## 3 Results and Discussion
### Analysis of Numerical Simulation Results
#### 3.2.2 Energy Analysis  
Under cyclic loading, sandstones with flaws can develop behaviors such as crack closure, sprouting, and expansion. These processes can significantly affect the energy response characteristics of sandstone [34]. There are no parameters in the PFC that directly characterize the energy changes. By comprehensively analyzing the changes in Porosity and Contact Force, we can approximate the complex energy evolution within the model. Therefore, during the simulation, monitoring points were set up inside the model to record the Porosity change and Contact Force change inside the sandstone. Figure 15 shows the schematic layout of the three monitoring points inside the model. The change curves of Porosity and Contact Force throughout the freeze-thaw process are plotted (Figure 16) based on the data from the monitoring points. Of the three monitoring points, measurement point 1 has a larger initial Porosity because it covers the prefabricated crack. Measurement point 2 and measurement point 3 more realistically reflect the initial pore state. As shown in Figure 16, the Porosity trend is related to the freeze-thaw cycles. At the beginning of the freeze-thaw period, the Porosity decreases slowly as the freeze-thaws increase. Subsequently, after reaching a certain point (50 th freeze-thaw), the Porosity increased rapidly. In addition, there are also small periodic fluctuations in the change of Porosity, a phenomenon closely related to the periodic change of temperature. When the model temperature increases, the particles squeeze each other, resulting in the Porosity decreasing. When the model temperature decreases, the inter-particle pore space increases, and the Porosity increases accordingly. The model's Porosity increased significantly between the 50 th and 55 th freeze-thaw, jumping from 15.9% to 17.2%, an amplification of 1.3%. However, from the 55 th to the 80 th freeze-thaw, Porosity increased from 17.2% to 17.5%, an amplification of 0.3%, much lower than the 1.3% amplification. It proves that that a significant structural change occurred within the model at around 50 freeze-thaw cycles, which occurred rapidly and stabilized over a short period. The Porosity fluctuation of measuring point 2 and measuring point 3 is more drastic. This indicates that the model's Porosity change close to the surface is more prominent, i.e., the region close to the surface is more affected by the freeze-thaw cycle. Figure 15: Layout of measurement points. The Contact Force increases continuously throughout the freeze-thaw process. As illustrated by the dotted coil labeling in Figure 15, substantial fluctuations in Contact Force and Porosity transpire in a near-simultaneous fashion. Specifically, an increase in temperature results in tighter squeezing of the particles against each other and an increase in Contact Force due to the squeezing. Conversely, when the temperature decreases, the extrusion between the particles reduces, and the Contact Force decreases. The Contact Force maintains minor periodic fluctuations during the first 50 freezes and thaws. However, at the 50 th freeze-thaw, there is a considerable fluctuation in the Contact Force, with the maximum value reaching 6310 Pa and the minimum value plummeting to 821 Pa, an abnormal change far beyond the previous fluctuation range. The Porosity and Contact Force shows that at the beginning of the freeze-thaw period, the freezing and swelling forces within the model did not destroy the cementation between the particles. The freezing and swelling effect gradually accumulates energy inside the model. When it reaches a certain threshold, the cementation between the particles begins to fail in a large amount, followed by a rapid release of energy. This process is manifested in the violent fluctuation of the Contact Force curve. Between the two sharp fluctuations in Contact Force, there was a relatively stable period of fluctuation, which was the accumulation stage before the next concentrated release of energy.