Archive for Rational Mechanics and Analysis

, Volume 180, Issue 2, pp 183–236

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


DOI: 10.1007/s00205-005-0400-7

Cite this article as:
Friesecke, G., James, R. & Müller, S. Arch. Rational Mech. Anal. (2006) 180: 183. doi:10.1007/s00205-005-0400-7


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 ∼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 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 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gero Friesecke
    • 1
    • 2
  • Richard D. James
    • 3
  • Stefan Müller
    • 4
  1. 1.Mathematical InstituteUniversity of WarwickCoventryUK
  2. 2.Centre for Mathematical SciencesTU MunichGarchingGermany
  3. 3.Department of Aerospace Engineering and MechanicsUniversity of MinnesotaMinneapolisUSA
  4. 4.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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