Construction of micromorphic continua by homogenization based on variational principles

The present contribution aims to revisit higher-order homogenization schemes towards micromorphic media based on variational principles and an extension of Hill macroho-mogeneity condition. Starting from the microscopic Cauchy balance equations, the local balance equations of the micromorphic continuum are formulated, highlighting the mi-cromorphic stress measures. Relying on both energy and complementary energy expres-sions combined with the extended Hill macrohomogeneity condition, the complete ho-mogeneous microscopic displacement field representative of the effective micromorphic continuum is obtained as a quartic expansion of the macroscopic micromorphic kinematic variables. This procedure leads to a higher-grade micromorphic theory, with the relative stress and hyperstress tensors including respectively second-order and third order polyno-mials of the relative position within the unit cell. The microscopic displacement is com-pleted by a fluctuating part evaluated from a variational principle and characterized by three unit cell boundary value problems. Numerical applications are done for inclusion based composite materials. The obtained results highlight that the higher-order moduli converge very quickly with unit cell size, due to the consideration of correction factors based on the higher-order moments of area. (c) 2020 Elsevier Ltd. All rights reserved.