Omnimodal topological polarization of bilayer networks: Analysis in the Maxwell limit and experiments on a 3D-printed prototype

Mohammad Charara, James McInerney, Kai Sun, Xiaoming Mao, Stefano Gonella

Research output: Contribution to journalArticlepeer-review


Periodic networks on the verge of mechanical instability, called Maxwell lattices, are known to exhibit zero-frequency modes localized to their boundaries. Topologically polarized Maxwell lattices, in particular, focus these zero modes to one of their boundaries in a manner that is protected against disorder by the reciprocal-space topology of the lattice’s band structure. Here, we introduce a class of mechanical bilayers as a model system for designing topologically protected edge modes that couple in-plane dilational and shearing modes to out-of-plane flexural modes, a paradigm that we refer to as “omnimodal polarization.” While these structures exhibit a high-dimensional design space that makes it difficult to predict the topological polarization of generic geometries, we are able to identify a family of mirror-symmetric bilayers that inherit the in-plane modal localization of their constitutive monolayers, whose topological polarization can be determined analytically. Importantly, the coupling between the layers results in the emergence of omnimodal polarization, whereby in-plane and out-of-plane edge modes localize on the same edge. We demonstrate these theoretical results by fabricating a mirror-symmetric, topologically polarized kagome bilayer consisting of a network of elastic beams via additive manufacturing and confirm this finite-frequency polarization via finite element analysis and laser-vibrometry experiments.

Original languageEnglish (US)
Article numbere2208051119
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number40
StatePublished - Oct 4 2022

Bibliographical note

Funding Information:
ACKNOWLEDGMENTS. This work was supported by NSF Grant EFRI-1741618 (to M.C. and S.G.) and the Office of Naval Research Multidisciplinary University Research Initiative N00014-20-1-2479 (to J.M., K.S., and X.M.) and leveraged the High Performance Computing systems at the Minnesota Supercomputing Institute.

Publisher Copyright:
Copyright © 2022 the Author(s). Published by PNAS.


  • flexural modes
  • mechanical metamaterials
  • topological mechanics

PubMed: MeSH publication types

  • Journal Article
  • Research Support, U.S. Gov't, Non-P.H.S.


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