## 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  
Taking the uniaxial compressive strength as rock integrity index, Equation (1) can be used to obtain decay index \(\lambda\):  
\[\lambda=\frac{1}{\text{N}}\ln\frac{I_{0}}{I_{\text{N}}}=\frac{1}{\text{N}}\ln \frac{\text{UCS}_{0}}{\text{UCS}_{\text{N}}} \tag{3}\]  
where, UCS\({}_{0}\) is the uniaxial compressive strength before freeze-thaw cycling; UCS\({}_{\text{N}}\) is the uniaxial compressive strength of the rock undergoing N freeze-thaw cycles. The measurement of uniaxial compressive strength of fine and coarse sandstone after the different number of freeze-thaw cycles and \(\lambda\) obtained by Equation (3) are listed in Table 5.  
It can be seen from Table 5 that with the increase in the number of freeze-thaw cycles, \(\lambda\) continues to change with a certain varying law. As far as the fine sandstone is concerned, \(\lambda\) continues to decrease, indicating that the integrity loss caused by the single freeze-thaw is rather serious in the initial freeze-thaw cycles; with the increase in the number of freeze-thaw cycles, the integrity loss caused by a single freeze-thaw gradually decreases and tends to become a certain constant. For the coarse sandstones, however, \(\lambda\) appears to decrease first and then increase with an increase in the times of freeze-thaw cycles. The following analysis focuses on fine sandstone. Substituting the obtained \(\lambda\) for freeze-thaw cycles of N = 30, 60, 90 and UCS\({}_{0}\) = 114.8 MPa into Equation (1), we can obtain the corresponding decay equations in the following:  
\[\text{UCS}_{\text{N}}=114.8e^{-0.005\text{N}} \tag{4}\]  
\[\text{UCS}_{\text{N}}=114.8e^{-0.0047\text{N}} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sandstone Variety** & **Time of F–T Cycles Cycles** & **UCS (MPa)** & **Peak Value Strain (\%)** & **Elastic Modulus (GPa)** & **Poisson Ratio** \\ \hline \multirow{5}{*}{Fine} & 0 & 114.8 & 0.573 & 19.28 & 0.21 \\  & 30 & 98.8 & 0.582 & 16.57 & 0.21 \\  & 60 & 86.7 & 0.649 & 12.08 & 0.2 \\  & 90 & 79.6 & 0.674 & 10.01 & 0.2 \\  & 120 & 72.3 & 0.741 & 8.16 & 0.21 \\ \hline \multirow{5}{*}{Coarse} & 0 & 104.1 & 0.672 & 18.2 & 0.29 \\  & 30 & 90.1 & 0.68 & 15.1 & 0.28 \\ \cline{1-1}  & 60 & 83.9 & 0.786 & 11.3 & 0.29 \\ \cline{1-1}  & 90 & 74.2 & 0.813 & 8.2 & 0.28 \\ \cline{1-1}  & 120 & 64.4 & 0.862 & 6.9 & 0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of sandstone specimen uniaxial compressive strength tests. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sandstone Variety** & **Time of F–T Cycles** &
\begin{tabular}{c} **Real Measured** \\ **UCS (MPa)** \\ \end{tabular} & **Decay Index \(\lambda\)** \\ \hline \multirow{5}{*}{Fine} & 0 & 114.8 \\  & 30 & 98.8 \\ Fine & 60 & 86.7 \\  & 90 & 79.6 \\  & 120 & 72.3 \\ \hline \multirow{5}{*}{Coarse} & 0 & 104.1 \\  & 30 & 90.1 \\ \cline{1-1}  & 60 & 83.9 \\ \cline{1-1}  & 90 & 74.2 \\ \cline{1-1}  & 120 & 64.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: \(\lambda\) corresponding to the different freeze–thaw cycles. \[\text{UCS}_{\text{N}}=114.8\text{e}^{-0.0041\text{N}} \tag{6}\]  
The above decay development by Equations (4)-(6) and the measured uniaxial compressive strength are illustrated in Figure 3.  
It can be seen from Figure 3 that if Equation (2) is used to calculate \(\lambda\) of 30, 60 and 90 freeze-thaw cycles, then the equation and these 3 different \(\lambda\) values are used to calculate the uniaxial compressive strength after 120 cycles, \(\text{UCS}_{120}\), we obtain three different values of 63 MPa, 65 MPa and 70.5 MPa, respectively. The differences from the measured \(\text{UCS}_{120}\) are 9.3 MPa, 6.8 MPa and 1.8 MPa, respectively.