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On the justification of plate models

by D. Braess and S. Sauter and Ch. Schwab

(Report number 2009-30)

Abstract
In this paper, we will consider the modelling of problems in linear elasticity on thin plates by the models of Kirchhoff--Love and Reissner--Mindlin. A fundamental investigation for the Kirchhoff plate goes back to Morgenstern [Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie. Arch. Rational Mech. Anal. 4, 145--152 (1959)] and is based on the two-energies principle of Prager and Synge. This was half a centenium ago. We will derive the Kirchhoff--Love model based on Morgenstern's ideas in a rigorous way (including the proper treatment of boundary conditions). It provides insights a) for the relation of the (1,1,0)-model with the (1,1,2)-model that differ by a quadratic term in the ansatz for the third component of the displacement field and b) for the role of the shear correction factor. A further advantage of the approach by the two-energy principle is that the extension to the Reissner--Mindlin plate model becomes very transparent and easy. Our study includes plates with reentrant corners.

Keywords:

@Techreport{BSS09_411,
  author = {D. Braess and S. Sauter and Ch. Schwab},
  title = {On the justification of plate models},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-30.pdf },
  year = {2009}
}

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