On the justification of plate models

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.