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.
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},
pages = {941--964},
year = {2011},
publisher = {Elsevier},
volume = {28},
number = {6},
doi = {10.1016/j.anihpc.2011.07.002},
mrnumber = {2859933},
zbl = {1256.35004},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.anihpc.2011.07.002/}
}
TY - JOUR AU - Gloria, Antoine AU - Neukamm, Stefan TI - Commutability of homogenization and linearization at identity in finite elasticity and applications JO - Annales de l'I.H.P. Analyse non linéaire PY - 2011 SP - 941 EP - 964 VL - 28 IS - 6 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.anihpc.2011.07.002/ DO - 10.1016/j.anihpc.2011.07.002 LA - en ID - AIHPC_2011__28_6_941_0 ER -
%0 Journal Article %A Gloria, Antoine %A Neukamm, Stefan %T Commutability of homogenization and linearization at identity in finite elasticity and applications %J Annales de l'I.H.P. Analyse non linéaire %D 2011 %P 941-964 %V 28 %N 6 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.anihpc.2011.07.002/ %R 10.1016/j.anihpc.2011.07.002 %G en %F 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, Tome 28 (2011) no. 6, pp. 941-964. doi: 10.1016/j.anihpc.2011.07.002
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