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

Gero Friesecke, Richard D. James, and Stefan Müller
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Submission date: 10. Jan. 2002
Pages: 40
published in: Communications on pure and applied mathematics, 55 (2002) 11, p. 1461-1506 
DOI number (of the published article): 10.1002/cpa.10048
MSC-Numbers: 74K20, 49J45, 53C24
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Abstract:
The energy functional of nonlinear plate theory is a curvature functional for surfaces firstproposed on physical grounds by G. Kirchhoff in 1850. We show that itarises as a -limit of three-dimensional nonlinear elasticitytheory as the thickness of a plate goes to zero.A key ingredient in the proof is a sharp rigidity estimate for maps . We show that the distanceof from a single rotation matrixis bounded by a multiple of the distance from the group SO(n) of all rotations.