On the Justification of Plate Models

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 rôle 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.