## 3 Experimental Results and Discussions
### Variation Law of Characteristic Stress
It can be seen from Figure 9 that the initial crack stress (\(\sigma_{\rm ci}\)), crack damage stress (\(\sigma_{\rm cd}\)), and peak strength (\(\sigma_{\rm p}\)) of sandstone samples gradually decreased with the increase in the number of freeze-thaw cycles and the decrease in the freeze-thaw temperature. In addition, there was no obvious variation law of crack closure stress (\(\sigma_{\rm cc}\)). In order to further study the variation law of characteristic stress of the sandstone samples, the variation trend of the characteristic stress with the damage variable \(D_{NT}\) was analyzed, where the results can be seen in Figure 10.  
It can be seen from Figure 10 that the initial crack stress (\(\sigma_{\rm ci}\)), crack damage stress (\(\sigma_{\rm cd}\)), and peak strength (\(\sigma_{\rm p}\)) of the sandstone samples decreased significantly with the increase in the damage factor \(D_{NT}\). The law of characteristic stress of the sandstone samples as a function of \(D_{NT}\) can be described as follows:  
\[\sigma_{\rm ci}=-33.675\rm{D}_{NT}+29.609\atop\sigma_{\rm cd}=-48.690\rm{D}_{ NT}+45.191\atop\sigma_{\rm p}=-39.671\rm{D}_{NT}+54.590 \tag{8}\]  
Combining the Equations (6) and (8), we obtained the characteristic stresses of the sandstone samples as a function of the number of freeze-thaw cycles \(N\) and the lowest freeze-thaw temperature \(T\), which can be seen as follows:  
\[\sigma_{\rm ci}=33.675\rm{e}^{(3.6\times 10^{-3}T-1.2\times 10^{-2}N)}-4.066\atop \sigma_{\rm cd}=48.690\rm{e}^{(3.6\times 10^{-3}T-1.2\times 10^{-2}N)}-3.499\atop \sigma_{\rm p}=39.671\rm{e}^{(3.6\times 10^{-3}T-1.2\times 10^{-2}N)}+14.919 \tag{9}\]  
Figure 10: The relationship between the characteristic stress of the sandstone samples and \(D_{NT}\): (**a**) initial crack stress \(\sigma_{\rm ci}\); (**b**) crack closure stress \(\sigma_{\rm cd}\); (**c**) peak strength \(\sigma_{\rm p}\). From the experimental results of Tan [4], the uniaxial compression strength of granite samples with freeze-thaw cycles can be expressed by  
\[V_{N}=55.28+80.15\text{e}^{-2\times 10^{-2}N} \tag{10}\]  
It can be seen that Equations (9) and (10) have similar expressions, which shows the validity of using the initial damage to fit the rock strength formula. In addition, the influence of the freeze-thaw temperature is incorporated into Equation (9). Figure 11 describes the mechanism of crack damage in the rock after the freeze-thaw treatment, from which we can see that the length and volume of internal microcracks of rock gradually increases. \(\sigma_{\text{cc}}\) is related to the closure of microcracks in the rock. It seems that the crack closure stress must gradually increase as the damage intensifies. However, the crack closure stress is also related to the shape of the crack. For example, for an elliptical crack, if the increase in crack volume is mainly along the long axis, and the increase in crack volume increases the ratio of the long axis to the short axis of the crack, then the increase in crack volume does not necessarily lead to an increase in crack closure stress. Studies on the initial crack stress \(\sigma_{\text{ci}}\) have been widely discussed. Based on experimental observations, [PERSON] et al. [43] obtained that in the middle stage of the compression test, only the lateral nonlinear strain of the rock changes rapidly, and the axial deformation of the cylindrical sample still maintains a linear change, indicating that the initial damage in the process of rock failure is caused by the extension of tensile cracks. [PERSON] et al. [30] pointed out that the initial damage of rock during tunnel excavation is caused by the propagation of a tensile wing crack. [PERSON] and [PERSON] [44] introduced in detail the propagation process of wing cracks in a rock under compression stress in their research and deduced the theoretical model of initial crack stress \(\sigma_{\text{ci}}\) under triaxial compression. The theory of [PERSON] and [PERSON], which can be seen in Figure 12, has been adopted by many subsequent studies [29; 30; 45].