Elasto-magnetic waves in metamaterials: physics and modelling

The present work is framed in the wide topic of metamaterials. New elements portray this investigation with respect to what already existing in the current worldwide scenario. Indeed, the attention is focused on elastic metamaterials, with the idea to control wave propagation within the structure in terms of wave targeting, wave stopping and wave absorption. This is a novelty, since the concept of metamaterials is usually related to electromagnetic applications, for which all the uncommon effects, as for instance the invisible cloaking, are mainly related to the negative refraction index. A simple one-dimensional structure is analysed. More than the short-range elastic constitutive relationship, a nonlocal new long-range elastic material is here considered. This deeply affects and changes the topology of the system, leading to unexpected propagation phenomena. A mathematical model, based on the nonlocal elasticity theory of Eringen, is considered. The long-range interaction term appears as an integral function that induces a nonlinear characteristic to the conventional equation of motion. Special types of forces are chosen to model the long-range term, as they not only accurately model the elastic forces, but also lend themselves to analysis by Fourier transform. Closed form analytical solutions are achieved and it is the first time that such type of problem is so exhaustively examined. Indeed, long-range interactions have already been investigated, first by V.E. Tarasov, then by Zingales. However, Tarasov developed an analysis on the purely static response of structures, whilst this work discusses its dynamic behaviour, and Zingales performed only numerical solutions, which prevent a thorough understanding, and is mainly interested in modal analysis. The results are discussed in terms of modal analysis, dispersion relationship and propagation. It can be seen how the introduction of unconventional connections affects the typical behaviour of the structure and new phenomena, as hypersonic and superluminal propagation and negative group velocity arise. The analysis has been extended to a twin system, composed by two identical waveguides, with no structural coupling, but mutually coupled only through the long-range characteristic. An experimental campaign concludes the work. A twin waveguide system has been realized by 3D printing; several magnet holders and metallic strips acting as springs are used so to reproduce the mathematical model. The magnetic coupling recreates the long-range interaction. Different types of excitations have been applied to the primary waveguide, so to retrace first the complex frequency response and secondly the dispersion relationship. First results, even though rough, exhibit a damped response on the main waveguide, and a more complex response in the secondary waveguide, in agreement with what analytically observed.