## 1 Introduction  
Due to the growing energy demand, unconventional oil and gas resources, such as shale gas, have become pivotal in the energy supply landscape [1, 2]. The combination of horizontal well technology and hydraulic fracturing represents a significant advancement in the efficient development of shale gas [3]. However, due to their unique sedimentary environment and geological processes, shales exhibit significant anisotropic characteristics [4, 5, 6, 7]. Among these, the anisotropic tensile mechanical behavior of shale has a significant impact on fracture pressure analysis during drilling and is closely related to fracture initiation and propagation during the reconstruction of reservoirs [8; 9; 10; 11; 12]. The bedding strength is obviously smaller than that of matrix. On the one hand, it affects the initiation and propagation behavior of induced fractures near wellbore wall, resulting in induced fracture lost circulation, which seriously affects drilling safety and cost. On the other hand, the bedding plane may capture hydraulic fractures and influence their direction of fracture propagation. Consequently, a comprehensive understanding of the anisotropic tensile mechanical behavior of shale is crucial for the exploration and development of underground shale oil and gas. Tensile strength is a crucial mechanical property of layered rocks. Currently, two primary methodologies are employed to test rock tensile strength: the direct tensile test (DTT) and the indirect tensile test [13]. Among these methods, the Brazilian splitting test (BST) has gained widespread popularity for indirect tensile strength testing due to its simplicity in specimen preparation and cost-effectiveness [14]. However, discrepancies often arise between the tensile strengths of the same rock measured by these two methods [15]. These variations become even more pronounced when anisotropy is considered. For instance, [PERSON] et al. [16] conducted BST on Mosel slate, investigating the influence of anisotropy on tensile mechanical behavior. They combined the discrete element method to analyze the effects of microparameters on the bearing capacity and failure modes of Brazilian disk specimens. Similarly, [PERSON] et al. [17] conducted BST on layered limestone at various loading angles, incorporating acoustic emission (AE) to study the relationship between tensile strength, tensile modulus, and loading angle. Based on their findings, they classified failure modes into three categories: \"perpendicular splitting\" in a roughly straight shape, \"shear splitting\" in an arc shape, and \"parallel splitting\" in a smoothly straight shape. Anisotropy also plays a significant role in direct tensile tests. [PERSON] et al. [18] conducted DTT on marine shale to investigate the impact of anisotropy. Their results showed that tensile strengths varied between 1.5 and 9.5 MPa across different loading angles, demonstrating strong anisotropic behavior of marine shale. Additionally, the failure modes varied significantly with anisotropy and were categorized into four types: tapered head failure, planar tensile failure, layer activation failure, and mixed-mode failure across bedding planes. Similarly, [PERSON] et al. [19] performed DTT on Kangding slate at five loading angles, integrating AE and the finite-discrete element method to reveal an S-shaped relationship between direct tensile strength and loading angle, and the observed failure modes were classified into three categories: layer activation failure, mixed failure, and rock matrix failure. While both direct and indirect tensile tests can provide insights into the anisotropic tensile behavior of rocks, a notable discrepancy exists between the tensile strengths measured by these two methods [20; 21]. Generally, the indirect tensile strength obtained from Brazilian disk specimens is higher than the direct tensile strength due to factors such as stress concentration [22].