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Hamilton's Principle in Continuum Mechanics

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Hamilton's Principle in Continuum Mechanics Hamilton’s Principle in Continuum Mechanics A. Bedford University of Texas at Austin This document contains the complete text of the monograph published in 1985 by Pitman Publishing, Ltd. Copyright by A. Bedford. 1 Contents Preface 4 1 Mechanics of Systems of Particles 8 1.1 The First Problem of the Calculus of Variations . 8 1.2 Conservative Systems . 12 1.2.1 Hamilton’s principle . 12 1.2.2 Constraints.......................... 15 1.3 Nonconservative Systems . 17 2 Foundations of Continuum Mechanics 20 2.1 Mathematical Preliminaries . 20 2.1.1 Inner Product Spaces . 20 2.1.2 Linear Transformations . 22 2.1.3 Functions, Continuity, and Differentiability . 24 2.1.4 Fields and the Divergence Theorem . 25 2.2 Motion and Deformation . 27 2.3 The Comparison Motion . 32 2.4 The Fundamental Lemmas . 36 3 Mechanics of Continuous Media 39 3.1 The Classical Theories . 40 3.1.1 IdealFluids.......................... 40 3.1.2 ElasticSolids......................... 46 3.1.3 Inelastic Materials . 50 3.2 Theories with Microstructure . 54 3.2.1 Granular Solids . 54 3.2.2 Elastic Solids with Microstructure . 59 2 4 Mechanics of Mixtures 65 4.1 Motions and Comparison Motions of a Mixture . 66 4.1.1 Motions............................ 66 4.1.2 Comparison Fields . 68 4.2 Mixtures of Ideal Fluids . 71 4.2.1 Compressible Fluids . 71 4.2.2 Incompressible Fluids . 73 4.2.3 Fluids with Microinertia . 75 4.3 Mixture of an Ideal Fluid and an Elastic Solid . 83 4.4 A Theory of Mixtures with Microstructure . 86 5 Discontinuous Fields 91 5.1 Singular Surfaces . 91 5.2 An Ideal Fluid Containing a Singular Surface . 98 Acknowledgments 101 Bibliography 102 3 Preface The good of Hamilton is not in what he has done but in the work (not nearly half done) which he makes other people do. But to understand him you should look him up, and go through all kinds of sciences, then you go back to him, and he tells you a wrinkle. James Clerk Maxwell In 1808, when he was two years old, William Rowan Hamilton was sent to live with an aunt and uncle, Elizabeth and James Hamilton, in Trim, County Meath. James Hamilton was a classics scholar and graduate of Trinity College Dublin, and was headmaster of a diocesan school for boys. He soon recognized that his nephew showed extraordinary promise, and gave him intensive training in languages and the classics. While he prepared for entrance to Trinity College, Hamilton became inter- ested in mathematics, particularly analytic geometry. At the age of seventeen he was reading Th´eorie des Fonctions Analytiques and Mecanique Analytique by Lagrange in addition to the books prescribed for the undergraduate science course at Trinity. At Trinity College, Hamilton pursued a dual course in science and the classics—although he found it increasingly difficult to maintain his interest in the latter—and also began independent research on geometric optics as a natural extension of his interest in analytic geometry. His work led to a paper, “Theory of Systems of Rays” [37], which he presented to the Royal Irish Academy in April, 1827. Primarily on the basis of his original researches in optics, he was elected to the position of Andrews Professor of Astronomy at Trinity College in June, 1827. Hamilton’s theory of ray optics was a variational theory. It was based on the principle, due to Fermat, that a light ray traveling between two points will follow the path that requires the least time. In the course of his work on optics, he also began to consider the possibility of developing an analogous theory for the 4 dynamics of systems of particles. This resulted, in 1834–1835, in two papers, “On a General Method in Dynamics” [38] and “Second Essay on a General Method in Dynamics” [39]. In the second paper he presented the result that is known today as Hamilton’s principle. Hamilton’s general and elegant work on dynamics was widely quoted but not extensively applied during the remainder of the nineteenth century. However, when quantum mechanics was developed, it was realized that Hamilton’s work was the most natural setting for its formulation. In fact, in retrospect Hamilton’s formal analogy between optics and classical mechanics was seen as a precursor of wave mechanics. A somewhat similar historical development has occurred in the field of con- tinuum mechanics. Although formulations of Hamilton’s principle for continua began to appear as early as 1839, with the exception of applications to struc- tural analysis variational methods in continuum mechanics were regarded as academic, because the same results could be obtained using more direct meth- ods. Some modern treatises on continuum mechanics do not mention variational methods. In recent years, however, interest in variational methods has increased markedly. They have been used to obtain approximate solutions, as in the finite element method, and to study the