Mathematical Models for Elastic Structures

MATHEMATICAL MODELS FOR ELASTIC STRUCTURES

MATHEMATICAL MODELS FOR ELASTIC STRUCTURES PIERO VILLAGGIO Universitd di Pisa

| CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521573245 ©PieroVillaggiol997 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1997 This digitally printed first paperback version 2005 A catalogue recordfor this publication is available from the British Library Library of Congress Cataloguing in Publication data Villaggio, Piero. Mathematical models for elastic structures/Piero Villaggio. p. cm. Includes bibliographical references and indexes. ISBN 0 521 57324 6 (hardcover) 1. Elastic analysis (Engineering) 2. Nonlinear mechanicsMathematical models. 3. Structural analysis (Engineering) I. Title. TA653.V55 1997 96-44172 624.T71 DC20 CIP ISBN-13 978-0-521-57324-5 hardback ISBN-10 0-521-57324-6 hardback ISBN-13 978-0-521-01798-5 paperback ISBN-10 0-521-01798-X paperback

Contents

Preface Introduction

ix 1

Chapter I Basic Concepts 1 Density, Motion, and Temperature 2 Balance Equations 3 Constitutive Equations 4 The Search for Solutions 5 The Notion of a Loading Device 6 An Example of Alternative Modeling

7 11 20 31 44 51

Chapter II Rod Theories: Three-dimensional Approach 7 Geometrical Properties of Rods 8 Approximation of Three-dimensional Equations 9 Rod Problems in Linear Elasticity 10 Uniformly Loaded Beams (Michell's Theory) 11 Timoshenko's Correction 12 Vlasov's Theory for Thin-walled Beams 13 The Influence of Concentrated Loads 14 The Influence of Curvature 15 The Influence of Initial Twist 16 The Energy of an Elastic Rod 17 The Influence of Large Strains 18 Justification of the One-dimensional Model

61 64 70 76 81 93 100 106 117 125 138 158

Chapter III Rod Theories: Director Approach 19 Rods as Oriented Bodies 20 Kirchhoff s Theory 21 Solutions for More General Loads 22 Nonclassical Problems for Rods 23 Initially Curved Rods 24 The Stability of Rods

162 167 175 184 202 218

vi

Contents

25 Dynamical Problems for Rods 26 Kirchhoff s Problem for Nonlinearly Elastic Rods 27 Dynamical Problems in Nonlinear Rods

248 267 289

Chapter IV Theories of Cables 28 Equilibrium Equations 29 Classical Solutions for Linearly Elastic Strings 30 Influence of Friction 31 Equilibrium States for Nonlinear Elastic Strings 32 Inverse Problems in Strings 33 Unilateral Problems for Strings 34 Dynamical Problems for Strings 35 Peeling and Slipping

303 306 316 322 328 336 340 357

Chapter V Theories of Membranes 36 Constitutive Properties of Membranes 37 Plane Membranes 38 Theories of Networks 39 Membranes Stretched against an Obstacle 40 Curved Membranes 41 Stability of Curved Membranes 42 Dynamic Behavior of Membranes 43 Detachment and Suturing of Membranes 44 Optimization of Membranes

366 367 385 392 404 414 422 426 433

Chapter VI Theories of Plates 45 The Equations of Linear Theory 46 Theories of Thick Plates 47 Influence of Varying Thickness 48 The Problem of Plate Optimization 49 Paradoxes in the Theories of Plates 50 Foppl-Karman Theory 51 Berger's and Reissner's Equations 52 The Wrinkling of Plates 53 Buckled States of Plates in Nonlinear Elasticity

451 460 477 485 498 501 522 531 538

Chapter VII Theories of Shells 54 Geometric Definitions and Mechanical Assumptions 55 Equations Deduced from Three-dimensional Theory 56 The Solvability of the Equations of Shell Theory 57 Estimates on Solutions 58 Closed Solutions for Shells 59 Buckling of Shells

