## 4 Cumulative Damage Theory and Numerical Model Establishment  
### Surrounding Rock Model  
The explicit finite element software LS-DYNA19.0 is widely used for simulating explosion processes. In this study, this simulation method is applied, and the Johnson-Holmquist II (JH-2) constitutive model was used to simulate the surrounding rock of the tunnel. In the JH-2 material model, the relationship between the equivalent stresses of the integrated (\(\hat{\sigma_{i}^{*}}\)), damaged (\(\hat{\sigma^{*}}\)), and fractured (\(\hat{\sigma_{t}^{*}}\)) materials with a change in standardized pressure (\(P^{*}\)) is shown in Figure 5[21]. The relationship between the equivalent stress and damage variable is described by the function shown in Equation (2).  
\[\hat{\sigma^{*}}=\hat{\sigma_{i}^{*}}-D\Big{(}\hat{\sigma_{i}^{*}}-\hat{\sigma _{t}^{*}}\Big{)} \tag{2}\]  
Figure 4: Vibration waveform monitored from tests. (**a**–**c**) correspond to the vibration velocity in the \(x,y,\) and \(z\) directions, respectively.  
Figure 3: Design and implementation process of alternative in situ testing: (**a**) Diagram of the testing; (**b**) the operating process of testing.  
where the dimensionless number \(D\) is the damage variable. This variable can sufficiently reflect the gradient damage process when the material gradually deteriorates from the complete to the final fractured material with increasing loading.  
\[D=\sum\frac{\Delta\varepsilon^{\text{P}}}{\varepsilon_{\text{f}}^{\text{P}}} \tag{3}\]  
\[\varepsilon_{\text{f}}^{\text{P}}=d_{1}\Big{(}p^{*}+t^{*}\Big{)}^{d_{2}} \tag{4}\]  
\[\sigma_{\text{f}}^{*}=b\Big{(}p^{*}\Big{)}^{m}\Big{(}1+c\ln\hat{\varepsilon}^{ *}\Big{)}\leq SF_{\text{max}} \tag{5}\]  
where \(\Delta\varepsilon^{\text{P}}\) is the increased equivalent plastic strain in the integral cycle during loading, \(d_{1}\) and \(d_{2}\) are the damage coefficients related to the material properties, and the main function of \(d_{1}\) is to control the rate of damage accumulation. \(SF_{\text{max}}\) represents the maximum normalized fracture strength of the material and \(\varepsilon_{\text{f}}^{\text{P}}\) is the equivalent plastic fracture strain value of the surrounding rock under the action of a constant hydrostatic pressure load \(p\). The equation of state of the JH-2 material model is given by Equation (6):  
\[p=k_{1}\mu+k_{2}\mu^{2}+k_{3}\mu^{3} \tag{6}\]  
\[\mu=\frac{\rho}{\rho_{0}}-1 \tag{7}\]  
where \(k_{1}\) is the material's bulk modulus; \(k_{2}\) and \(k_{3}\) are coefficients related to loading pressure; \(\mu\) is the volume strain rate, and it will gradually increase with the process of material; damage; \(\rho\) and \(\rho_{0}\) are the current density and the initial density values of the material.  
The rock strata in this area are composed primarily of hard granite. Based on an engineering geological survey report and the relevant literature [35], the values of each parameter are listed in Table 2. Parameters of rock material model [36]. The G in the table is the material's shear modulus. HEL represents the material's elastic limit. \(A\), \(N\), and \(C\) are constants related to the material properties.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\rho_{0}\) (kg/m\({}^{\text{3}}\)) & \(G\) (Gpa) & \(k_{1}\) (Gpa) & \(k_{2}\) (Gpa) & \(k_{3}\) (Gpa) & _HEL_ (Gpa) & \(A\) & \(N\) & \(C\) \\ \hline \(2.46\times 10^{3}\) & \(18.5\) & \(2.51\times 10^{2}\) & \(32.0\) & \(-4.5\times 10^{3}\) & \(3.0\times 10^{3}\) & \(9.7\times 10^{-3}\) & \(0.72\) & \(5.0\times 10^{-3}\) \\ \hline \(B\) & \(M\) & \(\sigma_{\text{FMax}}^{*}\) (Gpa) & \(T/\text{Gpa}\) & \(D_{2}\) & \(\beta\) & \(D_{1}\) & \(T_{\text{f}}\) (Gpa) & \(\text{G}_{\text{f}}\) (J-m\({}^{\text{2}}\)) \\ \hline \(0.32\) & \(0.72\) & \(25\) & \(-5.3\times 10^{-2}\) & \(0.7\) & \(0.5\) & \(5.0\times 10^{-3}\) & \(3.5\times 10^{-2}\) & \(70\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of rock material model.  
Figure 5: Relationship of different strength parameters of JH-2 material model.