Construction of Micropolar Continua from the Homogenization of Repetitive Planar Lattices

Abstract

The derivation of the effective mechanical properties of planar lattices made of articulated bars is herewith investigated, relying on the asymptotic homogenization technique to get closed form expressions of the equivalent properties versus the geometrical and mechanical microparameters. Considering lattice microrotations as additional degrees of freedom at both scales, micropolar equivalent continua are constructed from discrete lattices made of a repetitive unit cell, from an extension of the asymptotic homogenization technique. We will show that it is necessary to solve on two different orders a linear system of equations giving the kinematic variables, at both the first and second order. The effective strain and effective curvature appear respectively as the first and second order strain variables. In the case of a centrosymmetric unit cell, there is no coupling between couple stresses and strains nor between stress and curvature. The unknown kinematic variables are determined by solving the translational and rotational equilibrium for the whole lattice. This in turn leads to the expression of the stress vector and couple stress vector, allowing to construct the Cauchy stress and couple stress tensors. The homogenized behavior of the tetragonal and hexagonal lattices is determined in terms of homogenized micropolar moduli.