## 3 Mechanical Parameters and Energy Evolution Characteristics
### Characterization of Energy Evolution  
The deformation and failure of rocks under loading are primarily due to the release of energy. Assuming that the specimen is in a closed system with no energy dissipation and no energy exchange conditions with the external environment, according to the first law of thermodynamics, the work done by the external force is equal to the total energy released [37; 38; 39]:  
\[U_{i}=U_{ei}+U_{di} \tag{7}\]  
\[U_{i}=\int_{\epsilon_{O}}^{\epsilon_{A}}\sigma_{i}^{+}d\epsilon_{i} \tag{8}\]  
\[U_{ei}=\int_{\epsilon_{C}}^{\epsilon_{A}}\sigma_{i}^{-}d\epsilon_{i} \tag{9}\]  
\[U_{di}=U_{i}-U_{ei}=\int_{\epsilon_{O}}^{\epsilon_{A}}\left(\sigma_{i}^{+}- \sigma_{i}^{-}\right)d\epsilon_{i} \tag{10}\]  
In the above equation, \(U_{i}\), \(U_{ei}\) and \(U_{di}\) represent the total energy, elastic strain energy, and dissipated energy for the \(i\)-th loading cycle, respectively; \(\sigma_{i}^{+}\) and \(\sigma_{i}^{-}\) represent the stress during the loading phase and the stress during the unloading phase, respectively; and \(\epsilon_{O}\), \(\epsilon_{A}\), and \(\epsilon_{C}\) correspond to the abscissa coordinates of points O, A, and C in Figure 5, respectively. The dissipated energy generated by cyclic loading and unloading is further divided into damping energy A\({}_{R}\) and plastic damage energy U\({}_{ds}\). The damping energy is primarily the energy consumed by the hysteresis loop to overcome the viscosity of the rock, while the damage energy is associated with the initiation and development of microcracks within the rock, leading to plastic deformation, which is the dissipated energy minus the damping energy [40; 41]:  
\[U_{dsi}=U_{di}-A_{R}=\int_{\epsilon_{O}}^{\epsilon_{A}}\left(\sigma_{i}^{+}- \sigma_{i}^{-}\right)d\epsilon_{i}-\int_{\epsilon_{C}}^{\epsilon_{A}}\left( \sigma_{i+1}^{+}-\sigma_{i}^{-}\right)d\epsilon_{i} \tag{11}\]  
where \(U_{dsi}\) is the \(i\)-th loading cycle's damage energy. Figure 7 shows the variation in total energy, elastic energy, dissipated energy, damping energy, and damage energy with the increase in loading level. At the initial stage of loading, the overall energy change is relatively small, mainly due to the compaction of the original pores, which results in a smaller release of energy. As the loading level gradually increases, each part of the energy gradually rises, and the rate of increase accelerates. Near the peak (in the final loading stage), the elastic energy stored within the sandstone is suddenlyreleased and converted into forms such as thermal energy, frictional energy, and energy for fracture damage, leading to a sudden and rapid increase in dissipated energy during the final loading stage. Research has also found that the damage energy increases sharply in the final loading stage, further indicating that the stored energy is released in large quantities during the final stage of destruction. The damping energy does not exhibit a significant mutation during the destruction stage, maintaining an upward trend [42,43]. The dissipation energy release gradually increases at each level, so the dissipation energy release is used to reflect the cumulative damage within the rock, and the damage variable D is calculated as in Equation (12):  
\[D_{n}=\sum_{i=1}^{n}U_{i}/U_{d} \tag{12}\]  
where U\({}_{i}\) is the dissipated energy released per stage; U\({}_{d}\) is the dissipated energy released by the whole process; and n is the number of load rating stages. From Figure 8, it can be observed that as the loading cycle level gradually increases, the damage factor follows an exponential growth pattern, which further indicates that the damage caused by cyclic loading and unloading is a process that accumulates over time. Under low load conditions, the damage factor rises slowly, while it increases rapidly as it approaches the failure curve. This is primarily because, during this stage, microcracks extensively initiate, propagate, and connect, eventually forming macroscopic fractures.