  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.
