Damage detection using wavelet entropy of acoustic emission waveforms in concrete under flexure

Abstract

This work dives into the spectral realm of acoustic emission waveforms. The acoustic emission waveforms carry a footprint of source, its mechanism, and the information of the medium through which it travels. The idiosyncrasies of these waveforms cannot be visualized from the time-domain parameters. The complex fracture process of the heterogeneous composite, such as concrete, reflects in the spectral disorder of acoustic emission signals. The use of wavelet entropy is proposed to estimate the spectral disorder. To evaluate wavelet entropy, the relative energy distribution in frequency sub-bands is determined using the wavelet transform. The Shannon entropy formulation as a wavelet entropy is utilized for discriminating spatiotemporally distributed acoustic emission events according to their respective level of disorder. The possible twofold application of the wavelet entropy as a signal discriminator and a damage index is qualitatively demonstrated. The increase in the statistical variance of wavelet entropy distribution with the increase in stress level reveals the presence of multi-sources as well as multi-mechanistic fracture process.

Keywords

Concrete acoustic emission wavelet transform wavelet entropy

Introduction

The civil engineering structures are designed for the desired service loads. Due to the large size and the longer expected life of the structures, the risk of early failure is often elevated as service life progresses. The damage caused by varying load conditions and environmental effects as well as unforeseeable disasters, such as earthquakes and fire, may result in substantial loss of structural strength, instability, and lead to ultimate failure. The visual damage indicators, such as cracks and large deformations, may become visible at the critical stage of the damage progress. Hence, monitoring of growing cracks is the ultimate goal for structural health assessment to prevent irreparable damage in the structure.

Concrete being a major construction material is easy to manufacture and cast in situ. However, the heterogeneity of the materials makes its behavior deviate with a large margin making failure prediction difficult. The complexity of the fracture process in concrete is a result of the micro/macrocrack cluster known as the fracture process zone (FPZ) ahead of the crack tip. A variety of fracture mechanisms occur during damage progress, and each fracture mechanism has a different severity level in overall damage. Mechanisms, such as crack bridging, branching, closure, arrest along with deflecting, retarding, and accelerating cracks, are the few labels one can use to elucidate the crack growth within the FPZ of concrete. The structural health monitoring, on the critical mission, is discovering how to determine damage-sensitive features and how to extract information about them from the data measured by sensors, to detect damage or changes to systems. The acoustic emission (AE) technique is one of the structural health monitoring methods, which focuses on the non-destructive evaluation of transient damage progress in structures. It has seen an appreciable amount of development in the last few decades.^1,2 AEs are elastic stress waves produced by cracking of material which can be acquired by piezoelectric sensors mounted on the material surface. AE is the footprint left behind by the dynamic fracture process evolution. Each AE event is considered as an individual crack, although the possibility of multiple cracking cannot be denied. Rate of incoming signals, as well as their parameters, such as amplitude, energy, and frequency content, are usually correlated to the density, intensity, and severity of the cracks. For well-behaved materials, such as metals, AE monitoring is an efficient and accurate technique for source localization and damage characterization. However, due to inherent heterogeneity in cementitious materials, such as concrete, such accuracy is yet to be achieved. Crack initiation and propagation in concrete are caused by many complex phenomena such as cement matrix cracking, aggregate cracking, debonding of aggregates, debonding of reinforcement, and friction between fractured and interlocked surfaces. Therefore, crack mode classification and source identification are the most sought-after problems in the AE of concrete after the source localization problem. The solution to these problems will lead to ascertain the cause of damage to structural health assessment.

Investigation methods based on time-domain parameters of AE signals are competently developed for damage analysis in concrete fracture.^3–5 For crack mode classification, a parametric method uses the rise angle (RA—rise time/amplitude) and the average frequency (AF—counts/duration). Such crack mode classification is often conducted in studies on reinforced (steel and fiber) concrete using the RA–AF method.^6,7 Another crack classification technique known as moment tensor analysis uses a mathematical representation of moment tensor determined by linear inversion technique, both in time and frequency domain using Green’s function.⁸ However, determining Green’s function for concrete-like heterogeneous material is quite a tedious task, and therefore, despite being a reliable method, moment tensor analysis is not as popular as RA–AF analysis for crack classification in concrete.

