## 3 Results and Discussion
### Rock Integrity Decay Laws Based on Uniaxial Compressive Strength
#### 3.4.1 Decay Index \(\lambda\) under the Different Times of Freeze-thaw Cycles
A better prediction for rock integrity after a certain cycle number M can be made by \(\lambda\) obtained from another cycle N closer to M. If the decay equation is characterized with the obtained \(\lambda\) in the 30 cycles, the obtained uniaxial compressive strength in the extrapolation N = 120 freeze-thaw cycles is 13% less than the measured value, i.e., the integrity loss after undergoing 120 freeze-thaw cycles is considerably overestimated. Taking the uniaxial compressive strength of the marble [42] after different freeze-thaw cycles as an example, the results are illustrated in Figure 4.  
Figure 4: Different \(\lambda\) corresponding to marble integrity decay curves and measured UCS. Figure 3: Different \(\lambda\) corresponding to fine sandstone integrity decay curves and measured UCS. It can also be noticed from Figure 4 that better prediction for rock integrity after a certain cycle number M can be made by a \(\lambda\) obtained from another cycle N closer to M. If the decay equation is characterized with the obtained \(\lambda\) in the 50 times of freeze-thaw cycles, the obtained uniaxial compressive strength in the extrapolation N = 150 times of freeze-thaw cycles is 45% less than the measured value, i.e., the integrity loss after undergoing 150 times freeze-thaw cycles is considerably overestimated. The above analysis indicates that \(\lambda\) in the decay Equation (1) is not a constant but rather changes with the increase in the number of freeze-thaw cycles. When the rock integrity degree is characterized with the uniaxial compressive strength, \(\lambda\) (of fine sandstone and marble) decreases with the increase in number of freeze-thaw cycles, and gradually tend to be a constant. Therefore, when the obtained \(\lambda\) from a small number of freeze-thaw cycles is substituted into the decay equation, there will be a larger discrepancy between the calculated and measured results for the integrity after a larger number of cycles. The loss of the integrity is overestimated. The decay parameter that in previous studies is considered as a constant might only be reasonable for small number of freeze-thaw cycles. What is worth noticing is that changes in the laws of \(\lambda\) with the number of freeze-thaw cycles vary with different rocks and, accordingly, micro-structures. In our testing program, for coarse sandstone, \(\lambda\) decreased first and then increased; while the \(\lambda\) of andestite [11] basically remains unchanged in the process of freeze-thaw cycles. The varying law can be different for different rocks, but it is also influenced by factors such as the structure components, grain shapes, pores and micro-cracks, etc. 4.2 Relationship of the Decay Parameter versus Porosity and Longitudinal Wave Velocity of the Sandstone  
Taking UCS as the integrity index, it is necessary to perform freeze-thaw cycling and compression tests to obtain \(\lambda\); however, it is rather time- and labor-intensive and costly. It is therefore desirable to relate \(\lambda\) with some easily obtained indexes, so that the decay equation can be used to predict the UCS of rocks subjected to certain freeze-thaw cycles. The fine sandstone \(\lambda\) decreases with the increase in the number of freeze-thaw cycles, which implies that greater integrity loss is caused by a single freeze-thaw cycle in the initial cycles; with the increase in number of freeze-thaw cycles, the integrity loss caused by a single cycle gradually decreases until it becomes a constant. At the same time, it can be seen from Tables 2 and 3 that, with the increase in freeze-thaw cycles, the porosity continues to increase and the longitudinal wave velocity decreases. This implies certain connections between these indexes.