# Quantitative Description for Sand Void Fabric with the Principle of Stereology
## 4 Novel Void Fabric Tensors Definitions
### Transversely Isotropic Void Fabrics  
In this paper, the definition of orthotropic void fabric is based on the measurement method of stereology, which describes the orthotropic distribution of voids. When the scan line distribution meets certain geometric conditions, it is simplified into the form of transversely isotropy and isotropy naturally. In Equations (14) and (15), when \(\beta_{m}=\pi/4\), then \(b_{1}=b_{2}\), i.e., the void parameters in the two directions are equal. At this point, the material shows as transversely isotropic, and Equation (19) is simplified into 3D fabric and reads:  
\[M_{ij}=\frac{1}{3-b}\left[ \begin{array}{ccc}1+b&0&0\\ 0&1-b&0\\ 0&0&1-b\end{array} \right] \tag{22}\]where:  
\[b=\sqrt{\left[\frac{\sum\limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m},\beta_{m})( \cos^{2}\alpha_{m}-\frac{1}{2}\sin^{2}\alpha_{m})}{\sum\limits_{m=0}^{N-1}L_{ \mathrm{L}}(\alpha_{m},\beta_{m})}\right]^{2}+\frac{1}{2}\left(\frac{\sum \limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m},\beta_{m})\sin 2\alpha_{m}}{\sum \limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m},\beta_{m})}\right)^{2}} \tag{23}\]  
where \(b\) is the amplitude parameter of the transversely isotropic void tensor, and its theoretical value range is [0,1], which is obtained by solving the invariant of the planar void fabric tensor from any one of the two homogeneous planes. \(b_{1}=b_{2}=b\) indicates that the void tensor invariants of the two planes are equal. Equation (22) is used to describe the transversely isotropy of the voids in the sand by an amplitude parameter. The physical geometric meaning of transversely isotropy is legible. According to Equations (9) and (22), \(N_{ij}\) is obtained and reads:  
\[N_{ij}=\frac{5}{3-b}\left[ \begin{array}{ccc}2b&0&0\\ 0&-b&0\\ 0&0&-b\end{array} \right] \tag{24}\]  
The average porosity \(n_{m}\) in Equation (9) is easy to determine. Thus, when \(N_{ij}\) is determined, the distribution of transversely isotropic void space is determined. Similarly, when \(\alpha_{m}=\pi/2\) in Equation (14) and \(\beta_{m}=0\) in Equation (16), then \(b_{1}=b_{3}\) and Equation (20) is simplified into the form of transversely isotropy and reads:  
\[M_{ij}=\frac{1}{3+b}\left[ \begin{array}{ccc}1+b&0&0\\ 0&1+b&0\\ 0&0&1-b\end{array} \right] \tag{25}\]  
where \(b\) is the following:  
\[b=\sqrt{\left[\frac{\sum\limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m},\beta_{m} )\cos 2\alpha_{m}}{\sum\limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m},\beta_{m})} \right]^{2}+\left(\frac{\sum\limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m}, \beta_{m})\sin 2\alpha_{m}}{\sum\limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m}, \beta_{m})}\right)^{2}} \tag{26}\]  
where \(b\) is the parameter of transversely isotropic fabric. The variable of Equation (14) after degenerating is \(\beta_{m}\). The variable of Equation (16) after degenerating is \(\alpha_{m}\). For convenience, the variable \(\alpha_{m}\) is used here. When \(\beta_{m}=\pi/2\) in Equation (15) and \(\alpha_{m}=0\) in Equation (16), then \(b_{2}=b_{3}\). Equation (19) is simplified into the following equation. \[M_{ij}=\frac{1}{3+b^{2}}\left[ \begin{array}{ccc}\left(1+b\right)^{2}&0&0\\ 0&1-b^{2}&0\\ 0&0&\left(1-b\right)^{2}\end{array} \right] \tag{27}\]  
where the equation of \(b\) is the same as the form of Equation (26), but the variable \(\beta_{m}\) is different. Although the amplitude parameter is simplified into a transversely isotropic form, the simplified fabric tensor (Equation (27)) is orthotropic. To sum up, when the orthotropic void fabrics are simplified into transversely isotropy, Equation (22) is applied to the detection and application of transversely isotropy void fabric directly. Although Equation (25) has some contradictions in the degradation process, it is consistent with the description results of the existing void fabric. Hence, under certain conditions, Equation (25) still has certainly applicable scope. Equation (27) is still in the orthotropic form after degradation and cannot describe transversely isotropic material als.