## 5 Conclusions  
A comprehensive study on the tensile properties of rock materials containing fractures and voids has been conducted using the finite element method (FEM) combined with the cohesive zone model (CZM). In the numerical simulations, all initial finite mesh discretizations utilized zero-thickness cohesive elements globally, which facilitated the description of combined continuous and discontinuous properties. Based on this method, the following main conclusions can be drawn:  
1. When the crack angle is small, the crack typically initiates at a specific distance from the initial crack. As the external load increases, the crack propagates and  
Figure 13: Statistics of damaged cohesive elements: (**a**) effect of fissure angle; (**b**) effect of fissure length.  
intersects along the radial direction, potentially leading to specimen failure. The crack propagation mode exhibits local closure. With an increase in the crack inclination angle, the initiation location shifts, becoming more pronounced as the angle further increases. As crack length increases, the initiation location may shift to other parts of the crack or near the hole. Shorter cracks tend to penetrate more easily but do not necessarily exhibit closure, whereas longer cracks may show local closure with a more complex penetration pattern.
2. The load-displacement curves for samples with different fracture angles coincide during the tensile and stability failure stages but diverge in the shear failure stage due to varying fracture angles. The fracture angle has a negligible effect on the overall trend of the load-displacement curve. Similarly, the load-displacement curves for samples with different crack lengths exhibit characteristics of an initial straight rise, followed by three gradual declines, a rapid rise to the peak, a rapid decline, and eventual stabilization. While the fracture length significantly influences the peak value of the load-displacement curve, it has minimal impact on the overall trend.
3. As displacement increases, the proportion of tensile failure undergoes a process of rapid decline, slow rise, another rapid decline, and finally stabilizes. Specimens with crack inclinations of 0\({}^{\circ}\) and 15\({}^{\circ}\) have the highest tensile failure ratio at stability, while those with a 90\({}^{\circ}\) inclination have the lowest. Crack inclination significantly affects the tensile failure ratio. Although fracture length influences the specific value of the tensile failure ratio, it does not affect the overall change trend.
4. As the crack angle and crack length increase, the damage quantity of cohesive force units also increases.  
Conceptualization, methodology, writing--review and editing, H.L. and S.Z.; writing--original draft, visualization, K.L., Y.Z. and L.W.; conceptualization, methodology, funding acquisition, supervision, D.G.; investigation, methodology, W.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Fundamental Research Funds for the Central Universities (3142023005) and the Science and Technology Support Project of Langfang (2023013149).  
Not applicable.  
Data Availability Statement: Data associated with this research are available and can be obtained by contacting the corresponding author upon reasonable request.  
Author [PERSON] was employed by the company Shandong Energy Group Co. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.