## 5 Conclusions  
This paper investigates the geometric relationship between principal stress and fabric and proposes a hypoplastic constitutive model which considers the rotation of the principal stress axis.  
(1) To describe the mechanical characteristics of orthogonal anisotropic fabric relative to the rotation of principal stress, a new anisotropic state variable, denoted as \(A\), is defined. As the principal stress axis rotates, the angle between the fabric direction and stress direction also changes. When fabric amplitude parameters \(a_{1}\) and \(a_{2}\) are nonzero, the material becomes anisotropic, and \(\|A\|\) increases monotonically, within the 0 to 90 degrees range, with the rotation angle. Accordingly, the failure strength changes with the rotation angle, and the effect of \(a_{1}\) on \(A\) is greater than that of \(a_{2}\).  
Figure 16: The anisotropic differences in strength characteristics: (**a**) peak stress; (**b**) residual stress.  
Figure 17: The anisotropic differences in deformation characteristics.  
(2) \(A\) is embedded in the critical state line, and the impact of fabric amplitude parameters and principal stress axis rotation on \(f(e)\) is analyzed. The results show that the value of \(f(e)\) increases with the increase in \(a_{1}\), \(a_{2}\), and the rotation angle.  
(3) An anisotropic hypoplastic constitutive model is proposed, which incorporates the critical state line that takes into account principal stress axis rotation. By using the results of a series of Toyoura sand plane strain tests, the influences of void ratio, confining pressure, and rotation angle on the anisotropic strength and deformation characteristics have been simulated. The proposed model is also compared with the transversely isotropic hypoplastic model by [PERSON]. The proposed model exhibits a balanced performance when simulating the variation of anisotropy in both strength and deformation with respect to rotation angle, without being overestimated within a certain range of rotation angles. The prediction results demonstrated a certain degree of validity and effectiveness of the proposed model.  
(4) We have only conducted simulations of drained anisotropic experiments in this paper, and have not yet addressed undrained anisotropic experiments, which will be the main focus of our future research. Additionally, we plan to introduce the constitutive model into finite element software, such as ABAQUS, through the UMAT subroutine to enable numerical modeling and serve the engineering practice field.  
Conceptualization, X.L. and G.F.; methodology, G.F.; software, X.L. and G.F.; validation, Y.H.; formal analysis, G.F. and Y.H.; investigation, G.F. and Y.H.; resources, Y.H.; data curation, Y.H.; writing--original draft preparation, G.F. and Y.H.; writing--review and editing, X.L. and G.F.; visualization, G.F. and Y.H.; supervision, X.L.; project administration, X.L.; funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript.  
This work was financially supported by the Projects for Leading Talents of Science and Technology Innovation of Ningxia (No. KJT2019001), the National Natural Science Foundation of China (No. 12162028), and The innovation team for multi-scale mechanics and its engineering applications of Ningxia Hui Autonomous Region (2021), and these supports are gratefully acknowledged.  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
The authors declare no conflict of interest.