Variational Methods in Theoretical Mechanics 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 7 2. MATHEMATICAL FOUNDATIONS OF CLASSICAL VARIATIONAL THEORY 7 2.1 Introduction . . . . . . . . Tale of Contents 1. Introduction 1.1 The Role of Variational Theory in Mechanics 1.2 Some Historical Comments 1.3 Plan of Study 2. Mathematical Foundations of Classical Variational Theory 2.1 Introduction 2.2 Nonlinear Operators 2.3 Differentiation of Operators 2.4 Mean Value Theorems 2.5 Taylor Formulas 2.6 Gradients of Functionals 2.7 Minimization of Functionals 2.8 Convex Functionals 2.9 Potential Operators and the Inverse Problem 2.10 Sobolev Spaces 3. Mechanics of Continua — A Brief Review 3.1 Introduction 3.2 Kinematics 3.3 Stress and the Mechanical Laws of Balance 3.4 Thermodynamic Principles 3.5 Constitutive Theory 3.6 Jump Conditions for Discontinuous Fields 4. Complementary and Dual Variational Principles in Mechanics 4.1 Introduction 4.2 Boundary Conditions and Green’s Formulas 4.3 Examples from Mechanics and Physics 4.4 The Fourteen Fundamental Complementary Dual Principles 4.5 Some Complementary Dual Variational Principles of Mechanics and Physics 4.6 Legendre Transformations 4.7 Generalized Hamiltonian Theory 4.8 Upper and Lower Bounds and Existence Theory 4.9 Lagrange Multipliers 5. Variational Principles in Continuum Mechanics 5.1 Introduction 5.2 Some Preliminary Properties and Lemmas 5.3 General Variational Principles for Linear Theory of Dynamic Viscoelasticity 5.4 Gurtin’s Variational Principles for the Linear Theory of Dynamic Viscoelasticity 5.5 Variational Principles for Linear Coupled Dynamic Thermoviscoelasticity 5.6 Variational Principles in Linear Elastodynamics 5.7 Variational Principles for Linear Piezoelectric Elastodynamic Problems 5.8 Variational Principles for Hyperelastic Materials 5.9 Variational Principles in the Flow Theory of Plasticity 5.10 Variational Principles for a Large Displacement Theory of Elastoplasticity 5.11 Variational Principles in Heat Conduction 5.12 Biot’s Quasi Variational Principle in Heat Transfer 5.13 Some Variational Principles in Fluid Mechanics and Magnetohydrodynamics 5.14 Variational Principles for Discontinuous Fields 6. Variational Boundary Value Problems Monotone Operators and Variational Inequalities 6.1 Direct Variational Methods 6.2 Linear Elliptic Variational Boundary Value Problems 6.3 The Lax Milgram Babuska Theorem 6.4 Existence Theory in Linear Incompressible Elasticity 6.5 Monotone Operators 6.6 Existence Theory in Nonlinear Elasticity 6.7 Variational Inequalities 6.8 Applications in Mechanics 7. Variational Methods of Approximation 7.1 Introduction 7.2 Several Variational Methods of Approximation 7.3 Finite Element Approximations 7.4 Finite Element Interpolation Theory 7.5 Existence and Uniqueness of Galerkin Approximations 7.6 Convergence and Accuracy of Finite Element Galerkin Approximations References.