A parametric class of composites with a large achievable range of effective elastic properties

In this paper, we study an instance of the G-closure problem for two-dimensional periodic metamaterials. Specifically, we consider composites with isotropic homogenized elasticity tensor, obtained as a mixture of two isotropic materials. We focus on the case when one material has zero stiffness, i.e., single-material structures with voids. This problem is important, in particular, in the context of designing small-scale structures for metamaterials that can be manufactured using additive fabrication. A range of effective metamaterial properties can be obtained this way using a single base material. We demonstrate that two closely related simple parametric families based on the structure proposed by Sigmund in [26] attain good coverage of the space of isotropic properties satisfying Hashin–Shtrikman bounds. In particular, for positive Poisson's ratio, we demonstrate that Hashin–Shtrikman bound can be approximated arbitrarily well, within limits imposed by numerical approximation: a strong evidence that these bounds are achievable in this case. For negative Poisson's ratios, we numerically obtain a bound which we hypothesize to be close to optimal, at least for metamaterials with rotational symmetries of a regular triangle tiling.