Contrary to crack mode classification, source identification is mainly focused through spectral analysis, since individual constitutive material components will have their respective signature in the frequency domain of the AE signals. In this regard, notable development has been achieved for source identification through spectral analysis and statistical techniques in AE of composite materials.^9–12 However, the inertia of spectral analysis compared to other application fields has gradually grown in the last few decades for AE of concrete. An early study on spectral analysis of concrete proposed by Rossi et al.¹³ has suggested the partition of signal energy of linear spectrum obtained by Fourier transform in some frequency bands and their respective ratios for spectra classification which can be used for explaining fracture mechanisms. Burud and Chandra Kishen¹⁴ compared AE signals from plain and reinforced concrete using frequency sub-band energy ratios obtained by wavelet transform (WT). Yoon et al.¹⁵ have studied AE signals in corroded reinforced concrete using WT and reported high-energy signals near the failure stage of damage which are composed of several short closely spaced events. A similar observation of AE signal transition from burst type to continuous type when failure approaches in concrete under compression is reported by Wang et al.¹⁶ Sagaidak and Elizarov¹⁷ have concluded that crack growth decreases the dominant frequency in AE spectra for concrete under flexure. Wang et al.¹⁸ have used peak frequency as an AE feature to investigate the influence of aggregate size and stress level on AE in mortar specimens. Farnam et al.¹⁹ analyzed AE waveforms from concrete with varying water-to-cement ratio using the frequency center of gravity as a spectral parameter. Carpinteri et al.²⁰ studied the influence of damage on AE parameters along with AE wave attenuation and scatter and cautioned the use of time-domain AE parameters due to strong distortion and attenuation caused by the heterogeneity of material.

Concrete, a multi-phase composite at the mesoscale, is well known for its inherent multi-scale fracture process. The dispersed energy dissipation through multi-scale cracking bestows the quasi-brittle nature of the material. The AE technique has helped to find historical evidence of the multi-scale cracking, and a power law (Gutenberg–Richter law) model is now being extensively used to describe the frequency–magnitude relationship in concrete. However, multi-scale cracking is the only one aspect of the complex fracture process in concrete. Certainly, more often enunciated fracture mechanisms (crack bridging, crack closure, crack arrest, etc.), fracture modes (modes I and II), and compositional heterogeneity stimulate the multi-scale crack growth. Therefore, the concrete fracture process can also be termed as multi-source and multi-mechanistic phenomena along with being multi-scale. It is hypothesized that the diverse cracking phenomena, occurring in various hydration products of the cement matrix, coarse aggregates, and at the interfacial transition zone, along with applied energy flux may cause the disparity in the AE spectral response. Consequently, the complexity of the fracture process will reflect in AE spectral disorder as damage progresses. Moreover, similar material sources emanate AE signals of roughly similar spectral content, resulting in similar spectral disorder as far as low noise level and the same effect of measuring instruments are maintained. The available literature also confirms the variation in AE spectra as damage passes through transient stress change in concrete. However, the often-studied spectral features, such as bandwidth, dominant frequency, and spectral centroid, does not represent the intricacy of multi-frequency energy distribution present in the AE spectra.

This work is motivated by the need to find the traces of multi-source and multi-mechanistic fracture process in the spectra of AE signals which can ameliorate the understanding of the damage process. In this regard, the disorder of the spectral AE energy distribution is quantified using a measure of wavelet entropy. The wavelet entropy is determined from the distribution of energy in frequency sub-bands evaluated using WT. Many researchers have applied WT to AE for signal detection, denoising, and classification.^21–25 In this work, the WT is utilized for multi-resolution spectral analysis. An experimental study on plain concrete beams under flexure is conducted and AE signals are acquired. The possible twofold application of the wavelet entropy as a signal discriminator and a damage index is qualitatively demonstrated.

