## 3 Energy Characteristics of SHIR Specimens with Different \(\alpha\)
### The Evolution Characteristics of Energy Ratios
Additionally, it is evident that for medium \(\alpha\) (\(\alpha\) = 45\({}^{\circ}\)-60\({}^{\circ}\)), the range of \(R_{\rm ED}\) values is significantly smaller comparedto other rocks. This suggests that rocks with these \(\alpha\) store relatively little elastic energy, further influencing the release of energy during rock failure. **4. Failure Characteristics of SHIR Specimens with Different \(\alpha\)**  
_4.1. Fracture Process and Failure Precursors of SHIR Specimens_  
To further investigate the fracture process and crack propagation characteristics of SHIR specimens with varying \(\alpha\), the concept of Energy Dissipation Ratio (EDR) has been introduced, with the calculation formula as follows [33]:  
\[\text{EDR}=\frac{{U_{t_{2}}}^{d}-{U_{t_{1}}}^{d}}{t_{2}-t_{1}} \tag{10}\]  
In the formula, \(t_{1}\) and \(t_{2}\) represent two adjacent time points, and \({U_{t1}}^{d}\) denote the dissipation energy corresponding to these adjacent time points. Through studying the variation patterns of the EDR curve and combining the aforementioned analysis of energy ratio evolution characteristics of SHIR specimens with different \(\alpha\) at various stages, this section employs a CCD camera to observe in real-time the crack status on the rock surface, thereby characterizing the extent of crack propagation at each stage. To streamline and avoid redundancy, typical specimens S-0, S-45, and S-90 were selected for analysis. The EDR and energy ratio (\(R_{\text{DT}}\) and \(R_{\text{ED}}\)) curve variations under uniaxial compression conditions, along with the evolution of surface crack processes, are depicted in Figure 7. T, S, and S\({}_{\text{S}}\), respectively, represent tensile cracks, shear cracks, and surface spalling cracks. The red line indicates the crack propagation path, while \(\sigma_{pd}\), \(\sigma_{pi}\), and \(\sigma_{p}\) denote the compaction stress, yield stress, and peak stress during the rock loading process, with \(\sigma_{f}\) representing the stress at complete rock failure. In addition, the turning point where \(R_{\text{ED}}\) starts to decline and \(R_{\text{DT}}\) begins to rise is defined as the precursor point, and the point where EDR first exhibits high amplitude is defined as the high EDR point. Figure 7: Evolution of stress, EDR, \(R_{\text{DT}}\), \(R_{\text{ED}}\), and cracks in SHIR specimens: (**a**) S-0; (**b**) S-45; (**c**) S-90. Figure 7a shows the EDR and energy ratio (\(R_{\rm DT}\) and \(R_{\rm ED}\)) curve variations of specimen S-0 and the process of surface crack propagation. In Section 3.4, the energy ratio curves of specimens with different \(\alpha\) have been analyzed in detail, so it will not be repeated here. At Stage I, the energy dissipation in the rock is minimal. The EDR fluctuates near zero with a low amplitude, and there are essentially no visible cracks on the surface of the rock. With increasing load, entering Stage II, the amplitude distribution of EDR remains mostly unchanged, and the rock undergoes slight deformation, characterized by minor surface spalling near the bedding planes of the rock. When the stress reaches \(\sigma_{pi}\) (11.3 MPa), the amplitude of EDR gradually increases and remains at a high level, indicating continuous generation and expansion of internal cracks within the rock. When the stress reaches \(\sigma_{p}\) (20.8 MPa), the distribution range of surface cracks (S\({}_{\rm S}\) cracks) on the rock expands further. Subsequently, as the stress begins to decrease, there is a sudden increase in the amplitude of EDR at 152.0 s, resulting in a significant positive value. This reflects the intensification of damage and fracture in the rock, leading to severe destruction of its original structure. Subsequently, EDR continues to fluctuate continuously, with cracks continuously expanding and intersecting with each other, ultimately leading to complete instability and failure. As the stress decreases to \(\sigma_{f}\) (8.5 MPa), the rock exhibits numerous surface spalling cracks and tensile fractures.