Table Of ContentSTUDIES IN APPLIED MECHANICS
1. Mechanics and Strength of Materials (Skalmierski)
2. Nonlinear Differential Equations (Fuöik and Kufner)
3. Mathematical Theory of Elastic and Elastico-Plastic Bodies
An Introduction (Neöasand Hlavaöek)
4. Variational, Incremental and Energy Methods
in Solid Mechanics and Shell Theory (Mason)
5. Mechanics of Structured Media, Parts A and B (Selvadurai, Editor)
6. Mechanics of Material Behavior (Dvorak and Shield, Editors)
7. Mechanics of Granular Materials: New Models and Constitutive
Relations (Jenkins and Satake, Editors)
8. Probabilistic Approach to Mechanisms (Sandier)
9. Methods of Functional Analysis for Application in Solid
Mechanics (Mason)
STUDIES IN APPLIED MECHANICS 9
M e t h o ds of Functional
Analysis f or Application
in Solid Mechanics
Jayme Mason
Federal and Catholic Universities, Rio de Janeiro, Brazil
Amsterdam — Oxford — New York — Tokyo 1985
ELSEVIER SCIENCE PUBLISHERS B.V.
1 Molenwerf,
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC.
52, Vanderbilt Avenue
New York, NY 10017
ISBN 0-444^42436-9 (Vol. 9)
ISBN 0-444^1758-3 (Series)
© Elsevier Science Publishers B.V., 1985
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or
transmitted in any form or by any means, electronic, mechanical, photocopying, recording or other
wise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./Science &
Technology Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands.
Special regulations for readers in the USA — This publication has been registered with the Copyright
Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about
conditions under which photocopies of parts of this publication may be made in the USA. All other
copyright questions, including photocopying outside of the USA, should be referred to the publisher.
Printed in The Netherlands
V
Lo giorno se ne andava, e I'aere bruno
toglieva li animai ehe sono in terra
dalle fatiche loro; e io sol uno
m'apparechiava a sostener la guerra
si del cammino e si della pietate,
ehe ritrarrä la mente ehe non erra.
0 Muse, o alto ingegno, or m'aiutate;
o mente ehe scrivesti cio ch'io vidi,
qui si parrä la tua nobilitate.
INFERNO, Canto II
0 insensata cura de# mortali
quanto son difettivi sillogismi
quei ehe ti fanno in basso batter ralli!
PARADISO, Canto XI
Dante Alighieri — Divina Commedia
Das schönste Glück des denkenden Menschen ist,
das Erforschliche erforscht zu haben und das
Unerforschliche ruhig zu verehren.
Goethe - Natur und Wissenschaft
VI
IMPORTANT MATHEMATICAL SYMBOLS AND NOTATIONS
For reference purposes, some important symbols and notations used in the book
are summarized:
"is an element of"; "is in"; "belongs to"
"is not an element of"; "is not in"; "does not belong to"
"for each" or "for every"
"there exists"
"if and only if"
A <=B, "set inclusion"
A[J B, "set union"
A Π B, "set intersection"
"approaches", "converges to"; f: X*Y' = "function from X into
(onto) Y"
"implication"
"implication in two directions"
"real numbers"
"positive real numbers"
"Euclidean space of dimension n"
"modulus", "norm"
"norm notations
"supremum", "infinum"
weak convergence
"duality pairing", "outer product"
"inner product"
"dual of U", "conjugate of U "
"space of continuous functions"
"space of n-times continuously differentiable functions"
"identity"
VII
PREFACE
This book is an attempt to explain some important and uptodate mathematical
topics to engineers interested in research and recent developments in Solid
Mechanics. It was not written for professional mathematicians, but for engineers
interested in Mathematics, since the author himself belongs in this last
category. This is not a work of research, it is a work of synthesis.
There is little doubt that, in the near future, engineers working in the
fields of Solid Mechanics and Advanced Structures, will be forced to handle
abstract spaces with the same ease that they now handle tensor symbolism.
All those colleagues who have been long enough in the field, surely remember
the arguments advanced against such a symbolism by opponents, a couple of
decades ago. It was the natural reaction of all those who wanted to escape the
trouble of having to learn something they had done without until then.
