## 4 Conclusions  
In this study, by changing the number of freeze-thaw cycles and temperature, we carried out a series of experiments on coarse sandstone samples under uniaxial compression. The initial damage \(D_{NT}\) and the crack propagation characteristics of sandstone samples after freeze-thaw treatment were obtained, which may have a certain significance for understanding rock mechanics properties in cold regions. From this research, the following conclusions were drawn:  
1. The increase in the number of freeze-thaw cycles and the decrease in temperature aggravated the initial damage of the rock. \(D_{NT}\) increased exponentially with the increase in the number of freeze-thaw cycles and the decrease in freeze-thaw temperature, where the quantitative relationship between them is reflected by Equation (4).
2. The characteristic stress of the rock had a good corresponding relationship with the initial damage \(D_{NT}\). The initial crack stress (\(\sigma_{\text{ci}}\)), crack damage stress (\(\sigma_{\text{cd}}\)), and peak stress (\(\sigma_{\text{p}}\)) of the rock after the freeze-thaw treatment decreased linearly with the increase in \(D_{NT}\). The reason for this phenomenon was that as the initial damage \(D_{NT}\) increased, the length of the original microcracks in the rock increased.
3. In cold regions, such as tunnel excavation and other projects, rocks that have been damaged by freezing and thawing conditions for a long time are more prone to produce tensile cracks. Because the tensile strength of rocks is much smaller than the shear strength, the stability of such engineering rock masses needs to be paid more attention and certain measures should be taken to prevent potential disasters.  
Conceptualization, K.S. and Z.C.; methodology, Z.C.; software, K.S.; validation, K.S., Z.C. and R.P.; formal analysis, Z.C.; investigation, Z.C.; resources, R.P.; data curation, Z.C.; writing--original draft preparation, K.S.; writing--review and editing, Z.C.; visualization, R.P.; supervision, L.Z.; project administration, Y.Z.; funding acquisition, Z.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Second Tibetan Plateau Scientific Expedition and Research Program (STEP) (grant no. 2019 QZKK0904), the National Natural Science Foundation of China (grant no. 51727807), and the Fundamental Research Funds for the Central Universities (grant no. 2020 YSMT06).  
Not applicable.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Crack Propagation Stage** & **Tensile Crack (\%)** & **Mixed Crack (\%)** & **Shear Crack (\%)** \\ \hline Crack closure stage & 27.2 & 32.1 & 40.7 \\ Elastic stage & 36.6 & 16.7 & 46.7 \\ Stable crack growth stage & 38.0 & 21.1 & 26.7 \\ Unstable crack growth stage & 54.3 & 26.3 & 19.2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Different crack types in the process of rock failure of sample S-T1-0.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Crack Propagation Stage** & **Tensile Crack (\%)** & **Mixed Crack (\%)** & **Shear Crack (\%)** \\ \hline Crack closure stage & 24.4 & 30.6 & 45.0 \\ Elastic stage & 28.6 & 9.5 & 61.9 \\ Stable crack growth stage & 53.2 & 13.0 & 33.8 \\ Unstable crack growth stage & 68.1 & 10.1 & 21.8 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Different crack types in the process of rock failure of sample S-T1-20.