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A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

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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)3→ℝ3, 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.

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Authors and Affiliations

  1. Mathematical Institute, University of Warwick, Coventry, CV4 7AL, UK

    Gero Friesecke

  2. Centre for Mathematical Sciences, TU Munich, Boltzmannstr. 3, D - 85747, Garching, Germany

    Gero Friesecke

  3. Department of Aerospace Engineering and Mechanics, University of Minnesota, 107 Akerman Hall, Minneapolis, MN, 55455, USA

    Richard D. James

  4. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, D–04103, Leipzig, Germany

    Stefan Müller

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  1. Gero Friesecke
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  2. Richard D. James
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  3. Stefan Müller
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Friesecke, G., James, R. & Müller, S. A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence. Arch. Rational Mech. Anal. 180, 183–236 (2006). https://doi.org/10.1007/s00205-005-0400-7

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