Issue 29

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01 2 characterized by a periodic array of the rings connected by four or six elastic ligaments. Lakes [10], with reference to the planar isotropic case of hexachiral lattice, proposed this microstructure geometry firstly. Later Prall and Lakes [11] showed for this material a Poisson’s ratio of -1 under the hypothesis of ignoring the axial strain of the ligaments. Further studies on the homogenization of hexachiral auxetic materials have been carried out by Spadoni and Ruzzene [12] and by Liu et al. [13]. Liu et al. [14] proposed a hexachiral metacomposite by integrating a two-dimensional hexachiral lattice with elastic resonating inclusions to obtain low-frequency band gap. This metacomposite has been analysed through a numerical model where the ligaments are modelled as multi-beam elements and the inclusions as a two dimensional FEM model. In this paper, hexa- and tetra-chiral beam lattices are considered having local resonators at the nodes of the periodic array. The model is developed in closed-form and is based on a micropolar homogenization of the lattice. This approach partially relies on the results by Bacigalupo and Gambarotta [15], which developed and compared the results from the micropolar and a second displacement gradient homogenization for both the hexachiral and the tetrachiral periodic cells in order to evaluate the validity limits of the beam lattice model. 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 structures are obtained following Huang et al. [4] and Liu et al. [14], by the insertion of a circular mass connected through an elastic surrounding interface to each ring of the microstructure. The equations of motion are given within a micropolar continuum model and the overall elastic moduli and the inertia terms are obtained for both the hexachiral and the tetrachiral lattice. The constitutive equation of the beam lattice given in [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. (a) (b) Figure 1 : (a) Hexachiral lattice; (b) tetrachiral lattice. C HIRAL MASS - IN - MASS PERIODIC MATERIAL : M ICROPOLAR HOMOGENIZATION he periodic materials shown in Figure 1 are considered as beam-lattices made up of a periodic array of rigid rings, each one connected to the others through n elastic slender ligaments rigidly connected to the rings. In Figure 2.a the periodic hexachiral cell ( n =6) is shown, while in Figure 1.b is shown the tetrachiral cell ( n =4). Each ligament is tangent to the joined rigid rings and has length l measured between the connection points, with section width t, thickness d and Young’s modulus s E . Both the ligaments and the rings have mass density s  , so that the mass and the rotation inertia of the rings are 1 M 2 s rt   and 2 1 1 J M r  , respectively. The two dimensional composite materials are auxetic. The hexachiral lattice is isotropic and its Poisson’s ratio becomes negative when increasing the chirality angle  . On the other hand, the tetrachiral material is strongly anisotropic and presents a directional dependency of the chirality on the direction of the applied stress, as shown in [15]. To obtain low-frequency stop bands, a metamaterial inclusion consisting of a softly heavy disk is located inside the ring as shown in Figure 3.a. This inclusion plays the role of low-frequency resonator with mass and rotation inertia denoted with 2 M and 2 J , respectively. The motion of the rigid ring of the beam lattice is denoted by the displacement vector u and the rotation  , respectively (see Figure 3.b), while the motion of the mass of the resonator is denoted by the displacement vector v and the rotation  (see Figure 3.c). T