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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.
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 :
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 :
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) |
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. |
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.
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 |
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