## 2 Materials and Methods  
### Indoor Testing Program  
The rock is sandstone, taken from a mine. A crack is machined in the center of the entire rock sample. The crack runs through the sample, inclines 45\({}^{\circ}\), is 2 mm long, and 0.2 mm wide. The rock samples were then divided into 5 groups, namely, freeze-thaw 0 times (i.e., without freeze-thaw treatment), 20 times, 40 times, 60 times, and 80 times groups, to simulate sandstones with different degrees of freezing in the natural environment. A freeze-thaw tester is used to freeze-thaw sandstone specimens. The specific operation of freeze-thaw is as follows: the sandstone specimens are frozen in the air, and after the freezing is completed, the specimens are immersed in water to warm up and melt slowly. The freezing temperature is set to \(-20\)\({}^{\circ}\)C, and the melting temperature is set to 20 \({}^{\circ}\)C. There are 4 stages in a freeze-thaw cycle. In the first stage, the sandstone specimens were slowly cooled to \(-20\)\({}^{\circ}\)C. In the second stage, the specimens were frozen at \(-20\)\({}^{\circ}\)C for 4 h to ensure they were fully frozen. In the third stage, the water in the freeze-thaw tester tank was returned to the freeze-thaw box to submerge the sandstone specimens so that they were slowly warmed up to 20 \({}^{\circ}\)C. In the fourth stage, the temperature was kept at 20 \({}^{\circ}\)C for 4 h to allow the sandstone specimens to be fully thawed. The above process is a complete freeze-thaw cycle, repeated until the preset number of freeze-thaws is reached. The temperature change process is shown in Figure 1.  
Figure 2 shows the damage pattern of some sandstones after being subjected to freeze-thaw cycles. Figure 1: Temperature changes during freeze\(-\)thaw. Figure 2: Damage pattern of rock samples after freeze–thaw. After the freeze-thaw cycle test, the sandstones were dried using a drying oven. Then, the mass of the sandstone specimens was measured using an electronic balance, and the sandstones' wave velocity was measured using a wave velocity meter. The measuring instruments used are shown in Figure 3.  
The sandstone wave velocity was calculated by Equation (1). \[V=\frac{L}{t_{p}-t_{0}} \tag{1}\]  
where \(V\) is the longitudinal wave velocity, [m/s];  
\(L\) is the gap between the center points of the transmitting and receiving transducers, [m];  
\(t_{p}\) is the propagation time of the longitudinal wave in the specimen, [s];  
\(t_{0}\) is the zero delay time of the instrument system, [s]. To assess the effect of sandstone by the freeze-thaw cycles, two metrics, Mass Damage Factor (D\({}_{\text{m}}\)) and Wave Velocity Damage Factor (D\({}_{\text{v}}\)), were selected to measure the changes in sandstone specimens. These two indicators are calculated as shown in Equations (2) and (3). \[D_{m}=\frac{\overline{m_{0}}-\overline{m_{t}}}{\overline{m_{0}}} \tag{2}\]  
\[D_{v}=\frac{\overline{V_{0}}-\overline{V_{t}}}{\overline{V_{0}}} \tag{3}\]  
where \(D_{m}\) is the Mass Damage Factor;  
\(\overline{m_{0}}\) is the average mass of the sandstone without freeze-thaw and \(\overline{m_{t}}\) is the average mass after the freeze-thaw cycle, [kg];  
\(D_{v}\) is the Wave Velocity Damage Factor;  
\(\overline{V_{0}}\) is the average wave velocity of the sandstone without freeze-thaw and \(\overline{V_{t}}\) is the average wave velocity after the freeze-thaw cycle, [m/s]. Before the cyclic loading test starts, the first task is to clarify the cyclic loading methods. Through reading the references [21; 22; 23; 24; 25; 26; 27], we learned that there are various cyclic loading methods. The main methods are equal amplitude, graded, constant lower limit, variable lower limit cyclic loading and unloading, and so on. To simulate the change rule of the load  
Figure 3: Wave velocity measuring instruments. on the roadbed during regular use, we adopted the constant lower limit cyclic loading and unloading method to study sandstone's mechanical properties and damage mechanism.