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A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity
Article first published online: 23 AUG 2002
DOI: 10.1002/cpa.10048
Copyright © 2002 Wiley Periodicals, Inc.
Issue
1097-0312/asset/cover.gif?v=1&s=27e2b2132052ff78aa82eddfab7608281350adb9 "Communications on Pure and Applied Mathematics")
Communications on Pure and Applied Mathematics
Volume 55, Issue 11, pages 1461–1506, November 2002
Additional Information
How to Cite
Friesecke, G., James, R. D. and Müller, S. (2002), A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math., 55: 1461–1506. doi: 10.1002/cpa.10048
Publication History
- Issue published online: 23 AUG 2002
- Article first published online: 23 AUG 2002
- Manuscript Received: JAN 2002
Funded by
- TMR Networks. Grant Number: FMRX-CT98-0229
- AFOSR Multidisciplinary University Research Initiative. Grant Number: F49620-98-1-0433
- National Science Foundation. Grant Number: DMS-007-4043
- Office of Naval Research. Grant Number: MURI N000140110761
- Abstract
- References
- Cited By
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 Γ-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 : U → ℝn, U ⊂ ℝn. We show that the L2-distance of ∇v from a single rotation matrix is bounded by a multiple of the L2-distance from the group SO(n) of all rotations. © 2002 Wiley Periodicals, Inc.