## 3 Material Behavior Models for Numerical Analysis
### Calibration of Simulation Parameters  
To assess the precision of the RHT model, a comparison was made between the outcomes of physical testing carried out by [PERSON] [28] and the numerical simulation results. The physical test is shown in Figure 5a to comprise a cylindrical sample of Barre granite measuring 144 mm in diameter and 150 mm in height, with a centrally located blashhole of 6.45 mm in diameter. To prevent explosive gases from leaking, a copper tube (6.45 mm for outer diameter and 5.15 mm for inner diameter) was inserted into the blashhole. Air and polyethylene were employed as the coupling medium. As depicted in Figure 5b, an equivalent planar numerical model was constructed for validation with the test results. This model comprised five materials: the PETN explosive, polyethylene, air, copper, and Barre granite. The ALE algorithm was used to simulate the former three materials, while the copper and Barre granite were simulated using the Lagrangian descriptions. The copper tube was modeled with a plasticity kinematic material model, the parameters of which have been presented in [PERSON] et al. [29], as can be seen in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Density/(kg/m\({}^{3}\))**} & \multirow{2}{*}{
\begin{tabular}{c} **Velocity of** \\ **Detonation/(m/s)** \\ \end{tabular} } & \multirow{2}{*}{\(P_{\text{ef}}\)/GPa} & \multirow{2}{*}{\(A\)/GPa} & \multirow{2}{*}{\(B\)/GPa} & \multirow{2}{*}{\(R_{\text{1}}\)} & \multirow{2}{*}{\(R_{\text{2}}\)} & \multirow{2}{*}{\(\omega\)} & \multirow{2}{*}{\(E_{\text{0}}\)/GPa} \\ \hline
1320 & 6690 & 16 & 575 & 22.4 & 5.73 & 1.81 & 0.276 & 7.42 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material and JWL EOS parameters for PETN.  
Figure 5: Samples of Barre granite that are cylindrical in shape: (**a**) blast testing; (**b**) numerical simulation.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Density/(kg/m\({}^{3}\))**} & **Young’s** & **Poisson’s** & **Yield** & **Tangent** & & & \\  & **Modulus/GPa** & **Ratio** & **Stress/MPa** & **Modulus/MPa** & **P** & **C/s\({}^{-1}\)** & \(\beta\) \\ \hline
8330 & 138 & 0.36 & 390 & 110 & 5.286 & 1.346 \(\times\) 10\({}^{6}\) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Material parameters for copper.  
According to [PERSON] et al. [16], the damage threshold was set to 0.2. To ensure a more accurate comparison between the experimental and numerical results, the rock elements that were severely damaged beyond a certain threshold were excluded from the analysis. Figure 6 presents the crack pattern of the Barre granite after blasting. Figure 6a,b shows the final image obtained from pictures taken under high-intensity UV light and the experimental crack pattern based on [PERSON]'s results [28], respectively. Figure 6c illustrates the spatial distribution of cracks at a damage level of D = 0.2, and Figure 6d depicts the simulated cracks pattern. These images illustrate that the high-intensity shock waves generated by the explosive detonation formed a compressive crushed zone near the blasthole. The shock waves dissipated into stress waves during propagation, with a strength exceeding that of the rock's tensile strength, thereby forming a cracked zone. Reflection of the compressive stress waves at the free boundary of the specimen led to the generation of tensile stress waves and the consequent formation of a spalling damage zone. Overall, the simulation outcomes concurred with the experimental results, which served to authenticate the RHT model.