## 3 Results and Analysis  
### Freeze-Thaw Strain  
Figure 2a shows the curve of the maximum and minimum strains measured in the freeze-thaw cycles as a function of the number of cycles. It can be seen from the figure that the strain values differ significantly between samples subjected to unidirectional freeze-thaw and those subjected to all-directional freeze-thaw. During unidirectional freeze-thaw, as the number of cycles increases, both the maximum and minimum values of freeze-thaw strain initially decrease before stabilizing. The minimum value corresponds to the freeze shrinkage strain of rock during the freeze-thaw cycle, which gradually stabilizes after approximately 25 unidirectional freeze-thaw cycles. At this stage, the influence of freeze-thaw cycling on the rock's frost shrinkage diminishes, and the water within the rock is redistributed during the cycles. The maximum value corresponds to the combined effect of pore-water frost-heave strain and rock frost shrinkage during the freeze-thaw cycle. When the unidirectional freeze-thaw cycle count reaches 30, the maximum freeze-thaw strain value stabilizes. At this point, pore expansion and micro-crack development within the rock matrix, influenced by pore water, are complete, and the rock sample's porosity increases, enabling it to accommodate the volume expansion of pore water during freeze-thaw cycles. In contrast, with an increasing number of all-directional freeze-thaw cycles, the maximum freeze-thaw strain value increases while the fluctuation in the minimum freeze-thaw strain decreases. After about 30 freeze-thaw cycles, both the maximum and minimum strains of the specimen increased significantly and the specimen cracked due to frost heaving. Following frost-heaving cracking, the maximum strain of the sample remains nearly constant with further freeze-thaw cycles. At this stage, frost heaving has induced macro-cracks due to freeze-thaw damage, and the expansion from the water-ice phase change during frost heaving cannot fill the cracks once macro-cracks have formed in the rock. As mentioned above, the measured freeze-thaw strain represents the total strain of rock samples during the freeze-thaw process. This total strain comprises four components (as illustrated in Figure 3): the temperature-induced strain of the protective coating, the temperature-induced strain of waterproof silicone rubber sealant, the temperature-induced strain of the rock itself, and the frost-heave strain. It can be expressed by the following equation:  
\[\varepsilon=\varepsilon_{\mathrm{s}}+\varepsilon_{\mathrm{c}}+\varepsilon_{ \mathrm{r}}+\varepsilon_{\mathrm{P}}\]  
where \(\varepsilon_{\mathrm{s}}\) is the temperature strain of structural adhesive, \(\varepsilon_{\mathrm{c}}\) is the temperature strain of waterproof silicone rubber, \(\varepsilon_{\mathrm{r}}\) is the temperature strain of rock matrix, and \(\varepsilon_{\mathrm{P}}\) is the strain caused by the participation of pore water. As mentioned above, the strain of dry red sandstone with a known thermal expansion coefficient can be measured using the quarter-bridge strain gauge configuration [21]. The frost-heaving strain during the freeze-thaw process is shown in Figure 2b. As can be seen from the figure, the minimum strain under unidirectional freeze-thaw is significantly reduced compared with the measured total strain, indicating that the frost shrinkage observed during unidirectional freeze-thaw is primarily caused by the temperature-induced strain of rock matrix and the structural adhesive. The maximum frost-heave strain under all-directional freeze-thaw is generally greater than that under unidirectional freeze-thaw throughout the 40 freeze-thaw cycles. With an increase in the number of freeze-thaw cycles, the maximum frost-heave strain of the sample fluctuates significantly in magnitude. After 30 freeze-thaw cycles, both the maximum and minimum strain values gradually stabilized following a brief period of fluctuation.