Cauchy and the modern mechanics of continua.pdf

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M CLIFFORD A. TRUESDELL Cauchy and the modern mechanics of continua In: Revue d'histoire des sciences. 1992, Tome 45 n°1. pp. 5-24. Résumé RÉSUMÉ. — On démontre que quelques-uns des travaux de Cauchy ont fourni les bases et l'inspiration pour le développement moderne de la mécanique rigoureuse des milieux continus par Walter Noll et ceux qui suivirent ses concepts et méthodes. Abstract SUMMARY. — Some of Cauchy's major works are shown to have served as bases and inspiration for the modern development of rigorous continuum mechanics by Walter Noll and his followers. Citer ce document / Cite this document : TRUESDELL CLIFFORD A. Cauchy and the modern mechanics of continua. In: Revue d'histoire des sciences. 1992, Tome 45 n°1. pp. 5-24. doi : 10.3406/rhs.1992.4229 http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1992_num_45_1_4229

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Page 1: Cauchy and the modern mechanics of continua.pdf

M CLIFFORD A. TRUESDELL

Cauchy and the modern mechanics of continuaIn: Revue d'histoire des sciences. 1992, Tome 45 n°1. pp. 5-24.

RésuméRÉSUMÉ. — On démontre que quelques-uns des travaux de Cauchy ont fourni les bases et l'inspiration pour le développementmoderne de la mécanique rigoureuse des milieux continus par Walter Noll et ceux qui suivirent ses concepts et méthodes.

AbstractSUMMARY. — Some of Cauchy's major works are shown to have served as bases and inspiration for the modern developmentof rigorous continuum mechanics by Walter Noll and his followers.

Citer ce document / Cite this document :

TRUESDELL CLIFFORD A. Cauchy and the modern mechanics of continua. In: Revue d'histoire des sciences. 1992, Tome 45n°1. pp. 5-24.

doi : 10.3406/rhs.1992.4229

http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1992_num_45_1_4229

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Cauchy and the modern mechanics

of continua

RÉSUMÉ. — On démontre que quelques-uns des travaux de Cauchy ont fourni les bases et l'inspiration pour le développement moderne de la mécanique rigoureuse des milieux continus par Walter Noll et ceux qui suivirent ses concepts et méthodes.

SUMMARY. — Some of Cauchy's major works are shown to have served as bases and inspiration for the modern development of rigorous continuum mechanics by Walter Noll and his followers.

I. — General background

My title makes it clear that I shall write little about Cauchy himself. Neither shall I discuss Cauchy's work as it is reflected in the textbooks of the nineteenth century and in most engineering books today. By "the modern mechanics of continua" I mean the development, beginning in the late 1940s, of rigorous mathematical theories of large deformation of material bodies. Several limitations follow.

a) Those who created modern continuum mechanics were not historians. They read the works of Cauchy for what they could get out of them, what would serve as something to build upon. They treated them as mathematicians have treated, over and over again, the works of Euler and Gauss and Riemann. Historians of science have often remarked that mathematicians are the most unhistorical of scientists because they tend to regard mathematicians of old as if they were colleagues today. For a mathematician, a mistake is a mistake no matter how old it be, or what great man made it. If it turns out that some

Rev. Hist. ScL, 1992, XLV/1

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obscure, forgotten school teacher or trash collector named John Smith or William Jones discovered the theorema egregium before Gauss did, he is not dismissed because he was "out of the main stream" or "without historical influence"; rather, if mathematicians learn of the facts, they recognize the said John Smith or William Jones and regularly attribute his discovery to him, perhaps with a hyphen: "Smith-Gauss" or "Jones-Gauss". On the other hand, some mathematicians are the most historical of scientists in that they study old sources for what they can learn from them and use.

b) Such students are not likely to search out manuscripts left unpublished, letters, and the circumstances of persons. They study the contents of published works. Therefore I will mention only such writings of Cauchy as are published and hence available in his Œuvres, and of those, only a few.

