Ondes élastiques guidées

La dispersion des ondes élastiques guidées est très dépendante des propriétés élastiques du matériau. C’est en particulier le cas des modes à vitesse de groupe nulle (mode ZGV). Ces modes sont remarquablement bien observés par technique ultrason-laser et leurs applications aux contrôle non destructif des matériaux (mesures d’épaisseur, évaluation des propriétés élastiques) demeurent un sujet actif . Ils font l’objet de développement de méthodes numériques puissantes [1]. En collaboration avec l’Université d’Hokkaido, un mode ZGV engendré par impulsion femtoseconde a été filmé au GigaHertz dans une strucutre bi-couche nanométrique [2]. Avec l’Université Technique de Hambourg, les modes ZGV ont été utilisés pour caractériser des matériaux nano-poreux. Jouant un rôle grandissant dans la conception de matériaux fonctionnels, une meilleure connaissance des propriétés mécaniques de ces matériaux est recherchée. Les méthodes usuelles en contact n’étant pas adaptées, les modes guidés engendrés et détectés par laser, ont été eploités dans des membranes de silicium nano-poreux, ayant typiquement une épaisseur de 100 μm et une porosité de 55%. En particulier, l’effet significatif sur les fréquences ZGV, d’un chargement en fluide a été démontré. [3].

Les modes de ruban résultant du couplage entre mode de plaque et mode de cisaillement (modes in-plane) ont fait l’objet de différents travaux. Par des mesures US laser dans des rubans métalliques, nous avons confirmé que la dispersion de ces modes était indépendante de l’épaisseur et similaire à celles des ondes de Lamb et nous avons mis en évidence les résonances ZGV associées [4]. Ayant révélé l’existence d’un cône de Dirac dans les rubans incompressibles (coefficent de Poisson proche de 0.5), cette étude nous a conduits à étudier des ondes guidées dans des solides mous. Nous avons montré l’impact des pertes sur la dispersion des modes in-plane dans des membranes et des rubans visco-élastiques [5] [6] et modélisé l’effet conjoint de la rhéologie et de l’acousto-élasticité [7]. Ces avancées ouvrent la voie vers une rhéologie plus large bande des solides mous et vers une élastographie quantitative plus robuste.

Par ailleurs, à l’aide des techniques ultrasons laser, nous avons poursuivi l’étude des modes à dispersion négative existant au voisinage des modes ZGV dans les plaques [8]. Lors de l’ANR COPOLA portée par Alexandre Aubry, les effets de réflexion et de réfraction négative de ces modes ont été exploité pour créer différents systèmes de guidage : coin à double réfraction négative pour piéger l’onde, milieux complémentaires qui annihilent la propagation ou super lentille plate [9].

Enfin, si les US laser sont utiles pour observer finement les ondes de Lamb, les transducteurs multi-éléments présentent l’avantage de la simplicité d’utilisation. Nous avons développé un échographe à ondes de Lamb destiné à caractériser en temps réel des plaques anisotropes. L’analyse de la matrice de réflexion d’une sonde multi-éléments au contact de la plaque fournit les courbes de dispersion des ondes guidées. Ces courbes alimentent une méthode d’inversion rapide qui permet d’évaluer jusqu’à 5 constantes élastiques [10]. Cette méthode ouvre la voie à une caractérisation temps réel de milieux très anisotropes tels que les super-alliages de nickel ou les matériaux à matrice céramiques.

Contacts : Claire Prada, Sylvain Mezil, François Bruno, Fabrice Lemoult, Alexandre Aubry

Notes

[1]

Computing zero-group-velocity points in anisotropic elastic waveguides: Globally and locally convergent methods Kiefer, D. A., B. Plestenjak, H. Gravenkamp, and C. Prada The Journal of the Acoustical Society of America 153, no. 2, 1386-1398 (2023) Résumé: Dispersion curves of elastic waveguides exhibit points where the group velocity vanishes while the wavenumber remains finite. These are the so-called zero-group-velocity (ZGV) points. As the elastodynamic energy at these points remains confined close to the source, they are of practical interest for nondestructive testing and quantitative characterization of structures. These applications rely on the correct prediction of the ZGV points. In this contribution, we first model the ZGV resonances in anisotropic plates based on the appearance of an additional modal solution. The resulting governing equation is interpreted as a two-parameter eigenvalue problem. Subsequently, we present three complementary numerical procedures capable of computing ZGV points in arbitrary nondissipative elastic waveguides in the conventional sense that their axial power flux vanishes. The first method is globally convergent and guarantees to find all ZGV points but can only be used for small problems. The second procedure is a very fast, generally-applicable, Newton-type iteration that is locally convergent and requires initial guesses. The third method combines both kinds of approaches and yields a procedure that is applicable to large problems, does not require initial guesses and is likely to find all ZGV points. The algorithms are implemented in GEW ZGV computation (doi: 10.5281/zenodo.7537442).

