Elastic properties of chiral, anti-chiral, and hierarchical honeycombs: A simple energy-based approach

• Effects of chirality and hierarchy on elastic response of honeycombs are studied.
• Closed-form relations are derived for elastic moduli and validated using finite element method (FEM).
• Chirality always decreases the stiffness and Poisson’s ratio.
• Hierarchical refinement increases the stiffness in hexagon based honeycombs.
• Anti-tetra-chiral structure shows anisotropy, auxeticity, and low shear stiffness.

The effects of two geometric refinement strategies widespread in natural structures, chirality and self-similar hierarchy, on the in-plane elastic response of two-dimensional honeycombs were studied systematically. Simple closed-form expressions were derived for the elastic moduli of several chiral, anti-chiral, and hierarchical honeycombs with hexagon and square based networks. Finite element analysis was employed to validate the analytical estimates of the elastic moduli. The results were also compared with the numerical and experimental data available in the literature. We found that introducing a hierarchical refinement increases the Young’s modulus of hexagon based honeycombs while decreases their shear modulus. For square based honeycombs, hierarchy increases the shear modulus while decreasing their Young’s modulus. Introducing chirality was shown to always decrease the Young’s modulus and Poisson’s ratio of the structure. However, chirality remains the only route to auxeticity. In particular, we found that anti-tetra-chiral structures were capable of simultaneously exhibiting anisotropy, auxeticity, and remarkably low shear modulus as the magnitude of the chirality of the unit cell increases.