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 (Arch. Ration. Mech. Anal. 4:145–152, 1959) and is based on the two-energies principle of Prager and Synge. This was half a century ago. We will derive the Kirchhoff–Love model based on Morgenstern’s ideas in a rigorous way (including the proper treatment of boundary conditions). Our derivation provides insights (a) into the relation of the (1,1,0)-model with the (1,1,2)-model which differs by a quadratic term in the ansatz for the third component of the displacement field and (b) into the rôle of the shear correction factor. A further advantage of the approach by the two-energies principle is that the extension to the Reissner–Mindlin plate model becomes very transparent and easy. Our study includes plates with reentrant corners with any interior opening angle <2π.