## 4 Discussion
### The Comparison Between Direct and Indirect Tensile Strength  
In this study, DTT and BST were conducted on anisotropic shales to investigate the effects of anisotropy on the peak load, direct and indirect tensile strength, and failure modes of Longmaxi layered shales, as well as to compare the differences between these two testing methods. The results reveal that anisotropy significantly influences both direct and indirect tensile strength, with a particularly pronounced effect on direct tensile strength. Additionally, the indirect tensile strengths calculated using Equations (2) and (3) are significantly higher than the direct tensile strength. One contributing factor is the stress concentration caused by the linear loading at both ends of the BST specimen. Generally, both the direct and indirect tensile strengths calculated in this study represent apparent tensile strengths and may not accurately reflect the true tensile strength of the rock.  
To further analyze the relationship between direct and indirect tensile strengths, in addition to the test data from this study, direct and indirect tensile strengths of seven additional rock types were identified from the literature [13; 15; 22; 23; 32; 33; 34; 35; 36; 37; 38]. As shown in Figure 10, the results indicated that the indirect tensile strength is higher than the direct tensile strength, typically ranging from one to three times the direct tensile strength. The difference between direct and indirect tensile strength does not appear to be directly related to lithology but rather to the testing method or specimen geometry. [PERSON] and [PERSON] [39] suggest that the indirect tensile strength can be approximated as 1.11 times the direct tensile strength. As shown in the figure, the strength data for most rocks align well with this relationship.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mean Error for BST** & **Mean SD for BST** & **Mean Error for DTT** & **Mean SD for DTT** \\  & **(\%)** & **(MPa)** & **(\%)** & **(MPa)** \\ \hline H-B criterion & 6.62 & 0.29 & 15.14 & 0.53 \\ N-Z criterion & 2.74 & 0.12 & 13.26 & 0.45 \\ SPW criterion & 7.55 & 0.32 & 14.11 & 0.49 \\ L-P criterion & 2.35 & 0.11 & 10.36 & 0.36 \\ CPA criterion & 3.23 & 0.14 & 9.50 & 0.33 \\ Phenomenological criterion & 7.96 & 0.37 & 24.35 & 0.87 \\ SS criterion & 2.35 & 0.11 & 10.36 & 0.36 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Accuracy comparison of seven anisotropic tensile strength criteria based on experimental results.  
Figure 10: The direct and indirect tensile strength of eight kinds of rocks.  
In summary, the indirect tensile strength is higher than the direct tensile strength due to various factors, including anisotropy, testing methods, and specimen geometry. For isotropic rocks, simple statistical methods such as linear regression can be used to establish the relationship between direct and indirect tensile strength. However, for anisotropic rocks, when rock failure is influenced by bedding or foliation structures and fractures deflect, the peak load can vary significantly. As a result, the direct and indirect tensile strengths of anisotropic rocks cannot be accurately quantified using linear regression. Instead, a more precise tensile strength should be determined by either establishing a calculation equation that accounts for anisotropy [24; 29] or developing a testing method that minimizes or eliminates the influence of anisotropy.