# Indices to Determine the Reliability of Rocks under Fatigue Load Based on Strain Energy Method
## 3 Test Results  
### The Evolution of Strain Energy  
Based on the strain energy types mentioned above and the laboratory test results, we obtained the strain energy evolution of red sandstone samples under different confining pressures. It should be noted that the data of these kinds of strain energies in the post-peak stages cannot be obtained in uniaxial compression tests, and two sets of post-peak strain energy data are obtained in the confining pressure tests of 5 MPa. A set of post-peak strain energy data is obtained in the confining pressure tests of 10 MPa and 20 MPa, respectively. According to the results of evolution of strain energy shown in Figures 4 and 5, it can be found that the evolution of strain energies has some characteristics as follows:  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\sigma_{\text{3}}\) (MPa) & \(E\) (GPa) & \(\sigma_{\text{cd}}\) (MPa) & \(\varepsilon_{\text{cd}}\) & \(\sigma_{\text{p}}\) (MPa) & \(\varepsilon_{\text{p}}\) \\ \hline
0 & 2.79 & 18.69 & 0.0064 & 24.92 & 0.0093 \\
5 & 4.42 & 30.95 & 0.0068 & 38.15 & 0.0109 \\
10 & 5.27 & 38.23 & 0.0071 & 49.75 & 0.0118 \\
20 & 8.75 & 51.84 & 0.0083 & 67.87 & 0.0161 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The stress and strain at each stage of red sandstone sample’s deformation under different confining pressures.  
Figure 3: A representative cyclic loading and unloading stress-strain curve under 5 MPa confining pressure.  
1. The characteristics of \(u_{i}^{p}\). The \(u_{i}^{p}\) grows in quadratic form during the whole deformation process. With the increase in strain, the numerical growth rate of \(u_{i}^{p}\) is accelerated especially as it enters the yield stage. The value of \(u_{i}^{p}\) is affected by the confining pressure, the higher the confining pressure the higher the value of \(u_{i}^{p}\).
2. The characteristics of \(u_{i}^{e}\). In the crack closure stage, the growth rate of \(u_{i}^{e}\) is accelerated, and the growth rate in elastic stage is almost constant. After entering the yield stage, the growth rate of \(u_{i}^{e}\) decreases, but the value of \(u_{i}^{e}\) still increases. In the post-peak stage, the growth rate of \(u_{i}^{e}\) becomes negative. With the increase of confining pressure, the values of \(u_{i}^{e}\) increase, and the attenuation effect in the post-peak stage is also inhibited.
3. The characteristics of \(u_{i}^{S}\). The value of \(u_{i}^{S}\) increases in S-shape during the deformation process and decreases after peak stress, which is similar to the stress-strain curve of rocks. The growth rate of \(u_{i}^{S}\) decreases once the sample enters the yield stage. For some samples, the value of \(u_{i}^{S}\) will still increase slightly due to confining pressure in the early post-peak stage. The value of \(u_{i}^{S}\) will also increase with the increase of confining pressure.  
Figure 4: The evolution of \(u_{i}^{p}\) and \(u_{i}^{e}\) of red sandstone samples under different confining pressures: (**a**) Confining pressure is 0 MPa; (**b**) Confining pressure is 5 MPa; (**c**) Confining pressure is 10 MPa; (**d**) Confining pressure is 20 MPa.  
(4) The characteristics of \(u_{i}^{c}\).  
The value of \(u_{i}^{c}\) increases slightly with the increase of strain before the peak point. After entering the post-peak stage, the \(u_{i}^{c}\) of each sample is highly discrete. For most samples, the \(u_{i}^{c}\) decrease in the post-peak stage and even becomes negative, only a small number of them increase. This may be as a result of the effects of confining pressures.  
The results of Figures 4 and 5 show that under the same test conditions, the same kind of strain energy of different samples shows a highly concentrated behavior, and the rock yielding or failure is because the strain energy reaching a certain value. However, due to different confining pressures, different strain energy is required for rock entering different deformation stages, which is relatively limited in application.