PROPAGATION OF LAMB WAVES ON AN IMMERSED CORRUGATED PLATE.
Nadia Harhad(1), Mounsif Ech-Cherif El-Kettani(2), Hakim Djelouah(3), Jean-Louis Izbicki (2), Mihai Predoi(4)
(1)Scientific Center of Research on welding and Control, CSC, Algiers, Algeria. (2) LOMC UMR CNRS 6294, University of Le Havre, Le Havre, France. (3) U.S.T.H.B, Faculty of Physics, Algiers, Algeria, (4)Department of
Mechanics, University POLYTECHNICA of Bucharest, Bucharest, Romania
nharhad@yahoo.fr
Abstract
In this paper the propagation of Lamb waves in an aluminum plate with a roughness on only one side is studied. The interaction between the incident Lamb wave and the grating gives rise to reflected converted waves. This phenomenon is studied experimentally in the case of an immersed plate in water. Our objective is to show that retro-converted waves radiating into the water are detectable although their energies are small. The damping coefficient of the propagating Lamb wave in the plate is evaluated. Preliminary numerical simulation by using a finite elements method is performed in order to help experiments.
Key words: Rough surface, Lamb waves, conversion modes, phonon relation.
1 Introduction
Rough liquid-solid interfaces have been the object of several theoretical and experimental studies especially in the field of nondestructive testing and evaluation (NDTE) and in optics. In the case of a plate with roughness, Lamb waves can be used [1-2]. Their interaction with the rough surface gives rise to converted backward modes. In the case of a periodic grating, incident and retro-converted modes wave numbers and the grating period Λ obey to the phonon relation [3]. For some particular frequencies, the incident mode is reflected into itself or into a different mode (mode conversion) in the first Brillouin zone [4]. This work is an experimental study of the propagation of Lamb waves on an immersed plate containing a phononic grating with a periodic geometry. Two questions arise: is it possible to detect experimentally the roughness on an immersed structure? Is the study of the scattering by the rough plate sufficiently sensitive? These problems arise during the underwater detection of rough surface as for example in the case of an immersed corroded structure.
In this paper, after a brief description of the studied sample, results from numerical simulations are given in order to help experiments. The experimental dispersion curves are presented and compared to the theoretical ones. The damping coefficient of the incident Lamb wave is also measured. The attenuation introduced by the grating is shown and a comparison is made between experimental and theoretical results.
2 Numerical Simulation
2.1 Description of the Studied Sample
The studied sample is an aluminum plate of 5 mm thickness. Longitudinal and shear velocities are
respectively CL = 6320 m/s and CT = 3115 m/s. Aluminum density is 2700 Kg/m3. Only the middle part of one side of the plate is grooved with 20 identical rectangular grooves. The grooves have a depth p = 0.1 mm and a spatial periodicity Λ = 6 mm (Figure 1).
Figure 1. Geometry of the studied sample
2.2 Numerical Processing
To identify Lamb waves that compose the reflected backward waves, a numerical simulation is made with COMSOL in transient analysis. The plate described in part (II-A) is considered in vacuum in order to avoid a long time of computation. An analytical formulation is used for the displacements which are applied on the left edge of the plate in order to generate a single S0 Lamb mode at a frequency of 320 kHz. To represent the time-space image of the propagating waves in the plate, the normal surface displacements on the face not grooved are recorded along 245 mm propagation distance.
x(m)
t(s)
0 0.05 0.1 0.15 0.2
0
1
2
3
4
5
6 x 10-5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x 101 -15
1
2
Figure 2: Time-Space (x,t) representation ofsignal. The levels of the incident wave (1) are saturated in order to visualise the reflected one (2).
In Figure 2, a time-space (x,t) representation is given, part (1) corresponds to the incident wave and part (2) to the reflected wave from the grating.
