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

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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.
Funding Information
  • TMR Networks (FMRX-CT98-0229)
  • AFOSR Multidisciplinary University Research Initiative (F49620-98-1-0433)
  • National Science Foundation (DMS-007-4043)
  • Office of Naval Research (MURI N000140110761)

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