# Quantitative Description for Sand Void Fabric with the Principle of Stereology
## 1 Introduction  
The void ratio is one of the most important, earliest, and most widely used material parameters for controlling the mechanical response of the soil. In critical state soil mechanics, it is the only material physical parameter that describes the critical state. However, [PERSON] and [PERSON] [1], [PERSON] and [PERSON] [2] and [PERSON] et al. [3] have recently confirmed that the critical state of soil was not unique due to the existence of fabric anisotropy, i.e., the relationship between the void ratio of critical state and hydrostatic pressure was not unique, which was a challenge to the uniqueness hypothesis for the critical state theory of soil mechanics. According to the anisotropic characteristics of soil meso-structure, [PERSON] et al. [4] studied the meso-structure of particles under one-dimensional consolidation conditions and found that the particle structure had a horizontal arrangement trend, and the void structure showed a similar orientation trend. [PERSON] et al. [5] and [PERSON] et al. [6] found that the void fabric tensor and particle sphericity, relative density, and coordination number have a strong functional relationship through the image analysis of loose and dense sand specimens. [PERSON] et al. [7] presented a fabric tensor-based bounding surface model accounting for the anisotropic behavior of granular materials. [PERSON] et al. [8] established the relations between the macroscopic volumetric strain, shear strain, and void fabric combined with the micromechanics theory, and linked with void fabric tensor. [PERSON] et al. [9] consideredthe effects of initial anisotropy, porosity, stress ratio, loading direction, and intermediate principal stress ratio, and established the anisotropic evolution law for fabric. Although the void fabric has a substantial impact on the mechanical properties of sand, the relative lag in the development of mesoscopic testing technology, experiment and description theory of void fabric has led to a lack of understanding of void fabric and has restricted the study of the mechanical properties by the method of macro and micro combination. The three main causes of the meso-fabric of sand are the anisotropic distribution of contact normal particles, void space, and the arrangement of the nonspherical particles [10]. It is difficult to consider the three causes simultaneously because they have different physical and geometric meanings and applications. Therefore, the different fabric theories and their description methods have been formed. For example, [PERSON] [10] and [PERSON] [11] defined the fabric tensor by considering the arrangement direction of the long axes of sand particles and the distribution of contact stress, which laid the foundation of the fabric theory and has been widely applied. However, the determination of single-particle direction and the direction of contact normal between particles has always puzzled scholars, who believed that these two factors were hardly measured accurately [12; 13]. Thus, there are more qualitative studies than quantitative studies for fabric, which is the biggest obstacle to be applied to soil constitutive relationships. In the author's work of establishing strength criterion [14], constitutive model [15] and potential theory [16] with fabric to describe anisotropy, the fabric description has always been at the core of thinking. Comparatively speaking, the description of void in three kinds of fabric is more intuitive and easier to accept, but the bottleneck is the development of its theory and measurement technology. Void fabric is a mathematical relationship describing the spatial distribution of voids in soil and there are three main theories to describe void fabric. First, [PERSON] [17] proposed the 2D void fabric theory and its measurement method; then [PERSON] and [PERSON] [18] used it in an experimental study. This method was difficult to popularize because it involved segmentation and image recognition, such as the center of gravity, boundary, shape, arrangement, and movement of particles.