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 ReissnerMindlin. 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 ReissnerMindlin plate model becomes very transparent and easy. Our study includes plates with reentrant corners.

Some solutions of minimaxmax problems for the torsional displacements of rectangular plates

“…We refer to [, (2.40)] for the justification of these boundary conditions, see also [] for full details on how to derive them for the rectangular plate Ω under study. The behavior of rectangular plates subject to a variety of boundary conditions is studied in []. The solution u of –– represents the vertical displacement of the plate under the action of f and, since the boundary conditions – satisfy the complementing condition [, Lemma 4.2] so that elliptic regularity applies, u is a strong solution of whenever f belongs to suitable spaces.…”

Estimates of the modeling error for the Kirchhoff–Love plate model

“…Indeed, it is natural to define W such that the value of M 5 be minimal, what leads to a singularly perturbed variational problem which was used in [3]. Hence, our analysis shows that this singularly perturbed problem follows from the functional a posteriori estimate if we define the correction function W as the function that minimizes the majorant and select γ and λ in a special form.…”

“…We assume that f and F belong to L 2 (ω). The exact solution of the 3D elasticity problem in question is presented by the displacement vector u and the stress tensor σ (x) = (σ ij (x)) 3 i,j=1 that satisfy the equilibrium equation…”

“…Our goal is to show that majorant decreases with a faster rate. Our analysis of asymptotic properties is based upon the following result (see [1,3]). …”

“…In [3], generalizations of Lemma 5.4 have been established. They are related to cases where φ has lower regularity.…”