Optimal 2D auxetic micro-structures with band gap

A systematic investigation is presented that explores band gap properties of periodic micro-structures architected for maximum auxeticity. The design of two-dimensional auxetic cells is addressed using inverse homogenization. A non-convex optimization problem is formulated that is solved through mathematical programming. Different starting guesses are used to explore local minima when distributing material and void or two materials and void. The same numerical tool succeeds in capturing re-entrant, chiral and anti-chiral layouts with negative Poisson's ratio, retrieving solutions originally found through other approaches as well as generating variations. A Floquet-Bloch approach is then applied to the achieved periodic cells to investigate possible band gaps characterizing the in-plane wave propagation. Directional and full band-pass filters are found in the case of micro-structures whose auxetic behavior comes from the arising of a rotational deformation of the periodic cell. Such kind of topologies could be exploited to design tunable wave guides and tunable phononic crystals, respectively.