Preprint 7/2005

A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence

Gero Friesecke, Richard D. James, and Stefan Müller

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Submission date: 19. Jan. 2005
Pages: 70
published in: Archive for rational mechanics and analysis, 180 (2006) 2, p. 183-236 
DOI number (of the published article): 10.1007/s00205-005-0400-7
Bibtex
MSC-Numbers: 74K20, 49J45
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Abstract:
We derive a hierarchy of plate models from three dimensional nonlinear elasticity by formula7-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume formula9, where h is the thickness of the plate. This is in turn related to the strength of the applied force formula13. Membrane theory, derived earlier by Le Dret and Raoult, corresponds to formula15, nonlinear bending theory to formula17, Föppl von Kármán theory to formula19, formula21 and linearized vK theory to formula23. Intermediate values of formula25 lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [31] that for maps formula27, the formula29 distance of formula31 from a single rotation is bounded by a multiple of the formula29 distance from the set SO(3) of all rotations.

23.06.2018, 02:11