/MAT/LAW41 (LEE_TARVER)
Block Format Keyword This material law describes detonation products using an ignition and growth model of a reactive material.
The LeeTarver model is based on the assumption that the ignition starts at local hot spots in the passage of shock front and grows outward from these sites. The reaction rate is controlled by the pressure and the surface area as in a deflagration process.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW41/mat_ID/unit_ID or /MAT/LEE_TARVER/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  ${\rho}_{0}$  
Ireac  
${A}^{r}$  ${B}^{r}$  ${R}_{1}^{r}$  ${R}_{2}^{r}$  ${R}_{3}^{r}$  
${A}^{p}$  ${B}^{p}$  ${R}_{1}^{p}$  ${R}_{2}^{p}$  ${R}_{3}^{p}$  
${C}_{\nu}^{r}$  ${C}_{\nu}^{p}$  ${E}_{Q}$  
itr  $\epsilon $  check  
r_{ki}  ex  r_{i}  
r_{kg}  y_{g}  z_{g}  ex_{1}  
k  X  tol  
grow_{2}  ex_{2}  y_{g2}  z_{g2}  
ccrit  fmxig  fmxgr  fmngr  
G  T_{i} 
Definitions
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  Unit Identifier. (Interger, 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 = ${\rho}_{i}$ (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
Ireac  Ignition and growth model flag.
(Integer) 

${A}^{r}$  Reagents JWL
parameter. (Real) 
$\left[\text{Pa}\right]$ 
${B}^{r}$  Reagents JWL
parameter. (Real) 
$\left[\text{Pa}\right]$ 
${R}_{1}^{r}$  Reagents JWL
parameter. (Real) 

${R}_{2}^{r}$  Reagents JWL
parameter. (Real) 

${R}_{3}^{r}$  Reagents JWL parameter. 4 (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ 
${A}^{p}$  Product JWL
parameter. (Real) 
$\left[\text{Pa}\right]$ 
${B}^{p}$  Product JWL
parameter. (Real) 
$\left[\text{Pa}\right]$ 
${R}_{1}^{p}$  Product JWL
parameter. (Real) 

${R}_{2}^{p}$  Product JWL
parameter. (Real) 

${R}_{3}^{p}$  Product JWL
parameter. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ 
${C}_{\nu}^{r}$  Heat capacity
reagents. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ 
${C}_{\nu}^{p}$  Heat capacity
product. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ 
${E}_{Q}$  Heat of reaction. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$ 
itr  Maximum number of iterations for the
mixing law. Default = 80 (Integer) 

$\epsilon $  Precision on hydrodynamic
balance. Default = 10^{3} (Real) 

check  Limiter of the mass fraction of
products. Default = 10^{5} (Real) 

r_{ki}  Chemical kinetic coefficient of the
starting phase (LeeTarver and Dyna2D). (Real) 
$\left[{s}^{1}\right]$ 
ex  Chemical kinetic coefficient of the
starting phase (LeeTarver and Dyna2D). (Real) 

r_{i}  Chemical kinetic coefficient of the
starting phase (LeeTarver and Dyna2D). (Real) 

r_{kg}  Chemical kinetic coefficient of the
growing phase (LeeTarver and Dyna2D). (Real) 

y_{g}  Chemical kinetic coefficient of the
growing phase (LeeTarver and Dyna2D). (Real) 

z_{g}  Chemical kinetic coefficient of the
growing phase (LeeTarver and Dyna2D). (Real) 
$\left[{s}^{1}P{a}^{{Z}_{g}}\right]$ 
ex_{1}  Chemical kinetic coefficient of the
growing phase (Dyna2D). (Real) 

k  Numerical limiters coefficient
(LeeTarver and Dyna2D). Default = 99.0 (Real) 

