Theoretical and experimental investigation of Lamb waves excited by partially debonded rectangular piezoelectric transducers

The paper proposes a new hybrid approach technique to simulate acousto-ultrasonic wave excitation and propagation due to operation of the partially debonded piezoelectric transducer attached to a plate-like structure. The semi-analytical boundary integral equation method is applied to calculate guided waves propagation in the unbounded structures and to separate different guided waves in the piezo-induced wave-fields. The obtained model is verified experimentally using the scanning laser Doppler vibrometry. Eigenfrequencies are calculated and analysed for various sizes of the transducer and for different bonding conditions between the transducer and the waveguide. The impact of the transducer’s height, size and debonding area on symmetric and antisymmetric Lamb waves excitation is analysed. The paper demonstrates that one-sided debonding of the transducer exerts intense influence on the distribution of the wave energy among the excited Lamb wave modes, while center debonding has a sizable impact only at relatively high frequencies.


Introduction
Structural health monitoring (SHM) is known as the continuous and automated method for monitoring and evaluating the condition of a load-carrying structure based on data acquisition, post-processing and analysis of the measured data [1]. The reliable long-term attachment or integration of sensors on structures, which should be inspected, is a major prerequisite. The main goal of the SHM system is to provide adequate information on the state of the inspected structure to find structural damages before they reach critical levels and provide adequate information for condition-based maintenance [2,3]. The acquired data is used to identify incipient damage, providing low-cost inspection compared to a conventional manual examination and reduction of the total system Smart Materials and Structures Smart Mater. Struct. 29 (2020) 045043 (19pp) https://doi.org/10.1088/1361-665X/ab75a1 5 Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. downtime. Moreover, nowadays novel techniques employing data acquired by sensors are proposed to determine the damage location and size as well as to estimate the remaining life-time before the system failure due to the presence of the defect.
The health monitoring of structures using ultrasonic waves has been proved to be a reliable and cost-effective tool [4]. If plate-like structures are considered, Lamb waves are widely used for the inspection due to their capability to propagate over long distances without significant attenuation and with sensitivity to all types of structural defects [5]. Many model-based studies of guided wave propagation in elongated structures confirm the convenience and relevance of the use of guided waves for SHM and nondestructive testing (NDT) applications. It was demonstrated that Lamb waves' sensitivity to notches depends on the defect depth to plate thickness ratio [6]. Cho and Lissenden [7] showed the potential of piezo-generated guided waves in the monitoring of fatigue crack growth based on the transmission coefficient depending on the crack size. Experimental and finite element-based studies of Lamb wave interaction with defects in an aluminium plate were provided [5,8]. Recently, Shen and Cesnik [9] investigated nonlinear scattering and mode conversion of Lamb waves interacting with breathing cracks. Besides the above-mentioned papers, a large number of studies of guided waves interaction with defects has been published in the last decades (for more references see reviews on guided waves applications in SHM and NDT [10][11][12]).
Debonding between a piezoelectric wafer active sensor (PWAS) and a host structure could occur due to different factors such as bonding defects, impact loading, highest forces at the bonding interface, environment effects, fatigue effects and others [13][14][15][16][17]. The defective sensor can significantly influence the damage detecting algorithms output and lead to false diagnostic reports. Investigation of the effects of the PWAS debonding on vibration control of smart beams performed in [13] showed that a debonding located at the end of the piezoelectric transducer can significantly worsen the control of the first several modes of the beam. Kumar et al [14] analysed the performance of an active control system with healthy and unhealthy actuators. They ascertained that debonding in actuator reduces both its loadcarrying ability and electro-mechanical potential and influences active damping and active stiffening effects. In [17], authors employed improved layerwise theory and the FEM to investigate sensor partial debonding influence on the active vibration control of a smart composite laminate. A finite element model for a piezoelectric plate with edge debonded actuators was developed in [18]. Based on this model, the authors showed that the edge debonding of actuators results in a considerable degradation in actuation authority and vibration control performance. The effects of a bonding layer including its possible degradation on the coupled electromechanical behaviour of piezoelectric actuators subjected to high-frequency electric loads were analysed in [19].
There have been developed several methods to detect sensors' failure. In [20], a technique for sensor debonding monitoring based on the change in voltage relations between the segmented electrodes of a piezoelectric patch was suggested. It was numerically shown that the developed technique is applicable to detect one-sided debonding and also central debonding. In [21], the sensor error function was derived using the measurements from the health sensor, which was later on employed to detect and isolate the instants of sensor failure. It was also shown that debonding affects the electrical impedance of the piezoelectric sensor [22] and, therefore, it might be used for the inspection of piezoelectric transducers themselves [16].
