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

Abstract

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. Under convexity and p-growth conditions (p > 1), it is proved that all such possible effective energy densities obtained by a Γ -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 in terms 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. RésuméCet article est consacré à la caractérisation de toutes les limites effectives possibles de matériaux composites en terme d'homogénéisation périodique. Ce problème est connu sous le nom de G-fermeture. Il est démontré, sous des hypothèses de convexité et de croissance p > 1, que toutes ces densités d'énergies effectives, obtenues lors d'une analyse par Γ -convergence, peuvent être localement vues comme la limite ponctuelle d'une suite de densités d'énergies périodiquement homogénéisées avec une fraction de volume fixée. Un résultat plus faible est obtenu sans aucune hypothèse de convexité et l'ensemble des zéros de ce type de densités d'énergies effectives est caractérisé en terme de mesures d'Young. Un résultat similaire est donné pour des intégrandes cellulaires, permettant ainsi de proposer de nouveaux contre-exemples quant à la validité de la formule de cellule dans le cas non convexe, ainsi qu'à la continuité du déterminant par rapport à la convergence à double échelle.

A Quasistatic Evolution Model for the Interaction Between Fracture and Damage

“…In this section we recall well known facts about the homogenization of (convex) integral functionals, and then we will specialize our study to the case of mixtures for which local properties of effective materials can be obtained (see [7]). Given f ∈ F(Q, α, β, p), we denote by f hom (ξ) its homogenized energy density defined by…”

“…The first one is that the densities W 1 and W 2 are restricted to be convex functions. The reason is that the local character of G-closure proved in [7], and used many times here, has only been proved in the convex case. To our knowledge, it is not known yet whether it holds or not in the general quasiconvex case.…”

“…We denote by G θ (W 1 , W 2 ) the closure of P θ (W 1 , W 2 ) for the pointwise convergence. For any θ ∈ L ∞ (Ω; The localization result proved in [7] states that if W 1 and W 2 have suitable growth and coercivity conditions, then f ∈ G θ (W 1 , W 2 ) if and only if f (x, ·) ∈ G θ(x) (W 1 , W 2 ) for a.e. x ∈ Ω.…”

“…In fact, a localization property proved in [7] for energies resulting from a mixture between two arbitrary materials will be instrumental because it will enable us to obtain an alternative formula for the relaxed energy. The local character of G-closure states that, at least for convex energies, any effective energy obtained by an arbitrary mixture between two materials can be locally recovered as the pointwise limit of a sequence of effective energies obtained as a periodic mixture between the same materials with the same local volume fraction.…”

New Counterexamples to the Cell Formula in Nonconvex Homogenization

“…A counterexample due to Stefan Müller in [12] shows that in general, for quasiconvex nonconvex energy densities, the inequality W cell ≥ W hom can be strict. More recently, Jean-François Babadjian and the first author gave another such example in [3].…”