Ultrasonic reflection and material evaluation

Transcription

1 Ultrasonic reflection and material evaluation W.G. Mayer To cite this version: W.G. Mayer. Ultrasonic reflection and material evaluation. Revue de Physique Appliquee, 1985, 20 (6), pp < /rphysap: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1985 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 (a) An Revue Phys. Appl. 20 (1985) JUIN 1985, PAGE 377 Classification Physics Abstracts Ultrasonic reflection and material evaluation W. G. Mayer Physics Department, Georgetown University, Washington D.C , U.S.A. (Reçu le 7 novembre, accepté le 9 novembre 1984) Résumé On donne dans l article un panorama des principes physiques de base concernant la réflectivité ultrasonore en relation avec la caractérisation de matériaux solides. Les exemples donnés se rapportent à la réflexion critique, aux ondes de Rayleigh et de Lamb et à l orientation cristalline effectuée au moyen de mesures de reflexion Abstract overview is given describing the physical basis of ultrasonic reflectivity as it relates to solid material characterization. The examples given illustrate critical angle reflection, Rayleigh and Lamb waves, and crystal orientation via reflection measurements. 1. Introduction. The reflection coefficient of ultrasonic waves for a flat boundary formed by a solid and a fluid depends very strongly on the angle of incidence and the mechanical properties of the materials involved. In particular, reflectivity shows very pronounced maxima and minima as well as other changes at critical angles of incidence. Critical angles are defined as those angles of incidence (in the fluid) where a refracted wave (in the solid) travels parallel to the boundary between the media as indicated in figure 1. The angle of refraction for the longitudinal wave, a, or for the shear wave, j8, in the solid is in general related 10 the angle of incidence, 0, in the liquid by Snells law sin 03B8/vf = sin 03B1/v1 = sin 03B2/vs, (1) where vf is the sonic velocity in the fluid and v, and v. are the velocities of the longitudinal and shear waves, respectively, in the solid. Therefore, the critical angle of incidence for the longitudinal wave, 6L, will be defined as the particular 0 for which a 900 in = equation (1); correspondingly Os is defined as the 0 for which P 900 in equation (1). = Snell s law only yields the angles of reflection and refraction but not the amount of reflectivity or transmissivity into the solid These latter quantities were first calculated by Knott [1], a seismologist, in 1899 for plane waves. Rewriting his results yields a reflection coefficient for the energy which is of the form where Fig. 1. Snells law refraction for ultrasonic waves at infinite half space boundary. (b) Conversion to plate modé vibration. The density of the solid is given by p and that of the fluid by 03C1f. An examination of equation (2) shows that if 0 = 03B8L or 0 Os, the ratio of reflected to incident = energy is unity while for all other angles of incidence (R/I)2 is less than unity except when 0 > Os in which case total reflection occurs. Based on this simple analysis one can make use of reflection measurements to determine a number of properties of the reflecting solid. Although there exist a great number of variants, only the fundamental features of three distinct combi Article published online by EDP Sciences and available at

