Spiral-Based Phononic Plates: From Wave Beaming to Topological Insulators

André Foehr, Osama R. Bilal, Sebastian D. Huber, and Chiara Daraio

Phys. Rev. Lett. 120, 205501 – Published 15 May 2018

Abstract

Phononic crystals and metamaterials can sculpt elastic waves, controlling their dispersion using different mechanisms. These mechanisms are mostly Bragg scattering, local resonances, and inertial amplification, derived from ad hoc, often problem-specific geometries of the materials’ building blocks. Here, we present a platform that ultilizes a lattice of spiraling unit cells to create phononic materials encompassing Bragg scattering, local resonances, and inertial amplification. We present two examples of phononic materials that can control waves with wavelengths much larger than the lattice’s periodicity. (1) A wave beaming plate, which can beam waves at arbitrary angles, independent of the lattice vectors. We show that the beaming trajectory can be continuously tuned, by varying the driving frequency or the spirals’ orientation. (2) A topological insulator plate, which derives its properties from a resonance-based Dirac cone below the Bragg limit of the structured lattice of spirals.

Received 21 November 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.205501

Physics Subject Headings (PhySH)

Acoustic phonons

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

André Foehr^1,2, Osama R. Bilal^2,3,*, Sebastian D. Huber³, and Chiara Daraio^2,†

¹Department of Mechanical and Process engineering, ETH Zurich, 8092 Zurich, Switzerland
²Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
³Institute for theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

^*bilal@caltech.edu
^†daraio@caltech.edu

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Vol. 120, Iss. 20 — 18 May 2018

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Images

Figure 1
Spiral-based structured plates. (a) An Archimedean spiral with its parameters. (b) Structured plate composed of a periodic array of spirals. White areas indicate void. (c) Variations of the spiral geometries with the corresponding flexural band gaps shaded in green. The band gaps span two decades of frequencies, while modifying the plate thickness or the lattice spacing. Unit cells exhibiting Bragg scattering are shown in purple and local resonances in red.
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Figure 2
Band gap opening mechanisms. Normalized dispersion curves with both the attenuated (orange) and propagating (blue for in-plane and green for out-of-plane) waves. The corresponding unit cells are in the inset. Band gaps are shaded in green. (a) Bragg scattering, (black line) flexural dispersion of the virgin material, (b) local resonance, (c) inertial amplification, and (d) Bragg scattering.
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Figure 3
Wave beaming. (a) Dispersion relation for elastic waves of the inset unit cell. (b) Isofrequency plot for the first Brillouin zone. The dashed gray lines indicate the unit cell edges in reciprocal space. (c) Frequency response of a finite plate, under out-of-plane harmonic excitation at 530 Hz and (d) 1570 Hz with absorbing boundary conditions.
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Figure 4
Topological insulators. (a) Dispersion curves of the hexagonal unit cell in the inset with a flexural subwavelength Dirac cone highlighted in red and in-plane band gap highlighted in gray. Frequencies above the Bragg limit are hashed in purple. (b) Left: Supercell that includes the original hexagonal unit cell (blue) with additional neighboring spirals (red hexagon) in real space. Right: Folded symmetry line ( $Γ − N − L − Γ$ ) for the modified spiral patterns in reciprocal space. (c) Two dispersion curves of two supercells with small variations between the blue and red spirals (supercell I, $n_{blue} = 1.02 n$ ) and (supercell II, $n_{blue} = 1.03 n$ , $w_{blue} = w_{red} = 1.068 w$ ). The insets show the mode shapes of each unit cell at the edge of the topological gaps for super cells. (d) Dispersion curves for a quasifinite (periodic in one direction and finite with 27 spiral unit cells in the other) line of material I and II. In addition to trivial edge bands (gray), the previously observed band gap is closed by two topologically protected counterpropagating modes. The mode shape corresponding to the circled dispersion point (red), with the interface between the materials indicated by a black dashed line. (e) The mode shape at frequency $f = 857 Hz$ for a finite sample composed of materials I and II.
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