# Assessment of the Sympathetic Detonation of Blasting Caps
## 3 Numerical Approach
### Material Models  
The basic properties of any substance, explosives included, are usually identified through a mathematical relation that correlates pressure, volume, and internal energy/ temperature. The relation is called an equation of state (EOS).  
In the process of numerical analysis of the sympathetic detonation, it was found that the choice of the equation of state that describes the behavior of the donor and acceptor charges plays a very important role.  
Figure 4: Materials location and gauges position.  
Figure 3: Typical mesh example.  
Over time, numerous attempts to define an EOS that accurately predicts the behavior of explosive gas products have been made. In fact, the number of proposed equations was high enough to classify them into two distinct categories: one considers the chemistry explicitly and the other does not [15].  
Unfortunately, all of the above-mentioned equations have limited the applicability and variable accuracy. Despite this shortcut, their use in numerical calculus is extremely beneficial since different particular situations can be investigated in a reasonable time frame and an almost costless manner.  
When explosive detonation applications are numerically investigated, one EOS is usually involved, namely Jones-Wilkins-Lee (JWL) [16]. In fact, the use of the JWL EOS is so common that by now almost all hydrocodes have implemented it and several forms can be identified in the literature. However, the most known form of equation is the form of a family of isentropes [17], which is illustrated in Equation (3).  
\[p(S,V)=Ae^{-R_{1}V}+Be^{-R_{2}V}+C^{*}(S)V^{-(\omega+1)} \tag{3}\]  
where \(p\) is the pressure; \(S\) refers to the entropy per unit initial volume (\(s/v_{0}\)); \(V\) is the volume relative to the undetonated state (\(r/v_{0}\)); \(A\), \(B\), \(R_{1}\), and \(R_{2}\) are constant fitting parameters; \(\omega\) is an assumed-constant material parameter (Gruneisen function); and \(C^{*}(S)\) is a parameter dependent only upon the entropy \(S\).  
Based on the previously mentioned JWL EOS and ideal gas EOS assumptions, the pressure in the front of a traveling blast wave can be accurately evaluated when hydrodynamic simulations are employed. Even though the equation's versatility is impressive, the application of JWL EOS by itself cannot deliver crucial data regarding the initiation of the explosive when subjected to blast wave stimulus. Thus, in order to investigate the blasting cap sympathetic detonation susceptibility, a slightly different EOS was chosen, the Lee-Tarver equation of state [18]. In fact, the Lee-Tarver EOS is basically a JWL EOS that has been upgraded with a supplementary equation, Equation (4), that allows the evaluation of the burning fraction based on the pressure level acting on/inside the explosive [18].  
\[\frac{\partial F}{\partial t}=I(1-F)^{b}\left(\frac{\rho}{\rho_{0}}-1-a\right) ^{x}+G_{1}(1-F)^{c}F^{d}p^{y}+G_{2}(1-F)^{c}Fg^{p} \tag{4}\]  
where \(F\) is the explosive burning fraction which has a value between 0 and 1.  
The importance of JWL EOS and Lee-Tarver EOS for the current blast cap sympathetic detonation study is correlated with the Chapman-Jouquet pressure level that can be used as the Go/No-Go criterion.  
The materials used in the numerical simulation are TNT for the donor charge, PETNJJ1 for the acceptor charge, aluminum for the blasting cap walls, and air for the space between the donor and the acceptor. All the equations of state (EOS) and strength models of the materials were adopted from the library of the Autodyn 2021(r) software.