# Crack-Length Estimation for Structural Health Monitoring Using the High-Frequency Resonances Excited by the Energy Release during Fatigue-Crack Growth

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## Abstract

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## 1. Introduction

^{−4}mm/cycle and 1 × 10

^{−3}mm/cycle. Four resonant sensors and four wideband sensors were used to record the AE signals. Via waveform inspection, crack events were separated from grip events. It was found that the use of wideband sensors can enhance the ability to make the necessary distinction of AE signals during a fatigue experiment. Roberts and Talebzadeh [1] discussed the correlation between acoustic emission count rates and crack propagation rates. Various signal processing methodologies and clustering techniques have been used to differentiate AE signals caused by various activities [12]. Correlations between crack characteristics and AE signal features have also been analyzed using different techniques [13,14,15,16]. Recently, a novel Savitzky–Golay filter (SGF) was used for damage location and quantification in a bridge under a moving load [17]. A vibration-based structural damage identification method for structures under the influence of temperature and noise was developed by Huang et al [18]. Various techniques have been developed to detect damage in engineering structures based on the interaction of waves with damage [19,20,21,22,23]. Researchers have effectively used clay boundaries to prevent the reflection of waves from the plate boundaries during ultrasonic experiments. Clay boundaries have been used for reducing reflections from AEs during a low cycle fatigue experiment [8]. A FEM numerical study was also conducted of the effects of providing damping at the boundaries of a plate to prevent boundary reflection [24].

## 2. PWAS Sensor and Sensing Equation

_{3}direction, the constitutive piezoelectric equations are [41]:

- $\begin{array}{l}{S}_{p}\u2014\mathrm{Strain}\mathrm{on}\mathrm{the}\mathrm{PWAS}\end{array}$
- $\begin{array}{l}{\mathrm{s}}_{pq}^{E}\u2014\mathrm{Compliance}\end{array}$
- $\begin{array}{l}{\sigma}_{q}\u2014\mathrm{Stress}\mathrm{on}\mathrm{the}\mathrm{PWAS}\end{array}$
- $\begin{array}{l}{d}_{iq}^{},{d}_{kp}^{}\u2014\mathrm{Electrical}-\mathrm{mechanical}\mathrm{coupling}\end{array}$
- $\begin{array}{l}{\epsilon}_{ik}^{T}\u2014\mathrm{Electric}\mathrm{permittivity}\end{array}$
- $\begin{array}{l}{E}_{k}\u2014\mathrm{Electric}\mathrm{field}\end{array}$

_{31}operation mode, was used for AE sensing. Thus, Equation (2) was the analytical equation used to obtain the voltage sensed by the PWAS from the in-plane strain.

## 3. FEM Analysis of Fatigue-Crack AE

^{3}) were considered for the analysis. Thirty-millimeter non-reflective boundaries (NRBs) were applied at the edges of the model using COMBIN14 spring-damper elements in ANSYS to eliminate the reflections from the boundaries of the plate. Figure 2 presents the application of NRBs at the boundaries. On the top and bottom surfaces, and at both ends of the plate, the COMBIN14 elements were implemented. Starting from zero in a linear pattern on the plate, the damping coefficients of the elements were varied gradually. The maximum stiffness and damping values were applied at the edges of the plate. The detailed information can be found in Ref. [19].

_{11}dipole excitation was defined using equal and opposite nodal forces. The time profile of the excitation was defined as a cosine-bell function with 0.5 µs as the rise time of the excitation [42]. Figure 3c,d presents the time domain of the excitation and the frequency spectrum. The acoustic waveforms generated due to the dipole excitation were obtained by performing the finite element simulation.

_{xx}+ ε

_{yy}) extracted from the FEM simulation was plotted. Figure 4a represents the wave propagation pattern in a non-cracked specimen and Figure 4b shows the wave propagation pattern in a specimen with an 8 mm crack. As can be observed, although the excitation was the same for all simulations, the wave propagation patterns differed due to the existence of the crack. This difference is due to the resonance of AE signals originating at the crack. AE energy generated at one crack tip travels to the other tip and generates propagating waves. This causes some additional resonance and acts as an additional wave source, causing the difference in the AE wave propagation pattern compared to the no-crack situation.

