## 3 Experimental Results Analysis
The area enclosed by the curve OABO in Figure 6 represents the magnitude of the total energy input to the rock. The area enclosed by the curve CABC represents the magnitude of elastic strain energy. The area enclosed by the curve OACO represents the magnitude of dissipative energy. The area enclosed by the curve OABCDO represents the magnitude of plastic deformation energy. The area enclosed by the curve CDEC represents the magnitude of hysteresis energy. (This experiment excludes the influence of temperature and radiation). Note: In the translation above, I assumed that the curve labels \"OABO\", \"CABC\", \"OACO\", \"OABCDO\", and \"CDEC\" are labels specific to the diagram and are not common terms or acronyms. If they hold significant meaning beyond the diagram, please let me know, and I can adjust the translation accordingly. The relationships between elastic strain energy, dissipative energy, total input energy, hysteresis energy, and plastic deformation energy are expressed by the following equations:  
\[U=U_{d}+U_{e} \tag{1}\]  
\[U_{e}=U_{se}+U_{c} \tag{2}\]  
\[U=\int_{e_{0}}^{e_{2}}\sigma d\varepsilon \tag{3}\]  
\[U_{d}=\int_{e_{1}}^{e_{2}}\sigma d\varepsilon \tag{4}\]  
In the above equations, \(U\), \(U_{d}\), \(U_{e}\), \(U_{se}\), and \(U_{c}\) represent the total input energy, elastic strain energy, dissipative energy, hysteresis energy, and plastic deformation energy, respectively, in the cyclic loading and unloading process. To enhance computational efficiency, the energy values for each cycle can be obtained using the \"Polygon Area\" function within the Analysis feature of Origin software. From Figure 7, it can be concluded that in the graded cyclic loading and unloading process, elastic strain energy, dissipative energy, total energy, hysteresis energy, and plastic deformation energy are positively correlated with the number of cycles. In the graded cyclic loading and unloading path, the majority of the input energy is converted into elastic strain energy stored within the rock. Dissipative energy exists in the form of hysteresis energy, and the energy consumed by the rock's plastic deformation is relatively small. As a result, there are few vertical cracks on the macroscopic surface of the rock, and its integrity is relatively good when it undergoes failure. From Figure 8, it can be observed that in the variable lower-bound cyclic loading and unloading process, elastic strain energy, dissipative energy, total energy, and plastic deformation energy are positively correlated with the number of cycles. Along the variable lower-bound cyclic loading and unloading path, as the number of cycles increases, after 5 cycles, dissipative energy gradually exceeds the elastic strain energy. Hysteresis energy decreases gradually, while the plastic deformation of the rock increases. The internal cracks of the rock continue to expand, and numerous cracks appear on the macroscopic surface of the rock when it undergoes failure. Comparative Analysis of Energy Calculation Results for Different Cyclic Lading and Unloading Scenarios  
Studying the energy distribution patterns of rocks under different cyclic loading and unloading paths can provide a visual representation of the energy evolution patterns of rocks under different cycling paths. By utilizing the energy results calculated in the previous section, the energy storage rate and energy dissipation rate are plotted against the number of cycles on a dual \(Y\)-axis curve, as shown in Figure 5.0  
From Figure 9, it can be observed that in the graded cyclic loading and unloading path, the energy storage rate during the 9 cycles ranges between 81% and 85%, indicating that prior to rock failure, the majority of energy is stored in the rock as elastic strain energy. However, the variable lower-bound cyclic loading and unloading path exhibits a different pattern. The energy storage rate initially decreases and stabilizes at around 50%, while the energy dissipation rate increases initially and stabilizes at around 50%.