## 3 Finite Element Model of Near-Ground Explosion
### Material Models and Equation of State  
Air is commonly modeled by *MAT_NULL with a linear polynomial equation of state, EOS_LNIEAR_POLYNOMIAL. *MAT_NULL is material type 9 in LS-DYNA 970 code. This material allows equations of state to be considered without computing deviatoric stresses. Optionally, a viscosity can be defined. Also, erosion in tension and compression is possible. Air and water usually use this model. This model avoids the calculation of stress and strain. EOS_LNIEAR_POLYNOMIA defines the pressure by  
\[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+\Big{(}C_{4}+C_{5}\mu+C_{6}\mu^{2} \Big{)}E_{0} \tag{11}\]  
where, the parameter \(\mu\) is defined as \(\mu\)=\(\rho\)/\(\rho_{0}\), \(\rho\) is the current density, and \(\rho_{0}\) is a nominal or reference density; \(C_{0}\)-\(C_{6}\) are the equation coefficients; and the parameter \(E_{0}\) is the initial internal energy of reference specific volume per unit. The parameters used are shown in Table 3[37; 38]. TNT is modeled by the high explosive material model, MAT-HIGH-EXPLOSLVE-BURN, it allows the modeling of the detonation of a high explosive. In addition, an equation of state must be defined. Burn fractions, \(F\), which multiply the equations of states for high explosives, control the release of chemical energy for simulating detonations. The Jones-Wilkins-Lee (JWL) equation of state defines pressure by  
\[P=A\bigg{(}1-\frac{\omega}{R_{1}V}\bigg{)}e^{-R_{1}V}+B\bigg{(}1-\frac{\omega} {R_{2}V}\bigg{)}e^{-R_{2}V}+\frac{\omega E}{V} \tag{12}\]  
where, \(A\), \(B\), \(R_{1}\), \(R_{2}\), \(\omega\) are the equation coefficients and \(V\) is the initial relative volume. Parameters selected are shown in Table 4[39]. The concrete ground is modeled using MAT_PLASTIC_KINEMATIC[40]. This model can describe the isotropic hardening and follow-up hardening plastic models, and can also consider the influence of strain rate. It is applicable to beam, shell, and solid units, and the calculation efficiency is very high. In the plastic kinematic material model, the mechanical properties are characterized by the material yield strength (\(\sigma_{0}\)), the Young's modulus (\(E\)), and the tangent modulus (\(E_{t}\): The slope of the stress-strain curve in the plastic region). The yielding is defined by [PERSON] yield criterion. In the plastic kinematic plasticity algorithm, the flow stress (\(\sigma\)) is given as  
\[\sigma_{y}=\Bigg{[}1+\bigg{(}\frac{\dot{\varepsilon}}{C}\bigg{)}^{\frac{1}{ \beta}}\Bigg{]}\bigg{(}\sigma_{0}+\beta E_{P}\varepsilon_{p}^{eff}\bigg{)} \tag{13}\]  
where, \(\dot{\varepsilon}\) is the strain rate, \(C\) and \(P\) are the strain rate parameters, \(\beta\) is the hardening coefficient and \(\varepsilon_{p}^{eff}\) and \(E_{P}\) are the effective plastic strain and plastic hardening modulus, respectively. The plastic hardening modulus is calculated by using the following relation  
\[E_{P}=\frac{EE_{t}}{E-E_{t}} \tag{14}\]  
The parameters of the concrete used are shown in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(\rho\) (g/cm\({}^{3}\)) & \(v_{\text{D}}\) (km/s) & \(P_{\text{CJ}}\) (GPa) & \(A\) (GPa) & \(B\) (GPa) & \(R_{1}\) & \(R_{2}\) & \(\omega\) & \(E_{0}\) (J/m\({}^{3}\)) & \(V_{0}\) \\ \hline
1.63 & 6.93 & 27 & 371 & 7.43 & 4.15 & 0.95 & 0.3 & \(7\times 10^{3}\) & 1.0 \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of TNT charge. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\rho\) (g/cm\({}^{3}\)) & \(C_{0}\) & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(C_{4}\) & \(C_{5}\) & \(C_{6}\) & \(E_{0}\) (J/m\({}^{3}\)) & \(V_{0}\) \\ \hline
1.29 & \(-1.0\times 10^{-6}\) & 0 & 0 & 0 & 0.4 & 0.4 & 0 & 25 & 1.0 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters of the air. The soil ground is modeled using MAT_SOIL_AND_FOAM [42]. Due to its simplicity and added flexibility, this model has a shear failure surface that is pressure dependent, which is a basic property of geo-material. It is more fluid-like under many conditions, which is an idea for soft soil.