Homogenization of periodic hexa- and tetrachiral cellular solids
  The homogenization of auxetic cellular solids having periodic hexachiral and tetrachiral microstructure is dealt with two different techniques. The first approach is based on the representation of the cellular solid as a beam-lattice to be homogenized as a micropolar continuum. The second approach is developed to analyse periodic cells conceived as a two-dimensional domain consisting of deformable portions such as the ring, the ligaments and possibly an embedded matrix internally to these. This approach is based on a second displacement gradient computational homogenization proposed by the Authors (Bacigalupo and Gambarotta, 2010). The elastic moduli obtained by the micropolar homogenization are expressed in analytical form from which it appears explicitly their dependence on the parameter of chirality, which is the angle of inclination of the ligaments with respect to the grid of lines connecting the centers of the rings. For hexachiral cells, the solution of Liu et al., 2012, is found, showing the auxetic property of the lattice together with the elastic coupling modulus between the normal and the asymmetric strains; a property that has been confirmed here for the tetrachiral lattice. Unlike the hexagonal lattice, the classical constitutive equations of the tetragonal lattice turns out to be characterized by the coupling between the normal and shear strains through an elastic modulus that is an odd function of the parameter of chirality. Moreover, this lattice is found to exhibit a remarkable variability of the Young's modulus and of the Poisson's ratio with the direction of the applied uniaxial stress. The properties of the equivalent micropolar continuum are qualitatively detected also in the equivalent second-gradient continuum. 
Andrea Bacigalupo and Luigi Gambarotta
cond-mat.mtrl-sci cond-mat.soft