## 4 Numerical Simulation
### Numerical Analysis of Charge Structure
#### 4.1.2 Numerical Model Parameters
Next, in the RHT model, the equation of state for the P-\(\alpha\) compaction of the rock mass is given as follows [41; 42]:  
\[P_{\rm R}=\frac{1}{\alpha_{0}}((B_{0}+B_{1}\mu)\alpha_{0}\rho e+A_{1}\mu+A_{2 }\mu^{2}+A_{3}\mu^{3}) \tag{8}\]  
where \(P_{\rm R}\) is the pressure for the equation of state, GPa; \(\alpha_{0}\) is the initial porosity; \(e\) is the internal energy per unit mass; \(\mu\) is the volumetric strain; the parameters for the polynomial EOS are determined by \(B_{0}=B_{1}=2s-1\), \(T_{1}=\rho c_{0}^{2}\), and \(T2=0\); \(s\) is the empirical constant of the material; \(c_{0}\) is the sound speed at ambient pressure and temperature, m/s; the Hugonid polynomial coefficients are determined by \(A_{1}=\rho c_{0}^{2}\), \(A_{2}=\rho c_{0}^{2}(2s-1)\), and \(A_{3}=\rho c_{0}^{2}(3s^{2}-4s+1)\); finally, the crush pressure is denoted by \(P_{\rm crush}=2f_{\rm c}/3\). The damage variable \(D\) in the RHT model is defined as the accumulation of plastic strain \(\epsilon_{\rm P}\). \[D=\sum\frac{d\epsilon_{\rm P}}{\epsilon_{\rm P}^{\rm f}} \tag{9}\]  
\[\epsilon_{\rm P}^{\rm f}=\left\{ \begin{array}{cc}D_{1}[P_{0}-(1-D)P_{\rm t}^ {*}]^{D_{2}}&P_{0}^{*}\geq(1-D)P_{\rm t}^{*}+(\epsilon_{\rm P}^{\rm m}/D_{1}) ^{D_{2}}\\ \epsilon_{\rm P}^{\rm m}&P_{0}^{*}<(1-D)P_{\rm t}^{*}+(\epsilon_{\rm P}^{\rm m }/D_{1})^{D_{2}}\end{array} \right. \tag{10}\]  
where \(\epsilon_{\rm P}^{\rm f}\) is the plastic strain at failure; \(\epsilon_{\rm P}^{\rm m}\) is the minimum damaged residual strain; \(P_{\rm t}^{*}\) is the failure cutoff pressure; \(D_{1}\) and \(D_{2}\) are the damage constants, with values of 0.04 and 1.0, respectively. Based on the basic physical and mechanical properties of coal and rock, combined with theoretical calculations and derivations, parameters \(\rho_{0}\), \(f_{c}\), \(G\), \(\beta_{c}\), \(\beta_{\rm f}\), \(\hat{\epsilon}_{0}^{\rm c}\), \(\hat{\epsilon}_{0}^{\rm c}\), \(\hat{\epsilon}_{0}^{\rm c}\), \(\hat{\epsilon}^{\rm c}\), \(\hat{\epsilon}^{\rm t}\), \(\hat{\epsilon}_{0}\), \(B_{0}\), \(B_{2}\), \(T_{1}\), \(T2\), \(A_{1}\), \(A_{2}\), \(A_{3}\), \(P_{\rm crush}\), \(D_{1}\), and \(D_{2}\) can be determined, totaling 20 parameters. The remaining 14 parameters are obtained based on mechanical test results and using calculation methods from the literature [42; 43]. For instance, for the determination of failure surface parameters, when \(3P^{*}\geq\sigma_{\rm f}^{*}\), the failure surface is expressed as [41]  
\[\sigma_{f}^{*}(P^{*},F_{\rm rate})=A(P^{*}-F_{\rm rate}/3+(A/F_{\rm rate})-1/ N)N\ \ 3P^{*}\geq F_{\rm rate} \tag{11}\]  
where \(\sigma_{f}^{*}\) is the normalized strength, \(\sigma_{\rm f}^{*}=\sigma_{\rm f}/f_{\rm c}\), \(\sigma_{\rm f}=\sigma_{1}-\sigma_{3}\); \(\sigma_{1}(\sigma_{3})\) is the first (third) principal stress, MPa; \(P_{0}^{*}\) is the normalized pressures, \(P_{0}^{*}=P/f_{\rm c}\); \(A\) and \(N\) are failure surface parameters, which are determined through triaxial compression tests. The ratio of the radii of the tensile and compressive meridians is given by [43]. \[Q=Q(P^{*})=Q_{0}+BP^{*} \tag{12}\]  
where \(Q_{0}\) and \(B\) are lode angle dependence factors. The relative tensile (shear) strength \(F_{\rm t}^{*}(F_{\rm s}^{*})\), compressive yield surface parameter \(G_{\rm c}^{*}\), tensile yield surface parameter \(G_{\rm t}^{*}\), shear modulus reduction factor \(\xi\), shear modulus reduction factor \(\epsilon_{\rm P}^{\rm m}\), residual surface parameter (index number) \(A_{\rm f}(N_{\rm f})\), compaction pressure \(P_{\rm CO}\), porosity exponent \(N_{\rm P}\), and other parameters not explicitly defined or calculated are summarized using relevant parameters from concrete [40; 44]. (2) Explosive Model and State Equation  
The \"MAT_HIGH_EXPLOSIVE_BURN material model was selected for the explosives, and the \"EOS_JWL equation of state was utilized to describe the relationship between the relative volume \(V\) and the specific initial internal energy \(E_{0}\) under the pressure of the detonation products.