Two-scale homogenization of elastic layered composites with interfaces oscillating in two directions

Abstract

In a variety of situations of practical interest, the interface between two phases in a composite cannot be reasonably assumed to be smooth but has to be taken as being rough at the microscopic scale. How to determine the effective properties of such a composite remains a largely open problem in micromechanics. The present work is concerned with layered composites in which the interface between two neighboring layers oscillates quickly and periodically along two directions in the plane normal to the layering direction. In this case, the classical homogenization theory of layered composites is no longer applicable, since the interfacial oscillations prevent the layered composite in question from being homogeneous in the plane perpendicular to the layering direction. To overcome this difficulty, a two-scale homogenization method is proposed in the present work. First, at the mesoscopic scale, each zone in which an interface oscillates is homogenized as an interphase by an asymptotic analysis. The effective elastic properties of this interphase are determined by using a numerical method based on the fast Fourier transform (FFT) or estimated by applying the generalized self-consistent scheme (GSCS). Then, at the macroscopic scale, the effective elastic moduli of the composite made of the resulting plane layers and interphases are calculated with the help of the classical homogenization theory of layered composites. Finally, numerical examples are provided to illustrate the results for the effective elastic moduli of a layered composite obtained by the two-scale homogenization method proposed and to compare them with the corresponding numerical results given by the finite element method (FEM).