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1985 | Book

Essential Notions and Propositions of Functional Analysis Read chapter Read first chapter

Inequality Problems in Mechanics and Applications

Convex and Nonconvex Energy Functions

Author: P. D. Panagiotopoulos

Publisher: Birkhäuser Boston

Included in: Professional Book Archive

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About this book

In a remarkably short time, the field of inequality problems has seen considerable development in mathematics and theoretical mechanics. Applied mechanics and the engineering sciences have also benefitted from these developments in that open problems have been treated and entirely new classes of problems have been formulated and solved. This book is an outgrowth of seven years of seminars and courses on inequality problems in mechanics for a variety of audiences in the Technical University of Aachen, the Aristotle University of Thessaloniki, the University of Hamburg and the Technical University of Milan. The book is intended for a variety of readers, mathematicians and engineers alike, as is detailed in the Guidelines for the Reader. It goes without saying that the work of G. Fichera, J. L. Lions, G. Maier, J. J. Moreau in originating and developing the theory of inequality problems has considerably influenced the present book. I also wish to acknowledge the helpful comments received from C. Bisbos, J. Haslinger, B. Kawohl, H. Matthies, H. O. May, D. Talaslidis and B. Werner. Credit is also due to G. Kyriakopoulos and T. Mandopoulou for their exceptionally diligent work in the preparation of the fmal figures. Many thanks are also due to T. Finnegan and J. Gateley for their friendly assistance from the linguistic standpoint. I would also like to thank my editors in Birkhiiuser Verlag for their cooperation, and all those who helped in the preparation of the manuscript.

Table of Contents

Frontmatter

Introductory Topics

Frontmatter
Chapter 1. Essential Notions and Propositions of Functional Analysis
Abstract
The aim of Chapter 1 is to provide some notions and propositions of functional analysis which will be necessary in the next chapters for the study of inequality problems in mechanics. Commencing with the notion of topological vector spaces and the corresponding notion of duality, we give some properties of certain function spaces. Particular attention is paid to Sobolev spaces and spaces of functions of bounded deformation for which the trace theorems and some imbedding properties are given. Korn’s inequalities and the Green-Gauss theorem are also presented. Elements of the theory of vector-valued functions and distributions as well as of differential calculus close this chapter.
P. D. Panagiotopoulos
Chapter 2. Elements of Convex Analysis
Abstract
The purpose of this chapter is to provide some notions and fundamental results of convex analysis which will be used throughout this book. Starting with the notion of convexity, some propositions on convex and lower semi-continuous functionals as well as on the minimization of functionals on convex sets are given. The notion of subdifferential is introduced and its relation to one-sided Gâteaux-differentiability is illustrated. There follows the definition of the conjugate functional and some propositions on the conjugacy operation. The chapter closes with some elements of the theory of maximal monotone operators. Our attention is concentrated on the maximal monotone operators on ℝ as they allow a compact formulation of general classes of variational inequalities. In the present chapter, the relation between convex analysis and the theory of variational inequalities becomes clear.
P. D. Panagiotopoulos

