# Quantitative Description for Sand Void Fabric with the Principle of Stereology
## 6 Determination and Application Analysis of Void Fabric
### The Fabric of Orthogonal Plane Determined by Experiment  
Experimental images are selected from reference [26]. [PERSON] et al. [26] used SEM (Scanning Electron Microscope) to collect images of sand in three orthogonal planes from solidifying sand specimens. The image processing technology is used to obtain a clear picture of the void distribution, and then the image analysis technology is used to extract the void information. The distribution of voids is obtained by annular scan line, of which the number is 180. The inner diameter R\({}_{1}\) is 40 pixels and the outer diameter R\({}_{2}\) is 230 pixels. The calculation results are presented in Figure 8b. Figure 9 shows the image analysis results of void distribution and the simulation results in this paper. The polar diameter is the amplitude parameter, and the polar angle is the rotation angle per 1\({}^{\circ}\) of the plane scan line on the test surface. \(\beta_{m}=0^{\circ}\), \(\beta_{m}=45^{\circ}\), and  
Figure 8: SEM binary diagrams of sand specimens. (**a**) \(x_{2}\)-\(x_{3}\) plane. (**b**) Analysis of image. (**c**) \(x_{1}\)-\(x_{3}\) plane. (**d**) \(x_{1}\)-\(x_{2}\) plane. Figure 7: The procedures to determine voids with the annular scan line. (**a**) Scan line painting. (**b**) Binarization. (**c**) Annular scan line. (**d**) Analysis of image. \(\beta_{m}=90^{\circ}\) in the legend represents the amplitude parameters measured by \(0^{\circ}\), \(45^{\circ}\) and \(90^{\circ}\) spatial scan lines, respectively. The discrete points in Figure 9 are the measured data from image analysis, which shows the anisotropy with different degrees. For the measured value (\(b_{1}\), \(b_{2}\), and \(b_{3}\)) of each plane (plane \(x_{1}\)-\(x_{3}\), \(x_{1}\)-\(x_{2}\), and \(x_{2}\)-\(x_{3}\)) in the figure, the simulated data of three geometric conditions are given, i.e., the plane void distribution is measured when the angle of the spatial scan line is \(\beta_{m}=0^{\circ}\), \(\beta_{m}=45^{\circ}\), and \(\beta_{m}=90^{\circ}\), respectively. The crucial data to be calculated are as follows: the average porosity \(n_{0}\) and amplitude parameter \(b\) varied with \(\beta_{m}=0^{\circ}\), \(45^{\circ}\), \(90^{\circ}\). In Figure 9, \(n_{0}=0.365\), \(b_{1}=0.012\), \(0.053\) and \(0.169\), \(\theta_{m}=92.9^{\circ}\). In Figure 9, \(n_{0}=0.308\), \(b_{2}=0.128\), \(0.067\) and \(0.008\), respectively, and the direction \(\theta_{m}\) is \(-4^{\circ}\). In Figure 9, \(n_{0}=0.348\), \(b_{3}=0.249\), and \(\theta_{m}=0^{\circ}\), \(-45^{\circ}\), \(0^{\circ}\), respectively. \(\beta_{m}\) defined in this paper is another independent variable that is a possible description of the spatial distribution of voids. When \(\beta_{m}=0^{\circ}\) in Figure 9, \(\beta_{m}=45^{\circ}\) in Figure 9 and \(\beta_{m}=90^{\circ}\) in Figure 9, the calculation of the horizontal plane tensor is not affected, i.e., the plane tensor is simplified into the form proposed by [PERSON] [23]. Under such conditions, the method defined in this paper will better simulate the planar void experiment and determine the distribution. Under other conditions, the void distribution is affected by the measured value of voids and \(\beta_{m}\). Figure 9: Rose diagrams of amplitude parameters. (**a**) Results of \(b_{1}\). (**b**) Results of \(b_{2}\). (**c**) Results of \(b_{3}\). According to the results of the void fabric on the orthogonal plane in Figure 9, the test laws of different spatial scan lines for the same plane image are analyzed. Figure 10 shows all possible spatial distributions with the different \(\beta_{m}\) in three planes. It is concluded from the figure that amplitude parameters on different planes have different effects on \(\beta\). With the variation of the spatial position of the scan line, the trend of \(b_{1}\) and \(b_{2}\) is reversed, while \(b_{3}\) remains unchanged, which is consistent with the theoretical definition in this paper.