Analytical formulation of a discrete chiral elastic metamaterial model

By embedding appropriately designed chiral local resonators in a host elastic media, a chiral metamaterial with simultaneously negative effective density and bulk modulus can be achieved. In this work, an two dimentional (2D) ideal discrete model for the chiral elastic metamaterial is proposed. The discrete dynamic equation is derived and then homogenized to give the continuous description of the metamaterial. The homogenization procedure is validated by the agreement of the dispersion curve of the discrete and homogenized formulations. The form of homogenized governing equations of the metamaterial cannot be classified as a traditional Cauchy elastic theory. This result conforms the conscience that the Cauchy elasticity cannot reflect the chirality, which is usually captured by higher order theory such as the non-centrosymmetric micropolar elasticity. However when reduced to a (2D) problem, the existing chiral micropolar theory becomes non-chiral. Based on reinterpretation of isotropic tensors in a 2D case, we propose a continuum theory to model the chiral effect for 2D isotropic chiral solids. This 2D chiral micropolar theory constitutes a hopeful macroscopic framework for the theory development of chiral metamaterials.