Inequality Problems

Frontmatter
Chapter 3. Variational Inequalities and Superpotentials
Abstract
The aim of this chapter is to explain the origins of the inequality problems encountered in mechanics. To do this we introduce certain notions of convex analysis into mechanics; more precisely, we consider material laws and boundary conditions involving subdifferentials of convex functionals.
P. D. Panagiotopoulos
Chapter 4. Variational Inequalities and Multivalued Convex and Nonconvex Problems in Mechanics
Abstract
In the first section of this chapter two general types of variational inequalities are studied and a general scheme is proposed for the derivation of variational inequality “principles”. The second section deals with the study of problems of coexistent phases, using the notion of subdifferentiability. Also minimum problems are derived for Gibbsian states. In the third section we attempt to generalize the notion of superpotential for nonconvex energy functionals through the concepts of the generalized gradient of Clarke and of the derivate container of Warga. Hemivariational inequalities are derived, substationarity properties are proved and certain classes of material laws and boundary conditions leading to such problems are discussed. This chapter closes with a section concerning the study of inequality problems in terms of multivalued integral equations.
P. D. Panagiotopoulos
Chapter 5. Friction Problems in the Theory of Elasticity
Abstract
In this chapter we shall concentrate our attention on two typical inequality problems, one static and one dynamic, formulated within the framework of linear elasticity. They are the friction B.V.P.s, which were first studied by G. Duvaut and J. L. Lions [83] [84]. Because of the boundary conditions expressing the friction phenomenon, variational inequalities arise both in the static and dynamic cases. After deriving these variational inequalities, we shall study the existence and the uniqueness of the solution in an appropriate functional setting. For the static case, the propositions of minimum potential and complementary energy are derived and their duality is proved. Finally, some other types of friction B.V.P.s are also considered, and the corresponding variational inequalities are derived.
P. D. Panagiotopoulos
Chapter 6. Subdifferential Constitutive Laws and Boundary Conditions
Abstract
In this chapter, the following three problems are studied regarding a deformable body undergoing small deformations.
P. D. Panagiotopoulos
Chapter 7. Inequality Problems in the Theory of Thin Elastic Plates
Abstract
This chapter is devoted to the study of certain classes of static and dynamic inequality problems pertaining to the theory of thin elastic plates. Here we consider the model proposed by von Kármán for plates undergoing large deflections relative to their thickness. Kirchhoff’s theory for plates with small deflections constitutes a special case of von Kármán’s theory. The latter permits a rational treatment of the problem of plate buckling and therefore is preferred in this chapter. The first section, in which some static unilateral problems are formulated and studied, is based on papers by G. Duvaut and J. L. Lions [87], J. Franču [102], P. Hess [140], O. John [154], J. Naumann [225], M. Potier-Ferry [268], [269] among others. In the second section, to study the buckling problem, the corresponding eigenvalue problem is discussed. With reference to this problem, we give some general propositions concerning the eigenvalue problem for variational inequalities. This is a new area, only recently developed, and for this reason many questions are still open; we rely herein on the papers of A. F. Abeasis, J. P. Dias and A. Lopes-Pinto [1], M. Berger [22], M. Berger and P. Fife [23], A. Cimetière [43], [44], C. Do [71], [72], [73], [74], M. Kučera, J. Nečas and J. Souček [165], E. Miersemann [203], J. Naumann and H. Wenk [226], and M. Potier-Ferry [270], [271]. The third section concerns dynamic unilateral problems of von Kármán plates and is based on [88] and [248].
P. D. Panagiotopoulos
Chapter 8. Variational and Hemivariational Inequalities in Linear Thermoelasticity
Abstract
This chapter deals with the study of certain unilateral B.V.P.s which are formulated in linear thermoelasticity. It is assumed that on the boundary of the body under consideration subdifferential relations hold, first between temperature and the heat flux vector and second between velocity and the stress vector. After deriving the corresponding variational inequalities, we prove some propositions on the existence and uniqueness of the solution. These two problems were first studied by G. Duvaut and J. L. Lions [85]. Finally, by the method of Sec. 4.1.3, some general variational inequalities are formulated. They stem from the assumption that the linear elasticity law and the well-known Fourier’s law of heat conduction are replaced by nonlinear laws described by superpotentials. This chapter closes with the study of certain hemivariational inequalities.
P. D. Panagiotopoulos
Chapter 9. Variational Inequalities in the Theory of Plasticity and Viscoplasticity
Abstract
In this chapter, we first study B.V.P.s arising in the framework of the geometrically linear theory of elastic viscoplastic and elastic perfectly plastic solids, and second B.V.P.s concerning rigid viscoplastic flows. We formulate the problems in terms of variational inequalities, and then discuss the existence and uniqueness of their solutions.
P. D. Panagiotopoulos

Numerical Applications

Frontmatter
Chapter 10. The Numerical Treatment of Static Inequality Problems
Abstract
The aim of the two final chapters is to illustrate certain applications in the engineering sciences of the theory which has been developed up to this point. This chapter deals with the numerical solution of some static inequality problems.
P. D. Panagiotopoulos
Chapter 11. Incremental and Dynamic Inequality Problems
Abstract
This chapter is devoted to the numerical treatment of incremental and dynamic inequality problems. The first Section deals with the elastoplastic analysis of cable structures, where we assume that the cables may become slack. With respect to this problem, we illustrate the application of multilevel decomposition techniques to the analysis of inequality problems with many unknowns. The second Section contains the incremental elastoplastic analysis of structures presenting geometric nonlinearities and physical destabilizing effects. Within every load increment a variational inequality is formulated which gives rise to a minimization problem. The last Section concerns the dynamic unilateral contact problem. The resulting variational inequality is discretized with respect to time, and within every time step a minimization problem arises which is solved by an appropriate optimization alogrithm.
P. D. Panagiotopoulos
Backmatter
Metadata
Title
Inequality Problems in Mechanics and Applications
Author
P. D. Panagiotopoulos
Copyright Year
1985
Publisher
Birkhäuser Boston
Electronic ISBN
978-1-4612-5152-1
Print ISBN
978-0-8176-3094-2
DOI
https://doi.org/10.1007/978-1-4612-5152-1