## 4 Numerical Simulation Analysis of the Hole-by-Hole blasting and Three-Hole Simultaneous Blasting
### Numerical Simulation Model
#### 4.2.2 Material Model  
The Riedel-Heirmaier-Thomas (RHT) model is a tension compression damage constitutive model proposed by [PERSON], [PERSON], and [PERSON] based on the traditional Holmquist-Johnson-Cook constitutive model [29]. The model introduces three limit surfaces: the elastic limit surface, the failure surface, and the residual strength surface. On this basis, it can better reflect the dynamic mechanical behavior, strain rate sensitivity, strain hardening, and damage softening characteristics of rocks under different stress states. Overall, the RHT constitutive model is widely used in numerical simulations of explosive shocks and other related fields. In this paper, the numerical simulation of the rock material models employed the * MAT_RHT model. The material parameters of the rock model were obtained through physical and mechanical tests, references, and SHPB shock tests. A series of physical and mechanical tests were conducted on sandstone from the Xin-guang Open-pit Mine. The density of sandstone is 2440 kg/m\({}^{3}\). The uniaxial compressive strength was 104 MPa. The elastic shear modulus was 16.94 GPa. The uniaxial tensile strength was 10.0 MPa. According to reference [30], the strain rate, failure surface, and damage parameters in the constitutive model have been determined. Through parameter sensitivity analysis, it was found that the tensile yield surface parameters \(g_{c}^{*}\) and \(g_{f}^{*}\), failure surface parameters \(A_{f}\) and \(n_{f}\), and shear compression strength ratio \(f_{s}^{*}\) have a significant impact on the numerical results. Based on the simulation results, these parameters have been adjusted and optimized. The parameters that are not sensitive to simulation results were taken from the values given by [PERSON] et al. [31]. Figure 10: Model grid division. Figure 9: Schematic diagram of the numerical model. The remaining parameters that were difficult to determine were calibrated using SHPB shock tests. The sandstone samples were taken from the Xinguang Open-pit Mine. After polishing, the diameter of the sample was 50 mm, and the height was 25 mm. The SHPB test apparatus is shown in Figure 11. Among them, the length of the bullet was 30 cm. The length of the incident rod was 240 cm. The length of the transmission rod was 140 cm. The bullets and all rods were made of high-strength alloy steel. Its density is 7900 kg/m\({}^{3}\). The wave speed was 5172 m/s. The elastic modulus was 210 GPa. The reasonable range of projectile velocity was determined by the pre-impact velocity, and the final test projectile velocity was 15.61 m/s. Based on the actual dimensions of the incident rod, transmission rod, and specimen of the SHPB testing system, the model was established in a 1:1 ratio. The rigid rod adopted the elastic constitutive law, and the specimen adopted the RHT constitutive law. After repeated trial and error, we selected a set of parameters that highly matched the simulation results with the experimental data, as shown in Table 2.  
The comparison between the stress-strain curve obtained from the numerical simulation and experimental results is shown in Figure 12. From the graph, it can be seen that the peak stress and strain of both were almost the same. Under dynamic impact loads, the specimen fractures into numerous fragments. It was relatively close to the results of the experiment, as shown in Figure 13.