Elastic wave propagation in hierarchical lattices with convex and concave hexagons stacked vertexes.

The paper investigates the in-plane elastic wave propagation in hierarchical lattices. The hierarchical organization is obtained by removing a certain amount of cells successively from a general hexagonal lattice that involves the re-entrant configuration with negative Poisson's ratio. The symmetry reduction caused by the alteration of the internal angle and the formed vertexes with stacking hexagons gives rise to a significant effect on the dispersion properties of the structure materials. Results show that the lattices with different hierarchy levels possess a stable behavior of the band gap in the position near the resonant frequency of the cell walls, and the gap width has an evident increase in the re-entrant hierarchical structures. In addition, band structure in the low frequency range has an unchanged performance for the varying parameters of cell walls, and the reason is owing to the steady of the structure vibration pattern for the Bloch modes. The authors' findings of the dispersion behaviors in the hierarchical lattices provide a broad design space for the lightweight materials in the wave manipulation and vibration control.