## 4 Results and Discussion
### Parameters Analysis
#### 4.2.3 Case 3--Nonlinear Shear Strength  
The impact of soil strength nonlinearity on tunnel cross-section stability, based on the nonlinear strength equations from Table 1 and various soil characteristics from Table 2, is depicted in Figure 9, in which Figure 9a-d represent the nonlinear shear strengths of sandy soil, silt, clay, and ultra-fine grained soil, respectively. In this section, specific parameters were preselected: \(\varphi=34^{\circ}\), \(D=5\) m, and \(C/D=1\).  
As depicted in Figure 9, varying assessing equations for nonlinear strength and soil kinds result in various impacts for matric suction in soil on the \(FS\) of tunnel face, as shown in Figure 9. In the case of sandy soil, the curves for [PERSON] et al. (1996) ([PERSON] et al., 1996) and [PERSON] et al. (1998) ([PERSON] et al., 1998) exhibit a flat and overlapping trend, indicating that these two models can be considered for saturated conditions. Regarding the other three models, the pattern of the \(FS\) varies nonlinearly within the range of \(u_{a}-u_{w}=40\) kPa, and subsequently, it linearly increases with the augmentation of matric suction. In comparison to the case where the suction angle is constant (\(\varphi^{b}=15^{\circ}\)), under the nonlinear equations, suction has a relatively minor effect on the stability of sandy soil tunnel faces.  
Figure 8: The factor of safety (\(FS\)) for tunnel faces under linearly increasing matric suction conditions.  
For fine-grained soils, the curves of the [PERSON] et al. (1998) [38] and [PERSON] et al. (1996) [35] models exhibit a similar pattern, both increasing nonlinearly at low suction, before gradually decreasing and ultimately degenerating into a horizontal and overlapping configuration. The curves for [PERSON] (1996) [36] and [PERSON] (2006) [39] indeed exhibit a close resemblance, both eventually converging to stability. Additionally, the curve for [PERSON] and [PERSON] (1998) [37] is notably higher than the other nonlinear curves. When compared to sandy soil, the nonlinearity of fine-grained soil has a more pronounced effect on the stability of the tunnel face.  
In the case of clay, when matric suction remains within 40 kPa, the trends of variation for all nonlinear curves, except for the Vilar model, are largely consistent with \(\varphi^{b}=\varphi^{\prime}=34^{\circ}\). Subsequently, the growth trends of the curves become more gradual. Only the curve for [PERSON] et al. (2006) [39] falls below \(\varphi^{b}=15^{\circ}\) and ultimately levels off. For ultra-fine grained soil, apart from the Vilar model, the safety factor curves are essentially consistent with the safety factor curve at \(\varphi^{b}=\varphi^{\prime}=34^{\circ}\), and the nonlinearity can be considered negligible. This aligns with the findings of [PERSON] et al. (2014) [46]\(\varphi^{b}=\varphi^{\prime}=34^{\circ}\) regarding  
Figure 9: The factor of safety (\(FS\)) for tunnel faces under nonlinear shear strength conditions: (**a**) sandy soils; (**b**) fine-grained soils (silt); (**c**) clays; (**d**) extremely fine-grained soils [35; 36; 37; 38; 39].  
larger AEV values. These curves are all significantly higher than the curve at \(\varphi^{b}=15^{\circ}\). From the comparison in Figure 9, it can be concluded that matric suction has the most significant impact on ultra-fine grained soil, followed by clay and silt, while its effect on sandy soil is the least pronounced.