Calculus of Variations and Geometric Measure Theory

J. F. Babadjian - M. Barchiesi

A variational approach to the local character of G-closure: the convex case

created by barchiesi on 14 Mar 2007
modified on 12 Jan 2012


Published Paper

Inserted: 14 mar 2007
Last Updated: 12 jan 2012

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
Volume: 26
Number: 2
Pages: 351-373
Year: 2009


This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a G-closure problem. It is proved that, in the convex case, all such possible effective energies obtained by a $\Gamma$-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized by means of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.

Keywords: Young Measures, Homogenization, Gamma-convergence, Two-scale convergence, quasiconvexity, G-closure, convexity