The piezoelectric elements used in this study are spherical or cylindrical portions which radiate directly into water, and thus form an aberration free surface that focuses the emitted acoustic beam into a diffraction limited spot (or line) at the centre of curvature of the transducer. This acoustic configuration is advantageous because the elimination of the buffer rod removes the systematic noise due to internal reverberations that tend to overlap in time with the reflected signal from the object.
where C is a normalization factor, P(θ) the pupil function of the transducer, R(θ) the reflectance function of the material, z the distance between the focal plane and the object plane and (θ) _{o} the half angle aperture of the transducer. In the most general case, P(θ) takes into account the non uniform excitation of the transducer and any aberration effects. In our computation, an approximation of P(θ) has been experimentally evaluated by measuring optically the displacement of a thin membrane placed in the aperture plane^{4}. Without going into further detail, it is important to point out that this formulation doesn't take into account any diffraction effect, neither attenuation for the surface acoustic waves. The acoustic signal S(z,f) received by the transducer is assumed to be defined in the frequency domain by : S(z,f) = A_{o}(f)x V(z,f) (2) where A_{o}(f) is the signal spectrum at the focal point. The second step consists in Fourier transforming of S(z,f) that gives s(z,t) in the time domain.
The computation of the theoretical signal has been carried out for a ceramic material at several z distances. Numerical results are now compared with a set of experimental data. To validate the modeling, the experimental and computed signals are plotted at a distance z=-4.5 mm (Fig.1). This distance has been chosen because both signals are sufficiently resolved in time in order to compare them clearly. Both consist of several components : the specular wave which first reaches the transducer, then skimming longitudinal and shear waves of relative small amplitude and eventually the Rayleigh wave echo. The two main echoes have a close amplitude and have their polarity inverted, both in computation and experiment. It must be emphasised, however, that the relative amplitude of the echoes are quite well computed by taking into account a measured pupil function. | Fig1 : theoretical and experimental signals for z=4.5mm |
Two sets of ASCAN for a z displacement are also given in figure 2 for theoretical and experimental signals. Acoustic signals are described for z=- 2.58mm to z=-4.98 mm. As illustrated by these figures, the computed results well match with experiments: both amplitude and shape of the theoretical signals are consistent with the observations at any z position. The polarities of specular and Rayleigh waves are always inverted. Other experimental signals of very low amplitude occurring in the same time of the reflected waves are not taken into account. |
Fig 2a: theoretical, and Fig 2b: experimental, set of ASCAN. for z=-2.58 mm to z=-4.98 mm |
V_{o} and V_{R} are the velocity of longitudinal wave in the couplant and leaky surface wave on the sample. The slopes of expressions (3) and (4) give V_{O} and V_{R} respectively. In order to improve the accuracy on the time measurement, several techniques of signal processing have been performed and the intercorrelation technique has been selected. By this way, the signal stored at the first position z. is taken as a reference. Then, for each z position the calculation of the time t_{o}(z)-t_{o}(z0) is first evaluated between the position z and z_{o}. Next, after the computation of V_{o} V_{R} is evaluated in the same way on the Rayleigh echo.
A relatively large number of Rayleigh velocity measurements was made on machined surface and polished surface of ceramic specimen. Some samples were cross sectioned perpendicular to the surface being studied so that the depth of damage could be measured. Roughness evaluation was also performed on these samples. Examples of typical signals on machined and polished ceramics are shown in figure 3.
Fig 3: machining effects on signals for 3 samples of
different roughness .
We observe that the Rayleigh-wave arrival-time increases with the roughness whereas its amplitude decreases. The results of a number of these measurements are summarised in table 1. These results demonstrate that the presence of machining induced damage in ceramic surface can be non destructively detected with good confidence.
specimen | velocity VR (m/s) | roughness Ra (µm) | damage depth (µm) |
polished | 3132 +/- 4 | 0.28 | 4 |
fine machining | 3112+/-12 | 0.6 | 12 |
rough machining | 3065 +/- 18 | 2.03 | 20 |
A cylindrical transducer is still excited by a pulse signal and moved vertically. For each z position, a Fast Fourier Transform is performed. Then, the amplitude of the spectrum is sampled for each frequency component of the transducer bandwidth, versus z. 1t gives curves similar to V(z). For the positive z distances, the V(z) curve depends only on the transducer geometry and the water temperature. Therefore, it provides a reference for any sample. In contrast, for negative z, the V(z) curve takes into account the surface wave contributions. Next, the Rayleigh wave velocity is computed through the V(z) analysis developed by Kushibiki^{5}. Figure 4 shows two V(z) curves corresponding to a ceramic and to an ideal reflector (without interference for z<0).
Fig 4 : V(z) curves for an ideal reflector and a
ceramic material at 10 MHz.
Experiments have to be performed, exactly in the same conditions (position, coupling temperature ... ), on an ideal reflector (R(θ) =l) and on the sample of interest. The two V(z) curves are stored (Fig.4) and subtracted each other (the reference subtraction is given by the positive z region). The Rayleigh wave velocity is deduced by measuring the oscillation period of the resultant curve by Fast Fourier Transform. In spite of the fact that two experiments have to be performed for a Rayleigh velocity measurement, an attractive characteristic of this method is that it enables to discriminate the Rayleigh and specular components which are not necessarily resolved in the time domain. Moreover, in contrast to traditional acoustic microscopy, this technique indicates if the material is dispersive or not with only one measurement instead of several at any frequency. As a typical case, we examined the variation of leaky surface-wave velocity due to machining. The three ceramic samples previously investigated were analysed. A cylindrical piezocomposite transducer was used in this study. The radius of curvature was 10 mm, the half angle aperture 33° and the central frequency was 10 MHz. The frequency dependent velocity was calculated and plotted in figure 5.
Fig 5: frequency dependent velocity on ceramic. polish specimen (1), fine machined specimen (2), rough machined specimen (3). The linear regression is indicated by a solid line. |
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