Prediction of fiber breakage and matrix cracking in polymeric composites under low-cycle fatigue regimes by fuzzy and wavelet clustering of acoustic emission signals

One of methods for detecting cracks and estimating their growth in materials such as composites is the acoustic emission technique. The detection of damages, cracks and their growth in industrial composite structures, under static and dynamic loads, has a significant importance, in order to prevent any damages and increase the reliability. Therefore, achieving required technical knowledge in this field, can be helpful in repairing and the maintenance of the part in industries. The prediction of the damage in polymeric composites under static loads has been already investigated by researchers; however, under cyclic loadings, researches about this behavior are still rare. In this study, by acoustic emission sensors and analyzing experimental data, the damage, including matrix cracking, the fiber breakage and other damages (debonding, fiber pull-out and delamination) during dynamic loading was investigated. At the first stage, standard specimens were made by the pure resin epoxy and the pure carbon fiber, subjected to monotonic tensile loading and then, the frequency of the failure was extracted. Then, composite specimens were loaded in the low-cycle fatigue regime. Mechanical test results and acoustic emission data were analyzed by fuzzy C-Means and wavelet transform methods and then compared to each other to find the percentage of failures in first, mid- and last cycles by the differentiation of failure types. Results clearly indicated that the acoustic emission approach is useful and an effective tool for identifying and detecting damages in polymeric composites.

In this study, according to the reviewed sources, a perforated composite sample of carbon fibers and the epoxy resin in two directions and low-cycle fatigue loading were selected for testing. The wavelet transform method and fuzzy clustering for acoustic emission data analysis was used. Moreover, due to the fact that less research has been done in the field of monitoring composite structures with acoustic emission under low-cycle fatigue loads; therefore, this type of loading was selected for carbon fiber composite samples. The results were tested and analyzed. The type of loading and dimensions of the sample were completely in accordance with the standard [25]. The conditions of data collection and the expression of results in different cycles could be considered as a part of the innovation of this research.

Materials and Tests
In this study, composite specimens were manufactured of single-sided carbon fiber manufactured by the Sika Company (Sika Wrap-230 C) with a specific gravity of 230 gr/m 2 and the epoxy resin manufactured by the Sika called CR-80. The CH-80-2 hardener was used to harden the resin, which was combined with the resin in a weight ratio of 30%. The thickness of each layer of single-sided fabric was equal to 0.13 mm, which after making the samples and combining with the resin, the thickness of each layer was approximately 0.22 mm. After layering, in order to improve the quality of the manufactured samples and to control the volume percentage of the resin and fibers, as well as to control the final thickness, the layers were pressed very carefully by two polished steel plates and baked at room temperature. The produced composite samples plates were removed from the mold after 24 hours and placed in the room temperature, 20 to 25°C, for one week before cutting. Then, the composite sheet was made with the dimension of 250 × 350 mm and the thickness of 3.2 mm, in the [03/902/02]s layout. Then, standard specimens were cut from this sheet according to ASTM D5766 [26] using the water-jet approach. Finally, a hole with a diameter of 6 mm was drilled in the center of the samples. Using the water-jet during cutting operations minimized damages such as the delamination to near zero.
To investigate the damages in the samples by the acoustic emission approach, the characteristics of the acoustic emission signals related to each of the damages must first be determined. To this purpose, samples of the pure resin and pure fibers were made. To make pure resin samples, after hardening was added to the resin, they were poured into the mold and removed from the mold after one week. To make pure fiber samples, carbon fiber fabric with a width of 20 mm and a length of 150 mm was cut and glued to aluminum plates.
To obtain the acoustic emission signals related to the resin and fiber failures, first the pure resin and pure fiber tensile tests were performed at constant rates of 2, 5, 10 and 20 (mm/min). For all samples, two sensors were connected to the samples to record the acoustic emission signals. For pure fiber samples, the sensors were attached to aluminum tubes and all samples were subjected to quasi-static loading. The signals were amplified with a factor of 40 dB by two pre-amplifiers and the appropriate threshold was set at 30 dB. Three samples were loaded for the reproducibility at each rate. According to ASTM D5766, the layers must have at least two different angles in manufacturing. Moreover, the sample length should be in the range of 200 to 300 (mm) and the sample width should be 36 (mm). The hole should also be in the center of the specimen with a width to diameter ratio of 6.0 [25].
For repetitive tensile loading, it was first necessary to extract the maximum deformation of the specimens during damage. Therefore, the samples were loaded at a lower range. To this purpose, a sample was first subjected to tensile loading and the maximum amount of the deformation during damage was obtained. The loading case of the samples was in the form of the controlled displacement. It was done in two steps. In the first stage, the samples were subjected to the constant displacement and different frequencies. In the next stage, the samples were loaded with the constant rate and different displacements. The threshold of acoustic emission signals was set at 37 dB by using trial and error methods. In these tests, the acoustic emission signals were recorded by two sensors, each was located 2 cm from the center of the hole. Acoustic emission signals were amplified 40 dB by two 2/4/6 preamplifiers. In order to calibrate the sensors, the pencil lead break (PLB) method was utilized, according to the ASTM-E976 standard [26,27].