2.3 Results
In order to identify the incident and the reflected waves, only the displacements before the grating are studied with a two-dimensional Fast Fourier Transform (2D FFT). These results are superimposed on the theoretical dispersion curves of an infinite plane plate, which allows to identify the Lamb waves: a part of S0 incident mode at 320 kHz Figure 3 (a), is converted into A0 mode around 290 kHz Figure 3
(b).
f (Hz)
k (1/m)
1 2 3 4 5
x 105 0
500 1000 1500 2000 2500
1 2 3 4 5 6 7 8 9 10 11 x 10-18
A0
S0 A1
f (Hz)
k (1/m)
1 2 3 4 5
x 105 -2500
-2000 -1500 -1000 -500 0
0.5 1 1.5 2 2.5 x 10-19
A1
S0
A0
(a) (b)
Figure 3: Representation in dual space (f,k): (a) S0 incident wave and (b) A0 retro-converted wave.
At this frequency the corresponding wave numbers of S0 incident mode and A0 retro-converted mode are respectively kS0 = 306 m-1 and kA0 = 735 m-1. The wave number related to the grating is 2π/ Λ = 1447 m-1. With these values, the phonon relation:
kS0+kA02/Λ
is satisfied with a relative error of 0.6%.
The conversion coefficient is defined by the ratio between the energy of A0 retro-converted mode ΦA0 and the energy of S0 incident mode ΦS0 : RA0 = ΦA0/ ΦS0. At 290 kHz frequency, RA0 = 0.8%. The A0
mode is reflected by the grating with a small energy lower than 1% of the energy S0 incident mode.
The wave numbers of S0 and A0 modes deduced by simulation are used in the experimental study to determine the angles values of the emitting and the receiving transducers.In an experiment involving the scattering of the waves in water, is it possible to detect A0 retro-converted mode?
3. E
XPERIMENTS3.1 Experimental setup
The experimental setup is represented in Figure 4. All measurements are performed in an immersion tank. The generation and the reception of Lamb waves are made thanks to two identical piezoelectric transducers of 500 kHz central frequency. The transducers are adjusted so that the incident beam, the normal at the point of incidence and the reflected acoustic beam are in the same plane, orthogonal to the surface of the plate. The propagation distance from the plate to the emitter and to the receiver transducers are not necessarily equals but are sufficiently far from the plate to emit and to receive in the farfield conditions. The incidence and reception angles depend on the wave number of respectively the generated and received Lamb wave. The values of the angles are deduced from the Snell-Descartes relation [5]:
sin-1(CLiq / CLamb)
CLiq is the ultrasonic velocity in water and CLamb is the phase velocity of the Lamb wave, which is deduced from the theoretical Lamb waves dispersion curves. A generator delivers a burst with eight periods to the emitting transducer. The amplitude of the signal is about 10 V. In order to observe and to verify that the generated Lamb wave is well the wished one, emission and reception are done at the same angle θi. The receiving transducer is maintained in the same direction as the emitter and is
translated parallel to the surface of the plate with a step of 0.5 mm as shown on Figure 4 (a). The received signal V(x,t) is recorded for several positions, located by their abscissa x. This scan starts when the receiving transducer is out of the limits of the direct emitted field and the end is determined by the attenuation of the propagated Lamb wave. To detect the backward wave as predicted by the numerical simulation, the receiving angle θt is corresponding to the angle of remission of the expected retro-converted mode as shown on figure 4 (b). The receiving transducer is translated parallel to the plate in the opposite direction of the propagation of the incident Lamb wave. The scan in this case starts at the beginning of the first groove and finished when the retro-converted Lamb wave is totally attenuated.
Figure 4:(a) Scan path to detect the incident propagating Lamb wave.(b) Scan path to detect the backward (converted/reflected) Lamb modes.
