Experimental and Theoretical Analysis of Lamb Wave Generation by Piezoceramic Actuators for Structural Health Monitoring

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1 Experimental Mechanics DOI /s Experimental and Theoretical Analysis of Lamb Wave Generation by Piezoceramic Actuators for Structural Health Monitoring J. Pohl & C. Willberg & U. Gabbert & G. Mook Received: 24 January 2011 / Accepted: 17 April 2011 # Society for Experimental Mechanics 2011 Abstract Piezoceramic transducers, acting as actuators and sensors, are attractive for generation and reception of Lamb waves in Structural Health Monitoring (SHM) systems. To get insight into the source-mechanisms of Lamb waves, the vibrations of piezoceramic actuators are analyzed for the free and bonded state of the piezoceramic by analytical and finite element (FEM) calculations. Mode shapes and spectra of piezoceramic actuators and Lamb wave fields are experimentally recorded by scanning laser vibrometry. The analytical solutions for bending modes are shown to be valid for large diameter-tothickness-relations of a free piezoactuator (D/H>10) only. For thicker piezoceramics, a FEM-solution gives better results. Calculated frequencies for radial modes of vibration are confirmed by 3-D-laser-vibrometry and measurements of electrical impedance. The bonded case of a piezoactuator exhibits a broad resonance peak resulting from the strong coupling between radial and bending modes. The assumption that optimal excitation of Lamb modes occurs for a matching of the wavelengths to the diameter of the piezoceramic holds only for thin ceramics. Otherwise the distinct modes of out-of-plane J. Pohl (*) Department of Electrical, Mechanical and Industrial Engineering (EMW), Anhalt University of Applied Sciences, Bernburger Str. 55, Köthen, Germany j.pohl@emw.hs-anhalt.de C. Willberg : U. Gabbert Institute of Mechanics (IFME), Otto-von-Guericke-University, PF 4120, Magdeburg, Germany G. Mook Institute of Materials and Joining Technology (IWF), Otto-von-Guericke-University, PF 4120, Magdeburg, Germany and in-plane vibrations control the excitation of the Lamb modes more than the wavelength matching. Keywords Lamb waves. Structural health monitoring. Piezoceramic actuator Introduction Structural Health Monitoring (SHM) is a new, rapidly developing technology for the monitoring of engineering structures [1 6]. The main features are permanent and automatic assessment of the structural integrity by built-in devices such that non-destructive testing becomes an integral part of the structure. Therefore, SHM aims at damage detection and loads monitoring by intrinsic means of the structure. Consequently, a SHM-system requires built-in sensors to provide this new functionality, and if necessary, for the generation of diagnostic signals, actuators as well. The tasks can be defined similar to those of conventional non-destructive testing: damage detection, localisation and further characterisation. The final goal is an estimation of the remaining life time of the structure. The new monitoring approach of SHM replaces conventional maintenance concepts with fixed inspection intervals by more effective condition-based maintenance concepts where actions are taken only if required [6]. Among different approaches, the use of ultrasonic Lamb waves is an attractive method for structural health monitoring. Lamb waves are able to propagate over large distances, thus wide areas of a structure can be monitored. Because Lamb waves sensitively interact with defects, they offer a chance for defect detection and further characterization. SHM-systems using Lamb waves require effective means for generating and receiving Lamb wave modes. The use of embedded or surface-attached piezoceramic

