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Archive for Rational Mechanics and Analysis

, Volume 201, Issue 2, pp 465–500 | Cite as

On the Commutability of Homogenization and Linearization in Finite Elasticity

  • Stefan MüllerEmail author
  • Stefan Neukamm
Article

Abstract

We consider a family of non-convex integral functionals
$$\frac{1}{h^2}\int_\Omega W(x/\varepsilon,{\rm Id}+h\nabla g(x))\,\,{\rm d}x,\quad g\in W^{1,p}({\Omega};{\mathbb R}^n)$$
where W is a Carathéodory function periodic in its first variable, and non-degenerate in its second. We prove under suitable conditions that the Γ-limits corresponding to linearization (h → 0) and homogenization (\({\varepsilon\rightarrow 0}\)) commute, provided W is minimal at the identity and admits a quadratic Taylor expansion at the identity. Moreover, we show that the homogenized integrand, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the second variation of W.

Keywords

Strong Convergence Strong Topology Quadratic Domain Bound Lipschitz Domain Quadratic Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Hausdorff Center for Mathematics, Institute for Applied MathematicsUniversität BonnBonnGermany
  2. 2.Zentrum Mathematik/M6Technische Universität MünchenGarching bei MünchenGermany
  3. 3.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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