Waves in one-dimensional quasicrystalline structures: dynamical trace mapping, scaling and self-similarity of the spectrum

Morini, Lorenzo ORCID: https://orcid.org/0000-0001-7155-5036 and Gei, Massimiliano ORCID: https://orcid.org/0000-0003-3869-7504 2018. Waves in one-dimensional quasicrystalline structures: dynamical trace mapping, scaling and self-similarity of the spectrum. Journal of the Mechanics and Physics of Solids 119 , pp. 83-103. 10.1016/j.jmps.2018.06.007

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Abstract

Harmonic axial waves in quasiperiodic-generated structured rods are investigated. The focus is on infinite bars composed of repeated elementary cells designed by adopting generalised Fibonacci substitution rules, some of which represent examples of one-dimensional quasicrystals. Their dispersive features and stop/pass band spectra are computed and analysed by imposing Floquet–Bloch conditions and exploiting the invariance properties of the trace of the relevant transfer matrices. We show that for a family of generalised Fibonacci substitution rules, corresponding to the so-called precious means, an invariant function of the circular frequency, the Kohmoto’s invariant, governs self-similarity and scaling of the stop/pass band layout within defined ranges of frequencies at increasing generation index. Other parts of the spectrum are instead occupied by almost constant ultrawide band gaps. The Kohmoto’s invariant also explains the existence of particular frequencies, named canonical frequencies, associated with closed orbits on the geometrical three-dimensional representation of the invariant. The developed theory represents an important advancement towards the realisation of elastic quasicrystalline metamaterials.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Engineering
Publisher:	Elsevier
ISSN:	0022-5096
Funders:	European Commission, Welsh Government
Date of First Compliant Deposit:	20 July 2018
Date of Acceptance:	6 June 2018
Last Modified:	02 Dec 2024 00:30
URI:	https://orca.cardiff.ac.uk/id/eprint/113244

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