MAT_EXPLOSIVE_INITIATION
Material properties
"Optional title"
mid, $\rho_0$, $G$
$K$, $\gamma$, $A$, $B$, $n$, $\omega$, $e_0$, $\eta$
$\xi_1$, $p_1$, $t_1$, $\alpha_1$, $\xi_2$, $p_2$, $t_2$, $\alpha_2$
$l_{ref}$, $\beta$, $\delta$, $b$, $F_{init}$
Parameter definition
Description
This material model is used to predict explosive deflagration and the transition to detonation. The triggering mechanism is pressure. Undetonated explosive is treated as an elasto-plastic iso-tropic material with a J2 flow stress:
$\displaystyle{\sigma_y = \left( 1 - \left( 1 - \delta \right) F \right) \cdot \left( A + B(\varepsilon_{eff}^p)^n \right)}$
Here $F$ is the burn fraction and $(A,B,n)$ are classical power law hardening parameters. $0 \leq \delta \leq 1$ is an optional numerical parameter that can be set to keep some shear strength after reaching full deflagration. The purpose of this parameter is to reduce mesh distortions and to delay the need for element erosion.
The deflagration burn rate $\dot{F}$ is defined as:
$\displaystyle{ \dot{F} = (1 - \eta) \dot{F}_1 + \eta \dot{F}_2 }$
$\displaystyle{ \dot{F}_i = \left\{ \begin{array}{ccc} 0 & : & p \lt p_i \\ \frac{\xi_i + F_i(1 - F_i)}{t_i} \cdot \left( \frac{p}{p_i} - 1 \right)^{\alpha_i} \cdot \mathrm{max} \left( 1, \frac{l_c}{l_{ref}} \right)^\beta & : & p \geq p_i \end{array} \right. }$
By splitting the burn fraction into two terms $(F_1,F_2)$ it is possible to model different levels of detonation. That is, depending on how the charge is initiated, different detonation and energy release processes can follow. $F$ starts at $0$ and full deflagration is reached when $F=1$.
$p_i$ is a threshold pressure for the deflagration process, $t_i$ is a deflagration time parameter and $\alpha_i$ determines the pressure sensitivity of the deflagration process. $\xi_i$ is a small (dimensionless) parameter that controls the rate of the ignition phase. $l_{ref}$ (reference element size) and $\beta$ are optional numerical parameters that can adjust the burn rate $F_i$ to the characteristic element size $l_c$. A correction is necessary if the model parameters are calibrated on a fine mesh $(l_c \leq l_{ref})$ and then used on a finite element grid where the elements are too large to resolve the pressure profile at the burn front.
The combustion process generates thermal energy, and the energy release rate per unit volume is:
$\displaystyle{ \frac{\partial e}{\partial F} = \frac{\rho}{\rho_0} e_0 }$
Pressure $p$ is a function of both density $\rho$ and internal energy per unit volume $e$. As long as the deflagration process has not started (i.e. while $F=0$) the pressure level is defined as:
$\displaystyle{ p = \frac{K}{\gamma} \left( \left( \frac{\rho}{\rho_0} \right)^\gamma - 1 \right) + \frac{\omega e}{1 - \rho b} }$
Once the deflagration process has started ($F \gt 0$) negative elastic pressures are no longer allowed:
$\displaystyle{ p = \frac{K}{\gamma} \left( \left( \frac{\mathrm{max}(\rho_0, \rho)}{\rho_0} \right)^\gamma - 1 \right) + \frac{\omega e}{1 - \rho b} }$
