## 4 Numerical Analysis of the Slope Stability of Coarse-Grained Soil Based on COMSOL Multiphysics
### The Temperature Variation of the Slope's Soil Under the Action of FT Cycles  
The temperature variation of the surface soil under FT cycles and the affected depth are of significant importance for the study of coarse-grained soil slopes. A variable temperature with 24 h FT cycles is applied to the surface of the slope model with an initial temperature of 10 \({}^{\circ}\)C. The temperature extremes are the same as the previous experimental conditions. The temperature distribution contour maps of the slope under different FT cycle counts are shown in Figures 15-18.  
Figure 14: Model of the coarse-grained soil slope.  
Figure 15: Temperature of the slope model after 15 FT cycles. (**a**) High environmental temperature state and (**b**) low environmental temperature state.  
According to Figure 15, after 15 FT cycles, the minimum temperature of the slope soil reaches \(1.48~{}^{\circ}\)C under high environmental temperature conditions and \(-2.16~{}^{\circ}\)C under low environmental temperature conditions. At this point, the depth of the soil affected by temperature changes does not exceed \(1~{}\mathrm{m}\), and the affected layers of soil are completely thawed, with the water in the soil transforming into liquid form.  
Figure 16: Temperature of the slope model after 35 FT cycles. (**a**) High environmental temperature state and (**b**) low environmental temperature state.  
Figure 17: Temperature of the slope model after 54 FT cycles. (**a**) High environmental temperature state and (**b**) low environmental temperature state.  
Figure 18: Temperature of the slope model under high environmental temperature conditions at different FT cycle counts. (**a**) FT = 55 and (**b**) FT = 150.  
According to Figure 16, after 35 FT cycles, the minimum temperature of the slope soil is \(-0.95\)\({}^{\circ}\)C under high environmental temperature conditions and \(-17.97\)\({}^{\circ}\)C under low environmental temperature conditions. At this point, the depth of the soil affected by temperature changes is approximately 1 to 2 m. Additionally, certain areas of the affected soil layers cannot fully thaw, and some water within the soil cannot transform into liquid form. However, most areas can still undergo normal FT cycles.  
According to Figure 17, after 54 FT cycles, the minimum temperature of the slope soil is \(-9.01\)\({}^{\circ}\)C under high environmental temperature conditions and \(-20.00\)\({}^{\circ}\)C under low environmental temperature conditions. At this point, the depth of the soil affected by temperature changes is approximately 2 to 3 m. Moreover, most areas of the affected soil layers cannot fully thaw, and some water within the soil cannot transform into liquid form. However, some regions can still undergo normal FT cycles.  
As shown in Figure 18, under the aforementioned FT cycle conditions, when the number of FT cycles reaches 55, the minimum temperature of the slope soil under high environmental temperature conditions remains at \(-20.00\)\({}^{\circ}\)C. This indicates that at this point, the upper layer of the slope soil is no longer significantly affected by the increase in external environmental temperature. The soil layers affected by temperature changes have likely reached a state where they can no longer thaw, and much of the moisture in the soil has turned into ice. As the number of FT cycles increases, the impact of environmental temperature changes on the upper layer of the slope becomes more pronounced, with the affected soil depth increasing. When the number of FT cycles reaches 150, the depth of the temperature-affected zone in the slope soil can reach 4-5 m. In the model test, this depth will continue to increase with the number of FT cycles. However, considering that the number of FT cycles that actually occur in the study area within a year will not exceed 150, this issue will not be further discussed.