Numerical Implementation of Higher-Order Homogenization Problems and Computation of Gradient Elasticity Coefficients

A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of tridimensional elasticity, but, since the assumption of strict separation of scale is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which, higher-order terms (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. All these terms are then obtained by solving a hierarchy of higher-order elasticity problems with prescribed body forces and eigen-strains whose expression depends on the solution at the lower-order. An energy based micro-macro transition is then proposed for the change of scale and constitutes, in fact, a generalization of the Hill-Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. It is rigorously proved that the macroscopic quantities derived from this homogenization procedure comply with the equations of strain gradient elasticity. As an illustration, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed whose efficiency is shown in the particular case of composites reinforced by long fibers.