# Numerical Analysis and Experimental Test for the Development of a Small Shaped Charge
## 2 Materials and Methods
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Shaped Charge Liner** & **Steel Target** \\ \hline Density & \(\rho\) & Kg/m\({}^{3}\) & 8940 & 7860 \\ Shear modulus & G & GPa & - & 81.8 \\ Young modulus & E & GPa & 126 & 209 \\ Poisson’s ratio & \(\
u\) & - & 0.335 & 0.28 \\ Yield stress & A & MPa & 99.7 & 792 \\ Hardening constant & B & MPa & 262.8 & 510 \\ Hardening exponent & N & - & 0.23 & 0.26 \\ Strain rate constant & C & - & 0.029 & 0.014 \\ Thermal softening exponent & M & - & 0.98 & 1.03 \\ Room temperature & Tr & K & 293 & 300 \\ Melting temperature & Tm & K & 775 & 1790 \\ Ref. strain rate & EPSO & s\({}^{-1}\) & 1.0 & 1.0 \\ Specific heat & Cp & J/kgK & 875 & 477 \\ Johnson Cook failure & & & & \\ Failure parameter & D\({}_{1}\) & - & 0.13 & 0.05 \\ Failure parameter & D\({}_{2}\) & - & 0.13 & 3.44 \\ Failure parameter & D\({}_{3}\) & - & \(-1.5\) & \(-2.12\) \\ Failure parameter & D\({}_{4}\) & - & 0.011 & 0.002 \\ Failure parameter & D\({}_{5}\) & - & 0 & 0.61 \\ EOS\_LINEAR\_POLYNOMIAL & & & & \\  & C\({}_{0}\) & - & 20.790 & - \\  & C\({}_{1}\) & - & 1.337 \(\cdot\)\({}^{1}\)\(0^{5}\) & - \\  & C\({}_{2}\) & - & 1.256 \(\cdot\)\(10^{5}\) & - \\  & C\({}_{3}\) & - & 1.454 \(\cdot\)\(10^{5}\) & - \\  & C\({}_{4}\) & - & 1.940 & - \\  & C\({}_{5}\) & - & 0.585 & - \\  & C\({}_{6}\) & - & 1.125 & - \\ EOS\_GRUNESEN & & & & \\  & C & m/s & - & 4570 \\  & S\({}_{1}\) & - & - & 1.49 \\  & \(\gamma_{0}\) & - & - & 1.93 \\  & A & - & - & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material data for cumulative insert and steel target [20; 21; 22]. whereas for the expanded material as:  
\[p=\rho_{0}C^{2}\mu+(\gamma_{0}+a\mu)E \tag{12}\]  
\(C\)--bulk speed of sound,  
\(\gamma_{0}\)--Gruneisen gamma,  
\(S_{1}\)--linear coefficient,  
\(S_{2}\)--quadratic coefficient,  
\(S_{3}\)--cubic coefficient,  
\(a\)--first order volume correction to \(\gamma_{0}\),  
\(\mu\)--volume parameter, expressed as \(\mu=(\rho/\rho_{0})-1\),  
\(\rho\)--actual density,  
\(\rho_{0}\)--initial density,  
\(E\)--internal energy per unit of mass. The MAT_HIGH_EXPLOSIVE_BURN material model was selected to describe the octogen (HMX) explosive. The material data for the HMX explosive and presented in Table 2.  
The Jones-Wilkins-Lee (JWL) equation of state was used to describe the relationship between the parameters of the thermodynamic system for the explosive. The equation of state of gaseous products of detonation of condensed explosives takes the following form:  
\[p=A\bigg{(}1-\frac{\omega}{R_{1}V}\bigg{)}e^{-R_{1}V}+B\bigg{(}1-\frac{\omega} {R_{2}V}\bigg{)}e^{-R_{2}V}+\frac{\omega E}{V}, \tag{13}\]  
\(A\), \(B\) and \(E\) have units of pressure. \(R_{1}\), \(R_{2}\), \(\omega\), and \(V_{0}\) are dimensionless. \(E\)--internal energy per unit volume and \(\omega\)--the relative volume of the explosive. The air domain was modeled using the MAT_NULL material model and the EOS_LINEAR_POLYNOMIAL equation of state, for which the material data summarized in Table 3 were used. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Value** \\ \hline MAT_HIGH_EXPLOSIVE\_BURN & & & \\ Density & \(\rho\) & Kg/m\({}^{3}\) & 1890 \\ Detonation velocity & D & m/s & 9110 \\ Chapman–Jouget pressure & PCJ & GPa & 42 \\ EOS\_JWL & & & \\ A & GPa & 778.3 \\ B & GPa & 7.071 \\ R\({}_{1}\) & - & 4.2 \\ R\({}_{2}\) & - & 1 \\ \(\omega\) & - & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pressed octogen (HMX) explosive data with equation of state [23]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Value** \\ \hline MAT\_NULL & & & \\ Density & \(\rho\) & Kg/m\({}^{3}\) & 1.29 \\ EOS\_LINEAR\_POLYNOMIAL & & & \\ \(C_{4}\) & GPa & 0.4 \\ \(C_{5}\) & GPa & 0.4 \\ \(E_{0}\) & GPa & \(2.5\times 10^{-4}\) \\ \(V_{0}\) & - & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Air material data with the equation of state [24].