Negative refraction in quasicrystalline multilayered metamaterials

Inspired by some recent results in elastodynamics of layered composites, we address here the problem of an antiplane elastic wave obliquely incident at the interface between a substrate and a periodic laminate with a quasicrystalline structure (generated by the Fibonacci substitution rule). The angles of refraction of the transmitted modes are computed by combining the transfer matrix method with the normal mode decomposition and evaluating the direction of the average Poynting vector. It is shown that, with respect to a periodic classical bilayer, on the one hand, beyond a certain frequency threshold, high order Fibonacci laminates can provide negative refraction for a wider range of angles of incidence, on the other, they allow negative wave refraction at lower frequencies. The outcome strongly relies on the Floquet–Bloch dynamic analysis of this class of laminates that is performed thoroughly. It is revealed that the corresponding spectra have a self-similar character linked to the specialisation of the Kohmoto's invariant, a function of the frequency that was recently studied by the authors for periodic one-dimensional quasicrystalline-generated waveguides. This function is able to explain two types of scaling occurring in dispersion diagrams. The attained results represent an important advancement towards the realisation of multilayered quasicrystalline metamaterials with the aim to control negatively refracted elastic waves.