Elastic Wave Propagation in Hierarchical Honeycombs with Woodpile-Like Vertexes

This paper investigates the dispersion behavior of elastic wave propagation in hierarchical honeycombs using the finite element method in conjunction with the Bloch's theorem. The hierarchical honeycomb is constructed by replacing each vertex of a regular hexagonal honeycomb with smaller hexagons stacked in a woodpile pattern. Band structure analysis reveals that, in the considered range of frequency, the maximum band gap for the hierarchical honeycomb is localized in the frequency corresponding to the natural vibration frequency of the cell strut, and moreover, the width of this particular gap is significantly broadened as the order of hierarchy increases. In addition, for the hierarchical honeycombs satisfying an invariable ratio between the thickness and squared length of the cell strut, which is extracted from the expression of the natural frequency of the simply supported element beam, a coincidence among dispersion curves (or contours) for the hierarchical configurations with the same scale order occurs. The resulting identical band gaps as well as the quasi-static phase wave velocities provide an advantage or the hierarchical honeycombs in the manipulation of vibration and associated multifunction designs.