# Applied Sciences Free Full-Text Localization and Imaging of Micro-Cracks Using Nonlinear Lamb Waves

# Localization and Imaging of Micro-Cracks Using Nonlinear Lamb Waves with Imperfect Group-Velocity Matching

Nonlinear Lamb waves have attracted attention for the detection and identification of microscopic structural changes in structural health monitoring. However, most identification methods that determine the damage location based on the intersection of elliptical loci inevitably result in positioning errors due to the change in group velocity before and after interaction with damage. In this study, we propose a method focusing on elliptical rings to detect and image microclusters in three-dimensional structures using nonlinear Lamb waves with imperfect group velocity matching. The width of the elliptical ring can be determined by the degree of group velocity mismatch of the nonlinear S0 function. The S0-S0 function pair that satisfies approximate group velocity matching is mainly introduced by the interaction with microcracks. The effectiveness of the proposed method for defect detection was verified by experimental tests and numerical simulations. Although the length of the tested microcrack (about 1 mm) is shorter than the wavelength of the baseline Lamb wave (about 20 mm), the microcrack can be well identified and localized using nonlinear Lamb waves. The experimental results show that the proposed method can more reliably detect microcracks with the following intersection regions: Keywords

To reduce the risk of catastrophic destruction and extend the life span of structures, structural health monitoring (SHM) is widely applied in aerospace, transportation, civil engineering, mechanical structures, etc., to detect defect-related changes in real time with sensitivity. In SHM technology, ultrasonic bear waves have been considered as a promising method to detect damage in plate-like structures due to their attractive advantages of high sensitivity to various types of defects and low attenuation and wide inspection range [1]. Damage detection of linear cracks and holes has been achieved by the ellipsis method based on the time flight of damage waves using a spatially distributed array of sensors connected to a homogeneous slab structure [2, 3]. In addition, efforts have been made to characterize the damaged areas, such as their size and shape, by various characterization algorithms [4, 5]. Qualitatively or quantitatively detecting multiple damage cases [6, 7] in mechanical structures seems to be a challenging task, but there is growing interest. In general, the aforementioned methods based on the linear properties of Lamb waves (such as velocity and attenuation) are usually sensitive to gross defects and open-type defects (such as circular holes, notches, and cracks), and insensitive to small changes in the microstructure (such as precipitates, inclusions, and micro-bioparadots) and contact-type defects (such as dissolution, delamination, and microcracks). If we want to improve the sensitivity of linear Lamb waves to small defects, an effective solution is to increase the excitation frequency, but this will bring additional complexity to the signal processing due to the multimodal and dispersive characteristics of Lamb waves at high frequencies. Recently, another way to overcome this drawback is to utilize the nonlinear properties of Lamb waves [8]. There are basically two types of acoustic nonlinearities, classical and nonclassical, which are brought about by the coupling of nonlinear ultrasound and damage. Classical nonlinearities are mainly associated with the intrinsic nonlinearities of materials, such as contact atomization and imperfections in atomic lattices and the Luxembourg-Gorky effect. [9].## 1. Introduction

As a typical nonlinear lamb wave, highe r-order harmonic wave output is closely related to material nonlinear behavior. Over the past 20 years, the cumulative secondary hig h-conditioning waves of lamb waves have attracted a lot of attention because of the high sensitivity to material nonlinearity. The phase speed consistency and the no n-zero power flow are considered to be the required conditions for the cumulative second hyperonic waves of the lamb wave. [10, 11]. The pair of the mode that satisfies these conditions (for example, S 1-s2, A 2-s4, S 2-s4) is no n-strolled plasticity detection [12], fatigue [13], creep [14], holes [15], and fine. Used for structural damage [16].

Recently, in order to quantitatively evaluate the early stage of material deterioration, an S0-S0 pair was proposed in a low-frequency area that satisfies the approximation phase-speed matching. [17, 18, 19]. Furthermore, the S0-S0 mode pair was also used to detect a closed defect or incomplete connection interface that opens and closes when the waves pass. Mori et al. [20] examined the nonlinear interaction between the S0 mode wave wave and the incomplete substrate by using the intake analysis and numerical simulation. Zhao et al. [21] performed a numerical simulation for the propagation of the S0 lamb wave in a thin plate with a microscopic small crack that is randomly distributed to study the behavior of nonlinear lamb waves. Wang et al. [22] examined the scattered pattern of micro cracks. Wang et al. [22] studied the scattered patterns of S0 lamb waves and fatigue cracks from an analytical, numerical, and experimental perspective. Wang et al. [22] researches the scattered patterns of the S0 lamb waves and fatigue cracks from an analytical, numerical, and experimental perspective, and accurately oriented the fatigue cracks generated in the aluminum plate guide pipe. I found something I could. Yang et al. [23 [23] examined the effects of the length of fatigue and incident waves on the amplitude of the second hyperplas generated by the S0 mode. As a typical no n-linear lamb wave, highe r-order harmonic output is closely related to the behavior of material nonlinearity. Over the past 20 years, the cumulative secondary hig h-conditioning waves of lamb waves have attracted a lot of attention because of the high sensitivity to material nonlinearity. The phase speed consistency and the no n-zero power flow are considered to be the required conditions for the cumulative second hyperonic waves of the lamb wave. [10, 11]. The pair of the mode that satisfies these conditions (for example, S 1-s2, A 2-s4, S 2-s4) is no n-strolled plasticity detection [12], fatigue [13], creep [14], holes [15], and fine. Used for structural damage [16].

