Localized modes on a metasurface through multiwave interactions

Martin Lott, Philippe Roux, Léonard Seydoux, Benoit Tallon, Adrien Pelat, Sergey Skipetrov, and Andrea Colombi

Phys. Rev. Materials 4, 065203 – Published 29 June 2020

Abstract

In this paper, we describe the manifestation of localized states of a diffuse elastic wavefield inside a two-dimensional metamaterial made of a collection of vertical long beams glued to a thin plate. Through mesoscopic physics, we demonstrate that localized states arise due to multiwave interactions at the beam-plate attachment when the beams act as coupled resonators for both compressional and flexural resonances on the metasurface. In practice, when the compressional resonance of the beams clamps the plate on a large frequency band gap, thus preventing wave propagation inside the metamaterial area, the flexural resonance of the same beams permits evanescent waves to be diffused on a narrow frequency band between clusters of neighboring beams. This experiment physically highlights a tight-binding-like coupling in the localized regime for this two-dimensional metamaterial.

Received 2 December 2019
Accepted 9 June 2020

DOI:https://doi.org/10.1103/PhysRevMaterials.4.065203

Physics Subject Headings (PhySH)

Acoustic measurements Acoustic metamaterials Acoustic modeling Acoustic signal processing & noise Acoustic wave phenomena Ballistic transport Continuum mechanics Diffusion Elasticity Localization Phonons

Statistical Physics & ThermodynamicsGeneral PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Martin Lott ¹, Philippe Roux¹, Léonard Seydoux¹, Benoit Tallon¹, Adrien Pelat ², Sergey Skipetrov ³, and Andrea Colombi ⁴

¹University Grenoble Alpes, University Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France
²Laboratoire d’Acoustique de l’Université du Mans, UMR CNRS 6613, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 09, France
³University Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
⁴Department of Civil, Environmental and Geomatic Engineering, Institute of Structural Engineering, Swiss Federal Institute of Technology, Zurich, Switzerland

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Vol. 4, Iss. 6 — June 2020

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Images

Figure 1
Scheme of the experiment. The cluster of beams (a) occupies a small area (20 cm × 20 cm) of the 2-mm-thick plate. The three laser heads (b) extract the motions at the tops of the beams as well as on the opposite side of the plate surface, through positioning of the robot arm, depicted with another scanning position in (c). A central computer performs the signal generation and recording, as well as driving the arm motion. Two piezoelectric sources, one inside and one outside the beam cluster (d), excite the $A_{0}$ Lamb mode into the plate. The multireverberated recorded wavefield (e) includes the plate and beam three-component motion (f).
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Figure 2
Metasurface dynamics. (a) Average motion recorded at the top of the beams for compressional (red) and flexural (blue) motions for the source located on the free plate. (b) Transmitted intensity (out-of-plane component) inside the metasurface from the source located outside of the beam cluster. (c,d) Modal representations of the beam motion under flexural (c) and compressional (d) resonance for two wave vectors and their corresponding resonant frequencies. The resulting stress applied to the plate is either dipolar (bending) or monopolar (compression).
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Figure 3
Complex dispersion curves for the effective medium. (a) Real (blue) and imaginary (red) parts of the wave number extracted through the two-point correlation method. (b) Ioffe-Regel “ $k l$ ” parameter over frequency, interpreted as the ratio between the real and imaginary parts of the wave number.
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Figure 4
Intensity distribution: experimental results (b,c) and theoretical fit (a) for the source located on the free plate. (a) The “resistivity” 1/ $g$ ( $g$ is defined as the Thouless conductance) is plotted with respect to the frequency. The three red crosses indicate the flexural resonance locations, and the color bar indicates the average horizontal motion amplitude of the beams. The two arrows refer to two nearby frequencies with low and high resistivity for which the intensity distribution $I$ /⟨ $I$ ⟩ is plotted in (b) and (c). (b,c) Blue, the curves that correspond to the Rayleigh-like intensity distribution measured outside of the metamaterial area; red, the curves of the intensity distribution measured inside the metamaterial. The conductivity $g$ is extracted from a theoretical fit from Eqs. (1) and (2). Part (b) is obtained at 2045 Hz, inside the band gap, and (c) is obtained for flexural resonance at 2205 Hz.
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Figure 5
Normalized intensity maps of the late coda of the signals measured for the source located inside the metamaterial. The horizontal ( $x - y$ ) beam motion $I_{b}$ is plotted with the white-to-black color map, and the out-of-plane ( $z$ ) plate motion $I_{p}$ is plotted with the green-to-yellow color map. (a)–(c) The three maxima of the resistivity $〈 〈 1 / g 〉 〉$ [shown in Fig. 4 as red crosses, and in Fig. 6 as red arrows]. (d) The close frequency of the first maximum where the intensity is leaking outside of the metamaterial area [Fig. 6, black arrow].
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Figure 6
Singular value decomposition of the cross-spectral density matrix (CSDM) of the three-component motion of the 98 beams. The Von Nuemann entropy (a) is computed from the distribution of the CSDM eigenvalue at each frequency. The three red arrows (1–3) indicate the three minima of the entropy that are associated with the three flexural resonances, which correspond to Figs. 5. (b) The hyper angle, which comes from the scalar product between two of the first eigenvectors extracted at neighboring frequencies, shows the robustness of the modal excitation, with magnification around the first flexural resonance (2200 Hz) shown in (c). The blue and red shaded areas highlight the flexural resonances of the beams inside the band gap [see Fig. 2]. (c) The black arrow (4) indicates another mode for the beam cluster, which corresponds to (d) in Fig. 5. (d) The average three-component intensity $I_{t}$ measured at the top of the beam.
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