547 555 563 577 593 607

Contents

vii

60 Asymptotic Analysis of Shells 61 The Influence of Constitutive Nonlinearities 62 Dynamical Problems for Shells

620 625 647

References

654

Index of Authors Cited

667

Index

671

Preface

Few words are used with so many different meanings as the term "model." In everyday language the word "model" can be applied in a moral, fashion, economic, linguistic, or scientific context; in each case it means something completely different. Even if we restrict ourselves to the category of scientific models, the notion is ambiguous, because it could signify the reproduction in miniature of a certain physical phenomenon, and at the same time present a theoretical description of its nature that preserves the broad outline of its behavior. It is the theoretical aspect of models that we wish to consider; in order to emphasize this, we describe this type of model as "mathematical" (Tarski 1953). Formulating a mathematical model is a logical operation consisting in: (i) making a selection of variables relevant to the problem; (ii) postulating statements of a general law in precise mathematical form, establishing relations between some variables said to be data and others unknown; and (iii) carrying out the treatment of the mathematical problem to make the connections between these variables explicit. The motivations underlying the use of mathematical models are of different types. Sometimes a model is the passage from a lesser known theoretical domain to another for which the theory is well established, as, for example, when we describe neurological processes by means of network theory. In other cases a model is simply a bridge between theory and observation (Aris 1978). The word "model" must be distinguished from "simulation." The simulation of a phenomenon increases in usefulness with the quantity of specific details incorporated, as, for example, in trying to predict the circumstances under which an epidemic propagates. The mathematical model should instead include as few details as possible, but preserve the essential outline of the problem. The "simulation' is concretely descriptive, but applies to only one case; the "mathematical model" is abstract and universal. Another special property of a good mathematical model is that it can isolate only some aspects of the physical fact, but not all. The merit of such a model is not of finding what is common to two groups of observed facts, but rather of indicating their diversities. A long-debated and important question is that of how to formulate a model in its most useful form. The answer is clearly not unique, because there are examples in which the same observed phenomenon can be described equally well by two completely different models. However, in order to initiate the formulation of a good model, six precepts have been proposed (Hammersley 1973): (i) notation IX

x

Preface

should be clarified; (ii) suitable units should be chosen; (iii) the number of variables should be reduced, whenever possible; (iv) rough sketches should be made and particular cases examined; (v) rigor should be avoided at this stage; and (vi) equations should be adjusted to have roughly the same number of terms on each side. Among these rules the third is by far the most important. The essential elements needed to describe a physical phenomenon should initially be isolated. The description is not improved by adding new terms to them. All the theories that have contributed decisively to the progress of physics are remarkable for their simplicity. If we look at the history of mechanics we see that the most important advances conform to the Baconian criterion of dissecare naturam, that is to strive to retain only those ingredients in a model that give the answer we require from a specific physical event. The theory of elastic structures is, by definition, the collection of all reasonable models, proposed during almost three centuries, concerned with simplifying the solutions of problems involving elastic bodies. The equations describing the motion and equilibrium of a three-dimensional elastic body were formulated in full generality during the first half of the nineteenth century, but their solutions are known only in a few cases. From the beginning of the theory of elasticity the interest of many scientists was focused on the solution of the problems of the bending of a beam, the vibrations of bars and plates, and the stability of columns. Later, other problems were formulated and solved, as, for example, those concerning the torsion of a beam, the equilibrium and vibrations of thin shells, and the longitudinal impact of rods. The progress of the theory is not uniform, because we can find frequent retrogressions on the part of experimentalists in adopting hypotheses that had not been properly established, or on the part of mathematicians in using approximate methods beyond the limits of their validity. However, in the case of linear elasticity, a satisfactory degree of knowledge about the range of applicability of each of the theories has been achieved. Nevertheless, things are radically different when the hypothesis of small strains and small displacements in elastic materials is removed. We may think that the equations of elastic structures with large deformations can be simply obtained from those valid with small strains and displacements by replacing the linear constitutive equations of classical elasticity with the constitutive equations of nonlinear elasticity, while referring the data and the unknown to the reference configuration. This is what is customarily done in deriving the equations of three-dimensional finite elasticity. In the theory of structures, however, a new aspect arises. When formulating the model of a structure, such as a rod or a plate, we try to describe it without using the three-dimensional equations. The aim is to find certain reduced equations capable of representing the essential features of the state of strain and stress in simplified form. To do this we make some conjecture about the form of the solutions, as, for example, the conservation of the planarity of the cross-sections in a bent beam, and we write certain averaged equations for the simplified unknowns. The procedure is perfectly rigorous, provided that the hypotheses are coherent and physically plausible. Unfortunately, the range of applicability of assumptions of this kind is very narrow. It may happen that it succeeds if the strains are infinitesimal, although not when there are large strains, and vice versa. One of the reasons why the

