## 3 Experimental Results Analysis  
Considering the length of the article, one set of test data is selected from each different cyclic loading path for analysis. Undesirable data points are eliminated, and different cyclic loading and unloading stress-strain curves are plotted using Origin software 2022, as shown in Figure 4.  
From Figure 4, it can be observed that no unstable failure occurred during 9 cycles for both the cyclic loading and unloading paths of the rock samples. However, during the 10 th cycle, when the stress exceeded the peak strength, the stress-strain curve exhibited a rapid drop, indicating the typical brittle failure of the rock. The peak strength of the graded cyclic loading and unloading test on the rock sample was 98.8 MPa with an axial strain of 5.12, while the peak strength of the variable lower limit cyclic loading and unloading test was 99.87 MPa with an axial strain of 4.87. The rock exhibited elastic-plastic characteristics, with the unloading curve not coinciding with the loading curve, forming a hysteresis loop. As the number of cycles increased, the hysteresis loop gradually expanded and shifted towards increased strain. The slope of the stress-strain curve during each cycle of loading and unloading gradually increased, indicating that cyclic loading and unloading improved the elastic modulus of the rock. The rock failure under different cyclic loading-unloading tests is illustrated in Figure 5. In the graded cyclic loading-unloading test, the sound emitted during rock failure is relatively minimal, and crack propagation on the macroscopic surface of the rock occurs from bottom to top. In contrast, violent rock failure is observed in the lower limit cyclic loading-unloading test, with significant macroscopic surface cracks indicating tensile failure. It is evident from Figure 5 that the rock failure under lower limit cyclic loading-unloading is relatively severe. Figure 4: (**a**). Stress-strain curve under graded cyclic loading. (**b**). Stress–strain curve under variable lower limit cyclic action. **4. Study on the Evolutionary Patterns of Energy in Different Cyclic Loading and Unloading**  
_4.1. Methods for Energy Calculation_  
From an energy perspective, the cyclic loading and unloading process of rock materials conforms to the first law of thermodynamics, which states that different forms of energy remain conserved during energy transfer and conversion [11]. In the cyclic loading and unloading process, the work applied by the testing machine on the rock is the total input energy. Part of this energy is converted into elastic strain energy stored internally within the rock, which can be fully released during the unloading process. Another portion of the energy is irreversibly dissipated, including hysteresis energy and plastic deformation energy [7]. Some researchers argue that hysteresis energy exists in both the loading and unloading stages of the rock and is a specific type of irreversible dissipation energy. The energy consumed for the expansion of internal cracks and macroscopic failure within the rock is known as plastic deformation energy. Each point on the stress-strain curve of the rock corresponds to a unique energy state, and the magnitude of hysteresis energy can be represented by the area enclosed by the i-th unloading curve and the i + 1-th loading curve. The energy consumed by rock failure during the loading process can be determined using Formula (1), while the plastic deformation energy during the loading process can be calculated using Formula (2). The total energy input to the rock, represented by the area enclosed by the stress-loading curve and the strain axis, can be obtained using Formula (3). The elastic strain energy of the rock can be calculated by the area enclosed by the unloading curve and the strain axis, utilizing Formula (4). The energy schematic is illustrated in Figure 6.  
Figure 5: (a). Rock failure under graded cycling. (b). Rock failure under variable lower limit cyclic action. Figure 6: Energy schematic diagram.