## 2 The Critical State Line Considering the Effect of Principal Stress Axis Rotation
### The Effect of the Rotation of Principal Stress Axis on the Critical State Line  
The original model, i.e., without introducing a critical state line, cannot reflect the influence of density and stress levels on the constitutive response, nor can it demonstrate the strain softening characteristics of strength. The introduction of a critical state line can improve the above shortcomings, and a single set of parameters can be used to describe the constitutive relationship of soil within a wide range of density and stress levels.  
The critical state line definition by [PERSON] et al. (1996) and the author's prior research (2015, 2016) guide the introduction of anisotropic state variables into it [10,11,44]. This paper adopts the critical state line form as follows:  
\[f(e)= \left(\frac{e-e_{min}}{e_{crit}-e_{min}}\right)^{a}. \tag{5}\]  
The dimensionless parameter, \(a\), controls the degree of stress softening and is assigned a value of 0.1 based on actual conditions. \(E\) represents the current void ratio, while \(e_{min}\), the minimum void ratio, is set at 0.32 in this paper. \(e_{crit}\), the critical void ratio, is expressed as \(e_{crit}=e_{\Gamma}-\lambda_{c}(P/P_{a})^{\xi}+tA\), which is a function of \(A\) and the average principal stress \(P\)  
Figure 2: The influence of the fabric magnitude parameter on the anisotropic state variables: (**a**) \(a_{1}=0.1\); (**b**) \(a_{1}=0.5\); (**c**) \(a_{2}=0.1\); (**d**) \(a_{2}=0.5\).  
Atmospheric pressure is denoted by \(P_{a}\), while \(e_{\Gamma}\), \(\lambda_{c}\), and \(\zeta\) are material parameters that determine the critical state line and \(t\) is a state parameter [10]. Recommended parameter values from reference [10] include \(e_{\Gamma}=0.934\), \(\lambda_{c}=0.019\), \(\zeta=0.7\), and \(t=0.26\).  
After the rotation of the principal stress axis, \(A\) undergoes changes that, in turn, affect \(e_{\text{{crt}}}\) and consequently influence the critical state line, also altering the stress history and thereby causing a change in the strength during the loading process.  
Figure 3 depicts the impact of principal stress rotation angle on the critical state line under conditions of \(a_{1}=a_{2}\). The outcomes evince that for anisotropic materials (\(a_{1}\), \(a_{2}\
eq 0\)), the critical state line undergoes variation with respect to the rotation angle, displaying a monotonically ascending tendency within the 0 to 90 degrees range. As \(a_{1}\) and \(a_{2}\) increase, the critical state line's variation range also increases.  
In Figure 4, the influence of the principal stress rotation angle on the critical state line is presented for \(a_{1}\
eq a_{2}\). Specifically, Figure 4a,b shows the impact of varying \(a_{2}\) on the critical state line under a constant \(a_{1}\) (e.g., \(a_{1}\) values of 0.1 and 0.5). Likewise, Figure 4c,d portrays the impact of varying \(a_{1}\) on the critical state line under a constant \(a_{2}\) (e.g., \(a_{2}\) values of 0.1 and 0.5). Our analysis reveals that an increase in \(a_{1}\) and \(a_{2}\) results in a higher value of the critical state line, while \(a_{1}\) exerts a more pronounced effect compared to \(a_{2}\).  
Figure 4: The influence of fabric magnitude parameter on the critical state line _f(e)_: (**a**) \(a_{1}=0.1\); (**b**) \(a_{1}=0.5\); (**c**) \(a_{2}=0.1\); (**d**) \(a_{2}=0.5\).  
Figure 3: The influence of rotation angle on the anisotropic state variables on _f(e)_.  
Furthermore, the essential reason for the influence of the principal stress axis rotation on the behavior of soils can be summarized as follows: as the rotation angle of the principal stress changes, the components of the principal stress decomposed into the normal and tangential directions of the bedding plane will also change, thus affecting the stress state of the soil and causing differences in strength and deformation characteristics, leading to an impact on the soil's behavior.