## 3 Methodology
### Simulation of Explosive Detonation
The values of the relevant physical and mechanical parameters for the RHT material model of rock are provided in Table 3.  
The air material is modeled using the \"MAT_NULL material model available in LS-DYNA software. The pressure variation process is described using the \"EOS_LINEAR_POLYNOMIAL equation of state. \[P=C_{0}+C_{1}V_{\varepsilon}+C_{2}V_{\varepsilon}^{2}+C_{3}V_{\varepsilon}^{ 3}+\left(C_{4}+C_{5}V_{\varepsilon}+C_{6}V_{\varepsilon}^{2}\right)E_{0} \tag{3}\]  
Here, \(C_{0}\)-\(C_{6}\) represent the parameters of the equation of state, with \(C_{4}\) = \(C_{5}\) = 0.4, \(E_{0}\) = 2.5 \(\times\) 10\({}^{9}\) J/m\({}^{3}\), and an air density of 1.29 kg/m\({}^{3}\). In the computational setup, ALE (arbitrary Lagrange-Eulerian) elements are used to model explosives and compressible fluids such as air, while Lagrange elements are employed for solid materials, including stemming and rock. The interaction between these domains is handled through a fluid-solid coupling algorithm. Similar to air, water, as a fluid material, requires both a constitutive equation and an equation of state to define its thermodynamic state and pressure evolution. The material behavior of water is defined using the \"MAT_NULL keyword, with its pressure response described by the \"EOS_GRUNIESEN formulation. The parameters for the Gruneisen equation of state are set as follows: \(\rho\) = 1000 kg/m\({}^{3}\), \(C\) = 1480 m/s, \(S_{1}\) = 2.56, \(S_{2}\) = -1.986, \(S_{3}\) = 0.227, GAMAO = 0.5, and \(A\) = 1.3937. Additionally, \(E_{1}\) = 256 J/kg specifies the initial internal energy, and \(V_{0}\) = 1.0 defines the initial relative volume. Notably, parameters \(C\), \(S_{1}\)-\(S_{3}\), GAMAO, and \(A\) are solver-specific constants in LS-DYNA, while \(E_{1}\) and \(V_{0}\) are physical initial conditions. The stemming material was modeled using the MAT_SOIL_AND_FORM keyword. Figure 2 illustrates the detonation process in a coupled charge. The detonation wave, initiated at the detonation position, propagates along the direction of energy flow, influenced by the directional and temporal limitations of detonation velocity in cylindrical charges [35]. The detonation position dictates both the propagation direction of the detonation wave and the flow direction of the detonation products. As the detonation wave propagates, the blast hole expands due to the interaction between the detonation products and the hole wall. This expansion results in explosion pressure being transmitted to the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Density/(kg m\({}^{-3}\))** & **Young’s Modulus/GPa** & **Poisson’s Ratio** & **Yield Strength/MPa** & **Shear Modulus/GPa** & **Hardening Index \(\beta\)** \\ \hline
2060 & 2.5 & 0.25 & 245 & 10.5 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of rock materials used in the RHT model. surrounding rock, causing compressive deformation and fracture development. The expansion process creates space that is eventually occupied by the detonation products [36]. As shown in Figure 2d, when detonation waves transition from the explosive to the rock, the impedance mismatch between the two media leads to partial reflection and transmission at the interface. This impedance contrast causes wave superposition phenomena that significantly influence the pressure distribution in the surrounding rock medium. Figure 3 illustrates the blasting load curve on the blast hole wall under bottom detonation conditions. Five measurement points (L1-L5) were positioned 0.2 m apart from the detonation point in a straight line (Figure 1) to analyze the blasting load during detonation. The simulation results show that the blasting load gradually increases from the initiation point to the top of the explosive, eventually stabilizing at its maximum value. This behavior is attributed to the high length-to-diameter ratio of the columnar explosive, which induces both directional and time effects during the detonation process.