Multi-band topological valley modes of flexural waves in micro-perforated phononic plates

To develop phononic topological insulators (PTIs) with time reversal invariance, phononic crystals (PCs) are elaborately configurated to ensure the lattice symmetry and meanwhile to open the complete band gap. Particularly, a multi-band PTI requires more complex architecture than its single-band counterpart, leading to extra difficulty in implementation and poor manufacturability of PCs. Here, we present a straightforward and generic design strategy, to realize multi-band PTIs in micro-perforated phononic plates. Rather than using complex surface-mounted or embedded scatterers, the novel phononic plate is micro-perforated with a type of fan-shaped split-ring resonators (SRRs), the multi-modal resonances of which under constrained lattice symmetry induce multiple Dirac degeneracies in flexural-wave dispersions at different frequency scales. The multi-band topological phase transitions are observed, and triple topological band gaps are caused when rotating micro-perforated slots properly. Experimental results agree well with numerical simulation results, both demonstrating triband valley edge-state propagation of flexural waves. In addition, we also prove that the developed valley PTIs enable the second-order corner modes of flexural waves within the corresponding topological band gaps. Parametric studies further reveal that the emergence of the valley-shaped Dirac degeneracy bands in the developed PTI is insensitive to the plate thickness and slots parameters, and more than three Dirac cones exist in flexural-wave dispersions. The developed micro-perforated plates with SRRs provides a convenient platform for enabling multi-band PTIs with various symmetries, facilitating applications of multi-frequency topological effects of elastic waves in a cost-effective way.