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 -convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume , where h is the thickness of the plate. This is in turn related to the strength of the applied force . Membrane theory, derived earlier by Le Dret and Raoult, corresponds to , nonlinear bending theory to , Föppl von Kármán theory to , and linearized vK theory to . Intermediate values of 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 , the distance of from a single rotation is bounded by a multiple of the distance from the set SO(3) of all rotations.