Variational methods for the homogenization of periodic heterogeneous media

In this paper, several variational principles for the evaluation of the overall properties of composite materials with periodic microstructure are introduced. The two classical homogenization problems corresponding to assigned average strain or assigned average stress are considered. The periodicity of the variables governing the problem is enforced either by considering special representations (i.e. Fourier series) of the periodic part of the displacement and stress fields or by adopting appropriate boundary conditions on the unit cell. In particular, the boundary conditions ensuring the periodicity of the governing variable are introduced in the functionals by using Lagrangian multipliers. Once the variational principles are introduced, the Fourier series technique and the finite element method are adopted to obtain rational approximation procedures. Finally, numerical applications are carried out in order to assess the performances of the proposed methods in the computation of estimates or bounds on the overall elastic properties of a composite material, and in the determination of the displacement and stress distribution in the unit cell.