Topological edge states of quasiperiodic elastic metasurfaces

In this work, we investigate the dynamic behavior and the topological properties of quasiperiodic elastic metasurfaces, namely arrays of mechanical oscillators arranged over the free surface of an elastic half-space according to a quasiperiodic spatial distribution. An ad-hoc multiple scattering formulation is developed to describe the dynamic interaction between Rayleigh waves and a generic array of surface resonators. The approach allows to calculate the spectrum of natural frequencies of the quasiperiodic metasurface which reveals a fractal distribution of the frequency gaps reminiscent of the Hofstadter butterfly. These gaps have nontrivial topological properties and can host Rayleigh-like edge modes. We demonstrate that such topologically protected edge modes can be driven from one boundary to the opposite of the array by a smooth variation of the phason, a parameter which modulates the geometry of the array. Topological elastic waveguides designed on these principles provide new opportunities in surface acoustic wave engineering for vibration control, energy harvesting, and lossless signal transport, among others.