2 elements as actuators and sensors for generation and reception of Lamb waves is an attractive way for designing smart SHM-structures [1, 2, 4-6]. The paper contributes to a better understanding of the generation process of Lamb waves which are excited by piezoelectric patch actuators bonded to the surface of a lightweight structure. For creating effective SHM-systems it is necessary to understand in detail the generation process of Lamb waves. The multimodal, frequency-dependent properties of Lamb waves [7], the complexity of modal behaviour of piezoceramic elements and the influence of coupling conditions are investigated experimentally as well as theoretically. It is shown that the natural vibration behaviour of a piezoelectric patch coupled by a bonding layer to the structure can be used for a more effective wave excitation. It is also shown that the classical rule of thumb, which relates the optimal excitation frequency to the geometry of a piezoelectric patch actuator, only holds for very thin patches and fails for more realistic thicker patches. The paper is organized as follows. At first the free vibration of circular piezoelectric patches are investigated theoretically as well as experimentally. Then the dynamic behaviour of circular patches bonded to the structure is studied and the relation between the geometry of the patch actuator and the applied frequency is investigated in detail. Here it is also shown that directional wave effects can occur even if all involved materials are isotropic. The paper finishes with a brief conclusion. Vibrations of Free Piezo-Actuators We start with theoretical and experimental investigations of vibrations of free circular piezoceramic plates. Out-of-plane and in-plane vibration modes of the piezoceramic plate will activate the different Lamb modes [1, 2, 8 10]. Out-ofplane modes are represented by modes designated as transverse, bending or flexural modes and thickness modes of the plate, in-plane modes by radial and torsional ones. Due to coupling effects nearly all modes will show out-of plane as well as in-plane components of particle displacement. Because thickness modes for the piezoceramic plates with thicknesses less than 2 mm considered here occur at frequencies higher than 1 MHz, these modes are not taken into consideration. Torsional modes are excluded as well because of their weak signal strength. Free modes are generally a valuable starting point for the treatment of the vibration modes of piezo-plates that are bonded to a structure. Figure 1 schematically shows the coordinate system and the geometric parameters. Investigations of different modes of vibration of free piezoceramic discs comparing analytical and FEM solutions with experimental results were done by Huang et al. Fig. 1 Coordinate system of a circular plate with radius r max, plate thickness h and amplitude W j,p [10]. Following this work, the analytical solution (plate theory) for the resonance frequencies of bending modes of the piezoceramic disc under a constant electrical field is given by equation (1). The 1 2 directions match with the isotropic plane in radial direction, shown in Fig. 1. The direction 3 is perpendicular to that plane and relates to the polarization of the piezoceramic material. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 n 12 Þk12 2 h f j;p ¼ 24r S11 E 1 n k p rmax 2 b 2 j;p with ð1þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 12 ¼ 2d 2 31 " 33 S11 E ð 1 n 12Þ Here S E 11 is the compliance in the isotropic plane, ν 12 the Poisson s ratio, d 31 the electromechanical coupling coefficient, ε 33 the permittivity and ρ the density. The values of β j,p are the natural eigenvalues of the solution for a free-free circular piezoceramic plate. The values j, p are defined as the number of nodal circles and nodal diameters respectively. It was shown that the error between the experimental result, the analytical solution and the FEM solution for the Table 1 Material properties of PIC-181 PIC-181 ceramics S E 1 [10 12 m 2 /N] S E S E S E S E S E S66 E d 31 [10 10 m/v] 1.08 d d " T 11 =" 0 [ ] 1224 " T 33 =" ρ [kg/m 3 ] 7850

3 from PI Ceramics), given by the manufacturer, are illustrated in Table 1. l j;p r l j;p r W j;p ðr; 6Þ ¼A j;p J p þ C j;p I P cos p6 ð3þ r max r max Fig. 2 Principal setup of laservibrometric measurements transversal vibrations is small for a diameter D to thickness h ratio of D/h>10 [10]. For our investigations no constant electrical field applies. Then equation (1) becomes the pure elastic description of equation (2). This equation, given by Itao [11] and used by Giurgiutiu and Huang et al. [2, 10] shows the analytical solution for bending modes of a circular free-free plate without including the piezoelectric behaviour. These assumptions are also used for our FEM simulation. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h f j;p ¼ 12S11 E r 1 n2 12 2p rmax 2 l 2 j;p ð2þ The values of 1 j,p are the natural eigenvalues of the solution for a free-free circular elastic plate. To compare the measurements with the theoretical solutions of the eigenfrequencies, the mode shapes have to be considered. Equation (3) gives the displacement W j,p in x 3 -direction showninfig.1. With the Bessel functions J and I, the amplitude A j,p, and the mode shape parameter C j,p,the correlated eigenform to the frequency f j,p from equation (2) can be calculated [12]. The material properties of a typical piezoceramic material under investigation (PI-181 For experimental validation of theoretical results the mode shapes and spectra of free and bonded piezoceramic actuators and the resulting Lamb wave fields were recorded with 1D- and 3D-scanning laser vibrometers (Polytec PSV 300 and Polytec PSV 400). Scanning laser vibrometry represents an interferometric method with high sensitivity for non-contact detection of small displacements of wave fields or vibrations [13 16]. The principal setup of the laservibrometric measurements is displayed in Fig. 2. The specimen in the case of vibration measurements represented by the piezoceramic transducers under investigation was scanned point-bypoint to resolve the mode shapes. The electrical excitation of the piezoceramic was performed with chirp signals, covering successively a frequency region of 0.5 to 300 khz (in some cases up to 500 khz). Different circular discs from PI Ceramics and Marco, made from hard and soft lead zirconate titanate materials (PIC 151, PIC 181, FPM 100 and FPM 202), were used as piezoceramic actuators. The diameters ranged from 10 to 40 mm, the thicknesses from 0.5 to 2 mm. All discs had wrapped electrodes providing electrical contacts at the free side of the transducer. To reveal the in-plane modes of vibration more unambiguously, investigations with a 3D-laser vibrometer (PSV 400 3D) were made for some piezo-actuators in addition to two-dimensional scans of the out-of-plane displacements with the PSV-300, allowing the recording of all three components of displacement. Figure 3 shows the calculated mode shapes of the first eight modes in comparison to measured ones, demonstrating a very good correspondence of model and experiment. The comparison of the analytically determined frequencies by using equation (2) with the measured natural ones, based on the identification of corresponding modes, in Fig. 4 suggests good agreement for thin piezoceramics. Therefore equation (2) holds well for small h-values (D/h> Fig. 3 Mode shapes of the first eight out-of-plane (bending) modes of a free piezoceramic disc (D=10 mm, thickness 1 mm, PIC181) calculated (top) and experimentally detected (bottom)

4 Fig. 4 Comparison of theoretical solutions (pure elastic) and experimental results of the nth order frequencies f of free piezoceramic discs (D=10 mm, PIC181) with different thicknesses h 10). For thicker piezoceramics, a FEM-solution calculated with the FEM-program ABAQUS gives better results and fits very well with the experimental results. To calculate the free radial (in-plane) modes analytically, equation (4) is used [2, 10]. The values z n are the eigenvalues of the analytical in-plane solution. The solution is independent from the electromechanical coupling terms [10]. Here z n are the non trivial solutions of the general equation for axial vibration of the circular plate [2]. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 z f n ¼ S11 E r 1 n ð4þ n2 12 2p r max 3D-laser scanning measurements show that these in-plane modes also remarkably produce secondary out-of-plane components of displacement. It was observed that even in 1D-laser vibrometric generated out-of-plane spectra radial mode peaks became visible. In those cases it was not possible to correlate a fundamental bending mode shape shown in Fig. 3 (and higher ones too) with the measured peaks. FEMsimulations show that many bending modes coexist around the higher resonance frequencies of the first radial mode. This was validated by the experimental results. In Fig. 5, for example, the position of the first radial mode marked by an arrow is surrounded by several bending modes in this frequency region. A splitting of the mode types by this 1D (out-of-plane) measurements is not possible. As the right part of Fig. 5 shows, 3D-measurements allow an unambiguous identification of in-plane modes even in these complicated situations. Following Huang et al. [10] only in-plane and thickness modes cause resonance peaks in the spectrum of electrical impedance. To additionally validate the resonance values, the impedance was measured with a network analyzer Advantest R3751 A. Table 2 compares the calculated frequencies of the first radial mode with the 3D-measured ones and values obtained with impedance measurements. As equation (4) states, a change of the height of the piezoceramic disc has no influence on the resonance frequencies. Vibrations of Bonded Piezo-Actuators To investigate the case of bonded piezoceramic actuators, the situation was modelled with FEM. The adhesive layer Fig. 5 Spectrum of free vibrations of a piezoceramic disc (D=10 mm, h=1 mm, PIC 151), measured with 1D-laser vibrometer (left) and first radial mode shape, recorded with 3D-vibrometer (right) Table 2 Calculated and measured frequencies for the first radial mode Diameter [mm] Material Height [mm] Calculated frequency [khz] Measured frequency (3-D vibrometer) [khz] Measured frequency (impedance) [khz] 10 PIC PIC PIC PIC PIC FPM FPM

5 Fig. 6 Second calculated radial mode of a coupled piezoceramic disc (D=10 mm, h=0.5 mm, PIC 181) for a PMMA plate was considered as an isotropic material with different heights and Young s moduli. By including the adhesive layer, effects connected with its properties, e.g. the shear lag effect, are taken into account [17]. To initially consider isotropic media for Lamb waves, the piezo-actuators were bonded to a polystyrene (PS) plate with 4.75 mm thickness and to a plate of polymethylmethacrylate (PMMA) with 3 mm thickness. To achieve a reversible coupling to the structure, the bonding was performed by paraffin. The thickness of the bonding layer was between 10 and 40 μm for all plates. Material properties of the bonding layer were taken from the literature (E=1.02 GPa, ν=0.37, ρ=910 kg/m 3 )[18] and also determined directly by measurements of ultrasonic wave velocities (E=1.2 GPa, ν=0.4). Figure 6 displays the 15th calculated eigenmode of a bonded piezoceramic plate, which is the second mode with a significant contribution of in-plane displacements in relation to the out of plane displacements. So, this mode is designated second bending-radial eigenmode. Due to the coupling of the piezoceramic actuator with the plate by an adhesive layer no pure radial modes occur as in a free circular plate. The error of solutions of the radial eigenfrequencies is around 4%. The bending axis shifts in direction of the plate and a bending moment will be induced. If the frequency of a bending mode is close to the frequency of a radial mode, both modes will be activated. The simulations of the coupled piezoceramic (D=10 mm, h= 1 mm, PI 181, h bond =40 μm) have shown the second radial mode at 254 khz. Five axially symmetric bending modes are in a frequency range of plus minus ten percent. This expresses the strong coupling between radial and bending modes in such cases. The 3D-laser vibrometer measurements confirm this but don t reveal the radial modes clearly. As Fig. 7 (left) shows, the strong peak between 200 and 300 khz is caused by a mode which dominantly assumes a bending eigenform in the mode shape image (right). The measurement of the in-plane displacement confirms the coupling of the in-plane and out-of-plane vibration and shows the in-plane components of the displacements to be higher than in the free vibration state. For these increased values two reasons are possible. The first is a shift of the bending axis from the middle of the disc towards the plate due to the bonding. As a result, the displacement at the measured top plane of the disc must be higher. The radial modes which exist in that range are the second reason for an elevation of the in-plane components. Figure 7 (left) shows a wide peak between 205 and 270 khz. The bending mode, shown in Fig. 7 (right) is the same for the whole frequency range of the peak. The area of the shadowing electrical wires is removed to avoid disturbances of the spectrum. The mode properties clearly show the mixture between the bending and the radial