To develop effective damage detection algorithms, a detailed understanding of Lamb waves excitation and interaction with various structural defects is required. Related phenomena can be studied in laboratory experiments, but their understanding can be achieved only with the help of a combination of experimental, numerical and analytical approaches. Thus, a reliable mathematical model simulating Lamb wave excitation, propagation and scattering in elongated plate-like structures is needed. To date, a variety of pure numerical [23][24][25] and semi-analytical [26][27][28][29] approaches have been applied to model Lamb waves excitation by a perfectly bonded piezoelectric transducer. Thus, the finite element method (FEM) was applied in [23] to simulate three-dimensional Lamb wave propagation excited by a phased array transducer. The spectral element method was employed in [24] to model wave propagation induced by a built-in piezoelectric actuator. In [26], the authors used a one-dimensional model to simulate the dynamic interaction of a piezoelectric actuator with an elastic half-plane. Additionally, a modelling approach based purely on the spectral element method was shown in [30]. The semianalytical approach was suggested in [27] for PWAS-structure interaction, where the PWAS was simulated using a lamina model excluding vertical displacements and shear strains [27]. Another implementation of the semi-analytical approach was proposed in [28], where the boundary integral equation method (BIEM) was used to model perfectly bonded PWAS. For practical applications detailed understanding of the waves propagating in structure is essential, therefore, several hybrid methods for simulating piezo-induced ultrasonic guided waves were proposed. In [31], the FEM was employed to model piezoelectric actuator and afterwards the calculated displacement field was used in the local interaction simulation approach to model guided waves. Another hybrid approach was presented in [32], where the authors modelled a perfectly bonded PWAS using expansion via Chebyshev polynomials, while the solution in the host structure was obtained via the BIEM. Though this approach provides an analytical solution in the waveguide, it does not suit for faulty sensor modelling. In the semi-analytical hybrid approach (SAHA) [29], a piezoelectric actuator is simulated by the frequency domain spectral element method (FDSEM) [33], while the propagation of the excited Lamb waves in the laminate structure is modelled via the semi-analytical BIEM [34].
It is evident that investigation of the dynamic behaviour of the debonded PWAS as well as alteration of the excited guided waves due to debonding is needed for refinement of the existing ultrasonic NDT and SHM methods, the development of the reliable self-diagnosis methods and the estimation of the probability of detection, which is crucial for successful implementation of the SHM systems. However, while many studies are devoted to the sensor diagnosis methods [16,17,35,36], a detailed investigation of Lamb wave excitation and propagation in the waveguide due to debonded PWAS is strikingly scarce. For this purpose, the SAHA proposed by the authors in [29], is employed here to simulate and analyse the dynamic interaction of a debonded PWAS with a layered elastic waveguide.
The SAHA allows separating different guided waves in the wave-field excited by PWAS and to calculate complexvalued eigenfrequencies, which provides a tool for studying the influence of PWAS debonding on Lamb wave excitation. A good agreement between experimental and numerical results is demonstrated, which allows to verify the model. Based on the SAHA model, the interrelation between eigenfrequencies of the structure, i.e. a waveguide with a surface bonded PWAS, and Lamb waves excitation (A 0 and S 0 ) is investigated for perfectly bonded, one-side debonded and centrally debonded PWASs. It is demonstrated that debonding between a PWAS and a plate causes considerable changes in the amount of energy transferred into the waveguide and its distribution among Lamb waves. Of course, some other situations (asymmetric two-sided debondings, three or more contact spots etc) could be also studied. Though variations in contact/debonded area may lead to essential changes in wave-fields and wave energy distribution, no additional physical phenomena are expected to be revealed in these cases [37]. Accordingly, only three cases mentioned above are considered here as the most common. However, the discrepancy between bonded and centrally debonded PWASs is relatively small even at higher frequencies and for moderate severity of the damage. For one-side debonded PWASs only, strong resonances are observed at the resonance frequencies calculated as the real part of a certain eigenfrequency of the plate with PWAS. It can be concluded that information about wave energy flux distribution between A 0 and S 0 modes can be used to identify debonding of a PWAS.

Mathematical model
In this study, debonding of a rectangular PWAS is considered, therefore, instead of a full three-dimensional model, a two-dimensional assumption is used. If the PWAS is accurately attached and the excited wave-front is plane then the assumption of plane state with no influence of x 3 according to figure 1 is valid and a two-dimensional model can be used for the analysis of the effects of debonding (see also [38] for the problem with surface-bonded rectangular block). Therefore, the two-dimensional implementation of the SAHA is discussed in this section, and as it is shown in the next section it provides an accuracy, which is good enough to be in a good agreement with experimental observations.

