Table Of ContentUn iversitext
J. T. Oden J. N. Reddy
Variational Methods in
Theoretical Mechanics
Springer-Verlag
Berlin Heidelberg New York 1976
Prof. J. T. Oden
The University of Texas at Austin
Prof. J. N. Reddy
The University of Oklahoma
AMS Subject Classification (1970): 35 A 15; 49 H 05; 73 B 99; 73 E 99;
73F99; 73G05; 73K25; 76005.
ISBN-13: 978-3-540-07600-1 e-ISBN-13: 978-3-642-96312-4
001: 10.1007/978-3-642-96312-4
Library of Congress Cataloging in Publication Data. Oden, John Tinsley, 1936-Variational methods in theoretical
mechanics. (Universitext). Bibliography: p. Includes index. 1. Mechanics. 2. Continuum mechanics. 3. Calculus of variations.
I. Reddy, Junuthula Narasimha, 1945-II. Title. QA80S.03. 531'.01'5157. 75-45099.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for
other than private use, a fee is payable to the publisher. the amount of the fee to be determined by agreement with
the publisher.
© by Springer-Verlag Berlin Heidelberg 1976.
To Walker, Lee, Anita, and Anil
Preface
This is a textbook written for use in a graduate-level course for
students of mechanics and engineering science. It is designed to cover
the essential features of modern variational methods and to demonstrate
how a number of basic mathematical concepts can be used to produce a
unified theory of variational mechanics. As prerequisite to using this
text, we assume that the student is equipped with an introductory course
in functional analysis at a level roughly equal to that covered, for
example, in Kolmogorov and Fomin (Functional Analysis, Vol. I, Graylock,
Rochester, 1957) and possibly a graduate-level course in continuum mechanics.
Numerous references to supplementary material are listed throughout the
book.
We are indebted to Professor Jim Douglas of the University of Chicago,
who read an earlier version of the manuscript and whose detailed suggestions
were extremely helpful in preparing the final draft. He also gratefully
acknowledge that much of our own research work on variational theory was
supported by the U.S. Air Force Office of Scientific Research. He are
indebted to Mr. Ming-Goei Sheu for help in proofreading. Finally, we
wish to express thanks to Mrs. Marilyn Gude for her excellent and pains
taking job of typing the manuscript.
