Effective impedance of a locally resonant metasurface

M. Lott and P. Roux

Phys. Rev. Materials 3, 065202 – Published 7 June 2019

Abstract

We study here a mesoscopic metasurface made of a randomly distributed set of long vertical metallic rods attached to a thin elastic plate. The $A_{0}$ Lamb wave propagation is strongly affected by the local change in apparent stiffness of the plate induced by the low-quality factor resonance of the rods. At the resonance, the plate-plus-rods system is allowed to move freely, and plate waves can penetrate into the metamaterial. At the antiresonance, the plate behaves in terms of waves as if it was clamped by the rods in the metamaterial region, which induces large-frequency band gaps. Between the resonant and antiresonant frequencies, the continuous change in effective rigidity results in a continuous change in reflectivity. In the present paper, we aim at measuring the corresponding complex impedance of the metasurface in terms of amplitude and phase. Experimental data are presented to estimate the effective impedance of a locally resonant metasurface, in agreement with theoretical prediction and numerical simulation.

Received 4 January 2019

DOI:https://doi.org/10.1103/PhysRevMaterials.3.065202

Physics Subject Headings (PhySH)

Acoustic measurements Acoustic metamaterials Acoustic wave phenomena Continuum mechanics Diffusion

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsGeneral Physics

Authors & Affiliations

M. Lott^* and P. Roux

Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France

^*martin.lott@univ-grenoble-alpes.fr

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 3, Iss. 6 — June 2019

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Experimental setup at the laboratory scale. The metamaterial (a) is made of 100 vertical aluminum rods that are glued to the underside of a plate of the same material [(b), black square]. The particle velocity is measured using an out-of-plane laser Doppler velocimeter (c) that is connected to a PC-controlled (e) motorized robot arm (d). The recorded signal (f) is strongly dispersed due to the weak intrinsic attenuation of the plate. Using the late part of the reverberated coda (g) and a set of a few sources (h), homogeneous focal spots are reconstructed from correlations inside (i) and outside (j) the beam cluster.
Reuse & Permissions
Figure 2
(a) Real part of the averaged two-point correlation function (normalized) measured at 5 kHz for all of the receiver pairs located inside the metamaterial region (blue). The modeled plate Green's function [Eq. (3)] is in red. (b), (c) Averaged two-point correlation versus frequency measured inside the metamaterial (b) and outside the metamaterial (c). The black line in (b) corresponds to the averaged intensity versus frequency measured inside the metamaterial.
Reuse & Permissions
Figure 3
(a) Metamaterial band structure. The blue and red curves (yellow and purple) are computed from the averaged two-point correlation function using Eq. (3) for the real and imaginary parts of the effective wave number, respectively, inside (outside) the metamaterial region. The black solid line shows the real part of the analytical effective wave number from Williams et al. (2015). The $A_{0}$ dispersion curve is plotted as a dashed line. (b) Impedance ratio obtained through different methods. Purple and yellow, from the time reversal focal spot amplitude (in the passband only); blue and red, from the two-point correlation method and using Eq. (3); black and grey, theoretical values from Williams et al. (2015), as the effective medium formulation and using Eq. (9). The impedance value of the free plate is plotted as the dashed line. The background colors highlight the bandwidth of the stop band (red) and passband (blue).
Reuse & Permissions
Figure 4
Metamaterial simulation in two dimensions. (a) Simulation box: A random set of 100 beams attached to a shell component with absorbing boundaries. (b) Response at 6400 Hz, inside the stop band. (c) Response at 4200 Hz, inside the passband. The black square represents the metamaterial region. In the two left panels, the arrows indicate the incident-wave direction.
Reuse & Permissions
Figure 5
Metamaterial simulation in one dimension. (a) Frequency domain simulation with lateral periodic conditions and absorbing layers at the boundaries of the medium. Inset: Metamaterial dimensions. (b) Simulation result. Blue, reflection coefficient at the beam interface calculated from the spatial Fourier transform in the free plate region; red, theoretical results obtained with Eqs. (9) and (1); green, lateral motion recorded at the top of the first beam. (c) Band structure derived from the spatial Fourier analysis of the wave field inside the metamaterial region.
Reuse & Permissions

Physical Review Materials