Poisson’s ratio for an anisotropic linear elastic material depends on two orthogonal directions $n$ and $m$. Materials with negative Poisson’s ratios for all $(n,m)$ pairs are called completely auxetic while those with positive Poisson’s ratios for all $(n,m)$ pairs are called nonauxetic. Simple necessary and sufficient conditions on elastic compliances are derived to identify if any given material of cubic or hexagonal symmetry is completely auxetic or nonauxetic. When these conditions are not satisfied, the medium is auxetic for some $(n,m)$ pairs. Several simple necessary conditions for completely auxetic or nonauxetic media are derived for a general anisotropic elastic material.

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