## 2 Study on the Compressive and Tensile Mechanical Properties of Shale
### Study on the Tensile Properties of Shale  
Figure 4 shows the crack characteristics after the Brazil splitting test of parallel bedding and vertical bedding shale samples. It can be observed that all shale samples developed a distinct main crack that is primarily aligned with the central axis, demonstrating a clear tensile failure. The crack patterns of the vertical bedding and parallel bedding shale samples are not significantly different. Figure 5 presents the axial load-axial displacement curves for the Brazil splitting test of parallel bedding shale samples (SC-H-1, SC-H-2, SC-H-3), which exhibit similar compaction, elastic, and failure stages as observed in the uniaxial compression tests. Utilizing Equations (1) and (2) in conjunction with Figure 5, the Brazil splitting test results for parallel bedding shale samples are shown in Table 4. The failure limit load of the parallel bedding shale samples ranges from 2.20 to 4.51 kN with an average value of 3.62 kN; the tensile strength ranges from 3.68 to 7.54 MPa with an average of 6.04 MPa; the tensile modulus ranges from 4.52 to 5.19 GPa with an average of 4.81 GPa; and the peak energy rate ranges from 0.10 to 0.32 kJ/m\({}^{2}\) with an average of 0.23 kJ/m\({}^{2}\). Figure 4: Split shale samples after Brazilian test. Referencing the calculation of shale tensile strength from Ref. [51] and combining the ultimate load from Figure 5 for parallel bedding, Formula (1) can be used to derive the tensile strength calculation formula for the sample as follows:  
\[\sigma_{t}=-\frac{2P_{max}}{\pi DH}, \tag{1}\]  
where \(\sigma_{t}\) is the tensile strength of shale (unit: MPa). Referring to the relevant research on anisotropy of disks by [PERSON] et al. [52] and [PERSON] et al. [53], an expression (2) for the tensile modulus \(E_{t}^{h}\) of shale has been derived:  
\[\frac{\pi DH}{P_{max}^{h}}\left\{ \begin{array}{c}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array} \right\}=\left[ \begin{array}{ccc}1/E_{t}&-v/E_{t}&0 \\ -v/E_{t}&1/E_{t}^{h}&0\\ 0&0&2(1+v)/E_{t}\end{array} \right]\left\{ \begin{array}{c}-2\\ 6\\ 0\end{array} \right\}, \tag{2}\]  
where \(\varepsilon_{x}\), \(\varepsilon_{y}\), and \(\gamma_{xy}\) represent the strain values, while \(E_{t}\) and \(E_{t}\) are the tensile modulus and Poisson's ratio, respectively. Figure 6 illustrates the load-displacement curves for the Brazil splitting test of vertical bedding shale samples, which are similar to those of the parallel bedding shale samples. By applying Equations (1) and (3) in conjunction with Figure 6, the Brazil splitting test results for the vertical bedding shale samples are presented in Table 5. The failure limit load for the vertical bedding shale samples ranges from 7.91 to 8.98 kN with an average value of 8.52 kN; The tensile strength ranges from 13.01 to 14.65 MPa with an average of 13.98 MPa; the tensile modulus ranges from 3.52 to 3.75 GPa with an average value of 3.64 GPa. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample Number** & **Destruction Load \(P_{max}^{h}\) (kN)** & **Tensile Strength \(\sigma_{t}^{h}\) (MPa)** & **Tensile Modulus \(E_{t}^{h}\) (GPa)** \\ \hline SC-H-1 & 4.15 & 6.91 & 4.71 \\ SC-H-2 & 4.51 & 7.54 & 5.19 \\ SC-H-3 & 2.20 & 3.68 & 4.52 \\ Average Value & 3.62 & 6.04 & 4.81 \\ Variance & 1.03 & 2.86 & 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of Brazilian splitting test for parallel bedding shale samples. Figure 5: Stress-strain curve of parallel bedding shale samples under the Brazilian splitting test. By comparing the results presented in Figure 4 and Table 5, it can be observed that the failure load of the vertical bedding shale samples is greater than that of the parallel bedding shale samples. The tensile strength of the parallel bedding shale is lower than that of the vertical bedding shale samples, while the tensile modulus of the parallel bedding shale samples is slightly higher than that of the vertical bedding shale samples, indicating a noticeable anisotropy in the tensile properties of shale.