Jean-François de Belleval

Propagation in an anisotropic periodically multilayered medium

An anisotropic multilayered medium is studied using the method of transfer matrices, developed by Thomson [J. Appl. Phys. 21, 89 (1950)] and Haskell [Bull. Seismol. Soc. Am. 43, 17 (1953)]. The propagation equations in each layer of the multilayered medium use the form developed by Rokhlin et al. [J. Acoust. Soc. Am. 79, 906–918 (1986); J. Appl. Phys. 59 (11), 3672–3677 (1986)]. Physical explanations are given, notably when a layer is made up of a monoclinic crystal system medium. The displacement amplitudes of the waves in one layer may be expressed as a function of those in another layer using a propagation matrix form, which is equivalent to relating the displacement stresses of a layer to those in another layer. An anisotropic periodically multilayered medium is then studied by using a propagation matrix that has particular properties: a determinant equal to one and eigenvalues corresponding to the propagation of the Floquet waves. An example of such a medium with the axis of symmetry of each layer perpendicular to the interfaces is then presented together with the associated reflection coefficients as a function of the frequency or of the incident angle.

The use of inhomogeneous waves in the reflection–transmission problem at a plane interface between two anisotropic media

A mathematical form is proposed for the study of the problem of reflection–transmission of monochromatic ultrasonic plane waves at the interface between two arbitrary anisotropic semi‐infinite media. The method used leads to a complete determination of all characteristics of the studied waves for all possible configurations, i.e., their directions of propagation, polarizations, and magnitudes. In particular, cases are taken into account where the waves generated by the existence of the interface are evanescent or more generally inhomogeneous. Applying this method to the calculation of several practical cases typically encountered in ultrasonic nondestructive evaluation fields, this paper points out the existence of phenomena that cannot be interpreted by classical methods usually used for the resolution of this kind of problem.

Characterization of composite materials by ultrasonic methods: modelization and application to impact damage

This paper presents some ultrasonic methods to detect and to characterize defects, possibly obtained after damage caused in composite materials. Firstly, a two-dimensional ultrasonic cartography, performed section by section, at different positions from the impact point, allows the participation of the delamination mechanisms which took part through the thickness of a pultruded glass fiber-reinforced plastic composite beam, to be analyzed. A very good agreement has been found with destructive testings. Furthermore, some examples are given on the use of an ultrasonic propagation model which has been developed. This model permits optimum experimental configurations to be determined, by the use of transmission and reflection coefficients or of Lamb waves. In addition, experimental and modelized time signals have been compared.

Acoustic propagation in anisotropic periodically multilayered media: A method to solve numerical instabilities

Acoustic propagation through thick composites has become a subject of intensive study due to their application to nondestructive evaluation. The anisotropic multilayered media are now usually studied by the propagator matrix formalism. Though this formalism is very convenient, it leads to numerical instabilities for thick composites at high frequencies. These numerical instabilities come from the combination of very high exponential terms which reduces the dynamics of the calculation. A very interesting case is the one of anisotropic periodically multilayered media. The method developed in this paper uses Floquet waves which correspond to the modes of an infinite periodically multilayered medium. They are linear combinations of the real waves propagating in each layer of the medium. The Floquet wave numbers are the eigenvalues of the propagation matrix of one period of the medium. The anisotropic periodically multilayered medium can then be considered as a dummy medium in which the Floquet waves propagate...

Floquet waves and classical plane waves in an anisotropic periodically multilayered medium: Application to the validity domain of homogenization

The aim of this paper is to better understand the correspondence between classical plane waves propagating in each layer of an anisotropic periodically multilayered medium and Floquet waves. The last are linear combinations of the classical plane waves. Their wave number is obtained from the eigenvalues of the transfer matrix of one cell of the medium. A Floquet polarization which varies with its position in the periodically multilayered medium has been defined. This allows one to define a Floquet wave displacement by analogy with the displacement of classical plane waves, and to check the equality of the two displacements at any interface separating two layers. The periodically multilayered medium is then an equivalent material, considered as homogeneous, and one can draw dispersion curves and slowness surfaces which are dispersive. In the low‐frequency range, when the relation between the Floquet wave numbers and the frequency is linear, the multilayered medium can be homogenized; the Floquet polarizati...

