Asymptotic homogenization approach for anisotropic micropolar modeling of periodic Cauchy materials

Abstract

A micropolar-based asymptotic homogenization approach for the analysis of composite materials with periodic microstructure is proposed. The upscaling relations, conceived to determine the macro-descriptors (macro displacement and the micropolar rotation fields) as a function of the micro displacement field, are consistently derived in the asymptotic framework. In particular, the micropolar rotation field is expressed in terms of the microscopical infinitesimal rotation tensor and perturbation functions. The micro displacement field is, in turn, obtained by choosing a third order approximation of the asymptotic expansion, in which the macroscopic fields are expressed as a third order polynomial expansion. It follows that the macro descriptors are directly related to both perturbation functions and micropolar two-dimensional deformation modes. Furthermore, a properly conceived energy equivalence between the macroscopic point and a microscopic representative portion of the periodic composite material is introduced to derive the consistent overall micropolar constitutive tensors. It is pointed out that these constitutive tensors are not affected by the choice of the periodic cell. Moreover, in the case of vanishing microstructure the internal-length-scale-dependent constitutive tensors tend to zero, as expected. Finally, the capabilities of the proposed approach are shown through some illustrative examples.