# Dynamically Triggered Damage Around Rock Tunnels: An Experimental and Theoretical Investigation
## 6 Conclusions  
This study utilized a combination of the DIC technique and the SHPB test system to perform impact tests on sandstone specimens with arched holes of varying inclination angles. The transient dynamic stress distributions around the arched holes under different incident conditions were theoretically obtained by employing the conformal mapping and complex function methods. Based on the experimental and theoretical findings, the following conclusions can be drawn:  
1. The initial damage in the vicinity of the arched hole occurs in areas where there is a concentration of circumferential stress, leading to a strain concentration. As the loading continues, macroscopic cracks develop, resulting in overall specimen damage. The early destruction of the specimen is accompanied by the ejection of rock fragments, which serves as an indication of the dynamic destruction of the rock mass. 2. The complex function and conformal mapping method are capable of handling the transient dynamic stress distribution around the arched hole or tunnel, and the theoretical results are consistent with the experimental results. Under transient dynamic loading, the stress distribution around the arched hole/tunnel is highly influenced by the angle of incidence, and both the overall strength and the elastic modulus of the specimen are at their lowest at an incident angle of 60\({}^{\circ}\). In the direction of perpendicular incidence, a significant concentration of compressive stress is generated, which is the primary cause of tunnel damage under dynamic disturbance, and the tensile stress concentration occurs in the direction of incidence. 3. When there is initial stress, dynamic disturbance introduces additional stress and energy, altering the original stress and energy state of the surrounding rock. As the initial stress increases, the surrounding rock becomes more susceptible to damage under dynamic disturbance. The intensity of the damage becomes more dramatic, accompanied by a considerable strain energy release. Overall, the dynamic damage modes of arched tunnels in engineering practice are well reproduced by indoor experiments, and the damage mechanisms of arched tunnels under different incidence angles are well revealed by stress analysis adopting the theoretical  
Figure 17: Numerical simulation results of the dynamic failure of tunnels under initial stresses. (a) RFPA results [42]; (b) LS-DYNA results [21]. method. The theoretical results and experimental results, as well as the field observations, corroborate each other, which provides insights into the damage pattern and damage mechanism of arched tunnel surroundings under dynamic disturbances. However, this study also has some limitations, such as the use of an isotropic elastic assumption model and the small size of the samples, which do not account for the anisotropy of the rock mass and the regular shape of the boundaries. In future work, we will develop a new conformal transformation algorithm to map the real boundaries in engineering and incorporate the anisotropy of the rocks into our considerations. Conceptualization, R.Z. ; methodology, W.W. and W.D. ; software, W.D. ; validation, W.D. ; formal analysis, W.W.; investigation, R.Z. ; resources, M.T. ; data curation, M.T. ; writing--original draft preparation, W.W. and W.D. ; writing--review and editing, R.Z. and M.T. ; visualization, M.T. and W.D. ; supervision, M.T. ; project administration, W.D. ; funding acquisition, M.T. 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 (Grant No. 52274105; 12302505) and the Yunnan Provincial Phosphorus Resource Technology Innovation Center Project (202305 AK340002). The raw data supporting the conclusions of this article will be made available by the authors on request. The authors would like to thank the editors and reviewers for reviewing this paper and for their valuable comments! Authors [PERSON] and [PERSON] were employed by the company Yunnan Phosphating Group Co., Ltd.