## 3 Results and Discussions
### Cluster Analysis of AE Signals  
K-means is an effective pattern recognition technique which distinguishes superimposed AE signals and is performed using MATLAB. Before the multi-parameter analysis of the AE signals, it is necessary to select the representative AE signals and confirm the optimal cluster number. To ensure the efficiency of the algorithm's execution, according to the reference [32], five AE parameters such as AE rise time, duration, energy, peak amplitude, and frequency are selected to carry out the clustering task [46]. In addition, the optimal number of clusters is determined by the silhouette value which is defined as:  
\[s=\frac{1}{n}\sum_{i=1}^{n}\frac{[\min(a(x,k)-b(x))]}{\max[b(x),\min(a(x,k))]} \tag{2}\]  
Here, in cluster \(k\), \(a(x\), \(k)\) and \(b(x)\) represent the average distance of inter-class and intra-class, respectively. The range of the silhouette value is from \(-1\) to \(1\). The number of clusters evaluated by the silhouette index is shown in Figure 10. It can be found that the maximum silhouette value is 0.95 at \(k=3\) and this results in a better clustering quality. Therefore, indicating that \(k=3\) is optimal for cluster analysis. Figure 9: Progressive amplitude and peak frequency distribution at 30%, 60%, and 90% of the failure load. (**a**) amplitude distribution; (**b**) peak frequency distribution. The cluster results for each class are shown in Figure 11 and three clusters, CL-1, CL-2, and CL-3, are obtained. The irregular pentagons are interpreted for the average values of five AE parameters and linked by the red line. Obviously, there are the differences in the statistical dispersion for clusters, which means the unsupervised pattern recognition algorithm leading to a better separation of the AE data. The results indicate that the K-means algorithm is suitable for the purpose of classifying the AE data and distinguishing between the different damage modes. More importantly, it is necessary to establish the relationship between clusters and damage modes. The peak frequency can accurately describe information on fault-specific AE signals and characterize the change in damage mode associated with progressive failure. Therefore, according to the fundamental assumptions, it is possible to associate specific failure mechanisms with three clusters. The mechanism-based analysis of AE signals to establish the relationship between clusters and damage modes is shown in Figure 12. The clusters referred to as CL-1, CL-2, and CL-3 are clearly separated by peak frequency as shown in Figure 12a. It is found that class CL-1 and CL-2 appear simultaneously as the applied load under 50% of the failure load. In addition, the SF trend is shown in Figure 12b and indicates that the combination of S\({}_{1}\) and S\({}_{2}\) corresponds to the loss of the strain energy  
Figure 11: Average properties of three clusters with regard to the five AE parameters. (**a**) Class CL-1, (**b**) CL-2, and (**c**) CL-3. Figure 10: Silhouette value as the function of the number of clustering class, \(k\). storing capability. According to the literature [38], these typical AE signals are associated with matrix cracking and fiber/matrix debonding. The possibility to discriminate class CL-1 and CL-2 depends on the different process of visco-elastic relaxation for matrix cracking and the delamination of composites. The relation between peak frequencies \(f\), the relaxation times \(t\), elastic module \(E\), and densities \(\rho\) can be defined as:  
\[f\sim\frac{1}{t}\sim\sqrt{\frac{E}{\rho}} \tag{3}\]  
Therefore, the matrix cracking occurs in a lower frequency band than the fiber/matrix debonding. Class CL-1 corresponds to the dominant frequency from 0 to 100 kHz which is lower than that of CL-2 with a frequency range from 150 to 300 kHz. Additionally, class CL-1 and CL-2 are associated with matrix cracking and fiber/matrix debonding, respectively. However, the matrix cracking corresponding to the frequency range in this paper is higher than that reported in [46].