Elastic valley Hall edge wave in a hierarchical hexagonal lattice

•Elastic waveguides are designed by exploiting acoustic valley Hall effect.•Valley-dependent edge modes in hierarchical hexagonal lattice are demonstrated.•Valley Hall edge wave propagates without backscattering at sharp bends.•Valley Hall edge waves can be split into different directions.•Frequency filter is numerically demonstrated using gapped, gapless edge modes. By exploiting topological phases, a new approach can be taken to design elastic waveguides with robust wave propagation against sharp bends or defects. In particular, the topological phase based on the quantum valley Hall effect, or valley topological phase, is an attractive candidate for elastic waveguide because its mechanism can be realized in a relatively simple way. In this work, we report a waveguide design with in-plane dynamics based on the acoustic valley Hall effect, leveraging a hierarchical hexagonal lattice built by replacing the vertices of a regular hexagonal lattice with smaller hexagons. These are purposely introduced because they provide better structural performance than the regular hexagonal lattice. We modified the size of the adjacent smaller hexagons using two different values to break spatial inversion symmetry, which leads to topologically distinct lattices depending on the sign of the modification parameter and the emergence of the valley-dependent edge modes at the interface of two lattice structures. We numerically demonstrated the robust propagation of the valley-dependent edge states through sharp bends. Notably, we aggressively exploited the ease of tuning the geometric parameters to design frequency-tailorable edge modes, to demonstrate wave splitting and frequency filtering with the waveguide. The proposed structure can be a pathway for practically utilizing the elastic topological phase, and can also widen the scope of multifunctional (elastic wave control and load-bearing) structures to include the topological phase.