# Experimental Study on Anisotropic Mechanical Characteristics of Shale under Triaxial Loading
## 4 Analysis of Anisotropic Mechanical Characteristics of Shale
### Compressive Strength Anisotropy of Shale
On the other hand, with the increase in confining pressure, the bedding angle \(\theta\) corresponding to the maximum value of \(\sigma_{\text{cs}}\) tends to change from \(0^{\circ}\) to \(90^{\circ}\). When confining pressure \(\sigma_{3}\) is between 0 MPa and 30 MPa, the bedding angle corresponding to the maximum value of \(\sigma_{\text{cs}}\) is obtained at \(0^{\circ}\), while at the higher confining pressure of 40 MPa, the maximum value is obtained at a bedding angle of \(90^{\circ}\). It can be seen that the sensitivity of \(\sigma_{\text{cs}}\) to the confining pressure under different bedding angles is different. In addition, the compressive strength \(\sigma_{\text{cs}}\) of shale samples at the same bedding angle \(\theta\) mostly increases with the confining pressure \(\sigma_{3}\), which conforms to the effect of confining pressure in the traditional rock triaxial test, as shown in Figure 6b. Moreover, within the confining pressure \(\sigma_{3}\) range of 0 MPa to 40 MPa in the test, the compressive strength \(\sigma_{\text{cs}}\) and confining pressure \(\sigma_{3}\) show an approximate linear relationship. Therefore, the curve between compressive strength \(\sigma_{\text{cs}}\) and confining pressure \(\sigma_{3}\) in Figure 6b is fitted by Equation (1), and the coefficient \(A\) and \(B\) in Equation (1) represent the sensitivity of compressive strength of confining pressure at this bedding angle and the uniaxial compressive strength, respectively. \[\sigma_{\text{cs}}=A\sigma_{3}+B \tag{1}\]  
Table 3 shows the values of coefficient \(A\) and \(B\) obtained by linear fitting of the experimental data according to Equation (1). It is worth noting that some abnormal data in Figure 6b were excluded in the fitting process. The reason why these data are considered abnormal is that they violate the trend of rock compressive strength increasing with the confining pressure in traditional rock triaxial tests, which is also followed by most of the data in the triaxial tests in this paper. Therefore, in order to ensure that the fitted equation can be applied to most test data, and to increase the accuracy of linear fitting (that is, to improve the value of R-square) and the safety of subsequent strength criteria, some abnormal data are removed, as shown in Table 3, and the fitting index R-square before and after abnormal data are removed is also given in the table. Figure 6: Variation of shale compressive strength \(\sigma_{\text{cs}}\) with bedding angle and confining pressure: (**a**) Bedding angle \(\theta\); (**b**) Confining pressure \(\sigma_{3}\). As can be seen from Table 3, \(R\)-square values are all greater than 0.95, indicating that the linear relationship between shale compressive strength \(\sigma_{\text{cs}}\) and confining pressure \(\sigma_{3}\) is very significant after removing some abnormal data. The fitting curve and the measured data are drawn in Figure 7, and it is not difficult to find that due to the difference in the sensitivity coefficient of bedding angle \(\theta\), the critical point of confining pressure is about 35 MPa between the shale compressive strength at a 90\({}^{\circ}\) bedding angle and the shale compressive strength at a 0\({}^{\circ}\) bedding angle. When the confining pressure \(\sigma_{3}\) is near the critical point, the bedding angle corresponding to the maximum shale strength will be transformed. According to the linear relationship between compressive strength \(\sigma_{\text{cs}}\) and confining pressure \(\sigma_{3}\) of shale, it can be considered that the shale samples with different bedding angles basically obey the Mohr-Coulomb strength theory under the minimum principal stress of nearly 0-40 MPa, and the principal stress expression of Mohr-Coulomb strength theory is as shown in Equation (2). \[\sigma_{\text{cs}}=\frac{1+\sin\varphi}{1-\sin\varphi}\sigma_{3}+\frac{2c\cdot \cos\varphi}{1-\sin\varphi} \tag{2}\]  
where \(c\) is cohesion and \(\varphi\) is the internal friction angle.