[2]

Imaging gigahertz zero-group-velocity Lamb waves Xie, Q., S. Mezil, P. H. Otsuka, M. Tomoda, J. Laurent, O. Matsuda, Z. Shen, and O. B. Wright Nature Communications 10, no. 1 (2019) Résumé: © 2019, The Author(s). Zero-group-velocity (ZGV) waves have the peculiarity of being stationary, and thus locally confining energy. Although they are particularly useful in evaluation applications, they have not yet been tracked in two dimensions. Here we image gigahertz zero-group-velocity Lamb waves in the time domain by means of an ultrafast optical technique, revealing their stationary nature and their acoustic energy localization. The acoustic field is imaged to micron resolution on a nanoscale bilayer consisting of a silicon-nitride plate coated with a titanium film. Temporal and spatiotemporal Fourier transforms combined with a technique involving the intensity modulation of the optical pump and probe beams gives access to arbitrary acoustic frequencies, allowing ZGV modes to be isolated. The dispersion curves of the bilayer system are extracted together with the quality factor Q and lifetime of the first ZGV mode. Applications include the testing of bonded nanostructures.

[3]

Laser-excited elastic guided waves reveal the complex mechanics of nanoporous silicon Thelen, M., N. Bochud, M. Brinker, C. Prada, and P. Huber Nature Communications 12, no. 1 (2021)

[4]

In-plane backward and zero group velocity guided modes in rigid and soft strips Laurent, J., D. Royer, and C. Prada The Journal of the Acoustical Society of America 147, no. 2, 1302 (2020) Résumé: Elastic waves guided along bars of rectangular cross sections exhibit complex dispersion. This paper studies in-plane modes propagating at low frequencies in thin isotropic rectangular waveguides through experiments and numerical simulations. These modes result from the coupling at the edge between the first order shear horizontal mode SH0 of phase velocity equal to the shear velocity VT and the first order symmetrical Lamb mode S0 of phase velocity equal to the plate velocity VP. In the low frequency domain, the dispersion curves of these modes are close to those of Lamb modes propagating in plates of bulk wave velocities VP and VT. The dispersion curves of backward modes and the associated zero group velocity (ZGV) resonances are measured in a metal tape using noncontact laser ultrasonic techniques. Numerical calculations of in-plane modes in a soft ribbon of Poisson's ratio ν≈0.5 confirm that, due to very low shear velocity, backward waves and ZGV modes exist at frequencies that are hundreds of times lower than ZGV resonances in metal tapes of the same geometry. The results are compared to theoretical dispersion curves calculated using the method provided in Krushynska and Meleshko [J. Acoust. Soc. Am. 129, 1324-1335 (2011)].

[5]

Dirac cones and chiral selection of elastic waves in a soft strip Lanoy, M., F. Lemoult, A. Eddi, and C. Prada Proceedings of the National Academy of Sciences, 202010812 (2020) Résumé: We study the propagation of in-plane elastic waves in a soft thin strip, a specific geometrical and mechanical hybrid framework which we expect to exhibit a Dirac-like cone. We separate the low frequencies guided modes (typically 100 Hz for a 1-cm-wide strip) and obtain experimentally the full dispersion diagram. Dirac cones are evidenced together with other remarkable wave phenomena such as negative wave velocity or pseudo-zero group velocity (ZGV). Our measurements are convincingly supported by a model (and numerical simulation) for both Neumann and Dirichlet boundary conditions. Finally, we perform one-way chiral selection by carefully setting the source position and polarization. Therefore, we show that soft materials support atypical wavebased phenomena, which is all of the more interesting as they make most of the biological tissues.

