Hierarchy of generalized continua issued from micromorphic medium constructed by homogenization

Abstract

The present contribution provides a classification of generalized continua constructed by a micromechanical approach, relying on an extension of the Hill macrohomogeneity condition. The virtual power of equilibrium for a micromorphic effective medium is derived from the microscopic Cauchy balance equations, highlighting the classical and higher-order macroscopic stress tensors. The so-called homogeneous displacement associated with the micromorphic effective medium is derived from variational formulations. It allows establishing the extended Hill macrohomogeneity condition that prevails for the micromorphic continuum, wherein the higher-order stress tensors arise as the static variables conjugated to the selected macroscopic degrees of freedom. Suitable projections of the introduced kinematic micromorphic variables into degenerated kinematic variables lead to various subclasses of generalized continua: microstretch, micropolar, couple stress, microdilatation, microstrain, microshear, and strain gradient. An asymptotic ranking of the formulated generalized continua versus a small-scale parameter is formulated in the last part of the paper to quantify their relative importance. The micromorphic homogenization scheme is validated by comparing the predictions of the homogenized response at the macroscale for a double shear test to a reference exact solution. The proposed micromorphic homogenization method remedy most of the limitations of the existing schemes of the literature.