## 4 Discussion
### Damage Characterization Analysis Based on Axial Strain  
At the beginning of cyclic loading, the strain hardening phenomenon in rock is evident as a decrease in the axial strain increment, even at a constant stress level, suggesting that the rock possesses damage resistance. However, as the loading cycles increase, the cumulative effect of damage begins to dominate the rock's behavior, leading to the strain softening phase. In this stage, the axial strain increment increases at the same stress level, indicating an accelerated damage accumulation within the rock and a degradation of its macroscopic mechanical properties. This transition from hardening to softening [39] is a critical feature of the damage evolution process in rocks under cyclic loading, enhancing the understanding and prediction of rock fatigue life. In this study, rock damage is quantified in terms of axial strain using the following equation:  
\[\mathrm{D}=\frac{\varepsilon_{\mathrm{P}}}{\varepsilon}\frac{\varepsilon- \varepsilon_{0}}{\varepsilon_{\mathrm{P}}-\varepsilon_{0}} \tag{4}\]  
where \(D\) is the damage variable, \(\varepsilon\) represents the strain corresponding to the minimum unloaded stress in a single cycle, \(\varepsilon_{0}\) is the axial initial strain in the first cycle, and \(\varepsilon_{p}\) is the strain at the axial minimum unloaded stress in the last cycle. Based on the calculation results in Equation (4), Figure 9 illustrates the trend of damage evolution in skarn specimens under fatigue loading. The S-shaped curve in the figure demonstrates the correlation between the damage value and the relative number of cycles, defined as the ratio n/N of the number of cycles n to the total number of cycles N. In the early stages of cyclic loading, compaction of the rock's internal pores leads to a rapid increase in the damage value. As the number of cycles accumulates, the expansion of cracks and further damage accumulation become the predominant processes within the rock. When damage accumulates to a certain extent, the growth of the damage value significantly accelerates in the late stages of cycling, typically indicating the rock's critical state prior to undergoing significant damage. As shown in Figure 9a, within a single cyclic group, the damage effects on the rock vary with different loading rates. The data points marked by rectangular boxes show a trend of increasing damage with the number of cycles, with higher damage values observed under low-rate loading conditions, suggesting the impact of the loading rate on the damage accumulation process. Comparing adjacent cycle groups, a loading rate of 6 kN/s versus 2 kN/s shows that the latter induces a more significant damage increment, highlighting the role of the loading rate in rock damage behavior. Additionally, the effect of an increasing loading rate on rock damage in the cyclic group shown in Figure 9d aligns with the previously described trends. As the stress amplitude increases, the damage increment exhibits a decreasing trend, possibly due to the rock's greater deformation capacity and damage tolerance at higher stress levels, indicating that stress amplitude is another crucial factor influencing rock damage evolution. In this study, we develop a mathematical model based on the stress-strain relationship of rocks to describe the damage evolution of rocks under cyclic loading. The model is derived through parameter fitting and can quantify and digitally characterize the damage state of rocks. The expression for the proposed mathematical model of rock damage evolution is presented as follows:  
\[\mathrm{D}=1-\left(1-\left(\frac{\mathrm{n}}{\mathrm{N}}\right)^{\mathrm{a}} \right)^{\mathrm{b}} \tag{5}\]  
where \(n\) is the number of cycles, \(N\) is fatigue life, and \(a\) and \(b\) are material-related parameters. Figure 10 details the relationship between maximum damage and the relative number of cycles for four specimens across different cycle sets, as well as the evolutionary trend of damage accumulation versus the relative number of cycles.