## 2 Numerical Study
### Establishment of Numerical Model  
The numerical model is shown in Figure 3, which is a rectangular board with a length of 350 mm and a height of 150 mm. We set a hole with a radius of 7 mm to simulate a blast hole and generate cylindrical waves. The distance from the center of the hole to the left boundary is 55 mm and is the same from the center to the top edge. On the right side of the hole, there is a crack with an angle of \(\theta\) with a length of L. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Material** & **Density** & **Bulk Modulus** & **Elastic Modulus** & **Poisson** & \(V_{s}\) & \(V_{p}\) \\  & **Kg/m\({}^{3}\)** & **GPa** & **GPa** & **Ratio** & **m/s** & **m/s** \\ \hline Sandstone & 2380 & 9.4624 & 17.60 & 0.19 & 2549.4 & 1459.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of sandstone used in this study. Figure 2: An acoustic velocimeter (Sonic Viewer-SX). Considering that the object of this study is deep rock, it can be approximated as an infinite rock plane, but the modeling scope is limited. So, when establishing the model, the boundary is set to transmit boundary, so that the stress waves generated by the blashole at the boundary will not have any effect on the reflected waves in the subsequent process, greatly reducing the impact of tensile and compressive waves in the reflected P-wave on the model research. This can transform the research object from a finite model to an infinite region. Taking the model with \(\theta\) = 45\({}^{\circ}\) and L = 75 mm as the first exploratory research example, the velocity vector excited by stress waves was collected as shown in Figure 4. We set the loading condition to a peak of 50 Mpa with a loading time of 2 us. The advantage of designing such a shorter wavelength is that it creates a good separation between different waves, making it easier to study the propagation and characteristics of different waves. By observing the images, it was found that both the P-wave and S-wave of the stress wave diffracted near the crack, especially when observed in the vector plots. And the P-wave and S-wave diffracted by excitation have different effects on the side of the crack near the blast hole and the side away from the blast hole. On the side that is the farther from the explosion hole, due to diffraction, the conventional diffraction P-wave and S-wave are generated, and the conventional lower Rayleigh wave propagates along the direction of the crack. On the side near the blast hole, due to the tensile wave inside the P-wave, its reflected wave propagates together with the diffracted P-wave, forming the first P-arc wave with stress concentration. The diffraction and reflection of the S-waves are mainly composed of shear waves that combine to form a second S-arc wave with stress concentration. It is worth mentioning that the two newly generated arc-shaped waves have the same properties as the diffraction wave at the tip that generates them, namely, the P-arc-shaped wave is also a tensile wave, and the S-arc-shaped wave is also a shear wave. However, the reflection wave still plays the main role in the two types of arc waves, so in future research on crack initiation, only the reflected wave can be considered as the main factor. Figure 3: Numerical model. After the above operations, in this study, the distance from the crack's center to the center of the circular hole is called the center distance, and the fixed center distance is 55 mm. The crack is rotated to change the angle for multiple sets of numerical model research, as shown in Figure 5.  
Figure 4: Stress-wave-induced particle velocity vectors. Figure 5: _Cont._  
As the angle increases, it can be observed that the Rayleigh waves on both sides of the crack, which play a dominant role in horizontal cracks, no longer become the main factor affecting the crack. At the same time, the arc wave formed by the combination of reflected and diffracted waves on the side of the crack passing through the blast hole becomes a high-stress concentration area.