## 4 Failure Modes  
As a non-destructive monitoring method, DIC technology is unaffected by the environment and loading during the monitoring process, enabling accurate recording of crack initiation, propagation, and coalescence. DIC calculates the displacement field of the specimen by comparing the corresponding speckle areas in the original (unloaded) and deformed (during loading) stages, further allowing the computation of the principal strain field on the speckle surface [28]. The present study selects three representative specimens based on the failure surface morphology and crack coalescence patterns to analyze the evolution of the principal strain field, accompanied by corresponding stress levels. The naming convention for joints follows the description in Figure 1b, with the harder layer designated as the lower joint (joint 1) and the softer layer as the upper joint (joint 2). For ease of analysis, the lower joint is defined as joint 1 and the upper joint as joint 2. The DIC technology captures the evolution patterns of macro-cracks based on variations in the principal strain field. Therefore, DIC technology can only monitor the macro-failure behavior of specimens. However, the development of macro-cracks is comprised of numerous micro-cracks. Micro-fracture events within the specimen propagate in the form of acoustic waves. Consequently, this experiment also employs the AE technique to record micro-fracture events, enabling an accurate analysis of the micro-failure mechanisms. Key parameters frequently utilized for AE analysis encompass AE counts, \(b\)-value, RA-AF distribution, and cumulative crack ratio. Among them, AE counts are directly obtained  
Figure 2: Mechanical properties of defective specimens: (**a**) S-30 group, (**b**) S-60 group, (**c**) S-90 group, (**d**) S-120 group, and (**e**) S-150 group; and (**f**) peak strength under various upper joint angles. through monitoring; RA-AF distribution and cumulative crack ratio (shear or tensile) are derived from the functional relationship between the AE rise time and amplitude; and the b-value, originally proposed by [PERSON] and [PERSON] [29], is employed to evaluate the relationship between earthquake magnitude and amplitude. The \(b\)-value can also represent the slope of the amplitude distribution. Its calculation formula is as follows [30]:  
\[\text{lg}N=a-bM \tag{1}\]  
where \(a\) is constant, varying from region to region, and \(b\) is the \(b\)-value, which represents the ratio of the number of small events to large events in earthquakes [31]. \(M\) denotes the earthquake magnitude, and \(N\) is the number of earthquakes with magnitudes falling within the interval [\(M\), \(M+\Delta M\)]. Subsequently, this formula is adapted to assess the degree of rock failure, transforming Equation (1) into the following:  
\[\text{lg}N=a-b(A_{dB}/20) \tag{2}\]  
\(A_{dB}\) denotes AE amplitude. An increase in the \(b\)-value indicates more small failures, while its drop points to more significant failures. The variation trend of the \(b\)-value serves as an indicator to discern whether the specimen is experiencing a severe or progressive failure stage. Furthermore, since the AE signals generated by micro-fractures are recorded in the form of waves, the types of micro-fractures can be identified through the AE parameters \(RA\) and \(AF\)[32]. \(RA\) and \(AF\) can be calculated by utilizing the following equations [33]:  
\[RA=RT/AEA \tag{3}\]  
\[AF=AEC/AED \tag{4}\]  
where \(RT\) is the rise time, \(AEA\) is the AE amplitude, \(AEC\) is the AE counts, and \(AED\) is the duration. The units for \(RA\) and \(AF\) are ms/V and kHz, respectively. Tensile micro-cracks generate longitudinal waves with low \(RA\) values, while shear micro-cracks produce transverse waves with low \(AF\) values. Therefore, based on the \(AF/RA\) ratio, it is possible to discern whether the AE signal corresponding to a micro-crack is tensile or shear in nature. [PERSON] et al. [33] suggest a threshold value of 200 for cement mortar.