Commutability of homogenization and linearization at identity in finite elasticity and applications

Antoine Gloria and Stefan Neukamm

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Submission date: 30. Nov. 2010
Pages: 23
published in: Annales de l'Institut Henri Poincaré / C, 28 (2011) 6, p. 941-964 
DOI number (of the published article): 10.1016/j.anihpc.2011.07.002
Bibtex
MSC-Numbers: 35B27, 49J45, 74E30, 74Q05, 74Q20
Keywords and phrases: homogenization, nonlinear elasticity, linearization, Gamma-closure
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Abstract:
In this note 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 -convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the -closure is local at identity for this class of energy densities.