Complex dispersion analysis of topologically protected interface states in two-dimensional viscoelastic phononic crystals

Abstract

In this paper, the viscoelastic effect on the topologically protected interface states in two-dimensional (2D) solid phononic crystals (PnCs) is investigated. The valley topological interface states for the 2D in-plane and out-of-plane modes are generated on the interfaces between two PnCs with opposite topological phases. The complex band structures are calculated by the ω-k approach based on the weak formulation of the governing equations of wave motion, which is solved numerically by the finite element method. From the complex band structures, it is demonstrated that even though the material viscoelasticity is introduced into the systems, the topological interface states still exist. However, for the viscoelastic case, the topological interface states becomes the complex wave modes, which means that the interface states inevitably attenuate as they propagate in the viscoelastic PnCs. The amplitude of the elastic waves traveling along the interfaces exhibits an exponential decay, which can be analytically predicted based on the imaginary part of the wave number. Despite suffering from the attenuation due to the material loss, the topological interface states in the viscoelastic PnC structures also exhibit their robustness against the sharp bends or local disorders. In practice, the material loss is ubiquitous, and hence these results are relevant to the PnC devices based on the topological states.