## 1 Introduction  
The presence of anisotropy in the microstructure of sand causes notable variations in the mechanical response regarding the rotation and immobilization of the principal stress axis, particularly under complex loads such as earthquakes, traffic, and waves. Due to this, it has been the subject of extensive research by scholars globally [1; 2; 3]. Rotation of the principal effective stress axes with respect to the anisotropic fabric causes anisotropic behavior in sand. Research studies conducted by [PERSON] (1972), [PERSON] (1972, 1999), and [PERSON] (1979) have found that the stochastic arrangement of non-spherical sand particles in space exhibits statistically significant characteristics known as microstructures which can be described using a second-order symmetric tensor [4; 5; 6; 7]. The preferred distribution of contact direction, particle morphology, and pore space of sand particles results in different anisotropic microstructures. Thus, [PERSON] et al. (2010, 2016, 2017) comprehensively investigated the normal contact fabric, porosity fabric, and particle morphology fabric [8; 9; 10; 11; 12; 13; 14; 15]. Furthermore, the analytical method has also demonstrated its efficiency in describing the crack fabric of rocks [16]. The rotation effect of the principal stress axis can be attributed to the existence of anisotropic microstructures, and the principal stress axis rotation causes the development of sand particle contact, morphology, and pore space, contributing to the anisotropic evolution of the microstructure [17]. Consequently, an increasing number of researchers have become interested in these characteristics. [PERSON] (2010), [PERSON] (2008), and the author (2010) previously considered the micro-characteristics of particles and proposed using the fabric tensor to establish an anisotropic strength criterion [8; 18; 19]. In the research of [PERSON] et al. (2008), the authors demonstrated the significant effect of sand fabric rearrangement on its shear strength [20]. Moreover, the authors showed that when the principal stress axis rotates, there is a more significant relationship between the strength of the sand soil and its variations. [PERSON] et al. (2021) utilized finite element analysis to investigate the variation in strength parameters caused by the rotation of the principal stress axis during tunnel excavation [1]. The results indicate that the principal stress rotation angle gradually increases during soil construction. Additionally, [PERSON] et al. (2022) investigated the deformation properties of fiber-reinforced aeolian sand under traffic loads using a small-strain hollow cylindrical torsion shear device [2]. The authors found that continuous rotation of the principal stress axis leads to non-coaxial deformation. Since the establishment of the soil mechanics theory regarding the critical state, the non-coaxial behavior of soil has attracted significant attention from numerous scholars [21; 22; 23]. [PERSON] (2022) examined the seismic settlement deformation of soft soil foundations, considering the effect of principal stress axis rotation, and developed a soft soil model under earthquake loading [3]. As a result, it can be observed that actual engineering practices often involve the effect of the principal stress axis rotation. This effect can lead to anisotropic differences in the mechanical properties of soil in different directions, which are specifically involved in engineering problems such as slope stability and foundation bearing capacity. The importance of this effect is evident. Neglecting this characteristic while establishing the constitutive model of soils and applying them to engineering practice may lead to an improper evaluation of the actual stress state in soils, which can further potentially cause engineering disasters. In addition to the rotation of the principal stress axis, plane strain problems are also common in the process of engineering construction, such as retaining wall instability, foundation instability, slope slippage [24; 25; 26], etc.