WT

The raw signals from any analog or digital instrument are usually acquired in the time-domain, which needs to be transformed for further processing. The sole purpose of the mathematical transformations is to perceive the information in some other function space for deciphering the generative process of the signal. The transformation of a time-domain signal to the frequency domain using the Fourier transform is a routinely followed technique to extract the spectral information of a signal. Fourier transform decomposes the signal into complex exponentials which can be represented by the sum of sinusoids. The representation of the signal in the sinusoidal form offers excellent resolution in the frequency domain due to the well-localized frequency of sinusoids. Although the application of Fourier transform for amplitude–frequency analysis of stationary signals is elementary, its application for time–frequency analysis of non-stationary signal is complicated. Short-time Fourier transform (STFT) is often used to localize frequencies in the time-domain by windowing with the assumption of local periodicity. Despite the guideline provided by the uncertainty principle for window length, the dilemma of window selection remains subjective. On the shortfall of Fourier transform, the WT is an alternate tool that gained dazzling popularity for time–frequency localization in non-stationary signals.

The concept of scaling at the crux of the WT epitomizes a mathematical microscope to scroll through a time series spectra.²⁶ The theory of WT introduced by Grossmann and Morlet²⁷ proposes scaling of a basis function by dilation and translation to facilitate time–frequency signal representation by multi-frequencies band. One of the pioneers in wavelet analysis, a French geophysicist Jean Morlet²⁸ introduced the term wavelet to describe arbitrary square-integrable real-valued functions which he used for seismic time series analysis. Today, wavelet analysis is being used not only for time–frequency analysis but also for signal and image smoothing and denoising, data compression, speech recognition, and so on.

The continuous wavelet transform (CWT) of a signal X(t) with respect to a prototype or mother wavelet $ψ (t)$ can be defined as follows

W_{ψ} X (n, m) = \frac{1}{\sqrt{m}} \int_{- \infty}^{\infty} X (t) ψ (\frac{t - n}{m}) dt

(1)

where a and b are dilation and translation parameters, respectively. Equation (1) is the simple convolution of the signal X(t) with the scaled wavelet $ψ (t)$ and in discrete form, the equation produces K sub-signals of length N for the N samples of the signal X(t) and K level of wavelet scales. The CWT does not necessarily require orthogonal wavelet, and the transformed data are highly correlated due to use of arbitrary scales. However, use of continuous translation and scale parameters cause oversampling which makes CWT computationally expensive. Furthermore, CWT is more suitable for analysis of local differentiability and singularity detection in a signal. It is also impractical to use CWT for discrete signals.

Discrete wavelet packet transform (DWPT) overcomes the impracticality of CWT by a discrete set of the parameters (a, b) such that the signal is decomposed into a mutually orthogonal set of wavelets. Practically, DWPT is implemented as the filter bank representing a cascade of high-pass and low-pass filters.

DWPT

DWPT is an appropriate choice to decompose the discrete AE signals for multi-resolution study. DWPT can be expressed using the same equation (1) except that the DWPT uses scale and translation values based on powers of two. The value of m and n are $m = 2^{k}, n = j \cdot 2^{k}$ , where k is the number of different levels and j is for different translation at each level. The essence of DWPT is to filter the signal through a series of dyadic filters (low-pass and high-pass filters with complementary bandwidths also known as a quadrature mirror filter pair) with the use of downsampling and upsampling, respectively. At any level of signal decomposition, the low-pass signal component is referred to as the approximation, while the high-pass component of the signal is referred to as the detail of the signal. Therefore, the approximation is the high-scale, low-frequency components of the signal. The smallest scale used at the first level of decomposition corresponds to the highest frequency of the signal. The signal is consecutively downsampled as the required sample size reduces according to the Nyquist rate for the increasing decomposition levels with decreasing frequencies. DWPT results in sub-band filtering of the input signal into finer equal-width intervals due to the equipartition of the input signal spectra at each level of decomposition. The resulting components can be consecutively numbered and referred as nodes.

Relative wavelet energy

In this work, discrete WT is used to discretize signals into sub-bands. If $X (t)$ is a AE signal, then it is possible to decompose it into components $X_{j 1}, X_{j 2}, \dots X_{jk}$ , where j is the decomposition level and k is the node number. A node represents a decomposed component (child) of the signal such that the addition of all the node components at a decomposition level gives the original signal. The tree of the hierarchically signal decomposition by DWPT up to third level is shown in Figure 1. Energy of each component can be defined as $E_{j 1}, E_{j 2}, . . ., E_{jk}$ at jth level. Therefore, the normalized energy at each level with respect to total energy gives relative wavelet energy.