[6]

Soft elastomers: A playground for guided waves Delory, A., F. Lemoult, M. Lanoy, A. Eddi, and C. Prada The Journal of the Acoustical Society of America 151, no. 5, 3343-3358 (2022) Résumé: Mechanical waves propagating in soft materials play an important role in physiology. They can be natural, such as the cochlear wave in the inner ear of mammalians, or controlled, such as in elastography in the context of medical imaging. In a recent study, Lanoy, Lemoult, Eddi, and Prada [Proc. Natl. Acad. Sci. U.S.A. 117(48), 30186-30190 (2020)] implemented an experimental tabletop platform that allows direct observation of in-plane guided waves in a soft strip. Here, a detailed description of the setup and signal processing steps is presented as well as the theoretical framework supporting them. One motivation is to propose a tutorial experiment for visualizing the propagation of guided elastic waves. Last, the versatility of the experimental platform is exploited to illustrate experimentally original features of wave physics, such as backward modes, stationary modes, and Dirac cones.

[7]

Guided elastic waves in a highly-stretched soft plate Delory, A., F. Lemoult, A. Eddi, and C. Prada Extreme Mechanics Letters, 102018 (2023)

[8]

Negative reflection of elastic guided waves in chaotic and random scattering media Gérardin, B., J. Laurent, F. Legrand, C. Prada, and A. Aubry Scientific reports 9, no. 1, 2135 (2019) Résumé: The propagation of waves in complex media can be harnessed either by taming the incident wave-field impinging on the medium or by forcing waves along desired paths through its careful design. These two alternative strategies have given rise to fascinating concepts such as time reversal or negative refraction. Here, we show how these two processes are intimately linked through the negative reflection phenomenon. A negative reflecting mirror converts a wave of positive phase velocity into its negative counterpart and vice versa. In this article, we experimentally demonstrate this phenomenon with elastic waves in a 2D billiard and in a disordered plate by means of laser interferometry. Despite the complexity of such configurations, the negatively reflected wave field focuses back towards the initial source location, thereby mimicking a phase conjugation operation while being a fully passive process. The super-focusing capability of negative reflection is also highlighted in a monochromatic regime. The negative reflection phenomenon is not restricted to guided elastic waves since it can occur in zero-gap systems such as photonic crystals, chiral metamaterials or graphene. Negative reflection can thus become a tool of choice for the control of waves in all fields of wave physics.

[9]

Cloaking, trapping and superlensing of lamb waves with negative refraction Legrand, F., B. Gérardin, F. Bruno, J. Laurent, F. Lemoult, C. Prada, and A. Aubry Scientific Reports 11, no. 1 (2021) Résumé: We report on experimental and numerical implementations of devices based on the negative refraction of elastic guided waves, the so-called Lamb waves. Consisting in plates of varying thickness, these devices rely on the concept of complementary media, where a particular layout of negative index media can cloak an object with its anti-object or trap waves around a negative corner. The diffraction cancellation operated by negative refraction is investigated by means of laser ultrasound experiments. However, unlike original theoretical predictions, these intriguing wave phenomena remain, nevertheless, limited to the propagating component of the wave-field. To go beyond the diffraction limit, negative refraction is combined with the concept of metalens, a device converting the evanescent components of an object into propagating waves. The transport of an evanescent wave-field is then possible from an object plane to a far-field imaging plane. Twenty years after Pendry’s initial proposal, this work thus paves the way towards an elastic superlens.

[10]

Towards real-time assessment of anisotropic plate properties using elastic guided waves Bochud, N., J. Laurent, F. Bruno, D. Royer, and C. Prada Journal of the Acoustical Society of America 143, no. 2, 1138-1147 (2018) Résumé: © 2018 Acoustical Society of America. A method to recover the elastic properties, thickness, or orientation of the principal symmetry axes of anisotropic plates is presented. This method relies on the measurements of multimode guided waves, which are launched and detected in arbitrary directions along the plate using a multi-element linear transducer array driven by a programmable electronic device. A model-based inverse problem solution is proposed to optimally recover the properties of interest. The main contribution consists in defining an objective function built from the dispersion equation, which allows accounting for higher-order modes without the need to pair each experimental data point to a specific guided mode. This avoids the numerical calculation of the dispersion curves and errors in the mode identification. Compared to standard root-finding algorithms, the computational gain of the procedure is estimated to be on the order of 200. The objective function is optimized using genetic algorithms, which allow identifying from a single out-of-symmetry axis measurement the full set of anisotropic elastic coefficients and either the plate thickness or the propagation direction. The efficiency of the method is demonstrated using data measured on materials with different symmetry classes. Excellent agreement is found between the reported estimates and reference values from the literature.

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