## 3 Experimental Results and Theoretical Analysis
### Acoustic Emission Parameters
#### 3.3.2 Distribution of Acoustic Emission Energies  
The acoustic emission energies of AE events are determined by fast numerical integration of the square voltage of signals [17], as  
\[Q=\frac{1}{R}\int_{t_{1}}^{t_{2}}U^{2}(t)dt, \tag{5}\]  
where \(Q\) and \(U\) are the acoustic emission energy and voltage, respectively, and \(R\), \(t_{1}\) and \(t_{2}\) are the reference electrical resistance and the starting and ending times of the voltage transient record, respectively. Figure 7: The time evolution of the AE counts, cumulative AE counts and axial stress: (a) B-30, (b) B-45, (c) B-60. Firstly, we used the histogram statistics method to analyze the acoustic emission energies during the whole process of compression failure of the specimens. The probability distribution function is P(\(Q\)) and is calculated by appropriate linear binning of acoustic emission energies. Figure 8 shows the energy distributions in log-log plots for selected samples with different flaw inclination angles. From Figure 8, the probability distribution function in log-log plots is quite linear. Therefore, we suggest that the distributions of acoustic emission energies of specimens containing en echelon flaws follow a power law, as  
\[P(Q)dQ\sim\frac{Q^{-r}}{Q_{\min}^{1-r}}dQQ>Q_{\min}, \tag{6}\]  
where \(Q_{\min}\) and \(r\) are the lower cutoff needed for normalization and the optimum exponent, respectively. The optimum exponent is the absolute value of the straight slope in Figure 8. Figure 8 shows that the optimum exponent of the acoustic emission energy distribution of the specimens changes between 1.62 and 1.68. All previous uniaxial compression experiments have identified the power law exponent as ranging between 1.33 and 1.97 [18]. Therefore, the optimum exponent in this study lies within the range of power law exponents from previous studies, which suggests that this result is reliable. Secondly, to use the histogram statistics method to further analyze the distributions of acoustic emission energies in the two stages of compression failure of the specimens, the large number of acoustic emission energy data analyses required the construction of histograms and the choice of the number of bins. To improve the accuracy of the power law exponent, we used the maximum likelihood method to estimate the power law exponent for the two stages during the uniaxial compression tests, as  
\[r^{\prime}(x_{\min})=1+n[\sum_{i=1}^{n}\ln\frac{x_{i}}{x_{\min}}]^{-1}, \tag{7}\]  
\[\sigma=\frac{r^{\prime}(x_{\min})-1}{\sqrt{n}}+O(r-1), \tag{8}\]  
where \(x_{i}\), \(i=1,2,3,\ldots,n\) are the observed values of \(x\) such that \(x_{i}\geq x_{\min}\). In this study, \(x\) represents the AE energy. \(\sigma\) is the standard error. Here and elsewhere, we use \"prime\" symbols such as \(r^{\prime}\) to denote estimates derived from data. A no-prime symbol such as \(r\) denotes the optimum value, which is often unknown in practice [24]. Figure 9 shows the results for the two stages during the uniaxial compression test. This analysis leads to a plateau that defines the optimum exponent; while the plateau is not as clean as possible, the optimum exponent can be determined from the vertical position of the shoulder revealed by this curve [18]. From Figure 9, \(r_{1}\) is the optimum exponent of the first stage, and \(r_{2}\) is the optimum exponent of the second stage. The optimum exponents of the first stage of selected samples with  
Figure 8: Log–log plots of the energy distribution of the AE events: (**a**) B-30, (**b**) B-45, (**c**) B-60. different flaw inclination angles are greater than those of the second stage in Figure 9. At the same time, combining Figures 8 and 9, the optimum exponent of the second stage is equal to that of the whole process of compression failure. Therefore, we determine that the optimum exponent of the acoustic emission energy distribution changes in different stages with increasing axial stress and gradually decreases as failure is approached, which acts as a warning signal for the impending major collapse.