## 4 Sandstone Failure Criterion  
The relationship curve between the confining pressure and peak elastic strain energy density \(U_{EP}\) obtained from the test results is shown in Figure 7.  
The peak elastic strain energy density of sandstone under conventional triaxial compression increases with an increase in the confining pressure and exhibits a good linear relationship where R\({}^{2}\) is 0.987, indicating that the confining pressure can effectively improve the energy storage limit of sandstone. To further verify the generality of this rule, the peak elastic strain energy densities U\({}_{EP}\) of seven different types of rocks under conventional triaxial compression failure were statistically collected, and the results are presented in Table 2.  
Figure 6: Relationship between confining pressure and \(U_{FP}\). Figure 7: Relationship between confining pressure and \(U_{EP}\). of sandstone. Based on the data in Table 2, the curves of the confining pressure and peak elastic strain energy density of the different rock types are shown in Figure 8.  
As shown in Figure 8, the peak elastic strain energy density and confining pressure of the seven different types of rocks satisfy the linear growth relationship, and the fitting formula is as follows:  
\[U_{EP}=a+b\sigma_{3} \tag{6}\]  
where \(a\) represents the peak elastic strain energy density of the rock under uniaxial compression. Combined with the boundary condition, \(\left.\sigma_{1}\right|_{\sigma_{3}=0}=\sigma_{c}\), \(a=\dfrac{\sigma_{c}^{2}}{2E}\), and \(\sigma_{c}\) represent the uniaxial compressive strength of the rock, and \(b\) is the slope of the peak elastic strain energy density of the rock with increasing confining pressure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**References** & **Rock Types** & \(\sigma_{3}\)**/MPa** & \(\sigma_{c}\)**/MPa** & \(\text{\boldmath$U_{EP}$/(MJ\cdot m^{-3})}\) \\ \hline \multirow{4}{*}{[16]} & & 0 & 34.57 & 0.0408 \\  & & 5 & 51.51 & 0.0608 \\  & Chlorite schist & 20 & 78.67 & 0.1062 \\  & & 30 & 97.36 & 0.1539 \\  & & 40 & 115.11 & 0.2138 \\ \hline \multirow{4}{*}{[17]} & & 0 & 76.07 & 0.1027 \\  & & 10 & 108.23 & 0.1796 \\  & & 20 & 132.95 & 0.2439 \\  & sandstone & 30 & 148.19 & 0.2759 \\  & & 40 & 169.31 & 0.3324 \\ \hline \multirow{4}{*}{[18]} & & 0 & 20.20 & 0.0358 \\  & Argillaceous & 5 & 67.80 & 0.2155 \\  & dolomite & 10 & 107.50 & 0.3694 \\  & & 15 & 132.00 & 0.4792 \\  & & 20 & 171.00 & 0.6431 \\ \hline \multirow{4}{*}{[19]} & & 0 & 39.87 & 0.0757 \\  & & 4 & 65.63 & 0.1677 \\  & Jinping & 8 & 88.79 & 0.2041 \\  & greenschist & 20 & 133.92 & 0.3499 \\  & & 40 & 173.03 & 0.4385 \\  & & 50 & 188.68 & 0.5577 \\ \hline \multirow{4}{*}{[20]} & & 0 & 140.36 & 0.2143 \\  & Huashan granite & 15 & 272.36 & 0.5994 \\  & & 25 & 313.82 & 0.7263 \\  & & 35 & 376.00 & 0.9854 \\ \hline \multirow{4}{*}{[21]} & & 0 & 82.31 & 0.1831 \\  & & 5 & 146.12 & 0.3466 \\  & & 10 & 187.21 & 0.5091 \\  & & 20 & 246.34 & 0.8211 \\  & & 30 & 290.78 & 1.0484 \\  & & 40 & 331.74 & 1.0899 \\ \hline \multirow{4}{*}{[22]} & & 0 & 164.62 & 0.2596 \\  & & 1 & 176.79 & 0.3025 \\  & & 2.5 & 203.88 & 0.4043 \\ \cline{1-1}  & Granodiorite & 5 & 217.59 & 0.4418 \\ \cline{1-1}  & & 10 & 289.88 & 0.7263 \\ \cline{1-1}  & & 15 & 310.43 & 0.8377 \\ \cline{1-1}  & & 20 & 347.14 & 1.0591 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Peak elastic energy density of seven types of rocks. During the compression of rock samples, the elastic strain energy gradually increases with an increase in the external force. However, because the energy input by the system gradually transforms into dissipated energy with the plastic deformation of the samples, the elastic strain energy cannot continue to increase; that is, there is an elastic strain energy storage limit for rock samples. When the elastic strain energy reaches the storage limit, the sample is damaged.