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

Gero Friesecke, Richard D. James, Stefan Müller

Research output: Contribution to journalArticle

237 Scopus citations

Abstract

We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume ∼ h β , where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼ h α . Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Kármán theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)33, the L 2 distance of ∇. v from a single rotation is bounded by a multiple of the L 2 distance from the set SO(3) of all rotations.

Original languageEnglish (US)
Pages (from-to)183-236
Number of pages54
JournalArchive For Rational Mechanics And Analysis
Volume180
Issue number2
DOIs
StatePublished - May 1 2006

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