## 3 Results and Discussion
### Damage Evolution
#### 3.2.3 Damage Variable  
To quantitatively analyze the damage suffered by the specimen during cyclic loading, it is necessary to introduce a damage factor that monotonically increases with the number of cycles. In this study, the damage variable is defined on the basis of the dissipation energy in the following form:  
\[D_{n}=\frac{\sum\limits_{i=1}^{i=n}u_{d,i}}{\sum\limits_{i=1}^{i=N}u_{d,i}} \tag{8}\]  
where \(D_{n}\) denotes the damage level of the specimen after \(n\) cycles; and \(i\) and \(N\) denote the \(i\)th cycle and the total number of cycles, respectively. During the loading process, a proportion of the total energy absorbed by the rock from the testing machine is transformed into the elastic energy, while the remainder is dissipated within the rock in order to facilitate the development of cracks. Once the cracks within the rock have absorbed sufficient energy, they begin to link up with each other, eventually forming a fracture surface, which leads to the destruction of the specimen. Since the development of the crack is irreversible, the dissipated energy increases throughout the process, thus ensuring  
Figure 14: The relationship between \(u_{d}\) and stress level for specimens at (**a**) cyclic stress step of 20 MPa; (**b**) \(\beta=0^{\circ}\); (**c**) \(\beta=30^{\circ}\); and (**d**) \(\beta=90^{\circ}\). Figure 15: The relationship between \(u_{d}/u\) and stress level for specimens at (**a**) cyclic stress step of 20 MPa; (**b**) \(\beta=0^{\circ}\); (**c**) \(\beta=30^{\circ}\); and (**d**) \(\beta=90^{\circ}\). that the damage variable increases monotonically with the number of cycles. Since there is no complete unloading process in the last cycle, the elastic-energy density of the last unloading process is estimated here in terms of \(u_{c,n}=\frac{\sigma_{p}^{2}}{2E_{u}}\), where \(\sigma_{p}\) is the maximum stress reached in the last cycle, and \(E_{\text{u}}\) is the elastic modulus of the ideal unloading curve, approximated by the elastic modulus of the unloading curve in the previous cycle [42]. The calculated values of \(u_{N}\), \(u_{e,N}\) and \(u_{d,N}\) for different specimens under cyclic loading are listed in Table 4.  
In addition, to facilitate the comparison between the results, the number of cycles is normalized. The evolution of the damage variable for specimens at different cyclic stress steps and loading-foliation angles are shown in Figure 16. The results indicate that as the normalized cycle increases, \(D\) increases, with the rate of increase growing gradually. The rates of increase of the damage variables of specimens at different loading-foliation angles are different. Within the same normalized cycle, the damage variable of the specimen generally increases when \(\beta\) increases from \(0^{\circ}\) to \(45^{\circ}\), and then decreases when \(\beta\) increases from \(45^{\circ}\) to \(90^{\circ}\). Furthermore, the influence of the cyclic stress step on the evolution of the damage variable of the specimen is significant, in that increasing the cyclic stress step can generally accelerate the evolution of the damage variable, but its degree of influence is related to the loading-foliation angle. To be specific, the degree of influence of the cyclic stress step on the evolution of the damage variable of the specimen is minimal when \(\beta=0^{\circ}\), and then increases to the maximum when \(\beta=30^{\circ}\), and finally decreases when \(\beta=90^{\circ}\). \begin{table}
\begin{tabular}{c c c c} \hline
**Specimen Name** & \(u_{N}\) **(kJ/m3)** & \(u_{e,N}\) **(kJ/m3)** & \(u_{d,N}\) **(kJ/m3)** \\ \hline C-0-20 & 382.58 & 306.05 & 76.53 \\ C-0-40 & 453.46 & 255.23 & 198.23 \\ C-30-20 & 359.82 & 292.96 & 66.85 \\ C-30-40 & 247.14 & 215.93 & 31.21 \\ C-45-20 & 147.91 & 115.80 & 32.11 \\ C-90-20 & 482.01 & 331.05 & 150.97 \\ C-90-40 & 512.15 & 396.38 & 115.77 \\ \hline \end{tabular}
\end{table} 
Table 4: The calculated values of \(u_{N}\), \(u_{e,N}\) and \(u_{d,N}\) for different specimens under cyclic loading.