## 3 Experimental Results and Analyses
### Rock Fatigue Life Analysis  
Owing to the different loading frequencies, the time-axial load curves behaved differently, as shown in Figure 5. The loading path curve indicated that the samples subjected to different loading frequencies went through different fatigue loading stages, including failure. For the frequencies of 0.5, 0.2, 0.1, and 0.05 Hz, failure occurred at the fifth, sixth, sixth, and seventh stages, respectively. The curve also shows that the loading cycle number varied owing to the differences in the loading frequencies. The total number of loading cycles for the four tested frequencies were 228, 214, 184, and 159, respectively. The fatigue life of the rock decreased accordingly. As the loading frequency increased, the loading cycle number showed an increasing trend. The peak stress of the failure for the four samples were 21.3, 23.9, 26.1, and 27.6 MPa, respectively, indicating that the peak stress increased with the frequency.  
Further analyses showed that the increase in the fatigue life occurred for two reasons. First, the changes in the strain and stress were not synchronized, which usually indicated that the changes in the strain damage lagged behind the changes in the stress, (i.e., the so-called elastic after-effect). For the different loading frequency disturbances, the stress loading stage led to an accumulation of the damage strain. However, for the higher loading frequencies, the strain accumulation in the loading stage was incomplete, and the stress change entered the unloading stage, which caused the plastic strain to decrease with the increase in the frequency under a single cycle load. However, the maximum plastic strain generated when the rock sample was damaged remained similar. Therefore, the sample underwent additional cyclic loads after it was damaged (i.e., it had a longer fatigue life). Second, in the test frequency range, especially when f = 0.5 Hz, the average strain rate was \(10^{-3}\). At this rate, the dynamic strength was higher than the static strength. However, the fatigue test involved the incremental variable amplitude loading to destroy the sample without considering the improvement in the strength, and the strain rate for the uniaxial compressive strength (\(10^{-6}\)) was still used as a reference for the comparison. This confirmed that the fatigue life increased with the loading frequency.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Group**} & **Sample** & **L \(\times\) d** & **Mass** & **Peak** & **Loading** & **Loading** & **Lower Limit** & **Loading and Unloading** \\  & **ID** & **(mm \(\times\) mm)** & **(g)** & **Strength** & **Cycles** & **Frequency** & **Stress (MPa)** & **Conditions** \\ \hline \multirow{2}{*}{I} & UC-1 & 100.25 \(\times\) 49.45 & 399.8 & 24.5 & / & / & / & Uniaxial, monotonous \\  & UC-2 & 100.03 \(\times\) 49.94 & 401.4 & 27.1 & / & / & / & compression, 0.06 mm/min \\  & UF-1 & 99.58 \(\times\) 49.88 & 401.8 & 27.6 & 228 & 0.5 & 3 & \\  & UF-2 & 100.01 \(\times\) 49.61 & 402.5 & 27.1 & 219 & 0.5 & 3 & \\  & UF-3 & 99.66 \(\times\) 49.77 & 399.9 & 26.1 & 214 & 0.2 & 3 & \\  & UF-4 & 100.05 \(\times\) 49.95 & 402.3 & 25.8 & 211 & 0.2 & 3 & \\  & UF-5 & 99.73 \(\times\) 49.98 & 398.5 & 24.1 & 184 & 0.1 & 3 & 3 MPa, 30 cycles were performed \\  & UF-6 & 100.72 \(\times\) 49.82 & 400.3 & 23.5 & 175 & 0.1 & 3 & for each fatigue loading stage \\  & UF-7 & 100.02 \(\times\) 50.01 & 397.1 & 21.3 & 159 & 0.05 & 3 & \\  & UF-8 & 100.08 \(\times\) 49.91 & 401.2 & 22.1 & 146 & 0.05 & 3 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Testing scheme for the argillaceous shale samples.