Study of tunable locally resonant metamaterials: Effects of spider-web and snowflake hierarchies

Abstract

The present research is an effort to enhance and control locally resonant bandgaps in periodic metamaterials of hexagonal and triangular topologies by introducing spider-web and snowflake-inspired hierarchies to the conventional geometries. Due to their low connectivity number, hexagonal lattice structures are unable to exhibit locally resonant bandgaps. By introducing the spider-web hierarchy, the connectivity number of the structure is increased, and the constituting beams act as local resonators. Thus, the hexagonal lattice structure can attenuate propagating waves in desirable frequency ranges by localizing the wave energy in the constituent beams. As a result, better wave-filtering performance is achieved by introducing hierarchy to hexagonal lattices. Furthermore, the snowflake hierarchical triangular lattice structures exhibit new tunable attenuation zones for which the locations depend on the mechanical and geometrical parameters. The finite element method and Bloch’s theorem are applied to analyze the wave propagation in the considered architected structures. Moreover, theoretical formulations are presented to predict the locations of each locally resonant bandgaps. It is shown that by changing the mechanical and geometrical parameters of the added hierarchical structures, the locally resonant bandgaps can be perfectly tuned to the desired frequencies in the long-wavelength region. Finally, two theoretical diagrams are proposed to represent initial design concepts of tunable acoustic/elastic metamaterials of hexagonal and triangular unit cells with maximum bandgap widths in low frequencies. To present a more comprehensive analysis of the wave propagation in hierarchical structures, the effects of imaginary wave-vector components on the dispersion curves and attenuation behavior of each topology are also investigated. The results of the current study enable the engineers to design architected periodic structures with the ability to present bandgaps in desired frequencies, which has been the goal of structural engineers for years.