## 4 Acoustic Emission Characterization
### Characterization of Rock Damage Precursors  
Many scholars have adopted the b-value of acoustic emission (AE) to study the precursors of rock failure, believing that researching the b-value can help to profoundly understand the mechanisms of rock failure and has significant practical importance for the prevention of instability and failure in rock engineering [47; 48]. It is calculated as follows:  
\[b=\frac{lge}{\overline{M}-M_{min}} \tag{15}\]  
where \(\overline{M}\) is the average magnitude; \(M_{min}\) is the minimum magnitude; and \(\mathrm{e}\) is a constant approximately equal to 2.72. The magnitude/20 is generally used in place of the magnitude, modifying the above equation to [49; 50]:  
\[lg[N(A/20)]=a-b\times(A\times 20) \tag{16}\]  
where \(A\) is the acoustic emission amplitude; \(N(A/20)\) is the cumulative frequency of amplitude \(\geq\mathrm{A}/20\) acoustic emission events; \(a\) is an empirical constant; and \(b\) is the acoustic emission b-value. The acoustic emission b-value of Equation (14) was modified according to the literature [51] to:  
\[b=\frac{20 lg\mathrm{e}}{\overline{A}-A_{min}} \tag{17}\]  
where \(\overline{A}\) is the average amplitude; and \(A_{min}\) is the minimum amplitude. Figure 11 shows the variation in the b-value over time. From the figure, we can observe the following: Compaction of original cracks stage: The b-value shows a short-term increase, mainly due to the compaction of original microcracks, which generates minor fluctuations. Elastic stage: The b-value remains almost stable with no significant fluctuations, indicating a steady development. Unstable crack propagation stage: As the stage progresses, the b-value starts to fluctuate, signaling the initiation of microcracks within the sandstone and the first warning sign. With the continued increase in load, the sandstone enters the unstable crack propagation stage. In this stage, the b-value experiences a sudden and larger fluctuation than at the first warning point, suggesting the formation of larger micro-fractures within the sandstone. At the peak stage, the b-value increases dramatically, indicating the appearance of a large number of macro-cracks on the 45\({}^{\circ}\) sandstone surface, which are rapidly expanding. This suggests that the sandstone is about to fail, marking the third warning point. From these observations, it is noted that the first warning point is not very obvious, the second warning point is accompanied by a faint rockburst sound during the experiment, and the third warning point already implies an impending failure, which  
occurs rapidly. Therefore, the author believes that the second warning point should be considered as the best predictive point for the precursor of 45\({}^{\circ}\) sandstone failure to prevent instability in rock engineering. The Felicity effect refers to the phenomenon where acoustic emission (AE) activity significantly increases during the cyclic loading process when the load experienced by the specimen is less than the maximum load it has previously endured. The Felicity effect characterizes a kind of irreversibility in the AE process and is defined as:  
\[FR_{i}=\frac{P_{i+1}}{P_{max}} \tag{18}\]  
In the formula, \(FR_{i}\) represents the Felicity ratio for the \(i\)-th cycle; \(P_{max}\) is the highest load level previously experienced; \(P_{i+1}\) is the load level at which effective acoustic emission is resumed during the (\(i+1\))-th loading process. However, there is no unified standard for identifying effective acoustic emission due to limited research findings. To ensure a consistent analysis of the experimental data, this study adopts the guidelines recommended by the Composites Acoustic Emission Monitoring Committee (CARP) in the United States, which is that acoustic emission is considered effective when it exceeds 20 ring counts as the load increases by 10% [13,14]. Figure 12 shows that the FR value decreases with the increase in load cycle level, and the rate of decrease accelerates over time.