The Theory of Pseudo-rigid Bodies

verfasst von: Harley Cohen, Robert G. Muncaster

Verlag: Springer New York

Buchreihe : Springer Tracts in Natural Philosophy

Enthalten in: Professional Book Archive

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This monograph concerns the development, analysis, and application of the theory of pseudo-rigid bodies. It collects together our work on that subject over the last five years. While some results have appeared else where, much of the work is new. Our objective in writing this mono graph has been to present a new theory of the deformation of bodies, one that has not only a firm theoretical basis, but also the simplicity to serve as an effective tool in practical problems. Consequently, the main body of the treatise is a multifaceted development of the theory, from foundations to explicit solutions to linearizations to methods of approximation. The fact that this variety of aspects, each examined in considerable detail, can be collected together in a single, unified treat ment gives this theory an elegance that we feel sets it apart from many others. While our goal has always been to give a complete treatment of the theory as it now stands, the work here is not meant to be definitive. Theories are not entities that appear suddenly one day and thereafter stand as given. Rather, they must mature and grow with time and experience. Our development is more correctly a beginning, tempting others to explore, appraise, and modify its features so as to produce something better.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Prolegomena to the Theory

Abstract

The theory of pseudo-rigid bodies focuses on the large-scale motions of deformable bodies. It provides a convenient framework, like classical rigid-body mechanics, for the analysis of gross changes in the position and orientation of a body. As such it represents a generalization of that classical theory. At the same time, the theory of pseudo-rigid bodies concerns deformation, treating changes in the shape of a body by use of certain gross measures of strain. As such it represents a restriction, or coarse version, of many theories now commonplace in continuum mechanics. Between these two extremes, the modern and the classical, the theory of pseudo-rigid bodies takes a middle road, focusing on problems that exhibit a high degree of interplay between deformation and rigid-body motion.

Harley Cohen, Robert G. Muncaster

Chapter 2. Foundations of the Theory of Pseudo-rigid Bodies

Abstract

A pseudo-rigid body, seen from one perspective, is a generalization of a rigid body that accounts for the effects of deformation. In order to lay the foundations for a theory of such bodies, it is useful to begin by revising our view of rigid bodies. While physically we perceive a rigid body to be an object of finite size, analytically we characterize it only through the position of its center of mass and the orientation of its remaining points relative to a set of axes fixed in space. More concisely, a rigid body may be viewed as a moving point to which is attached a time-varying measure of orientation. In this way, we may view the mechanics of rigid bodies as a theory of a special directed continuum. As an immediate generalization, we define a pseudo-rigid body here to be a moving point to which is attached a measure of both orientation and deformation, and we base our formulation of a theory for it on the general principles that have been developed in recent years for the study of directed continua.

Harley Cohen, Robert G. Muncaster

Chapter 3. Consistency with Other Continuum Theories

Abstract

The theory of pseudo-rigid bodies represents a deformable body in terms of a single point moving in three-dimensional space and a tensor measuring changes in orientation and features of deformation. This is an extremely coarse description of a real body, especially when cast against the sophistication of most modern studies of deformable media. It is simply a reflection, however, of the class of motions that we choose to consider, and it must be viewed in relation to the tractability of the theory. We choose to consider only those motions of real bodies that are characterized largely by (1) a transplacement of the mass center, (2) a change of orientation similar to that appearing in the mechanics of rigid bodies, and (3) overall measures of extension-compression and shear. This admittedly limited description of motion must be weighed against the theory itself, in which the basic equations of motion, given by (2.3.9) or (2.3.13), form a system of ordinary differential equations. By restricting the class of motions of interest, we have obtained a theory much more tractable than ones based on initial-boundary-value problems for systems of partial differential equations.

Harley Cohen, Robert G. Muncaster

Chapter 4. Explicit Motions of Pseudo-rigid Bodies

Abstract

The preceding chapters lay the foundations of the theory of pseudo-rigid bodies and result in governing differential equations. Here we examine some special pseudo-rigid motions by seeking solutions to these equations. Special solutions provide one means of appraising the quality of our developments: they permit comparisons not only with more familiar results but also with our intuition and experience concerning particular problems. In order to obtain a variety of explicit pseudo-rigid motions, we proceed by the semi-inverse method. Namely, we speculate about the general structure or form of motions in a given class, informed by experience from rigid-body mechanics and nonlinear elasticity, and then we seek conditions under which this structure prevails in time.

Harley Cohen, Robert G. Muncaster

Chapter 5. Variational Formulation of the Theory

Abstract

Scholars of mechanics generally agree that variational principles provide one of the most elegant, systematic, and satisfying approaches to the study of finite-dimensional dynamical systems. The strength of the variational method lies in its generality. Systematic schemes of approximation, once developed for one problem, are easily adapted to others. Symmetries can be incorporated and exploited in a variety of problems in an essentially uniform and straightforward manner. And deep theorems on the behavior of dynamical systems can readily be applied to a host of otherwise intractable problems.

Harley Cohen, Robert G. Muncaster

Chapter 6. Approximations for Almost-rigid Bodies

Abstract

If for all motions in a certain class the deformations of a body are small enough to be ignored, then relative to this class we say the body is rigid. The assumption of rigidity is an idealization that has given rise to a large body of work in mechanics. Nevertheless, there are also important problems in which the effects of deformation, while still quite small, cannot be ignored. For motions of this type we say the body is nearly rigid. Approximations for nearly rigid bodies have technological implications both old and new. Historically, the gyroscopic motion of the earth has always fascinated and challenged astronomers and geophysicists. This motion exhibits special complexities owing to the fact that the earth is not exactly rigid. An indication of the scope of work in this area can be found in the treatise of Lambeck [1980]. More currently, problems arise in space technology where the control of orbital motions of space vehicles and satellites is paramount. Complexities arise here from the flexibility of the satellite and accompanying elastic oscillations and rotational perturbations. The analysis of such problems involves rather ingenious modeling and large-scale numerical computations. The work of Williams [1976] provides an indication of the state of the art in the field.

Harley Cohen, Robert G. Muncaster

Backmatter

Titel: The Theory of Pseudo-rigid Bodies
verfasst von: Harley Cohen
Robert G. Muncaster
Verlag: Springer New York
Electronic ISBN: 978-1-4613-9589-8
Print ISBN: 978-1-4613-9591-1
DOI: https://doi.org/10.1007/978-1-4613-9589-8

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Prolegomena to the Theory

Chapter 2. Foundations of the Theory of Pseudo-rigid Bodies

Chapter 3. Consistency with Other Continuum Theories

Chapter 4. Explicit Motions of Pseudo-rigid Bodies

Chapter 5. Variational Formulation of the Theory

Chapter 6. Approximations for Almost-rigid Bodies

Backmatter