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