## 2 Materials and Methods  
### Blast Loading  
The blast wave is propagated through the explosion source due to a sudden release of energy during a short time. This wave applies an incident pressure on each of the surfacessurrounding the explosion [52; 53; 54]. A typical incident pressure-time history is divided into positive and negative phases as described in [PERSON]'s illustration of time history of incident pressure and expressed using Equation (1) [54]. \[P(t)=+P_{so}(1-\frac{t}{t_{d}})e^{-bt/t_{d}} \tag{1}\]  
where \(P(t)\) is the incident pressure; \(P_{0}\) is the ambient air pressure; \(P_{so}\) is the peak incident pressure; \(t_{d}\) is the positive phase duration; and the coefficient \(b\) represents the extension of the negative phase. Various empirical equations have been proposed for peak incident pressure [55; 56; 57]. These empirical equations predict peak incident pressure as a function of a scaled distance parameter. This scaled distance is defined by Equation (2) [54]. \[Z=\frac{R}{W^{1/3}} \tag{2}\]  
where \(R\) is the standoff distance from the explosion source in m, and \(W\) is the equivalent charge of TNT in kg. [PERSON] and [PERSON] [58] proposed Equations (3) and (4) as the earliest empirical equations for peak incident pressure [58]. \[P_{so}=\frac{6.7}{Z^{3}}+1\ (\text{bar})\qquad for\ P_{so}\geq 10\ (\text{bar}) \tag{3}\]  
\[P_{so}=\frac{0.975}{Z}+\frac{1.455}{Z^{2}}+\frac{6.7}{Z^{3}}-0.019\ (\text{bar}) \qquad for\ 0.1\ \text{bar}\leq P_{so}\leq 10\ \text{bar} \tag{4}\]  
The pressure induced by blast wave propagation in a soil medium is different from the incident pressure. Many studies have investigated the blast-induced vibration parameters for a soil medium [4; 5; 6]. The peak pressure in different soil types can be obtained by Equations (5) and (6) [15]. \[p_{o}=\rho.C.u \tag{5}\]  
\[u=48.8f_{c}(\frac{2.52R}{W^{1/3}})^{-n} \tag{6}\]  
where \(p_{o}\) is the peak pressure in N/m\({}^{2}\); \(\rho\) is the density of soil in kg/m\({}^{3}\); \(u\) is the peak particle velocity in m/s; \(f_{c}\) is a dimensionless coupling factor that depends on the scaled depth of explosion; \(R\) is the distance of the measuring point from the center of charge in m; \(n\) is a dimensionless attenuation coefficient; and \(C\) is the loading wave velocity in m/s, which is determined by Equation (7) [13]. \[C= \begin{cases}c&\text{for fully saturated clays}\\ 0.6c+\left(\frac{n+1}{n-2}\right)u&\text{for saturated clays}\\ c+\left(\frac{n+1}{n-2}\right)u&\text{for sand}\end{cases}  \tag{7}\]  
where \(c\) is the seismic velocity in m/s. The coupling factor is determined as 0.4 for surface explosions and increases with the depth of the explosion according to the scaled depth of the explosion, based on TM 5-855-1 [15]. The attenuation coefficient depends on the soil type, as presented in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Soil Type** & **Attenuation Coefficient, N** \\ \hline Very loose sand, dry or wet & 3.25 \\ Loose sand, dry or wet & 3.0 \\ Dense sand, dry or wet & 2.75 \\ Very dense sand, dry or wet & 2.5 \\ Partially saturated clay and silt & 2.5 \\ Saturated clay & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Attenuation coefficient for different soil types [13]. Another parameter that has received attention by many researchers in the last decades is the blast-induced crater [14; 59; 60]. The effect of sand-tire shred mixtures on the blast-induced crater is also evaluated in this research. The analysis of the explosion scenarios has been performed based on the coupled Eulerian-Lagrangian analysis where the Jones-Wilkins-Lee (JWL) equation of state [61] is used to simulate the explosion materials as shown in Equation (8).