Hierarchical metamaterials with tunable flat bands, zero-frequency, and wavenumber gaps

TL;DR

The paper tackles the challenge of simultaneously realizing wavenumber band gaps, flat bands, and zero-frequency band gaps in a tunable metamaterial. It introduces a passive, hierarchical unit cell with embedded magnets and a tunable magnetic boundary to adjust lattice periodicity, enabling these exotic dispersion features as shown by analytical Bloch theory, nonlinear numerical simulations, and experimental validation on an air-bearing setup. Key findings include the coexistence of a wavenumber gap, a flat band, and a zero-frequency gap within designs A–D, with phenomena like degenerate flat bands and orbital angular momentum localization demonstrated in circular configurations. The work provides design guidelines for magnetic-tunable hierarchy and paves the way for advanced acoustic and mechanical devices that exploit controlled dispersion engineering.

Abstract

Metamaterials are arrangement of basic building blocks that repeat in space, time, or both. These material systems serve as an excellent platform for controlling waves, such as engineering wavenumber band gaps, flat bands, and zero-frequency band gaps. However, combining one or more of these exotic features within the same unit cell design remains a challenge. Moreover, once a metamaterial is realized, its dispersive properties are usually fixed. In this work, we present a tunable passive hierarchical metamaterial capable of exhibiting wavenumber band gaps, flat bands, and zero-frequency band gaps within the same dispersion curve. Our metamaterial is composed of magnetic elements confined within a fixed magnetic boundary. The metamaterial can be tuned by adjusting the magnetic boundary, which in turn can alter the lattice periodicity. We open wavenumber band gaps by incorporating magnetic coupling within the unit cell elements, resulting in negative physical stiffness. The tunability of the magnetic coupling also enables complete flattening of the dispersion bands. Moreover, the ground stiffness within our unit-cell design causes the opening of zero-frequency band gaps. We present our approach through a combination of analytical, numerical, and experimental methods. The analytical framework provides a blueprint for obtaining each of these exotic dispersion characteristics. The numerical analysis, using both linear and nonlinear models, validates our analytical predictions, which we further confirm through experimental demonstrations. Our work opens the door to exploring magnetic tunability and hierarchy in engineering metamaterial systems with exotic properties that can be harnessed in advanced acoustic and mechanical devices.

Figures (11)

• Figure 1: Hierarchical metamaterial unit cell designs, orientation, and dispersive properties. (a) Schematic of the tunable metamaterial designs (rectangular and circular frames with inner resonators). (b) The minimum energy positions of both the rectangular frames (Design A and Design B). The minimum energy position for the circular moving frame and its inner resonator (Design C and Design D) in the case of (c) small lattice constant. (d) large lattice constant. (e) By varying the periodicity of the lattice, we can achieve dispersion curves with wavenumber band gap, flat bands, and zero frequency band gap.
• Figure 2: A study of the rectangular system's parameters in Design A (a) A schematic of the rectangular system indicating the directions of the longitudinal and shear excitations. Three-dimensional parametric sweep of (b) shear modes and (e) longitudinal modes, showing the effect of varying both the lattice width and periodicity on the frequency range of transmission. Dispersion frequency band width as a function of lattice periodicity with unit cell width = 40 $\mathrm{mm}$ for (c) shear modes and (f) longitudinal modes. Dispersion relation (frequency vs. wavenumber) for unit cell width and length $a = b = 40 \, \mathrm{mm}$ for (d) shear modes and (g) longitudinal modes.
• Figure 3: A study of the rectangular system's parameters in Designs C and D A schematic of the circular system with (a) double circular design and (h) triple circular nested design. Three-dimensional parametric sweep of (b& e) Design C and (i& l) Design D in both shear and longitudinal modes, respectively, showing the effect of varying both the lattice width and periodicity on the frequency range of transmission. Dispersion frequency band width as a function of lattice periodicity with unit cell width = 50 $\mathrm{mm}$ for (c& f) Design C and (j& m) Design D for shear and longitudinal modes, respectively. Dispersion relation (frequency vs. wavenumber) for unit cell width and length $a = b = 50 \, \mathrm{mm}$ for (d& g) Design C and (k& n) Design D for shear and longitudinal modes, respectively.
• Figure 4: Degenerate flat band and polarization independent localization. (a) Unit-cell schematic (left), dispersion diagram (center) showing two degenerate flat bands associated with the inner resonator at $f = 11.7$ Hz, and a zoomed view (right) showing $\Delta f = 0.05$ Hz. Time responses of the inner resonators under longitudinal polarization (b) in the $x$ direction and (c) in the $y$-direction. Time responses of the inner resonators under shear polarization (d) in the $x$ direction and (e) in the $y$-direction. Time responses of the inner resonators under tilted (45$^\circ$) polarization (f) in the $x$- direction and (g) in the $y$-direction. Displacement trajectories of (h) the first and second inner resonators and (l) the first and second main masses under tilted polarization.
• Figure 5: Numerical validation of analytical dispersion curves with flat bands, zero-frequency band gaps and wavenumber band gaps. (a) A schematic of the rectangular system Design A with indicated shear and longitudinal direction excitations. Analytical shear dispersion curves (black lines) with superimposed 2D-FFT contours from the numerical excitation of the shear modes with periodicity (b) $a = 45 \, \mathrm{mm}$ and (f) $a = 40 \, \mathrm{mm}$. The time -displacement history of the metamaterial frame and resonator with a periodicity (c) $a = 45 \, \mathrm{mm}$ and (g) $a = 40 \, \mathrm{mm}$. Analytical longitudinal dispersion curves (black lines) with superimposed 2D-FFT contours from the numerical excitation of the longitudinal modes with periodicity (d) $a = 45 \, \mathrm{mm}$ and (h) $a = 40 \, \mathrm{mm}$. The time -displacement history of the metamaterial frame and resonator with a periodicity (e) $a = 45 \, \mathrm{mm}$ and (i) $a = 40 \, \mathrm{mm}$.