A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence
by
Gero Friesecke, Richard D. James & Stefan Müller
in
Arch. Rational Mech. Anal., 180, 183–236, 2006.
Category: Journal Article
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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 ∼ hβ, where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼ hα. 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 toα > 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)3 → R 3, the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations.
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