## 4 Discussion
### Damage Characterization Analysis Based on Dissipative Energy  
The damage process in rocks is usually accompanied by energy dissipation, released in the form of heat, acoustic energy, etc. [40], indicating the degradation of the internal structure of the rock. Under cyclic loading conditions, the damage variable of a rock accumulates with increasing loading cycles, and its growth is closely related to the initiation, extension, and penetration of microcracks within the rock. The accumulation of dissipated energy, a key indicator [41; 42] for evaluating the evolution of rock damage, reflects the intensification of damage within the rock. By monitoring changes in dissipated energy, the damage state of rocks can be quantitatively analyzed, providing a scientific basis for predicting fatigue life and assessing the engineering stability of rock materials. To quantify the degree of damage in rock samples, many early researchers adopted the concept of dissipated energy density. Based on the cumulative change characteristics of dissipated energy density, the damage variable D is defined as the ratio of dissipated energy density to the cumulative dissipated energy density, expressed as follows:  
\[\mathrm{D_{e}}=\frac{\sum_{i=1}^{n}\mathrm{E_{D}}}{\sum_{i=1}^{n}\mathrm{E_{D}}} \tag{6}\]  
where \(D_{e}\) denotes the dissipated energy damage variable, \(n\) denotes the nth cycle in cyclic loading and unloading, and \(E_{D}\) is the dissipated energy in a single cycle. Using the method defined in Equation (6), the dissipated energy damage variables of the four rock specimens were calculated in this study, and the results are displayed in Figure 11. The trends of the dissipated energy damage variables under different amplitude intervals were consistent, showing an increasing trend with the increase in the relative number of cycles. When comparing the same cycle groups, lower loading rates resulted in higher levels of dissipated energy, which indicated an increase in the degree of rock damage. Additionally, for a given relative number of cycles, the increase in dissipated energy damage variables is more pronounced at higher amplitude intervals, as shown in Figure 11d, indicating the contribution of higher stress amplitude intervals to the accumulation of rock damage. By integrating the relationship between dissipated energy and rock damage, a model can be developed for predicting rock damage, which is used to calculate the cumulative damage of rocks under multi-stage cyclic loading and unloading. The model can quantify the cumulative process of rock damage and predict the evolution of rock damage over the number of cyclic loading cycles. The damage accumulation model is expressed mathematically as follows:  
\[\mathrm{D}=\mathrm{a}\Big{(}\frac{n}{\mathrm{N}}\Big{)}^{2}+\mathrm{b}\Big{(} \frac{n}{\mathrm{N}}\Big{)}+\mathrm{D_{0}} \tag{7}\]  
Figure 10: Cumulative damage evolution model for rocks: (**a**) specimen X54; (**b**) specimen X48; (**c**) specimen X53; (**d**) specimen X43. where \(n\) is the number of cycles, \(N\) is fatigue life, \(a\) and \(b\) are the fitting coefficients, and \(D_{0}\) is the initial damage. In this study, Equation (7) was utilized to fit and quantify the dissipated energy damage variable, enabling the quantification of the cumulative damage of rocks under multi-stage cyclic loading and unloading. As shown in Figure 11, the fitting results of this equation exhibit high consistency with the experimental data for specimens of different amplitude intervals, and the coefficients of determination of all the fitted curves exceed 0.98, indicating a very high goodness of fit. This result validates the applicability of Equation (7) in describing the variation of dissipated energy damage variables and provides a reliable mathematical tool for the numerical analysis of rock damage evolution. Using this model, the damage accumulation process of rocks under cyclic loading can be more accurately predicted.