On the adoption of canonical quasi-crystalline laminates to achieve pure negative refraction of elastic waves

A way to achieve negative refraction of elastic anti-plane shear waves is a transmission across an interface between a homogeneous substrate and a periodic transverse laminate. To achieve pure negative refraction, the frequency of the source should be lower than the upper limit of the second transition zone (TZ) of the harmonic spectrum of the laminate. An effective way to control the location of TZ is to consider a canonical configuration for the laminate, a concept that originates from the properties of quasi-crystalline sequences among which the Fibonacci one is a particular case. Based on the universal structure of frequency spectrum, we provide a method based on the reduced torus to study the effect of a change in canonical ratio on the limits of the TZ. A further contribution consists in the analytical estimate of the angle of refraction for a linear relationship between frequency and longitudinal wavenumber. This is achieved by determining the components of the in-plane Poynting vector. The outcome provides a tool for the selection of a suitable laminate-substrate combination to accomplish a particular angle of the refracted wave. Finally, it is shown that for some particular configurations, the transmitted energy displays a peak that can be exploited to maximize the amount of energy travelling across the laminate. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.