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2014 | Buch

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Variational Inequalities and Frictional Contact Problems

verfasst von: Anca Capatina

Verlag: Springer International Publishing

Buchreihe : Advances in Mechanics and Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Variational Inequalities and Frictional Contact Problems contains a carefully selected collection of results on elliptic and evolutionary quasi-variational inequalities including existence, uniqueness, regularity, dual formulations, numerical approximations and error estimates ones. By using a wide range of methods and arguments, the results are presented in a constructive way, with clarity and well justified proofs. This approach makes the subjects accessible to mathematicians and applied mathematicians. Moreover, this part of the book can be used as an excellent background for the investigation of more general classes of variational inequalities. The abstract variational inequalities considered in this book cover the variational formulations of many static and quasi-static contact problems. Based on these abstract results, in the last part of the book, certain static and quasi-static frictional contact problems in elasticity are studied in an almost exhaustive way. The readers will find a systematic and unified exposition on classical, variational and dual formulations, existence, uniqueness and regularity results, finite element approximations and related optimal control problems. This part of the book is an update of the Signorini problem with nonlocal Coulomb friction, a problem little studied and with few results in the literature. Also, in the quasi-static case, a control problem governed by a bilateral contact problem is studied. Despite the theoretical nature of the presented results, the book provides a background for the numerical analysis of contact problems.

The materials presented are accessible to both graduate/under graduate students and to researchers in applied mathematics, mechanics, and engineering. The obtained results have numerous applications in mechanics, engineering and geophysics. The book contains a good amount of original results which, in this unified form, cannot be found anywhere else.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Nowadays, the expression Variational Inequalities and Contact Problems can be considered as a syntagm since the variational methods have provided one of the most powerful techniques in the study of contact problems and, on the other hand, the variational formulations of the contact problems are, in most cases, variational inequalities.

Anca Capatina

Preliminaries

Frontmatter

Chapter 2. Spaces of Real-Valued Functions

Abstract

This chapter is a brief background on spaces of continuous functions and some Sobolev spaces including basic properties, embedding theorems and trace theorems. We recall some classical definitions and theorems of functional analysis which will be used throughout this work. These results are standard and so they are stated without proofs; details and proofs can be found in to many references. We only deal with real valued functions. We assume that the reader is familiar with the basic concepts of general topology and functional analysis.

Anca Capatina

Chapter 3. Spaces of Vector-Valued Functions

Abstract

In this chapter we will introduce additional tools which are fundamentals for the study of evolutionary problems studied later in this book. We consider here spaces of functions defined on a time interval \(I \subset \mathbb{R}\) with values into a Banach or Hilbert space X. The results are presented without proofs and for details we refer to the bibliography.

Anca Capatina

Variational Inequalities

Frontmatter

Chapter 4. Existence and Uniqueness Results

Abstract

This chapter deals with existence and uniqueness results for variational and quasi-variational inequalities. With the intention of focusing the differences among the proofs of results, we first consider elliptic variational inequalities of the first and second kind with linear and continuous operators in Hilbert space or monotone and hemicontinuous operators in Banach space. Next, we deal with elliptic quasi-variational inequalities involving monotone and hemicontinuous or potential operators. The last section concerns the study of a class of evolutionary quasi-variational inequalities. The results presented here will be applied, in the last part of the book, in the study of frictional contact problems.

Anca Capatina

Chapter 5. Some Properties of Solutions

Abstract

In this chapter one studies some properties of solutions of various variational inequalities of the first and second kind. We first consider a class of variational inequalities of the first kind and we emphasize a property of solutions, namely a maximum principle. We illustrate it by a problem which models the flow of fluids through a porous medium and an obstacle problem. Next, using the method of the translation, local and global regularity results of solutions of a class of variational inequalities of the second kind are derived. In the last part of this book, these results will be applied to a frictional contact problem.

Anca Capatina

Chapter 6. Dual Formulations of Quasi-Variational Inequalities

Abstract

The aim of this chapter is to derive dual formulations for quasi-variational inequalities. First, we present a brief background on convex analysis and, then, we recall the main ideas of the Mosco, Capuzzo-Dolcetta, and Matzeu duality theory in its form adapted for implicit variational inequalities.

Anca Capatina

Chapter 7. Approximations of Variational Inequalities

Abstract

This chapter is devoted to the discrete approximation of abstract elliptic and implicit evolutionary quasi-variational inequalities. Convergence results for internal approximations in space of elliptic quasi-variational inequalities together with a backward difference scheme in time of implicit evolutionary quasi-variational inequalities are proved. The results obtained in this chapter, representing generalizations of the approximations of variational inequalities of the first and second kinds, can be applied to a large variety of static and quasistatic contact problems, including unilateral and bilateral contact or normal compliance conditions with friction. In particular, static and quasistatic unilateral contact problems with nonlocal Coulomb friction in linear elasticity will be considered in last part of this book.

Anca Capatina

Contact Problems with Friction in Elasticity

Frontmatter

Chapter 8. Static Problems

Abstract

In this chapter we study, in an almost exhaustive way, a contact problem with friction which models the contact between an elastic body and a rigid foundation. The contact is modeled upon the well-known Signorini conditions and the friction is described by a nonlocal Coulomb friction law. The classical formulation of the model is described, and a variational formulation of the problem is derived. Under appropriate assumptions on the data, existence, uniqueness and regularity results are provided. We also derive two dual formulations of this problem. Numerical analysis is carried out and convergence results are proved. Finally, a related optimal control problem is studied.

Anca Capatina

Chapter 9. Quasistatic Problems

Abstract

This chapter deals with the study of quasistatic contact problems with a nonlocal Coulomb friction law. We first consider that the unilateral contact is modeled by the Signorini conditions. In this case, a variational formulation involves two inequalities with the simultaneous presence of the displacement field and of the velocity field. An existence result is provided and convergence results, for a space finite element approximation and an implicit time discretization scheme of this problem, are proved. Next, we study a boundary control problem related to a quasistatic bilateral contact problem with nonlocal Coulomb friction.

Anca Capatina

Backmatter

Titel: Variational Inequalities and Frictional Contact Problems
verfasst von: Anca Capatina
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-10163-7
Print ISBN: 978-3-319-10162-0
DOI: https://doi.org/10.1007/978-3-319-10163-7