/MAT/LAW5 (JWL)

Format

Definitions

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)
   $$P_{jwl}=A\left(1-\frac{\omega }{R_{1}V}\right)e^{-R_{1}V}+B\left(1-\frac{\omega }{R_{2}V}\right)e^{-R_{2}V}+\frac{\omega \left(E+Q\right)}{V}$$
   Radioss then outputs:(2)

   $$P=B_{frac}\cdot P_{jwl}+(1-B_{frac})(P_0+B_{unreacted}.\mu )-P_{sh}$$
   Where,

   $V=\frac{V}{V_0}$

   Relative volume

   $E=\frac{E_{\mathrm{int}}}{V_0}$

   Internal energy per unit initial volume

   $\omega =\gamma -1$ with $\gamma =\frac{C_p}{C_V}$

   Adiabatic constant

   As usual (3)

   $$\mu =\frac{\rho }{{\rho }_0}-1$$

   B_frac

   Burn fraction of explosives 3

   $Q=\lambda \cdot E_{add}$

   Optional afterburning energy

   $\lambda$

   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 ( $\gamma$ ) 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 (P_CJ) are used in the burn fraction calculation $\left(B_{\mathit{frac}}\in \left[0,1\right]\right)$ . It controls the release of detonation energy and corresponds to a factor which multiplies JWL pressure.

   For a given time: $P(V,E)=B_{\mathit{frac}}{P}_j{}_{\mathit{wl}}(V,E)$

   A lighting time, $T_{\det }$ , is computed by the Starter from the detonation velocity. During the simulation the burn fraction is computed as: (4)

   $$B_{\mathit{frac}}=\text{min}\left(1,\text{max}(\text{}{\text{B}}_{\text{f}1},{\text{B}}_{\text{f}2})\right)$$

   Where, $B_{f1}=\{\begin{array}{cc}\frac{T-T_{\text{det}}}{1.5\text{Δ}x}& T\ge T_{\text{det}}\\ 0& T<T_{\text{det}}\end{array}$

   $B_{f2}=\frac{1-V}{1-V_{\mathit{CJ}}}=\frac{{\rho }_0{D}^2}{P_{\mathit{CJ}}}\left(1-V\right)$

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

   Burn fraction calculation can be changed defining I_BFRAC flag:

   I_BFRAC = 0: $B_{frac}=\min \left(B_{f1},B_{f2}\right)$ is the default value

   I_BFRAC = 1: $B_{frac}=\min \left(1,B_{f1}\right)$

   I_BFRAC= 2: $B_{frac}=\min \left(1,B_{f2}\right)$

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

   $T_{\text{det}}=-\mathrm{f}\left(0\right)$

   $B_{\mathit{frac}(t)}=\{\begin{array}{c}0,\mathrm{f}(t)<0\\ \mathrm{f}(t),\mathrm{f}(t)\ge 0\end{array}$

5. Detonation times can be written in the Starter output file for each JWL element. The printout flag (I_pri) 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 E_add = 0, then there is no afterburning model and material law becomes a standard JWL EOS. If E_add > 0, then the afterburning model is enabled with the default formulation Q_OPT = 0.

   Table 1. Available Afterburning Models in Case of E_add > 0

   Modeling Type	QOPT	Reaction Rate ( $\frac{d\lambda }{dt}$ )
   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 $\frac{d\lambda }{dt}=a{\left(1-\lambda \right)}^m{\left(P_{unit}P\right)}^n$

   Afterburning energy released is then $Q=\lambda \left(t\right)\cdot E_{add}$ where $\lambda \in \left[0,1\right]$ . 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, $\mu s$ which results in the pressure unit of Mbar. The ‘a’ parameter is also provided with the unit $\mu s^{\text{-1}}Mba{r}^{-n}$ 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, $\mu s$ 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 B_unreacted 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 B_unreacted can be estimated with the following formula:

    ${{\displaystyle B}}_{unreacted}= {\rho }_0\cdot {\left(c_0^{unreacted}\right)}^2$

    Where, $c_0^{unreacted}$ is the speed of sound in the unreacted explosive and an estimation for TNT is 2000 m/s.

11. The B_unreacted parameter is the same parameter as the $C_1^{mat\_4}$ parameter in /MAT/LAW51, I_form=10 and 11.