## 3 Computational Modeling
### Model Setup
#### 3.2.2 Material Formulation  
There are three parts to consider: air, explosive, and the projectile. As far as the air and the explosive are concerned, two equations must be defined. They require each a material model and an equation of state (EOS). The air is considered as an ideal gas and is modeled using (*MAT-NULL). The latter allows an EOS to be used without computing the deviatoric stresses [22]. Therefore, the gamma law EOS, given by Equation (2), with the keyword (*LINEAR-POLYNOMIAL) is used. \[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+(C_{4}+C_{5}\mu+C_{6}\mu^{2})E \tag{2}\]  
The parameters \(C_{0}\), \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), \(C_{5}\), and \(C_{6}\) are constants, \(\mu=(\rho/\rho_{0})-1\), \(\rho\) is the current density, \(\rho_{0}\) is the initial density having a value of 1.225 kg.m\({}^{-3}\), \(E\) refers to the specific internal energy. By setting \(C_{0}=C_{1}=C_{2}=C_{3}=C_{6}=0\), and \(C_{4}=C_{5}=\gamma-1\), the ideal EOS is obtained. Figure 15: Schematic representation of the 2D axi-symmetric model in LS-DYNA: geometry, dimensions, materials, and boundary conditions. \[P=(\gamma-1)E(\rho/\rho_{0}) \tag{3}\]  
\(\gamma\) represents the ratio of the specific heat for air and is equal to 1.4. The initial specific internal energy is set as \(E_{0}=0.2534\) kJ.kg\({}^{-1}\) to give an atmospheric pressure of 0.101325 MPa. The initial relative volume is set as \(V_{0}=1\)[22]. The corresponding numerical parameters are collected in Table 2.  
(*MAT-HIGH-EXPLOSIVE-BURN) is used for the C4 explosive material. The hydrodynamic behavior is given by the semi-empirical (*)ONES-WILKINS-LEE) EOS in Equation (4). \[P=A\bigg{(}1-\frac{\omega}{R_{1}V}\bigg{)}exp^{-R_{1}V}+B\bigg{(}1-\frac{ \omega}{R_{2}V}\bigg{)}exp^{-R_{2}V}+\frac{\omega E}{V} \tag{4}\]  
The parameters \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) are constants which depend on the type of explosive and which can be determined experimentally [38]. \(V\) is the volume and \(E\) is the specific internal energy. The Chapman-Jouguet pressure \(P_{cj}\) and the detonation velocity \(D_{v}\) are also needed to control the programmed detonation of the explosive. All values for the explosive properties and the EOS are taken as the parameters published by [PERSON] and [PERSON] [39] and are summarized in Table 3.  
It is experimentally observed that the projectile exhibits considerable deformation, especially during the initial blasting but with little effect on the inflight velocity characteristics. Taking this into account together with the fact that the projectile is made out of rolled homogeneous armored (RHA) steel, the (*SIMPLIFIED-JOHNSON-COOK) material model is used in the simulation. The general expression of the equivalent yield stress is given by Equation (5):  
\[\sigma_{eq}=(A+B\epsilon^{n})(1+C\ln\frac{\dot{\epsilon}}{\dot{\epsilon_{0}}}) \tag{5}\]  
where \(\epsilon\) is the equivalent plastic strain, \(\dot{\epsilon}\) is the strain rate, \(\dot{\epsilon_{0}}\) is a reference strain rate. \(A\), \(B\), \(n\), and \(C\) are the constant parameters for the material model. Table 4 summarizes the different mechanical parameters and the Johnson-Cook model constants for RHA steel. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Property Name** & **Symbol** & **Value** & **Unit** \\ \hline \multirow{3}{*}{**Elastic constants**} & Mass density & \(\rho\) & 7850 & [kg/m\({}^{3}\)] \\  & [PERSON]’s modulus & \(E\) & \(1.97\times 10^{5}\) & [MPa] \\  & [PERSON]’s ratio & \(\
u\) & 0.3 & [-] \\ \hline \multirow{3}{*}{**Johnson–Cook parameters**} & Yield stress & \(A\) & 1225 & MPa \\  & Strain-hardening constant & \(B\) & 1575 & [MPa] \\ \cline{1-1}  & Strain-hardening coefficient & \(n\) & 0.768 & [-] \\ \cline{1-1}  & Strengthening coefficient of strain rate & C & 0.0049 & [-] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Density, elastic constants, and Johnson–Cook parameters for the RHA steel sphere [40].