A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity
Article first published online: 23 AUG 2002
Copyright © 2002 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 55, Issue 11, pages 1461–1506, November 2002
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
- Issue published online: 23 AUG 2002
- Article first published online: 23 AUG 2002
- Manuscript Received: JAN 2002
- 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
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.