# Experimental Study on Mechanical Properties of Anisotropic Slate under Different Water Contents
## 4 Analysis of Anisotropic Properties of Slate Parameters  
### Compressive Strength Anisotropy  
Predicting strength using rock mass strength criteria ranks among the most crucial issues in engineering. An essential standard for testing the strength theory's applicability is whether the predicted strength corresponds to the actual strength [23]. [PERSON] and [PERSON] introduced the Hoek-Brown strength criterion in 1980 [24; 25], but this criterion shows limited applicability to anisotropic rock masses. Various scholars have enhanced and refined it by incorporating the anisotropic parameter [26; 27], which accounts for the effects of stress and fabric tensors' relative orientations. This enhancement better represents the failure strength of foliated rock masses when combined with the Hoek-Brown criterion. The improved formula requires only a few known parameters to yield results consistent with actual measurements, offering broad applicability. The layered structure's variation within the structure is expressible via the fabric tensor. When material coordinate axes do not align with the principal value components, the fabric tensor \(F\) is representable as follows:  
\[F=Q_{ki}Q_{ij}F_{kl} \tag{1}\]  
\[Q_{ij}=e_{i}\cdot e_{j} \tag{2}\]  
where \(Q_{ij}\) is the cosine of the angle between \(e_{i}\) and \(e_{i}\), \(e^{\prime}_{i}\) is the \(i\)-th base vector of the original coordinate system, and \(e_{j}\) is the \(j\)-th base vector of the transformed coordinate system. Let \(b=(\sigma_{22}-\sigma_{33})/(\sigma_{11}-\sigma_{33})\), then \(\sigma_{22}=b\sigma_{11}+(1-b)\sigma_{33}\), \(p=c_{ii}/3=[(1+b)\sigma_{11}+(2-b)\sigma_{33}]\), deviatoric stress \(s_{ij}=\sigma_{ij}-p\); after a series of derivation, the unit deviatoric stress tensor \(n\) can be obtained. \[[n]=\frac{1}{\sqrt{6(b^{2}-b+1)}} \begin{bmatrix}2-b&0&0\\ 0&2b-1&0\\ 0&0&-1-b\end{bmatrix}  \tag{3}\]  
Based on \(n\), the anisotropy parameter can be defined as \(A=F_{ij}n_{ij}=F^{\pi}_{\ ij}n_{ij}\), where \(F^{\pi}_{\ ij}=F_{ij}-F_{kk}\delta_{ij}/3\). For the torsion test, the expression of \(A\) can be derived based on the same coordinate system of \(\sigma_{11}\), \(\sigma_{22}\), and \(\sigma_{33}\), combined with Equations (1)-(3), after definingthe parameter \(D\), where \(\theta\) is the rotation angle of the maximum principal stress, and \(\Delta\) is the rock's degree of anisotropy. When the rock mass is isotropic, \(\Delta=0\). \[\left. \begin{array}{l}A=\frac{D}{\sqrt{\
u_{\Delta}^{2}-b+1}}(1+b-3\cos^{2} \theta)\\ D=\frac{\sqrt{6}\Delta}{3(3+\Delta)}\end{array} \right\}\Rightarrow A=D(1-3 \cos^{2}\theta) \tag{4}\]  
The uniaxial compressive strength \(f_{c}\) of foliated rock mass, as a function of parameter \(A\), can be formulated as follows:  
\[f_{c}=ft\Big{\{}1+D(1-3\cos^{2}\theta)+d_{1}[D(1-3\cos^{2}\theta)]^{2}+d_{2}[D (1-3\cos^{2}\theta)]^{3}+\ldots+d_{m}[D(1-3\cos^{2}\theta)]^{m+1}\Big{\}} \tag{5}\]  
where \(f\), \(d_{1}\), \(d_{2}\), \(\ldots\), \(d_{m}\) are undetermined coefficients. The compressive strength of each sample is given in Table 1. This experimental research tested the uniaxial compressive strength of slates with five distinct bedding angles. Adhering to the solution principle that undetermined coefficients should not surpass known parameters, a third-power polynomial was chosen to predict test values to ensure result convergence. The resultant relationship curve between the slate's compressive strength and the changing bedding angle appears in Figure 10, where \(R^{2}\) is the correlation coefficient indicating the experimental data's conformity level with the fitting function. An \(R^{2}\) value closer to 1 signifies higher agreement. Figure 10 shows that the compressive strength of slate in natural, saturated, and dry states generally exhibits a U-shaped trend, initially decreasing and then increasing with the bedding angle's growth. Specifically, it reaches its maximum at \(0^{\circ}\) or \(90^{\circ}\) and its minimum around \(60^{\circ}\).