Recently, in order to quantitatively evaluate the early stage of material deterioration, an S0-S0 pair was proposed in a low-frequency area that satisfies the approximation phase-speed matching. [17, 18, 19]. Furthermore, the S0-S0 mode pair was also used to detect a closed defect or incomplete connection interface that opens and closes when the waves pass. Mori et al. [20] examined the nonlinear interaction between the S0 mode wave wave and the incomplete substrate by using the intake analysis and numerical simulation. Zhao et al. [21] performed a numerical simulation for the propagation of the S0 lamb wave in a thin plate with a microscopic small crack that is randomly distributed to study the behavior of nonlinear lamb waves. Wang et al. [22] examined the scattered pattern of micro cracks. Wang et al. [22] studied the scattered patterns of S0 lamb waves and fatigue cracks from an analytical, numerical, and experimental perspective. Wang et al. [22] researches the scattered patterns of the S0 lamb waves and fatigue cracks from an analytical, numerical, and experimental perspective, and accurately oriented the fatigue cracks generated in the aluminum plate guide pipe. I found something I could. Yang et al. [23 [23] examined the effects of the length of fatigue and incident waves on the amplitude of the second hyperplas generated by the S0 mode. As a typical nonlinear linear wavy, highe r-order harmonic wave output is closely related to the behavior of material nonlinearity. Over the past 20 years, the cumulative secondary hig h-conditioning waves of lamb waves have attracted a lot of attention because of the high sensitivity to material nonlinearity. The phase speed consistency and the no n-zero power flow are considered to be the required conditions for the cumulative second hyperonic waves of the lamb wave. [10, 11]. The pair of the mode that satisfies these conditions (for example, S 1-s2, A 2-s4, S 2-s4) is no n-strolled plasticity detection [12], fatigue [13], creep [14], holes [15], and fine. Used for structural damage [16].

Recently, in order to quantitatively evaluate the early stage of material deterioration, an S0-S0 pair was proposed in a low-frequency area that satisfies the approximation phase-speed matching. [17, 18, 19]. Furthermore, the S0-S0 mode pair was also used to detect a closed defect or incomplete connection interface that opens and closes when the waves pass. Mori et al. [20] examined the nonlinear interaction between the S0 mode wave wave and the incomplete substrate by using the intake analysis and numerical simulation. Zhao et al. [21] performed a numerical simulation for the propagation of the S0 lamb wave in a thin plate with a microscopic small crack that is randomly distributed to study the behavior of nonlinear lamb waves. Wang et al. [22] examined the scattered pattern of micro cracks. Wang et al. [22] studied the scattered patterns of S0 lamb waves and fatigue cracks from an analytical, numerical, and experimental perspective. Wang et al. [22] researches the scattered patterns of the S0 lamb waves and fatigue cracks from an analytical, numerical, and experimental perspective, and accurately oriented the fatigue cracks generated in the aluminum plate guide pipe. I found something I could. Yang et al. [23 [23] examined the effects of the length of fatigue and incident waves on the amplitude of the second hyperplas generated by the S0 mode. No n-linear wave

