On the justification of Koiter's model for generalised membrane shells of the "first kind'' confined in a half-space

TL;DR

This paper rigorously justifies Koiter's model for generalised membrane shells of the first kind confined in a half-space, showing convergence from 3D to 2D models under asymptotic analysis.

Contribution

It provides a rigorous asymptotic analysis confirming Koiter's model's validity for confined membrane shells and identifies conditions for the equivalence of solution sets.

Findings

3D obstacle problem solutions converge to 2D variational inequalities

Koiter's model accurately describes the limit behavior of membrane shells

Conditions are identified for the solution sets of 2D models to coincide

Abstract

In this paper we justify Koiter's model for linearly elastic generalised membrane shells of the first kind subjected to remaining confined in a prescribed half-space. After showing that the confinement condition considered in this paper is in general stronger, under the validity of the Kirchhoff-Love assumptions, than the classical Signorini condition, we formulate the corresponding obstacle problem for a three-dimensional linearly elastic generalised membrane shell of the first kind, and we conduct a rigorous asymptotic analysis as the thickness approaches to zero on the unique solution for one such model. We show that the solution to the three-dimensional obstacle problem converges to the unique solution of a two-dimensional model consisting of a set of variational inequalities that are posed over the abstract completion of a non-empty, closed and convex set. We recall that the two-dimensional limit model obtained departing from Koiter's model is characterised by the same variational inequalities as the two-dimensional model obtained departing from the variational formulation of a three-dimensional linearly elastic generalised membrane shell subjected to remaining confined in a prescribed half-space, with the remarkable difference that the sets where solutions for these two-dimensional limit models are sought do not coincide, in general. In order to complete the justification of Koiter's model for linearly elastic generalised membrane shells of the first kind subjected to remaining confined in a half-space, we give sufficient conditions ensuring that the sets where solutions for these two-dimensional limit models are sought coincide.