Obstacle Problems in Linearised Elasticity

Abstract

Unilateral contact problems arise in many fields such as medicine, engineering, biology and material science. For instance, the description of the motion inside the human heart of three valves of the aorta, which can be regarded as linearly elastic shells, is governed by a mathematical model built up in a way such that each valve remains confined in a certain portion of space without penetrating, or being penetrated, by the other two valves. Many practitioners succeeded in performing numerical simulations of contact problems but, so far, no general mathematical proofs have been given yet.

In this dissertation, we study static unilateral contact problems for linearly elastic shells confined in a half-space. The confinement condition gives rise to a system of variational inequalities posed over a nonempty closed convex subset of a suitable functional space where the solutions are to be sought. We will first study an obstacle problem for linearly elastic elliptic membrane shells. Secondly, we will establish the convergence of the numerical approximation scheme for the static obstacle problem for shallow shells, the model of which has already been justified by A. Leger and B. Miara.

The first objective of this dissertation is to establish the existence and uniqueness of the solution to the variational inequalities modelling the three-dimensional problem and the existence and uniqueness of the solution to the variational inequalities associated with the two-dimensional Koiter's model for linearly elastic elliptic membrane shells.

In the first chapter we identify and justify, by means of a rigorous asymptotic analysis as the thickness approaches zero, the corresponding "limit" two-dimensional variational problem, which takes, as expected, the form of a set of variational inequalities posed over a convex subset of a suitable Sobolev space.

In the second chapter we prove that the same limit model can also be obtained by performing a rigorous asymptotic analysis on the corresponding Koiter's model for linearly elastic elliptic membrane shells. Koiter's model, introduced for the first time by W. T. Koiter, possesses the remarkable feature of mimicking the behaviour of the solution to three-dimensional problem as the thickness approaches zero. For this reason, it is deemed adequate for modelling and approximating a variety of situations where three-dimensional equations would be otherwise inconvenient to use.

In the third chapter we present a sufficient condition that ensures the validity of the "density property", which plays a crucial role in the asymptotic analyses presented in Chapters 1 and 2.

In the fourth and last chapter we present an overview on shallow shells, for which similar asymptotic analyses have already been performed, and we present some new recent results concerning the convergence of the numerical scheme approximating the obstacle problem for static shallow shells.

In the final part of this dissertation, we provide an appendix, which is devoted to the proof of useful results in integration theory whose proofs are quite long and, customarily, omitted in the literature, and to the proof of some inverse trace inequalities that play a crucial role in the numerical analysis in Chapter 4.

The confinement condition considered here substantially departs from the Signorini condition usually considered in the existing literature, where only the "lower face" of the shell is required to remain above the "horizontal" plane. Such a confinement condition renders the asymptotic analysis considerably more difficult, however, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field.

In the study of the numerical scheme for the obstacle problem involving shallow shells, the main difficulty arises from the fact that we cannot assume the third component of the displacement field to be more regular than in H³(ω). This is the main difficulty which shall be overcome by means of ad hoc operators introduced in the late nineties by S. Brenner and her collaborators.

Date of Award	20 May 2019
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Philippe G CIARLET (Supervisor)