stability of solutions to problems in fluid and solid mechanics. Variational formulations have also been used to develop generalizations of the classical theories of fluid and solid mechanics. The objective of this monograph is to give a comprehensive account of the use of Hamilton’s principle to derive the equations that govern the mechanical behavior of continuous media. The classical theories of fluid and solid mechanics are discussed as well as two generalizations of those theories for which Hamilton’s principle is particularly suited—materials with microstructure and mixtures. These topics are brought together for the first time to acquaint readers who are new to this subject with an interesting and powerful alternative approach to the formulation of continuum theories. Persons interested in fluid and solid mechanics will gain a broadened perspective on those subjects as well as learn the fundamental background required to read the large literature on variational methods in continuum mechanics. For readers who are familiar with these methods, a number of recent results are presented on applications of Hamilton’s principle to generalized continua and materials containing singular surfaces. These results are presented in a setting that could encourage generalizations and extensions. Hamilton’s principle was originally expressed in terms of the classical me- chanics of systems of particles. The concepts and the terminology involved in 5 applying Hamilton’s principle to continuum mechanics are quite similar, and some familiarity with the applications to systems of particles is very helpful in understanding the extension to the case of a continuum. The application of Hamilton’s principle to systems of particles is therefore briefly discussed in Chapter 1. This subject provides a simple context in which to introduce the variational ideas underlying Hamilton’s principle as well as the method of La- grange multipliers and the concept of virtual work. Chapter 2 provides a brief survey of the mathematics and elements of con- tinuum mechanics that are required in the subsequent chapters. Most of this chapter can be skipped by persons familiar with modern continuum mechanics; however, even those who are acquainted with variational methods in continuum mechanics should briefly examine Section 2.3 before proceeding to the following chapters. Applications of Hamilton’s principle to a continuous medium are described in Chapter 3. Ideal fluids and elastic solids are treated in Sections 3.1.1 and 3.1.2. The general case of a continuum that does not exhibit microstructural effects is presented in Section 3.1.3. Section 3.2 presents applications of Hamilton’s principle to two particular theories of materials with microstructure. These applications illustrate the use of Hamilton’s principle to generalize the ordinary theories of fluid and solid mechanics. Persons who are new to this subject may choose to omit this section and the following chapter in a first reading. As another example of the use of Hamilton’s principle to develop generalized continuum theories, applications to mixtures are described in Chapter 4. The fact that the sum of the volume fractions of the constituents of a mixture must equal one at each point can be introduced into Hamilton’s principle using the method of Lagrange multipliers. As a result of “wrinkles” such as this, Hamil- ton’s principle provides a simple and elegant way to derive continuum theories of mixtures. A mixture of ideal fluids is discussed in Section 4.2. The case of a liquid containing a distribution of gas bubbles is treated as an example, includ- ing the microkinetic energy associated with bubble oscillations. In Section 4.3, a mixture of an ideal fluid and an elastic material is considered, and it is shown that the equations obtained through Hamilton’s principle are equivalent to the Biot equations. A theory of mixtures of materials with microstructure in which the constituents need not be ideal or elastic is presented in Section 4.4. In Chapter 5 a discussion is given of the application of Hamilton’s princi- ple to a continuous medium containing a surface across which the fields that characterize the medium, or their derivatives, suffer jump discontinuities. The fundamental results required to include a singular surface in a statement of 6 Hamilton’s principle are presented in Section 5.1. An elastic fluid is treated as an example in Section 5.2, and it is shown that Hamilton’s principle yields the jump conditions of momentum and energy across the surface. The results presented in this monograph are expressed in a modern frame- work. Persons wishing to gain an impression of Hamilton’s research in its origi- nal form should consult his collected works [40], [41]. The definitive references on Hamilton’s life are Graves [32] and Hankins [42]. In Chapters 1-3, the sources that have been used are cited, but no attempt is made to give complete or original references except for results that are relatively recent.
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