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

2002/08/23 by Gero Friesecke, Richard D. James, Stefan Müller · 69 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #Elasticity and Material Modeling

paper · pdf · doi:10.1002/cpa.10048

Abstract

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 → 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.

Citations

• Convexity conditions and existence theorems in nonlinear elasticity

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