c) While Cauchy himself often acknowledged work by others, for example Fresnel and Navier and Poisson, the modern student in search of useful achievements from Cauchy' s period will read mainly if not only Cauchy, because Cauchy is explicit, brief, and usually clear. Thus I will not discuss priorities. The case here is much like Euler's. We know that Huygens, Newton, James and John Bernoulli, Taylor, Daniel Bernoulli, Clairaut, and d'Alembert made great discoveries in mechanics, but they are authors hard for a modern student to read, and so, to learn what was known in the late eighteenth century, we read Euler, or at least we read Euler first.

d) Mathematics necessarily is cultivated by many persons, usually professors, but the great discoveries have been made by a few, lonely men. The concepts introduced by the discoverers were new, and so they seemed unnecessary if not arcane and repelled the ordinary practitioners of their times, especially if new notations were needed to present them. The hoipolloi of mathematics rejected those very men who to us seem the heroes of their times. An example is the treatment of Cauchy himself by his paedagogic boss at the Ecole polytechnique, who forbade him to waste the time of incipient engineers by trying to make them think rather than parrot. Another, recently described in a paper by Mr Bottazzini, is the reception given Cauchy on his visit to northern Italy, where the mathematicians clung to Lagrange's

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formal manipulations, which Cauchy attacked, and rejected all that unnecessary rigor he attempted to promote. Modern continuum mechanics likewise developed on a small scale, first in Britain and the United States, then in Italy, and later in some other countries, but still now is nevertheless widely ignored, set aside for the very reasons that Cauchy' s rigorous proofs were set aside during his lifetime. Consequently my subject refers to work done by just a few persons, though not so few as those who were followers of Cauchy.

e) Few of those in the small group developing modern continuum mechanics went back to Cauchy 's papers. A lot of fluid can flow through one small pipe in a short time if the pressure be great enough. In this instance the pipe is a treatise called The Classical Field Theories (1), published in 1960 in Fliigge's Encyclopedia of Physics.

Thus the first part of my subject today is in effect reduced to motivation of The Classical Field Theories and the direct influence of Cauchy's works upon the contents of that treatise.

I remark also that Cauchy was the principal and most successful creator of theories of bodies modelled as assemblies of small masses, often called molecules, especially when taken in arrangements intended to model crystal lattices. That work, too, has been studied again, resurrected, and extended, but it does not come within my scope here. (In passing I remark that some historians of science, especially the modern ones indoctrinated more in sociology or political science than in mathematics and logical thought, seek to discern what a particular scientist believed, and so they find Cauchy particularly troublesome, for in one and the same volume he published papers that regard bodies as assemblies of tiny molecules and others that take a body as being a plenum. I have never found a word in Cauchy's mathematical works indicating which type of theory he thought truer.) Newton may have believed in the molecular nature of matter, at least sometimes; nevertheless, in most of his Principia his treatments fall in better with continuum mechanics. (I have substantiated this claim else-

(1) C. Truesdell and R. A. Toupin, The Classical Field Theories, with an appendix by J. L. Ericksen, in Fliigge's Encyclopedia of Physics, vol. III/l (Berlin : Springer- Verlag, 1960), 236-902.

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where (2) by citing chapter and verse.) In science Cauchy seems not ever to have been a believer. I do not find that at all strange. While not making any comparisons, I mention that I have more than once published in the same year researches both on continuum mechanics and on the statistical mechanics of systems of mass-points. I feel neither commitment nor guilt.

II. — Background, composition AND INTENDED FUNCTION OF THE CLASSICAL FŒLD THEORIES

In the late 1940s I was directed to study and develop the flow of rarefied gases, especially air at moderately high altitudes. Of course I consulted such modern works as there were ; I found them altogether unsystematic, jumbled, neither clear nor convincing. In a course on tensor analysis when I was an undergraduate I had studied a systematic and clear development, then recent, of the theory of large elastic deformations, and I thought that it might be extended to cover fluids with non-linear viscosity, but that was not true. After writing an overview of continuum theories of large deformation I submitted it to an international congress to be held in 1948; it was rejected, and I now see that it deserved to be. In it I had mentioned a very brief survey which Richard v. Mises had delivered as an invited lecture (3) for a similar congress in 1930. Though I thought his collection superficial and inconclusive, I sent him my manuscript and asked his advice. On December 23, 1948, he replied by inviting me to write a more extensive survey of the mechanics of deformable masses for a volume he was then assembling. He allowed me five months for the writing. Kurt Frie- drichs also saw my manuscript. He told me I had not given sufficient attention to earlier studies. He was right. As I began to revise my work, I tried to study contributions from the nineteenth century because I had learned that in that period some mathematicians