modes. The in-plane spectrum of the 3D-measurements displays the same behaviour as shown in Fig. 7. The associated shapes of the radial modes are not unambiguously detectable in the area plot of the in-plane displacements but the data of measurements of electrical impedance resonances in Table 3 reveal radial modes at the edges of the peak region, e.g. the second radial mode at 262 khz. A shift to lower frequencies occurs for increasing thickness of the piezoceramic. In summary, vibrometry as well as impedance measurements indicate the coupling effects of bending and radial modes producing a wide resonance region with strong amplitudes. Fig. 7 Out-of-plane spectrum of a piezoceramic disc (D=10 mm, h=1 mm PI 181) (left) bonded to a PMMA plate, and mode shape (right), measured with 3D laser vibrometer

6 Table 3 Frequencies of the first three radial mode resonances (bonded to PMMA), determined by impedance measurement Diameter [mm] Material Height [mm] Measured frequencies [khz] 10 PIC PIC PIC The experimental spectra and mode shapes of a piezoceramic disc for the free and for the bonded case on PS measured by 1D-vibrometry (out-of-plane displacement) are depicted in Fig. 8. The radial modes become detectable only by their secondary out-of-plane components of displacement in these 1D-measurements. For the piezoceramic displayed in Fig. 8 the first radial mode for the bonded case is situated at 152 khz as determined by impedance measurement. The displayed mode shapes exclusively represent bending modes. So again, a strong resonance region with pronounced coupling of bending and radial modes appears around the location of the radial mode. For the bonded case, the conductor terminals of the piezoceramic and the electrical wires provide inevitable disturbances of the images. FEM and experimental results show that the bending modes are more influenced by the bonding of the piezoceramic to the structure than the radial modes. Figure 8 indicates the typical significant shift to higher frequencies for bending modes in comparison to the free case. Due to the change from free to bonded conditions certain small resonance peaks are extinguished. The corresponding modes shift in frequency and their shapes are not anymore indicated by distinct peaks in all cases. Excitation of Lamb Wave Modes The design of the piezoceramic actuator has to match the conditions for most effective Lamb wave generation. Following models including only shear stress transfer in the bonding layer [2, 4, 19], the conditions for effective generation of Lamb waves are fulfilled, when the actuator length (or diameter) 2a corresponds to 2a ¼ lðn þ 1=2Þ ð5þ with the wave length 1 and n=0, 1, 2,... Considering the situation for an isotropic model material and a typical structural material, the frequencies for optimal Lamb wave generation of the S 0 - and the A 0 -mode by a piezo-actuator with 10 mm diameter on a PMMA plate with 3 mm thickness and a carbon-fibre-reinforced plastic (CFRP) plate with a ½ð0=90Þ f =þ45= 45=ð0=90Þ f Š S layup with 2 mm thickness were calculated for wavelengths derived Fig. 8 Spectra (gray = free, black = bonded) and mode shapes (free top, bonded bottom) of a piezoceramic disc (D=16 mm, h=1 mm, PIC 181) bonded to a PS plate

7 from equation (5) and are given in Table 4 for frequencies up to 500 khz. Measured dispersion curves for Lamb waves [20] were used for the determination of the frequencies for the given wavelength values. Figures 9, 10, 11 and 12 display the measured areaaveraged spectra of the A 0 -ands 0 -mode in the PMMA and CFRP plates for a propagation distance of 200 mm. The Lamb waves were generated by actuators with the same diameter of 10 mm and different heights of 0.5 to 2 mm. The positions of frequencies of optimal excitation calculated with equation (5) are marked by gray lines and arrows. The frequencies with pronounced amplitude do not correlate in all cases with the calculated ones. This holds true for the other piezoceramics (not displayed here). In the lower frequency domain up to about 100 khz A 0 is produced with high amplitudes by thin piezoceramics. This is explained by the bending modes which are dominant in producing the A 0 mode with high amplitudes. These bending modes are seen with relatively high amplitudes in the spectra of the