/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
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/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 $  F_{tol}  
I  b  x  
G_{1}  d  y  c  
kappa  khi  tol  
G_{2}  e  g  z  
a  F_{igmax}  F_{G1max}  F_{G2min}  
G  T_{i} 
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 = ${\rho}_{i}$ (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
Ireac  Ignition and growth model flag. 1 (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. 2
(Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ = $\left[\frac{\mathrm{Pa}}{\mathrm{K}}\right]$ 
${A}^{p}$  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. 2 (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ = $\left[\frac{\mathrm{Pa}}{\mathrm{K}}\right]$ 
${C}_{\nu}^{r}$  Volumetric heat capacity for
reagents. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ = $\left[\frac{\mathrm{Pa}}{\mathrm{K}}\right]$ 
${C}_{\nu}^{p}$  Volumetric heat capacity for
product. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ = $\left[\frac{\mathrm{Pa}}{\mathrm{K}}\right]$ 
${E}_{Q}$  Heat of reaction. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$ 
itr  Maximum number of iterations. 2 Default = 80 (Integer) 

$\epsilon $  Convergency tolerance. 2
Default = 10^{3} (Real) 

F_{tol}  Burn fraction threshold. 3 Default = 10^{5} (Real) 

I  LeeTarver
parameter. (Real) 
$\left[{s}^{1}\right]$ 
b  LeeTarver
parameter. (Real) 

x  LeeTarver
parameter. (Real) 

G_{1}  LeeTarver
parameter. (Real) 
$\left[{s}^{1}P{a}^{{Z}_{g}}\right]$ 
d  LeeTarver
parameter. (Real) 

y  LeeTarver
parameter. (Real) 

c  LeeTarver
parameter. (Real) 

kappa  Numerical limiters. 6 Default = 99.0 (Real) 

khi  Numerical limiters (extended model).
5 Default = 99.0 (Real) 

tol  Numerical tolerance (extended model).
7 Default = 0.0 (Real) 

G_{2}  LeeTarver
parameter. (Real) 
$\left[{s}^{1}P{a}^{{Z}_{g2}}\right]$ 
e  LeeTarver
parameter. (Real) 

g  LeeTarver
parameter. (Real) 

z  LeeTarver
parameter. (Real) 

a  LeeTarver parameter (extended
model). (Real) 

F_{igmax}  Ignition term limiter (extended model).
1 (Real) 

F_{G1max}  Growth term #1 limiter (extended model).
1
(Real) 

F_{G2min}  Ignition term #2 limiter (extended
model). 1
(Real) 

G  Shear modulus. 4 (Real) 
$\left[\text{Pa}\right]$ 
T_{i}  Initial
temperature. (Real) 
$\left[\text{K}\right]$ 
Example (COMPB)
#RADIOSS STARTER
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW41/1
Military CompB (unit g,mm,µs,Mbar) ; Ref:UCRLJC111975,May 1993 (submittal 10th Det. Symposium)
# RHO_0
1.630
# Ireac
2
# Ar Br R1r R2r R3r
1479.0 0.05261 12.0 1.2 2.268144E5
# Ap Bp R1p R2p R3p
5.5748 0.0783 4.5 1.2 0.34E5
# Cvr Cvp Eq
2.487E5 1E5 .081
# iter eps Ftol
0 0 0
# I b x
44.0 0.22222222222222 4
# G1 d y c
514.0 0.66666666666666 2 0.22222222222222
# kappa khi tol
0 0 0
# G2 e g z
0.0 0.0 0.0 0.0
# a Figmax FG1max FG2min
0 .30 1.0 1.0
# G Ti
0.0354 298
#12345678910#ENDDATA
/END
#12345678910
Comments
 This material describes a mixture of reagents (unreacted explosive) and products (gas from detonation) for which the burn fraction is dictated by function $F\left(t\right)$ . There are two possible models to describe function $F$ depending on Ireac value:
 Both reagents and products
are described with a JWL equation of state. The temperature dependency form is used
$P\left(\nu ,T\right)=A{e}^{{R}_{1}\nu}+B{e}^{{R}_{2}\nu}+{R}_{3}T/\nu $
where,
${R}_{3}=\omega {c}_{\nu}$
.Consequently, a set of JWL parameters must be defined for both reagents and products:
 Reagents:$${P}_{r}\left(\nu ,T\right)={A}_{r}{e}^{{R}_{{}_{1}}^{r}\nu}+{B}_{r}{e}^{{R}_{{}_{2}}^{r}\nu}+{R}_{{}_{3}}^{r}T/\nu $$
 Products:$${P}_{p}\left(\nu ,T\right)={A}_{p}{e}^{{R}_{{}_{1}}^{p}\nu}+{B}_{p}{e}^{{R}_{{}_{2}}^{p}\nu}+{R}_{{}_{3}}^{p}T/\nu $$
For reagents, ${B}_{r}$ is negative so that the solid undergoes tension and ${\omega}_{r}$ is set to the initial Gruneisen coefficient.
An iterative solver is used to ensure equilibrium between reagents and products: ${P}_{r}={P}_{p}$ .
itr is the maximum number of iteration and eps is the convergency tolerance $\left(\left{P}_{r}{P}_{p}\right<eps\right)$ .
 Reagents:

${F}_{tol}$
is such as:
 $F<{F}_{tol}$ : reaction has not started
 $F>1{F}_{tol}$ : reaction has finished
 Shear modulus is used for
sound speed calculation: $${c}^{2}=\frac{1}{{\rho}_{0}}\left(\frac{dP}{d\mu}+\frac{4}{3}G\right)$$

$khi$
is a numerical limiter such as: $$F\left({t}_{n+1}\right)F\left({t}_{n}\right)\le khi$$

$kappa$
is a numerical limiter such as reaction is not calculated if
$Q\ge kappa\cdot P$
.
Where, $Q$ = Pseudo viscosity (from shock front).

$tol$
is a tolerance parameter.
$dF/dt$ is updated using $F+{F}_{tol}$ instead of $F$ in the right hand term of the 3termmodel.
 This material is not yet compatible with ALE.