## 4 Cumulative Damage Theory and Numerical Model Establishment
### Calculation Model Design and Verification  
The dimensions of the computational model were 20 m in width, 18 m in height, and 16 m in depth, as shown in Figure 8. The cut, relief, peripheral, and bottom holes were laid out on the tunnel excavation face, and the hole parameters were consistent with the actual construction. The element size of the model boundary is 30 cm, and the element size around the borehole is refined to 1 cm. The number of elements after division was more than 3.5 million. Displacement constraints were set on the two boundary surfaces before and after the excavation. To replicate the infinite domain condition, infinite domain boundary control conditions were set on the remaining four surfaces. The total calculation time of the simulated explosion using the software was set to 24,000 \(\upmu\)s.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Type of Hole**} & \multirow{2}{*}{**Part**} & **Hole** & **Line Charge** & **Detonator** & **Equivalent Load** \\  & & **Number** & **Coefficient** & **Series** & **Pressure (Mpa)** \\ \hline Cut holes & \(H_{1}\) & 1–4 & 0.70 & MS1 & 193.11 \\  & \(H_{2}\) & 5–13 & 0.55 & MS3 & 151.74 \\ Relief holes & \(H_{3}\) & 14–25 & 0.55 & MS5 & 151.74 \\ Peripheral holes & \(H_{4}\) & 26–44 & 0.40 & MS7 & 110.35 \\ Bottom holes & \(H_{5}\) & 45–59 & 0.60 & MS9 & 165.53 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Equivalent load of hole wall of different blast holes.  
Figure 7: Explosion stress wave pulse function curve.  
To ensure the accuracy of the material parameters and algorithm settings, field-test monitoring data were used for verification. The field monitoring data obtained from the alternative in situ test described in Section 3.2 is presented in Figure 4. In the numerical model, the same three blast holes were first detonated separately. The velocity response of the bottom surface of the pipeline at its nearest location was extracted and compared with the field blasting test data, as shown in Figure 9--the comparison between in situ test results and calculated results. If the same delay times as in the field are used in numerical simulations, namely MS1, MS3, and MS5, the time cost will be very high. Through trial calculation, when the delay time between two shot holes was more than 500 us, the influence of superposition interference was found to be small. Approximately 1300 us after the detonation, the blasting seismic wave reached the bottom of the pipeline lining. In the numerical model calculation of this study, the delay time of a single blast hole was set to 2000 us for subsequent study calculations.  
Because complex geological conditions such as joints and fissures were not considered in the numerical simulation, there was a certain error between the numerical simulation results and the measured results. The calculated PPV values in the x-, y-, and z-directions were compared with the measured PPV values, as shown in Figure 9. The comparison between in-situ test results and calculated results, with errors of 9.8%, 6.7%, and 4.9%, respectively. The relative error was less than 10%, verifying the accuracy of the numerical model. Therefore, this numerical simulation research method can be used as an alternative to study the impacts on pipelines under explosion effects.  
Figure 8: Numerical model. (**a**) Size parameters and (**b**) grid division.