bonded piezoceramic actuators in Figs. 13 and 14. Due to the governing role of bending stiffness a frequency shift toward higher values occurs and the effectivity of the A 0 -generation for lower frequencies decreases with increasing thickness of the actuator. The low frequency maximum of A 0 excitation agrees fairly well with the predicted value for n=0 (Table 4) for thin piezoceramics. For thicker piezoceramic actuators equation (5) does not hold true. The other calculated maxima of A 0 generation in Table 3 are not apparent or only correspond to peaks of low amplitude. As Figs. 13 and 14 show, the dynamics of the actuator cause strong amplitudes between 200 and 300 khz which do not match the values of equation (5) but are reflected by the spectra of A 0 in Figs. 9 and 11. In sum, the results display a more effective generation of A 0 by only the bending modes at lower frequencies and of coupled bending and radial modes at higher ones corresponding to the highest resonance peaks in the spectra of the piezoceramics. As Figs. 13 and 14 display, a wide peak with strong amplitudes occurs between 200 and 300 khz. At these Fig. 9 Spectra of the A 0 -mode for piezoceramic discs with 10 mm diameter, (a) h=0.5 mm, (b) h=1.0 mm, (c) h=2.0 mm bonded to a PMMA-plate Fig. 10 Spectra of the S 0 -mode for piezoceramic discs with 10 mm diameter, (a) h=0.5 mm, (b) h=1.0 mm, (c) h=2.0 mm bonded to a PMMA-plate Fig. 11 Spectra of the A 0 -mode for piezoceramic discs with 10 mm diameter, (a) h=0.5 mm, (b) h=1.0 mm, (c) h=2.0 mm bonded to a CFRP-plate Table 4 Optimal frequencies for Lamb mode generation in a PMMA and a CFRP plate up to 500 khz n 1 [m] PMMA CFRP f of A 0 [Hz] f of S 0 [Hz] f of A 0 [Hz] f of S 0 [Hz] Fig. 12 Spectra of the S 0 -mode for piezoceramic discs with 10 mm diameter, (a) h=0.5 mm, (b) h=1.0 mm, (c) h=2.0 mm bonded to a CFRP-plate

8 Fig. 13 Spectra of piezoceramic discs (D=10 mm, PIC 181) bonded to a PMMA plate (a) h=0.5 mm, (b) h=1 mm, (c) h=2 mm higher frequencies radial modes and coupled bending modes (e.g. Fig. 6) are dominant. The spectra of S 0 excitation in Figs. 10 and 12 accordingly confirm the effective generation of this Lamb wave mode. Again an agreement of the maxima of the S 0 spectra with the predicted values of equation (5) forn=0 becomes visible for thin transducers. Thicker transducers generally better fulfill the conditions for the radial mode with its maximum at about 230 khz, best visible in comparison with the spectra of the piezoceramics bonded to CFRP in Fig. 14 with the spectra of S 0 in Fig. 12. In this region a shift to lower frequencies occurs with increasing thickness. The corresponding spectra for PMMA exhibit the same tendency. All in all, the assumption that optimal excitation of Lamb modes occurs when the wavelengths are matched to the diameter of the piezoceramic according to equation (5) does not hold for all circumstances. Only for thin ceramics the lowest order (n=0) matching is valid. In other cases the dynamics, i.e. amplitude and direction at the resonances of the piezoceramics control the generation of the Lamb modes more than the wavelength matching. To sum up, the bending and radial modes produced by a bonded piezo-actuator control the generation of A 0 and S 0 in detail. The distinct modes of vibration, represented by the peaks in the spectra are responsible for the strength of generation of the wave modes with their corresponding wave fields. In addition to material properties and geometric parameters of the piezoceramic plate (diameter and thickness), the influence of the plate material properties and the bonding to the structure with its thickness, uniformity and material properties are factors of main influence [8, 20 25] and have to be included in this consideration. The vibration modes of the piezo-actuator do not only influence the general strength of Lamb wave emission, but also have consequences for the wave field formation. Figure 15 demonstrates the control of the activated Lamb wave field by the mode shape of the piezo-actuator for an isotropic material (PMMA), an effect reported by Huang et al. [9]. Here a distinct mode shape produces a corresponding radiation pattern of the wave field with five preferred directions. This effect is strongly frequency-controlled due Fig. 14 Spectra of piezoceramic discs (D=10 mm, PIC 181) bonded to a CFRP plate (a) h=0.5 mm, (b) h=1 mm, (c) h=2 mm Fig. 15 Mode shape and Lamb wave field of a piezoceramic (D= 15 mm, h=2 mm), bonded to a PMMA plate and excited at khz Fig. 16 Orientation of the piezoceramic transducer