Statement of the boundary value problem
In order to model piezo-induced ultrasonic elastic waves, it is considered that a piezoelectric transducer of the width w and height h occupies a domain Ω 2 and is bonded to an infinite isotropic plate Ω 1 of thickness H, see S c is the perfect contact area between the layer and PWAS and S d is the debonded area where stressfree surfaces are assumed for the layer and the PWAS. Hereinafter, all the wave-fields related to the domain Ω i are denoted by the upper index i, with i=1 for the layer and i=2 for the PWAS.
The electro-mechanical coupling for piezoelectric materials is described by where σ ij is the stress, D i is the electric displacement, s kl is the strain and C ijkl , e kij , ε ik are elastic, piezoelectric and dielectric constants respectively. The components of the electric field vector are expressed in terms of the electric potential f. The problem under consideration is to be treated within the limits of the plane theory of elasticity taking into consideration plain strain assumption. The two-component displacement vector t u x, 1 ( ) ( ) satisfies the Lame equations Material properties of the elastic isotropic layer are given by the Lame constants λ, μ and mass density ρ (1) . The equations of motion for the piezoelectric media Ω 2 are written as follows: An electric input impulse p(t) is applied at the upper surface of the PWAS at the moment t 0 , while the bottom surface is grounded: For convenience the vector of normal and tangential stresses t s s = , 12 22 { } is introduced. The faces ±H of the layer are free of stress except for the contact area S c : Electric displacements D t x, ( ) are equal to zero on the side boundaries of the PWAS: Displacements and stresses are continuous at the contact area S c : Normal and tangential stresses are equal zero in the debonded area S d : (3) and (4) the Laplace transform is used in order to exclude time derivatives.

Due to linearity of equations
In order to simulate coupling between the layer Ω 1 and the PWAS Ω 2 the unknown function x q 1 ( ) is introduced at the contact area Sc:

Solution of the stated problem
The described boundary value problem is solved here using the SAHA [29]. Based on the SAHA, we show how to extract information about eigenfrequencies of the system of a PWAS and an elastic layer. Moreover, we show the calculation of energy fluxes for separate guided wave modes, which we will use later for the detailed description of effects, resulting from the debonding of a transducer.
According to this method, the solution strategy is firstly to consider two separated problems and secondly to model the coupling at the surface S c . The first problem corresponds to the description of the displacements induced in the layer due to dynamic load function applied on its upper surface. The second problem is to describe the mechanical and electrical state of the piezoelectric transducer under given boundary conditions. Displacements in the elastic layer with a given surface load can be constructed using the BIEM [34]. In accordance with this approach, harmonic wave-fields u x 1 ( ) ( ) are represented as the inverse Fourier transform with respect to horizontal the coordinatex 1 as follows: Here, the integrand is a multiplication of the Fourier transform of the Green's matrix K(α, x 2 ) and a Q( ) which is the Fourier transform of x q 1 ( ) introduced in (11), α is the Fourier transform parameter. If the poles ±ζ k of the Green's Matrix K (ζ k are assumed in the upper complex half-plane) are known, then the integral representation (12) can be evaluated in terms of the Cauchy's residue theorem and Jordan lemma as follows [29]: The FDSEM based on a variational formulation is used to simulate time-harmonic motion of the PWAS. The variational formulation of the boundary value problem for the piezoelectric transducer occupying Ω 2 is then rewritten: { }are test functions, see more in [29]. The FDSEM requires the boundary of the body Ω 2 to be discretised into finite elements. The displacements u i 2 and the electric potential j are approximated by Lagrange interpolation polynomials in both coordinates x 1 and x 2 on each element. The solution vector f = u u y , , 1 2 { }of the system (13) is written as follows: is the specially determined index function, depending on the element and node number.
The SAHA is based on the coupling of the two different methods: the BIEM and the FDSEM in the contact area S c with the boundary conditions (9)- (10). The unknown function x q 1 ( ) is to be found in order to solve the coupled mathematical problem. Therefore, the vector of unknowns is enlarged f = u u q q y , , , , In [29] the authors have described two different methods of the x q 1 ( ) approximation: using first order splines and using Chebyshev polynomials. The general idea is to substitute the traction vector x q 1 ( ) into equations (12) and (13) taking into account boundary conditions (9)- (10). The substitution leads to the system of linear algebraic equations, see more in [29] = A y f.
The obtained mathematical model can be used to calculate complex-valued eigenfrequencies f n for the considered boundary value problem. The eigenfrequencies are calculated as roots of the characteristic equation detA( f n )=0 obtained from system (15).
