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A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity
Author(s) -
Friesecke Gero,
James Richard D.,
Müller Stefan
Publication year - 2002
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10048
Subject(s) - mathematics , rigidity (electromagnetism) , nonlinear elasticity , nonlinear system , mathematical analysis , elasticity (physics) , bounded function , curvature , rotation matrix , geometry , physics , quantum mechanics , thermodynamics
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 → ℝ n , U ⊂ ℝ n . We show that the L 2 ‐distance of ∇ v from a single rotation matrix is bounded by a multiple of the L 2 ‐distance from the group SO( n ) of all rotations. © 2002 Wiley Periodicals, Inc.