# A Novel Continuous-Discontinuous Multi-Field Numerical Model for Rock Blasting
## 3 Numerical Example
### Blasting of the PMMA Plate  
We used the proposed method to simulate the blasting test of the PMMA plate [40], see Figure 5. This case is also plane stress. We used Gmesh software to build a two-dimensional square model with a side length of 300 mm and set a blasting hole with a radius of 2 mm in the center of the model. We divided the model into 21,896 triangular elements and 32,972 interface elements, and the size of elements is 1-5 mm. To prevent the explosive \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material Properties** & **Units** & **Value** \\ \hline Density \(\rho\) & (kg/m\({}^{3}\)) & 2660 \\ Modulus of elasticity \(E\) & (GPa) & 51 \\ Poisson’s ratio \(\mu\) & (\(\cdot\)) & 0.16 \\ Cohesive force C & (MPa) & 25 \\ Tensile strength \(f_{t}\) & (MPa) & 7.3 \\ Angle of internal friction & \(\circ\) & 56.4 \\ The dilatancy angle & \(\circ\) & 28.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Granite model main parameters [39]. Figure 4: Cracking pattern of the single-hole blasting test. \begin{table}
\begin{tabular}{l c} \hline \hline
**Material Properties** & **Value** \\ \hline Charge density \(\rho_{w}\) (kg/m\({}^{3}\)) & \(1770\) \\ Specific internal energy of explosive U (J/m\({}^{3}\)) & \(1.010\times 10^{10}\) \\ C–J pressure \(P_{CJ}\) (GPa) & 3.2 \\ Detonation velocity V (m/s) & 8300 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PETN main paremeters [38]. stress wave reflection, we set a non-reflective boundary condition around the model. The material properties are listed in Table 4, which is determined based on [41]. The explosive is \(PbN_{3}\), the properties of which are listed in Table 5.  
According to [40], two conditions are considered as (i) blocked blasting hole and the (ii) unblocked blasting hole, where the gas permeation in the second condition can be ignored in the experiment. The obtained results are shown in Figure 6.  
Figure 6a,b show the numerically-obtained results, and Figure 6c,d show the experimental results. It can be seen that, when the hole is not blocked, there is almost no radial main crack. Only dense cracks are generated near the hole, resulting in a crush zone caused by the blasting wave. When the hole is blocked, several radial main cracks are generated, which are caused by the gas permeation. The numerical results agree well with the experimental results, indicating the reliability of the proposed model. We would like to emphasize that we did not use special techniques to simulate explosion crushing area such as element dissolution/deletion because such techniques do not follow mass and energy conservation, which may bring other problems. Hence, in our results, a large number of micro-cracks appeared on the edge of the explosion crushing area in Figure 6. \begin{table}
\begin{tabular}{l c} \hline \hline
**Material Properties** & **Value** \\ \hline Density \(\rho\) (kg/m\({}^{3}\)) & 1180 \\ Modulus of elasticity E (GPa) & 6.1 \\ Poisson’s ratio \(\mu\) (-) & 0.31 \\ Cohesive force C (MPa) & 66 \\ Tensile strength T (MPa) & 66 \\ Angle of internal friction \({}^{\circ}\) & 35 \\ The dilatancy Angle \({}^{\circ}\) & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: PMMA model main parameters. Figure 5: Numerical model. The damage region at \(t=30\)\(\upmu\)s in Figure 7 is similar to the final damage region of the case with an unblocked blasting hole, indicating that the effect of blasting wave has finished. At \(t=30.22\)\(\upmu\)s, the explosive gas begins to permeate into the crush zone with  
Figure 6: Cracking pattern. Figure 7: Pressure of explosive gas. maximum gas pressure around 20 MPa. When \(t=39.82\) us, the maximum gas pressure exceeds the strength of the PMMA plate. At \(t=56.32\) us, the main cracks appear, and the gas begins to escape. The maximum gas pressure also drops. Finally, at \(t=58.12\) us, the cracks become smooth, and the whole process ends.