In the application of rum waves to SHM, an appropriate fault identification algorizm is usually adopted to give an interpreted and intuitive image in the 3D structure and obtain a fault position. One of the most common damage imaging algorithms is based on the s o-called erypsis method [28, 29], and is determined to detect damage by the elliptical orketage based on the flight time (TOF), and the interaction with the damage. It is assumed that the group speed does not change. In this case, the group speed changes before and after the interaction with the damage, so that a positioning error occurs inevitably occurs. When using S0-S0 function pair and quasi-group speed matching, candidates for other positioning methods are applied to position damage. Zhou et al. [24] experimentally studied global focusing methods based on tim e-i n-decomposition and identified two fatigue damage. [24] was experimentally studied to identify the two fatigue fissures on the aluminum plate. In addition, Li et al. [25] also applied a stochastic algorithm based on a probability scan matrix to identify the position of the small lint of the thin plate, but it has a very hig h-density transducer configuration. Then, it was thought that a microscopy would occur with the highest probability. Furthermore, damage is generally not a point, but has a specific dimension. [26] is usually adopted an appropriate fault identification algorizm in the application of lamb waves to < Span> SHM to give an interpreted and intuitive image in the 3D structure and obtain a fault position. One of the most common damage imaging algorithms is based on the s o-called erypsis method [28, 29], and is determined to detect damage by the elliptical orketage based on the flight time (TOF), and the interaction with the damage. It is assumed that the group speed does not change. In this case, the group speed changes before and after the interaction with the damage, so that a positioning error occurs inevitably occurs. When using S0-S0 function pair and quasi-group speed matching, candidates for other positioning methods are applied to position damage. Zhou et al. [24] experimentally studied global focusing methods based on tim e-i n-decomposition and identified two fatigue damage. [24] was experimentally studied to identify the two fatigue fissures on the aluminum plate. In addition, Li et al. [25] also applied a stochastic algorithm based on a probability scan matrix to identify the position of the small lint of the thin plate, but it has a very hig h-density transducer configuration. Then, it was thought that a microscopy would occur with the highest probability. Furthermore, damage is generally not a point, but has a specific dimension. [26] is usually adopted in the application of the ram wave to the SHM to give an interpreted and intuitive image in the 3D structure and obtain a fault position. One of the most common damage imaging algorithms is based on the s o-called erypsis method [28, 29], and is determined to detect damage by the elliptical orketage based on the flight time (TOF), and the interaction with the damage. It is assumed that the group speed does not change. In this case, the group speed changes before and after the interaction with the damage, so that a positioning error occurs inevitably occurs. When using S0-S0 function pair and quasi-group speed matching, candidates for other positioning methods are applied to position damage. Zhou et al. [24] experimentally studied global focusing methods based on tim e-i n-decomposition and identified two fatigue damage. [24] was experimentally studied to identify the two fatigue fissures on the aluminum plate. In addition, Li et al. [25] also applied a stochastic algorithm based on a probability scan matrix to identify the position of the small lint of the thin plate, but it has a very hig h-density transducer configuration. Then, it was thought that a microscopy would occur with the highest probability. Furthermore, damage is generally not a point, but has a specific dimension. [26]

Therefore, the purpose of this study is to propose a modified erypsis method for accurate positioning and imaging in a thre e-dimensional structure using a no n-linear lamb wave with incomplete group speed matching. The configuration of this study is as follows. In section 2, the concept of the revision method is introduced, and a mathematical formula is established for localization of damage to damage using no n-linear S0 lamb waves with quas i-group speed consistency. Section 3 shows the details of the experimental settings to evaluate the performance of the method of localizing a small house on an aluminum plate. Discuss the experimental results of small slots based on linear and no n-linear characteristics, and compare the results of the proposed method with the results of the conventional elliptical method. Section 4 compares the limited element simulation of the bluring crack and the experimental verification. The proposal method is also verified when the blurring crack is not in the propagation route. Finally, guide the conclusion in section 5.

## 2. Modified Ellipse Method

When the lamb waves encounter a defect, the propagation wave is reflected or partially transmitted. These injured waves transmit sufficient information on damage. For example, the flight time (TOF) and the amplitude can be used to identify damage. TOF is defined as a time for a certain wave to move at a certain distance. In general, TOF shows damage parameters such as position, size, shape, and some linear correlation.

In order to explain the principle of damage detection using TOF, detection route (I, j = 1, 2, ... n, i ≠ ≠ (i ≠ ... n, ... n, ... n, ... n, ... n, ... n, n, n, ... n, ... n, ... n, ... n, ... n, ... n J) is schematically shown in Fig. 1A. Assuming that the actuator T I, the sensor T J, and the failure center are located (x I, y i), (x j, y j), and (x m, y m), using an orthogonal coordinates. These relationships can be described as follows using the wel l-known erypsis method. < SPAN> So, the purpose of this study is to propose a modified erypsis method for accurate positioning and imaging a microscopic crack in a thre e-dimensional structure using a no n-linear lamb wave with incomplete group speed matching. It is. The configuration of this study is as follows. In section 2, the concept of the revision method is introduced, and a mathematical formula is established for localization of damage to damage using no n-linear S0 lamb waves with quas i-group speed consistency. Section 3 shows the details of the experimental settings to evaluate the performance of the method of localizing a small house on an aluminum plate. Discuss the experimental results of small slots based on linear and no n-linear characteristics, and compare the results of the proposed method with the results of the conventional elliptical method. Section 4 compares the limited element simulation of the bluring crack and the experimental verification. The proposal method is also verified when the blurring crack is not in the propagation route. Finally, guide the conclusion in section 5.

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