## 3 Application Examples and Comparative Analysis
### Calculation Results and Comparative Analysis  
The peak stress points A-F are shown in Table 2. The uncertainty in the peak stress is \(\pm\)0.02. The results indicate that during blasting at the excavation face, the peak stresses in the rock mass follow the order: radial stress > axial stress > hoop stress. To further examine stress evolution, Figure 8 shows the triaxial stress variations of radial points A-F along the tunnel axis, based on the data in Table 2. The results indi \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Calculation Point**} & \multirow{2}{*}{**Distance to the Cut Hole Center (m)**} & \multicolumn{4}{c}{**Peak Stress (MPa)**} \\ \cline{3-6}  & & **Radial** & **Axial** & **Hoop** \\ \hline A & 11.42 & 1.60 & 0.93 & 0.76 \\ B & 12.38 & 1.37 & 0.78 & 0.65 \\ C & 13.35 & 1.18 & 0.68 & 0.56 \\ D & 14.32 & 1.02 & 0.59 & 0.49 \\ E & 15.29 & 0.90 & 0.52 & 0.43 \\ F & 16.26 & 0.80 & 0.46 & 0.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stress calculation results. Figure 6: Schematic diagram of cutting hole. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Density** (kg\(\cdot\)m\({}^{-3}\))**} & **Compressive** & **Tensile Strength** & **Elastic Modulus** & **P-Wave Velocity** & **S-Wave Velocity** & **Poisson’s** \\  & **Strength (MPa)** & **(MPa)** & **(GPa)** & **(m/s)** & **(m/s)** & **Ratio** \\ \hline
2650 & 54.50 & 7.50 & 30.70 & 3036 & 1704 & 0.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical and mechanical parameters of tunnel surrounding rock. Figure 7: Schematic diagram of stress calculation points. cate that radial stress decays inversely with distance and attenuates more rapidly than axial stress. Hoop stress remains consistently lower than both radial and axial stresses. Based on Table 2, the time history curve of radial stress at Point A is plotted in Figure 9. This curve shows that radial stress decreases over time and exhibits a double peak. The peak occurs after the arrival of the S-wave, as the superposition of P- and S-waves causes a sudden increase in stress, indicating their combined effect on the surrounding rock particles. According to Table 2, radial, axial, and hoop stresses decay with distance following the power law \(\sigma=\sigma_{0}\mathrm{d}^{\alpha}\), where \(\sigma_{0}\) is the stress at d = 0, d is the distance from the cut hole center, and \(\alpha=-2.004\pm 0.01063\) is the common power coefficient, identical for all stresses (see Figure 7). The initial stresses are \(\sigma_{\mathrm{r}0}\) = 215.427 \(\pm\) 5.89566 MPa, \(\sigma_{\mathrm{a}0}\) = 123.68921 \(\pm\) 3.36189 MPa, and \(\sigma_{\mathrm{h}0}\) = 102.10915 \(\pm\) 2.85727 MPa. To compare the stress results, parameters from the engineering case were applied to Equations (20)-(22) and (25) from this study--which account for the instantaneous axial pressure from short-cylinder charges and the rock mass damage factor D--and to Equations (23) and (24) from previous studies. The computed stress values are shown in Figures 10 and 11. Figure 8: Stresses decrease as a power law of the distance to the cut hole center. Figure 9: The radial stress–time history curve of point A. Comparative analysis of instantaneous detonation axial pressure (\(q(t)\)): As shown in Figure 10, radial stress is higher when instantaneous detonation axial pressure is considered. The maximum difference reaches 12.42%, with an average difference of 11.25% across all calculation points. Similarly, Figure 10 shows that axial stress is also higher with this factor included, with a maximum difference of 10.36%, and an average difference of 9.68%. Comparative analysis of rock mass damage (D): As illustrated in Figure 11, incorporating the rock mass damage factor D reduces all the stress components. Stress attenuation follows a nonlinear trend as D increases. At D = 0.4, radial stress decreases by 44.58%, axial stress by 29.55%, and hoop stress by 27.18%, compared to the undamaged case (D = 0).