3 Classical eliminating 378 nations of parameters will be discussed here. These three groups are : fluidsolid boundaries (both media are considered to be infinite half spaces) ; solid plates in fluids; anisotropic solidfluid boundary. 2. Infinite half space boundaries. When the reflecting solid is thick enough so that the product frequency (of the incident ultrasound, in MHz) times sample thickness (in mm) is greater than approximately 15, one can, to a very good approximation, use equation (2) to experimentally locate a reflection peak at 03B8L. If the sample is immersed in water (vf 1.49 km/s) then the reflection coefficient = yields information about the longitudinal velocity of the solid. Plots of the reflection coefficient are shown in figure 2 for a ceramic material (alphasiliconcarbide, v km/s, vs 7.5 km/s), a typical = = metal (brass, vl 4.43 km/s, Vs 2.12 km/s), and = = a plastic (plexiglas, v km/s, vs = = 1.12 km/s). It is seen that in all cases the reflection peak at the longitudinal critical angle is very pronounced and well defined, making it possible to determine v, simply by observing experimentally the angular location of the reflection peak the need to propagate ultrasound through the sample. Fig.2. reflection coefficient for boundary between water and plexiglas (solid line), brass (dashdot), and alpha silicon carbide (dotted line). It should be noted that the curves in figure 2 show a sharp rise to 100 % reflectivity at the shear wave critical angle, except for plexiglas for which no real Os exists because vf > Vs and therefore sin 0 > sine in equation (1). Some of the earliest experimental investigations of critical angle reflectivity [2] confirmed the general validity of the theory. The 03B8L peak is relatively easy to detect although it does not usually reach 100 % due to the fact that attenuation is not considered [3] in equation (2). However, the predicted total reflection for Os (when a real Os exists) is ordinarily unobservable. Instead, near Os the reflectivity decreases [2] to values of 50 % or less at a 0 slightly greater than Os. Just as 0L is reached when oc becomes 900, this new critical angle is reached when another possible wave travels along the surface. A reflection dip occurs whenever this wave, a Rayleightype wave, is generated. Rayleigh waves are surface waves whose velocity is less than the shear wave velocity, and in a first approximation this velocity is given [4] by where J is the Poisson s ratio of the solid. One could add one more term to Snells law, equation (1), namely... sin y/vr, but since this Rayleightype wave only travels along the surface, sin y = = 1 always, giving rise to another critical angle of incidence. This critical angle, usually called the Rayleigh angle, OR, denotes the incident angle under which the Rayleightype wave is generated. This wave travels along the interface and radiates its energy back into the liquid with the angle of radiation being equal to the angle of reflection. The maximum of radiation is occasionally «displaced» along the surface as shown in figure 3a. This phenomenon of nonspecular reflection was first observed by Schoch [5] for Rayleigh angle incidence. Nonspecular reflections were later investigated by Bertoni and Tamir [6] for ultrasonic beams of finite width incident at the Rayleigh angle and later by Ngoc et al. [3] for other angles, showing the existence of nonspecular reflection effects other than a simple «displacement». Among them the most noticible effect is a splitup of an incident ultrasonic beam into a doublepeaked reflected beam as shown as an example in figure 3b. The occurrence of nonspecular reflection at critical angles is useful in material characterization, particularly for surface homogeneity determinations. Once the Rayleigh angle of a material has been determined [through use of equation (3) or other formulae] and the transducer has been positioned over the interface so that Rayleigh angle nonspecular reflections are observable, the reflected beam should not change its characteristic beam profile when the sample is moved parallel to itself If, however, any portion of the sample surface has an inhomogeneity which changes the elastic properties, and thus v1 and v., the Rayleigh wave velocity also changes in this isolated location. This in turn makes the local value of v, different from the rest of the sample and the fixed incident beam no longer strikes at the Rayleigh angle when the local inhomogeneity is irradiated. As a consequence, the nonspecular reflection profile changes. This change in the beam profile can easily be seen in Schlieren images similar to figure 3. Local changes in the value of vr of about 1 % can be detected and an inhomogeneity can thus be localized. Thus, reflectivity studies can reveal small local inhomogeneities. Another application of reflectivity studies is concerned with the determination of the thickness of a plate one side of which may not be accessible. This aspect is discussed in the next section.

4 Examples Section 379 Fig. 3. of nonspecular reflection. (a) lateral shift with location of reflected beam indicated by dashed lines if reflection would be specular, (b) doublepeaked nonspecular reflection. The applied force may well be an ultrasonic beam striking the plate which is immersed in a fluid as indicated in figure 1 b. If 03BBp is the wavelength of one of the possible plate modes there will be a corresponding 0 that will match the wavefronts at the interface and a plate mode will be generated When this condition is met the reflection coefficient for the incident beam, in a first approximation, goes to zero. However, if a bounded ultrasonic beam is used nonspecular reflection effects become important, similar to the case of Rayleigh angle incidence for infinite half spaces. In the case of solid plates there will be a number of critical angles of incidence, corresponding to the generation of the various plate modes (or Lamb modes). The computation of the velocities of plate modes is somewhat involved [8], and the calculations are quite cumbersome [9] when incident bounded ultrasonic beams are considered. The use of the results, on the other hand, is quite simple. As an illustration of a technique to determine the thickness of a plate when only one side of the plate is accessible, consider figure 4. It represents a portion of the Lamb mode velocity dispersion curves for a brass plate immersed in water. The abscissa is labelled fd (in MHz. mm) and the ordinate in terms of Lamb wave velocities and in angle of incidence in water, the two being related through Snells law. 3. Solid plates in a floid When a flat, homogeneous solid plate is subjected to an external locally applied periodic force the plate may be in resonance with the force and the disturbance may propagate along the plate. It was shown by Lamb [7] that more than one resonance is possible and thus one may observe different normal modes of vibration, depending on the thickness of the plate, its elastic properties, and the frequency of excitation. The defining equations for Lamb modes, in their simplest forms, are given by Fig. 4. modes. of velocity dispersion curves for Lamb 4 LS coth (03C0Sfd/w) (1 + S 2)2 coth (nlfd/w) = ix 4 LS tanh (03C0Sfd/w) (1 + S2)2 tanh(lfd/w) = ix where The unknown, the Lamb mode velocity, w, has more than one possible value ; the possible magnitudes of w depend very much on the product fd, the bulk wave velocities and to a lesser extent on the ratio of the liquid and solid densities. (4) Assume now that a 1 mm thick brass plate is used and that the ultrasonic beam has a frequency of 4 MHz. This locates the product fd at 4, the same as for a 4 mm brass plate and a frequency of 1 MHz. One notes that Lamb modes can exist when the angle is 4660, 35.8, 24.4, 18.3 or 15.9 because the possible Lamb mode velocities are 2.05, 2.55, 3.6, 4.75 and 5.45 km/s. Under these conditions reflections should be nonspecular. Some modes are easier to excite than others and the form of the nonspecular reflection effects may well differ even for the same beam being used, with attenuation of the modes and beam width being the determining factors [10]. Suppose now that a small section of the plate is 5 % thinner than the rest of the plate. In this thin section

