## 3 Results  
### Mechanical Data Analysis  
Uniaxial compression tests were conducted on specimens A-1 to A-3, with an average uniaxial compressive strength of about 87.097 MPa, and cyclic loading and unloading tests were conducted on specimens A-4 to A-10. Of these, A-5 specimen failed the test due to a data acquisition error, and the rest of the specimens had obvious stress-strain hysteresis loops with an average cyclic loading and unloading with a compressive strength of 84.862 MPa, as shown in Table 2 and Figures 2-4.  
The loading deformation modulus \(E_{1}\) is defined as the ratio of the stress difference between the end and the beginning of the loading phase to the corresponding strain difference during the same loading interval. The unloading deformation modulus \(E_{2}\) is defined as the absolute value of the ratio between the stress difference at the end and beginning phases of unloading and its corresponding strain difference. The calculation formula is shown in Equation (1), and the results are shown in Tables 3 and 4, Figures 5 and 6.  
\[E(i)=\frac{\sigma_{\max}(i)-\sigma_{\min}(i)}{\varepsilon_{\max}(i)-\varepsilon _{\min}(i)} \tag{1}\]  
\(i\)--Number of cycles;  
\(E(i)\)--Deformation modulus per cycle of loading and unloading, GPa;  
\(\sigma_{\max}(i)\), \(\sigma_{\min}(i)\)--Maximum and minimum stresses per cycle of loading and unloading, GPa;  
\(\varepsilon_{\max}(i)\), \(\varepsilon_{\min}(i)\)--Load and unload maximum and minimum strains per cycle.  
Figure 4: Stress–time diagram of rock specimen A-4.  
Figure 5: Stress–strain curves under cyclic loading and unloading.  
Figure 3: Stress–time diagram of rock specimen A-1.  
As shown in Figures 5 and 6, the first cycle of loading and unloading \(E_{1}\) of the red sandstone is the smallest, and its stress-strain diagram exhibits the smallest stress-strain slope for the first cycle of loading; the second cycle \(E_{1}\) has the largest increment, indicating that the mechanical properties of the rock mass are significantly enhanced after the previous loading. Each cycle \(E_{1}\) increases with the number of cycles, except for the high-stress phase where a decrease occurs. The decrease in the loaded deformation modulus of specimen A-4 at the fifth loading was due to the rock rupture sound of specimen A-4 when it was loaded at the upper stress of the fifth cycle. Specim A-4 showed a greater degree of internal damage during the fifth cycle, so the loaded deformation modulus of specimen A-4 showed a decrease in the fifth cycle. The loading deformation modulus \(E_{1}\) of the remaining specimen exhibited a decline during the sixth loading cycle, indicating that the progressive deterioration of rock mass damage occurred with increasing cycle number and upper stress limits, ultimately leading to structural failure. From the perspective of mechanical parameter analysis, this phenomenon corresponds to the gradual slowing of  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Times**} & \multicolumn{5}{c}{**Deformation Modulus E\({}_{2}\) at Different Unloading Stages/GPa**} \\ \cline{2-7}  & **A-4** & **A-6** & **A-7** & **A-8** & **A-9** & **A-10** \\ \hline
1 & 20.70 & 23.66 & 12.78 & 23.81 & 12.71 & 12.84 \\
2 & 21.84 & 22.84 & 13.16 & 25.07 & 13.32 & 12.71 \\
3 & 21.77 & 23.81 & 14.45 & 25.13 & 14.40 & 13.68 \\
4 & 22.70 & 24.08 & 15.50 & 25.26 & 15.38 & 14.94 \\
5 & 22.93 & 24.30 & 16.34 & 25.94 & 16.09 & 15.94 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Calculation results of deformation modulus at each unloading stage.  
Figure 6: Loading deformation modulus–number of cycles.  
growth rate in modulus \(E_{1}\) followed by eventual decrease, which can serve as a critical failure characteristic indicator in rock mechanics. The unloading deformation modulus \(E_{2}\) shows a gradual increase with the number of cycles, indicating the enhanced elastic recovery capacity of the rock during unloading.