## 3 Results and Analysis
### \(\beta_{\text{t}}\)-Value
#### 3.4.2 \(\beta_{t}\)-Value Analysis  
Previous research has revealed that when \(\beta_{t}\) is less than 1, the specimen is in a stable state; when \(\beta_{t}\) equals 1, it is in a metastable state; and when \(\beta_{t}\) is greater than 1, it is in an unstable state [27]. To process the acquired AE signals using the calculation method described in Section 3.4.1, we calculate the \(\beta_{t}\)-value for different stress stages. Based on the stress distribution characteristics, the data are segmented for analysis.  
Before the initial stress drop, due to compaction or linear elastic behavior, the specimen generates fewer cracks, resulting in fewer AE signals. After the stress drop, the specimen's damage increases, leading to more AE signals. Therefore, we use the initial stress drop as the dividing point, dividing the entire loading stage into three phases: Stage 1 extends from the beginning to the first stress drop, Stage 2 spans from the first stress drop to the final failure, and Stage 3 encompasses the entire loading stage, as illustrated in Figure 3.  
This paper uses a 0-1 specimen as an example to illustrate the changes in \(\beta_{t}\)-value at different AE stages, as shown in Figure 9. From Figure 9, it can be observed that in Stage 1, the \(\beta_{t}\)-value is 1.72. This phase includes a stress drop, during which the rock releases a significant amount of energy, resulting in a higher occurrence of AE signals, indicating an unstable state. In Stage 2, the \(\beta_{t}\)-value is 1.14, which is close to 1. In this phase, AE signals are relatively dense, indicating that the specimen is in a metastable state. In Stage 3, the \(\beta_{t}\)-value is 1.95, encompassing the entire loading process of the specimen, and it ultimately experiences a catastrophic failure.  
\(\beta_{t}\)-value calculations were performed for all specimens, and the results are shown in Table 3. In Stages 1 and 3, the \(\beta_{t}\)-value is greater than 1, indicating that these two stages are both unstable. The experimental results for these two stages  
Figure 8: Stress and cumulated event number versus time (**a**); The relationship between the cumulated event number and time, with a slope value of \(\beta_{t}\) (**b**).  
Figure 9: Distribution of \(\beta_{t}\)-value across different stages, 0–1 specimen. stages 1 (**a**), stage 2 (**b**), stage 3 (**c**).  
stress drops, confirming the consistency between the calculated results and the experimental data. As for Stage 2, the \(\beta_{t}\)-value for the 0-1, 1-2, and 1-3 specimens are all near 1, while the 0-2 specimen's \(\beta_{t}\)-value is greater than 1. This variation is related to the different AE behaviors of the specimens.  
For the 0-1, 1-2, and 1-3 specimens, damage continually intensifies after the first stress drop, leading to a continuous generation of AE signals. In contrast, the 0-2 specimen only produces signals at the moment of collapse after the initial stress drop.  
According to Equation (11), it is evident that the value of \(\beta_{t}\)-value is solely dependent on time and the number of AE events. It reflects the relative rate of AE signal generation concerning time. Within a specific time interval, if AE signals continue to be generated, and considering signal amplitude, time is also continuously changing, resulting in a \(\beta_{t}\)-value approximately equal to 1. However, it is clear that during the initial loading stage of any specimen, if there is a continuous generation of AE signals within a certain time period, the calculation based on Equation (11) would also yield a \(\beta_{t}\)-value close to 1. However, in this stage, it is actually a stable state.  
Therefore, the criterion for determining the instability of material damage and failure based on \(\beta_{t}\)-value is as follows:  
* \(\beta_{t}\) < 1: The specimen structure is in a stable state.
* \(\beta_{t}\) \(\approx\) 1: The specimen structure can be in both stable and unstable states.
* \(\beta_{t}\) > 1: The specimen structure is in an unstable state.