PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Guided Lamb and edge wave excitation by piezoelectric transducers
in elastic plates
Mikhail V. GOLUB1; Artem A. EREMIN2; Maria V. WILDE3; Alisa N. SHPAK4; Inka MUELLER5
1,2,4

Institute for Mathematics, Mechanics and Informatics, Kuban State University, Russian Federation
35Faculty of Mathematics and Mechanics, Saratov State University, Russian Federation

Department of Civil and Environmental Engineering, Ruhr-University Bochum, Germany

ABSTRACTIn this contribution, the results of theoretical and experimental studies of guided waves excitation by
piezoelectric transducers attached at an edge or a face of an elastic plate are reported. The semi-analyticalmethods relying on integral transform and modal expansion techniques are applied for numerical analysis.
Wave energy distribution amongst excited guided waves (Lamb and edge waves) is carefully analysed anddiscussed to improve the quality of damage identification with techniques based on guided waves. The
influence of partial debonding of a piezoelectric transducer on guided waves excitation is investigated. The
potential of edge waves for practical applications in non-destructive evaluation and structural healthmonitoring is discussed. Namely, it is demonstrated that edge waves excited in plates and scattered bysurface-breaking cracks can be useful for detection of defects located at the edges.
Keywords: Guided waves, Edge wave, Lamb wave, Piezoelectric transducer, Wave excitation

1. INTRODUCTION
Elastic guided waves (GWs) are widely employed in ultrasonic NDT/NDE for damage
characterization due to their capabilities for long-range propagation and sensitivity to defects. Forexcitation and sensing of GWs, thin piezoelectric transducers adhesively attached to the structure areamong commonly utilized tools [1]. Debonding or imperfect contact between a transducer and an
inspected structure can result in the failure of the whole SHM system. Thus, the objective of themonitoring applies to the piezoelectric wafer active sensors (PWASs) as well [2]. The complexity ofwave propagation and excitation in laminates conditioned by, e.g., the multimodality of GW motion,
dynamics of piezoelectric transducers and the quality of their contact with the host structure should beproperly addressed for better performance of GW-based NDT/NDE systems. The aim of the present
study is to apply semi-analytical mathematical approaches in order to study Lamb wave excitation by
fully bonded and partially debonded piezoelectric transducers in plates. The potential of GWs
propagating along plate edges for practical applications is discussed and experimentally justified.

2. MATHEMATICAL MODELLING
2.1 Guided waves
To consider wave propagation in an elastic plate of thickness H, let us introduce the Cartesian
coordinates as shown in Figure 1. Two locations of PWAS for GW generation are considered: arectangular PWAS with dimensions 30 mm x 5 mm x 0.25 mm adhesively a ttached to the face and at
the edge of the plate (see Figure 1). In the numerical simulations, vibrations of an elastic plate are1

m_golub@inbox.ru
eremin_a_87@mail.ru
3
mv_wilde@mail.ru
2458164alisashpak7@gmail.com
inka.mueller@aol.dedescribed by the Lame equations of the linear theory of elasticity, which are formulated in terms of thedisplacement vector, while vibrations of the piezoelectric transducer are expressed in terms of thedisplacement vector and the electric potential (for more details see [3]). The faces of the plate except
the contact area S c between the PWAS and the plate (bottom of the PWAS in this case), where thecontinuity of traction and displacement vectors is assumed, are modelled via stress-free boundary
conditions. If the PWAS is partially debonded, the bottom of the PWAS is split into two parts: thecontact area Sc and the debonded domain S d, where stress-free boundary conditions are assumed.
Severity C of PWAS’s damage is defined as the ratio between the area of the debonded domain S d andthe total area of the bottom of the PWAS.

Figure 1 – Geometry of the problem considered
Along with the Lamb waves, guided edge waves (EW) can propagate in elastic plate shown in
Figure 1 [4,5]. Figure 2 demonstrates slownesses of propagating GWs (both, LWs and EWs) in arectangular aluminium elastic plate with Lame constants ߣ ൌ ͷͳǤͳ GPa, ߤ ൌ ʹ͸Ǥ͵ GPa and massdensity ߩ ൌ ʹ͸ͶͲ kg/m3 . At lower frequencies, the wave energy generated by a PWAS mounted at asurface is to be distributed among two fundamental Lamb waves (‫ Ͳܣ‬and ܵͲ), two fundamental EWs(‫ Ͳܣܧ‬and ‫ )Ͳܵܧ‬and ܵ‫Ͳܪ‬.

