Negative refraction for anti-plane elastic waves in canonical quasicrystalline laminates

Elastic anti-plane shear waves can be refracted negatively when they are transmitted across an interface between a homogeneous substrate and a transverse periodic laminate. To achieve pure negative refraction, the frequency of the source should be lower than the upper limit of the second transition zone of the harmonic spectrum of the laminate. An effective way to control the location of transition zones is to consider a canonical configuration for the laminate, a concept that originates from the properties of quasicrystalline sequences among which the Fibonacci one is a particular case. We give a detailed account of the classification in three families of canonical configurations and the role of canonical frequency. We exploit the knowledge of the scaling factor of the self-similar structure of the layout of transition zones for laminates of this kind to provide a quantitative tool to predict the relevant frequencies to accomplish negative refraction. We also investigate how the change of other parameters of the elementary cell may affect the values of those frequencies. The obtained results show that the features of quasicrystalline sequences may be profitably exploited for the realisation of elastic metamaterials.