# Quantitative Description for Sand Void Fabric with the Principle of Stereology
## 3 Novel Void Fabric Tensors Definitions
### Definition of Void Fabric  
The void fabric describes the distribution form of the three orthogonal planes where the value of trace is equal to 1. Thus, according to the characteristic and combining with normalized ideas, the equation of the void fabric tensor is defined and reads:  
\[M_{i_{1}i_{2}\ldots i_{n}}=\frac{\int_{\Omega}n_{i_{1}i_{2}\ldots i_{n}}n_{in} L_{\mathrm{L}}(\alpha,\beta)\mathrm{d}\overrightarrow{n}}{\int_{\Omega}L_{ \mathrm{L}}(\alpha,\beta)\mathrm{d}\overrightarrow{n}} \tag{7}\]  
where \(M_{i_{1}i_{2}\ldots i_{n}}\) is the void fabric tensor, which is used for the microscopic quantitative detection of the void tensor directly.  
The tensor determined by Equation (7) accurately describes the variation of void space distribution in theory. However, in practice, the second-order fabric tensor has been able to describe the variation of void distribution. Hence, the influence of higher-order tensors is ignored. Equation (7) describes the second-order tensor as follows.  
\[M_{ij}=\frac{\int L_{\mathrm{L}}(\alpha_{m},\beta_{m})n_{i}n_{j}\mathrm{d} \overrightarrow{n}}{\int L_{\mathrm{L}}(\alpha_{m},\beta_{m})\mathrm{d} \overrightarrow{n}} \tag{8}\]  
where \(M_{ij}\) is the second-order symmetric fabric tensor. \(\alpha_{m}\) and \(\beta_{m}\) are the description angles of the \(m\)-th scan line. \(n_{i}n_{j}\) is the direction component of the second-order fabric tensor.  
According to Equation (5), a tensor that's trace is 0 is obtained directly and reads:  
\[N_{ij}=\frac{15}{2}(M_{ij}-\frac{1}{3}\delta_{ij}) \tag{9}\]  
where \(N_{ij}\) is the symmetric fabric tensor that's trace is 0 describing the void distribution. \(\delta_{ij}\) is the Kronecker tensor.  
According to Equation (4), the void distribution equation of the second-order tensor is as follows.  
\[N(\overrightarrow{n})=n_{m}(1+N_{ij}n_{i}n_{j}) \tag{10}\]  
where \(n_{m}\) is the form described by Equation (2).  
From Equation (8), the description method in this paper improved the fabric definition. According to the definition of measurement voids, the measurement process is expressed as the following equation by series.  
\[M_{ij}=\frac{\sum\limits_{m=0}^{N-1}\frac{L_{\mathrm{L}}(\alpha_{m},\beta_{m})n_{ i}n_{j}}{N}}{\sum\limits_{m=0}^{N-1}\frac{L_{\mathrm{L}}(\alpha_{m},\beta_{m})}{N}} =\frac{\sum\limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m},\beta_{m})}{ \sum\limits_{m=0}^{N-1}L_{\mathrm{L}}(\alpha_{m},\beta_{m})} \tag{11}\]  
From Figure 2, the components of the unit vector \(\overset{\rightarrow}{n}\) in the three orthogonal directions are \(n_{1}=\cos\alpha\), \(n_{2}=\sin\alpha\sin\beta\), and \(n_{3}=\sin\alpha\cos\beta\), i.e., the tensor defined by Equation (11) is the following equation.  
\[M_{ij}=\left[\begin{array}{ccc}M_{11}&M_{12}&M_{13}\\ M_{21}&M_{22}&M_{23}\\ M_{31}&M_{32}&M_{33}\end{array}\right] \tag{12}\]  
From Equations (11) and (12), the trace of the novel void tensor \(M_{ij}\) is equal to 1, so the trace of the tensor described in Equation (9) is equal to 0, which provides great convenience for the derivation of the void distribution equation and the determination of the test tensor.