Preface

xi

theory of structures is exposed to serious criticisms is that, very often, a set of equations valid under certain restrictions has been illegitimately extended to different situations. In the history of mechanics of structures the use of models incapable of giving results for suitably large extensions is very frequent. Theories perfectly sound for thin rods have been applied to thick rods or to thin-walled beams; the theory of Kirchhoff s plate has been extrapolated to thick and sandwich plates; and methods of solution well established for curved membranes and shells in small deformations have been employed in the presence of large deformations, with arbitrary adjustments that invalidate the results. Those who have proposed these generalizations have often realized the weakness of some arguments. However, instead of setting up new, consistent, formulations of problems, they have limited themselves to adding small corrections to the old models, with the consequence that they have proposed dubious theories from both physical and logical aspects. This kind of attitude has been common for several decades, save for a few praiseworthy exceptions, although remarkable change has occurred in the last twenty years. Though the scientific literature is still abundant in old-style articles, we can now find a number of novel contributions. These are important in two respects: they pay more attention to the formulation and to the treatment of the mathematical problem, drawing on developments in modern mathematics; and they tackle new problems arising in the fields of technology or everyday life. An example of the first type of advance is seen in the application of bifurcation theory. This allows the characterization of solutions of a nonlinear problem even when they are very far from those of the corresponding linearized problem. An example of the second development comes in the extended application of the theory of structures to new fields, such as those of geomechanics, biology, and medicine. The purpose of this book is to give an account of the advances in the theory of structures in both directions. The number of papers actually published on subjects involving the theory is so vast that it is physically impossible to enumerate them. The selection of the most significant contributions is naturally conditioned by the taste and prejudices of the author. It is therefore natural to ask what criterion has been used in selecting or rejecting papers. The papers that were rejected were judged on their feature of offering only "slight generalizations." Very often the essential features of an important mechanical problem were isolated and formulated in mathematical equations more than a century ago. Since then much work has been put into adding small corrections to improve the physical validity of the system. The result has been an immediate complicating of the mathematics. In general, there are two types of development in a certain field. Sometimes the problem is an old one, but treated now by a new mathematical technique that allows the properties of solutions to be illustrated. In other cases an unsatisfactory analytical treatment of an important mechanical problem has prompted questions normally considered outside the domain of engineering. There are also papers that have the distinction of encompassing both these aspects, resulting in new, original, problems treated with elegant mathematics. A book of this kind is the result of many direct and indirect contributions from friends, correspondents, and the authors I have read. However, one person deserves

xii

Preface

special mention. Ernest Wilkes has been a continual source of support throughout the preparation of this book, and the extraordinary care with which he has offered suggestions for improvements to the entire manuscript has been decisive. I wish to express my warmest thanks to him. Piero Villaggio Universitd di Pisa

