# Experimental Study on Anisotropic Mechanical Characteristics of Shale under Triaxial Loading
## 4 Analysis of Anisotropic Mechanical Characteristics of Shale
### Anisotropic Index of Shale's Mechanical Parameters  
Anisotropic index \(R_{\mathrm{c}}\) is an important parameter to quantitatively evaluate the anisotropic degree of the mechanical properties of materials. [PERSON] et al. [49] proposed a formula for calculating the anisotropic index \(R_{\mathrm{c}}\) of rock materials, which is the uniaxial compressive strength of the materials at a 90\({}^{\circ}\) bedding angle divided by the minimum uniaxial compressive strength of the rock materials within 0\({}^{\circ}\)-90\({}^{\circ}\) bedding angles, and the classification standard of anisotropic degree is also given, as shown in Table 8.  
The calculation method of the anisotropic index \(R_{\mathrm{c}}\) used in this paper adopts the calculation formula proposed by [PERSON] et al. [17], as shown in Equation (4), and the anisotropic \(R_{\mathrm{c}}\) is calculated by the ratio of the maximum value to the minimum value of the physical and mechanical parameters of shale at different bedding angles. In this paper, the compressive strength \(\sigma_{\mathrm{cs}}\) and elastic modulus \(E_{\theta}\) of shale samples are selected for calculating anisotropic indexes, and Table 9 shows the anisotropic indexes calculated by the \(\sigma_{\mathrm{cs}}\) and \(E_{\theta}\) of shale samples in Tables 2 and 5. \[R_{c}=\frac{\left(parameter\right)_{\mathrm{max}}}{\left(parameter\right)_{ \mathrm{min}}} \tag{4}\]  
According to the anisotropic indexes of the compressive strength \(\sigma_{\mathrm{cs}}\) of the shale, combined with the anisotropic grading standard in Table 8, it is determined that the shale has a moderate degree of anisotropy. In the confining pressure \(\sigma_{3}\) range of 0-40 MPa, the bedding angle \(\theta\) corresponds to the extreme values of the compressive strength \(\sigma_{\mathrm{cs}}\) and elastic modulus \(E_{\theta}\) of shale changes, which increases the complexity of analyzing the anisotropy. The fitting curves of the anisotropic indexes of \(\sigma_{\mathrm{cs}}\) and \(E_{\theta}\) with \(\sigma_{3}\) are sorted out, as shown in Figure 13, and the functional expressions of the fitting curves are shown in Equations (5) and (6). \begin{table}
\begin{tabular}{c c} \hline
**Grade** & **Range of Anisotropy Index \(R_{\mathrm{c}}\)** \\ \hline Isotropy & \(1.0<R_{\mathrm{c}}\leq 1.1\) \\ Low anisotropy & \(1.1<R_{\mathrm{c}}\leq 2.0\) \\ Moderate anisotropy & \(2.0<R_{\mathrm{c}}\leq 4.0\) \\ High anisotropy & \(4.0<R_{\mathrm{c}}\leq 6.0\) \\ Very high anisotropy & \(6.0<R_{\mathrm{c}}\) \\ \hline \end{tabular}
\end{table} 
Table 8: Classification criteria for anisotropy degree of rock materials proposed by [PERSON] et al. [49]. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Selected Mechanical** & \multicolumn{5}{c}{**Confining Pressure \(\sigma_{3}\) (MPa)**} \\ \cline{2-6}
**Parameters** & **0** & **10** & **20** & **30** & **40** \\ \hline Compressive strength \(\sigma_{\mathrm{cs}}\) & 2.811 & 2.928 & 2.158 & 2.242 & 2.302 \\ Elastic modulus \(E_{\theta}\) & 1.751 & 1.836 & 1.834 & 1.853 & 1.914 \\ \hline \end{tabular}
\end{table} 
Table 9: Shale anisotropic indexes under different confining pressures \(\sigma_{3}\). For compressive strength \(\sigma_{\text{cs}}\), the fitting formula is as follows:  
\[\left\{ \begin{array}{l}R_{c}=\frac{2.599x_{0}+175.273}{1.722x_{0}+62.453},(0 \text{MPa}\leq\sigma_{3}\leq 35.314\text{MPa}),\\ R_{c}=\frac{4.282x_{0}+114.427}{1.72x_{0}+62.453},(35.314\text{MPa}<\sigma_{3} \leq 40\text{MPa}).\end{array} \right. \tag{5}\]  
For elastic modulus \(E_{\theta}\), the fitting formula is as follows:  
\[\left\{ \begin{array}{l}R_{c}=\frac{0.393x_{0}+47.964}{0.172x_{0}+27.182},(0 \text{MPa}\leq\sigma_{3}\leq 33.261\text{MPa}),\\ R_{c}=\frac{0.393x_{0}+47.964}{0.126x_{0}+28.712},(33.261\text{MPa}<\sigma_{3} \leq 40\text{MPa}).\end{array} \right.