Command manual

This is the History Variable Reactive Burn (HVRB) model, for predicting initiation of explosive materials. It is currently only implemented for Finite Elements.

The material pressure $p$ is defined as:

$\displaystyle{ p = (1 - F) \cdot p_s + F \cdot p_g }$

where $0 \leq F \leq 1$ is burn fraction, $p_s$ is the pressure of the solid material and $p_g$ is the pressure of generated gas products.

Note that if equilibrium=0 then $\rho_s \equiv \rho_g$ and $p_s \not\equiv p_g$. equilibrium=1 activates an iterative scheme where the densities are balanced $(\rho_s \not\equiv \rho_g)$ such that $p_s \equiv p_g$. Temperature is not balanced. The thermal process is assumed to be slower and heat generated in the deflagration process is assumed to stay in the gas phase. Hence, $e_s \not\equiv e_g$.

The solid material is elasto-plastic with (J2) flow stress:

$\displaystyle{\sigma_y = \sigma_0 + Q \left( 1 - \mathrm{exp} (-C \varepsilon_p)\right)}$

The shear stregth is assumed to be lost as soon as the deflagration process starts (i.e. if $F \gt 0$).

Solid phase speed-of-sound

The relationship between sound speed $c$ and particle $u_p$ is divided into two regions.

$\displaystyle{ c = \left\{ \begin{array}{ccc} c_0 + s_0 \cdot u_p & : & u_p \lt u_p^* \\ c_1 + s_1 \cdot u_p & : & u_p \geq u_p^* \end{array} \right. }$

where:

$\displaystyle{ u_p^* = \frac{c_0 - c_1}{s_1 - s_0} }$

Solid phase pressure

The solid phase pressure (Gruneisen EOS) is:

$\displaystyle{ p_s = \frac{K \eta}{(1-s \eta)^2} \cdot \left( 1 - \frac{\Gamma \eta}{2} \right) + \Gamma e_s }$

$\displaystyle{ \eta = 1 - \rho_0 / \rho_s }$

Here $K$ is the bulk modulus, $\rho_0$ is the initial density and $\rho_s$ is the current density of the solid phase. $e_s$ is the sepcific internal energy per unit volume in the solid phase and $s$ is:

$\displaystyle{ s = \left\{ \begin{array}{ccc} s_0 & : & u_p \lt u_p^* \\ s_1 & : & u_p \geq u_p^* \end{array} \right. }$

Gas phase pressure

$\displaystyle{ p_g = A \left( 1 - \frac{\omega}{R_1 V} \right) \mathrm{e}^{-R_1 V} + B \left( 1 - \frac{\omega}{R_2 V} \right) \mathrm{e}^{-R_2 V} + \omega e_g}$

where:

$\displaystyle{ V = \rho_0 / \rho_g}$

$e_g$ and $\rho_g$ are the specific internal energy and density in the gas phase.

Deflagration

The burn fraction $0 \leq F \leq 1$ is defined according to:

$\displaystyle{ F = 1 - \left(1 - \mathrm{min} \left( 1, \frac{\phi^M}{X} \right) \right)^X}$

$\displaystyle{ \phi(t) = \frac{1}{\tau_R} \int_0^t \left( \frac{\mathrm{max}(0, p - p_I)}{p_R} \right)^Z \mathrm{d}\tau }$

The combustion process generates thermal energy, and the energy release rate per unit volume is:

$\displaystyle{ \frac{\partial e}{\partial F} = \frac{\rho}{\rho_0} e_0 }$