The total energy of the signal is

E_{Total} (t) = \sum E_{jk} (t)

(2)

The ratio of energies at different nodes to the total energy is considered to determine energy distribution at different components. Hence, the relative wavelet energy can be written as

p_{jk} = \frac{E_{jk} (t)}{E_{Total} (t)}, k = 1, . . ., 2^{j}

(3)

where $p_{jk}$ is the normalized energy ratio at kth node for jth level decomposition. In this study, four-level decomposition is performed which resulted in 16 sub-band components of signal. The frequency spacing is 31.25 kHz for each sub-band with maximum 500 kHz frequency. Table 1 lists the frequency ranges for each node at fourth level of decomposition.

Table 1.

Frequency range with respect to node of decomposition.

Node	Frequency range (kHz)
1	0–31.25
2	31.25–62.5
3	62.5–93.75
4	93.75–125
5	125–156.25
6	156.25–187.5
7	187.5–218.75
8	218.75–250
9	250–281.25
10	281.25–312.5
11	312.5–343.75
12	343.75–375
13	375–406.25
14	406.25–437.5
15	437.5–468.75
16	468.75–500

Wavelet entropy

The energy ratio satisfies $\sum_{1}^{N} p_{jk} = 1$ ; therefore, it can be considered as a probability-like distribution. Consequently, Shannon entropy can be used to measure the information of energy distribution in the signal with respect to frequency/scale.^29,30 The wavelet entropy can be defined as

S_{W T_{j}} = - \sum p_{jk} (p_{jk})

(4)

where $S_{W T_{j}}$ is the wavelet entropy at $j$ th level of decomposition and $p_{jk}$ is wavelet energy ratio.

The physical meaning of the wavelet entropy is the same as the Shannon entropy. Shannon entropy is known as a measure of average information, uncertainty, or surprise. Similarly, wavelet entropy measures the energy dispersal in various frequency sub-bands of the signal. From the physics point of view, the externally applied energy flux stimulates internal stress concentration at the microdefects in the material, resulting in the rupture of several atomic bonds known as fracture. These ruptured bonds emit energetic vibrations (elastic waves), thereby exciting certain frequencies. The vibrations from multiple elastic waves from several bonds go through scattering, reflections, interference, and other wave phenomena, resulting in a blended waveform. The oscillations captured at the sensors resemble a skyline of a city or mountain ranges in the spectral domain, spiking at various frequency values, disordered and chaotic. The segregation of the signal energy in frequency sub-bands using a multi-resolution WT consolidates the spectral disorder from numerous spikes to fewer columns of a histogram. The energy of these stress waves also show variation of several orders in their magnitude. Therefore, the normalization of the frequency sub-band energies by the total signal energy brings all the signals to the same scale on which a probability-like function can be assumed. Eventually, the wavelet entropy evaluates the disorder of the assumed probability distribution of relative wavelet energy over the spectra. Application of wavelet entropy is well recognized in the field of biomedical, power systems, mechanical vibrations and fault detection, artificial intelligence, and so on.

Experimental setup

Three notched beam specimens of plain concrete under three-point bending are tested in laboratory conditions. The mix design of concrete is done using the American Concrete Institute (ACI) 211.1³¹ method and the mix proportion of the cement, fine aggregate, and coarse aggregate obtained is 1:1.86:2.61 by weight. The maximum size of coarse aggregate is 12 mm and the fine aggregates are passing through 4.75 mm sieve. A water-to-cement ratio of 0.5 is used for preparing the concrete mix. The dimensions of the beams are shown in Figure 2. A computer-controlled servo-hydraulic machine was employed for testing under crack mouth opening displacement (CMOD) control. Loading rate was set to 1 µm/s for all the specimen. Center point deflection of beams was measured using a linear variable differential transformer (LVDT) of 5 mm range, while load was recorded using load cell of 40 kN capacity. A Physical Acoustic Corporation (PAC) system was used to monitor the AE throughout the test. Four resonant type R6D AE sensors were mounted on the beams, as shown in Figure 2. AE sensors with resonant frequency around 55 kHz was used to acquire AE waveforms. Due to weak strength of AE signals, preamplifier with 40 dB gain was set for signal amplification. A threshold limit was set to 45 dB for background noise reduction. The reference level for the decibel scale used in the AE measurement was 1 µV. Sampling rate of 1 MHz was used to ensure good frequency and time resolution of the signals. All waveforms crossing threshold level were stored digitally. Waveforms below the threshold level were neglected. The load versus CMOD curve obtained for one of the specimens is shown in Figure 3.

