Computation of the homogenized nonlinear elastic response of 2D and 3D auxetic structures based on micropolar continuum models

We develop mechanical and numerical homogenization models for the large strains response of auxetic structures in order to derive their effective elastic response accounting for large changes of the geometry. We construct a strain driven nonlinear scheme for the computation of the stress-strain relation of these repetitive networks over a reference representative unit cell (abbreviated as RUC). Homogenization schemes are developed to evaluate the effective nonlinear mechanical response of periodic networks prone to an auxetic behavior, in both planar and 3D configurations, which are substituted by an effective micropolar continuum at the intermediate mesoscopic level. The couple stress part of the homogenized constitutive law takes into account the impact of the local rotations at the mesoscopic level and allows computing the bending response. This methodology is applied to four planar auxetic periodic lattices (the re-entrant hexagonal honeycomb and alternative honeycomb topologies, including the arrowhead, Milton and hexachiral structures), and to the 3D re-entrant and pyramid-shaped lattices. We have considering successively the in-plane and out-of-plane responses of these structures. The transition to an auxetic behavior is shown to be triggered by the imposed strain over the unit cell boundary. The computed evolutions of Poisson's ratio versus the imposed stretch traduce an enhanced auxetic response as the stretch is increased. A satisfactory agreement is obtained between the homogenized stress-strain responses and the responses computed numerically by finite element simulations performed over a repeating unit cell.