# Mechanical and Energy Evolution Characteristics of Sandstone under True Triaxial Cyclic Loading
## 3 Test Results and Analysis
### Analysis of Deformation and Failure Characteristics of Specimens
#### 3.2.1 Definition of Rock Energy Parameters in the True Triaxial State
Total energy \(U_{j}\) is inputted by the radial stress of the loading part, total energy \(U_{je}\) is released by the unloading part, and total energy \(U_{jd}\) is dissipated during the cycle. The formula is as follows:  
\[U_{j}=\sum_{i=1}^{z}u_{j,i}=\sum_{i=1}^{z}\Bigl{[}\sigma_{j}\Bigl{(} \varepsilon_{j,i}^{e}-\varepsilon_{j,i}\Bigr{)}\Bigr{]}, \tag{6}\]  
\[U_{je}=\sum_{i=1}^{z}u_{j,i}^{e}=\sum_{i=1}^{z}\Bigl{[}\sigma_{j}\Bigl{(} \varepsilon_{j,i}^{u}-\varepsilon_{j,i}^{e}\Bigr{)}\Bigr{]}, \tag{7}\]  
\[U_{jd}=\sum_{i=1}^{z}u_{j,i}^{d}=\bigl{|}U_{j}\bigr{|}-\bigl{|}U_{je}\bigr{|}, \tag{8}\]  
where \(\sigma_{j}\) (\(j=2\), 3) is the \(j\) principal stress, \(\varepsilon_{j,i}\) is the strain corresponding to the \(j\) principal stress at the starting point of the \(i\)-cyclic loading, \(\varepsilon_{j,i}^{e}\) is the strain corresponding to the \(j\)  
Figure 9: Energy principle diagram of true triaxial cyclic loading and unloading. (**a**) Axial direction, (**b**) circumferential directions. principal stress at the upper limit of the \(i\)-cyclic loading, and \(e_{j,i}^{u}\) is the strain corresponding to the \(j\) principal stress at the lower limit of the \(i\)-cyclic unloading. 2.2 Energy Evolution Law of Maximum Principal Stress Direction under Different \(\sigma_{2}\) Conditions  
Deformation and failure occur when a rock is subjected to external force in a true triaxial state accompanied by energy accumulation and dissipation. The specific values of energy in each principal stress direction of the rock under true triaxial cyclic loading can be obtained using the abovementioned energy parameter definition method. In this study, by comparing the energy accumulation and dissipation in the rock in each principal stress direction under true triaxial cyclic loading, the energy evolution law of the rock under different \(\sigma_{2}\) conditions was analyzed. A true triaxial cyclic loading test was conducted using the energy analysis method in the true triaxial compression and uniaxial graded loading and unloading test of [PERSON] et al. [38]. With the linear energy storage law proposed by [PERSON] et al. [39; 40; 41], the energy evolution law in the true triaxial cyclic loading test of sandstone was analyzed. In addition, to facilitate the analysis, energy density (\(\mathrm{J}\cdot\mathrm{m}^{-3}\)) was utilized to represent the energy per unit volume of rock. The energy calculation results for each principal stress direction under true triaxial cyclic loading and unloading under different \(\sigma_{2}\) conditions are shown in Table 2.  
The axial input energy-stress, elastic energy-stress, and dissipated energy-stress curves under different \(\sigma_{2}\) conditions are plotted in Figure 10, with the axial loading upper limit stress of single-cycle loading and unloading as the abscissa. \[U=U_{1}+U_{2}+U_{3}, \tag{2}\]0 Axial energy-stress curves under different \(\sigma_{2}\) conditions. (**a**) Axial input energy, (**b**) axial elastic energy, (**c**) axial dissipated energy. With the increase in stress, the axial input energy, dissipation energy, and elastic energy of sandstone generally showed nonlinear growth, and the larger \(\sigma_{2}\) was, the smaller the axial input energy, elastic energy, and dissipation energy were, indicating that the different \(\sigma_{2}\) values had a great influence on the energy storage and dissipation of the rock in unit volume under the true triaxial state. Under the same \(\sigma_{2}\), the elastic energy of the rock at each loading upper limit of axial stress was much larger than the dissipation energy, revealing that the elastic energy of the rock was higher than the dissipation energy during cyclic loading. Under different \(\sigma_{2}\) values, the larger \(\sigma_{2}\) was, the smaller the dissipation energy that corresponded to the upper limit points of axial stress at all levels, and the gentler the change in unit volume dissipation energy with stress.