Introduction

What does it mean to solve a problem in classical elasticity? The question may appear trivial, but, if we ask scholars working in the field, we receive surprisingly different answers. Let us assume, in order to make the subject more explicit, that the problem concerns the impact between two elastic bodies. For an experimental physicist solving the problem means interpreting those crucial experiments that make it possible to decide which are the important variables in the phenomenon: in this specific case, the densities and elastic moduli of the materials are important, but the temperature and atmospheric pressure, for instance, are not. A theoretical physicist will say instead that the solution consists in formulating the general equations of the problem, having inserted all the significant variables. For a mathematician it will be obvious that solving the problem meansfindingan existence, uniqueness, and possibly regularity, theorem for the equations of elastic impact. Yet another answer will be given by an engineer, who will require an explicit formula giving the stress components within the two bodies at each point and at each instant. Confronted with such a variety of answers, a typical student feels disoriented, being immediately aware of the basic ambiguity in the way in which the question itself has been posed. Ludwig Wittgenstein would say in explanation that the confusion arises from the vague use of the verb "to solve." For the same word has been used in different contexts with different meanings. Though dissimilar, the four answers have a common characteristic. They represent four attempts at describing the same phenomenon by abstraction, setting aside the unessential details, with the purpose not merely of illustrating but also of predicting. We say that they propose four models for the elastic impact. At this point we immediately ask whether there is a rational criterion for deciding which model is preferable, provided that all satisfy the three necessary requisites of being realistic, logically coherent, and simple. It is evident that a model must not be in obvious conflict with the physical data, nor must it be self-contradictory or too complicated. However, unfortunately, there is no incontrovertible way of establishing that one model is better than another. Setting up a model means creating conceptual conditions suitable for posing a particular question about the problem. The choice of these conditions then depends on the kind of answer we wish to obtain. For instance, if the question we ask ourselves about the elastic impact is one concerning the qualitative interdependence of some quantities, then the rough tests of the experimentalist represent the right method; if, alternatively, we want to know the detailed 1

2

Introduction

distribution of stresses on the surface of contact of the two colliding bodies, then the formulae proposed by the engineer work better. However, even if a rigorous science for formulating good models is lacking we have at least the comfort of returning to the traditional analysis of those models that have been most deeply involved in the development of continuum mechanics, as, for example, the Euler equations of motion of perfect fluids, or the Navier equations of classical elasticity. Besides the three necessary properties of realistic representation, coherence, and simplicity, these models possess two other qualities. The first is that there is a surprising harmony between the originality of the physical problem that must be described and the novelty of the mathematical method by which the problem is treated. Newton's second law is a model for explaining the motions of heavenly bodies, but, at the same time, it requires the solution of ordinary differential equations of second order, a problem which was just at that time beginning to be studied. On the other hand, there are several examples of physically interesting problems treated with primitive mathematics, or of insignificant problems accompanied by a brilliant mathematical manipulation. Using an adjective coined by Nietzsche, this first quality may be called the "Apollonian" attribute of a good model. But this is not all. The great models of classical mathematical physics are characterized by a second quality which may be called "reproductive," because we devise from them the methodological example for tackling new problems. A typical instance of this is found in the classical theory of elasticity, which, as Love (1927) says in the Historical Introduction to his treatise, is not only important for its contribution to the material advance of mankind, but also, and more importantly, for the light thrown on other branches of the physics of other areas, such as optics and geomechanics. It would appear that, at this point, the question of finding a good model does not necessarily depend on these terms. We cannot hope to find an infallible recipe for suitably formulating and treating all the problems proposed by physics, but, at least, we have inherited from earlier work a significant number of examples suggesting means of dealing with those that interest us. However, in spite of this, it is hardly likely that these examples will provide satisfactory solutions to fresh problems. We must also consider that the great men of the past, when unable to find a general closed solution, have often reformulated their problem in more manageable terms, even at the cost of convoluted approximations. This is the so-called path of "reasonable" approximations. When confronted with a problem of too great a complexity, simplification is necessary; we can achieve this, for instance, by eliminating the residual non-linear terms, which jeopardizes the application of classical procedures, or by changing the shape of the original domain into one of more regular form. All this implies the substitution of the primitive model with a new one, and means that the new model must be reconsidered ab initio from the point of view of physical completeness and mathematical consistency. Models thus generate submodels, thereby creating the possibility of multiplicity, and deciding on their suitability can cause confusion. But here again examples from the past give us encouragement. Workers such as Kelvin (1848), Hertz (1881) and others, have passed into history for their contributions to the general principles of mechanics, but are not always appreciated for their formidable capacity in making