Wavelet selection and signal processing

Absolute criteria for wavelet selection are difficult to specify as wavelet selection also depends on the type of signal to be processed. Symmetry, compact support, vanishing moments, and regularity are the few properties of wavelets generally considered for various applications. The study of energy distribution in the frequency domain is the aim of this work; therefore, orthogonal wavelets are the appropriate choice due to their energy preserving property. Haar, Daubechies, Symlet, Fejér-Korovkin, and Coiflet are the widely used orthogonal wavelets for various structural health monitoring applications.^32–36 Daubechies, Symlet, and Coiflet are the most used regular orthogonal wavelets for AE studies in metals, composites, and ceramics.^37–42 A preliminary investigation, on mother wavelets suggested in AE literature of concrete and rock fracture studies,³⁸ was performed using energy-to-Shannon entropy ratio criteria.⁴³ Daubechies (db4) wavelet was selected based on maximum energy-to-Shannon entropy ratio, being computationally less expensive. Inappropriate selection of mother wavelet may cause improper representation of the signal in frequency sub-bands which may result in erroneous wavelet entropy values.

According to Nyquist’s sampling theorem, the maximum frequency which can be acquired by 1 MHz sampling rate is about 500 kHz (i.e. half of the sampling frequency). The effect of the narrow-banded sensors used in this work can be deconvoluted using the frequency response of the sensors provided by the manufacturer. The sensors have high sensitivity in the specified frequency range, therefore, flattening of the sensor effect might also flatten important features of the signals and eventually, alter the spectral disorder. Therefore, the wavelet entropy is determined without altering the acquired signals. Denoising in post-processing is also avoided and the signals were processed as raw as possible to avoid biases. The mechanical test was performed in isolated laboratory conditions which helped in noise reduction from surroundings resulting in good signal-to-noise ratio. The signals were further decomposed into four levels using “Daubechies db4” wavelet. The DWPT produces $2^{j} = 2^{4} = 16$ sub-bands at fourth-level decomposition with frequency spacing of $F_{s} / (2^{j}) = 31.25$ , where $F_{s} = 1 MSPS$ is the sampling frequency.

Results and discussion

CMOD control allowed stable crack growth in the notched concrete beam specimens. The tested plain concrete beams exhibited post-peak softening behavior until failure, as shown in Figure 3. The average load carried by the beams is 3.30 kN. A notch is provided to ensure the propagation of single dominant crack at the center of the beam specimens which is easier to monitor using AE. One of the tested specimens is shown in Figure 4.

Spectra of AE waveforms by Fourier transform

A typical AE signal obtained from the concrete fracture is shown in Figure 5(a). The spectral distribution of signal amplitude computed using the Fourier transform for the same signal is shown in Figure 5(b). The lowest frequency of the signal around 0.95–1 kHz is not visible in Figure 5(b); therefore, it is shown in Figure 5(c) by expressing frequency on the logarithmic scale. There are multiple pronounced peaks present in the spectra of the signal. The dominant frequency is located in the range of 105–110 kHz with the highest peak. The second dominant frequency appears at around35–40 kHz. A distinct small peak is visible around 145–150 kHz. No significant frequencies are present beyond 200 kHz. The Fourier transform of the signal infers the lack of periodicity in the AE signals. Apart from the above information, the Fourier transform does not convey any other meaningful information about the signal and therefore WT is used as a tool for further investigation.

Evaluation of relative wavelet energy and wavelet entropy by DWPT

The non-stationary and non-periodic AE signals require the WT tool for detailed spectral analysis. All the AE signals acquired from concrete fracture may not be useful for damage analysis. Processing all the signals is also computationally expensive and therefore we have used only those signals which contribute to events. The hits are multiple copies of the same signal originated from a source and acquired at different locations by multiple sensors. The four hits of an event in the time-domain are shown in Figure 6. Figure 6 demonstrates the approximate similarity of the voltage variation over the duration of the signal for all hits of an event. One cannot assure this similarity just by looking at signals; therefore, correlation and coherence analysis between two signals need to be performed to define the degree of the similarity between signals.⁴⁴ Figure 7 shows the scalogram of the same four hits computed using CWT. The scalogram is the absolute amplitude of CWT expressed as a function of time and frequency. The meaning of scale in the WT is similar to the meaning of frequency in Fourier transform. However, an inverse relationship exists between scale and frequency (i.e. low scale corresponds to high frequency and vice versa). Therefore, the equivalent of the scalogram from the frequency perspective is the spectrogram produced by the STFT. The non-stationary character of the signals can be observed from the scalogram. The frequency content of non-stationary signal changes with respect to time. The change in the magnitude of frequencies over the duration of the signals is evident from Figure 7, which justifies the use of WT for the spectral study of AE signals instead of Fourier transform.