Recently, a mathematical area which belonged hitherto to the province of
professional mathematicians becomes of great value in obtaining a deeper
knowledge of well known problems, as well as an essential tool for the solution
of many new questions in Solid Mechanics, posed by technological advances. This
area includes among other topics, classical Functional Analysis, the Theory of
Variational Methods for Linear and Nonlinear Operators and Convex Analysis.
Among the concrete problems that have favoured such a development, we may
quote the one for establishing acceptable theoretical foundations for the Finite
Element method, as well as the necessity of solving a host of new problems
termed "unilateral" and involving variational inequalities.
The number of problems demanding the use of such new tools is increasing
every day. The time will come when a research engineer lacking a minimum
knowledge in these areas will be seriously hampered in his pursuits or in the
study of a considerable body of literature.
If anyone with reasonable prerequisites could master tensor calculus in a
matter of a year, a working knowledge of Functional Analysis cannot be gained
without several years of painstaking efforts, as I can testify from my own
experience.
The subject of Functional Analysis, with its abstract character and sweeping
generalizations is not easy for untrained minds to master, since it departs
considerably from the usual offhand engineering approach to mathematics, but
once one succeeds in learning some of it, the dividends are very rewarding.
VIII
The existing literature is mainly written by experts for experts and texts
oriented in the interest of engineering applications are still exceptions. Thus,
the object of this book is to provide engineering scientists and graduate
students with a general introduction to the methods of Functional Analysis, as
it is applied in Solid and Computational Mechanics.
The choice of the topics has been directed to make the book self-contained,
covering as much material as possible, with a good deal of illustrative examples
and some topics devoted to applications in Elasticity, Plasticity, variational
principles, inequalities and eigenvalue problems. A great deal of effort has
been put towards providing motivation for abstract concepts.
The text sometimes goes into considerable detail. A professional
mathematician would consider many explanations or remarks as obvious, but I am
well aware of the help that some type of comment can be to engineers.
I believe that this book can be helpful to graduate students, researchers
and engineers interested in mathematical methods in Solid Mechanics. I feel
sure that it can decrease considerably the time necessary to become familiar
with many new principles and methods scattered throughout the specialized
1iterature.
A word of caution should also be uttered. It has not been intended to produce
complete mathematical proofs in all instances, but to emphasize the most
relevant aspects for those interested in general ideas and applications.
A reasonable list of literature has been appended at the end, with comments
throughout the text. Among the sources that have most influenced the shaping and
the contents of this book, I would like to include the treatise on Nonlinear
Functional Analysis by E.Zeidler, the books by Vainberg, Necas, Rektorys, J.L.
Lions and Ekeland-Temam. In the field of applications, I am mostly indebted to
the standard treatise on variational inequalities by Duvaut and Lions and to
the several books and contributions by Prof. J.T. Oden and his associates.
To conclude this preface, some acknowledgements must be made.
First of all, I owe a great debt to Dr. Ricardo S. Kubrusly for a thorough
and competent mathematical review of the whole manuscript, the full
responsibility of which, however, rests with the author.
To Maria Cristina Lima Raymundo goes my gratitude for the expert typing of
the camera-ready manuscript.
Editora Campus, from Rio de Janeiro, provided invaluable assistance in the
typographical shaping of the manuscript, in preparing headings and other
technical details. In conclusion, I shall consider myself well rewarded if this
book will be of good use to those who read it and shall be grateful for any
constructive criticism.
Rio de Janeiro, 1 985
J. Mason
XVII
INTRODUCTION
The main purpose of this monograph is to give a tentative exposition of
important mathematical concepts, techniques and methods which are being
increasingly used in Solid and Computational Mechanics.
In recent years, a major development has taken place in the field of Solid
and Computational Mechanics, with the increasing importance of methods based on
the techniques of Functional Analysis, in response to the needs of applications
in Finite Element theory.
The method of Finite Elements, invented by engineers in order to solve
structural problems, proved to be, in the hands of mathematicians, a much
stronger and general tool for the solution of variational boundary value
problems. A great deal of effort has been spent lately, in order to frame the
method into a consistent and rigorous mathematical foundation.
This effort has proved very fruitful, both in opening new avenues of approach
to the solution of many new problems, in exposing the paramount importance of
Functional Analysis and in dealing with the corresponding questions.