Another useful tool for simulation-based investigations is the calculation of the elastic wave energy distribution in the waveguide and the amount of wave energy carried by each Lamb wave mode. Such ananalysis is based on the timeaveraged power density vector e x ( ) or Umov-Poynting vector: Im . 17 The time-averaged energy flux  P m transferred by the mth Lamb waves in both directions is calculated by integrating the component e 1 of the time-averaged power density vector along a certain cross-section: The total amount of the wave energy flux or power input transferred from the PWAS into the waveguide is calculated as follows [39]: In this paper, we also use another well-known method for comparison purposes. The problem under consideration can be evaluated using the standard FEM simulation software COMSOL Multiphysics, including electro-mechanical coupling. The solution is constructed at the finite elements employing interpolation polynomials of the first and second order. This method is purely numerical and therefore demands great computational power. Although, it can be applied to a problem of any geometry, but is limited to a finite size.
One of the methods to simulate piezo-induced load is the use of a pin-force model (PFM) [40] where the contact stresses are replaced with the pin forces concentrated at the tips of the transducer. The load function in this way takes the form: Here δ is the Dirac delta function and the value of τ A is determined within the PFM.

Experimental setup
To verify the obtained mathematical model and its numerical solution an experiment has been conducted. The scheme of the experiment is shown in figure 2. A rectangular PWAS with dimensions 70 mm×10 mm×0.2 mm has been glued on the surface z=0 of an aluminium plate of the dimensions 500 mm by 500 mm and a thickness of H=1 mm. An electric voltage signal p(t) with the amplitude of V 0 =12 V is applied to the piezoelectric transducer and Lamb waves are excited at the surfaces of the plate (z=0, z=−H). In this study, only N c =5 cycles of Hann window are used to determine the input voltage (20). As a measurement quantity, the velocity perpendicular to the plate surface is measured at several points according to figure 2 in the middle of the 70 mm direction of the transducer. These measurements have been conducted with a 1D helium-neon laser Doppler vibrometer (LDV) from Polytec. The device is composed of a CLV700 measurement head with a CLV800 laser unit, connected with a controller CLV1000. This controller is equipped with modules CLVM002, CLVM030 and module CLVM200. This setup is limited in the frequency range up to 250 kHz, already with reduced amplitudes in this frequency area. The laser is mounted on a two-axial measurement table to be able to measure at all positions on the plate. The data acquisition is realized via Matlab. As digital oscilloscope and arbitrary waveform generator, a TiePie Handyscope HS3 was used, both controlled via Matlab, see more details in section 6 in [16].
In order to obtain experimental data of two different bonding conditions (states), two experiments have been conducted. Firstly the PWAS has been glued only partly. For this state the contact area is = - After the measurements of the velocities of motion excited by the partially debonded PWAS were performed, the transducer has been properly glued in the For the second state, velocities of the motion have been measured as well. In this way, two states in the experimental set-up were examined. To take into account the effects of fluctuation within setting up the experiment and especially the process of partial bonding, the experiments have been conducted twice. Figure 3 shows velocities of the motion measured on the back surface of the plate z=−H during two independent experiments. Registered data is almost identical in case of the perfectly glued PWAS. However, for the case of the halfdebonded transducer, experimental results differ. In the bonded direction < -  transducer: it continues wave excitation for some time after the input signal (20) had been applied.

Comparison of the theoretical and experimental results
To verify the obtained mathematical model, a comparison between the experiment and calculated signals has been performed.  Figure 6 shows velocities of the motion measured on the surface z=0 in the points x=±30 mm and x=±95 mm with the perfectly glued PWAS and calculated with three different models: the standard FEM model (COMSOL), the PFM [3] and with the help of the SAHA. The PFM has proven to be valid at rather low frequencies and when transducer's to waveguide's thicknesses ratio tends to be less than one. Therefore, one can see in figure 6 that the PFM allows getting acceptable results in case of accurate contact conditions, though local maxima do not fully coincide with the experimental signal. Wave-fields calculated with the FEM coincide with the SAHA model completely, though amplitudes of the signals differ from the experiment data with factor 1.2. Figure 7 shows the comparison of the calculated and measured signals over spatial value x with the central frequency of 80 kHz at the fixed time t. Three independent models have been used for calculation. Data from the first experiment measured at the front surface of the plate (z=0) is taken for comparison. It should be marked that there is some constant time-shift due to the experimental setup. First of all, the results obtained with the SAHA model completely agree with the same values calculated with the standard two  dimensional FEM model. The measured signal has the same waveform, though its amplitudes differ from the calculated signals. One of the reasons for such effect is that