Variational Methods in Theoretical Mechanics (Universitext)

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. We also gratefully acknowedge that much of our own research work on va ri at i ona 1 theory was supported by the U. S. Ai r Force Offi ce of Scientific Research. We 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 painstaking job of typing the manuscript. This revised edition contains only minor revisions of the first. Some misprints and errors have been corrected, and some sections were deleted, which were felt to be out of date.

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.- The Principle of Conservation of Mass.- The Principle of Balance of Linear Momentum.- The Principle of Balance of Angular Momentum.- 3.4 Thermodynamic Principles.- The Principle of Conservation of Energy.- The Clausius-Duhem Inequality.- 3.5 Constitutive Theory.- Rules of Constitutive Theory.- Special Forms of Constitutive Equations.- 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.- Linear (Coupled) Thermoelasticity.- 5.6 Variational Principles in Linear Elastodynamics.- 5.7 Variational Principles for Linear Piezoelectric Elastodynamic Problems.- 5.8 Variational Principles for Hyperelastic Materials.- Finite Elasticity.- Quasi-Static Problems.- 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.- Non-Newtonian Fluids.- Perfect Fluids.- An Alternate Principle for Invicid Flow.- Magnetohydrodynamics.- 5.14 Variational Principles for Discontinuous Fields.- Hybrid Variational Principles.- 6. Variational Boundary-Value Problems, Monotone Operators, and Variational Inequalities.- 6.1 Direct Variational Methods.- 6.2 Linear Elliptic Variational Boundary-Value Problems.- Regularity.- 6.3 The Lax-Milgram-Babuska Theorem.- 6.4 Existence Theory in Linear Incompressible Elasticity.- 6.5 Monotone Operators.- 6.6 Variational Inequalities.- 6.7 Applications in Mechanics.- 7. Variational Methods of Approximation.- 7.1 Introduction.- 7.2 Several Variational Methods of Approximation.- Galerkin’s Method.- The Rayleigh-Ritz Method.- Semidiscrete Galerkin Methods.- Methods of Weighted Residuals.- Least Square Approximations.- Collocation Methods.- Functional Imbeddings.- 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.