X  Numerical limiters coefficient
(Dyna2D). Default = 99.0 (Real) 

tol  Numerical limiters coefficient
(Dyna2D). Default = 0.0 (Real) 

grow_{2}  Growing phase 2 coefficient
(Dyna2D). (Real) 
$\left[{s}^{1}P{a}^{{Z}_{g}}\right]$ 
ex_{2}  Growing phase 2 coefficient
(Dyna2D). (Real) 

y_{g2}  Growing phase 2 coefficient
(Dyna2D). (Real) 

z_{g2}  Growing phase 2 coefficient
(Dyna2D). (Real) 

ccrit  Starting threshold (for compression)
(Dyna2D). (Real) 

fmxig  Starting threshold (mass fraction)
(Dyna2D). (Real) 

fmxgr  Coefficient (Dyna2D). 5 (Real) 

fmngr  Coefficient (Dyna2D). 5 (Real) 

G  Shear modulus. (Real) 
$\left[\text{Pa}\right]$ 
T_{i}  Initial
temperature. (Real) 
$\left[\text{K}\right]$ 
Example (LX17)
#RADIOSS STARTER
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW41/1
LX17 (unit Mgmms)
# RHO_I RHO_0
1900 0
# Ireac
2
# Ar Br R1r R2r R3r
4930000000000 166000000000 7.44 3.72 3.3337E5
# Ap Bp R1p R2p R3p
696000000000 2500000000 4.4 .94 4.3E6
# Cvr Cvp Eq
2781 1000 .088
# iter EPS check
0 0 0
# rki ex ri
100000000 1 4
# rkg yg zg ex1
1000000000 .371 3 .191
# K X tol
0 0 0
# grow2 ex2 yg2 zg2
0 1 1 1
# ccrit fmxig fmxgr fmngr
0 .25 1 100
# G Ti
75000000 298
#12345678910
#ENDDATA
/END
#12345678910
Comments
 If f is
the mass fraction of the products and p is the reduced
pressure:Ireac = 1: "Ignition and growth" according to Lee/Tarver
(1) $${\frac{df}{dt})}_{i}={r}_{ki}\cdot {\left(1f\right)}^{ex}\cdot {\left(\frac{\rho}{{\rho}_{0}}1\right)}^{{r}_{i}}$$(2) $${\frac{df}{dt})}_{g}={r}_{kg}\cdot {\left(1f\right)}^{ex}\cdot {f}^{{y}_{g}}\cdot {p}^{{z}_{g}}$$Ireac = 2: "Ignition and growth" according to the formulation introduced in Dyna2D(3) $${\frac{df}{dt})}_{i}={r}_{ik}\cdot {\left(fmxigf\right)}^{ex}\cdot {\left(\frac{\rho}{{\rho}_{0}}1ccrit\right)}^{{r}_{i}}$$(4) $${\frac{df}{dt})}_{g1}={r}_{kg}\cdot {\left(1f\right)}^{ex1}\cdot {f}^{{y}_{g1}}\cdot {P}^{{z}_{g1}}$$(5) $${\frac{df}{dt})}_{g2}=gro{w}_{2}\cdot {\left(1f\right)}^{ex2}\cdot {f}^{{y}_{g2}}\cdot {P}^{{z}_{g2}}$$  Coefficient grow_{1} is initialized by r_{kg}
 Coefficients y_{g1} and z_{g1} are respectively initialized by y_{g} and z_{g}.
 Coefficients R_{3} and $\omega $ are linked by the relation: ${R}_{3}=\omega {C}_{\nu}$
 Coefficients fmxgr and fmngr are the limiters of the growth rate according to the mass fraction of products.
 This material law is not compatible with ALE.
 Heat of reaction ${E}_{Q}$ is supposed to be constant whatever the value of F is.
 Reagent pressure and
detonation products pressure are computed using a modified JonesWilkinsLee equation of
state:In terms of relative volume $v$ :
(6) $$\mathrm{P}\left(v,T\right)=A{e}^{{R}_{1}v}+B{e}^{{R}_{2}v}+{R}_{3}Tv$$Where, $v=\frac{V}{{V}_{0}}$
In terms of $\mu $ :(7) $$\mathrm{P}\left(\mu ,T\right)=A{e}^{{R}_{1}/\left(1+\mu \right)}+B{e}^{{R}_{2}/\left(1+\mu \right)}+{R}_{3}T/\left(1+\mu \right)$$Where, $\mu =\frac{\rho}{{\rho}_{0}}1=\frac{1}{v}1$ .