Commutability of homogenization and linearization at identity in finite elasticity and applications (Q654167)

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.