J. T. ODEN
J. N. REDDY
Table of Contents
PREFACE
1. INTRODUCTION
1.1 The Role of Variational Theory in Mechanics. 1
1.2 Some Historical Comments ......... . 2
1.3 Plan of Study ............... . 5
2. MATHEMATICAL FOUNDATIONS OF CLASSICAL VARIATIONAL THEORY 7
2.1 Introduction . . . . . . . . 7
2.2 Nonlinear Operators ..... 9
2.3 Differentiation of Operators 17
2.4 Mean Value Theorems .... . 25
2.5 Taylor Formulas ...... . 28
2.6 Gradients of Functionals .. 32
2.7 Minimization of Functionals. 35
2.8 Convex Functionals ..... 39
2.9 Potential Operators and the Inverse Problem. 41
2.10 Sobo 1e v Spaces . . . . . . 45
3. MECHANICS OF CONTINUA- A BRIEF REVIEW. 52
3.1 Introduction .•........ 52
3.2 Kinematics .......... . 53
3.3 Stress and the Mechanical Laws of Balance. 57
The Principle of Conservation of Mass .... 59
The Principle of Balance of Linear Momentum. 60
The Principle of Balance of Angular Momentum 62
3.4 Thermodynamic Principles .... 63
The Principle of Conservation of Energy. 63
The Clausius-Duhem Inequality. 66
3.5 Constitutive Theory ..... . 68
Rules of Constitutive Theory . 68
Special Forms of Constitutive Equations. 72
3.6 Jump Conditions for Discontinuous Fields 77
IX
4. COMPLEMENTARY AND DUAL VARIATIONAL PRINCIPLES
IN MECHANICS . • • .. .. . . 82
4.1 Introduction.. .•.•.. .. 82
4.2 Boundary Conditions and Green's Formulas 87
94
4.3 Examples from Mechanics and Physics.
4.4 The Fourteen Fundamental Complementary-Dual
Principles . .•... ... 104
4.5 Some Complementary-Dual Variational Principles of
Mechanics and Physics . .. . .. 114
4.6 Legendre Transformations . . • . 123
4.7 Generalized Hamiltonian Theory.. .. 128
4.8 Upper and Lower Bounds and Existence Theory. 131
4.9 Lagrange Multipliers. .• .. 136
5. VARIATIONAL PRINCIPLES IN CONTINUUM MECHANICS. 139
5.1 Introduction.. . . .. . ... 139
5.2 Some Preliminary Properties and Lemmas. 140
5.3 General Variational Principles for Linear Theory
of Dynamic Viscoelasticity ....•.... 143
5.4 Gurtin's Variational Principles for the Linear
Theory of Dynamic Viscoelasticity. • 153
5.5 Variational Principles for Linear Coupled
Dynamic Thermoviscoelasticity. 158
Linear (coupled) Thermoe1asticity. . 161
5.6 Variational Principles in Linear E1astodynamics. 162
5.7 Variational Principles for Linear Piezoelectric
E1astodynamic Problems . . .. .•.• 169
5.8 Variational Principles for Hypere1astic Materials. 173
Fi nite Elasticity. . . 175
Quasi-Static Problems .• . . 182
5.9 Variational Principles in the Flow Theory of
Plasticity . . . • . . .. .. .• 184
5.10 Variational Principles for a" Large Displacement
Theory of E1astop1asticity . . . • • • •. . 186
5.11 Variational Principles in Heat Conduction. • . 189
5.12 Biot's Quasi-Variational Principle in Heat Transfer. 191
5.13 Some Variational Principles in Fluid Mechanics and
Magnetohydrodynamics 195
Non-Newtonian Fluids. . 197
Perfect Fluids ..... . . . . . . . . 198
An Alternate Principle for Invicid Flow. 199
Magnetohydrodynamics 200
x
5.14 Variational Principles for Discontinuous Fields. 202
Hybrid Vari ati ona 1 Princi pl es. • • • . • 208
6. VARIATIONAL BOUNDARY-VALUE PROBLEMS, MONOTONE OPERATORS,
AND VARIATIONAL INEQUALITIES • • • • . . . • • • • • . • 215
6.1 Direct Variational Methods. . . . . . • . . . . . . 215
6.2 Linear Elliptic Variational Boundary-Value Problems. 216
Regularity • . . • • . • . . • • • . . . • • • • . . 222
6.3 The Lax-Milgram-Babuska Theorem. • . . . • . . . . . 223
6.4 Existence Theory in Linear Incompressible Elasticity 227
6.5 Monotone Operators • . . . • . . • . • . 235
6.6 Existence Theory in Nonlinear Elasticity 248
6.7 Variational Inequalities. . • 253
6.8 Applications in Mechanics. ". • 258
7. VARIATIONAL METHODS OF APPROXIMATION 262
7.1 Introduction .•..........•... 262
7.2 Several Variational Methods of Approximation 262
Galerkin's Method .•..•.. 265
The Rayleigh-Ritz Method ... 266
Semidiscrete Galerkin Methods. 266
Methods of Weighted Residuals. 267
Least Square Methods • . . • • 268
Collocation Methods ....•• 268
Funct i ona 1 Imbedd i ngs. . . • • 269
7.3 Finite-Element Approximations. . 270
7.4 Finite-Element Interpolation Theory. . • • . • . 273
7.5 Existence and Uniqueness of Galerkin Approximations. 281
7.6 Convergence and Accuracy of Finite-Element Galerkin
Approximations 285
REFERENCES . . . . . • . . . • . . . . • • . . . . • . . . . . 291
1. Introduction
1.1 The Role of Variational Theory in Mechanics. Variational principles
have always played an important role in theoretical mechanics. To most
students of mechanics, they provide alternate approaches to direct appli
cations of local physical laws. The principle of minimum potential energy,
for example, can be regarded as a substitute to the equations of equili
brium of elastic bodies, as well as a basis for the study of stability.
Hamilton's principle can be used in lieu of the equations governing dynami
cal systems, and the variational forms presented by Biot displace certain
equations in linear continuum thermodynamics.
However, the importance of variational statements of physical laws,
in the general sense of these terms, goes far beyond their use as simply
an alternate to other formulations. In fact, variational or weak forms
of the laws of continuum physics may be the only natural and rigorously
correct way to think of them. The idea that they are only equivalent
substitutes for local statements of these laws is an all too common mis
conception. The fundamental principles of mechanics are global principles;
they may require local integrability of certain fields, but not local
differentiability. Hence, we can generate local forms of these laws only
if we endow all physical field quantities with a possibly unnatural degree
of smoothness. This done, we rule out all traces of point sources, dis
continuities, and their derivatives, and we restrict ourselves to a rather
2
unrealistic view of the universe. While all sufficiently smooth fields
lead to meaningful variational forms, the converse is not true: there
exist physical phenomena which can be adequately modeled mathematically
only in a variational setting; they are nonsense when viewed locally.
Aside from this basic observation, the use of variational statements
of physical laws makes it possible to concentrate in a single functional
all of the intrinsic features of the problem at hand: the governing equations,
the boundary conditions, initial conditions, conditions of constraint, even
jump conditions. Variational formulations can serve not only to unify
diverse fields but also to suggest new theories, and they provide a
powerful means for studying the existence of solutions to partial differ
ential equations. Finally, and perhaps most importantly, variational methods
provide a natural means for approximation; they are at the heart of the
most powerful approximate methods in use in mechanics, and in many cases
they can be used to establish upper and/or lower bounds on approximate
solutions.
1.2 Some Historical Comments. In modern times, the term "variational
theory" applies to a wide spectrum of concepts having to do with weak,
generalized, or direct variational formulations of boundary- and initial
value problems. Still, many of the essential features of variational
methods remain the same as they were over 200 years ago when the first
notions of variational calculus began to be formulated.
Actually, the most primitive ideas of variational theory are present
in Aristotle's writings on "virtual velocities" in 300 B.C., to be revived
again by Galileo in the sixteenth century and finally to be formulated
into a principle of virtual work by John Bernoulli in 1717. The development