the twodimensional mathematical model is based on a series of assumptions, which can lead to a discrepancy in amplitudes of the measured and calculated signals. Moreover, this discrepancy is also frequency-dependent. It is clearly seen if the three-dimensional COMSOL model is calculated. In this case, the amplitudes of the obtained signal coincide with the experimental data. The other reason for the difference in amplitudes lies in the scattering of the data due to imperfections in the experimental setup. Moreover, it appears that such an elongated transducer (70 mm×10 mm) excites a wavefront propagating mainly along the x axis, therefore amplitudes of the measured signal heavily depend on the line of the measurements. Thus, figure 8 illustrates the difference in amplitudes measured within the perfectly glued transducer at different measurement lines y=0 mm and y=10 mm during first and second experiments. It is clear that signals measured with two experiments at the same line y=0 mm differ by 10%-12%. At the same time, a 10 mm shift of the measurement line   results in a decrease of the amplitudes by 15%-18%. However, the waveforms of the excited signals remain similar; therefore it can be concluded that the simulated wave-fields are trustworthy and the designed experimental setup can be used to verify a two-dimensional model. Figure 9 illustrates the measured and calculated signals with a central frequency of f 0 =180 kHz over the spatial value x at the time t=0.06 ms. Again, the SAHA and 2D COMSOL models provide similar signals, which waveforms coincide with the one measured during experiment conduction. Though the amplitudes of the measured signal are considerably lower, the correcting factor for simulation is 0.4. The signal, calculated with the 3D COMSOL model, has lower amplitudes, which are closer to those observed in the experiment. Nevertheless, even the results of the 3D COM-SOL model are twice as high as the experimental results. The reason for this effect could be that amplitudes are reduced as the frequency f 0 =180 kHz is close to the limit of 250 kHz up to which the laser vibrometer can be used. Also, if the line of measurements is taken y=±15 mm then amplitudes of the measured signal are higher. Nevertheless, the waveforms of the experimental and calculated signals agree very well.
In order to verify the mathematical model for the case of a debonded PWAS , a comparison with experimental data has been made with the central frequencies f 0 =80 kHz and f 0 =180 kHz. Figure 10 illustrates measured and calculated velocities of the motion of the plate surface in two points: from the glued side of the PWAS x= −113 mm and from the debonded side x=113 mm. The comparison shows that waveforms of the calculated and measured signals agree very well, but while the time of signal arrival matches perfectly on the right (from the glued side) there is some difference in the time of arrival on the left (from the debonded side). Although, if the debonded area is changed a little S d =[4.5, 10] mm, the resulting agreement in the time of signal arrival becomes better. Figure 11 shows the comparison of the signals obtained during the experiment and calculated with the help of the SAHA and the standard FEM method over the spatial value x with a central frequency of80 kHz. One can see that the amplitudes of the velocities of the motion are higher from the glued side. This effect is completely predicted by the mathematical model. The agreement of experimental and calculated signals is becoming worse for higher frequencies, see   The comparison of transient signals, calculated with the help of the SAHA, with experimental data verifies that the obtained two-dimensional mathematical model based on a semi-analytical hybrid approach is an efficient and reliable tool to simulate the dynamic interaction of a piezoelectric transducer on an elastic layer with different contact conditions. With the increase of the central frequency, additional effects appear and for its simulation, the use of threedimensional mathematical models is suggested. With the aim to experimentally realize the two-dimensional case, the results also show the difficulties, scattering and uncertainties of the experimental investigation focussing debonded transducers.

Numerical analysis of the wave phenomena
In this section, we provide a detailed parametric analysis of the influence of PWAS's dimensions and contact conditions between the actuator and the layer on Lamb wave excitation and resonance frequencies. The power density vector and corresponding energy streamlines are examined for specific debonding scenarios at frequencies corresponding to resonances or minima/maxima of the power input transmitted into the waveguide. Although the power input P 0 , calculated with (18), cannot be measured in the experiment, the results of the comparisons performed in section 3 verifies the obtained model and allows to assume the reliability of the calculated power input and corresponding energy distribution coefficients. For further analysis, let us introduce the following energy distribution coefficients indicating the contribution of the mth Lamb wave in the total amount of power P 0 (18) transferred into the waveguide or into a given direction P ± (for asymmetrically debonded PWASs). Here P ± is sum of the  P m obtained with (17) for all m. A 0 or S 0 mode excitation is considered and frequency ranges, where one Lamb wave is dominating for the particular PWAS's size, are revealed. Three different bonding scenarios are investigated: perfectly glued PWAS, damaged contact in the centre of the PWAS (the edges are still properly glued) and the one-side debonded PWAS.