For multi-resolution analysis of the signals, the DWPT is used as described in section “DWPT.” The resulting 16 components at the fourth level of signal decomposition are shown in Figure 8. Figure 8(a) shows high-scale (low-frequency) components from nodes 1 to 8 and Figure 8(b) shows low scale (high-frequency) components from nodes 9 to 16. The spectral energy distribution ratios of the signals are computed as discussed in section “Relative wavelet energy.” The energy ratios are illustrated in Figure 9 for an event. The height of the bars in Figure 9 indicates the fraction of energy content in each node of the fourth-level signal decomposition. Energy ratios are calculated only for the first hit as a representative of an event. The first hit as a representative waveform of an event is a reasonable assumption due to the similarity of hits belonging to the same event as discussed above.

The wavelet entropy is determined from relative wavelet energy ratios using equation (4). The entropy ranges from 0.2747 to 0.7324. The histogram of computed wavelet entropy for a specimen up to failure is shown in Figure 10. Negatively skewed long-tailed distribution of wavelet entropy is distinct from the distribution of other AE parameters (amplitude and energy show positively skewed distribution). The negatively skewed distribution indicates the presence of higher number of high entropy events. The physical meaning of high and low entropy events is elaborated in the following section.

Discrimination capability of wavelet entropy

The theoretical upper and lower limit for the wavelet entropy is 1.2041 (substituting p = 1/16 in equation (4) for 16 nodes) and 0 (p=1) resulting from the white noise and a sinusoid, respectively. The white noise has uniformly distributed energy overall frequencies ranges, while the sinusoid has a well-localized frequency. To demonstrate the signal discriminating capability of the wavelet entropy, we select two signals resulting in two extreme entropy values. 0.7324 and 0.2747 are the observed maximum and minimum wavelet entropies, respectively, in this work. The amplitude of the signal with entropy 0.7324 is 44 dB and the other is 63 dB. The signals of extreme entropy in time-domain are shown in Figure 11(a). The difference between signals is clearly visible as the high entropy signal having high-frequency noise-like oscillations. However, these high-frequency oscillations are certainly not noise and dominate the spectra as seen from scalograms in Figure 11(b). The scalograms in Figure 11(b) highlight the prominent difference between the spectral content of the signals. A striking difference between the dominant frequency components lasting in the signals is visible in the scalograms. The signal with minimum entropy has a highly localized magnitude in the frequency range of 32–64 kHz, which lasts over the middle one-third signal duration. However, for the signal with high entropy, two distinct spots of high magnitude frequency at 0.5 and 0.7 ms are visible in the same frequency range of 64–128 kHz. Figure 11(c) shows comparative spectral energy distribution for the maximum and minimum entropy events. The energy of the signal with minimum entropy is concentrated in nodes 2 and 4 and therefore it exhibits lower entropy. Most of the events constitute a larger part of energy contribution spread over nodes 1–8 exhibiting higher entropy values.

The exact source identification of the signals (whether the signals originated from aggregate cracking or cement matrix cracking) is difficult to make using the wavelet entropy parameter alone. The source identification deserves a separate detailed study with X-ray tomography like a microscopic instrument to pinpoint the source in the material volume as well as advanced machine learning tools to learn and automatically perform source identification over numerous AE signals originating from the complex fracture process of concrete. However, we suspect that, on statistical grounds, the lowest entropy event is possibly caused by coarse aggregate cracking. A comparatively higher proportion of cement matrix contributes to the majority of AE events; therefore, the events belonging to the distinguishable higher entropy range between 0.55 and 0.75 can be attributed to cement matrix cracking

Wavelet entropy and damage

The wavelet entropy does not show any correlation with amplitude and absolute AE energy of the signal, as evident from Figures 12 and 13, respectively, as well as the same conclusion can be drawn from Figure 14 for other parameters derived from AE waveforms. This implies that the wavelet entropy is a unique parameter of AE waveform which discriminates signals based on the spectral distribution of the energy.