Analysis Methods
In this research, the fast Fourier transform (FFT), the wavelet packet transforms, and fuzzy clustering methods were used to investigate the condition of structures by the acoustic emission signal processing and to determine the structural health monitoring (SHM) or the damage detection.

Fast Fourier Transform (FFT)
For the signal frequency analysis, one of the analysis methods is discrete Fourier transform (DFT), in which the fast Fourier transform is a method to reduce the amount of computations.
Fourier transforms transmit the wave from the temporal space to the frequency space. The DFT can be used in functions that are sampled in the same time space and the time interval [28,29].

If
is the values of the samples and is the number of these samples and is their number, which varies from 0 to − 1, can be written as a series as Relation (1) [29].
According to Equation (5), the calculations for the number of samples are in the order of 2 . In this case, is a power of two, and one of the FFT methods can be used to reduce the computational value. Data can be divided into even and odd categories at each stage. Instead of calculating for the whole data, binary Fourier transforms can be recursively combined to reach the original Fourier transform. The data is divided into two pairs, and , and Relation (6) is rewritten [29].
The first part of Equation (6) is the discrete Fourier transform of even elements and the second part is the discrete Fourier transform of odd elements multiplied by .

Wavelet Transform
A wavelet is a wave with a mean value of zero and a finite period. If the function with two conditions has a finite energy function and the integral of the function is in the range (−∞, ∞) equal to zero, a wavelet or mother wavelet is called as Relations (8) and (9) [24,30,31]: Relations (10) and (11) show the discrete wavelet transform (DWT) and inverse it, respectively [18]: Where, ( ) is the processed signal, ( , ) is the wavelet transform coefficient, is the decomposition level, is the transfer parameter and is the mother wavelet.
In this research, the wavelet packet transform (WPT) is used because the wavelet transform divides the signal into two parts, general and detailed, only in the first level. In the next levels, only generalities are decomposed, while in the wavelet transform, the packages of all components at all levels are decomposed into low-frequency segments, which are related to generalities, and high-frequency, which is related to details. This difference is shown in Figure 1, which has been used in previous research in recent years [32]. As mentioned, in transforming a packet wavelet at each level, in addition to the analysis of generalities, the details are also decomposed into two levels of generalities and details.
The frequency range of generalities and details at each level is calculated with Relations (12) and (13) [33]: Which is the vector data rate and the 2 −1 number of components at the level. The components of the desired wavelet pack are decomposed into at level . The energy of the components is also defined as level , which is expressed mathematically in Relation (14) [24,34]: And the total signal energy from the relation (15) can be calculated [24,34]: The ratio of the energies of each component to the total energy of the signal determines the energy distribution in each of the decomposed components. The energy distribution indicated by ( ) is shown in Relation (16) [24,34]:

Fuzzy Clustering Method
The fuzzy clustering method refers to a clustering technique, in which each data belongs to a cluster and has a degree. This method is one of the advanced clustering methods introduced by Bezdak. A set of data including data to different clusters want to be clustered, = { 1 , 2 , … , } and each data defined by characteristic. That's mean, = { 1 1 , 2 2 , … , } where is in the set ( is a next space). Since all -properties have different units, we must normalize each property with a single-scale scale before clustering [24,35].
The objective function method is used to cluster data into different clusters. The main purpose of this function is to bring the distance between data in a particular category and a point called the center point of that category to the lowest value and to create the maximum distance between the center points of each category relative to the other category [24,36].
For a point like , the function will be the value of the membership in the data and the cluster as a Relation (17) [36]: It should be noted that for a particular point in all clusters the sum of all membership values will be equal to one and the cluster cannot be empty of any data or contain all data in the form of Relations (18) and (19) [36]: The objective function used in fuzzy clustering will be in the form of Relations (20) and (21) [36]: Where is the distance between from the center and is the -value of the amine data membership in the cluster. Then, ́ is the amount of weight that determines the fuzzy value of clustering. Moreover, in Relation (22) is the center of the cluster [24,37].
When using fuzzy clustering, first the number of clusters is determined for data with values between zero and one as . Then, the matrix of the random membership function is selected.
These values are assigned to each of the data. The values associated with the centers of the clusters are then determined, and the operation continues again until the Relation (23) is established.
Where, is the number of iterative steps and is the error value that is calculated compared to the previous step and must be less than the specified value to stop the clustering process [38].

