Commutability of homogenization and linearization at identity in finite elasticity and applications
type: Item
instance of: scholarly article
title: Commutability of homogenization and linearization at identity in finite elasticity and applications
zbMATH Open document ID: 1256.35004
author: Antoine Gloria
author: Stefan Neukamm
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2011-12-28T00:00:00Z
review text: The purpose of the paper is to prove the commutability of linearization and homogenization at identity in a context of finite elasticity. Let \(D\) be an open, bounded and Lipschitz domain of \(\mathbb{R}^{d}\) and consider the family \(\mathcal{I}_{\varepsilon }:u\mapsto \int_{D}W_{\varepsilon }(x,\nabla u(x))dx\), where \(W_{\varepsilon }:D\times \mathbb{M} ^{d}\rightarrow [ 0,+\infty ]\) is a Borel function which is frame indifferent, minimal at identity, non-degenerate and admitting a quadratic expansion at identity. The quadratic term \(Q_{\varepsilon }\) of this expansion is thus associated to a functional \(\mathcal{E}_{\varepsilon }:u\mapsto \int_{D}Q_{\varepsilon }(x,\nabla g(x))dx\), where \( g(x)=h^{-1}(u(x)-x)\) for a displacement field \(u\) satisfying \(\left| \nabla u-Id\right| \sim h\ll 1\). The main result of the paper proves that the limits of \(\mathcal{G}_{h,\varepsilon }\) when \(h\rightarrow 0\) and when \(\varepsilon \rightarrow 0\) commute, where the functional \(\mathcal{G} _{h,\varepsilon }\) is defined through \(\mathcal{G}_{h,\varepsilon }(g)= \mathcal{I}_{\varepsilon }(\varphi Id+hg)/h^{2}\), the limits being taken in a \(\Gamma \)-convergence sense. The authors here assume that the energy densities \(W_{\varepsilon }\) belong to the space \(\mathcal{W}_{\alpha ,\rho }^{p}(D\times \mathbb{R}^{d})\) of continuous functions which satisfy a growth condition of order \(p\in [ 2,+\infty )\) and which have a modulus of approximation \(\rho \) at \(Id\). This extends a result obtained by \textit{S. Müller} and the second author in [Arch. Ration. Mech. Anal. 201, No. 2, 465--500 (2011; Zbl 1262.74029)]. The proof of this result is mainly based on the properties of the \(\Gamma \)-convergence and of the \(G\)-convergence. The paper ends with an application to stochastic homogenization within this context.
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 74E30
Mathematics Subject Classification ID: 74Q05
Mathematics Subject Classification ID: 74Q20
zbMATH DE Number: 5992248
zbMATH Keywords: homogenization
zbMATH Keywords: nonlinear elasticity
zbMATH Keywords: linearization
zbMATH Keywords: \(\Gamma \)-convergence
zbMATH Keywords: \(\Gamma \)-closure
zbMATH Keywords: \(G\)-convergence
zbMATH Keywords: Korn inequality
zbMATH Keywords: Meyers estimate
zbMATH Keywords: stochastic homogenization
MaRDI profile type: Publication
full work available at URL: j.anihpc.2011.07.002
OpenAlex ID: W1975342759
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DOI: 10.1016/J.ANIHPC.2011.07.002
description: scientific article; zbMATH DE number 5992248