## 4 Analysis of Energy Evolution Characteristics of Triaxial Unloading Confining Sandstone
### Analysis of Pre-Peak Energy Characteristics  
According to the energy evolution curve of sandstone unloading confining pressure under different confining pressures in Figure 6, the energy eigenvalue under the state of peak stress is statistically shown in Table 3. In addition, the relationship between confining pressure and energy eigenvalue is analyzed according to the energy eigenvalue in Table 2. Figure 7 shows the variation curves of energy eigenvalue under different confining pressures, and the absolute value of circumferential energy consumption \(U_{3}\) is taken. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Crack-Characteristic** \\ **Stress** \\ \end{tabular} } & \multirow{2}{*}{\(\sigma_{3}\) **(MPa)**} & \multicolumn{4}{c}{**Energy Characteristic Parameters**} \\ \cline{3-6}  & & & \(\mathbf{U_{1}}\) (\(\mathbf{k}\))-\(\mathbf{m^{-3}}\) & \(\mathbf{U_{3}}\) (\(\mathbf{k}\))-\(\mathbf{m^{-3}}\) & \(\mathbf{U^{e}}\) (\(\mathbf{k}\))-\(\mathbf{m^{-3}}\) & \(\mathbf{U^{d}}\) (\(\mathbf{k}\))-\(\mathbf{m^{-3}}\) \\ \hline \multirow{4}{*}{\(\sigma_{cc}\)} & \(5\) & \(18.04\) & \(-0.66\) & \(9.71\) & \(9.91\) \\  & \(10\) & \(39.65\) & \(-2.50\) & \(27.52\) & \(17.00\) \\  & \(15\) & \(146.13\) & \(-10.33\) & \(113.89\) & \(37.53\) \\  & \(20\) & \(256.85\) & \(-16.63\) & \(213.73\) & \(48.63\) \\ \hline \multirow{4}{*}{\(\sigma_{cl}\)} & \(5\) & \(94.42\) & \(-4.73\) & \(61.75\) & \(30.18\) \\  & \(10\) & \(163.32\) & \(-10.29\) & \(118.53\) & \(41.87\) \\  & \(15\) & \(391.10\) & \(-27.92\) & \(297.26\) & \(74.19\) \\  & \(20\) & \(526.80\) & \(-35.68\) & \(436.40\) & \(84.21\) \\ \hline \multirow{4}{*}{\(\sigma_{cl}\)} & \(5\) & \(239.16\) & \(-12.44\) & \(164.59\) & \(64.36\) \\  & \(10\) & \(405.93\) & \(-28.90\) & \(295.16\) & \(89.25\) \\ \cline{1-1}  & \(15\) & \(614.90\) & \(-48.75\) & \(449.66\) & \(114.52\) \\ \cline{1-1}  & \(20\) & \(801.76\) & \(-59.78\) & \(660.09\) & \(117.48\) \\ \hline \multirow{4}{*}{\(\sigma_{p}\)} & \(5\) & \(390.47\) & \(-24.25\) & \(235.99\) & \(132.87\) \\  & \(10\) & \(580.21\) & \(-63.26\) & \(355.21\) & \(169.12\) \\ \cline{1-1}  & \(15\) & \(848.59\) & \(-97.90\) & \(536.27\) & \(206.96\) \\ \cline{1-1}  & \(20\) & \(1003.50\) & \(-122.07\) & \(678.06\) & \(225.52\) \\ \hline \end{tabular}
\end{table} 
Table 3: Energy characteristic parameters under different characteristic stresses. As can be seen from Table 3 and Figure 7, under each crack-characteristic stress state, each energy characteristic value has a linear relationship with the confining pressure, reflecting an obvious confining pressure effect. Under the crack closure stress \(\sigma_{cc}\) state, the axial absorption energy \(U_{1}\) increases the most, from 18.04 kJ\(\cdot\)m\({}^{-3}\) to 256.85 kJ\(\cdot\)m\({}^{-3}\), with an increase of 238.18 kJ\(\cdot\)m\({}^{-3}\). This is followed by the increase in elastic energy \(U^{e}\), which increases from 9.71 kJ\(\cdot\)m\({}^{-3}\) to 213.73 kJ\(\cdot\)m\({}^{-3}\), with an increase of 240.02 kJ\(\cdot\)m\({}^{-3}\). The value of circumferential energy consumption \(U_{3}\) is negative, with the smallest value increasing from 0.66 kJ\(\cdot\)m\({}^{-3}\) to 16.63 kJ\(\cdot\)m\({}^{-3}\). The dissipation energy \(U^{d}\) increases slightly more than that of \(U_{3}\), from 9.91 kJ\(\cdot\)m\({}^{-3}\) to 48.63 kJ\(\cdot\)m\({}^{-3}\), with an increase of 38.72 kJ\(\cdot\)m\({}^{-3}\). The change value of energy consumption relative to elastic energy is very small, which indicates that the energy used for the internal consumption of rock is very small. As the rock sample is destroyed, most of the stored energy is released. In the subsequent states of \(\sigma_{ci}\), \(\sigma_{cd}\), and \(\sigma_{p}\), the increased amplitude gradually increases, which is shown by the increase in slope in the linear fitting formula of energy characteristic parameters.