## 3 Energy Characteristics of SHIR Specimens with Different \(\alpha\)
### The Bedding Effect of Energy at the Characteristic Points  
Table 1 presents the values of various types of strain energies at characteristic points during the failure process of SHIR specimens with different \(\alpha\), enabling analysis of the influence of \(\alpha\) on the energy evolution patterns of rocks (see Figure 5). It is observed that as the characteristic stress increases, the energy difference between different \(\alpha\) gradually increases. With increasing \(\alpha\), the \(U\), \(U^{e}\), and \(U^{d}\) decrease first and then increase at the \(\sigma_{pd}\) and \(\sigma_{pi}\) during the loading process of rocks, reaching a minimum with \(\alpha\) = 60\({}^{\circ}\) and a maximum with \(\alpha\) = 90\({}^{\circ}\), showing an overall V-shaped trend in the energy curves. As loading continues, the \(U^{e}\) of the rock at peak stress initially decreases and then increases. However, the \(U\) and \(U^{d}\) show an upward trend with the lower \(\alpha\) (\(\alpha\) = 0\({}^{\circ}\)\(\sim\)30\({}^{\circ}\)). As the \(\alpha\) increases, these two types of energy subsequently decrease and then suddenly increase, resulting in an overall N-shaped trend in the energy curves. The above-mentioned observations indicate that the presence of bedding planes affects the energy evolution characteristics of rocks. Compared to the horizontal bedding plane, the slightly inclined bedding planes (0\({}^{\circ}\) \(<\)\(\alpha\) \(<\) 30\({}^{\circ}\)) enhance the capability of rock to absorb and dissipate energy to a certain extent, while a medium \(\alpha\) (\(\alpha\) = 45\({}^{\circ}\)\(\sim\)60\({}^{\circ}\)) significantly diminishes the energy absorption capacity and storage limitation of rock, thereby reducing its overall strength and minimizing energy dissipation during the deformation and failure process. With increasing \(\alpha\), the resistance of the rock to energy-driven deformation and failure shows an overall evolution characterized by a \"high-low-high\" pattern.  
In addition, according to Table 1, the detailed ranges of total strain energy, elastic energy, and dissipated energy at peak stress can be determined. Calculating the proportion of various types of rock energies at the peak stress reveals that a different \(\alpha\) leads to variations of energy distribution at peak stress, reflecting the influence of \(\alpha\) on energy accumulation and dissipation in rocks. For specimens S-0 and S-30, the elastic energy at peak stress accounts for 41.67% and 25.32% of the total energy, respectively, indicating that specimens with a lower \(\alpha\) (\(\alpha=0^{\circ}\)-\(30^{\circ}\)) exhibit significant plastic deformation and damage before reaching peak stress, where more elastic strain energy is converted into dissipation energy. Conversely, the proportion of elastic energy at peak values exceeds dissipation  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Specimens**} & \multicolumn{2}{c}{\(\sigma_{pd}\)/(kJ/m\({}^{-3}\))} & \multicolumn{3}{c}{\(\sigma_{pi}\)/(kJ/m\({}^{-3}\))} & \multicolumn{3}{c}{\(\sigma_{p}\)/(kJ/m\({}^{-3}\))} \\  & \(\mathbf{U}\) & \(\mathbf{U^{c}}\) & \(\mathbf{U^{d}}\) & \(\mathbf{U}\) & \(\mathbf{U^{c}}\) & \(\mathbf{U^{d}}\) & \(\mathbf{U^{c}}\) & \(\mathbf{U^{c}}\) \\ \hline S-0 & 1.17 & 0.60 & 0.57 & 14.75 & 14.14 & 0.61 & 91.52 & 38.14 & 53.38 \\ S-30 & 0.54 & 0.28 & 0.26 & 11.03 & 10.69 & 0.34 & 98.00 & 24.80 & 73.20 \\ S-45 & 0.48 & 0.25 & 0.23 & 4.97 & 4.77 & 0.20 & 7.48 & 6.26 & 1.22 \\ S-60 & 0.47 & 0.24 & 0.23 & 1.70 & 1.54 & 0.16 & 1.82 & 1.60 & 0.22 \\ S-90 & 4.59 & 2.29 & 2.30 & 53.74 & 50.70 & 3.04 & 229.05 & 132.96 & 96.09 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Energy values of SHIR specimens at the characteristic points with different \(\alpha\).  
energy when \(\alpha\) = 45\({}^{\circ}\)-90\({}^{\circ}\), indicating that before reaching the peak strength, the energy input from the outside is mainly converted into elastic energy and stored.