## 3 Triaxial Cyclic Loading and Unloading Experiment on High-Strength Sandstone
### Energy Properties of Sandstone under Triaxial Cyclic Loading and Unloading  
According to the law of energy conservation, it is assumed that the action of external loads does not cause changes in the internal thermal energy of the sandstone sample. Therefore, the energy input from outside is mainly mechanical energy. In this process, mechanical energy is converted into elastic energy stored in the elastic deformation of the sample, plastic energy corresponding to plastic deformation, and energy released by dissipation of the destruction of the sample. During the pre-peak phase of the stress-strain  
Figure 9: Strain curves of sandstone samples under triaxial cyclic loading and unloading with different surrounding pressures. (**a**) 0 MPa; (**b**) 5 MPa; (**c**) 10 MPa; (**d**) 15 MPa; (**e**) 20 MPa. curve, most of the energy input into the testing machine is stored in the sample in the form of releasable elastic energy. The elastic energy density \(U^{e}\) can be expressed as Equation (2) and the elastic strain \(\varepsilon^{e}\) can be expressed as Equation (3). \[U^{e}=\frac{1}{2}\sigma_{1}\varepsilon_{1}^{e}+\frac{1}{2}\sigma_{2}\varepsilon_ {2}^{e}+\frac{1}{2}\sigma_{3}\varepsilon_{3}^{e} \tag{2}\]  
\[\left. \begin{array}{l}\varepsilon_{1}^{e}=\frac{1}{E_{1}}[\sigma_{1}-\mu_{1}( \sigma_{2}+\sigma_{3})]\\ \varepsilon_{2}^{e}=\frac{1}{E_{2}}[\sigma_{2}-\mu_{2}(\sigma_{1}+\sigma_{3})] \\ \varepsilon_{3}^{e}=\frac{1}{E_{3}}[\sigma_{3}-\mu_{3}(\sigma_{1}+\sigma_{2})] \end{array} \right\} \tag{3}\]  
where \(\sigma_{1}\) is the principal stress, \(\varepsilon_{1}^{e}\) is the elastic strain in the direction of the principal stress, \(E_{i}\) is the elastic modulus, and \(\mu_{i}\) is Poisson's ratio. It is assumed that the sandstone sample is homogeneous and isotropic, and the testing machine is loaded with equal surrounding pressure (\(\sigma_{2}=\sigma_{3}\)). Therefore, it can be simplified to obtain Equation (4). \[U^{e}=\frac{1}{2E}\left({\sigma_{1}}^{2}+2{\sigma_{3}}^{2}-4\mu\sigma_{1} \sigma_{3}-2\mu{\sigma_{3}}^{2}\right) \tag{4}\]  
The surrounding pressure \(\sigma_{3}\) is kept constant during the test loading. Equation (5) is obtained by integrating Equation (4). \[dU^{e}=\frac{\sigma_{1}-2\mu\sigma_{3}}{E}d\sigma_{1}-\frac{1}{2E^{2}}\left( {\sigma_{1}}^{2}+2{\sigma_{3}}^{2}-4\mu\sigma_{1}\sigma_{3}-2\mu\sigma_{3}^{2 }\right)dE \tag{5}\]  
During unloading, the rock undergoes mainly elastic deformation, so d\(E\)\(\approx\)0. \(d\varepsilon_{3}=-\mu d\varepsilon_{1}=-\frac{\mu}{E}d\sigma_{1}\), and Equation (5) can be reduced to  
\[dU^{e}=\frac{\sigma_{1}}{E}d\sigma_{1}-2\frac{\mu\sigma_{3}}{E}d\sigma_{1}= \sigma_{1}d\varepsilon_{1}+2\sigma_{3}d\varepsilon_{3} \tag{6}\]  
The elastic energy released by unloading can be expressed as  
\[U^{e}=\int_{\varepsilon_{1}}^{\varepsilon_{1}}\sigma_{1}d\varepsilon+2\sigma_ {3}\left(\varepsilon_{3}-\varepsilon_{3}^{d}\right) \tag{7}\]  
where \(\varepsilon_{1}\) and \(\varepsilon_{3}\) are the axial and circumferential strains at unloading, respectively. \(\varepsilon_{1}^{d}\), \(\varepsilon_{3}^{d}\) are the axial and circumferential strains after unloading, respectively. The energy consumed for crack extension, development, internal damage, etc., in a sample is called dissipation energy \(U^{d}\) and can be expressed as  
\[U^{d}=U-U^{e}=\int_{\varepsilon_{1}}^{\varepsilon_{1}}\sigma_{1}d\varepsilon- \int_{\varepsilon_{1}}^{\varepsilon_{1}}\sigma_{1}d\varepsilon+2\sigma_{3} \left(\varepsilon_{3}^{d}-\varepsilon_{3}^{0}\right) \tag{8}\]  
where \(\varepsilon_{1}^{0}\), \(\varepsilon_{3}^{0}\) are the axial and circumferential strains at the start of loading, respectively. In general, the energy evolution of loaded samples goes through the processes of energy input, accumulation, dissipation, and release. Considering the reversibility of the elastic energy and the irreversibility of the dissipation energy, only the axial elastic energy and dissipation energy are systematically investigated.