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On the Commutability of Homogenization and Linearization in Finite Elasticity

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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.

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Author notes

  1. Stefan Neukamm

    Present address: Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

Authors and Affiliations

  1. Hausdorff Center for Mathematics, Institute for Applied Mathematics, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Stefan Müller

  2. Zentrum Mathematik/M6, Technische Universität München, Boltzmannstraße 3, 85748, Garching bei München, Germany

    Stefan Neukamm

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  1. Stefan Müller
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Correspondence to Stefan Müller.

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Müller, S., Neukamm, S. On the Commutability of Homogenization and Linearization in Finite Elasticity. Arch Rational Mech Anal 201, 465–500 (2011). https://doi.org/10.1007/s00205-011-0438-7

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  • DOI: https://doi.org/10.1007/s00205-011-0438-7

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