Perfectly bonded piezoelectric transducer
First, the influence of the thickness h of a perfectly bonded PWAS on Lamb wave excitation is studied. Figure 13(a) illustrates the power P 0 ( f, h) transferred into an elastic waveguide of thickness H=1 mm due to the action of the PWAS of width w=10 mm. The real parts of complexvalued eigenfrequencies Re f n (h) are also drawn in figure 13(a), while the imaginary parts are depicted in figure 13(b). One can conclude that the PWAS's height has a small influence on the power at low frequencies (up to 300 kHz). With further frequency growth, narrow bands of high power occur, therefore, one can conclude that the PWAS thickness influence on the power P 0 increases. It should be noted that resonance frequencies, which are real and correspond to local extreme points, should be distinguished from complex-valued eigenfrequencies f n . The analysis shows that the smaller absolute value of the imaginary part of the eigenfrequency, the closer eigenfrequency to a certain real resonance is and, therefore, the greater is the influence of the eigenfrequency. Thus, the first five eigenfrequencies exhibited in figure 13 have relatively large absolute values of the imaginary parts ( > f Im 100 i | [ ]| kHz) and, therefore, the real parts do not affect the location of maxima and minima of the power P 0 . The imaginary part of the sixth eigenfrequency has a very small absolute value ( < f Im 20 6 | [ ]| kHz) if h> 1 mm, and its real part coincides with the local maximum of the power P 0 . In addition, a global minimum of the power surface P 0 ( f, h) is observed at f≈430 kHz and h=0.9 mm. The nature of the occurrence of this minimum is analysed below.
In order to study the influence of the PWAS's width w on wave excitation, surfaces P 0 ( f, w) and eigenfrequencies f n for transducers of height h=0.2 mm and h=1 mm are shown in figures 14(a), (b) and 14(c), (d) respectively. Again, imaginary parts of eigenfrequencies f n are drawn in the bottom of figure 14. For thicker PWASs, the number of the eigenfrequencies in a certain range increases and they are situated closer to the real axis Imf=0 in the complex plane f. In the case of thinner transducer (h=0.2 mm), the surface P 0 ( f, w) demonstrating energy flux dependence on the PWAS's width is substantially smoother compared with h=1 mm (see figures 14(a) and (c)). Naturally, the number of local maxima in the same frequency range is larger for thicker PWAS of height h=1 mm than for h=0.2 mm.
Nevertheless, figures 13 and 14 do not provide explanation of the fact that some eigenfrequencies with relatively small f Im n | | are revealed in P 0 ( f, w) and P 0 ( f, h) plots. For instance, eigenfrequencies f 6 and f 8 exhibit themselves in energy flux plots P 0 ( f, w) and P 0 ( f, h), while f 7 is not visible. To study this issue, eigenforms of these three eigenfrequencies have been calculated. Figure 15 demonstrate eigenforms (amplitudes of the displacement vector x x u , 1 2 | |( ) normalized by maximum value) for three eigenfrequencies f 6 ( figure 15(a)), f 7 ( figure 15(b)) and f 8 ( figure 15(c)). It is clearly seen that wave motion for f 6 and f 8 is dominantly localized in the PWAS and in the layer below the PWAS, i.e. in the region - Whereas wave maxima values of the displacement vector for eigenfrequency f 7 , which is not pronounced via peaks in energy flux plots, are in the regions situated in the waveguide, but to the right and to the left from the PWAS (not below it). Therefore, it can be concluded that two kinds of eigenfrequencies with relatively small values f Im n | | can be separated: localized in the vicinity of the PWAS (they cause sufficient changes in P 0 ) and localized outside the area below the PWAS (they are not visible in P 0 plot as peaks).
Energy flux P 0 ( f ) and energy distribution coefficient h A 0 in dependence on frequency is exhibited in figures 16(a), (b) for PWASs of height h=0.2 mm and h=1 mm computed with the SAHA [29] and calculated applying the PFM. In figure 16(c), squares (h=0.2 mm) and circles (h=1 mm) show eigenfrequencies in −150Imf n 0 kHz of the complex plane f. At frequencies up to 300 kHz, the PFM gives results quite similar to the results calculated by the SAHA for the PWAS of height h=0.2 mm, which is in a good agreement with theoretical predictions. The energy flux for the thicker PWAS (h=1 mm) discriminates significantly from the flux, actuated by the thinner PWAS (h=0.2 mm), see figure 16(a). This conclusion is also valid for the wave energy distribution among A 0 and S 0 modes. It can be seen in  For the three cases considered in figure 16, the first maxima of the energy flux P 0 ( f ) are very close (around 220 kHz), even for the thicker transducer of height h=1 mm. To compare and exhibit more detailed wave processes related to the interaction between the waveguide and PWASs of various thicknesses, wave energy distribution is considered at frequencies corresponding to local maxima of P 0 ( f ). The power density vector e and related energy streamlines are shown in figure 17 for the PFM(a), PWAS's height h=0.2 mm(b) and h=1 mm(c) at the frequencies corresponding to the first maximum of the P 0 for each