# Quantitative Description for Sand Void Fabric with the Principle of Stereology
## 3 Novel Void Fabric Tensors Definitions
### A Novel Definition of 2D Void Fabric  
As shown in the unit test sphere in Figure 1, the spatial distribution of voids cannot be obtained directly by spatial scan line but can be obtained by plane image analysis of void distribution and then can be carried out reasonably in 3D modeling. The void projection distribution on three orthogonal planes is obtained by plane image analysis, and the detailed quantitative method is shown in the reference [35]. Then the void projection described in Figure 1 on the three orthogonal planes is measured by the actual image, i.e., the fabric tensor on the three orthogonal planes \(x_{1}\)-\(x_{3}\), \(x_{1}\)-\(x_{2}\), and \(x_{2}\)-\(x_{3}\). According to Equation (12), the tensor on the plane \(x_{1}\)-\(x_{3}\) is the following equation. \[M_{ij}^{1-3}=\left[ \begin{array}{cc}M_{11}&M_{13}\\ M_{31}&M_{33}\end{array} \right] \tag{13}\]  
Figure 2: The coordinate used in the void fabric tensor analysis. From Equation (11), \(N_{ij}\) is the partial tensor of \(M_{ij}\), then \(N_{ij}^{1-3}\) is described equivalently by the second invariant and component angle of \(M_{ij}^{1-3}\), and reads:  
\[\left. \begin{array}{l}b_{1}=\sqrt{\small{2}}=\sqrt{\small{\left|\frac{\sum_{m= 0}^{N-1}L_{1}(\alpha_{m},\beta_{m})\left(\cos^{2}a_{m}-\sin^{2}a_{m}\cos^{2} \beta_{m}\right)}{\sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})}\right|^{2}+ \left(\frac{\sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})\sin 2a_{m}\cos\beta_{m}}{ \sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})}\right)^{2}}}\\ \theta_{1}=\frac{1}{2}\text{arctan}\frac{2M_{13}}{M_{11}-M_{33}}\end{array} \right\} \tag{14}\]  
where \(b_{1}\) is the void amplitude parameter and is the second invariant of the void tensor in plane \(x_{1}\)-\(x_{3}\), which is a scalar and describes the anisotropic degree. \(\theta_{1}\) is the angle between the major direction of the void fabric and the direction of the coordinate axis, which describes the plane direction of the void fabric anisotropic degree. Similarly, the invariants and directions in the planes \(x_{1}\)-\(x_{2}\) and \(x_{2}\)-\(x_{3}\) are the following equations. \[\left. \begin{array}{l}b_{2}=\sqrt{\small{2}}=\sqrt{\small{\left|\frac{\sum_{m= 0}^{N-1}L_{1}(\alpha_{m},\beta_{m})\left(\cos^{2}a_{m}-\sin^{2}a_{m}\sin^{2} \beta_{m}\right)}{\sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})}\right|^{2}+ \left(\frac{\sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})\sin^{2}\alpha_{m}\sin 2 \beta_{m}}{\sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})}\right)^{2}}}\\ \theta_{2}=\frac{1}{2}\text{arctan}\frac{2M_{13}}{M_{11}-M_{22}}\end{array} \right\} \tag{15}\]  
\[\left. \begin{array}{l}b_{3}=\sqrt{\small{\sqrt{\small{2}}}=\sqrt{\small{ \left|\frac{\sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})\sin^{2}\alpha_{m}\cos 2 \beta_{m}}{\sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})}\right|^{2}+\left(\frac{ \sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})\sin^{2}\alpha_{m}\sin 2\beta_{m}}{ \sum_{m=0}^{N-1}L_{1}(\alpha_{m},\beta_{m})}\right)^{2}}}}\\ \theta_{3}=\frac{1}{2}\text{arctan}\frac{2M_{23}}{M_{22}-M_{33}}\end{array} \right\} \tag{16}\]  
where \(b_{2}\) and \(b_{3}\) are the void amplitude parameters on planes of \(x_{1}\)-\(x_{2}\) and \(x_{2}\)-\(x_{3}\), which describes the anisotropic degree. \(\theta_{2}\) and \(\theta_{3}\) are the angles of the two corresponding planes. According to Equation (12), the distribution of the void tensor on three orthogonal planes is described by its invariant. \[N_{i}(n_{m}^{2D},\theta_{m})=n_{m}^{2D}(1+b_{i}\cos 2(\theta_{m}-\theta_{i})) \tag{17}\]  
where \(\theta_{m}\) is the angle of the unit vector in the plane coordinate system. \(n_{m}^{2D}\) is the average porosity of the plane, which is determined directly by image analysis. \(b_{i}\) (\(i\) = 1,2,3) are the void amplitude parameters defined in Equations (14)-(16), respectively. \(\theta_{i}\) (\(i\) = 1,2,3) are the angle invariants defined by the corresponding equation. Due to \(M_{kk}\) = 1, the values of all three amplitude parameters are between 0 and 1.