/MAT/LAW5 (JWL)
Block Format Keyword This law describes the JonesWilkinsLee 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  
${\rho}_{i}$  ${\rho}_{0}$  
A  B  R_{1}  R_{2}  $\omega $  
D  P_{CJ}  E_{0}  E_{add}  I_{BFRAC}  Q_{OPT}  
P_{0}  P_{sh}  B_{unreacted} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

T_{start}  T_{stop} 
(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) 

${\rho}_{i}$  Initial
density (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
${\rho}_{0}$  Reference density used in
E.O.S (equation of state) Default = ${\rho}_{0}={\rho}_{i}$ (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
A  A parameter of equation of
state (Real) 
$\left[\text{Pa}\right]$ 
B  B parameter of equation of
state (Real) 
$\left[\text{Pa}\right]$ 
R_{1}  R_{1} parameter of equation of
state. (Real) 

R_{2}  R_{2} parameter of equation of
state. (Real) 

$\omega $ 
$\omega $
parameter of equation of
state. (Real) 

D  Detonation
velocity. (Real) 
$\left[\frac{\text{m}}{\text{s}}\right]$ 
P_{CJ}  Chapman Jouguet
pressure. (Real) 
$\left[\text{Pa}\right]$ 
E_{0}  Detonation energy per unit
volume. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$ 
E_{add}  Additional energy per unit volume.
(Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$ 
I_{BFRAC}  Burn fraction calculation
flag. 3
(Integer) 

Q_{OPT}  Optional afterburning
model (if E_{add} > 0).
(Integer) 

P_{0}  Initial
pressure. (Real) 
$\left[\text{Pa}\right]$ 
P_{sh}  Pressure
shift. (Real) 
$\left[\text{Pa}\right]$ 
B_{unreacted}  Unreacted explosive bulk
modulus. 9
10
11 (Real) 
$\left[\text{Pa}\right]$ 
T_{start}  Start time for additional
energy (Q_{OPT} =
0,1,2). (Real) 
$\left[\text{s}\right]$ 
T_{stop}  Stop time for additional
energy (Q_{OPT} =
0,1,2). (Real) 
$\left[\text{s}\right]$ 
a  Optional Miller parameter
if Q_{OPT} =
3. (Real) 
$\left[{s}^{1}P{a}^{n}\right]$ 
m  Optional Miller parameter
if Q_{OPT} =
3. (Real) 

n  Optional Miller parameter
if Q_{OPT} =
3. (Real) 
Example (TNT)
#12345678910
/MAT/JWL/2/123
TNT  data from example 46 with unit: (gcmmus)  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
#12345678910
/UNIT/123
Miller’s extension unit system
g cm mus
#12345678910
Comments
 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
 The JonesWilkinsLee 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 multimaterial law (/MAT/LAW51 (MULTIMAT)) is recommended instead.
 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, ${T}_{\mathrm{det}}$ , is computed by the Starter from the detonation velocity. During the simulation the burn fraction is computed as:(4) $${B}_{\mathit{frac}}=\text{min}\left(1,\text{max}\text{\hspace{0.05em}}(\text{}{\text{B}}_{\text{f}1},{\text{B}}_{\text{f}2})\right)$$Where, ${B}_{f1}=\frac{1V}{1{V}_{CJ}}=\frac{{\rho}_{0}{D}^{2}}{{P}_{CJ}}\left(1V\right)$
${B}_{f2}=\{\begin{array}{cc}\frac{T{T}_{\mathrm{det}}}{1.5\text{\Delta}x}& T\ge {T}_{\mathrm{det}}\\ 0& T<{T}_{\mathrm{det}}\end{array}$
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: ${B}_{frac}=\mathrm{min}\left({B}_{f1},{B}_{f2}\right)$ is the default value
I_{BFRAC} = 1: ${B}_{frac}=\mathrm{min}\left(1,{B}_{f1}\right)$
I_{BFRAC}= 2: ${B}_{frac}=\mathrm{min}\left(1,{B}_{f2}\right)$
 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.
${T}_{\text{det}}=\mathrm{f}\left(0\right)$
${B}_{\mathit{frac}(t)}=\{\begin{array}{c}0,\text{\hspace{1em}}\mathrm{f}(t)<0\\ \mathrm{f}(t),\text{\hspace{1em}}\mathrm{f}(t)\ge 0\end{array}$
 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).
 If a detonation card is not linked to the material, then instantaneous detonation will be assumed.
 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 Q_{OPT} Reaction Rate ( $\frac{d\lambda}{dt}$ ) Time controlled 0 Instantaneous 1 Constant rate for energy release from T_{start} to T_{stop} 2 Linear rate for energy release from T_{start} to T_{stop} 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.
 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.
 When dealing with a multimaterial formulation (/MAT/LAW51 (MULTIMAT) or /MAT/LAW151 (MULTIFLUID)), it is mandatory to provide a nonzero 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.
 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.
 The B_{unreacted} parameter is the same parameter as the ${C}_{1}^{mat\_4}$ parameter in /MAT/LAW51, I_{form}=10 and 11.