5 Variation thus 380 the value of fd is reduced to 3.8 and the mode velocities increase except the lowest velocity mode (which is equal to the Rayleigh mode). Thus the previously determined Lamb angles are now different. If such a plate is scanned, with the angle of incidence fixed so that a distinct nonspecular reflection profile is observable, the nonspecular character of the reflection will collapse into a pure specular reflection profile when the thin portion of the plate is irradiated. Fortunately, the same principle holds when the plate is asymmetrically loaded (i.e., water on one side and another fluid or a gas on the other). Although equation (4) would have to be expanded to include the second fluid the velocity dispersion curves usually vary little from those for the symmetrically loaded case. Thus one can find the thickness of the plate even if only one side is accessible. It is evident that one can detect corroded or otherwise thinned sections of containers of hazardous liquids, or one can determine the thickness of a plate which cannot be easily reached from both sides, or the presence of flaws which inhibit the normal propagation of Lamb modes, etc. 4. Crystal orientation. In the previous sections the assumption was made that the solid is isotropic. The situation becomes much more complicated when anisotropic (single crystal) substances are considered. In this case the bulk wave velocities are orientation dependent as was shown by Green [11] as early as Just as the Rayleigh velocity in isotropic substances depends on the values of vl and vs [Eq. (3)], so does the Rayleigh wave velocity on the surface of a single crystal, except here v, is also orientation dependent. Comprehensive search methods were employed by Farnell [12] to prepare surface wave velocity curves for single crystals. Figure 5 shows the incident angle in water needed to excite a surface wave (outer portion of Fig. 5) or a pseudosurface wave [12] on the (100)face of copper. One notes that the incident angle has an extremum when one of the surface waves is to be propagated in the [110]direction on the (100)face. This fact allows one to find this direction on the crystal face and thus all the other crystallographic directions simply by making use again of nonspecular reflection phenomena. If the incidence is adjusted to correspond to the Rayleigh angle or the pseudosurface wave critical angle for the [110]direction, a nonspecular effect will occur only when the projection of the incident beam onto the sample surface is in the [110]direction. Fig. 5. of critical angle incidence for a watercopper (100)plane interface, based on data of reference [12]. Rotation of the sample under the incident beam will thus show nonspecular reflection only when the lineup between the incident beam and the [110]direction is satisfied the crystal has been oriented It should be pointed out that no theory exists which will predict precisely what type of nonspecular reflection will occur when different beam widths and frequencies are used in this application. Nevertheless, the method has been used successfully [13] for the orientation of crystals and for the angle of energy flow [14] in surface waves on single crystals. 5. Conclusioa Nonspecular reflectivity of ultrasonic beams is always connected with wave excitation at a critical angle. Observation of these phenomena yields information about these critical angles and thus enables one to determine important properties of the solid, as for instance wave velocities, inhomogeneities, changes in thickness and certain crystal orientation aspects. The topics discussed above are illustrations of some of the possibilities to characterize solids by using ultrasonic reflection measurements. Acknowledgments, The work on which ihis paper is based has been supported by the Office of Naval Research, U.S. Navy.

6 381 References [1] KNOTT, C. G., Philos. Mag. 48 (1899) 64. [2] ROLLINS, F. R., Material Eval. 24 (1966) 683. [3] NGOC, T. D. K. and MAYER, W. G., J. Appl. Phys. 50 (1979) [4] BERGMANN, L., Der Ultraschall, 6th ed. (Hirzel Verlag, Stuttgart) [5] SCHOCH, A., Acustica 2 (1952) 19. [6] BERTONI, H. L. and TAMIR, T., Appl. Phys. Lett. 2 (1973) 157. [7] LAMB, H., Proc. R. Soc. London, A 93 (1917) 114. [8] VIKTOROV, I. A., Rayleigh and Lamb Waves (Plenum Press, New York) [9] NGOC, T. D. K. and MAYER, W. G., IEEE Trans. SU27 (1980) 229. [10] NGOC, T. D. K. and MAYER, W. G., IEEE Trans. SU29 (1982) 112. [11] GREEN, G., Trans. Cambr. Philos. Soc. 7 (1839) 121. [12] FARNELL, G. N., in Physical Acoustics, W. P. Mason, ed., vol. 6 (Academic Press, New York) [13] DIACHOK, O. I. and MAYER, W. G., Acustica 26 (1972) 267. [14] BEHRAVESH, M. and MAYER, W. G., IEEE Ultras. Symp. Proc. 1974, p. 104.