## 3 Experimental Results and Discussions  
### Evolution Law of \(D_{\text{NT}}\)  
\(D_{\text{NT}}\) is defined as the initial damage before the loading process with a minimum freezing temperature \(T\) after \(N\) freeze-thaw cycles. If it is assumed that the rock is an isotropic material that is composed of a rock matrix and microcracks, the ultrasonic velocity can be used to define the damage variable [37,38]:  
\[D_{\text{NT}}=1-\left(\frac{V_{\text{NT}}}{V_{0}}\right)^{2} \tag{1}\]  
where \(V_{0}\) is the initial \(P\)-wave velocity of the rock, \(V_{\text{NT}}\) is the \(P\)-wave velocity of the rock at the lowest freezing temperature \(T\) after \(N\) freeze-thaw cycles, and \(D_{\text{NT}}\) is the damage variable of the rock. In order to study the effect of the number of cycles and the lowest freeze-thaw temperature on the \(P\)-wave velocity and damage after freezing and thawing, the \(P\)-wave velocity was measured after every 5 cycles, and the \(P\)-wave velocity at each freeze-thaw temperature was measured after 20 freeze-thaw cycles. The measurement results are shown in Tables 2 and 3. Due to the heterogeneity of the rock, in order to ensure the reliability of the experimental results, when the number of freeze-thaw cycles was \(N\) and the freeze-thaw temperature was \(T\) during the freezing and thawing process of the rock samples, the \(P\)-wave velocity values of at least three samples needed to be measured. Taking the freezing and thawing temperature of \(-20\)\({}^{\circ}\)C and 10 cycles as an example, using the average \(P\)-wave velocity of samples S-T1-10, S-T1-15, S-T1-20, S-T1-25, and S-T1-30 at the 10 th cycle as the \(P\)-wave velocity under this condition produced a relatively stable result. As for the condition of a freeze-thaw temperature of \(-20\)\({}^{\circ}\)C and 30 cycles, it was also necessary to measure the average of the \(P\)-wave velocity of three samples; therefore, samples S-T1-30-1 and S-T1-30-2 were added, and the sample with the \(P\)-wave velocity at the middle value was named S-T1-30. The measurement results are shown in Tables 2 and 3. In the subsequent uniaxial compression experiments, the samples that had the middle P-wave velocities were selected for the experiments. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample No. ** & **Lowest Freeze–Thaw Temperature (\({}^{\circ}\)C)** & **Total Cycles** & **Dry Weight** & **Saturated Weight** & **Saturation** \\  & & & **(g)** & **(g)** & **(\%)** \\ \hline S-T1-0 & – & 0 & 462.5 & 477.6 & 66.2 \\ S-T1-5 & \(-20\) & 5 & 462.7 & 477.7 & 65.8 \\ S-T1-10 & \(-20\) & 10 & 462.7 & 478.1 & 67.5 \\ S-T1-1  
According to Figure 4, the relationship between the P-wave velocity and the number of freeze-thaw cycles under the condition of the lowest freeze-thaw temperature of \(-20\)\({}^{\circ}\)C was found to be  
\[V_{N}=2.054\text{e}^{-6\times 10^{-3}N} \tag{2}\]  
According to Figure 5, the relationship between the \(P\)-wave velocity and the lowest freeze-thaw temperature when the number of freeze-thaw cycles was set to 20 was found to be  
\[V_{T}=1.864\text{e}^{1.8\times 10^{-3}T} \tag{3}\]  
Based on Equations (2) and (3), it can be roughly concluded that the \(P\)-wave velocity at any number of cycles and freeze-thaw temperature was  
\[V_{NT}=\frac{V_{N}}{V_{N=20}}V_{T}=\frac{2.054\text{e}^{-6\times 10^{-3}N}}{1.801 }1.864\text{e}^{1.8\times 10^{-3}T}=2.13\text{e}^{(1.8\times 10^{-3}T-6\times 10^{-3}N)} \tag{4}\]  
Equation (5) shows that with the increase in the number of freezing and thawing cycles, the \(P\)-wave velocity of the rock gradually decreases, which was also shown by many research results. In the study of [PERSON] [39], the \(P\)-wave velocity of granite gradually decreased with the increase in the number of freeze-thaw cycles. From the results, this attenuation trend roughly conformed to the exponential attenuation trend, but it did not give a specific fitting relationship. In the study of [PERSON] et al.