SURFACE WAVES IN AN ANISOTROPIC PERIODICALLY MULTILAYERED MEDIUM : INFLUENCE OF THE ABSORPTION

The influence of absorption upon behavior of the reflection coefficient in water for an anisotropic periodically multilayered medium is studied. From observing a trough of the experimental reflection coefficient of a carbon/epoxy composite immersed in water, the propagation modes of the Floquet waves in an infinite anisotropic periodically multilayered medium in vacuum have been studied. This was explained by a mode that we called the multilayered Rayleigh mode. This mode results from a combination of the Floquet waves which propagate in a periodically multilayered medium. This occurs when all the Floquet waves are inhomogeneous. Analyzing the results obtained in the isotropic case permits the more complicated cases to be explained. By analogy with the isotropic case, the multilayered Rayleigh wave is related to the poles and the zeros of the reflection coefficient. The existence of a critical attenuation becomes evident when the reflection coefficient in water for the multilayered medium reaches zero.

Experimental verification of the theory of multilayered Rayleigh waves

A phenomenon which has been termed “multilayered Rayleigh modes” has been presented in previous papers. This study aims to prove experimentally the existence of these waves in anisotropic periodically multilayered media. These modes result from a combination of Floquet waves which propagate in a periodically multilayered medium when all the Floquet waves are inhomogeneous. The experimental verification was done using an acousto-optic technique and a measurement of the reflected field, which was obtained with a hydrophone measurement system, on a carbon/epoxy composite plate. The experimental and calculated dispersion curves of the multilayered Rayleigh modes were then drawn. The coincidence of the curves was found quite good, thus confirming our theory. However, two modes were found by the acousto-optic technique not to fit into the theory. One experimentally detected mode was found to correspond to the Lamb mode of the plate and the other was not experimentally detected by the acousto-optic technique. Me...

Numerical modeling of the effects on reflected acoustic field for the changes in internal layer orientation of a composite

Abstract Acoustic propagation through anisotropic multilayered media has become a subject of intensive study, because of its application to nondestructive evaluation, geophysics etc. We consider the multilayered media created by stacking distinct anisotropic layers. A very interesting case of multilayered media is that of the composite material, in which we have different layers with different orientations from one another. A realistic calculation of reflected ultrasonic beam patterns from these composites requires treatment of the effects of refraction and reflection from finite and infinite media. Application of the angular spectrum analysis (ASA) to calculate the acoustic fields has become increasingly adopted, because the ASA is easy to be numerically implemented upon the basis of the Fast Fourier Transform and can be used effectively to solve the variety of complex problems. The developed numerical model can be applied to determine the effects of changes in orientation of the layers in a composite material on the reflected acoustic field. Here we compare the results for different layer orientations in a carbon/epoxy composite. From this it can also be concluded, with better reliability, that when using multilayered composites it is essential that the internal construction fulfils the specifications exactly, because the changes in internal layer orientation can change totally the characteristics of that composite due to the fact that the elastic characteristics of the composite medium depends also on the orientation angle of the internal layers. Moreover, in using this method, we can detect the changes in orientation of the internal layers, and their effects on the acoustic fields through/from the composites.

Interaction of a monochromatic ultrasonic beam with a finite length defect at the interface between two anisotropic layers: kirchhoff approximation and fourier representation

This paper presents a fast computation method to simulate the interaction between a bounded acoustic beam and a 2-layered anisotropic structure with a finite defect on the internal interface. The method uses the classical Fourier decomposition of the fields into plane waves, and the Kirchhoff approximation is introduced to calculate the diffusion by the defect. The validity of the approximation is estimated by comparison with the Keller Geometrical Theory of Diffraction and with results obtained by boundary element methods. The quickness of the method allows testing several geometrical configurations (varying incident angle, thickness of the layers or the physical nature of the defect). These studies may be used to foresee what experimental configurations would be adequate to have a chance to detect the defect.

Numerical and experimental deviation of monochromatic Lamb wave beam for anisotropic multilayered media

The interaction of an incident monochromatic bounded beam with an anisotropic multilayered plate immersed in an external fluid has been modelled, using the decomposition method into monochromatic plane waves and the transfer matrix method, in a 3D configuration. The subject of this paper is the study of the deviation of a monochromatic Lamb wave beam due to the anisotropy of the structure. Comparisons between some theoretical, numerical and experimental results are given.