International audienceExtensions of classical homogenization methods are presented that are used to replace a composite material by an effective generalized continuum model. Homogeneous equivalent second gradient and micromorphic models are considered, establishing links between the macroscopic generalized stress and strain measures and the fields of displacement, strain and stress inside a volume element of composite material. Recently proposed non-homogeneous boundary conditions to be applied to the unit cell, are critically reviewed. In particular, it is shown that such polynomial expansions of the local displacement field must be complemented by a generally non-periodic fluctuation field. A computational strategy is introduced to unambiguously determine this fluctuation. The approach is well-suited for elastic as well as elastoplastic composites
Several extensions of standard homogenization methods for composite materials have been proposed in the literature that rely on the use of polynomial boundary conditions enhancing the classical affine conditions on the unit cell. Depending on the choice of the polynomial, overall Cosserat, second gradient, or micromorphic homogeneous substitution media are obtained. They can be used to compute the response of the composite when the characteristic length associated with the variation of the applied loading conditions becomes of the order of the size of the material inhomogeneities. A significant difference between the available methods is the nature of the fluctuation field added to the polynomial expansion of the displacement field in the unit cell, which results in different definitions of the overall stress and strain measures and higher order elastic moduli. The overall higher order elastic moduli obtained from some of these methods are compared in the present contribution in the case of a specific periodic two-phase composite material. The performance of the obtained overall substitution media is evaluated for a chosen boundary value problem at the macroscopic scale for which a reference finite element solution is available. Several unsatisfactory features of the available theories are pointed out, even though some model predictions turn out to be highly relevant. Improvement of the prediction can be obtained by a precise estimation of the fluctuation at the boundary of the unit cell.
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