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This is the classical Lee-Tarver reactive burn model. The fraction of combusted material is denoted $F$ and it ranges from 0 to 1. The unreacted material is modeled as elasto-plastic with flow stress:

$\displaystyle{\sigma_y = A + B(\varepsilon_{eff}^p)^n}$

The pressure in the unreacted phase is defined as:

$\displaystyle{ p_u = A_u \left( 1 - \frac{\omega_u}{R_{1,u} V_u} \right) \mathrm{e}^{-R_{1,u}V_u} + B_u \left( 1 - \frac{\omega_u}{R_{2,u} V_u} \right) \mathrm{e}^{-R_{2,u}V_u} + \omega_u e_u}$

Here $V_u$ is the relative volume of the unreacted material and $e_u$ is the specific internal energy of the unreacted material.

In an equivalent way the pressure in reaction products is defined as:

$\displaystyle{ p_r = A_r \left( 1 - \frac{\omega_r}{R_{1,r} V_r} \right) \mathrm{e}^{-R_{1,r}V_r} + B_r \left( 1 - \frac{\omega_r}{R_{2,r} V_r} \right) \mathrm{e}^{-R_{2,r}V_r} + \omega_r e_r}$

There is assumed be a pressure equilibrium between the phases in a mixture of unburned and burned material $p \equiv p_u \equiv p_r$. This is achieved by adjusting the relative volumes of the phases while maintaining:

$\displaystyle{ V = F V_r + (1-F) V_u}$

where $V$ is the relative volume of the mixture, i.e. the ratio of initial to current density.

$\displaystyle{ V = \frac{\rho_0}{\rho}}$

The deflagration/detonation rate is described with a burn model consisting of three terms:

$\displaystyle{ \frac{\mathrm{d}F}{\mathrm{d}t} = \frac{\mathrm{d}F_1}{\mathrm{d}t} + \frac{\mathrm{d}F_2}{\mathrm{d}t} + \frac{\mathrm{d}F_3}{\mathrm{d}t} }$

The first term (stage I) models the initiation based on compression:

$\displaystyle{ \frac{\mathrm{d}F_1}{\mathrm{d}t} = \left\{ \begin{array}{ccc} I(1-F)^b (\frac{\rho}{\rho_0} - 1 - a)^x & : & F \leq F_1 \\ 0 & : & F > F_1 \end{array} \right. }$

The second term (stage II) describes the early stage of hot spot growth:

$\displaystyle{ \frac{\mathrm{d}F_2}{\mathrm{d}t} = \left\{ \begin{array}{ccc} G_1(1-F)^c F^d \left(\frac{p}{p_0}\right)^y & : & F \leq F_2 \\ 0 & : & F > F_2 \end{array} \right. }$

The third one (stage III) handles the rapid growth as hot spots coalesce:

$\displaystyle{ \frac{\mathrm{d}F_3}{\mathrm{d}t} = \left\{ \begin{array}{ccc} 0 & : & F \lt F_3 \\ G_2(1-F)^e F^g \left(\frac{p}{p_0}\right)^z & : & F \geq F_3 \end{array} \right. }$

This material model works with shock viscosity. This adds an extra pressure term $q$ to the total pressure. It is calculated as:

$\displaystyle{ q = \left\{ \begin{array}{ccc} \rho L (1.5 L \dot{\varepsilon}_{vol}^2 - 0.06 c \dot{\varepsilon}_{vol}) & : & \dot{\varepsilon}_{vol} \lt 0 \\ 0 & : & \dot{\varepsilon}_{vol} \geq 0 \end{array} \right. }$

Here $c$ is the speed-of-sound, $\dot{\varepsilon}_{vol}$ is volumetric strain rate and $L$ is the local element size.