Amplitude-dependent topological edge states in nonlinear phononic lattices

Raj Kumar Pal, Javier Vila, Michael Leamy, and Massimo Ruzzene
Phys. Rev. E 97, 032209 – Published 21 March 2018

Abstract

This work investigates the effect of nonlinearities on topologically protected edge states in one- and two-dimensional phononic lattices. We first show that localized modes arise at the interface between two spring-mass chains that are inverted copies of each other. Explicit expressions derived for the frequencies of the localized modes guide the study of the effect of cubic nonlinearities on the resonant characteristics of the interface, which are shown to be described by a Duffing-like equation. Nonlinearities produce amplitude-dependent frequency shifts, which in the case of a softening nonlinearity cause the localized mode to migrate to the bulk spectrum. The case of a hexagonal lattice implementing a phononic analog of a crystal exhibiting the quantum spin Hall effect is also investigated in the presence of weakly nonlinear cubic springs. An asymptotic analysis provides estimates of the amplitude dependence of the localized modes, while numerical simulations illustrate how the lattice response transitions from bulk-to-edge mode-dominated by varying the excitation amplitude. In contrast with the interface mode of the first example studies, this occurs both for hardening and softening springs. The results of this study provide a theoretical framework for the investigation of nonlinear effects that induce and control topologically protected wave modes through nonlinear interactions and amplitude tuning.

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  • Received 3 May 2017
  • Revised 31 October 2017

DOI:https://doi.org/10.1103/PhysRevE.97.032209

©2018 American Physical Society

Physics Subject Headings (PhySH)

Nonlinear DynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Raj Kumar Pal1,*, Javier Vila1, Michael Leamy2, and Massimo Ruzzene1,2

  • 1School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
  • 2School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

  • *raj.pal@aerospace.gatech.edu

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Issue

Vol. 97, Iss. 3 — March 2018

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