## 3 Experimental Results and Analyses
### Analysis of Fatigue Damage Evolution  
Under the fatigue loading, the cracks in the rocks underwent three stages, namely, sprouting, extension, and penetration, and the damage inflicted upon them was a progressive process. There were various ways to represent the fatigue damage variables, and it was more common to characterize the evolution of the damage that occurred during the rock fracture via the axial deformation. The plastic deformation led to the generation of hysteresis loops, and the plastic deformation was irreversible. Quantifying the accumulated damage deformation of the sample by analyzing the irreversible axial strain was a reliable process. Therefore, the concept of using damage mechanics to determine the law describing the evolution of the rock damage can now be introduced. The equation describing the damage caused by the uniaxial compression of the rock based on the elastic strain method is  
\[1-D=\frac{\sigma}{E\epsilon} \tag{1}\]  
where \(\sigma\), \(\epsilon\), and \(E\) are the stress, strain, and elastic modulus of the rock sample, respectively, and \(D\) reflects the degree of the damage within the material (if \(D\) = 0, the rock is intact and undamaged; if \(D\) = 1, the rock is fully damaged). For each loading stage, the amplitude of each stress cycle in Equation (1) was considered to be constant, which allowed for the relationship between the value of the cumulative damage variable \(D\) and the cumulative strain to be analyzed. The derivative of both sides of Equation (1) is  
\[dD=\frac{\sigma}{E\epsilon^{2}}d\epsilon \tag{2}\]  
Let \(\epsilon_{0}\) and \(\epsilon_{d}\) be the axial strain at the beginning and end of the cycle. Integrating Equation (2) from \(\epsilon_{0}\) to \(\epsilon_{d}\),  
\[\int_{0}^{D}dD=\frac{\sigma}{E}\int_{\epsilon_{0}}^{\epsilon_{d}}\frac{1}{ \epsilon^{2}}d\epsilon, \tag{3}\]  
Figure 13: Relationship between the axial stress, AE energy, AE accumulative energy, and elapsed time for the argillaceous shale samples subjected to the different frequencies. (**a–d**). The loading frequencies were 0.5 Hz, 0.2 Hz, 0.1 Hz, and 0.05 Hz, respectively.). allows the damage index \(D\) to be obtained as  
\[D=\frac{\sigma}{E}\bigg{(}\frac{1}{\epsilon_{0}}-\frac{1}{\epsilon_{d}}\bigg{)}+C \tag{4}\]  
when \(D=0\), \(\epsilon=\epsilon_{0}\) and \(C=0\), and when \(D=1\),\(\epsilon=\epsilon_{d}\). Substituting \(D=1\) into Equation (4) results in  
\[\frac{\sigma}{E}=\frac{\epsilon_{0}\epsilon_{d}}{\epsilon_{d}-\epsilon_{0}} \tag{5}\]  
Combining Equations (4) and (5), the damage index \(D\) of the argillaceous shale can be expressed as  
\[\frac{\sigma}{E}=\frac{\epsilon_{0}\epsilon_{d}}{\epsilon_{d}-\epsilon_{0}} \tag{6}\]  
The loading frequency had an important impact on the damage evolution. The damage variable jumped, reaching 0.5 or slightly higher for the different frequencies in the second loading stage. Subsequently, the growth rate changed, and the rock samples subjected to higher frequencies became damaged more quickly, with the damage occurring after fewer loading stages. Overall, the variation in the damage variable in the later stages was gradually reduced and became more continuous, indicating that the rock samples experienced the greatest damage during the first several loading stages. In the cycle loading mechanical tests, the damage exhibited two stages: (1) rapid growth and (2) stabilization. This was different from the results of the equal amplitude cycle loading test. The previous \"S\"-type damage evolution model was not suitable for this situation. The relationship between the cumulative damage factors and the loading stage (according to the damage factors defined in Equation (1)) is shown in Figure 14. The sample damage exhibited two stages. The damage accumulated quickly at the beginning, and after that, it became relatively more stable. Figure 14: (**a–d**) Evolution of the damage variable for the tested argillaceous shale samples subjected to the multi-level cyclic loading frequencies.