Complex dispersion relation and topologically protected flexural wave transport in viscoelastic periodic plates

Abstract

Periodic plates are regarded as a promising platform for topological insulator investigation. However, most studies on the topological insulator are conducted by the ω(k) method and the material viscosity is usually ignored. The weakness of the ω(k) method is that the imaginary part of wave vector (evanescent wave modes) cannot be obtained. In other words, the attenuation property cannot be revealed. On the other hand, the material viscosity has a major effect on the dynamic responses of periodic structures, and cannot be considered in the ω(k) method also. To overcome these limitations, the complex dispersion method or the k(ω) method based on FEM is developed here and extended to the investigation of topologically protected flexural wave transport in viscoelastic periodic plates. In detail, we propose a double-sided stubbed periodic plate, mimicking the quantum valley Hall effect. The complex dispersion relation is obtained that reveals all the wave modes of the present periodic plates, exhibiting the propagative and attenuation properties comprehensively. Through the combination of complex dispersion analysis and full-field simulation, the evolution of dispersion relation regarding to material damping is discovered. Topological wave transport is worsened up to stop by the material viscosity. However, the robustness and energy localized capacity of the topologically protected edge states is little affected by material viscosity under the condition of small loss. This study provides a new perspective for analyzing topologically protected edge states and promotes the practical application of topological insulator.