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

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.