## 3 Test Curve Analysis
### Energy Characteristics  
The cyclic loading process is accompanied by the input and output of energy. The energy input during the loading process is partly used for the storage of rock elastic energy and is partly dissipated due to the development of damage inside the rock. The relationship among the three can be obtained according to the first theorem of thermodynamics [24; 25; 26].  
\[U=U_{e}+U_{d} \tag{1}\]  
\[U_{e}=\int_{e_{1}}^{e_{2}}\sigma d\varepsilon \tag{2}\]  
\[U_{d}=\int_{0}^{e_{1}}\sigma d\varepsilon-\int_{e_{1}}^{e_{2}}\sigma d\varepsilon \tag{3}\]  
where \(U\) is the total input energy density, \(U_{e}\) is the elastic energy density and \(U_{d}\) is the dissipation energy density.  
According to the diagram's expression of the relationship between energy density values in Figure 6, the elastic deformation energy stored in the rock unloading process was released. The area under the unloading curve and the strain axis is the elastic energy density. According to the relationship between fracture mechanics and energy, the area enclosed by the loading curve and unloading curve is the dissipated energy density. Dissipation energy consists of friction energy dissipation and crushing energy dissipation [27], which causes the accumulation of rock internal damage, crack initiation and propagation, and the formation of a sharp leaf-like hysteresis loop between the upper unloading curve and the lower loading curve due to the multi-phase, multi-component and non-uniformity characteristics of the rock.  
Energy conversion was used to further analyze the damage process inside the rock under different cycle amplitudes. The results are shown in Figure 7. The analysis suggests that the energy transformation process under cyclic loading can be divided into four stages: the compaction stage, elastic stage, plastic stage and failure stage. The curve of the dissipation energy percentage and elastic energy percentage shows obvious groove and inverted groove evolution. In the compaction stage, due to the compaction of pores and cracks in the rock, the percentage of dissipated energy in the input energy was large, but in the compaction stage, the elastic stage and the plastic stage were still dominated by elastic energy storage. In the plastic stage, due to the increase in the unrecoverable plastic strain of the rock, the elastic energy of the rock decreased by 1.86%, and the dissipation energy increased by 1.86%. In the failure phase, the elastic energy of the rock was released, and the  
Figure 6: Rock energy density value relationship diagram.  
dissipation energy increased rapidly, resulting in the loss of the carrying stability of the rock. From an energy point of view, the sudden release of the elastic energy accumulated inside the rock was the internal cause and power source of rock failure.  
Figure 8 shows the hysteresis loop contours and the hysteresis energy density-cycle amplitude relationship curve. The analysis shows that, as the amplitude of the cycles increased, the plastic damage of the rock increased continuously, the hysteresis loop moved to the right, the area increased continuously and the hysteresis curve of the rock changed from dense to sparse. The nonlinear data fitting of the rock hysteresis energy density and cycle amplitude was carried out. The fitting results are as follows:  
\[y=-0.018+0.0096e^{\frac{x}{4\pi 0.10}} \tag{4}\]  
Figure 8: Hysteretic loop profile and hysteretic energy density–cycle amplitude relationship curve.  
Figure 7: Energy transformation during the cyclic loading process.