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MiS Preprint
5/2002

A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity

Gero Friesecke, Richard D. James and Stefan Müller

Abstract

The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a $\Gamma$-limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps $v \in W^{1,2}(U, \mathbb{R}^n),\ U \subset \mathbb{R}^n$ . We show that the $L^2$ distance of $\nabla v$ from a single rotation matrix is bounded by a multiple of the $L^2$ distance from the group SO(n) of all rotations.

Received:
Jan 10, 2002
Published:
Jan 10, 2002
MSC Codes:
74K20, 49J45, 53C24

Related publications

inJournal
2002 Repository Open Access
Gero Friesecke, Richard D. James and Stefan Müller

A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity

In: Communications on pure and applied mathematics, 55 (2002) 11, pp. 1461-1506