Introduction

3

compromises. In order to evaluate the effect of a force acting at a point in an indefinitely extended solid, Kelvin considered the case in which body forces act within a finite volume T and vanish outside it. He constructed the solution of the problem and then passed to a limit by diminishing T indefinitely and supposing that the resultant force with T has a finite limit. The procedure seems tortuous, but the final solution is very simple. Hertz, in his solution of the problem of the pressure between two bodies in contact, assumed that the compressed area, common to the two bodies, is an ellipse, and that, provided this area is small, each body may be regarded as a half-space loaded over the bounding plane. We are tempted to believe that, in mechanics, reductions like those of Stokes and Rayleigh are the outcome of long reflections, but it has not always been so. To be more precise, it was so until the end of the last century, but after that, those working in mechanics, pressed by the need to achieve manageable solutions, have preferred to accept the technical artifices, without submitting them under critical scrutiny. The treatises of the nineteenth century, such as those of Clebsch (1862), Kelvin and Tait (1867), and Love (1927), are full of mathematical development, but still maintain a critical grasp whenever any new approximations are introduced. More recent books, such as those by Sommerfeld (1944) and Landau and Lifschitz (1971), only occasionally state the postulates introduced during the process of deriving the equations. Books written for engineers are even less punctilious and all this leaves the reader in a state of permanent disorientation. There are, of course, some brilliant exceptions: for instance, a paradigm of deduction of the linearized equations of the motion of a string, with exemplary justification of all assumptions, can be found in the book by Weinberger (1965), which is not a book on elasticity, but a course on partial differential equations! Thus a sort of no-man's land has been created between the principles of elasticity and their applications. And this terrain is shaky, either because one fears having somewhere violated the basic principles of mechanics and thermodynamics in conjecturing some simplifying property of a solution, or, because, given the arbitraries of choice, one is constantly afflicted by the doubt that another much simpler but more elegant model ought to have been used instead. However, in spite of possible misapplications, the technical literature is rich in contributions proposing expedient methods of treating more general problems. There are theories of plates and shells derived from the three-dimensional equations of elasticity through expansions of the displacements as functions of the first, second and third order of the distance from the middle surface; theories offinitedeformations with unjustified linearizations; and semi-inverse solutions in elastostatics for domains having noncylindrical forms. A great number of these attempts are mediocre, but some have contributed greatly to the development of technical mechanics. On the other hand, books on general mechanics are reluctant to supply the general equations with some practical application, as if the analysis of the motion of a toothed wheel or of a bicycle would compromise the elegance of the book. An attempt to fill the gap has been made by Szabo (1963, 1964), whose work ranges from celestial mechanics to plasticity theory with surprising versatility, and above all with a taste for applications. But the work is too fragmentary: it does not develop a method, but rather offers only a collection of smart artifices in solving particular problems.

4

Introduction

The divisions in an intellectual discipline are always pernicious. They damage the foundations of mechanics because the objects being systematized, whenever they have lost any connection with experience, become the "golden mountains" of which Russel (1903) speaks. In applications, they are even lethal. For historical reasons, in continuum mechanics subdivisions have been created between, roughly, the mechanics of incompressible fluids, of compressible fluids, and of solids. This division is necessary because the specific problems of each sector are so particular that an ad hoc procedure must be employed in their solution, as, for instance, in the theory of surface waves or in that of elastic structures. But these methods are in some cases so cogent that they might well be employed in related sectors, where they are unknown. An objection made to this observation is that nowadays the use of numerical calculus has overcome the difficulty of describing, in the limit, small technical detail by solving the general equations. Unfortunately, however, even precise numerical solutions are too restrictive and do give no idea of how solutions behave as data vary. Many ambiguities arise from three tacit premises: that classical elasticity is a logical science; that logic can be raised to the level of awareness; and that thinking in mechanical terms can be refined by its intelligent application. The logic of mechanics is not a formal logic of deductive inference having the symmetric structure of Aristotelian syllogism, nor is it an inductive logic, like that of John Stuart Mill. Using a word introduced by Peirce (1931), it is a process of "abduction," that is, of formulating a hypothesis and of deducting what would be the case if the hypothesis were true. Usually the most important advances have been achieved by workers who had a particular problem in mind, the solution of which did not obviously follow by applying the general equations. They have then started to manipulate those equations and to extract some new ones, until the solution furnished by the latter conform to their expectations. The success of the method has been ensured by two ingredients: a prior intuition of how things should evolve and, consequently, of how the solutions must be; and a sort of esthetic taste in choosing the clearest route in working through the intermediate stages. In these processes, as there are no definite prescriptions for reaching the result, it is easier to try to identify the common ways in which errors have been made. Fischer (1969), in his brilliant essay on the logic of historical thought, has endeavored to indicate the most frequent occasion of mistake. Essentially, these are the fallacies resulting from excessive motivation; that is, the neglect of relevant quantities simply because they disturb the desired result. On the other hand, there are the fallacies of overgeneralization, which arise from the belief that a model improves, the greater the number of its constituent entities. It is clearly forgotten in this fallacy that a good model can never be exhaustive, and must be simple to be effective. There are several examples of bad models, some of which originated with illustrious authors as, for instance, the celebrated equation of F6ppl-v. Karman in describing the large transverse displacements of a thin plate subjected to pressure. The defects of the theory have been pointed out by Truesdell (1978), whose criticisms can be summoned up in the following terms: unnecessary geometric approximations, and unjustified assumptions about the way in which the stress varies over a cross-section.

