Issue 29

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01 4             11,1 12,2 1 1 1 1 21,1 22,2 2 2 1 2 1,1 2,2 21 12 1 1 1 2 1 2 2 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ d d d d k v u u k v u u m m k I k u v v k u v v k I                                                       (2) where the stiffnesses ˆ d d cell k k A  and ˆ cell k k A    are introduced, cell A being the area of the periodic cell, together with the mass densities M 1 1 cell A   and M 2 2 cell A   , and the micro-inertia terms J 1 1 cell I A  and J 2 2 cell I A  , respectively. In Eq. (2) ij  and i m are the overall stress components, namely the asymmetric stress components and the micro- couples, respectively. In case of hexachiral lattice, one obtains 2 2 2 3 cos cell l A   and the following inertial parameters are obtained   1 3 sin 2 24 s       2 2 1 3 tan sin 24 s I l       2 2 2 3 sin 24 i cell M A       2 2 2 2 2 3 sin tan 192 i cell J I l A       For the tetrachiral lattice one obtains 2 2 cos cell l A   and the resulting inertia parameters are   1 sin 2 24 s       , 2 2 1 sin 48 s I l      , 2 2 2 sin 48 i cell M A       , 2 2 2 2 2 sin tan 192 i cell J I l A       . To obtain the displacement formulation of the equations of motion, the compatibility equations involving the macrostrain components 11 1,1 u   , 22 2,2 u   , 12 1,2 u     ,     21 2,1 u and the curvatures 1 ,1    and 2 ,2    have to be considered together with the constitutive equation. M ICROPOLAR CONSTITUTIVE EQUATION FOR HEXA - AND TETRA - CHIRAL LATTICES he constitutive equation of hexachiral honeycomb corresponds to that obtained in [13, 15] and is written as follows 11 11 22 22 12 12 21 21 1 1 2 2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A A A A A A k k A A k k m S m S                                                                                      (3) in which the five elastic moduli T