/MAT/LAW41 (LEE_TARVER)

Block Format Keyword This material law describes detonation products using an ignition and growth model of a reactive material.

The Lee-Tarver 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)	(3)	(5)	(7)	(9)
/MAT/LAW41/`mat_ID`/`unit_ID` or /MAT/LEE_TARVER/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$	$ρ_{0}$
`Ireac`
$A^{r}$	$B^{r}$	$R_{1}^{r}$	$R_{2}^{r}$	$R_{3}^{r}$
$A^{p}$	$B^{p}$	$R_{1}^{p}$	$R_{2}^{p}$	$R_{3}^{p}$
$C_{ν}^{r}$	$C_{ν}^{p}$	$E_{Q}$
`itr`	$ε$	`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

Definition

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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
$ρ_{0}$	Reference density used in E.O.S (equation of state). Default = $ρ_{i}$ (Real)	$[\frac{kg}{m^{3}}]$
`Ireac`	Ignition and growth model flag. =1 (Default) for Lee_Tarver = 2 for Dyna (Integer)
$A^{r}$	Reagents JWL parameter. (Real)	$[Pa]$
$B^{r}$	Reagents JWL parameter. (Real)	$[Pa]$
$R_{1}^{r}$	Reagents JWL parameter. (Real)
$R_{2}^{r}$	Reagents JWL parameter. (Real)
$R_{3}^{r}$	Reagents JWL parameter. 4 (Real)	$[\frac{J}{m^{3} \cdot K}]$
$A^{p}$	Product JWL parameter. (Real)	$[Pa]$
$B^{p}$	Product JWL parameter. (Real)	$[Pa]$
$R_{1}^{p}$	Product JWL parameter. (Real)
$R_{2}^{p}$	Product JWL parameter. (Real)
$R_{3}^{p}$	Product JWL parameter. (Real)	$[\frac{J}{m^{3} \cdot K}]$
$C_{ν}^{r}$	Heat capacity reagents. (Real)	$[\frac{J}{m^{3} \cdot K}]$
$C_{ν}^{p}$	Heat capacity product. (Real)	$[\frac{J}{m^{3} \cdot K}]$
$E_{Q}$	Heat of reaction. (Real)	$[\frac{J}{m^{3}}]$
`itr`	Maximum number of iterations for the mixing law. Default = 80 (Integer)
$ε$	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 (Lee-Tarver and Dyna-2D). (Real)	$[s^{- 1}]$
`ex`	Chemical kinetic coefficient of the starting phase (Lee-Tarver and Dyna-2D). (Real)
`r`_i	Chemical kinetic coefficient of the starting phase (Lee-Tarver and Dyna-2D). (Real)
`r`_kg	Chemical kinetic coefficient of the growing phase (Lee-Tarver and Dyna-2D). (Real)	$[s^{- 1} P a^{- Z_{g}}]$
`y`_g	Chemical kinetic coefficient of the growing phase (Lee-Tarver and Dyna-2D). (Real)
`z`_g	Chemical kinetic coefficient of the growing phase (Lee-Tarver and Dyna-2D). (Real)
`ex`₁	Chemical kinetic coefficient of the growing phase (Dyna-2D). (Real)
`k`	Numerical limiters coefficient (Lee-Tarver and Dyna-2D). Default = 99.0 (Real)
`X`	Numerical limiters coefficient (Dyna-2D). Default = 99.0 (Real)
`tol`	Numerical limiters coefficient (Dyna-2D). Default = 0.0 (Real)
`grow`₂	Growing phase 2 coefficient (Dyna-2D). (Real)	$[s^{- 1} P a^{- Z_{g 2}}]$
`ex`₂	Growing phase 2 coefficient (Dyna-2D). (Real)
`y`_g2	Growing phase 2 coefficient (Dyna-2D). (Real)
`z`_g2	Growing phase 2 coefficient (Dyna-2D). (Real)
`ccrit`	Starting threshold (for compression) (Dyna-2D). (Real)
`fmxig`	Starting threshold (mass fraction) (Dyna-2D). (Real)
`fmxgr`	Coefficient (Dyna-2D). 5 (Real)
`fmngr`	Coefficient (Dyna-2D). 5 (Real)
`G`	Shear modulus. (Real)	$[Pa]$
`T`_i	Initial temperature. (Real)	$[K]$

Example (LX17)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW41/1
LX17 (unit Mg-mm-s)
#              RHO_I               RHO_0
                1900                   0
#    Ireac
         2
#                 Ar                  Br                 R1r                 R2r                 R3r
       4930000000000       -166000000000                7.44                3.72           3.3337E-5
#                 Ap                  Bp                 R1p                 R2p                 R3p
        696000000000          2500000000                 4.4                 .94              4.3E-6
#                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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

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)
${\frac{d f}{d t})}_{i} = r_{k i} \cdot {(1 - f)}^{e x} \cdot {(\frac{ρ}{ρ_{0}} - 1)}^{r_{i}}$
(2)
${\frac{d f}{d t})}_{g} = r_{k g} \cdot {(1 - f)}^{e x} \cdot f^{y_{g}} \cdot p^{z_{g}}$

Ireac = 2: "Ignition and growth" according to the formulation introduced in Dyna-2D(3)
${\frac{d f}{d t})}_{i} = r_{k i} \cdot {(f m x i g - f)}^{e x} \cdot {(\frac{ρ}{ρ_{0}} - 1 - c c r i t)}^{r_{i}}$
(4)
${\frac{d f}{d t})}_{g 1} = r_{k g} \cdot {(1 - f)}^{e x 1} \cdot f^{y_{g 1}} \cdot P^{z_{g 1}}$
(5)
${\frac{d f}{d t})}_{g 2} = g r o w_{2} \cdot {(1 - f)}^{e x 2} \cdot f^{y_{g 2}} \cdot P^{z_{g 2}}$
Coefficient grow₁ is initialized by r_kg
Coefficients y_g1 and z_g1 are respectively initialized by y_g and z_g.
Coefficients R₃ and $ω$ are linked by the relation: $R_{3} = ω C_{ν}$
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 Jones-Wilkins-Lee equation of state:
In terms of relative volume $v$ :(6)
$P (v, T) = A e^{- R_{1} v} + B e^{- R_{2} v} + R_{3} T v$

Where, $v = \frac{V}{V_{0}}$

In terms of $μ$ :(7)
$P (μ, T) = A e^{- R_{1} / (1 + μ)} + B e^{- R_{2} / (1 + μ)} + R_{3} T / (1 + μ)$

Where, $μ = \frac{ρ}{ρ_{0}} - 1 = \frac{1}{v} - 1$ .

¹ E.L. Lee and C.M. Tarver "Phenomenological model of shock initiation in heterogeneous explosives" Phy. Fluids Vol. 23, No. 12, December 1980.