## 5 Discussion  
### Macroscopic Failure Modes  
Because of the extremely rapid crack propagation during the failure stage, numerous cracks initiate nearly simultaneously, posing significant challenges in capturing the entire failure process, even with the assistance of high-speed cameras. Therefore, researchers have qualitatively analyzed the crack evolution behavior of post-failure specimens. In Figure 7, the crack evolution diagrams during the failure of specimens containing unparallel defects are recorded. Upon observing Figure 7, it is evident that when the angle of joint 1 is 90 deg, varying the angle of joint 1 results in the failure of the specimen being predominantly governed by joint 1. This is due to the fact that the crack initiates first at the tip of the vertical joint. As the coplanar strain band at the upper tip of the joint penetrates the bedding plane, the coplanar tensile strain band at the lower tip accelerates its propagation, leading to the dominant failure of the specimen. When joint 1 is vertical, consider specimen S-90-90 as an example. The tensile crack starts at the outer tip of joint 1 and moves towards the bottom of the specimen. Meanwhile, the crack at the inner tip extends towards and penetrates the bedding plane, forming another tensile crack. Joint 2 does not exhibit crack propagation. Specimens S-30-90, S-60-90, S-120-90, and S-150-90 all experience instability and failure due to the propagation of tensile cracks initiated at the tip of joint 1.  
When the angle of joint 2 is 30 deg, as the angle of joint 1 increases from 30 deg to 150 deg, the final failure mode of the specimens follows a pattern of double-joint dominance--joint 2 dominance--double-joint dominance. S-30-30 and S-150-30 exhibit instability and failure dominated by both joints, while the rest are dominated by joint 2. For S-30-30, the wing cracks at the outer tip of joint 1 propagate to the bottom of the specimen, causing tensile failure. The wing cracks at the outer tip of joint 2 propagate to the top of the specimen, and the wing cracks between the joints penetrate the bedding plane, leading to rock instability. Similarly, when the angle of joint 2 is 60 deg, as the angle of joint 1 climbs, the pattern of failure modes follows double-joint dominance--joint 2 dominance--double-joint dominance. S-90-60 and S-120-60 exhibit instability dominated by joint 2. For S-90-60, the coplanar secondary cracks initiated at the outer tip of joint 2 propagate to the top of the specimen, generating shear slip. The inclined secondary cracks initiated at the inner tip penetrate the bedding plane and propagate to the bottom of the specimen, dominating the tensile failure. When the angle of joint 2 is 120 deg, the pattern of failure modes caused by increasing the angle of joint 1 is consistent with that of a 60 deg angle for joint 2. Notably, when the angle of joint 2 is 150 deg, the increase in the angle of joint 1 leads to a pattern of joint 2 dominance--double-joint dominance. Specimens S-120-150 and S-150-150 exhibit instability dominated by both joints.  
To observe the variation pattern of specimen failure modes with joint angles, Table 2 is provided. In summary, as the angle of joint 1 increases, the failure modes exhibit a cyclical pattern when joint 2 is at 30 deg, 60 deg, or 120 deg. Initially, the failure is mainly influenced by the double-joint effects. It then shifts towards being dominated by joint 2, and eventually returns to being dominated by the double-joint effects again. This is because when the angle of joint 1 is small, the initiation stress of wing cracks is lower, resulting in faster crack coalescence between joints and faster crack propagation at the tips. As the angle of joint 1 approaches vertical, the initiation of wing cracks becomes more difficult, and the failure of the specimen gradually becomes dominated by joint 2. The study found that the failure modes of double-layer rocks are distinctly different from those of single-rock layers [35; 36; 37; 38].