## 1 Introduction  
Investigations conducted in mountainous areas such as the Alps and the South-North Polar regions [1; 2] have revealed the detrimental effects of long-term freeze-thaw cycling on rocks. This natural weathering process leads not only to the rupture of surface rocks and the accumulation of debris but also increases the occurrence of hazardous events such as rock falls and landslides. Over an extended period, in conjunction with other weathering processes, these phenomena contribute to ongoing changes in the topography and geomorphology of cold regions [3]. Furthermore, during engineering construction activities in cold areas, various rock elements, including rock side-slopes, tunnel-wall rocks, and rock materials used for construction purposes (e.g., carvings, wall surfaces, and ground linings), are susceptible to damage caused by freeze-thaw cycling [4; 5; 6; 7; 8; 9]. Therefore, gaining a comprehensive understanding of the alterations in the physical and mechanical properties of rocks subjected to freeze-thaw cycles is crucial not only for elucidating the evolution of topography and geomorphology but also for facilitating effective engineering construction practices in cold regions. Since the 1950s, many experiments of rocks under F-T cycling have been carried out [10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. The results further confirmed that the physical and mechanical properties of rocks deteriorate with F-T cycles. Based on the results of 10 different rocks subjected to F-T cycling, [PERSON] [20] found that if Shore hardness (SH) was the rock integrity index, there is a relationship between the rock integrity index and the number of F-T cycles. \[I_{N}=I_{0}e^{-\lambda N} \tag{1}\]  
where, \(\lambda\) is the decay index, \(I_{0}\) is the rock initial integrity, and \(I_{\text{N}}\) is the integrity after N cycles of freeze-thaw (F-T). Since then, scholars [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31] have adopted different physical and mechanical indices of various rocks under F-T cycles as integrity to validate this equation. Partial data are summarized in Table 1. Considering that UCS is a more meaningful mechanical indicator for engineering purposes, this study uses UCS as integrity. When determining \(\lambda\), the usual method is to measure the integrity before and after it is subjected to different F-T cycles, and then based on Equation (1), regression analysis is used to obtain the \(\lambda\)[21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. Another approach is to establish a relationship between the initial physical and mechanical indices of the rock and the decay index and use this to estimate the decay index. For example, [PERSON] [32] calculated the decay index based on the initial integrity and porosity, while [PERSON] et al. [33] calculated the decay index based on the initial porosity, elastic modulus, and Brazilian tensile strength (BTS).There is an implicit prerequisite, i.e., \(\lambda\) is considered as a constant in the whole process of F-T cycling, which does not conform to the observed phenomena in the experiments [12; 34]. It is reasonable to consider that \(\lambda\) varies with the number of F-T cycles. The relationships between \(\lambda\) and the F-T cycle numbers are likely to be different. To investigate the effects of long-term freeze-thaw cycles on the physical and mechanical properties as well as the attenuation trend of rocks, the present authors investigated the changes in physical and mechanical properties of two sandstones subjected to F-T cycling. Taking uniaxial compressive strength (UCS) as the integrity, the decay equation and the decay index of the rock integrity in terms of F-T cycling were studied. For fine sandstone, we observed that the decay index gradually decreases with an increase in the number of freeze-thaw cycles. Furthermore, it exhibits a good fitting relationship with the longitudinal wave velocity. This relationship provides a more straightforward way to evaluate the integrity of fine sandstones subjected to different numbers of freeze-thaw cycles.