Figure 2 – Slownesses of GWs propagating in an aluminium plate8165

2.2 Transducer at the face of an elastic plate
Dynamic interaction of a PWAS attached to a face of an elastic plate (PWAS centre is situated at(0, h,b) ) is simulated within the plane-strain assumption, i.e., a two-dimensional model is considered.
It is based on the semi-analytical hybrid approach (SAHA) [3], which employs the boundary integral
equation method [4] and the spectral element method [5]. According to the SAHA, the displacement
vector and the electric potential function in the PWAS are expanded with Lagrange interpolation
polynomials at Gauss–Lobatto–Legendre points. These expansions are substituted into the equationsof motion, the latter leads to the corresponding system of linear algebraic equations. The semi-analytic
nature of the SAHA enables to extract GWs from the total wave-field and, therefore to analyse energydistribution among GW propagation in different directions from the source of motion . It should bementioned that the SAHA can be applied to simulate guided waves scattered by defects.
In order to model dynamic interaction of the piezoelectric transducer with the plate, continuity ofthe displacements and stresses in the contact area is assumed. The traction vector -function isinterpolated with the splines and the solution is found in the nodal points. If the degradation of thecontact (debonding) is to be modelled, the traction vector function is assumed zero at the nodal points
belonging the area of debonding and, therefore, the continuity condition is disregarded. Detailed
description of the proposed mathematical method is given in [3].
Since the two-dimensional model is applied, only the following wave energy distribution
coefficients can be defined:

Kmr Emr / E ( 0 ) .

(1)Here E ( 0 ) is the whole amount of wave energy transferred by the PWAS along the x1 axis into therplate, whereas Em denotes the wave energy carried by m-th Lamb wave in the directions x3 o rf .
The amount of wave energy is determined in terms of the time-averaged power density vector orUmov-Poynting vector [6]Z§ 3·Im¨ ¦ uiV ij* ¸.
(2)2 ©i 1¹
In the case of the PWAS attached at the face of the plate, the following expressions are employed:
ej

h l/2

E

 ³ ³ e3 ( x1,0, x3 )dx1dx3 ,( 0) h l / 2³³ e  n dS , SEmrr
R(4)2{ x3  b  x12!

R 2 ,h d x2 d h, x3b}.

SRIf the PWAS is perfectly bonded, the wave-fields excited by the PWAS are symmetric with respect tothe x 3 axis, so Em Em and distribution coefficients Km Km  Km are more applicable. It shouldbe also noted that only Lamb wave excitation is studied within the model described since the PWAS is
located far from the edges, where EW can propagate.






2.3 Transducer at the edge of elastic plate
In [6], the semi-analytical method based on the application of the Fourier transform with the modal
expansion of the displacement vector was applied to describe the GW excitation in the plate with thePWAS attached at the edge x3 0 . This method is employed in the study to calculate the wave energy
distribution coefficients K m

Em / E (0) defined via

h l/2E(0) ³ ³ e3 ( x1 , x2 ,0)dx1dx2
 h l / 2,Em³³ e  n dS , SR

{x32  x12R 2 ,h d x2 d h}

(3)

SR

The ratio of EWs in the spectrum is of the greatest interest here since it is important to understand how
effectively EWs can be excited by piezoelectric transducers.

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3. GUIDED WAVES EXCITATION BY TRANSDUCER AT THE FACETo analyse the influence of the bonding conditions between the PWAS and the waveguide on theenergy distribution among GWs, wave energy distribution coefficients K m are used. Figures 3-6
demonstrate the influence of the frequency on K m showing ratio of the m-th mode in E . The
simulation have been performed for the aluminium plate of width H=4.85 mm with the PWAS (PIC155 PI Ceramics) of 5 mm width and 0.25 mm height, while Figure 7 shows the variation of theamount of energy transferred into the waveguide for all the cases considered in Figures 3-6. Figure 3
demonstrates the alteration of K m if the PWAS is debonded in the middle and the edges are still glued.
It is observed that relatively small debonding (C=20 %) in the middle (Figure 3c) has almost no
influence on the wave energy distribution. The changes due to debonding are also not very substantial
at higher frequencies. With the debonding area growth, the energy distribution changes significantly.
( 0)

Moreover, a number of extra sharp resonance peaks are visible in K m and E
debonded PWASs (Figures 3b and 3d).

( 0)plots for severer

Figure 3 – Wave energy distribution coefficients K m for PWASs with symmetric debonding in the centre

Figure 4 – Wave energy distribution coefficients K m for two-sided symmetrically debonded PWASs
Figure 4 depicts wave energy distribution coefficient dependence on the debonding area S d of twoedges of the PWAS are symmetrically debonded, while the middle part is glued. One can observe thateven small debonding of the edges (0.5 mm from each edge – Figure 4b) alters energy distribution

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( 0)

close to the frequency ݂=300 kHz, where a sharp peak is visible in E
plot. At this frequency, theenergy distribution of the perfectly glued PWAS K A0 :KS 0 =65:35 but if the 0.5 mm of the bottom
surface are not properly glued, the energy distribution changes to K A0 :KS 0 =5:95.
If debonding of a PWAS is asymmetric, amount of energy transferred in the directions x3 o rf
are different. The energy distribution among Lamb waves excited by PWASs with one-sided
asymmetric debonding is shown at the Figure 5. As expected, for the majority of frequencies more
wave energy is transferred to the side, where the PWAS is bonded. Additionally, sharp peaks
corresponding to the abrupt alteration in the modes’ distribution are visible and the number of thepeaks increases with the debonding area growth.

