## 3 Energy Characteristics of SHIR Specimens with Different \(\alpha\)
### The Evolution Characteristics of Energy Ratios  
Many researchers analyze energy ratios to study the complex energy conversion and deformation processes within rocks [30; 31]. This study analyzes three energy ratios of SHIR specimens with varying \(\alpha\); the formulas for calculation are as follows:  
\[R_{\rm DT}=\frac{U^{d}}{U} \tag{7}\]  
\[R_{\rm ET}=\frac{U^{e}}{U} \tag{8}\]  
\[R_{\rm ED}=\frac{U^{e}}{U^{d}} \tag{9}\]  
In the formulas above, the \(R_{\rm DT}\) represents the ratio of \(U^{d}\) to \(U\). The \(R_{\rm ET}\) denotes the ratio of \(U^{e}\) to \(U\). The \(R_{\rm ED}\) signifies the ratio of \(U^{e}\) to \(U^{d}\). The curves of energy ratios calculated against strain variations are shown in Figure 6, exhibiting similar trends for the same energy ratio curves of SHIR specimens with different \(\alpha\). During the compaction stage, \(R_{\rm ET}\) decreases while \(R_{\rm DT}\) increases. In the elastic stage, as stress continuously increases, \(R_{\rm ET}\) consistently rises while \(R_{\rm DT}\) decreases. Upon reaching the yield stress, as the rock enters the plastic stage, \(R_{\rm ET}\) plummets and \(R_{\rm DT}\) increases steadily. The sudden change of curves can be used as a precursor to the early failure point of rock [24]. Based on the analysis above, the \(R_{\rm ET}\) curve shows a trend of decrease-rise-decrease throughout the deformation and failure process, whereas the \(R_{\rm DT}\) curve exhibits an increase-decrease-increase trend. These changes occur simultaneously but in opposite directions, allowing the change trend of one value to be inferred by analyzing the other of the values. Furthermore, careful observation of Figure 6 reveals that after the closure and frictional sealing of inherent pores within the rock, it enters the elastic stage, where the \(R_{\rm DT}\) curve initially increases before starting to decrease. The rate of decrease in the \(R_{\rm DT}\) curve at this stage shows no clear pattern with changes in \(\alpha\), primarily because the elastic modulus of rock depends mainly on the rock matrix rather than the bedding planes [32]. Therefore, the energy storage capacity and elastic energy conversion rate of rock in this loading stage have no obvious change rule. When the rock enters the plastic stage, the slope of the \(R_{\rm DT}\) curve has a significant difference with the change of \(\alpha\). After the \(R_{\rm DT}\) curves of specimens with S-45 and S-60 increase slightly, the slope approaches 1 near the peak stress. This indicates that a medium \(\alpha\) accelerates the energy dissipation process of rock, leading to a sudden increase in the damage accumulation rate. After reaching peak stress, the \(R_{\rm DT}\) curves of specimens S-0 and S-30 continue to rise in the post-peak stage until they reach 1, whereas for specimens S-45, S-60, and S-90, the \(R_{\rm DT}\) values almost reach 1 upon reaching peak stress. This further indicates that with increasing \(\alpha\), the rock transitions from plasticity to brittleness. From Figure 6, it also can be observed that the \(R_{\rm ED}\) curve exhibits an increasing-decreasing trend throughout the entire deformation and failure process. Initially, the \(R_{\rm ED}\) curve shows slight fluctuations in the early loading stage, and from the compaction stage to the elastic stage, \(R_{\rm ED}\) steadily increases, with the slope of the curve continuously rising. As the stress approaches the yield strength, the \(R_{\rm ED}\) curve begins to fluctuate and rapidly decrease, indicating a decline in the rate of elastic energy accumulation, accompanied by a continuous increase in dissipation energy. The sudden decrease in \(R_{\rm ED}\) and the fluctuations in the curve also serve as signals for the early failure of the rock [30]. During the plastic and post-peak stages, \(R_{\rm ED}\) steadily decreases, indicating a continuous interaction between the accumulated elastic energy and dissipation energy, leading to the continuous expansion and propagation of cracks, ultimately resulting in rock fracture.