## 1 Introduction  
For centuries, scientists have discussed how to effectively carry out geotechnical engineering practices in the face of complex, non-uniform, and discontinuous geotechnical conditions. The discontinuity, inhomogeneity, and anisotropy of geotechnical materials pose great difficulties for engineering practices, such as collapses or landslides that may occur during underground excavation. In order to ensure the smooth progress of engineering practices, scientists and engineers have researched, discussed, and used many approaches to try to model and simulate complex geotechnical conditions. Many approaches in this area fall within the following categories: continuous or discontinuous numerical methods, such as the finite difference method (FDM), finite volume method (FVM), finite element method (FEM), discrete element method (DEM), and discrete fracture network (DFN), etc. In addition, the homogenization methods used to simplify complex heterogeneous materials into homogeneous materials with equivalent properties are also applied in this field. Homogenization methods offer a valuable means to simplify and characterize the mechanical properties of complex heterogeneous materials. By utilizing appropriate constitutive models and relevant data, these methods enable the representation of complex materials as equivalent, homogeneous materials with simplified properties. This simplification allows for the computationally efficient calculations of material responses. So far, scientists have discussed how to achieve the characterization of these complex materials and widely apply them to the research and development of composite materials. In the field of geotechnical engineering, complex and discontinuous geotechnical conditions add many difficulties to engineering practices. Due to the performance of the homogenization method when facing complex heterogeneous materials, it is also applied in the characterization of geotechnical mechanical parameters. For example, [PERSON] developed a meso-macro constitutive model for saturated frozen saline sand based on homogenization theory, predicting the deformation of frozen saline sand under different salt contents and confining pressures, and reproducing strain-hardening/softening, high-expansion, and pressure-melting characteristics [1]. [PERSON] proposed a new numerical homogenization method called HLA-Dissim which simulates the discontinuity network of real rock masses based on the International Society for Rock Mechanics (ISRM) scan line field survey method [2]. [PERSON] studied the impact of the dilatancy effect of layered rock with rough interfaces on its mechanical behavior, used a homogenization method to simplify the layered rock structure into an equivalent continuous medium, defined interfaces with different roughness to evaluate the dilatancy effect, and used a dilatancy model called the contact density model to estimate the stiffness characteristics of rough surfaces [3]. These studies suggest that homogenization methods exhibit excellent performance in the field of geomechanics and engineering. Overall, the utilization of homogenization methods in geomechanics and engineering demonstrates their exceptional performance in tackling the difficulties posed by complex geotechnical conditions. These methods present a powerful tool for researchers to gain insights, make accurate predictions, and effectively address challenges encountered in geotechnical engineering practices. However, the field of the homogenization of nonlinear materials, such as soil, is relatively underexplored compared to other areas of research in geotechnical engineering. While there have been significant advancements in the characterization of complex heterogeneous materials using homogenization methods, the application of these methods specifically to nonlinear materials like soil has received limited attention.