_{xx}and ε

_{yy}of the AE signal at the nodes where the PWAS was located were obtained from the FEM simulation. The nodal strain data were integrated numerically, and the resulting voltage response was calculated according to Equation (2). The PWAS voltage response was evaluated for the cases in which there was no crack, a 4 mm crack, and an 8 mm crack. The numerically calculated PWAS response for the no-crack case is presented in Figure 5. For 4 and 8 mm cracks, the resulting voltage response at the PWAS is presented in Figure 6 and Figure 7, respectively. Figure 5a,b, Figure 6a,b and Figure 7a,b present the nodal in-plane strain response (ε

_{xx}+ ε

_{yy}) 25 mm from the crack and the frequency spectrum. The frequency spectrum was obtained using a discrete Fourier transform with a sampling frequency of 10 MHz. Figure 5c,d, Figure 6c,d and Figure 7c,d present the 7 mm PWAS response 25 mm from the crack and the frequency spectrum. As observed in Figure 5d, Figure 6d and Figure 7d, the nodal response was modified by the PWAS resonance according to the tuning curve corresponding to the dimensions of the PWAS. The effect of the tuning curve weakened the AE signal nodal response peaks. The nodal response has specific peaks and valleys in its frequency spectrum. It should be noted that, up to 1500 kHz, the 4 mm crack has two peaks in the frequency spectrum. However, in the case of the 8 mm crack length, the nodal AE signal frequency spectrum has four peaks; that is, the number of peaks doubled as the crack length doubled. More precisely, the crack length and peaks in the frequency spectrum of the AE signal have a proportional relationship. This was also observed in the peaks of the integrated effect due to the finite-size 7 mm PWAS, which only showed a weakening effect on the peaks at higher frequencies. This proportional increment in the peaks in the frequency spectrum of the signal is due to the change in the resonance of the AE signal at the crack. Due to the change in the crack length, the resonance of the AE signal at the crack changes, which causes the variation in the frequency spectrum peak–valley pattern at the PWAS.

## 4. Experimental Setup

#### 4.1. Specimen Preparation

#### 4.2. AE Experimental Setup

## 5. AE Experiment Results and Discussion

## 6. Summary, Conclusions, and Future Work

#### 6.1. Summary

#### 6.2. Conclusions

#### 6.3. Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**An M

_{11}moment tensor excitation was applied at the crack tip for FEM simulation of AEs due to crack growth. (

**a**) Top view of the M

_{11}moment excitation generated using dipole forces (F

_{1}); (

**b**) thickness view of the M

_{11}moment excitation; (

**c**) waveform of smooth-step excitation; (

**d**) frequency spectrum of smooth-step excitation.

**Figure 4.**Wave propagation pattern of surface strain (ε

_{xx}+ ε

_{yy}) due to (

**a**) no crack and (

**b**) 8 mm crack.

**Figure 5.**FEM simulation AE signal sensed at 25 mm in the case of no-crack. (

**a**) Nodal in-plane strain response of AE signal; (

**b**) frequency spectrum of nodal in-plane strain response of AE signal; (

**c**) PWAS response of AE signal; (

**d**) frequency spectrum of PWAS response of AE signal.

**Figure 6.**FEM simulation AE signal sensed at 25 mm. (

**a**) Nodal in-plane strain response; (

**b**) frequency spectrum of nodal in-plane strain response; (

**c**) 7 mm PWAS response of AE signal (

**d**) frequency spectrum of 7 mm PWAS response.

**Figure 7.**FEM simulation AE signal sensed at 25 mm in the case of an 8 mm crack. (

**a**) Nodal in-plane strain response; (

**b**) frequency spectrum of nodal in-plane strain response; (

**c**) 7 mm PWAS response (

**d**) frequency spectrum of 7 mm PWAS response.

**Figure 8.**AE test specimen installed with the PWAS sensor. Non-reflective clay boundaries (NRBs) were provided on the specimen to avoid the reflection of AE signals from the specimen boundaries.

**Figure 10.**AE signal at a crack length of 4 mm: (

**a**) experimental signal recorded using the PWAS sensor; (

**b**) experimental frequency spectrum; (

**c**) simulation frequency spectrum.

**Figure 11.**AE signal for a crack length of 8 mm: (

**a**) experimental signal recorded using PWAS sensor; (

**b**) experimental frequency spectrum; (

**c**) simulation frequency spectrum.

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**MDPI and ACS Style**

Joseph, R.; Mei, H.; Migot, A.; Giurgiutiu, V.
Crack-Length Estimation for Structural Health Monitoring Using the High-Frequency Resonances Excited by the Energy Release during Fatigue-Crack Growth. *Sensors* **2021**, *21*, 4221.
https://doi.org/10.3390/s21124221

**AMA Style**

Joseph R, Mei H, Migot A, Giurgiutiu V.
Crack-Length Estimation for Structural Health Monitoring Using the High-Frequency Resonances Excited by the Energy Release during Fatigue-Crack Growth. *Sensors*. 2021; 21(12):4221.
https://doi.org/10.3390/s21124221

**Chicago/Turabian Style**

Joseph, Roshan, Hanfei Mei, Asaad Migot, and Victor Giurgiutiu.
2021. "Crack-Length Estimation for Structural Health Monitoring Using the High-Frequency Resonances Excited by the Energy Release during Fatigue-Crack Growth" *Sensors* 21, no. 12: 4221.
https://doi.org/10.3390/s21124221