## 3 Results and Discussion
### Damage Evolution  
#### 3.2.1 Deformation Evolution Process  
When rocks are subject to cyclic loading, their physical and mechanical parameters will change, and the changes in these parameters can often reflect the internal conditions of rocks. The secant modulus is a deformation parameter that responds to the relationship between the stress increment and strain increment of material over a certain interval, which \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\beta\) (\({}^{\circ}\)) & 0 & 30 & 45 & 90 \\ \hline \(E\) (GPa) & 20.39 & 16.15 & 15.49 & 13.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Elastic moduli of specimens at different loading–foliation angles. Figure 6: Stress–strain curves and peak strengths of specimens at different loading\(-\)foliation angles under cyclic loading: (**a**) stress\(-\)strain curves at cyclic stress step of 20 MPa; (**b**) stress\(-\)strain curves at cyclic stress step of 40 MPa and (**c**) peak strength of all specimens. can reflect the damage degree within the rock, to some extent [39]. In order to study the damage process of rock specimens under cyclic loading, we analyze the secant moduli at three different locations selected in each cycle of the stress-strain curve. They are the secant modulus \(E_{u}\) at 80% of the upper stress of each cycle, the secant modulus \(E_{20}\) for the part of the stress value below 20 MPa, and the secant modulus \(E_{s}\) at 20 MPa, respectively. The specific calculations are explained by the following equations:  
\[E_{u,n}=\frac{\sigma_{0,8u,n}-\sigma_{0,8u-5,n}}{\epsilon_{0,8u,n}-\epsilon_{0,8u-5,n}} \tag{2}\]  
\[E_{20,n}=\frac{\sigma_{20,n}-\sigma_{l,n}}{\epsilon_{20,n}-\varepsilon_{l,n}} \tag{3}\]  
\[E_{s,n}=\frac{\sigma_{20,n}}{\epsilon_{20,n}} \tag{4}\]  
where \(n\) denotes the nth cycle; \(\sigma_{0,8u,n}\) and \(\epsilon_{0,8u,n}\) denote the stress and strain values corresponding to the location of 80% of the upper stress in the stress-strain curve of the nth cycle (in the following, the statement about the nth cycle is omitted), respectively; \(\sigma_{0,8u-5,n}\) and \(\epsilon_{0,8u-5,n}\) denote the stress and strain at the point corresponding to 80% of the upper limit of stress minus 5 MPa in the curve; \(\sigma_{20,n}\) and \(\epsilon_{20,n}\) denote the values of stress and strain corresponding to the position of stress of 20 MPa; and \(\sigma_{l,n}\) and \(\epsilon_{l,n}\) denote the corresponding stress and strain at the location of the lower stress limit. In order to differentiate between the loading phase and the unloading phase, the mechanical parameters in the loading phase are denoted by a superscript \"+\", and those in the unloading phase are denoted by a superscript \"\(-\)\", e.g., \(E^{+}\) and \(E^{-}\). Figure 7 illustrates the evolution of \(E_{u}\)+ and \(E_{u}\)- of specimens at different loading-foliation angles under cyclic loading. Regardless of the loading-foliation angle and the cyclic stress step, \(E_{u}\)+ and \(E_{u}\)- show a similar variation trend, in that as the cycle number increases, they first increase, accompanied by the increments reducing gradually, and then level off or even start to decrease. This indicates that the specimen is stiffer at higher stress levels. With increasing cycle number, the damage inside the specimen accumulates to a certain value, and then \(E_{u}\)+ and \(E_{u}\)- begin to decrease. It is also suggested that the damage within the rock specimen has a weakening effect on \(E_{u}\)+ and \(E_{u}\)-. Apart from that, \(E_{u}\)+ is always smaller than \(E_{u}\)- in each cycle, which can be attributed to the fact that specimens are deformed both elastically and plastically during the loading process, whereas only the elastic part of the deformation recovers during the unloading process, macroscopically manifested by the accumulation of irreversible deformation at the end of each cycle. As shown in Figure 8, the evolutions of \(E_{20}\)+ and \(E_{20}\)- with the cycle number are very different.