Rayleigh waves in locally resonant metamaterials

The design of metamaterials for surface waves control is an emerging field of research which can impact several technical applications, from electronic devices based on surface acoustic waves (SAW) to wave barriers for seismic isolation. So far, studies on the interaction of surface waves with locally resonant metamaterials have been limited to the context of metasurfaces, i.e., thin resonant interfaces or structures attached to the free-surface of a waveguide. In this work, we remove this constraint by formulating an original dispersion relation for vertically polarized surface waves of the Rayleigh-type existing in resonant metamaterials. We consider the case of a resonant layer of thickness H coupled to a non-resonant half-space. To easily account for the resonant material while setting the dispersion relation we propose a mixed dynamic-static homogenization approach, valid in the long-wavelength regime. We show the existence of a band gap in the spectrum of the Rayleigh surface waves and relate its width to the region of negative effective density of the resonant metamaterial. We highlight how the thickness of the resonant layer affects the frequency stop-band and the magnitude of wave attenuation. We next derive and discuss the limit cases of a fully resonant half-space, i.e. a resonant layer with thickness H >> λ where λ is the wavelength of the Rayleigh wave, and of the metasurface for small values of H << λ. In the latter case, we prove that our formulation properly model metasurfaces recovering previous formulations. As case study, we analyse the propagation of seismic Rayleigh waves across a deep barrier of meter-size resonators embedded in the soil. The barrier is modeled using the proposed analytical approach. By means of high fidelity finite element simulations and Bloch analyses, we validate our novel findings including the homogenization approach and the derived dispersion relations. Our work extends the knowledge on mechanical waves in metamaterials providing an analytical framework for the study of vertically polarized surface waves in resonant and partially resonant waveguides.