Passive control of wave propagation in periodic anti-tetrachiral meta-materials

Periodic anti-tetrachiral materials are strongly characterized by a marked auxeticity, the unusual and fascinating mechanical property mathematically expressed by negative values of the Poisson's ratio. The auxetic behavior is primarily provided by pervasive rolling-up mechanisms developed by the doubly-symmetric micro-structure of the periodic cell, composed by a regular pattern of rigid rings connected by tangent flexible ligaments. Adopting a beam-lattice model to describe the linear free dynamics of the elementary cell, the planar wave propagation along the bi-dimensional material domain can be studied according to the Floquet-Bloch theory. Parametric analyses of the dispersion curves, carried out with numerical or asymptotic tools, typically reveal a highly-dense spectrum, with persistent absence of total band-gaps in the low-frequency range. The paper analyses the wave propagation in the meta-material developed by introducing rigid massive inserts, locally housed by all the rings and working as undamped linear oscillators with assigned inertia and/or stiffness properties. The elastic coupling between the cell microstructure and the oscillators, if properly tuned (inertial resonators), is found to significantly modify the Floquet-Bloch spectrum of the material. The effects of the resonator parameters (tuning frequency and mass ratio) on the low-frequency band structure of the metamaterial are discussed, with focus on the valuable possibility to (i) open total band gaps, by either the widening of an existing partial band gap or the avoidance of a crossing point between adjacent dispersion curves, (ii) finely control the total band-gap amplification, in order to assess the maximum achievable performance of the meta-material against the vibration propagation.