# Indices to Determine the Reliability of Rocks under Fatigue Load Based on Strain Energy Method
## 4 Verification and Extension of Strain Energy Index
### Collection of Experimental Results from PreviousStudies  
We selected 94 rock cyclic loading and unloading tests of five types of rocks from published data for analysis. The basic information of the samples and test conditions are presented in Table 2. Although some of the data did not perform cyclic loading and unloading at the post-peak stage, it was impossible to analyze the post-peak condition, but the analysis in the pre-peak stage was still effective. After analyzing the obtained data, it is found that the strain energy evolution of various samples is consistent with the strain energy evolution of red sandstone obtained in Section 3. For almost all samples, the proportion of grain strain energy will decrease after entering the yield stage. In this study, we choose the elastic modulus as abscissa axis because elastic modulus is affected by confining pressure, porosity, compactness of rock grains arrangement, and other comprehensive factors, which is the general embodiment of rock elastic properties. By analyzing the collected data, we observed that the ratio of grain strain energy to total strain energy increases with the increase in elastic modulus, different rocks show the distinct classification. The proportion of \(u_{i}^{g}\) to \(u_{i}\) of the same rock typeis linearly increasing with the elastic modulus. The relations between elastic modulus and proportion of \(u_{i}^{g}\) to \(u_{i}\) of different rocks while entering the yield stage are shown in Figure 7.  
Let y denote the ratio of \(u_{i}^{g}\) to \(u_{i}\) when yielding and x denote the elastic modulus. Linear fitting of the data of different rocks in Figure 7 are given in the following equations:  
The coal is  
\[\text{y}=0.08278\text{x}+0.38472,\text{ R}^{2}=0.99638 \tag{7}\]  
The sandstone is  
\[\text{y}=0.03025\text{x}+0.04219,\text{ R}^{2}=0.53828 \tag{8}\]  
The basalt is  
\[\text{y}=0.01229\text{x}+0.41971,\text{ R}^{2}=0.74461 \tag{9}\]  
The marble is  
\[\text{y}=0.00753\text{x}+0.12095,\text{ R}^{2}=0.7071 \tag{10}\]  
Finally, the granite is  
\[\text{y}=0.00603\text{x}+0.31933,\text{ R}^{2}=0.88499 \tag{11}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Rock Type** & **Number of Data** & \(\sigma_{3}\) **(MPa)** & **References** \\ \hline
1 & Marble & 6 & 0 & [43] \\
2 & Marble & 4 & 40 & [44] \\
3 & Granite & 4 & 10 & [32] \\
4 & Coal & 1 & 0 & [44] \\
5 & Coal & 1 & 10 & [21] \\
6 & Sandstone & 1 & 10 & [21] \\
7 & Granite & 1 & 10 & [21] \\
8 & Marble & 1 & 5 & [37] \\
9 & Sandstone & 1 & 0 & [45] \\
10 & Sandstone & 1 & 0 & [17] \\
11 & Sandstone & 2 & 0 & [46] \\
12 & Sandstone & 6 & 0 & [47] \\
13 & Sandstone & 1 & 0 & [48] \\
14 & Basalt & 5 & 0 & [49] \\
15 & Granite & 1 & 0 & [2] \\
16 & Sandstone & 37 & 0 & [23] \\
17 & Coal & 2 & 3 & [22] \\
18 & Marble & 1 & 20 & [50] \\
19 & Marble & 1 & 35 & [50] \\
20 & Marble & 1 & 50 & [50] \\
21 & Marble & 2 & 5 & [51] \\
22 & Marble & 1 & 20 & [51] \\
23 & Marble & 1 & 40 & [51] \\
24 & Granite & 3 & 2 & [52] \\
25 & Granite & 1 & 5 & [52] \\
26 & Granite & 1 & 7 & [52] \\
27 & Granite & 1 & 40 & [53] \\
28 & Granite & 1 & 1 & [54] \\
29 & Granite & 1 & 10 & [54] \\
30 & Granite & 1 & 20 & [54] \\
31 & Granite & 1 & 30 & [54] \\ \hline \end{tabular}
\end{table} 
Table 2: Basic information of the samples and test conditions from literature. From the results of linear fitting, it can be seen that the slope of linear equation from Equation (7) to Equation (11) decreases successively, while the uniaxial compressive strength of rock increases successively. It should be noted that the uniaxial compressive strength referred to here is not the uniaxial compressive strength specific to a particular rock, but the uniaxial compressive strength commonly used to classify rocks. This is because the uniaxial compressive strength of the sandstone, in some cases, is higher than that of the marble. The classification of rock uniaxial strength is stated in China's national standards [55; 56].