## 2 Materials and Methods
### Numerical Analysis  
A numerical analysis is used in this research to evaluate the behavior of sand-tire shred mixtures under surface explosion hazards and investigate whether this mixture can be introduced as an effective method to protect underground structures from peak blast pressure. Figure 1 shows the explosion scenarios investigated in this research. A sand layer is initially subjected to four surface explosions with 100, 500, 1000, and 5000 kg charges. The sand-tire shred mixture is then introduced to the model with 2, 4, and 6 m thicknesses and subjected to the same surface explosions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Material** & \(\rho\) **(kN/m3)** & \(G\) **(kPa)** & \(M\) & \(\varphi\) & \(C_{s}\) **(kPa)** & \(\alpha\) & \(\beta\) \\ \hline Sand & 17.5 & 1400 & 0.3 & 35\({}^{\circ}\) & 0 & 0.4534 & 0.0035 \\
90\% sand + 10\% rubber & 15.25 & 950 & 0.35 & 35\({}^{\circ}\) & 2 & 0.9163 & 0.0489 \\
80\% sand + 20\% rubber & 14.0 & 800 & 0.35 & 32\({}^{\circ}\) & 3 & 1.3791 & 0.0923 \\
70\% sand + 30\% rubber & 13.7 & 550 & 0.35 & 30\({}^{\circ}\) & 5 & 1.8420 & 0.1357 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Engineering properties of the soil and mixture [65; 66]. Figure 1: The schematic model used for developing explosion scenarios. The air is modeled using the ideal gas equation of state, where the internal energy depends on the temperature. The ideal gas equation is shown in Equations (9) and (10) [59]. \[p_{a}=(\gamma-1)\rho_{a}e \tag{9}\]  
\[\gamma=1+\frac{R_{a}}{c_{v}} \tag{10}\]  
where \(p_{a}\) is the hydrostatic pressure of the air; \(\gamma\) is the adiabatic exponent; \(\rho_{a}\) is the air density; \(e\) is the specific internal energy depending on temperature \(T_{0}\); \(R_{a}\) is the universal gas constant divided by the effective molecular weight of the gas; and \(c_{v}\) is the specific heat at constant volume. Table 3 presents the material properties for the air [59]. The JWL equation of state [61] is used to model the explosion with the parameters stated in Table 4 [59]. The stress-strain response of sand-tire shred mixtures is simulated using the Mohr-Coulomb constitutive model. The numerical analysis consists of several steps, including validation of the numerical simulation, studying the behavior of the sand-tire shred mixture under different explosion scenarios, investigating the effect of the thickness of the sand-tire shred mixture on peak blast pressure, and evaluating the impact of the rubber content on peak blast pressure, as discussed in the following sections. The ABAQUS software is used for the finite element simulation in this research, and the initial results are validated against the results reported by [PERSON] and [PERSON] [66] and the equation of peak blast pressure [15]. The finite element model is developed and analyzed using the ABAQUS Explicit module. This module is suitable for dynamic events such as explosions and efficiently solves highly nonlinear problems involving changing contact conditions. The coupled Eulerian-Lagrangian method is used to model the explosion, air, and sand-tire shred mixture as an Eulerian element. This method is suitable for analyzing problems with large deformations. As described in previous sections, the explosion material is simulated using the JWL equation of state. The blast scenario is a surface explosion with a charge of 500 kg. Figure 2 shows the results of numerical model simulation in ABAQUS software. The surface explosion produced a crater with a diameter of 4.24 m (Figure 2b) compared with a crater diameter of 4.22 m reported by [PERSON] and [PERSON] [66], which indicated an insignificant difference of 0.5%. Figure 2c shows the time history of blast pressure at a depth of 10 m obtained by this research, which offers a peak blast pressure of 0.4 MPa, with a slight difference of 4.7% compared with the corresponding value obtained from Equation (5), equal to 0.42 MPa.