## 3 Results and Discussion  
### Effect of the Minimum Temperature  
From the tunnel entrance to a certain depth, the temperature amplitude inside the tunnel gradually decreases. Based on the fitting function of the temperature near the tunnel site in Equation (9), the freezing cycle and distribution of frost heaving force are calculated under the four minimum temperatures of \(-8\)\({}^{\circ}\)C, \(-10\)\({}^{\circ}\)C, \(-12\)\({}^{\circ}\)C, and \(-14\)\({}^{\circ}\)C, with changing the temperature amplitude, and the frost heave ratio is 0.2%. Figure 4 shows the distribution characteristics of the tunnel temperature field on the coldest day under different minimum negative temperatures. The distribution characteristics of the temperature near the tunnel lining are basically the same, i.e., as the magnitude increases, the temperature of the lining and the surrounding rock on the coldest day decreases; but, simultaneously, the temperature of the surrounding rock around the lining is slightly higher due to the influence of the maximum temperature in the warm season. Figure 4: Cloud diagram of the tunnel temperature field on the coldest day under different minimum temperatures. The method for calculating the freezing cycle and the frost heaving force is as follows. First, the initial stress field is calculated, and the transient temperature is calculated with the temperature boundary condition to determine the thickness of the freezing cycle. Then, the given thermal expansion coefficient of the freezing cycle is comparable to the frost heave ratio, thus enabling the simulation of the frost heaving force. In thermo-mechanical coupling, sequential coupling is adopted; that is, the temperature field affects the stress field, and the stress field calculation is realised based on the temperature field. To explore the changes in the freezing cycle, the distribution of the freezing cycle on the coldest day extracted under different minimum temperatures is shown in Figure 5. The distribution of the freezing cycles around the tunnel lining is uneven; therefore, the inhomogeneity of the thickness distribution of the freezing cycle should be considered when utilising the overall frost heave model of the freeze-thaw cycle to reasonably determine the freezing range. Among them, the freezing cycle of the top arch is relatively uniform and has the largest thickness, while the freezing cycle of the inverted arch is uneven. From the smallest minimum temperature to the largest minimum temperature, the freezing depth of the tunnel vault is 0.95 m, 1.14 m, 1.25 m, and 1.39 m, and the freezing depth of the side wall is 0.79 m, 0.98 m, 1.09 m, and 1.23 m. As the temperature continues to decrease, the scope of the freezing cycle gradually expands, and from \(-8\,^{\circ}\)C to \(-14\,^{\circ}\)C, the average increase in the freezing depth of the arch top is 0.147 m/\({}^{\circ}\)C.  
The temperatures at the tunnel vault position along the outer normal direction are extracted under each minimum air temperature, and the results are shown in Figure 6. As the minimum air temperature decreases, the temperature at the same position near the lining  
Figure 5: Distribution of the freezing cycle on the coldest day under different negative minimum temperatures. gradually decreases, but this trend becomes increasingly less obvious as the depth increases; the temperatures are basically the same after approximately 3 m of depth. This indicates that the effect of external temperature changes on the surrounding rock temperature is limited. After a certain depth, the surrounding rock temperature is mainly controlled by the stable geothermal gradient and is no longer disturbed by external temperature fluctuations, thus remaining relatively constant. The frost heaving force acting on the periphery of the tunnel lining can be divided into two components: the normal stress and the tangential stress. However, in general, the normal stress plays an important role in the frost heaving force [31; 32].