The effect of the loading rate
The results of the force-displacement diagrams of the tensile test are shown in Figure 2. It shows that with increasing the loading rate, the maximum amount of the force increased and it was directly related to it. Moreover, there is no significant behavior or trend could be seen in the amount of the displacement corresponding to the maximum force, which was in an agreement with the results of other researchers [39][40][41][42][43][44]. Although Gilat et al. [45] showed that in E-862 and PR-520 resins, the strain rate had an effective change on the material behavior.
The resin type in this research was CR-80.

The effect of the loading rate
The results of the force-displacement diagrams of the tensile test are shown in Figure 5.
Unlike the pure resin tensile test, the pure fiber tensile test indicated that as the loading rate increased, the maximum force decreased and this trend was inversely related to it. Moreover, there was no effective difference in the amount of the displacement corresponding to the maximum force, as in the pure resin test. In other words, no obvious trend could be found for the displacement at the fracture, by enhancing the loading rate. Therefore, there was a good match between these results and the results of another research [43].

Fatigue Tests of Composite Samples
Perforated composite specimens were subjected to two different types of loading. In the first series, loads with a constant amplitude and different frequencies were applied. In the second series, loads with a constant frequency (or rate) and different amplitudes were applied. In the following, the lifetime of the samples in each type of loading was examined. The effect of the frequency and the amplitude was also discussed.

The effect of the loading rate on the lifetime
Perforated composite samples were each loaded separately with different frequencies, the results of which can be seen in Table 1. In these tests, the constant amplitude of 7.0 mm was considered. As can be seen from the results, with increasing the frequency, the lifetime decreased and the amount of the maximum stress increased. Figure 8 shows the maximum stress-lifetime diagram of the samples, which is mentioned in Table 1. Figure 9 indicates the behavior of the material under the fatigue load. The closer the part was to its final lifetime and the longer the loading cycle, the amount of the maximum stress decreased. Figure 9 can predict the lifetime of the part under the fatigue load with the constant amplitude and the variable frequency. In a fatigue load with a constant amplitude, with increasing the loading rate, the lifetime of the sample shortened. These results were matched with the previous research [48].     Figure 10 indicates the maximum stress-lifetime diagram of the samples, which is mentioned in Table 2. Figure 11 illustrates the occurrence of the softening phenomenon of the material. With increasing the number of load cycles and the closer the sample was to its final lifetime, the maximum stress decreased. From Figure 11, the fatigue lifetime could be predicted by changing the amplitude, which was matched with the previous research [48].

The classification of damage mechanisms by the wavelet method
After recording the acoustic emission signals data by the sensors, the wavelet packet conversion was used to determine the percentage of damage mechanisms in each test. To convert the wavelet packet, the acoustic emission signals were broken down into three levels and into eight components using a code written in the MATLAB software. For fatigue tests, data related to first, mid-and last cycles were also analyzed by the wavelet method. All related figures are depicted in the appendix, at the end of the article.

The classification of damage mechanisms by the wavelet method for the constant amplitude
The results of the analysis of the acoustic emission signals for the constant amplitude and the variable frequency fatigue tests were calculated by the wavelet method for the first, mid-and last cycles of each test. The percentage of each type of damage mechanisms such as the fiber breakage, matrix cracking and other damages in each cycle was expressed. Then, the total percentage of each damage mechanism was calculated according to the results obtained for these three cycles. Finally, an overall average was taken from the average of the damages. It represents the percentage of damages in a total of three cycles for four samples (Table 3). As can be seen in the results of Table 3, with increasing the loading frequency, the average percentage of the fiber damage increased, and the matrix cracking decreased and other damages were found. These results were matched with the previous research [49,50]. Such results were also represented in the literature [51], only for the fiber breakage failure mechanism.

The classification of damage mechanisms by the wavelet method for the constant rate
In this section, mechanisms of the damage with the wavelet method for fatigue loading with the constant frequency and the variable amplitude were investigated. The results were given for the first, mid-and last cycles. Then, the average percentage of each type of damage is calculated for each sample. Finally, an overall average is taken from the average of the damages. It represents the percentage of failures in a total of three cycles for four samples (Table 4). As can be seen in the results of Table 4, with increasing the loading amplitude, the average percentage of the fiber breakage decreased, and the matrix cracking and other damages increased. These results were in good agreement with previous researches [18,44,50].

