Thermal conductivity and Young's modulus of cubic-cell metamaterials

Abstract

It is shown that the popular quasi-laminate solutions for the relative (thermal) conductivity of cellular materials with cubic cells (closed or open) are in conflict with the upper Hashin-Shtrikman bound and thus inadmissible. However, numerical modeling can be used to estimate the porosity dependence of conductivity and leads to results that are below the upper Hashin-Shtrikman bound, as required for cubic and isotropic materials. The numerical results predict differences of up to 0.073 relative property units (RPU) between closed- and open-cell materials. For porosities approaching 100%, i.e. infinitely thin walls, both the analytical and the numerical solution for closed cubic cells approach the upper Hashin-Shtrikman bound. Moreover, the porosity dependence of the relative conductivity for closed cubic cells is very close to the upper Hashin-Shtrikman bound in the whole range of porosities and is correlated to the relative Young modulus via a cross-property relation based on the Hashin-Shtrikman bounds.