# Investigation on Cutting Blasting Efficiency of Hard Rock Tunnels under Different Charge Diameters
## 3 Numerical Simulation  
### Numerical Model Descriptions  
As an important research method, numerical simulation currently has extensive applications in the research field of rock blasting engineering [39, 40, 41]. As shown in Figure 4a, the explicit dynamic program ANSYS/LS-DYNA was used to construct three cutting blasting numerical models with the same dimensions of 2500 mm \(\times\) 2500 mm \(\times\) 3000 mm  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(d_{\mathbf{c}}\) **(mm)** & **Explosive Density (kg\(\cdot\)m\({}^{-3}\))** & \(D_{\mathbf{c}}\) **(m\(\cdot\)s\({}^{-1}\))** \\ \hline
27 & 1100 & 3175 \\
35 & 1100 & 3425 \\
42 & 1100 & 3665 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Detonation velocities under different cartridge diameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(d_{\mathbf{b}}\) **(mm)** & \(d_{\mathbf{c}}\) **(mm)** & \(K_{\mathbf{r}}\) & \(R\) **(mm)** \\ \hline
32 & 27 & 1.185 & 288 \\
42 & 35 & 1.200 & 396 \\
50 & 42 & 1.190 & 525 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fracture radii under different charge diameters.  
Figure 3: Relationship between detonation velocity and cartridge diameter.  
(length \(\times\) width \(\times\) depth). The hole layout parameters of the three numerical models were consistent with each other. Six cutting holes with a vertical depth of 2500 mm were placed symmetrically on the left and right sides of the blasting model, and the included angle between the cutting holes and the driving face was 76.5\({}^{\circ}\). The hole top spacing and hole bottom spacing between the left and right cutting holes were 1500 mm and 300 mm, respectively. In addition, the longitudinal spacings, vertical depths, and charge lengths of the cutting holes were 500 mm, 2500 mm, and 1320 mm, respectively. The main difference between these blasting models mainly lay in the charge diameters; the working conditions of the three groups of blasting models were \(\Phi\) 32 mm holes matching \(\Phi\) 27 mm cartridges, \(\Phi\) 42 mm holes matching \(\Phi\) 35 mm cartridges, and \(\Phi\) 50 mm holes matching \(\Phi\) 42 mm cartridges, respectively.  
In this simulation, the Lagrange algorithm was used to simulate solid media including the rock and the tamping plug, and ALE (Arbitrary Lagrange-Euler) was employed to simulate fluid media, including the explosives and standard air [42; 43]. The blasting dynamic load transfer between the two phase media could be realized by a fluid-solid coupling algorithm, which needed to employ the \"CONSTRAINED_LAGRANGE_IN_SOLID keyword [44]. In addition to the heading face, the non-reflecting boundaries were applied to the other surfaces of the rock model to eliminate the harmful influences of the stress wave reflections on the simulation accuracy [45; 46]. Moreover, the reverse detonation mode was adopted in all the cutting holes, that is the explosive detonation reaction started from the bottom of each cutting hole at 0 \(\upmu\)s. The specific detonation points and detonation times were defined by the \"INITIAL_DETONATION keyword.  
In addition, a hexahedron was used for the mesh generation of the numerical model in this study. However, some recent studies [47; 48] have suggested that a reasonable mesh size was extremely important to control the accuracy of the simulated results. In order to avoid the obvious distortion of the final results, the convergence tests were carried out to determine a reasonable mesh. During this process, the mesh size was continually reduced until the difference between the simulated results of two adjacent tests was reduced to 5% [49], and the evaluation index was the peak stress at the same position. Finally, the  
Figure 4: Numerical model and mesh division. (**a**) Numerical model of cutting blasting (unit: mm). (**b**) Mesh division of rock model.  
reasonable numbers of grids and nodes of the rock model were determined to be 1,441,92 and 1,512,152, respectively. The mesh division of the rock model is presented in Figure 4b.