Marc Deschamps

Elastic waves in anisotropic plates: short-wavelength asymptotics of the dispersion branches vn(k)

Abstract The dispersion spectra of elastic eigenwaves in infinite plates of unrestricted anisotropy are analytically studied in the short-wavelength approximation. The three boundary problems are solved, for the free (i), clamped (ii) and clamped/free (iii) surfaces. It is shown that generally the dispersion curves v n ( k ) with increasing wavenumber k are bunched in the vicinity of only two asymptotic levels of tracing velocity v. The first level relates to the Rayleigh velocity v R and the second level to the limiting velocity v—the lowest tracing velocity for a bulk wave with the energy flux parallel to the surfaces. It is proved that in case (i) there are always two (and only two) dispersion branches v f ± ( k ), which at growing k are exponentially fast approaching the asymptote v = v R , one from above, the other from below. In case (iii) at large k one (and only one) wave branch v c/f 0 ( k ) always exists close to v R , even closer than v f ± ( k ). And for case (ii) the level v = v R is absent in the spectrum v c n ( k ). The other branches for all three boundary problems are described by the common asymptotic dependence v n (k)= v +(An/k) 2 , where A is a known constant. For case (iii) in the particular case of limiting wave with vanishing traction one should replace above An by πA ( n +1/2). Additional asymptotic levels arising due to symmetry or a special choice of the wave propagation geometry are also discussed. It is shown that they disappear after any “triclinic” perturbation of orientations.

Energy velocity of complex harmonic plane waves in viscous fluids

Abstract This paper presents a theoretical investigation of the energy velocity of complex harmonic plane waves in viscous fluids. The complex harmonic plane wave, which is characterized by a complex wave vector and a complex frequency, may propagate in absorbing fluids. The initial nonconservative energy balance equation is modified into another energy equation, in which the new loss density is, on average, nil. It appears that the energy velocity, which is defined from this system that is conservative on average only, is not always oriented along the phase velocity direction. More precisely, the energy velocity may be interpreted as the phase velocity in the direction of the real part of the slowness bivector.

Reflection and refraction of a heterogeneous plane wave by a solid layer

Abstract Reflection and refraction of harmonic heterogeneous waves by a plane, isotropic viscoelastic layer are described in the framework of linear acoustics. The reflection and transmission coefficients are obtained by two different approaches: firstly, using the well-known model of Brekhovskikh for the case of a homogeneous plane wave, secondly, using an expansion into a geometrical Debyes series through a matrix sum. the energy balance is also established. Some particular cases are analyzed and the results are discussed. For instance, although obeying the energy conservation law, the solution may have an unexpected behaviour due to heterogeneity.

Viscoelasticity Influence on Frequency Dependence of the Ultrasonic Transmission through Plates of Composite Materials

Leaky Lamb waves (LLW) have been extensively considered in the direct problem of reflection of elastic waves from liquid-coupled orthotropic plates [1-4] and in the inverse problem to determine the elastic properties of composites [5] This determination is based on the measurement of the incidence angle (or frequency) at fixed frequency (or fixed incidence angle) corresponding to the generation of LLW from the measurement of the amplitude of transmitted (or reflected) waves through (or by) the sample.

Interaction of guided waves with defects in a multilayered cylinder by an asymptotic approach

This work deals with the problem of the propagation of an elastodynamic field radiated by a source in a cylindrical layered medium which interacts with and is diffracted by a defect. At low frequencies, where the defect size is much smaller than the wavelength, this interaction can be approximated by a point source, located at the defect position. This secondary source is expressed by the Green function and its derivative. The Green function, which describes the response of an undamaged cylinder, is calculated using the canonical form of the wave equation initially expressed as a function of the spatial and temporal variables. Performing the Laplace in time and Fourier transforms along the cylinder axis, this equation is written as an ordinary differential equation with respect to the radial position. The solution for the transversely isotropic case is obtained by adopting the partial wave formulation, expressed as a combination of the modified Bessels functions of the first and second kind. Having assembled the layers, numerical inverse transforms are performed to obtain the real wave fields. This technique allows for reduced computational costs and faster calculation times and could be used for the non destructive testing of embedded pipes and tubes.This work deals with the problem of the propagation of an elastodynamic field radiated by a source in a cylindrical layered medium which interacts with and is diffracted by a defect. At low frequencies, where the defect size is much smaller than the wavelength, this interaction can be approximated by a point source, located at the defect position. This secondary source is expressed by the Green function and its derivative. The Green function, which describes the response of an undamaged cylinder, is calculated using the canonical form of the wave equation initially expressed as a function of the spatial and temporal variables. Performing the Laplace in time and Fourier transforms along the cylinder axis, this equation is written as an ordinary differential equation with respect to the radial position. The solution for the transversely isotropic case is obtained by adopting the partial wave formulation, expressed as a combination of the modified Bessels functions of the first and second kind. Having assem...

Locating damages in a complex structure using one or two fixed ultrasonic transducers

Detecting and locating defects in a plate is usually performed using multiple transducers and methods that are based on the measurement of the wave travelling directly from the defect to the transducer. The acoustic signatures of the interaction between the interrogating wave with the defects and the structure itself, such as the reflections from the boundaries, are generally discarded from the signal. This paper deals with the experimental application of the topological imaging method that takes advantage of all the information included in the multiple reflections between the boundaries of the structure and the defects. The application of the method is based on a preliminary measurement of the impulse response of the inspected structure in its undamaged state. The impulse response measurement is restricted to a defined region of interest and the structure is excited with a single fixed transducer. This preliminary measurement can be considered as a general acoustic calibration of the inspected structure that is performed once and for all, in the present case with a Laser Doppler velocimeter. The structure is then inspected with the single or several transducers and, if any, damages are located in the region of interest. The experimental results obtained with the one-channel topological imaging method show accurate location of a single small defect and of multiple small defects. The resolving power obtained is close to the theoretical limit of half a wavelength. These results demonstrate that the method has great potential for complex structures in general and for SHM applications involving a limited number of embedded transducers.

A 3D semi-analytical model to predict the behavior of ultrasonic bounded beam traveling in cylindrical solid bar embedded in a solid matrix.

A 3D semi-analytical model for predicting the behaviour of ultrasonic bounded beam travelling in cylindrical solid bar is presented. The bar is embedded in a solid matrix and the beam is emitted by an off-axis source for generating nonaxisymmetric waves. Fourier series and Vector Hankel transform are combined to decompose the inside field into infinity of elementary cylindrical waves propagating into radial direction and planar waves propagating in axial direction. Global resolution method and Generalized Debye Series Expansion (GDSE) are used both to calculate the global reflection/transmission coefficients. A numerical model is implemented to demonstrate the validity of the present theoretical solution for predicting velocity signatures.