# Quantitative Description for Sand Void Fabric with the Principle of Stereology
## 7 Conclusions  
Based on the framework of the stereology to describe void fabric, a novel description method of fabric tensor for sand was presented. This method uses the idea of normalization to redefine the fabric tensor to ensure that the trace of the tensor is always one, which facilitates the determination of the tensor greatly. The planar second-order void tensor was described equivalently by the defined void amplitude parameter and component angle. Both parameters are scalar, which describes the degree and direction of the material's macroscopic anisotropy. Among the three amplitude parameters defined on the orthogonal plane, there are only two independent changes. Any two amplitude parameters can be used to derive the orthotropic fabric tensor. With the change of the geometric relationship of void distribution, orthotropy will degenerate into transversely isotropy and isotropy naturally, and transversely isotropy can be described by only one amplitude parameter.  
Two kinds of scan lines were used for image comparison analysis. Parallel scan line analysis showed that the scan line density had a greater impact on the test results, the density was small, the test fluctuations were large, and the measured value was constant after the density reached a certain value. Parallel scan lines at different angles were used to measure average porosity, but the distribution of planar voids could not be determined. However, the annular scan line would better describe the spatial distribution of voids. The SEM image analysis for sand showed that the method presented in this paper would better describe the plane distribution. The novel method converted the void information into two scalars to describe its plane distribution. Through the change of angle conditions in the void space, it naturally degenerates into the traditional description form. The simulation results verify the rationality of its description, so that the measured void tensor can be used as a material parameter to be applied to the macro-mechanical equation of sand directly.  
Compared with the existing methods such as Kanatani, the novel method in this paper adopted normalized ideas and described the fabric with tensor invariants, which made the quantitative determination easier and more flexible. The method derives from the expressions of three different forms of orthotropic fabric tensor, which describe the orthotropy and transversely isotropy of materials with different forms. Its measurement methods are diverse and have a wide range of applications.  
Conceptualization, X.L. and F.M.; methodology, X.L.; software, X.L.; validation, X.L., Z.M. and F.M.; formal analysis, X.L.; investigation, Z.M.; resources, X.L.; data curation, X.L.; writing--original draft preparation, F.M.; writing--review and editing, Z.M.; visualization, Z.M.; supervision, 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 (Nos. 12162028, 51669027), and the Key Research and Development Program of Ningxia Hui Autonomous Region Projects of International Cooperation and Exchanges (No. 2018 DWHZ0084). This support is gratefully acknowledged.  
Not applicable.  
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
Data Availability Statement: 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.