9 Fig. 17 Wave fields of a piezoceramic transducer (D=15 mm, h=2 mm, FPM 202) with different orientations at a PMMA plate excited at khz Fig. 18 Wave fields of a piezoceramic transducer (D=15 mm, h=2 mm, FPM 202) with different orientations at a PMMA plate at 400 khz to the frequency dependence of the amplitude and the mode shapes of the vibration modes. To verify the effect in more detail and prove the reproducibility, a piezoceramic disc was bonded in different directions at the same position of the plate. Figure 16 shows schematically the orientation of the piezoceramic transducer, defined by the angle between the axis through the electrical contacts of the transducer and the coordinate axes of the isotropic plate. Figures 17 and 18 show the wave fields for orientations of 0, 45 and 90 (grey line) of the bonded piezoceramic at a PMMA-plate for different frequencies of excitation. At both frequencies the shapes of the wave fields follow the orientation changes with remarkably good reproducibility. It must be noted here that efforts were made to produce equivalent bonding of the piezoceramic for the three directions regarding bonding thickness and uniformity. This was validated by mechanical thickness measurements of the bonding layers. Not in all bonding cases and for all frequencies are the results as clear. Directional effects of the coupling layer due to different thickness and non-uniformity of the coupling layer and effects caused by the electrical contacts of the piezoceramic transducer were shown to be superimposed to the described orientation effects. In anisotropic media the strong influence of direction-dependent properties of the medium is added. Figure 19 shows this for the case of wave propagation in a macroscopic quasi-isotropic plate of carbon fibrereinforced polymer with a ½ð0=90Þ f =þ45= 45=ð0=90Þ f Š S layup. Here, anisotropy effects of the A 0 -mode (shown by the small wavelength) occur, showing the influence of fibre orientations in embedded layers. Typical patterns of devia- Fig. 19 Wave field in a quasi-isotropic CFRP plate of thickness 2 mm at 150 khz

10 tions of phase directions and bending of energy transport directions are indicators for this whereas the basic symmetric mode S 0 shows no influence of fibre directions of different layers. In summary, the influence of coupling layers as well as the effects of anisotropy cannot be ignored and have to be included in the consideration of the Lamb wave generation process. Conclusions The characteristics of the vibrations of the piezo-actuator control the generated lamb wave field. The combination of theoretical considerations and laser-vibrometric observations of the piezo-actuator vibration and Lamb wave propagation helps to explain the complex situation. Impedance measurements are helpful to further describe radial modes of vibration. Starting from free vibrations of the piezoceramic actuator the bonded case exhibits a broad resonance peak with large amplitudes which results from the strong coupling between the radial and the bending modes. Bending modes are generally shifted to higher frequencies. Depending on the geometry of the actuator, the bending and radial modes of vibration describe the frequency dependent generation of Lamb waves. The assumption that optimal excitation of Lamb modes occurs for a matching of the wavelengths to the diameter of the piezoceramic does not hold for all circumstances. Only for thin ceramics the lowest order matching is confirmed. Otherwise the distinct modes with their amplitude and direction of vibration control the excitation of the Lamb modes more than the wavelength matching. In order to apply this knowledge for designing an efficient actuator configuration a detailed finite element simulation is required. A simpler extended rule of thumb could not be developed until know. A detailed investigation of the different modes of vibration helps to further reveal the radiation patterns of Lamb wave transducers. 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