The comparison of the average results of the wavelet method for the constant amplitude and the constant rate
In Table 5

The classification of damage mechanisms by the fuzzy clustering method
To analyze the acoustic emission signals by the fuzzy clustering method, six characteristics of the peak time, the multiplication, the energy, the amplitude, the average frequency and the maximum frequency were used. In order to select the most effective parameter of the acoustic characteristics and to reduce the volume of information, the principal component analysis (PSA) method was utilized. In this method, in addition to displaying the best part of acoustic emission data, the volume of information was reduced and the most effective features were selected. First, each signal that contained a 6-dimensional space was analyzed. With the help of this method, the covariance between data was calculated. Then, the Eigen-vectors were calculated. Next, the percentage of the variance for each of them was obtained. The two vectors with the highest percentage of the variance could transfer from 6D to 2D with the minimal data reduction.
After reducing the dimensions made by the main components, the data points were clustered by the fuzzy method. In this case, they were divided into three main mechanisms; the matrix cracking damage, the fiber breakage and other damages (debonding, fiber pull-out and delamination). In the fuzzy clustering method, first, the best center of each cluster was determined based on the lowest limit of the objective function and the iterative methods.
Then, acoustic emission data that were close to each of these centers were considered as data of that class. As mentioned, data were divided into three different clusters and have their own characteristics, one of which is the frequency distribution. Each data before clustering had its own feature. One of these features was the maximum frequency. Therefore, it is possible to determine the damage mechanism of each cluster with the help of this feature related to data of each cluster. In order to determine which failure mechanism each cluster belongs to; data of each cluster was plotted based on the frequency distribution. The frequency distribution was the use of the maximum frequency associated with each data. According to the tests of the pure resin and pure fibers and their frequency range, it was possible to determine which failure mechanism belongs to each cluster.
After classifying and clustering data, the percentage of each of the failure mechanisms of the two types of fatigue loading related to the fuzzy clustering method was calculated. The classification of failure mechanisms by the fuzzy clustering method for fatigue loads with the constant amplitude for test data of samples 1 to 4 is mentioned in Table 6. The results of Table   6 showed that under loading of the constant amplitude and the variable frequency, with increasing the loading frequency, the percentage of matrix cracking and other damages increased and the percentage of the fiber breakage decreased. The classification of damage mechanisms by the fuzzy clustering method for the constant frequency fatigue loading is given in Table 7 for test data of samples 4 to 7. The results of Table 7 indicated that in fatigue loads with the constant frequency and the variable amplitude, with increasing the loading amplitude, the percentage of the fiber breakage and other damages increases and the percentage of matrix cracking decreased. Table 8 compared the fuzzy clustering results for the constant amplitude and the constant frequency fatigue loads. The results of Table 8 illustrated that the largest difference in the damage percentage in fatigue loads with the constant amplitude and the constant frequency belong to matrix cracking, which was about 12.5%. The percentage difference of other damages was 6.6% and the percentage difference of the fiber damage was 5.9%. Such results were also represented in the literature [51], which showed the highest frequency ranges would be achieved during the fiber breakage [44,[50][51][52].

The comparison of the results of fuzzy and wavelet clustering methods
A comparison of the results of the wavelet method and the fuzzy clustering technique is shown in Table 9. As shown in Table 9, it could be concluded that the largest difference represented in the literature [51], it showed, there was a good concurrence in the results of the wavelet analysis. Both of these techniques were useful tools for monitoring and characterizing the failure modes in composite materials [51].

Conclusions
In this study, by acoustic emission sensors and analyzing experimental data, the damage, Therefore, with these components, damages occurring in structures in the service could be well monitored to prevent the damage or increase. The results of this study could be summarized as following; • The acoustic emission method was a very practical and useful tool for monitoring the health of various polymer-based composite structures.
• The effect of the loading rate for the pure fiber tensile test showed that as the loading rate increased, the maximum force decreased and was inversely related to it. The matrix cracking frequency range was 250-100 kHz, the fiber breakage frequency range was 420-500 kHz and the frequency of other damages was 420-250 kHz. The results of the effect of the loading frequency on the lifetime indicated that with increasing the frequency, the lifetime decreased and the amount of the maximum stress increased.
• The results of the effect of the loading range on the lifetime of samples indicated that when the loading frequency was constant and the amplitude increased, the lifetime of the samples decreased and the maximum stress increased. With increasing the number of load cycles and the closer the sample was to its final lifetime, the maximum stress decreased.
• The results of the analysis of the acoustic emission signals for the constant amplitude and the variable frequency fatigue tests illustrated that with increasing the loading frequency, the average percentage of the fiber damage increased, and the matrix cracking decreased and other damages were found.
• A comparison of the results of the wavelet and fuzzy clustering methods demonstrated that the largest difference between these techniques was related to estimate the damage rate of the fiber breakage in the constant frequency fatigue loading. Moreover, the lowest difference was related to determining the percentage of the fiber breakage damage in the constant amplitude fatigue loading. The wavelet closed method was more accurate in determining the percentage of each damage.
In further investigations, the non-destructive tests could be used before fatigue testing to find any initial defects through the manufacturing process. These known initial conditions in the specimen could increase the prediction model in a reliable process.

Appendix
In this part, the diagram for analyzing acoustic emission data by the wavelet and fuzzy methods is depicted.
Fatigue tests in the constant amplitude The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of Interest
For this research, there is no conflict of interest for all authors.