaaSbaaSqaaiaadAgacaaIYaaabeaaaOGaayjkaiaawMcaaaaa@4689@

  4. Time histories for detonation time and burn fraction are available through /TH/BRIC with keyword BFRAC. You can output a function, f , whose first value is detonation time (with opposite sign) and positive values corresponds to the burn fraction evolution.

    T det = f ( 0 )

    B frac ( t ) = { 0 , f ( t ) < 0 f ( t ) , f ( t ) 0

  5. Detonation times can be written in the Starter output file for each JWL element. The printout flag (Ipri) must be greater than or equal to 3 (/IOFLAG).
  6. If a detonation card is not linked to the material, then instantaneous detonation will be assumed.
  7. Afterburning can be modeled by introducing an additional Energy. If Eadd = 0, then there is no afterburning model and material law becomes a standard JWL EOS. If Eadd > 0, then the afterburning model is enabled with the default formulation QOPT = 0.
    Table 1. Available Afterburning Models in Case of Eadd > 0
    Modeling Type QOPT Reaction Rate ( d λ d t )
    Time controlled 0 Instantaneous
    1 Constant rate for energy release from Tstart to Tstop
    2 Linear rate for energy release from Tstart to Tstop
    Pressure dependent 3 Miller's extension

    d λ d t = a ( 1 λ ) m ( P u n i t P ) n

    Afterburning energy released is then Q = λ ( t ) E a d d where λ [ 0 , 1 ] . This term is added to JWL energy as described on Equation 1.

  8. In many publications, Miller’s parameters are often provided in the unit system g, cm, µ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyTaiaadohaaaa@383E@ which results in the pressure unit of Mbar. The ‘a’ parameter is also provided with the unit µ s -1 M b a r n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyTaiaadohapaWaaWbaaSqabeaacaqGTaGaaeymaaaak8qacaWG nbGaamOyaiaadggacaWGYbWdamaaCaaaleqabaWdbiabgkHiTiaad6 gaaaaaaa@3FBA@ and would require a unit translation if the input unit is different (/BEGIN). To avoid any unit translation, /MAT/LAW5 can be input with g, cm, µ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyTaiaadohaaaa@383E@ using the /UNIT option and then the input is automatically translated to the unit defined for the file in the /BEGIN line. Refer to Example (TNT) above for usage.
  9. When dealing with a multi-material formulation (/MAT/LAW51 (MULTIMAT) or /MAT/LAW151 (MULTIFLUID)), it is mandatory to provide a non-zero value for the bulk modulus Bunreacted of unreacted explosive. It is used to model a linear EOS for the unreacted explosive in order to ensure an equilibrium calculation and numerical stability.
  10. According to Hayes 1 Bunreacted can be estimated with the following formula:

    B u n r e a c t e d =   ρ 0   ( c 0 u n r e a c t e d ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaubeaKaaGeqajeaybaGaamyDaiaad6gacaWGYbGaamyzaiaadgga caWGJbGaamiDaiaadwgacaWGKbaajeaibeqcdasaaiaadkeaaaGccq GH9aqpcaGGGcGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaeyyXICTa aiiOamaabmaapaqaa8qacaWGJbWdamaaDaaaleaapeGaaGimaaWdae aapeGaamyDaiaad6gacaWGYbGaamyzaiaadggacaWGJbGaamiDaiaa dwgacaWGKbaaaaGccaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaik daaaaaaa@55D2@

    Where, c 0 u n r e a c t e d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ya8aadaqhaaWcbaWdbiaaicdaa8aabaWdbiaadwhacaWGUbGa amOCaiaadwgacaWGHbGaam4yaiaadshacaWGLbGaamizaaaaaaa@4081@ is the speed of sound in the unreacted explosive and an estimation for TNT is 2000 m/s.

  11. The Bunreacted parameter is the same parameter as the   C 1 m a t _ 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiiOaiaadoeapaWaa0baaSqaa8qacaaIXaaapaqaa8qacaWGTbGa amyyaiaadshacaGGFbGaaGinaaaaaaa@3D90@ parameter in /MAT/LAW51, Iform=10 and 11.
1 Hayes, B. "Fourth Symposium (International) on Detonation" Proceedings, Office of Naval Research, Department of the Navy, Washington, DC (1965): 595-601