Two-scale limits in some non-linear problems of engineering mechanics

Abstract

The paper deals with the homogenization of variational formulations of non-linear systems of PDEs with periodically oscillating coefficients, occurring in engineering mechanics, whose microstructure can include holes of complicated shapes and parts of different dimensions. The first section discusses some approaches to the mathematical formulation of physical and engineering problems where the advanced analysis of microstructural properties is needed to obtain numerical results of non-negligible validity; in most cases some kind of homogenization cannot be avoided. One of the most effective approaches, involving certain compensated compactness phenomenon, is based on the idea of the two-scale convergence, whose definition and basic properties (with respect to Radon measures) are presented in the second section. The last section is devoted to the microstructural and macrostructural formulation of a model elliptic problem; it is demonstrated how the notion of the two-scale homogenization explains and simplifies the complicated form of the macroscopic limit equation, thanks to the addition of a new microscopic hidden variable.