Equiauxetic Hinged Archimedean Tilings

Abstract

There is increasing interest in two-dimensional and quasi-two-dimensional materials and metamaterials for applications in chemistry, physics and engineering. Some of these applications are driven by the possible auxetic properties of such materials. Auxetic frameworks expand along one direction when subjected to a perpendicular stretching force. An equiauxetic framework has a unique mechanism of expansion (an equiauxetic mode) where the symmetry forces a Poisson’s ratio of $-1$. Hinged tilings offer opportunities for the design of auxetic and equiauxetic frameworks in 2D, and generic auxetic behaviour can often be detected using a symmetry extension of the scalar counting rule for mobility of periodic body-bar systems. Hinged frameworks based on Archimedean tilings of the plane are considered here. It is known that the regular hexagonal tiling, $\left\{6^3\right\}$, leads to an equiauxetic framework for both single-link and double-link connections between the tiles. For single-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found here to be equiauxetic: these are $\{3.{12}^2\}$, $\{4.6.12\}$, and $\{4.8^2\}$. For double-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found to be equiauxetic: these are $\{3^4.6\}$, $\{3^2.4.3.4\}$, and $\{3.6.3.6\}$.