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Technical Brief

Multilayered Inclusions in Locally Resonant Metamaterials: Two-Dimensional Versus Three-Dimensional Modeling

[+] Author and Article Information
Anastasiia O. Krushynska

Department of Physics,
University of Turin,
Turin 10125, Italy
e-mail: akrushynska@gmail.com

Marco Miniaci

Laboratoire Ondes et Milieux Complexes,
University of Le Havre,
Le Havre 76600, France
e-mail: marco.miniaci@gmail.com

Varvara G. Kouznetsova

Department of Mechanical Engineering,
Eindhoven University of Technology,
Eindhoven 5600MB, The Netherlands
e-mail: v.g.kouznetsova@tue.nl

Marc G. D. Geers

Professor
Department of Mechanical Engineering,
Eindhoven University of Technology,
Eindhoven 5600MB, The Netherlands
e-mail: m.g.d.geers@tue.nl

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 27, 2016; final manuscript received November 10, 2016; published online February 3, 2017. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 139(2), 024501 (Feb 03, 2017) (4 pages) Paper No: VIB-16-1239; doi: 10.1115/1.4035307 History: Received April 27, 2016; Revised November 10, 2016

Locally resonant metamaterials (LRMs) controlling low-frequency waves due to resonant scattering are usually characterized by narrow band gaps (BGs) and a poor wave filtering performance. To remedy this shortcoming, multiresonant metamaterial structures with closely located BGs have been proposed and widely studied. However, the analysis is generally limited to two-dimensional (2D) structures neglecting the finite height of any real resonator. The aim of this paper is the comparison of the wave dispersion for two- and three-dimensional (3D) metamaterial models and evaluation of the applicability ranges of 2D results. Numerical study reveals that dual-resonant structures with cylindrical inclusions possess only a single (compared to two in the 2D case) BG for certain height-to-width ratios. In contrast, the wave dispersion in metamaterials with multiple spherical resonators can be accurately evaluated using a 2D approximation, enabling a significant simplification of resource-consuming 3D models.

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Figures

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Fig. 1

Schematic representation of the 3D unit cell and its cross section for an LRM with multilayered cylindrical inclusions (the geometrical parameters are given in millimeters)

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Fig. 2

Band diagrams for 2D out-of-plane (a) and in-plane (b) modes in the LRM with infinite value of h. Vibration patterns of the 2D localized out-of-plane—(c) f = 110 Hz and (d) f = 212 Hz—and in-plane modes—(e) f = 185 Hz and (f)f = 425 Hz.

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Fig. 3

(a) Band diagram of a 3D slab of height h = 0.2a. (b)–(e) Vibration patterns of the localized modes for the slab with height h = 0.2a. (f) Complete band gap frequencies as a function of the unit cell height-to-width ratio h/a. (b) f = 105 Hz, (c)f = 185 Hz, (d) f = 206 Hz, and (e) f = 259 Hz.

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Fig. 4

Schematic representation of a 3D unit cell for an LRM with multilayered spherical inclusions

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Fig. 5

Band diagrams for the metamaterial shown in Fig. 4 with (a) periodic and (b) traction-free boundary conditions at the faces normal to the z axis, respectively

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