## 3 Experimental Results and Discussions
### Variation Law of Characteristic Stress  
A large number of studies on the failure mechanism of rock samples showed that the rock samples experienced four stages during the loading process before reaching the peak stress, that is:  
1. The crack closure stage, in which the original existing microcracks in the rock are closed and the stress-strain curve shows a concave shape. The crack closure stress is \(\sigma_{\text{cc}}\), which can be determined from the point where the crack volumetric strain starts to be zero. 2. The elastic deformation stage, in which the stress increases linearly with strain and there are no new cracks generated until the stress increases to the initial crack stress \(\sigma_{\text{ci}}\). \(\sigma_{\text{ci}}\) can be determined according to the point where the crack volumetric strain deviates from zero. 3. The stable crack growth stage in which crack propagation begins to appear inside the rock and \(\sigma_{\text{ci}}\) is taken as the characteristic stress at the beginning of rock damage. The nonlinear increase of lateral strain is the main characteristic of the rock in this stage, which indicates that the cracks produced in this stage are mainly tensile cracks. 4. When the stress exceeds \(\sigma_{\text{cd}}\), the rock reaches its unstable crack growth stage in which the crack propagation speed is obviously accelerated, accompanied by the interaction between multiple cracks. In many studies, \(\sigma_{\text{cd}}\) is seen as the long-term strength of rocks and its value can be determined from the point where maximum volumetric strain appears [29; 39]. In practical engineering, \(\sigma_{\text{ci}}\) and \(\sigma_{\text{cd}}\) are two important characteristic stresses, which are used to predict the short-term and long-term strength of a rock mass. The crack propagation process of rock and the characterization method of characteristic stress are shown in Figure 7.  
Figure 6 shows the specific method to determine the characteristic stress. It was demonstrated that using characteristic stress is a reliable method to judge the damage and stability of rocks. To determine the characteristic stress of rocks, it is first necessary  
Figure 7: Schematic diagram of a rock stress-strain curve and corresponding stress threshold [29; 40; 41; 42]. to calculate the volumetric strain and crack volumetric strain during the loading process. For rock under uniaxial compression, the volume strain and crack volume strain can be calculated using Equation (7):  
\[ \begin{array}{c}\varepsilon_{V}=\varepsilon_{\text{axial}}+2\varepsilon_{\text {lateral}}\\ \varepsilon_{V}^{\epsilon}=\frac{1-2\mu}{E}\sigma_{1}\\ \varepsilon_{V}^{\epsilon}=\varepsilon_{V}-\varepsilon_{V}^{\epsilon}\end{array}  \tag{7}\]  
where \(\sigma_{1}\) is the axial stress, \(\varepsilon_{\text{axial}}\) is the axial strain, \(\varepsilon_{\text{lateral}}\) is the lateral strain, \(\varepsilon_{V}\) is the volumetric strain, \(\varepsilon_{V}^{\epsilon}\) is the crack volumetric strain, \(\varepsilon_{V}^{\epsilon}\) is the elastic volumetric strain, \(E\) is the elastic modulus, and \(\mu\) is Poisson's ratio. After a series of uniaxial tests, the stress-strain curve of each sandstone sample was obtained, as can be seen from Figure 8.  
According to the method of determining the characteristic stress of the rock shown in Figure 7, using Equation (7) and the stress-strain curves of the sandstone samples, the characteristic stress of the sandstone samples after the treatment of different freeze-thaw cycles and the lowest freeze-thaw temperature was obtained, as shown in Figure 9.  
Figure 8: Stress–strain curves of sandstone samples: (**a**) with different freeze–thaw cycles (\(T=-20\)\({}^{\circ}\)C); (**b**) with different freeze–thaw temperatures (\(N=20\)). Figure 9: Characteristic stress of the sandstone samples: (**a**) with different freeze–thaw cycles (\(T=-20\)\({}^{\circ}\)C); (**b**) with different freeze–thaw temperatures (\(N=20\)).