Homogenization towards chiral Cosserat continua and applications to enhanced Timoshenko beam theories

摘要

A homogenization methodology for the construction of effective Cosserat substitution media for heterogeneous materials is proposed, combining a variational principle in linear elasticity with the extended Hill-Mandel lemma accounting for the introduced generalized kinematics. The proposed method is general and can be applied to a wide class of architected materials and composites prone to such micropolar effects. The microscopic displacement field of the initially heterogeneous continuum splits into a homogeneous part polynomial in the generalized kinematic measures and a fluctuation involving localization operators. The tensors of effective micropolar moduli are formulated as integrals over a representative unit cell utilizing of the displacement localizators, solution of classical, and higher-order unit cell problems. The proposed method has the chief advantage of delivering size-independent higher-order effective moduli and to remedy the deficiencies of most of existing higher order homogenization schemes towards generalized continua in the literature. Based on the developed homogenization method, the effective micropolar moduli of the tetrachiral lattice and composites made of a tetrachiral lattice reinforcement are computed to elaborate an enhanced Timoshenko microstructured beam model exhibiting couplings between different deformation modes induced by the response of its underlying tetrachiral microstructure.