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MiS Preprint
40/2013

Homogenization of the nonlinear bending theory for plates

Stefan Neukamm and Heiner Olbermann

Abstract

We carry out the spatially periodic homogenization of Kirchhoff’s plate theory. 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 Kirchhoff’s 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.

Received:
Apr 15, 2013
Published:
Apr 19, 2013
MSC Codes:
35B27, 74Q15, 74K20
Keywords:
homogenization, Kirchhoff plate theory, two-scale convergence, nonlinear differential constraint

Related publications

inJournal
2015 Repository Open Access
Stefan Neukamm and Heiner Olbermann

Homogenization of the nonlinear bending theory for plates

In: Calculus of variations and partial differential equations, 53 (2015) 3, pp. 719-753