Introduction

5

In the history of mechanics the separation between theory and application has never really existed. Workers in the nineteenth century have simply followed their seventeenth and eighteenth century predecessors, who could pass from abstract mathematics to technical solutions with extraordinary ease. It is known that James Bernoulli (1694) formulated the problem of the elastic line or elastica of a beam, and that Euler (1744) proposed a simple theory to explain the onset of buckling in a thin strut. But they are not exceptions, since eclecticism is a characteristic of scientists less famous than J. Bernoulli and Euler. For instance, in 1823, Lame and Clapeyron were called on to assess the stability of the dome of the cathedral of St Isaac in St Petersburg, and on which occasion they virtually re-invented the technique of slicing the dome into lunes, each of which has to have an independent equilibrium. At this point it might by naively asked which have been the most creative ideas in the development of the branch of classical elasticity oriented towards applications. The answer is clearly complicated because, in general, an ingenious result is not the outcome of a single author but the result of many refinements by a sequence of authors over a long period of time. Nevertheless, let us suppose that one is required to construct a list of outstanding results produced in the last three centuries, solely with the purpose of providing a picture of the evolution of interests over time. A tentative list might be as follows: James Bernoulli's (1705) equation of the elastica; Euler's (1744) integration of the equation of the elastica and Euler's (1757) definition of buckling load; Navier's (1827) equations of three-dimensional elasticity; Cauchy's (1829) definition of elastic material; Green's (1839, 1842) definition of hyperelastic material; Saint-Venant's theory (1855) of the torsion of prisms and the invention of the semi-inverse method of solution; Kirchhoff s (1850) formulation of the boundary conditions at the edge of a plate; Kelvin's (1848) solution for the elastic displacement due to a point force in an indefinite medium; Boussinesq's (1885) solution for the half-plane normally loaded by a point force; Hertz's (1881) theory of the contact between two bodies; MichelPs (1904) theory of trusses; Geckeler's (1926) approximate theory for evaluating edge effects in shells; Kolosov's (1909, 1914) solution of the biharmonic equation in two variables; the three functions Ansatz of Boussinesq, Papkovic, and Neuber (Papkovic 1932; Neuber 1934); Vlasov's (1958) theory of thinwalled beams; Ericksen and TruesdelFs (1958) director theory of rods; and Ericksen's (1973) concept of a loading device. The list may suggest that all efforts to convert the problems created by practical applications into simple and precise terms have succeeded. But this impression is illusory, as there have also been numerous failures. For instance, Euler's (1766) theory of axisymmetric shells is not accepable; Coulomb's (1787) theory of twisting is wrong; Cauchy's (1828) theory of "rari-constants" has been contradicted by experience and thermodynamics; and Greenhill's (1881) solution for the buckling of a heavy column is incorrect, because the equation he solved neglects terms of the same order of magnitude as those maintained in the equation. It is instructive to recognize that faults occur in the works of those who have decisively contributed to the progress of continuum mechanics: this means that mechanics is a perfectible science, and what seems a definite conquest today may be demolished tomorrow. An unknown Italian philosopher, Giovanni Vailati

6

Introduction

(1863-1909), pointed out the importance of error in the development of science. Vailati maintained that an error is worth as much as an ingenious discovery, if it contributes equally in orienting our intellectual faculties: "Every error is a reef to be avoided, while each discovery doesn't always indicate the path to follow." The present-day literature on mechanics abounds with papers proposing modifications of the classical models, either by insignificant extension or by eliminating terms that generate difficulty. These works are not wrong. They are, of course, absolutely correct, but as mathematical works, they are deplorably inelegant.

