## 1 Introduction  
In many engineering fields, the safety and stability of rock engineering are significantly influenced by layered jointed rock masses. The tensile failure characteristics of layered rock masses play a decisive role in the success of engineering projects. Currently, more than two-thirds of the Earth's land surface is covered by rocks with layered structures, and in China, this proportion exceeds 77%. These rocks are characterized by numerous weak structural planes, such as stratification, foliation, and fractures [1, 2, 3], which impart varying degrees of anisotropy to their mechanical properties and significantly affect their overall mechanical behavior and failure modes. Currently, the study of the splitting characteristics of layered rocks primarily employs the Brazilian splitting test method [4, 5, 6, 7]. For example, [PERSON] et al. [8] measured the tensile strength of shales from different regions using splitting tests, concluding that the tensile strength of rocks might follow a linear relationship with the stratification angle. [PERSON] [9; 10] and [PERSON] [11] observed that, at different inclinations, the tensile strength of layered sandstone decreases non-linearly with an increasing stratification angle. [PERSON] [12] and [PERSON] [13] summarized Brazilian splitting test results for various typical layered rocks, finding a non-linear decrease in tensile strength with increasing stratification angle, and developed corresponding failure mode diagrams. Many studies [14; 15; 16] have conducted tensile tests on shale, slate, gneiss, and chlorite schist, discovering that the tensile strength of layered rocks initially increases and then decreases with increasing stratification dip angle, and that the deformation characteristics of layered rocks also exhibit complex changes with varying stratification dip angles. This complexity is mainly due to differences in the forces experienced by the stratification planes at different stratification dip angles. [PERSON] et al. [17] carried out a multi-angle loading of layered gneiss specimens and found that, when the loading direction is parallel to the stratification plane (90\({}^{\circ}\)), the failure mode is primarily tensile failure between weak layers and within the mineral matrix. When the stratification dip angle is less than 90\({}^{\circ}\), due to the anisotropy and heterogeneity of the gneiss, the failure mode becomes a complex tensile-shear composite failure. [PERSON] et al. [18] combined Brazilian tests and AE technology to investigate the mechanical properties and fracture modes of shale under different loading directions. The changes in AE counts revealed the fracture response of shale at different stratification angles, particularly at low stratification angles, where internal fissures and pores were not significantly compacted, resulting in the accumulation of less AE activity. These above studies, through tensile tests on layered rocks, have deeply explored the impact of stratification direction on the tensile strength and failure modes of layered rock masses. In addition to splitting characteristics, the fracture characteristics of rocks allow for a more in-depth study of the formation and propagation of internal cracks. In engineering practice, tensile (Mode I) failure is the predominant fracture mode in rocks. Currently, beyond the four recommended methods and specimen configurations proposed by the ISRM for measuring Mode I fracture toughness [19; 20; 21], the centrally cracked Brazilian disk (CSTBD) specimen [22] has garnered substantial attention due to its straightforward configuration and ease of preparation. For example, [PERSON] et al. [23] conducted fracture tests on CSTBD specimens at different loading rates. [PERSON] et al. [24] and [PERSON] et al. [25] analyzed the relationship between rock fracture toughness and specimen size and crack size. [PERSON] et al. [26] investigated the effect of stratification angles on the Mode I fracture toughness of shale. [PERSON] et al. [27] and [PERSON] et al.