A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. (English) Zbl 1021.74024

Friesecke, Gero; James, Richard D.; Müller, Stefan

Commun. Pure Appl. Math. 55, No. 11, 1461-1506 (2002).

Summary: 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 \(\Gamma\)-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 \(v:U \to\mathbb{R}^n\), \(U\subset \mathbb{R}^n\). We show that the \(L^2\)-distance of \(\Delta v\) from a single rotation matrix is bounded by a multiple of the \(L^2\)-distance from from the group \(\text{SO}(n)\) of all rotations.

MSC:

74K20	Plates
74B20	Nonlinear elasticity
74G65	Energy minimization in equilibrium problems in solid mechanics
35Q72	Other PDE from mechanics (MSC2000)

Keywords:

geometric rigidity; nonlinear plate theory; energy functional; curvature functional; three-dimensional nonlinear elasticity theory; rigidity estimate; rotation matrix

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