## 3 Results and Discussion
### Damage Evolution
#### 3.2.2 Energy Evolution Process  
The energy in a complete cyclic loading-and-unloading process can generally be categorized as the total input energy \(U\), the elastic energy \(U_{e}\), and the dissipation energy \(U_{d}\), and the relationship among them can be obtained according to the principle of energy conservation [41]:  
\[U=U_{e}+U_{d} \tag{5}\]  
If the volume of the specimen is taken as \(V\), Equation (5) can be expressed as  
\[\int udV=\int u_{e}dV+\int u_{d}dV \tag{6}\]  
where \(u\), \(u_{e}\) and \(u_{d}\) represent the total energy density, elastic energy density and dissipated energy density, respectively. \(u\) and \(u_{e}\) can be derived directly from the integral equation, and \(u_{d}\) can be derived from the relationship equation between the three quantities. The relevant physical quantities are shown in Figure 11, and, according to the above calculation method, the \(u_{n}\), \(u_{e,n}\) and \(u_{d,n}\) of the specimen in the \(n\)th cycle can be obtained by  
\[\left\{ \begin{array}{l}u_{n}=\int_{\epsilon_{2}}^{\epsilon_{u}}\sigma_{+}d \epsilon\\ u_{e,n}=\int_{\epsilon_{2}}^{\epsilon_{u}}\sigma_{-}d\epsilon\\ u_{d,n}=u_{n}-u_{e,n}=\int_{\epsilon_{1}}^{\epsilon_{u}}\sigma_{+}d \epsilon-\int_{\epsilon_{2}}^{\epsilon_{u}}\sigma_{-}d\epsilon\end{array} \right\} \tag{7}\]  
where \(\sigma_{+}\) and \(\sigma_{-}\) represent the stress in the loading and unloading curves, respectively. Since the stress level reached in the last few cycles during the loading process is not consistent, the relationship between the energy density and the stress level reached in each cycle is analyzed to explore the energy evolution process of specimens under cyclic loading. The effects of the cyclic stress step and loading-foliation angle on the evolution of total energy density (\(u\)) of the specimen are illustrated in Figure 12. It is found that the relationship between \(u\) and stress level follows a quadratic function, the fitting parameters for which are listed in Table 3. The results in Figure 12a show that, regardless of stress level, with an increasing loading-foliation angle, \(u\) first increases until \(\beta\) exceeds 45\({}^{\circ}\), and then levels off, with the \(u\) of specimen at \(\beta\) of 45\({}^{\circ}\) coinciding with that at \(\beta\) of 90\({}^{\circ}\). Additionally, as shown in Figure 12b-d, regardless of loading-foliation angle, the curves of \(u\) versus the stress level for specimens at cyclic stress steps of 20 MPa and 40 MPa are almost overlapped. Figure 11: Schematic diagram of the \(n\)th loading-and-unloading process. This indicates that the cyclic stress step has little influence on the total external energy absorbed by the specimen during the loading process. Figure 13 shows the relationship between \(u_{e}\) and stress level for specimens at different cyclic stress steps and loading-foliation angles. It can be found that \(u_{e}\) increases significantly with increasing stress level and then slows down in the last cycle steps. The reason behind this decrease is that the damage accumulated in the specimen during cyclic loading weakens the elastic-energy storage capability of the specimen. Moreover, as illustrated in Figure 13b-d, at the same stress level, the \(u_{e}\) corresponding to the cyclic stress step of 40 MPa is larger than that corresponding to the cyclic stress step of 20 MPa when the cycle number is small, whereas an opposite trend occurs when the cycle number is large enough. \begin{table}
\begin{tabular}{c c c c c} \hline
**Specimen Name** & **Quadratic Coefficient** & **Primary Term Coefficient** & **Number of Fit Points** & **Correlation Coefficient** \\ \hline C-0-20 & 0.022 & 0.399 & 8 & 0.9998 \\ C-0-40 & 0.020 & 0.624 & 4 & 0.9999 \\ C-30-20 & 0.025 & 0.553 & 6 & 0.9998 \\ C-30-40 & 0.023 & 0.701 & 4 & 1.0000 \\ C-45-20 & 0.027 & 0.741 & 4 & 0.9999 \\ C-90-20 & 0.030 & 0.536 & 8 & 0.9998 \\ C-90-40 & 0.032 & 0.409 & 7 & 0.9981 \\ \hline \end{tabular}
\end{table}