The temporal evolution of wavelet entropy is shown in Figure 15, along with load–time history. At first, the wavelet entropy appears randomly distributed over the timeline. If the wavelet entropy is considered as a random variable, then its progressive distribution parameters may construe the damage process. The distribution of the wavelet entropy is difficult to fit in distribution models, such as Gaussian and Log-normal, due to its negatively skewed long tail characteristic. Special long-tailed distribution models to determine appropriate distribution parameters can be developed for further detailed analysis.

Momentarily, its geometric mean and variance are shown in Figure 16. The geometric mean of the entropy reaches its minimum near peak load. An increase in high entropy events result in mean entropy increase and it reaches a maximum around 400 s of test duration beyond which it starts to decrease due to low entropy events. Such a decrease, after an increase in entropy mean value, indicates active participation of those fracture mechanisms which generate low entropy events that were rare up to this damage state. The variance of the distribution measures the degree of the distribution spread. The variance of entropy distribution increases almost monotonically up to failure. A slight decrease in entropy variance can be observed between 200 and 400 s due to concentrated low entropy events. After 400 s of test duration, low entropy events are more frequent indicating an increase in the complexity of the fracture process. The increasing variance indicates enriched diversification of the fracture process through a variety of mechanisms and sources within the material. Therefore, it is the manifestation of a multi-mechanistic as well as multi-source fracture process.

As damage progresses, the increase in crack length reduces the uncracked ligament area of the concrete beam. The reduction in uncracked ligament area elevates stress level in damage-prone zone due to bending stresses. It is also surprising to note that the reduced uncracked ligament area of concrete is still capable of producing variety of high and low entropy events which are seldom in the initial phase of damage where uncracked ligament area is more. The increase in wavelet entropy variance also indicates that the concrete near failure is adept to energy dissipation in wide spectral ranges which is a result of stress concentration and redistribution taking place in the heterogeneous microstructure of concrete. Such energy dissipation in wide spectral range surely contributes to the quasi-brittleness of the concrete.

The spatial distribution of localized events is shown in Figure 17. The color mapping and size of the bubbles in the shown scatter plot are parameterized by wavelet entropy. We expected that the high entropy events might occur in the fracture core zone⁴⁵ to exhibit a higher variety of fracture mechanisms compared to the rest of the FPZ. The fracture core zone is a volume centrally located within the FPZ with densely distributed microcracks which further develops into a dominant crack. However, the random scatter of entropy in the FPZ volume displays no particular pattern with respect to the location of the events. This implies that the fracture process of concrete is equally diversified all over the FPZ volume and no distinct region, such as fracture core zone, is identifiable from the spatial distribution of wavelet entropy.

Summary and conclusion

In this work, the AE spectrum is considered as a representative of the multi-source as well as themulti-mechanistic fracture process of concrete which results in multi-scale cracking. The complexity of the fracture process may reflect in the disorder of the spectral domain of the AE signals. Consequently, the use of the wavelet entropy as a quantitative measure of AE spectral disorder is proposed. The acquired AE signals from the flexural test on plain concrete beams were decomposed using WT. The wavelet entropy is then defined on the basis of normalized energy distribution in the frequency sub-bands. The signals associated with extreme wavelet entropy values are elaborated through scalograms to manifest the signal discrimination capability of wavelet entropy based on spectral disorder. If the generation of a certain level of disorder in the AE signal spectra is the inherent property of a material component present in a composite, then such signals can be discriminated using wavelet entropy leading to source identification. The distribution of wavelet entropy values up to failure is observed to be long-tailed and negatively skewed. The distribution parameters can be used to infer the damage. Moreover, wavelet entropy is shown uncorrelated to other AE parameters; therefore, it adds an extra dimension to the information ameliorating the understanding of the fracture process. It is difficult to separately identify the effect of source and crack mode from the spectra of the signals. However, the increasing variance of the wavelet entropy distribution provides an evidence of the multi-source and multi-mechanistic fracture process together.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

JM Chandra Kishen

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