Issue 29

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01 1 Focussed on: Computational Mechanics and Mechanics of Materials in Italy A micropolar model for the analysis of dispersive waves in chiral mass-in-mass lattices A. Bacigalupo University of Trento andrea.bacigalupo@unitn.it L. Gambarotta University of Genoa luigi.gambarotta@unige.it A BSTRACT . The possibility of obtaining band gap structures in chiral auxetic lattices is here considered and applied to the case of inertial locally resonant structures. These periodic materials are modelled as beam-lattices made up of a periodic array of rigid rings, each one connected to the others through elastic slender ligaments. To obtain low-frequency stop bands, elastic circular resonating inclusions made up of masses located inside the rings and connected to them through an elastic surrounding interface are considered and modeled. The equations of motion are obtained for an equivalent homogenized micropolar continuum and the overall elastic moduli and the inertia terms are given for both the hexachiral and the tetrachiral lattice. The constitutive equation of the beam lattice given by the Authors [15] are then applied and a system of six equations of motion is obtained. The propagation of plane waves travelling along the direction of the lines connecting the ring centres of the lattice is analysed and the secular equation is derived, from which the dispersive functions may be obtained. K EYWORDS . Auxetic materials; Chirality; Cellular materials; Mass-in-mass dynamic systems; Dispersive waves. I NTRODUCTION n recent years a considerable interest in acoustic metamaterials was witnessed by several research (see for reference [1]), most of them focused to the design of artificial materials having periodic microstructure conceived to get complete sound attenuation for a certain frequency range, namely acoustic wave spectral gap. Sonic crystals with spectral gaps [2] have been developed on the realization that composites with locally resonant structural units may exhibit effective negative elastic constants at certain frequency ranges, as shown by Liu et al. [3] and by Huang et al. [4,5]. Recently, Bigoni et al. [6] proposed a periodic metamaterial with internal locally resonant structures that supports tunable low- frequency stop bands. This effect is associated with localized rotational modes obtained from a chiral microstructure of the periodic cell. Tee et al. [7] have recently obtained a band gap structure in periodic tetrachiral materials. Spadoni et al. [8] have obtained analogous results with reference to hexachiral lattices. These kind of materials are auxetics [9] and their structure is I