## 3 Results and Discussion
### Effect on the Mechanical Behavior
#### 3.2.2 Mechanical Behavior after F-T Action  
There have been many studies about the decline in the mechanical behavior with F-T damage in different rocks [33; 34]. This decreasing trend can be attributed to the development of cracks in the rocks that is caused by the F-T action. In this research, the peak stress for the tested sandstone sample also decreased with the increasing F-T action [35]. In Figure 11, there is a negative correlation between the peak stress and the F-T cycle number of the tested samples, with the sample under 5 F-T cycles showing a minor mechanical deterioration (about 6.9%), and the sample suffered 30 F-T cycles having a larger loss (about 49%) in the peak stress. [PERSON] and [PERSON] [36] argued that the combined effect of F-T action and loading aggravated the total damage of the rock with obvious non-linear characteristics, whereas its coupling effect weakened the total damage. Moreover, the variation in the stress path that was caused by the F-T cracks was vitally important on the peak strength decrease of the tested samples at high F-T damage.  
As shown in Figure 12, a schematic diagram of the freezing sequence of the'main-side branch' was developed based on the size of the pore structure. The F-T damage in sandstone is the result of a combination of capillary, crystal pressure, hydrostatic, and volumetric expansion mechanisms [13]. When the temperature drops below 0 \({}^{\circ}\)C, the main pores initially freeze as ice by heterogeneous nucleation due to their greater curvature [27]. At this time, the ice in the pores drives the unfrozen water to form seepage damage on the pore wall, which is known as the hydrostatic pressure mechanism. The volumetric expansion mechanism, on the other hand, is the damage that is caused by the volumetric expansion of the ice [9]. Since the chemical potential of the water in the small pores is higher than that of the pore ice in the main pores, the supercooled water from the small pores migrates through the pellicular water into the main pores, promoting further development of the structural damage in the pores [27]. The capillary mechanism plays an important role in this process. In addition, when the pore ice develops into secondary pores, the growth of the pore ice causes a hydraulic fracturing effect at the pore tips when the secondary pores are closed [5]. As the F-T effect increases, the damage to the pore structure in the sandstone increases. Thus, as mentioned above, there are significant differences in the physical and mechanical behavior of sandstone under loading.  
Figure 11: Change in the peak stress and loss rate with F-T action in the tested sandstone.  
Besides, the elastic modulus showed a negative correlation with the increasing number of F-T cycles (Figure 13a). This indicates a considerable reduction in the deformation resistance of the studied samples that was induced by the F-T process. Additionally, the F-T damage (\(D_{N}\)) was estimated using the empirical Equation [22]. The empirical Equation is as follows:  
\[D_{N}=1-\frac{E(N)}{E(0)} \tag{3}\]  
where \(D_{N}\) is the damage under the F-T action; \(E(0)\) is the elastic modulus of the fresh rock; and \(E(N)\) is the elastic modulus of the rocks that are subjected to \(N\) F-T cycles. The F-T damage for the tested samples exhibited an increase with increasing F-T cycle number (Figure 13b), and a prediction model was developed for damage changes.