case presented in figure 16(a). If the PWAS is substituted by point forces, a certain amount of wave energy is trapped in two large vortices located in the rectangular domain of the waveguide below the PWAS. Two large vortices are also observed for h=0.2 mm,    though they are circulating through the transducer and the energy trapped in these vortices is much smaller compared to the PFM case. At last, if the PWAS is thick (h=1 mm) large vortices disintegrate and the energy flux is transferred into the layer. In each case, vortices obstruct the waveguide below the transducer, and it can be concluded that the latter leads to maximum energy flux transmittance. For a complete analysis of P 0 ( f ) behaviour, maxima and minima should be compared. Figure 18 illustrates the power density vector and related energy streamlines excited in the layer by a thick PWAS (h=1 mm) at three frequencies, where the global minimum ( f=430 kHz) and global maximum ( f=486 kHz) of P 0 are achieved and for regular transmission ( f=594 kHz), but at f=Ref 7 . Four huge vortices circulating in the transducer and the layer are visible in the case where a minimum of P 0 is achieved (see figure 18(a)). These vortices accumulate part of the energy excited by the PWAS and, therefore, the total amount of the energy flux P 0 is substantially smaller. On the contrary, only two tiny vortices are visible in figure 18(b) for the frequency with maximum power at 486 kHz and the amplitudes e | | in the far-field zone are greater than below the PWAS. It can be observed, that amplitudes of the power density vector e | | are almost 10 3 larger compared to the case, where minimum energy flux from PWAS is considered. A regular wave excitation is observed in figure 18(c). Though the considered frequency is obtained as a real part of the eigenfrequency f 7 , the values of e | | are medium compared to two extremal cases discussed above.

One-side debonded PWAS
The parameter w 1 is introduced here to describe a one-side debonded PWAS with a debonding area S d =[w 1 , w]. First, let us consider the particular debonding area S d = [1.4, 5] mm. Figure 19(a) depicts the power P 0 ( f ) and  P f m ( ), while figure 19(b) illustrates the distribution of the energy flux between Lamb waves propagating to the right ( + P A 0 and + P S 0 ) and to the left ( -P A 0 and -P S 0 ) from the PWAS by means of the wave energy distribution coefficients h  f m ( ) defined by relation (21).
Except for three relatively narrow frequency ranges, the amount of the wave energy flux transferred to the left (from the glued side of the PWAS) is greater. A very deep local minimum at 420 kHz, and a substantial number of sharp peaks are observed in P 0 ( f ) plot in figure 19(a). In order to reveal their resonance nature, complex-valued eigenfrequencies f n are also shown in figure 19(c) as circles. It should be mentioned that the computation of eigenfrequencies of a debonded PWAS is more complicated: several eigenfrequencies in a certain domain of the complex frequency plane f arises with the increase of the debonding area S d . Analysis shows that a particular eigenfrequency can be matched approximately with each sharp peak in  P f m ( ) plot shown in figure 19(a). The majority of eigenfrequencies with < f Im 50 kHz n | | invoke peaks in  P f m ( ) plots. As in the case of perfectly bonded PWAS, eigenfrequencies with relatively large > f Im 50 kHz n | | are not well pronounced, and no influence on resonance properties of the structure is revealed. Figure 18. The power density vector e | | and related energy streamlines for the minimum of P 0 at f=430 kHz (a), the maximum of P 0 at f=486 kHz (b) and for regular excitation, but at f=Ref 7 =594 kHz (c).
In order to investigate the effect of the size w 1 of the debonded area S d on Lamb wave excitation, P f w ,   whereas others have narrow peaks. They are located denser if the debonding area increases.
The share of the energy flux κ − =P − /P 0 transferred along the x 1 axis in x→-¥ (the glued side) in dependence on the frequency and the size of debonding is illustrated in figure 20(b). It should be mentioned that only trajectories with narrower peaks visible in P 0 ( f, w 1 ) can be observed in κ − ( f, w 1 ). Especially for the combinations of frequency f and debonding tip w 1 laying on or in the vicinity of these traces, the amount of wave energy, which is transferred to the left, is much larger than the amount of wave energy, which is transferred to the right. Figure 21 illustrates the influence of the debonding size on the wave energy distribution k  f w , ) surface (see figure 20(b)). Here, A 0 mode excitation usually increases at the debonded edge (kf w , A 1 0 ( )) and decreases at the bonded edge (k + f w , It is well known, that PWAS excites waves mostly at its edges [3,40]. If the PWAS is one-side debonded, Lamb waves are generated at the edges of the contact area, i.e. in the middle part of one-side debonded PWAS, where it is still glued. Figure 22 showing power density vector and the corresponding energy streamlines for the debonded PWAS S d = [1.4, 5] mm at the frequency f=49 kHz, where a local maxima in P 0 ( f ) is achieved, illustrates this fact.

