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

In this paper we investigate numerically an instance of the problem of G-closure for two-dimensional periodic metamaterials. Specifically, we consider composites with isotropic homogenized elasticity tensor, obtained as a mixture of two isotropic materials, focusing on the case of a single material with voids. This problem is important, in particular, in the context of designing small-scale structures for metamaterials in the context of additive fabrication, as this type of metamaterials makes it possible to obtain a range of material properties using a single base material. We demonstrate that two closely related simple parametric families based on the structure proposed by O. Sigmund attain good coverage of the space of isotropic properties satisfying Hashin-Shtrikman bounds. In particular, for positive Poisson 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 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.