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Variational Methods in Theoretical Mechanics

Name: Variational Methods in Theoretical Mechanics
ISBN: 978-3-540-11917-3

verfasst von: Professor John T. Oden, Professor Junuthula N. Reddy

Verlag: Springer Berlin Heidelberg

Buchreihe : Universitext

Enthalten in: Professional Book Archive

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Über dieses Buch

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 Some misprints and errors have been corrected, and some sections were deleted, which were felt to be out of date.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

Variational principles have always played an important role in theoretical mechanics. To most students of mechanics, they provide alternate approaches to direct applications of local physical laws. The principle of minimum potential energy, for example, can be regarded as a substitute to the equations of equilibrium 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 dynamical systems, and the variational forms presented by Biot displace certain equations in linear continuum thermodynamics.

John T. Oden, Junuthula N. Reddy

2. Mathematical Foundations of Classical Variational Theory

Abstract

Near the end of the eighteenth century, Lagrange observed that a function $u*\left( x \right)\, \in \,C_0^1\left[ {0\,,\,1} \right]$ which minimizes the functional $K:\,C_0^1\,\left[ {0,1} \right]\, \to \,R,$ given by

$${\text{K}}\left( {\text{u}} \right) = \smallint _0^1\,{\text{F}}\left( {{\text{x,u}}\left( {\text{x}} \right){\text{,u prime}}\left( {\text{x}} \right)} \right){\text{dx}}$$

where u′ = du/dx, also makes the bivariate functional δK(u,η) vanish, where

$$\begin{array}{*{20}{c}} {\delta K(u,\eta )}{ = \mathop {\lim }\limits_{\alpha \to {0^ + }} \frac{{\partial K\left( {u + \alpha \eta } \right)}}{{\partial \alpha }}} \\ {}{ = \int_0^1 {\left( {\frac{{\left( {\partial F\left( {x,u,u'} \right)} \right.}}{{\partial u}}\eta + \frac{{\partial F\left( {x,u,u'} \right)}}{{\partial u'}}\eta '} \right)dx} } \end{array} $$

and η is an arbitrary element in ${\text{C}}_{\text{0}}^{\text{1}}\left[ {{\text{0,1}}} \right]$.

John T. Oden, Junuthula N. Reddy

3. Mechanics of Continua-A Brief Review

Abstract

The objective of this chapter is to review the major principles of continuum mechanics and to record the governing equations for future reference. It is well-known that the equations of continuum mechanics fall into four basic categories: 1) kinematics, 2) kinetics and the mechanical balance laws, 3) thermodynamics, and 4) constitution. Kinematics, of course, is a study of the geometry of motion and deformation without regard to the agents which caused them, and by kinetics we mean the collection of mechanical ideas that includes the notion of force and stress, plus the axioms of physics which have to do with conservation of mass and balances of momenta. The thermodynamics of continua gives us global laws very important to the development of a variational theory for problems in mechanics, and the equations of constitution, of course, involve relations between the kinematic variables (or temperatures) and their duals, and define the constitution of the material under study.

John T. Oden, Junuthula N. Reddy

4. Complementary and Dual Variational Principles in Mechanics

Abstract

In this chapter, we consider a theory of complementary and dual variational principles associated with a large class of linear boundary- and initial-value problems of mechanics. The mathematical setting for the class of problems we wish to examine is the following abstract linear problem: find an element u of a Hilbert space U such that

$$\Lambda {\text{u = F, F}} \in {\text{u'}}$$

(4.1)

where Λ is a linear operator from U into its dual U′ which is given as a product of three linear operators,

$$\Lambda = {\text{A*EA}}$$

(4.2)

Here A is a continuous linear operator from U into a Hilbert space V, E is a canonical isomorphism mapping V onto its dual V′, and A* is the adjoint of A which, by definition, maps V ′ into U′.

John T. Oden, Junuthula N. Reddy

5. Variational Principles in Continuum Mechanics

Abstract

This chapter is devoted to the presentation of a representative collection of variational principles of continuum mechanics. Our principal objective is to present a general method for developing such principles, and to illustrate, by examples, its application to a number of different areas of solid and continuum mechanics.

John T. Oden, Junuthula N. Reddy

6. Variational Boundary-Value Problems, Monotone Operators, and Variational Inequalities

Abstract

If K(u) is a differentiate functional on a Banach space u, and if P: U → U’ is its gradient, we have shown that the abstract problem of finding u ∈ U such that

$$\left\langle {p\left( u \right),n} \right\rangle _u = 0 \forall \eta \in u $$

(6.1)

is equivalent to finding critical points of K(u).

John T. Oden, Junuthula N. Reddy

7. Variational Methods of Approximation

Abstract

In this chapter, we describe several of the more popular variational methods for the approximate solution of boundary-and initial-value problems. For ease in presentation, we confine our attention to linear problems, and most of what we present has to do with elliptic boundary-value problems. In particular, we discuss interpolation properties of finite element methods, existence and uniqueness of solutions to the approximate problem, and convergence and error estimates of finite element methods for linear elliptic problems. More detailed accounts of linear elliptic problems can be found in [47, 109, 135].

John T. Oden, Junuthula N. Reddy

Backmatter

Titel: Variational Methods in Theoretical Mechanics
verfasst von: Professor John T. Oden
Professor Junuthula N. Reddy
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-68811-9
Print ISBN: 978-3-540-11917-3
DOI: https://doi.org/10.1007/978-3-642-68811-9

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Mathematical Foundations of Classical Variational Theory

3. Mechanics of Continua-A Brief Review

4. Complementary and Dual Variational Principles in Mechanics

5. Variational Principles in Continuum Mechanics

6. Variational Boundary-Value Problems, Monotone Operators, and Variational Inequalities

7. Variational Methods of Approximation

Backmatter