## 5 Conclusions  
The present study investigates the deformation behavior of red sandstone with cracks at different depths during freeze-thaw cycles, focusing on the variations in deformation with time and temperature. The freezing-thawing deformation evolution mechanism of cracked red sandstone is analyzed. The main findings are as follows:  
1. The strain variation process of cracked red sandstone during a single freeze-thaw cycle can be roughly divided into five stages: initial contraction stage, stable contraction stage, continuous contraction stage, thermal expansion stage, and stabilization stage. The strain variation trend can be described as follows: during the cooling and freezing process, the strain in the cracked red sandstone gradually decreases to a minimum value and then stabilizes. During the heating and thawing process, the strain gradually increases to a maximum value, after which it decreases and stabilizes, resulting in residual strain. 2. The damage to cracked red sandstone worsens during multiple freeze-thaw cycles. Residual strain, representing plastic deformation, is generated in each freeze-thaw cycle. As the number of freeze-thaw cycles increases, residual strain accumulates, leading to significant expansion deformation and irreversible freeze-thaw damage. The cumulative residual strain of the rock specimen after 45 freeze-thaw cycles was 40.69 times greater than the residual strain from the first cycle. This expansion deformation is severe, manifesting as the extension, increase, and interconnection of rock pores and microcracks. 3. The freeze-thaw strain characteristic values of cracked red sandstone include cold contraction amplitude and thermal expansion amplitude, and these values are clearly correlated with crack depth. For the monitoring point on the long edge of the 40 mm depth cracked rock specimen, the average cold contraction amplitude and average thermal expansion amplitude are 1.47 times and 1.65 times those of the 20 mm depth cracked rock specimen, respectively. During freeze-thaw cycles, the rock skeleton particles contract due to the cold. Since the pre-existing crack lacks part of the rock matrix internally, the points at the surface of the pre-existing crack have a slightly higher degree of freedom and are less constrained by the surrounding rock matrix. The deeper the crack, the less constraint at the surface, resulting in more pronounced deformation. 4. The freeze-thaw deformation evolution mechanism of cracked red sandstone can be summarized into three points: (i) body deformation caused by the thermal expansion and contraction of rock skeleton particles; (ii) structural deformation caused by frost heave force; and (iii) structural deformation caused by damage to the cementingmaterials. This freeze-thaw deformation mechanism provides a valuable reference for the study of the stability and durability of underground rock engineering in cold regions. This study presents a monitoring method for freeze-thaw deformation, which can be applied to the deformation monitoring of rock underground engineering in cold regions. Additionally, the results of this research provide data references and a mechanistic basis for freeze-thaw damage and deformation in practical underground engineering projects in cold regions. Conceptualization, X.H. and G.S. ; formal analysis, H.F.; data curation, X.H., C.T. and Z.H. ; investigation, P.Y. and Q.Z. ; supervision, W.F. and S.Y. ; writing-original draft preparation, X.H. ; writing--review and editing, G.S. ; funding acquisition, G.S. All authors have read and agreed to the published version of the manuscript.