Abstract

In ultrasonic non-destructive testing (NDT), the time needed for the detailed inspection of a part is often due to the necessity of measurements at various angles of incidence leading to use several transducers. In order to reduce the duration of such a test, electronically driven array transducers generating ultrasounds propagating in various directions are more and more often used. Indeed, the angle and the focal length can be modified by applying appropriate delays on the signal emitted by each elementary transducer. The inspected sample can thus be quickly scanned.

This paper describes how the acoustic field generated by a phased array coupled to a solid can be modelled. In a first step, a modelling of the transducer-solid system is presented and validated by accurate measurements of the acoustic field. Then, thanks to some approximations, a modelling of a contact phased array is proposed. Simulation results are compared to measurements of the acoustic field generated in a duralumin sample by the phased array.

Modelling of the transducer-solid system

In infinite isotropic solids, two kinds of independent waves are propagating : longitudinal and shear waves. The propagation phenomenon in solids through these two kinds of waves is significantly different from the pressure wave propagation in liquids. An elastic solid is characterized by its mass density and by the longitudinal and shear wave propagation velocities c_L and c_T.

The kind of source modelled here is a transducer directly coupled to a semi-infinite solid medium. This transducer made of a backed thin piezoelectric ceramic (with or without acoustic matching layers) works mainly in thickness mode and preferentially generates longitudinal waves. The proposed model takes into account the intrinsic, elastic properties of the solid and the diffraction effects associated to the shape and the acousto-electrical environment of the transducer. The method used to solve this complex problem consists in separating the different physical phenomena.

In a first step, the transducer is decomposed in a continuous sum of normal point load sources generating a force normal to the surface of the solid (Huygens's principle). As the time variation of this load is a priori unknown (for example, it depends on the electrical environment of the transducer), it is interesting to find solutions to the propagation equations corresponding to an impulse point load. This approach is justified by the linearity of the system.

These solutions, called Green's functions, provide the displacements at each point of the solid; they only depend on the intrinsic elastic properties of the solid. The boundary conditions are that there is no surface stress except at the source point which generates the normal load [1]. It is interesting to notice that solving the equations evidences the existence of an additional wave called head wave LT. This wave propagates along the solid surface at the longitudinal wave propagation velocity c_L and generates a shear wave in the solid (velocity c_T) along the rays at the given angle referred to as critical angle q _c. q _c (see Figure 2) is characteristic of the conversion mode in solids :



Moreover, the Green's functions are not reduced to impulse spherical waves as in liquids; they are time-dependent in a complex way and the directivity patterns of each wave is not omnidirectional. The directivity patterns of the longitudinal and shear waves are given in the monochromatic approximation [2] by the following formula :

Fig 1. Theoretical directivity patterns of a normal point load (monochromatic approximation) : radial component of the displacement corresponding to the longitudinal wave (full curve) and tangential component of the displacement corresponding to the shear wave (dashed curve).

The figure 1 represents these directivities for the duralumin sample.

In a second step, the Green's functions being assumed to be well-known, an integration on the surface of the source is thus performed, taking into account the transducer geometry. As a result of this integration, we obtain the diffraction impulse responses as in the case of the propagation in liquids [3], but considering the vector nature of the displacement describing the field in solids and the characteristics of each type of wave. For example, the head wave propagation velocity depends on the relative position of the elementary source point and of the observation point as shown in following equation (see also Figure 2) :

Fig 2. Representation of the three wavefronts existing in an isotropic solid : longitudinal and shear wavefronts (half-circle), head wave wavefront (dashed line).

Knowing the temporal shape of the load generated by the transducer (transduction model [4]), the displacements in the solid resulting from this excitation are given by the time-convolution of this load T.e(t) with the diffraction impulse responses defined above :

Application to the case of a single element transducer

This impulse diffraction model has been assessed by accurate measurements of the acoustic displacement field generated by an element of a phased array coupled to a duralumin sample (Fig. 3a). This validation has been made possible thanks to the use of a powerful measuring tool, i.e. an heterodyne interferometric probe [5] which allows to measure absolute displacement amplitude (in a range from about 1 to 200 ) due to continuous or transient waves. Figures 2b and 2c show the good agreement between the simulations and the measurements of the displacement radial component resulting from the superposition of the waves L, T and LT. It is worth noticing that longitudinal waves are preferentially generated. That is the reason why we generally choose to focus these waves in the case of a contact phased array.

			Fig 3. Angular measurement of the radial component of the displacement generated by one or several elements of a phased array (32 elements 15x0.8 mm, frequency 2 MHz) coupled to a duralumin sample (a), measured and computed wavefronts corresponding to the displacement radial component generated by one element of the phased array (b)and comparison of the directivity patterns for one element (c): measurements (marker curves) and simulation (line curves) for longitudinal L, shear T waves and head wave LT.
(a)	(b)	(c)

Application to the case of a contact phased array

The displacement field generated by a contact phased array is computed by adding the contributions of each element with the delays applied to each of these elements in order to achieve the longitudinal wave focusing. The model without approximations requires a long computation time. This is why analytical expressions of the Green's functions valid in the neighborhood of the wavefronts are generally used. This approximation has given very good results for the longitudinal waves ; figure 4 shows that the angular directivity patterns corresponding to this wave are accurately simulated for different focusing directions. The results for the shear waves are less good but they allow nevertheless to evaluate the amplitude of the "parasitic" focusing of these waves (Figure 5). The difference between the simulation and the measurements can be explained by the presence of head waves, visible in figure 5 (wavefront frame) which are not taken into account by this simulation.

Fig 4. comparison of directivity patterns for the phased array: measurements (marker curves) and simulations (line curves) for the displacement radial component of the longitudinal wave at different deflection angles : 0°, 30°, 45° et 60°

Fig 5. Illustration of a parasitic focusing of shear waves (a) in the case of the chosen longitudinal focusing (60°) and comparison of the directivity patterns (b) measured (marker curves) and computed (line curves) corresponding to the displacement radial component of the longitudinal and shear waves.

Next steps

The proposed model could be generalized to more complex transducers with a wedge shape shoe used to generate preferentially shear waves in solids. This allows to increase the spatial resolution of the acoustic measurement. Consequently such simulation tools prove to be precious for the definition of phased arrays which are becoming more and more used in NDT.

The results presented here have been obtained by P. Gendreu at "Laboratoires d'Electronique PHILIPS" (Limeil-Brévannes - France) in collaboration with the "Laboratoire Ondes et Acoustique" (ESPCI - CNRS - Université Paris 7 - Paris - France). Many thanks to both laboratories for their kind authorisation to publish this paper.

References :

1. D.C. GAKENHEIMER, J. MIKLOWITZ, Transient excitation of an elastic half space by a point load traveling on the surface, Journal of Applied Mechanics 36, p.505, 1969.
2. G.F. MILLER, H. PURSEY, The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid, Proc. Roy. Soc. London, Ser. A, v.233, p.521-541,1954
3. P.R. STEPANISHEN, Transient radiation from pistons in an infinite planar baffle. J. Acoust. Soc. Am. 49, p.1629-1638. 1971.
4. W.P. MASON, Electromechanical transducers and waves filters, Van Nostrand,1948
5. D. ROYER, E. DIEULESAINT, Optical detection of sub-angstrom transient mechanical displacements. IEEE Ultrason. Symp. Proc. pp. 527, 1986.

Authors

Philippe Gendreu and Rémi Berriet IMASONIC, 15 Rue Alain Savary, 25000 Besançon FRANCE E-mail: imasonic@imasonic.com Homepage: http://www.imasonic.com Web Page on NDTnet