The use of the rigorous and general techniques of Functional Analysis opened
an enormous range of new possibilities to investigate difficult and essential
problems, such as those related to approximation, convergence and stability of
numerical calculations. Without this type of development, we would almost be
reduced to a condition of relying entirely on empirical and comparative analysis
based on intuition alone. Everybody knows that intuition may often be deceptive
and lead to pitfalls.
With a great deal of certainty, it can also be said that several new
problems, such as those of unilateral constraints, friction and contact forces
imposed on deformable bodies, as well as the investigation of the qualitative
behaviour of solutions in Solid Mechanics, could hardly be approached without
the use of tools from Functional Analysis.
Since the difficulties of numerical calculations for all sort of problems
have practically been removed by the extensive use of computers, our attention
must nowadays be directed to other equally important and essential issues. We
must inquire if the problem is well posed, if there exist solutions and, if a
solution exists, whether it is unique or not. We must also be sure that an
approximate solution converges to the exact solution and that we can predict
the amount of error involved.
These types of question and others cannot usually be answered by means of
XVIII
elementary and pedestrian methods. Their answer depends on the knowledge of
topics of advanced analysis, such as operator theory, topology, differentiable
manifolds, nonlinear analysis, etc., to be found only in the specialized
mathematical literature.
The mastery of these mathematical areas is difficult and requires years of
effort, as well as a complete change in the engineer's outlook, since the
emphasis of his training is concentrated on formal techniques and procedures,
without stressing too much the value of rigour and abstraction.
Professional mathematicians, on the other hand, are on the other extreme.
They are very keen on rigour but do not worry too much about providing motivation
for physical scientists, who would greatly benefit from their findings. Attempts
to convey to potential users the modern and powerful methods of abstract
analysis are rare and can be viewed as exceptions.
It is hoped with this publication to contribute, even on a modest scale, to
the awakening of the interest of engineers and researchers in the field of Solid
Mechanics and its computational aspects, and to the appreciation of the power,
beauty and effectiveness of some recent mathematical developments in their
field. We feel sure that an effort towards acquiring at least a basic knowledge
on these areas will be greatly rewarding and may encourage the pursuit of
further knowledge from more specialized and advanced literature.
With this aim in mind, the material selected endeavours to cover the largest
possible area of interest, without being either too technical to scare off
prospective readers or too superficial to be of any service. The form of
presentation is therefore a compromise between a minimum of acceptable rigour
and a maximum of motivation for abstract concepts and applications.
To accomplish these aims, we start by recalling some important concepts of
analysis, such as sets, Lebesgue integration, differentiation, functions, etc.
A clear grasp of these concepts will be essential and provides the basis for
the understanding of more abstract and general ideas of Functional Analysis.
Next we deal with basic topics and illustrations in classical Functional
Analysis, by presenting a short account of the theory of abstract spaces
(topological, Banach and Hubert spaces), linear operators, functionals,
distributions, spectral theory, projections, approximations, etc.
A chapter on variational boundary value problems follows. In it, we deal with
the formulation of strong and weak solutions, basic ideas on Lebesgue and
Sobolev spaces, embeddings of spaces and traces of operators, which are
essential for the precise formulation of solutions of differential equations
and variational boundary value problems of Solid Mechanics.
A chapter is dedicated to variational methods, including vitally important
topics such as functional derivatives, potential and monotone operators, theory
of duality and saddle points and the concepts of subdifferentiability, indicator
XIX
functions and others in the so-called discipline of Convex Analysis.
These concepts play a central role in both modern and classical applications
of variational methods to Elasticity and Plasticity and are complemented by an
additional chapter on variational inequalities. Variational inequalities are
the basic tool in solving several recent problems involving one-sided
(unilateral) constraints and friction forces on elastic and inelastic deformable
bodies.
A chapter on basic aspects of the discretization of variational problems,
with special emphasis, on the role of finite elements is a complement to the
mathematical part of the book.
The remaining chapters refer to applications of the above material to a
selection of problems of modern Solid Mechanics, including Variational
Principles and Finite Element models in Elasticity and various special
applications of variational methods and inequalities to Elasticity and
Plasticity.
The book is concluded with a brief account of the variational theory of
linear and nonlinear eigenvalue problems and its relationship to the Theory of
Elastic Stability.
