## 4 Cumulative Damage Theory and Numerical Model Establishment
### Explosion Stress Wave Loading Theory and Simulation Method  
Considering the characteristics and conditions of the research object and time savings, the loading blast pressure on the blast hole wall was used to simulate the source of the explosion [40]. The key to acquiring the blast pressure on the blast wall is mainly to determine the peak value and change the historical parameters of the blast wave load. Currently, the isentropic expansion of detonation products and the Chapman-Jouguet model are commonly used to calculate peak explosion loads [41]. Equation (12) can be used to calculate the initial average pressure \(P_{\text{c}}\) of the gas on the wavefront following explosive detonation. \[P_{\text{c}}=\frac{\rho_{\text{e}}(VOD)^{2}}{2(\gamma+1)} \tag{12}\]  
where \(VOD\) represents the detonation velocity, \(\gamma\) represents the isentropic exponent, and \(\rho_{\text{e}}\) represents the explosive density. The uncoupled form of the cylindrical charge was adopted on-site. When the gas in the detonation product collides with the surrounding rock of the borehole wall, reflection superposition of the explosion wave occurs, resulting in an instantaneous increase in the pressure calculated according to Equation (13) [41]. \[P_{0}=n\frac{\rho_{\text{e}}(VOD)^{2}}{8}\bigg{(}\frac{d_{\text{c}}}{d_{\text {b}}}\bigg{)}^{6}\bigg{(}\frac{l_{\text{c}}}{l_{\text{b}}}\bigg{)}^{3} \tag{13}\]  
The \(l_{\text{c}}\), \(d_{\text{c}}\), \(l_{\text{b}}\), and \(d_{\text{b}}\) represent the charge length, charge diameter, borehole depth, and borehole diameter, respectively. \(N\) is the proportional coefficient. The value of \(n\) is taken as 10.0 in this study. Using the parameters of the #2 rock emulsion explosive in the study, the explosive had a density of 1000 kg/m\({}^{3}\) and a velocity of 3600 m/s. \(\gamma=3.0\), \(n=10\), \(d_{\text{c}}=32\) mm, \(d_{\text{b}}=42\) mm. Table 5--equivalent load of hole wall of different blast holes--lists the calculated pressure on the borehole wall for different holes, as per Equation (13). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Steel Type** & \(\rho_{\text{0}}\) **(g/cm\({}^{-3}\))** & \(E\) **(Gpa)** & \(v\) & \(f_{\text{y}}\) **(Gpa)** & _Src (s\({}^{-1}\))_ & _Srp_ \\ \hline Longitudinal bar & 7.85 & 206 & 0.3 & 335 & 40.5 & 5 \\ Stirrup & 7.85 & 206 & 0.3 & 235 & 40.5 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Steel skeleton material parameters. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\rho_{\text{0}}\) & \(A_{\text{1}}\) & \(A_{\text{2}}\) & \(A_{\text{3}}\) & \multirow{2}{*}{\(B_{\text{0}}\)} & \multirow{2}{*}{\(B_{\text{1}}\)} & \multirow{2}{*}{\(A\)} & \multirow{2}{*}{\(G\)} & \(p_{\text{crush}}\) & \(p_{\text{comp}}\) & \(f_{\text{c}}\) & \multirow{2}{*}{\(T_{\text{1}}\)} & \multirow{2}{*}{\(T_{\text{2}}\)} \\ **(kg/m\({}^{3}\))** & **(Gpa)** & **(Gpa)** & **(Gpa)** & & & & & & & \\ \hline
2.41 \(\times\) 10\({}^{3}\) & 35.27 & 39.58 & 12.04 & 1.22 & 1.60 & 1.60 & 16.7 & 0.017 & 44.00 & 0.02 & 35.27 & 0.00 \\ \hline \(g_{\text{c}}^{*}\) & \(g_{\text{t}}^{*}\) & \(\beta_{\text{c}}\) & \(B_{\text{t}}\) & \(f_{\text{s}}^{*}\) & \(f_{\text{t}}^{*}\) & \(D_{\text{2}}\) & \(\gamma_{0}\) & \(\xi\) & \(B\) & \(D_{\textIn order to accurately replicate the effects of the explosion, a time-dependent exponential decay function was used to approximate the change in the explosion load, as shown in Figure 7--explosion stress wave pulse function curve--and given by Equation [42]. \[P=P_{0}\Big{[}e^{-\alpha t}-e^{-\beta t}\Big{]} \tag{14}\]  
In the given Equation, \(P\) represents the borehole wall pressure at time \(t\), \(P_{0}\) is the peak pressure of the borehole, and \(\alpha\) and \(\beta\) are the attenuation constants. \(T_{\mathrm{r}}\) and \(t_{\mathrm{d}}\) represent the time of the maximum load and the total time of the load, respectively. By referring to the theoretical analysis results of [PERSON] et al.