Duality of a coiled phononic crystal enables reflectionless interfaces

Recently, it has been demonstrated that one-dimensional bar-based phononic crystals can exhibit subwavelength Bragg bandgaps by coiling the bars and locking the nodal rotational degrees of freedom to create what is termed a coiled phononic crystal (CPnC) [C. L. Willey et al., Phys. Rev. Appl. 18, 014035 (2022)]. Here, it is shown that the CPnC exhibits duality of its dispersion curves relative to its coiling/twist angle, meaning that the dispersion curves are symmetric about a particular coiling/twist angle defined configuration. An exciting implication of this finding is that under a certain set of constraints, segments of dual unit cells with perpendicular wave propagation directions can be connected such that their wave transmission is equivalent to a finite CPnC entirely composed of identical unit cells with parallel wave propagation directions. The ability to link unit cells with different wave propagation orientations, but the same dispersion/dynamic stiffness, is used to create an elastic hierarchically coiled phononic crystal based on a fractal space-filling curve design. The novelty of this work is that it numerically demonstrates reflectionless wave propagation in large fractal architectures (created from specific combinations of dual unit cells) such that regular phononic properties (i.e., passbands and bandgaps) are preserved, allowing for propagation of broadband signals and filtering.