Micropolar continuum modelling of bi-dimensional tetrachiral lattices

The in-plane behaviour of tetrachiral lattices should be characterized by bi-dimensional orthotropic material owing to the existence of two orthogonal axes of rotational symmetry. Moreover, the constitutive model must also represent the chirality inherent in the lattices. To this end, a bi-dimensional orthotropic chiral micropolar model is developed based on the theory of irreducible orthogonal tensor decomposition. The obtained constitutive tensors display a hierarchy structure depending on the symmetry of the underlying microstructure. Eight additional material constants, in addition to five for the hemitropic case, are introduced to characterize the anisotropy under Z₂ invariance. The developed continuum model is then applied to a tetrachiral lattice, and the material constants of the continuum model are analytically derived by a homogenization process. By comparing with numerical simulations for the discrete lattice, it is found that the proposed continuum model can correctly characterize the static and wave properties of the tetrachiral lattice.