Strain-gradient effects in the discrete/continuum transition via homogenization

A second-gradient elastic material has been identified as the equivalent homogeneous material of an hexagonal lattice made up of three different orders of linear elastic bars (hinged at each junction). In particular, the material equivalent to the lattice exhibits: (i.) non-locality, (ii.) non-centrosymmetry, and (iii.) anisotropy (even if the hexagonal geometry leads to isotropy at first-order). A Cauchy elastic equivalent solid is only recovered in the limit of vanishing length of the lattice’s bars. The identification of the second-gradient elastic material is complemented by analyses of positive definiteness and symmetry of the constitutive operators. Solutions of specific mechanical problems in which the lattice response is compared to the corresponding response of an equivalent boundary value problem for the homogeneous second-gradient elastic material are presented. These comparisons show the efficacy of the proposed identification procedure.