## 2 The Critical State Line Considering the Effect of Principal Stress Axis Rotation  
### Anisotropic State Variables  
Conventional methods for examining the effect of the principal stress axis rotation, such as rotating the failure planes to investigate soil anisotropy, fail to effectively describe the impact of principal stress axis rotation on the capacity for shear resistance. In this paper, we introduce a new anisotropic state variable based on the microscopic fabric [11; 13; 41], which provides a practical solution to this problem. Based on a literature review, [PERSON] et al. (2021) proposed a normalized stress ratio-dependent second-order fabric tensor, while [PERSON] et al. (2020) used a back stress-modified yield surface proportional to the fabric tensor, both achieving fabric anisotropic function [42; 43]. However, the updating of the fabric tensor in these methods must be correlated with plastic strain rate.  
The anisotropic state variables used in this paper are defined through the joint invariants of the stress tensor and the fabric tensor. In comparison to the previous approach, the concept of elastic-plastic differentiation is not required, and the number of parameters is reduced. This makes both the form and programming implementation relatively simple. However, it should be noted that this paper does not take into account the evolution of the fabric tensor, which may limit its ability to simulate complex stress paths. The concrete expression of the fabric tensor is as follows:  
\[\mathbf{F}=\begin{bmatrix}F_{11}&0&0\\ 0&F_{22}&0\\ 0&0&F_{33}\end{bmatrix}=\begin{bmatrix}\frac{1+a_{1}+a_{2}+a_{1}a_{2}}{3+a_{1} +a_{2}-a_{1}a_{2}}&0&0\\ 0&\frac{1+a_{1}-a_{2}-a_{1}a_{2}}{3+a_{1}+a_{2}-a_{1}a_{2}}&0\\ 0&0&\frac{1-a_{1}+a_{2}-a_{1}a_{2}}{3+a_{1}+a_{2}-a_{1}a_{2}}\end{bmatrix}. \tag{1}\]  
\(a_{1}\) and \(a_{2}\) represent the fabric magnitude parameters of anisotropic materials, which is a physical quantity used to describe the degree of anisotropy on the orthogonal plane [12; 15]. It is worth noting that axes 1 to 3 are defined based on the bedding plane and they exhibit an orthogonal relationship between any two axes. Axis 1 is oriented perpendicular to the bedding plane (i.e., normal direction), while axes 2 and 3 are oriented parallel to the bedding plane (i.e., tangential direction). Rotation of the principal stress axis causes a change in the angle between the principals of the effective stress and fabric tensors, leading to changes in the anisotropic state variables \(A\). At this time, the transformation tensor is given by [8; 9]  
\[C_{ki}=\begin{bmatrix}\cos\alpha&0&\sin\alpha\\ 0&1&0\\ -\sin\alpha&0&\cos\alpha\end{bmatrix}.. \tag{2}\]  
\(\alpha\) refers to the angle between the principal stress axis and the principal direction of the fabric tensor. The rotated fabric tensor \(\mathbf{F}^{\prime}\) can be expressed as follows:  
\[F^{\prime}_{ij}=C_{ki}C_{lj}F_{ij}=\begin{bmatrix}\frac{F_{11}+F_{32}}{2}+\frac {F_{11}-F_{32}}{2}\cos 2\alpha&0&\frac{F_{32}-F_{11}}{2}\sin 2\alpha\\ 0&F_{22}&0\\ \frac{F_{32}-F_{11}}{2}\sin 2\alpha&0&\frac{F_{11}+F_{32}}{2}-\frac{F_{11}-F_{32}}{2} \cos 2\alpha\end{bmatrix}. \tag{3}\]  
The corresponding anisotropic state variable \(A^{\prime}\) is transformed as follows:  
\[A^{\prime}=\frac{\sigma_{ij}}{\sigma_{m}}F^{\prime}_{ij}-A_{0}. \tag{4}\]  
\(A_{0}\) is the reference point and can be determined by the stress state at conventional triaxial compression.