## 4 Discussion  
A comprehensive analysis of the infrasonic wave time domain characteristics of the red sandstone under uniaxial compression and cyclic loading and unloading shows that the changes in infrasonic waves amplitude reflect the development and expansion of the internal fracture of the rock, and that higher infrasonic amplitude under cyclic loading and unloading conditions generally occurs at and around the upper limit stresses of each cycle before peak strength. In the low-stress stage, the infrasonic wave amplitudes are relatively low and smooth. With the increase in stress, the infrasonic wave amplitudes change significantly, indicating the rapid expansion of the fracture. At the high-stress stage,  
Figure 17: The characterization of damage variable curves for rock specimens based on infrasonic energy. Figure 18: Characterization of cumulative damage variable curve for rock specimens based on infrasonic energy. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Number of**} & \multicolumn{8}{c}{**Parameters**} \\ \cline{2-9}  & \(\mathbf{N_{i+}}\) & \(\mathbf{N_{i-}}\) & \(\mathbf{N_{i}}\) & \(\mathbf{\Sigma N_{i}}\) & \(\mathbf{D_{i+}}\) & \(\mathbf{D_{i-}}\) & \(\mathbf{D_{+}}\) & \(\mathbf{D_{-}}\) \\ \hline
1 & 0.1664 & 0.1356 & 0.3020 & 0.3020 & 0.0242 & 0.0197 & 0.0242 & 0.0439 \\
2 & 0.2439 & 0.2200 & 0.4639 & 0.7659 & 0.0354 & 0.0320 & 0.0793 & 0.1113 \\
3 & 0.4039 & 0.5237 & 0.9276 & 1.6935 & 0.0587 & 0.0761 & 0.1700 & 0.2461 \\
4 & 0.6718 & 1.1125 & 1.7843 & 3.4778 & 0.0976 & 0.1616 & 0.3437 & 0.5053 \\
5 & 0.9244 & 1.3398 & 2.2646 & 5.7420 & 0.1343 & 0.1947 & 0.6396 & 0.8343 \\
6 & 1.1406 & & 1.1406 & 6.8826 & 0.1657 & & 1 & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The infrasonic energy and damage variables of rock specimen A-10. the variation in infrasonic amplitude is most significant for both uniaxial compression and cyclic loading and unloading, with a maximum in infrasonic amplitude, a feature which is shown to be independent of the loading path and is related to the stress. Before peak strength, infrasonic waves amplitude appear in a low and flat state, and cyclic loading and unloading under this condition is more obvious and can be used as a precursor characteristics of rock rupture. The comprehensive analysis of infrasonic energy rate characteristics in red sandstone under uniaxial compression and cyclic loading-unloading conditions reveals distinct patterns across stress stages. During low-stress conditions, the infrasonic energy rate remains consistently low and stable, reflecting minimal internal microcrack activity. As the stress increases to an intermediate level, significant fluctuations emerge in the energy rate, with cyclic loading-unloading amplifying these variations due to cumulative damage effects from repeated stress cycles. The most pronounced changes occur in the high-stress stage, where the energy rate peaks immediately before rock failure--a critical precursor regardless of loading paths. Notably, these characteristic variations demonstrate stress-level dependency rather than sensitivity to specific loading sequences, indicating that internal stress states predominantly govern energy release dynamics. The significant variation in infrasonic energy rate typically manifests prior to peak strength, and the rate generally exhibits a gradually decreasing trend after its initial emergence. The infrasonic energy rate remains notably low prior to peak strength, and this characteristic becomes more pronounced under cyclic loading-unloading conditions. Therefore, the emergence of substantial infrasonic signals accompanied by a sustained increase in energy rate can serve as a precursor characteristic for rock failure. The infrasonic cumulative energy under cyclic loading and unloading conditions can be characterized as three phases: a relatively stable period of infrasonic cumulative energy, a period of rapid growth of infrasonic cumulative energy, and a period of damage precursor.