## 4 Numerical Simulation
### Numerical Analysis of Charge Structure
#### 4.1.2 Numerical Model Parameters  
(1) Constitutive Models and Parameters of Coal and Rock  
The *MAT_RHT constitutive model is a tensile-compressive damage model based on the modified HJC model, which accounts for the effects of confining pressure, strain rate, strain hardening, and damage softening on the failure strength of rock materials under blasting and dynamic loading. This model is utilized to simulate the damage of brittle materials and is widely applied to the tensile and compressive damage evolution of brittle materials such as concrete and rock under dynamic loads. The detailed mathematical formulation is presented in references [40,41]. The primary model parameters for coal and sandstone are listed in Table 5. Among these, some RHT parameters, such as density \(\rho_{0}\), compressive strength \(f_{\rm c}\), tensile strength \(f_{\rm t}\), and elastic shear modulus \(G\), are derived from the mechanical parameters of coal and rock. Then, in the RHT model, the relationship between strain rate parameters and strength is given by Equation (7) [41]. \[F_{\rm rate}(\hat{\epsilon}_{\rm p})=\left\{ \begin{array}{ll}(\hat{\epsilon}_ {\rm p}/\hat{\epsilon}_{0}^{3})^{\hat{\rho}_{\rm c}}&P\geq f_{\rm c}/3\\ \frac{P+f_{\rm t}/3}{f_{\rm c}/3+f_{\rm t}/3}(\frac{\hat{\epsilon}_{0}}{\hat {\epsilon}_{0}^{3}})^{\hat{\rho}_{\rm c}}-\frac{P-f_{\rm c}/3}{f_{\rm c}/3+f_ {\rm t}/3}(\frac{\hat{\epsilon}_{0}}{\hat{\epsilon}_{0}^{3}})^{\hat{\rho}_{ \rm t}}&\frac{-f_{\rm t}}{3}<P<\frac{f_{\rm c}}{3}\\ (\hat{\epsilon}_{\rm p}/\hat{\epsilon}_{0}^{3})^{\hat{\rho}_{\rm t}}&P\geq-f_ {\rm t}/3\end{array} \right. \tag{7}\]  
where \(F_{\rm rate}(\hat{\epsilon}_{\rm p})\) is the increase factor for the strain rate; \(\hat{\epsilon}_{\rm p}\) is the strain rate under load \(P\); \(P\) is the pressure, MPa; \(\hat{\rho}_{\rm c}\) and \(\hat{\rho}_{\rm t}\) are the compressive and tensile strain rate dependence  
Figure 8: Finite element calculation model and measurement point arrangement. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Value**} & \multicolumn{2}{c}{**Value**} & \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Value**} & \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Value**} \\  & **Rock** & & & & & **Rock** & & **Coal** & & **Rock** & **Coal** \\ \hline \(\rho_{0}\) (kg\(\cdot\)m\({}^{-3}\)) & 2300 & 1310 & \(\beta_{c}\) & 0.0368 & 0.0481 & \(Q_{0}\) & 0.68 & 0.61 \\ \(G\) (GPa) & 11.76 & 1.93 & \(G_{\rm c}^{*}\) & 0.53 & 0.35 & \(B\) & 0.01 & 0.0105 \\ \(f_{\rm c}\) (MPa) & 34.4 & 21.07 & \(G_{\rm t}^{*}\) & 0.7 & 0.45 & \(A\) & 2.1 & 1.6 \\ \(T_{1}\) (GPa) & 34.20 & 8.90 & \(F_{\rm t}^{*}\) & 0.0942 & 0.0944 & \(N\) & 0.637 & 0.61 \\ \(B_{0}\) & 1.22 & 1.68 & \(F_{\rm s}^{*}\) & 0.21 & 0.18 & \(N_{\rm f}\) & 0.6 & 0.45 \\ \(B_{1}\) & 1.22 & 1.68 & \(A_{1}\) (GPa) & 34.19 & 8.90 & \(A_{\rm f}\) & 1.61 & 1.25 \\ \(P_{\rm crush}\) (MPa) & 22.93 & 14.05 & \(A_{2}\) (GPa) & 41.72 & 14.96 & & & \\ \(\beta_{\rm t}\) & 0.0325 & 0.0487 & \(A_{3}\) (GPa) & 8.77 & 9.14 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of the RHT model for sandstone and coal. exponents, respectively, \(\beta_{c}=4/(20+3f_{c})\) and \(\beta_{\rm t}=2/(20+f_{c})\); \(\hat{\epsilon}_{0}^{\rm c}\) and \(\hat{\epsilon}_{0}^{\rm t}\) are the reference strain rates under compression and tension, respectively; \(\hat{\epsilon}_{0}^{\rm c}=3.0\times 10^{-5}\) s\({}^{-1}\) and \(\hat{\epsilon}_{0}^{\rm t}=3.0\times 10^{-6}\) s\({}^{-1}\); \(\hat{\epsilon}^{\rm c}\) and \(\hat{\epsilon}^{\rm t}\) are the failure strain rates under compression and tension, respectively. Their values are \(\hat{\epsilon}^{\rm c}=\hat{\epsilon}^{\rm t}=3.0\times 10^{25}\) s\({}^{-1}\).