Asymptotic approximation of the band structure for tetrachiral metamaterials

Tetrachiral materials are characterized by a cellular microstructure made by a periodic pattern of stiff rings and flexible ligaments. Their mechanical behaviour can be described by a planar lattice of rigid massive bodies and elastic massless beams. Therefore, the periodic cell dynamics is governed by a monoatomic structural model, conveniently reduced to the only active degrees-of-freedom. The paper presents an explicit parametric description of the Floquet-Bloch spectrum (or band structure) governing the propagation of elastic waves through the tetrachiral material. By virtue of multiparametric perturbation techniques, an analytical asymptotic approximation is achieved for the dispersion surfaces in the Brillouin zone. Since different optimization strategies tend to fail in opening low-frequency band gaps in the material spectrum, this specific design purpose is commonly pursued by introducing interring inertial resonators. The paper demonstrates that multiparametric perturbation methods can efficiently deal with the consequent enlargement of the parameter space, necessary to describe the resulting inertial metamaterial. Indeed, paying due attention to the doubling of internal resonance conditions, an accurate parametric approximations of the enriched band structure can be achieved. From the applicative perspective, the research findings furnish suited analytical tools for the optimal design of pass and stop bands.