Tetra-, Tri-, Di-, and Mono-mode Metamaterials with Cubic Symmetry

We report a comprehensive design guide for crystalline metamaterials with tetra-, tri-, di-, and mono-mode based on cubic elasticity - the simplest anisotropic case specified by 3 elastic constants only. To clarify some popular misconceptions, we first present a discussion about symmetry requirements. Next, we demonstrate a complete design protocol for periodic structures that are guaranteed to exhibit cubic elastic behaviors, with an emphasis on designs without any four-fold rotational symmetry or mirror symmetry. These extremal metamaterials can realize elastic tetra-, tri-, di-, and mono-mode in addition to the well-known penta-mode. Further, we show several examples with different extreme anisotropic cases as well as functionalities for wave manipulation.