Chapter I Basic Concepts

1. Density, Motion, and Temperature

Mechanics studies the conditions for the motion and equilibrium of natural objects. In order to treat these in a sufficiently general form, mechanics introduces the notion of the body, which is a mathematical concept designed to give an abstract representation of the most important properties of these physical objects in order to describe their mechanical behavior. By the term "body" we mean a regularly open set $ in some topological space. The elements of a body are called particles, or substantial points to avoid confusion with the term "particle" as used in physics (Truesdell 1991, Ch. I, p.3). Bodies are available in their configurations, which are the regions %{$) that they may occupy in Euclidean space at some time. It is commonly assumed that configurations are regularly open sets in Euclidean space and there is a one-to-one mapping between the particles of the body and a possible configuration. It is often convenient to select one particular configuration K{$) and to identify each particle of K(£S) by its position in this configuration, called the reference configuration. If, in the reference configuration, we choose Cartesian coordinates, the particle can be designated XA or X, Y, Z. At the time t varies the particles change their positions in space, and consequently the position of a particle XA at the instant t is characterized by a vector function xh or x, y, z, of the form Xi = Xi(*A,t),

(1.1)

which is called the motion of the body. The motion determines the shape of the body at each instant. The inertia of a particle is determined by a scalar function p = p(XA, t), called the density; and the hotness of a particle is determined by another scalar function 6 = 0(XA, t), called temperature. Motion, density, and temperature are regarded as primitive concepts: that is, they need not be defined in terms of other known quantities, and must be measurable, at least in principle. The aim of mechanics is to determine the fields associated with Xi, p, and 6. However, these quantities are not completely unrestricted, because both density and temperature must be positive-valued functions, and the functions // must be continuous and invertible so as to exclude the possibility of two particles assuming the same position.

8

1. Density, Motion, and Temperature

At time t the particle XA has the position xh and we may use this position, instead of XA, to identify the particle. Thus, provided that the mapping (1.1) is continuous and invertible, carrying a regularly open set into regularly open sets, it is possible to represent the fields of mechanics in the form xt = xt{i), p = p(xh i), 0 = 0{xh t), which is called the spatial description. The representation in terms of XA and t as independent variables is said to be the material description. In the spatial description we regard as the object of our investigation a knowledge of the density and temperature at all points occupied by the body, at all instants in time; in the material description we seek the history of every particle. In continuum mechanics Green's theorem is of central importance in establishing mechanical relations in terms of equations. It is thus necessary that all the possible configurations of bodies are regular enough for the application of Green's formula, whenever the fields are sufficiently smooth. As the assumption made about the configurations is too weak, it is convenient to impose the condition that they are fit regions (Noll and Virga 1988) that is, regular open sets, bounded by a finite perimeter, and with negligible boundary. The restriction may appear too strong, since it would exclude the treatment of infinite regions. However, including infinite regions in the class of fit regions can be done with some further specification (Truesdell 1991, Ch. II, p. 1). Besides density, temperature, and motion, in classical mechanics there are two other primitive quantities: namely, force and heat. Forces are vectors introduced to describe the purely mechanical actions exerted on the parts 0* of a body $ in a configuration %($). There are two kinds of forces: the body forces f, acting in the interior of x(^)l an< l tractions th acting on the surface of X(£?)' Body forces and tractions depend, in general, on xt and t, and also on & and ^ , but in classical mechanics attention is restricted to body forces that are unaffected by the presence or absence of other bodies in space. These forces therefore have the form fi=f(Xitl

(1.2)

and are commonly given per unit mass. Tractions tt at any place and time have a common value for all parts of the surface of x(^) having a common tangent plane and lying on the same side of it. This implies that ti = ti(xht,eil

(1.3)

where et is the outward normal to the surface of x(^)- The assumption embodied in (1.3) is called the Cauchy stress principle, which characterizes simple bodies. Heat supply is a scalar quantity representing the thermal actions exerted on the parts & of Jf. There are two kinds of heating: the internal production r, having the form r = r{xbt\

(1.4)

given per unit mass; and the normal heat flux h across the surface of x(^)- An assumption like Cauchy's principle is also made for h. That is, h depends on xh t, and eh the exterior normal to the boundary of #(^), in the form: et).

