## 3 Experimental Results and Analyses
### Sample Deformation Characteristics  
#### 3.3.1 Relationship between Deformation and the Number of Cycles  
Under the disturbance of the cyclic loads, the internal cracks in the rocks went through different stages of development, and the fatigue damage could be quantified by the axial and radial deformation measurements. Thus, the process by which the rocks were damaged could be characterized by the \"deformation\" parameter. The deformation increased as the loading period increased, and the rate was affected by the loading stage and frequency. The axial and radial relationships to the test time are shown in Figure 7. When the number of loading cycles was small, the axial, lateral, and volumetric strains were low. However, after a while, the growth of the axial, lateral, and volumetric strains accelerated until failure. The deformation increased steadily in several stages before failure, but it increased significantly when the amplitude changed (i.e., when the loading stage changed). In the last loading stage, the deformation growth rate was the largest, and the deformation of the sample increased until it was destroyed. From the perspective of the entire failure process, the deformation gradually became larger and the deformation rate gradually increased.  
Figure 7: (**a–d**) Deformation characteristics of the argillaceous shale samples during the cyclic loading conditions subjected to different loading frequencies. The loading frequencies were 0.5 Hz, 0.2 Hz, 0.1 Hz, and 0.05 Hz).  
By plotting the curve describing the relationship between the deformation and the number of fatigue cycles, the deformation characteristics of the sample could be described more intuitively. For example, the axial deformation is shown in Figure 8, where it can be seen that the growth rate of the axial strain increased with the number of fatigue cycles. In the last stage of the fatigue loading, the axial strain increased sharply. In the time required to change the loading stage, the deformation increased substantially as the stress suddenly increased. The rapid growth of the sample deformation led to a sudden increase in the stress, which caused greater damage to the rock. At the same stage of the initial loading, the deformation tended to be stable because both the loading stress and the change in amplitude were small. The corresponding stress adjustment was implemented for each material part of the rock, and the sample was in the adaptive adjustment state. In the failure stage, the internal cracks were connected to each other, the strength dropped sharply, and the deformation of the sample was large. The deformation and failure responses of the rocks under the different frequencies were characterized by the curves shown in Figure 8.  
To study the effect of the axial deformation (\(\epsilon_{1}\)) and radial deformation (\(\epsilon_{2}\)) on the variation in the argillaceous shale volume (\(\epsilon_{v}\)), the volume strain (\(\epsilon_{v}\)) of each sample was calculated via \(\epsilon_{v}=\epsilon_{1}-2\epsilon_{2}\). The volume strain reflected the joint effect of the axial and radial deformation on the deformation of the sample. The changes in the volume strain were consistent and uniform (i.e., it changed regularly) as the axial and radial strains changed.