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Homogenization of the nonlinear bending theory for plates

  • Stefan NeukammEmail author
  • Heiner Olbermann
Article

Abstract

We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of \(\Gamma \)-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a “cylinder” or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions of class \(W^{2,2}\), using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.

Mathematics Subject Classification

35B27 49J45 74E30 74Q05 

Notes

Acknowledgments

The authors would like to thank an anonymous referee for pointing out a mistake in an earlier version of this manuscript, cf. Remark 2. This work was initiated while the first author was employed at the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. The second author gratefully acknowledges the hospitality of the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Department of MathematicsTechnische Universität DresdenDresdenGermany
  3. 3.Hausdorff Center for Mathematics and Institute for Applied MathematicsUniversity of BonnBonnGermany

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