## 4 Cumulative Damage Theory and Numerical Model Establishment
### Vibration Response of Original Non-Electric Detonator Blasting  
The blast holes of the upper bench on-site were divided into five parts for detonation: \(H_{1}\), \(H_{2}\), \(H_{3}\), \(H_{4}\), \(H_{5}\). To reduce the vibration wave superposition effect caused by the small delay time between each part, the delay time of each part in the simulation model is set to 5000 \(\upmu\)s according to the actual situation in the field. The initiation delay interval of 5000 \(\upmu\)s can be set to realize the individual initiation of each borehole without affecting each other, and similar millisecond delay blasting results can be achieved as in the field test.  
As shown in Figure 10, after the initiation of the cutting holes, a deep color crushing area and a crack area with long cracks appeared around the blasting holes. After the blasting holes in each part were detonated, the cracks were interlaced. In addition, it was found that the unexploded blast hole produced stress concentrations and cracks under adjacent hole blasting. After blasting the bottom holes, the entire upper bench section was blasted. There were no excessive overbreaks or underbreaks, and the blasting effect of the numerical simulation was good.  
To use non-electric detonator blasting, the vibration (e) of the bottom of the pipeline (called the dangerous point) closest to the pipeline above the tunnel was extracted, as shown in Figure 11.  
Figure 11 leads to the following conclusions.  
1. The \(y\)-direction is the vertical direction of the model, while the \(x\)- and \(z\)-directions are the horizontal directions. The vibration response in the vertical direction (with a PPV of 79.12 cm/s) was significantly higher than that in the horizontal directions (with PPVs of 28.31 cm/s and 29.98 cm/s in the \(x\) and \(z\) directions, respectively). This is mainly because the blasting seismic wave generated by the tunnel blasting propagates  
Figure 9: The comparison between in situ test results and calculated results. (**a**–**c**) are the comparison of the blasting vibration velocity extracted from the field monitoring and numerical simulation respectively.  
upward and reaches the bottom of the pipeline, causing intensive vibration of the pipeline;
2. From the results of the integration PPV (Figure 11d), the integration PPV of \(H_{1}\), \(H_{2}\), and \(H_{4}\) detonators is higher than that of \(H_{3}\) and \(H_{5}\) after detonation. At this point, the distance between the pipeline and the source of the explosion is sufficiently small, and the impact of the distance factor on the outcome far outweighs the influence of other factors. Hence, the PPV result (69.03 cm/s) caused by the blasting of peripheral holes (\(H_{4}\)) is slightly higher than that (65.49 cm/s) caused by the blasting of cutting holes (\(H_{1}\));
3. According to the requirements of the current national standard [20] for hydraulic tunnels and other similar structures, a PPV exceeding 15 cm/s will cause damage to the structure. The maximum PPV, 79.12 cm/s, has significantly exceeded the allowable vibration velocity. Blasting using non-electric detonators will cause serious damage to the pipeline;
4. The vibration waveforms generated by the five-part detonations (\(H_{1}\)-\(H_{5}\)) based on the set delay time were relatively independent, and the adjacent vibration waves were not superimposed. This indicates that the delay time setting of the blast holes in the numerical simulation is reasonable and can better reflect the actual situation on-site during blasting on construction.  
Figure 10: Damage and crack development of surrounding rock of upper bench after the detonation of each part using non-electric detonators: (**a**) Cut holes H1 detonation; (**b**) relief holes H2 detonation; (**c**) relief holes H3 detonation; (**d**) peripheral holes H4 detonation; (**e**) bottom holes H5 detonation.