Free and forced wave propagation in a Rayleigh-beam grid: flat bands, Dirac cones, and vibration localization vs isotropization

TL;DR

This study investigates wave propagation in a Rayleigh-beam grid, revealing complex localization phenomena, Dirac cone degeneracies, and isotropization effects, with implications for designing wave-controlling metamaterials.

Contribution

It provides a detailed analysis of in-plane wave behavior in Rayleigh-beam grids, highlighting new localization patterns, the impact of rotational inertia, and conditions for flat bands and Dirac cones.

Findings

Localization patterns depend on beam slenderness and forcing type

Rotational inertia sharpens Dirac cone degeneracies

Special frequencies induce isotropization of wave response

Abstract

In-plane wave propagation in a periodic rectangular grid beam structure, which includes rotational inertia (so-called 'Rayleigh beams'), is analyzed both with a Floquet-Bloch exact formulation for free oscillations and with a numerical treatment (developed with PML absorbing boundary conditions) for forced vibrations (including Fourier representation and energy flux evaluations), induced by a concentrated force or moment. A complex interplay is observed between axial and flexural vibrations (not found in the common idealization of out-of-plane motion), giving rise to several forms of vibration localization: 'X-', 'cross-' and 'star-' shaped, and channel propagation. These localizations are triggered by several factors, including rotational inertia and slenderness of the beams and the type of forcing source (concentrated force or moment). Although the considered grid of beams introduces an orthotropy in the mechanical response, a surprising 'isotropization' of the vibration is observed at special frequencies. Moreover, rotational inertia is shown to 'sharpen' degeneracies related to Dirac cones (which become more pronounced when the aspect ratio of the grid is increased), while the slenderness can be tuned to achieve a perfectly flat band in the dispersion diagram. The obtained results can be exploited in the realization of metamaterials designed to control wave propagation.