Commutability of homogenization and linearization at identity in finite elasticity and applications
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 6, pp. 941-964.

Nous démontrons que linéarisation et homogénéisation commutent à lʼidentité sous des hypothèses générales sur la densité dʼénergie élastique (à savoir indifférence matérielle, minimalité à lʼidentité, non-dégénérescence et existence dʼun développement quadratique à lʼidentité). Ceci généralise un résultat récent de S. Müller et du second auteur au cas non-périodique. En particulier, nous étendons au cas de lʼhomogénéisation stochastique leur diagramme de commutation de la linéarisation et de lʼhomogénéisation au sens de la Γ-convergence. Par ailleurs, nous démontrons que la Γ-fermeture est locale à lʼidentité pour la classe de densités dʼénergie non convexes considérée.

We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Müller and the second author by dropping their assumption of periodicity. As a first application, we extend their Γ-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Γ-closure is local at identity for this class of energy densities.

DOI : 10.1016/j.anihpc.2011.07.002
Classification : 35B27, 49J45, 74E30, 74Q05, 74Q20
Mots clés : Homogenization, Nonlinear elasticity, Linearization, Γ-closure
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Gloria, Antoine; Neukamm, Stefan. Commutability of homogenization and linearization at identity in finite elasticity and applications. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 6, pp. 941-964. doi : 10.1016/j.anihpc.2011.07.002. http://www.numdam.org/articles/10.1016/j.anihpc.2011.07.002/

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