## 4 Failure Modes
### Failure Dominated by Double Joints: S-30-60  
For the S-30-60 specimen, the crack propagation patterns and AE parameters [34] are illustrated in Figure 4. When the stress is 17.67 MPa (Figure 4a), wing cracks initiate at the  
Figure 3: Schematic representation of AE parameters: (**a**) direct parameters; (**b**) indirect parameters.  
tip of joint 1, with the inner tip strain band exhibiting a higher strain value compared to the outer tip. Simultaneously, a red strain band appears at joint 2, with a light-colored coplanar strain band initiating at the outer tip and a coplanar strain band at the inner tip extending along the joint direction. At this point, a light-colored strain band with a relatively low strain value (\(\sigma\) = 17.67 MPa) emerges in the rock bridge area. Notably, when the loading reaches approximately 120 s, \(b\)-value exhibits alternating increases and decreases over a short period (Figure 4a), and the cumulative tensile crack proportion drops sharply from 97.31% to 86.85% (Figure 4b). This is attributed to the shear slip of the coplanar shear strain band at the outer tip of joint 2, inhibiting the development of tensile wing cracks at the outer tip of joint 1.  
Figure 4: Principal strain field and AE parameter profiles of the S-30-60 specimen: (**a**) principal strain field; (**b**) stress–strain curves, \(b\)-value, and AE counts; (**c**) shear-tension cumulative crack proportion and real-time tensile crack proportion.  
As the stress climbs to 22.89 MPa, the coplanar inclined cracks at the outer tip of joint 1 continue to propagate downwards, forming macro-cracks. A red coplanar strain band appears and extends upwards at the outer tip of joint 2, and inclined strain bands initiate on the right side of joint 2. The wing cracks initiated at the inner tip of joint 1, and the coplanar secondary cracks at the inner tip of joint 2 begin to coalesce. Observing the \(b\)-value of stage II, it alternates between increments and decrements within a brief period, indicating accelerated damage to the specimen (Figure 4b). Examining the real-time tensile cracks (stage II), it drops from 85.78% to 75.11% before rising to 84.74% (stage II in Figure 4b). It suggests that tension-dominated damage occurs during stage II, but the extension and shear slip of strain bands in the rock bridge area between joints and the expansion of the coplanar shear strain band at the outer tip of joint 2 lead to a decrease in real-time tensile cracks.  
Upon reaching a stress level of 23.95 MPa, which is proximate to the peak strength, an examination of the principal strain field during stage III reveals that the coplanar strain band located at the outer extremity of joint 2 extends upwards, ultimately giving rise to the formation of inclined shear cracks. The cracks present between joints coalesce, resulting in the emergence of macro-cracks, while the inclined strain band situated on the right side of joint 2 progresses into inclined shear cracks. Examining Figure 4b, the AE counts reach their peak subsequent to the stress attaining its peak strength. Additionally, the real-time tensile crack proportion during stage III drops from 84.71% to 79.51% (Figure 4c), indicating that accelerated shear slip affects the damage of the specimen. As the stress declines to 13.68 MPa, the coplanar strain band at joint 2's outer tip merges with the upward-extending inclined band on the right, forming tension-dominated cracks. Observing the stress and AE counts during stage IV (Figure 4b), the stress falls in a stepwise manner, and the AE counts spike to 12 \(\times\) 10\({}^{3}\) at time = 238 s. In Figure 4c, the real-time tensile crack proportion remains stable at around 80%, indicating that tension-dominated damage occurs during stage IV.  
Entering stage V, as the stress drops to 13.04 MPa, the tensile cracks at the outer tip of joint 1 propagate to the bottom of the specimen, and the coplanar strain band at the outer tip of joint 2 extends to the top, resulting in specimen failure.