Control and suppression of elastic waves using periodic metasurfaces and bridges

Abstract

This thesis discusses how wave propagation in continuous linearly elastic media can be controlled or suppressed using periodic structures. This involves controlling 2D waves on a membrane, 3D longitudinal and transverse waves in a linearly elastic bulk, and waves across the surface of a linearly elastic half-space, termed `Rayleigh waves'.
First, the concept of wave `bridging' is introduced, using an array of periodic materials to carry waves across a void for two different continuous media. In a 2D membrane, the void is bridged by a periodic array of strings. Two arrays are considered; the first is a simple array of parallel strings, while the second is a square based string lattice. For each, bridging is shown to be possible but with limitations. The lattice bridge is also shown to be capable of limited wave filtering.
A 3D linearly elastic bulk is then considered, with an array of membranes and thin, rigid sheets held in parallel, intended to bridge the out-of-plane and in plane wave motion respectively. For bulk waves, `perfect' wave bridging is shown to be possible for normally incident waves only, with any other incident angle causing wave conversion. For Rayleigh waves `perfect' bridging is shown to be possible, but requires the bridge parameters to have dependence on depth, indicating that this bridge cannot bridge both Rayleigh waves and bulk waves. Furthermore, it is not possible to construct a broadband Rayleigh bridge.
Next, different metasurfaces are designed and treated, consisting of a periodic array of vertical resonators attached to the surface of a half-space. These metasurfaces are intended to control and suppress Rayleigh waves. Earlier studies have considered the effect of compressional resonators. In this thesis, an asymptotic model is used to determine the approximate dispersion relation of the previously considered compressional metasurface, which is shown to be remarkably close to the full unimodal dispersion relation. The same asymptotic model is then used to consider a flexural metasurface, with resonators formed from Euler-Bernoulli beams. The dispersion relation is again compared with the full unimodal dispersion relation, again showing the same key behaviours. Special attention is given to the effect of different junction conditions, which are shown to significantly change the size and behaviour of any stop bands.
Finally, a second-order term is derived for the previously employed asymptotic model, verified by comparison to the Taylor expansion of the Rayleigh determinant. This new model is applied to three different systems; a point harmonic forcing, a moving load and a compressional metasurface. In each, the new model is shown to more closely represent the exact solution, at a cost of increased complexity.