## 4 Conclusions  
The purpose of this paper is to use the stress wave theory to study the dynamic propagation mechanism of the crack and the influence law of the dangerous area from the perspective of the precast crack, and to predict the possible dangerous area in the model with the help of the wave propagation.  
In terms of innovation, this study is based on the model of rock models with cracks in a large number of excavation and construction projects. On the one hand, the stress wave theory is considered to study the dynamic load conditions, and on the other hand, the crack propagation of prefabricated crack models with different inclination angles is studied in groups. Finally, the following conclusions are obtained:  
Figure 14: Models with main crack propagation and length–time curve; (**a**) 45\({}^{\circ}\), (**b**) 60\({}^{\circ}\), (**c**) 75\({}^{\circ}\), (**d**) 90\({}^{\circ}\).  
1. The action of the impact load causes the crack and the expansion of the crack behavior and hole around a pre-existing crack angle.
2. The stress wave theory can predict the crack propagation behavior under the ideal condition of the example in this paper, and has a good fit with the final crack propagation results.
3. When the inclination angle of the existing crack is 0-45\({}^{\circ}\), the equivalent principal stress effect generated by the reflected wave is enhanced, and it is easier to form a dangerous area of high stress in the area near the crack hole. At 60-90\({}^{\circ}\), the equivalent principal stress is reduced due to the secondary reflection of the reflected wave at the borehole position, and cracking behavior is less likely to occur at the far and near ends of the fracture. At this time, fracture behavior mainly occurs around the hole, and only a few cracks have a high propagation rate.
4. The reflected wave is the main cause of the crack, followed by a shear wave; compared with the other side of the crack, the reflected wave may lead to more branches near the burst crack, forming a fracture area.
5. The reflected waves generated by the presence of cracks at different angles will promote the propagation of cracks around the blasting hole. The crack with an angle of 45-90\({}^{\circ}\) has a particularly significant effect on the expansion rate of nearly parallel cracks (main cracks) around the blasting hole. Such rapidly expanding cracks will be more easily connected with the free surface or surrounding cracks, which may be one of the main reasons for regional instability.  
Conceptualization, J.Z. and W.G.; methodology, J.Z.; software, J.Z.; validation., W.G. and M.W.; formal analysis, [PERSON]; investigation, [PERSON]; resources, [PERSON]; data curation, [PERSON]; writing--original draft preparation, [PERSON]; writing--review and editing, [PERSON]; visualization, [PERSON]; supervision, W.G. and M.W.; project administration, M.W. and L.Z.; funding acquisition, M.W. and L.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (12272247), the Natural Science Foundation of Sichuan, China (2022 NSFSC1915), the National Key Project (GJXM92579), and the major research and development project of the Metallurgical Corporation of China LTD in the non-steel field (2021-5).  
Not applicable.  
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The authors declare