(1.5)

Basic Concepts

9

This equation is typical of thermally simple bodies. In contrast to density, motion, and temperature, which are unknown, body forces and heat production are taken as known. The body force/j is specified at every point of %{$)\ tractions tt are known at the boundary of ^ ( J ) , but not in its interior. Heat production r is considered to be given; the normal heat flux h is known on the boundary of #(^), but unknown in its interior. This distinction means that we assume a knowledge of how the body 31 interacts with its exterior, but we do not know how the parts of $ interact with one another. Although the assumption may appear very natural and useful, because it permits us to study a body as isolated, it is open to question because, in many cases, the quantities we consider as known must be found by considering 31 as enveloped by a wider body called the environment of $ and the interaction between these two bodies must be evaluated. (An attempt at making a rational definition of these interactions has been made by Ericksen (1973).) Once the primitive quantities have been introduced, other quantities, known as derived quantities, can be defined. For instance, the velocity v( or w, v, w of a particle XA is defined as the rate of change of its position: &)

= Xt.

(1-6)

) = vi = xi

(1.7)

The rate of change of the velocity

is called the acceleration of the particle XA. Formula (1.7) gives the acceleration whenever we have adopted the material description of the velocity. But, if we consider the spatial description of the velocity field, then vt must be considered as a function of xh t instead of XA, t. Applications of the chain rule thus yields =

toi

, tot SXJJXA, 0

+

=

tot

+

, dvt

^ -aT a ^ — S — -aT ^ -

(L8)

Here, and in the remainder of the text, summation is implied by repeated subscripts. If Xf is fixed, the acceleration reduces to dvf/dt, which is called the spatial time derivative of the velocity. In addition to the time derivatives of the motion, we can also introduce its partial derivatives with respect to the variables XA, and define the matrix with coefficients ^

)

(1.9)

as the deformation gradient. This matrix must be nonsingular; that is, its determinant / = det(i^ A ) must always be positive in order to ensure that the mapping (1.1) is invertible for all XA and t. This requirement is said to be the axiom of permanence of matter (Truesdell and Toupin 1960, sec. 16). The deformation gradient is a measure of deformation since it permits us to evaluate the distance vector dxt between two neighboring points in the present configuration %(3fi) in terms of their distance vectors dXA in the reference configuration through the formula

10

1. Density, Motion, and Temperature ^

,

t)dXA = FiA dXA.

(1.10)

But, if FiA is known, it is possible to evaluate both the change in length of dXA and its rotation. A theorem, known as polar decomposition (Ericksen 1960, p. 840), determines these two contributions to the deformation. The theorem states that there are two, and only two, decompositions of FiA\ FiA = RmUBA and FiA = VyRJA,

(1.11)

where RiB is an orthogonal matrix, and UBA and Vy are two symmetric positivedefinite matrices. The theorem of polar decomposition has a simple geometric interpretation: UBA and Vy characterize the pure changes in length along the three orthogonal axes, while RiB, as an orthogonal matrix, represents a pure rotation; thus the passage of dXA to dxx can be obtained either by a stretch followed by a rotation, as indicated by the first equation (1.11), or starting with a pure rotation of dXA followed by pure stretching in the directions of the axes, as indicated by the second equation (1.11). The set of nine components FiA can also be represented by the single matrix symbol F. The transpose of F is denoted by F T and the inverse by F" 1 . The generic component of F T has the form FT)iA = FAi,

(1-12)

( F - V = 7 Qe^ABcfyJ'k),

(1-13)

1

and that of F" can be written as

where eijk and e ABC are two matrices which are antisymmetric with respect to the permutation of any two indices so that erss = 0 and e123 is equal to unity, with J == det¥ = sijkFnFj2Fk3. From the definition of the deformation gradient it is also possible to express the resulting change in an element of area consequent upon the motion. In fact, two distance vectors dX^\ dX