Sub-wavelength topological boundary states and rainbow trapping of local-resonance phononic crystal plate

Based on the analogy of the quantum valley Hall effect, a ligament-type phononic crystal plate with local resonators is designed in this study to facilitate the valley state transport of low-frequency elastic waves. We analyze the key factors affecting the local resonance modes and reduce the frequency of the Dirac cone by changing the connection form of the structure's beams. The spatial inversion symmetry of the structure is broken to open a new band gap by introducing a mass difference in the resonator pair. The robustness of the designed structure's topological valley waveguide under defects and bends is verified. Based on this characteristic, we introduce the gradient heights into the supercell structure where elastic waves at different frequencies split and stop significantly on the supercell structure to achieve sub-wavelength topological rainbow trapping. This design provides a theoretical reference for exploring the low-frequency elastic topological mode and the application of topological rainbow capture in sub-wavelength structures.