Periodic cellular materials with nonlinear elastic homogenized stress-strain response at small strains

We investigate the effective stress-strain behavior of cellular elastomers, structured with a periodic pattern of elliptic holes by means of full scale simulations under small deformations. First, we show that the elastic response behaves non trivially with the pore geometry. In particular, we show that auxetic and anistropic responses arise for a broad range of parameters, when the microstructure becomes sufficiently porous. Second, we show that, in the limit of large and near-circular holes, the stress-strain nonlinearities become very large. Third, we adopt an effective theoretical description where the filaments between the holes are modeled by slender beams to predict the linear response, and by bars and pivots to capture the leading order nonlinear corrections. This approach fully captures the asymptotic observations and open pathways for an effective-beams based homogenization and the design of nonlinear cellular materials.