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

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.

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.

DOI : https://doi.org/10.1016/j.anihpc.2011.07.002
Classification:  35B27,  49J45,  74E30,  74Q05,  74Q20
Keywords: Homogenization, Nonlinear elasticity, Linearization, Γ-closure
@article{AIHPC_2011__28_6_941_0,
     author = {Gloria, Antoine and Neukamm, Stefan},
     title = {Commutability of homogenization and linearization at identity in finite elasticity and applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {6},
     year = {2011},
     pages = {941-964},
     doi = {10.1016/j.anihpc.2011.07.002},
     zbl = {1256.35004},
     mrnumber = {2859933},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_6_941_0}
}
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, Volume 28 (2011) no. 6, pp. 941-964. doi : 10.1016/j.anihpc.2011.07.002. http://www.numdam.org/item/AIHPC_2011__28_6_941_0/

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