Affiliation:
1. Imperial College London Department of Mathematics, , London SW7 2AZ, UK
2. Department of Civil and Environmental Engineering, Politecnico di Milano , Piazza Leonardo da Vinci, 32, 20133 Milano, Italy
Abstract
Summary
We derive analytical solutions based on singular Green’s functions, which enable efficient computations of scattering simulations or Floquet–Bloch dispersion relations for waves propagating through an elastic plate, whose surface is patterned by periodic arrays of elastic beams. Our methodology is versatile and allows us to solve a range of problems regarding arrangements of multiple beams per primitive cell, over Bragg to deep-subwavelength scales; we cross-verify against finite element numerical simulations to gain further confidence in our approach, which relies upon the hypothesis of Euler–Bernoulli beam theory considerably simplifying continuity conditions such that each beam can be replaced by point forces and moments applied to the neutral plane of the plate. The representations of Green’s functions by Fourier series or Fourier transforms readily follows, yielding rapid and accurate analytical schemes. The accuracy and flexibility of our solutions are demonstrated by engineering topologically non-trivial states, from primitive cells with broken spatial symmetries, following the phononic analogue of the Quantum Valley Hall Effect. Topologically protected states are produced and coexist along: interfaces between adjoining chiral-mirrored bulk media, and edges between one such chiral bulk and the surrounding bare elastic plate, allowing topological circuits to be designed with robust waveguiding. Our topologically protected interfacial states correspond to zero-line modes, and our topological edgestates are produced in accordance with the bulk-edge correspondence. These topologically non-trivial states exist within near flexural resonances of the constituent beams of the phononic crystal and hence can be tuned into a deep-subwavelength regime.
Publisher
Oxford University Press (OUP)
Subject
Applied Mathematics,Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
1 articles.
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