3.2 Generation of S
0mode at 320 kHz frequency
For θi = 17°, S0 Lamb wave is generated at 320 kHz frequency. To observe the incident mode, the angle of the receiving transducer is oriented under the same angle. The area scanned by the receiver is 50 mm from the grating, 118 mm under the grating (which is its length) and 50 mm away from the grating as shown on Figure 4 (a). To observe A0 retro-converted Lamb wave, the receiving transducer is adjusted around the expected theoretical angle which is 35° and is displaced in the opposite direction of propagation of the incident mode. The waveforms are measured on 60 mm propagation distance by step of 0.5 mm as shown on Figure 4 (b). In Figure 5 (a) and (c) the amplitude of respectively the incident and the backward signals are represented. By applying a 2D FFT the incident signal and the retro-converted modes are respectively identified in Figure 5 (b) and Figure 5 (d) by superimposing the 2D FFT results on the theoretical dispersion curves of the Lamb waves for a free plate.
x (m)
t (s)
0.05 0.1 0.15 0.2
0 1 2 3 4 5 6 7 8 9 10
x 10-5
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
f (Hz)
k (1/m)
1 2 3 4 5
x 105 0
500 1000 1500 2000 2500
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 105 -5
S0 A0
A1
(a) (b)
x (m)
t (s)
0 0.01 0.02 0.03 0.04 0.05 0.06 0
1 2 3 4 5 6 7 8 9 10
x 10-5
-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01
f (Hz)
k (1/m)
1 2 3 4 5
x 105 0
500 1000 1500 2000 2500
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10-5
A1 S0 A0
(c) (d)
Figure 5. Time-space (x,t) representation: (a) S0 incident wave (c) A0 retro-converted wave . Dual space (f,k) representation : (b) S0 incident wave, (d) A0 retro-converted wave.
At the frequency of 300 kHz, the corresponding wave numbers of S0 incident mode and A0 retro- converted mode are respectively kS0 = 379 m-1 and kA0 = 706 m-1. With these values, the phonon relation is satisfied with a relative error of 3.6%.
For a plane plate the damping coefficient of the Lamb waves corresponds to the reemission of the wave in the surrounding medium. The resolution of the dispersion equations of the Lamb waves propagating in an immersed plate without roughness permits to determine the theoretical value of the damping coefficient which is 8.90 m-1.
0 0.05 0.1 0.15 0.2 0.25 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
x (m)
Amplitude (V)
After the grating Under the
grating Before
the grating
Figure 6: Representation of the evolution of the amplitude of S0 incident mode versus propagation distance.
Experimentally the maximum amplitude of the signal corresponding to S0 Lamb wave is represented versus the propagation distance in Figure 6. To determine the values of the damping coefficient before, under and after the grating, three exponential fitting of the curves are used.
The damping coefficients values before and after the grating are respectively 9.05 m-1 and 9.27 m-1, which are nearly the same. These values are slightly greater than the theoretical value which can be explained by the geometrical beam diffraction, which not taken into account in theory. The measured damping coefficient under the grating is 10.46 m-1, greater than its values before and after the grating.
The supplementary attenuation introduced by the grating is then shown.
4. Conclusion
In this work, an experimental detection of converted backward Lamb waves by an immersed corrugated plate has been performed. The retro-converted mode is well detectable, even if its energy predicted by numerical simulations in vacuum is weak. Its angular position is deduced from the numerical simulation. The phonon relation has been verified for the immersed grating. Moreover, we show that the damping coefficient is more important under the grating. The supplementary attenuation introduced by the grating could then be evaluated.
Bibliography
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[3] D. Leduc, A-C. Hladky, B. Morvan, J-L. Izbicki and P. Pareige, “Propagation of Lamb waves in a plate with a periodic grating: Iterpretation by phonon,” J.Acoust.Soc.Am., Vol. 118, No.4, October 2005, pp. 2234-2239.
[4] M. Bavencoffe, A-C. Hladky-Hennion, B. Morvan and J.L. Izbicki, “Attenuation of Lamb waves in the vicinity of a forbidden band in a phononic crystal,” IEEE Tarns. Ultrs. Ferr. Frq. Contl., Vol. 56, N°. 9, Septembre 2009, pp. 1960-1967.
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