Centrally debonded PWAS
The influence of the central debonding of the PWAS on the waves excitation is analysed in this section. Parameter Δw is introduced so that debonding of the PWAS is = Thus, if Δw=0.5 mm, only 0.25 mm is bonded at each edge. Figure 23(a) shows the dependence of energy flux P 0 ( f, Δw) excited in the structure by centrally debonded PWAS on the severity of the debonding and frequency. It is evident from figure 23(a) that central debonding has a modest influence on the excited energy flux P 0 apart from several resonance frequencies. However, the number of such frequencies is significantly  reduced in comparison to one-sided debonding. Since excitation is symmetrical in the considered case, only h + A 0 is demonstrated in figure 23(b). One can see that A 0 is most excited in the following frequency ranges: 0-70 kHz, 400-550 kHz. Besides, if the PWAS is debonded less than 50% then the influence of debonding area on wave distribution among Lamb waves is relatively low even at higher frequencies.  A particular case of centrally debonded PWAS with w=10 mm, h=0.2 mm, S d =[−2.65, 2.65] mm is illustrated in figure 24 in the same manner as for one-side debonded PWAS above (circles in figure 24(c) correspond to eigenfrequencies). To simplify comparison with perfectly bonded PWAS, dashed lines corresponding to PWAS with w=10 mm, h=0.2 mm and S d =∅are added in figures 24(a) and (b). A mismatch between centrally debonded and bonded PWASs is quite small at lower frequencies below 230 kHz, whereas the effect of debonding becomes more sufficient at higher frequencies.
A very deep non-resonant (eigenfrequencies are absent in its vicinity) minimum of a smooth wide arch in P 0 ( f ) plot is reached at f=605 kHz; around f=605 kHzA 0 dominates in the excited wave-fields. Three sharp peaks are revealed in P 0 ( f ), P ± ( f ) and h  f m ( ) plots, but only two of them are situated closely to Re f n (25 and 125 kHz) and can be reported as resonant ones. In figure 25, the energy density vector and the corresponding energy streamlines are depicted for two frequencies: f=528 kHz corresponding to the third local minimum (figure 25(a)) and f=533 kHz (figure 25(b)) corresponding to the local maximum next to the minimum at f=528 kHz. The amplitudes of the excited wave-field in the far-field zone differ approximately 4 times, whereas the difference in frequency is 5 kHz only. The reason of the low excitation at the frequency f=528 kHz is that a considerable part of the wave energy is accumulated in the two huge vortices under the PWAS and it does not flow into the waveguide. At the same time, more wave energy is excited by the edges of the PWAS at the frequency f=533 kHz, which is not captured under the PWAS and is radiated into the waveguide.

Conclusion
The semi-analytical hybrid approach combining the BIEM and the FDSEM is applied here to analyse Lamb wave excitation by a debonded PWAS. This method allows calculating efficiently eigenfrequencies and Lamb wave distribution in an unbounded layer with a debonded piezoelectric transducer. The results obtained during the experiments with perfectly bonded and partially debonded PWASs glued to an aluminium plate are presented and compared with simulations. The measured signals using laser Doppler vibrometer are employed to verify the mathematical model in case of various bonding conditions between the PWAS and the waveguide.
It is shown, that only eigenfrequencies with a small absolute value of the imaginary part < f Im 50 kHz n | | correspond to the resonances for perfectly bonded PWAS and might influence the total output from the PWAS. The analysis shows that two different kinds of eigenfrequencies can be distinguished. Eigenvibrations corresponding to the first kind of eigenfrequencies are strongly localized in the PWAS and its vicinity. On the contrary, eigenforms corresponding to the second kind of eigenfrequencies reach their maxima values in the regions situated to the right and to the left from the PWAS. It should be noted that eigenfrequencies of the second kind are not related to real resonances, even if the absolute value f Im n | | is small. Energy flux and excitation of Lamb waves A 0 and S 0 are analysed in dependence on the PWAS sizes and the debonding area between the PWAS and the waveguide. It is shown, that distribution between the A 0 and S 0 modes alters significantly if the transducer is thick, therefore, for deep analysis, one needs to take into account the transducer's height. A one-sided debonding deeply influences the wavefields excited in the plate; the distribution of the power between Lamb waves alters significantly with minor debonding area sweep and becomes even more frequencydependent.
On the contrary, if central debonding occurs, wave energy distribution among Lamb waves almost does not change for frequencies up to 600 kHz. Moreover, in particular cases, central debonding might lead to an increase in energy flux compared to the perfectly bonded PWAS. The provided analysis might be useful for the development of the self-diagnosis methods for the PWASs and to improve the existing NDE and SHM algorithms employing piezoelectric transducers. It should be noted, that the presence of delamination in the vicinity of PWAS significantly affects the wavefields and eigenfrequencies, and the latter should be kept in mind if PWAS is tested.