Figure 5 – Wave energy distribution coefficients K m for one-side debonded PWASsrDistribution of the energy excited by PWASs with two-sided asymmetric debonding isdemonstrated in Figure 6. An interesting effect is observed in a narrow frequency range around
f=295 kHz, which is right below the cut-off frequency of A1 mode. In this frequency range, the amountof energy Em carried by Lamb waves in the positive direction abruptly increases and exceeds Em . It
is also visible that at lower frequencies (up to 200 kHz) ‫ Ͳܣ‬mode prevails in the excited wave-field
for debonded PWASs except for the areas near resonance frequencies.
Analysis of Figures 3-6 allows concluding that defect in contact zone between the PWAS and thewaveguide can cause significant changes in wave excitation even if the damaged zone is not large(C<20%). Debondings may alter amount of energy and its distribution among Lamb waves
substantially, especially if debonding is asymmetric. Nevertheless, if the defect does not touch edgesof the PWAS (i.e. edges of the PWAS are glued properly) this debonding causes changes at higher
frequencies. Even in this case, there are frequencies close to the resonance, where the energy
distribution alters significantly compared to the perfectly glued PWAS case.

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Figure 6 – Wave energy distribution K m for two-sided asymmetrically debonded PWASsr

Figure 7 presents the dependence of the excited energy ‫ ܧ‬଴ on the frequency if PWAS is debonded.
Energy plots shown at Figure 7a, 7b, 7c and 7d are calculated for the same contact conditions as
considered in Figures 3,4, 5 and 6 respectively. Alteration in the contact conditions leads to theappearance of additional maxima of in E

( 0)

plots, (initial maxima remain intact at the same frequency,( 0)changes). Accordingly, all the maxima for defectively bonded PWASs
though absolute value of E
are shifted to lower frequencies compared to the perfectly glued PWAS the maximum at ݂=580 kHz.

Figure 7 – Wave energy for partially debonded PWASs

4. GUIDED WAVES EXCITATION BY TRANSDUCER AT THE EDGE
In order to demonstrate importance of EW for SHM and NDE, EW and LW interaction with asurface-breaking defect at the edge of aluminium plate (notch 3 mm depth and 0.6 mm width) has been8169

investigated experimentally. The geometry of the specimen is shown in Figure 1. The PWAS at theedge has driven with a transient voltage in the form of a narrow-band Hann-windowed five-cycle sine
burst with central frequency f0 . The latter are generated by a Tektronix AFG 3022B arbitrary signal
generator and are pre-amplified to the range 37.5 V-pp by a NF HSA4101 external high-frequency
power amplifier before being applied to the PWAS. The out-of-plane velocities were measured bymeans of a Polytec PSV-500 one-dimensional scanning laser Doppler vibrometer.
It was demonstrated in [5] that fundamental mode ES0 is predominant at the edge if a PWAS isattached at the edge. Mode ES0 is not localized in the vicinity of the edge surface at lower frequencies,
and the amplitudes of ES0 decay faster with depth at higher frequencies. Accordingly, relatively low
reflection by surface-breaking defects is observed at central frequencies above f0=150 kHz.
Nevertheless, at the central frequencies above 200 kHz, scattering of EW becomes much stronger(more details can be found in [5,8]). In Figure 8, an example of such scattering is demonstrated for
f0=300 kHz. Figure 8 shows snapshots of the Hilbert-enveloped out-of-plane velocities
w wu2 ( x1 , h, x3 , t ) / wt measured on the face x2=h of the specimen. It can be seen that the crack
mainly reflects ES0 mode backward and reradiates it into Lamb waves. Transmitted ES0 waves are ofmuch lower amplitudes compared to the reflected ES0 waves. Therefore, EWs provide enoughinformation about presence of a surface-breaking defect for its identification.

Figure 8 –Hilbert-enveloped out-of-plane velocities measured on the surface of the plate with defect of
depth d=3 mm at central frequency 300 kHz at different moments of time t

5. CONCLUSIONS
A detailed analysis of Lamb waves excitation by perfectly bonded and partially debonded
piezoelectric wafer active sensors has been performed. The provided analysis shows the influence ofdebonding between the PWAS and the structure on Lamb wave excitation. The present study also
demonstrates that EW scattering by surface-breaking cracks can be used in GW-based NDT/NDE
systems. Laser Doppler vibrometry is employed to approve the developed mathematical models
experimentally. The authors believe that the results might be useful for practical applications.

ACKNOWLEDGEMENTS
The work is supported by the Ministry of Science and Higher Education of the Russian Federation(Project 9.1022.2017/4.6). The authors are thankful to Prof. Rolf Lammering (Institute of Mechanics,Helmut Schmidt University, Hamburg, Germany) for the comprehensive support of the experimental
investigations.

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