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Γ-convergence for incompressible elastic plates

  • Sergio Conti
  • Georg Dolzmann
Article

Abstract

We derive a two-dimensional model for elastic plates as a Γ-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The resulting model describes plate bending, and is determined from the isochoric elastic moduli of the three-dimensional problem. Without the constraint of incompressibility, a plate theory was first derived by Friesecke et al. (Comm Pure Appl Math 55:1461–1506, 2002). We extend their result to the case of p growth at infinity with p ϵ [1, 2), and to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies the nonlinear constraint pointwise. One main ingredient is the density of smooth isometries in W 2,2 isometries, which was obtained by Pakzad (J Differ Geom 66:47–69, 2004) for convex domains and by Hornung (Comptes Rendus Mathematique 346:189–192, 2008) for piecewise C 1 domains.

Mathematics Subject Classification (2000)

74K20 49J45 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Sergio Conti
    • 1
  • Georg Dolzmann
    • 2
  1. 1.Fachbereich MathematikUniversität Duisburg-EssenDuisburgGermany
  2. 2.NWF I - MathematikUniversität RegensburgRegensburgGermany

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