## 4 Discussion  
### Anisotropic Tensile Strength Criterion for Layered Shale  
Since many rocks exhibit varying anisotropic characteristics, the traditional isotropic tensile strength criterion cannot meet practical requirements. For this reason, several scholars have proposed various anisotropic tensile strength criteria, including the Hobbs-Barron (H-B) criterion [26,27], Nova-Zaninetti (N-Z) criterion [28], single plane of weakness (SPW) criterion [29], Lee-Pietruszczak (L-P) criterion [29], critical plane approach (CPA) criterion [29], and a phenomenological anisotropic tensile strength criterion (referred to as the phenomenological criterion) [30]. In addition, [PERSON] et al. [31] developed an anisotropic mode I fracture toughness criterion based on the stress-strain transformation rule (referred to as the SS criterion). [PERSON] et al. [25] found that this criterion seems to be equally applicable to predicting tensile strength through the Brazilian splitting tests of layered shale. To verify the accuracy and applicability of different anisotropic tensile strength criteria, the experimental results of this study were compared and validated. (1) H-B criterion  
\[\sigma(\beta)= \begin{cases}=\frac{2\rho_{\text{b}}}{\cos\beta(1+\cos\beta)}&0 ^{\circ}\leq\beta\leq\beta^{*}\\ \sigma_{\text{m}}&\beta^{*}\leq\beta\leq 90^{\circ}\end{cases}  \tag{4}\]  
\[\cos\beta^{*}(1+\cos\beta^{*})=\frac{2\sigma_{\text{b}}}{\sigma_{\text{m}}} \tag{5}\]  
where \(\sigma(\beta)\) is the tensile strength with different loading angles, MPa; \(\sigma_{\text{b}}\) is the tensile strength of bedding plane at \(\beta=0^{\circ}\), MPa; \(\sigma_{\text{m}}\) is the tensile strength of matrix at \(\beta=90^{\circ}\), MPa. (2) N-Z criterion  
\[\sigma(\beta)=\frac{\sigma_{\text{m}}\sigma_{\text{b}}}{\sigma_{\text{b}}\sin ^{2}\beta+\sigma_{\text{m}}\cos^{2}\beta} \tag{6}\]  
(3) SPW criterion  
\[\sigma(\beta)= \begin{cases}\frac{\sigma_{\text{b}}}{\cos^{2}\beta}&0^{\circ} \leq\beta\leq\beta^{*}\\ \sigma_{\text{m}}&\beta^{*}\leq\beta\leq 90^{\circ}\end{cases}  \tag{7}\]  
\[\beta^{*}=\cos^{-1}\sqrt{\frac{\sigma_{\text{b}}}{\sigma_{m}}} \tag{8}\]  
(4) L-P criterion  
\[\sigma(\beta)=\frac{\sigma_{\text{m}}+\sigma_{\text{b}}}{2}-\frac{\sigma_{ \text{m}}-\sigma_{\text{b}}}{2}\cos 2\beta \tag{9}\]  
(5) CPA criterion  
\[\sigma(\beta)=\frac{\sigma_{\text{b}}\exp(-\Omega_{0})-\sigma_{\text{m}}\exp (2\Omega_{0})}{\exp(-\Omega_{0})-\exp(2\Omega_{0})}+\frac{\sigma_{\text{m}}- \sigma_{\text{b}}}{\exp(-\Omega_{0})-\exp(2\Omega_{0})}\exp\Bigl{[}\Omega_{0} \Bigl{(}3\cos^{2}\beta-1\Bigr{)}\Bigr{]} \tag{10}\]  
where \(\Omega_{0}\) is the spatial distribution parameter of the tensile strength, dimensionless. (6) Phenomenological criterion \[\sigma(\beta)=\frac{\sigma_{\rm m}\sigma_{\rm b}}{\sigma_{\rm b}+(\sigma_{\rm m}- \sigma_{\rm b})\cos^{\lambda}\beta} \tag{11}\]  
where \(\lambda\) is the anisotropy ratio, defined by the ratio of \(\sigma_{\rm b}\) to \(\sigma_{\rm m}\), dimensionless. (7) SS criterion  
\[\sigma(\beta)=\sigma_{\rm b}\cos^{2}\beta+\sigma_{\rm m}\sin^{2}\beta \tag{12}\]  
Figure 9 shows a comparison of the theoretical results based on different anisotropic tensile strength criteria with the experimental results. For DTT and BST, the spatial distribution parameters of the tensile strength for the CPA criterion are \(0.15\) and \(-0.35\), respectively. As shown in Figure 9, the theoretical results based on different strength criteria exhibit significantly different distributions. Based on the results of BST and DTT, we calculated the error and standard deviation (SD) of the prediction results of seven anisotropic tensile strength criterion, and the results are shown in Table 1. Among them, the L-P criterion, CPA criterion, and SS criterion show a high degree of alignment with the experimental results, particularly for indirect tensile strength. However, due to the limited number of DTT results, there is some deviation between the experimental data and the theoretical predictions based on these three criteria.