Asymmetric topological state in an elastic beam based on symmetry principle
Résumé
The symmetric propagation of topological edge states in two-dimensional (2D) mechanical systems cannot be directly applied to a one-dimensional (1D) system such as a beam. In this work, we adopt symmetry principle of single- and double-sided pillared phononic beams to realize asymmetric transmission of a topological edge state. The design is constituted by assembling a two-sided pillared beam (two symmetry planes) with a one-sided pillared beam (one symmetry plane). With symmetry plane consideration, we find that: 1) in a single-sided pillared beam, the torsional and the shear waves on one hand, and the longitudinal and the flexural waves on the other hand, are mixed together. 2) in a double-sided pillared beam, all the four beam's modes are decoupled from each other so that an appropriate choice of the parameters can produce for each pair (torsional-shear and longitudinal-flexural) a Dirac cone for one of the modes and a bandgap for the other mode in the same frequency range. The Dirac cone is lifted to form a trivial or nontrivial bandgap with band inversion by tuning the interval between pillars, and a topological shear/flexural edge state can be achieved consequently at the interface between two topologically different phases. Asymmetric topological edge state of the torsional/longitudinal incident wave can be achieved by assembling these two beams. It is further demonstrated a high robustness of the topological edge state against the disorder perturbation in pillar's position/frequency and the defects. The asymmetric topological state can be extended to other beam's motions following the symmetry principle and opens opportunities for designing topological solid devices with high performance, such as robust filter.