(2) Cf. § 8 of Suppesian Stews in C. Traesdell, An Idiot's Essays on Science, Second printing, revised and augmented (New York : Springer-Verlag, [1987]), 503-579.

(3) R. v. Mises, Uber die bisherigen Ansàtze in der klassischen Mechanik der Kontinua, in Proceedings of the 3rd International Congress of Applied Mechanics (Stockholm, 1930), vol. 2, 1-9.

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and physicists of the first rank had devoted part of their attention to continuum mechanics, while more recent studies, such as those v. Mises reviewed, came after most major mathematicians and physicists had abandoned classical continuum mechanics in favor of pure mathematics, relativity, or quantum mechanics. I tried to read some historical works, notably Todhunter and Pearson's History of the Theory of Elasticity (4), but I found them long on approximations and ugly formulae, opaque and tedious, short on fundamental thinking. For example, Todhunter has a chapter on Cauchy, at the end of which he writes of the "Mémoire sur les dilatations, les condensations et les rotations produites par un changement de forme dans un système de points matériels" in volume II (1841) of the Exercices d'Analyse et de Physique Mathématique (5), that it "contains various theorems demonstrated with clearness and simplicity; but with regard to our subject of elasticity they may be considered as analytical superfluities." This statement reflects Todhunter's view, illustrated again and again in his book, that elasticity to him meant linearized elasticity. My view was the opposite. I searched for the foundations of continuum theories of all kinds, in kinematics, in dynamics, in variety and nature of response to strain, and in classification that a student could use in creating and appraising new theories of materials. When I came, later, to read that paper by Cauchy (his last major contribution to continuum mechanics), I thought it the finest study of the geometry of deformation ever written. Getting ahead of my story, I mention that in The Classical Field Theories conclusions Cauchy derived in that paper are presented (though not always with the original proofs) in 10 of the 140 sections on kinematics. It introduces the Cauchy-Green tensors for arbitrary strains, the strain ellipsoids, the mean local rotations, and the relative local spin. By geometrical reasoning it calculates the proper numbers and determines the principal axes of a symmetric tensor, and it essentially proves what are now called the polar decompositions of an inver- tible tensor. (Some of the conclusions here had appeared in Cauchy's earlier papers.)

(4) I. Todhunter, A History of the Theory of Elasticity and of the Strength of Materials from Galilei to Lord Kelvin, edited and completed... by Karl Pearson, 2 vol. (Cambridge : Cambridge Univ. Press, I : 1886 and II : 1893).

(5) A.-L. Cauchy, ОС (2), XII, 343-377.

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On the survey v. Mises had asked me to write, I worked very hard; I read extensively in literature going back to Hooke's essays, including quite a bit from the eighteenth century, and referring to twelve of Cauchy's papers, with many sources before and after his time, and ending with half a dozen papers of my own that were still in press in 1952/1953. My "New Definition of a Fluid", which I had presented in Paris in 1949, had appeared (6). Influenced by my first teacher, Harry Bateman, in my survey article I wished to cite the original sources, but I was beginning to see that those sources, particularly the works of Cauchy and Euler, contained more than the origins of things commonly known in the middle of the twentieth century. They included good theorems and good ideas that had not gotten into the textbooks and were not mentioned in recent articles. I wished I had time to read and study more.

I met the term of five months, but I had to condense the article to keep within little more than twice the number of pages allowed me. After a year of silence the publisher told v. Mises that my article was too long, used too many symbols and equations and footnotes, and wasted space on old references, which could not be useful to a modern scientist. Meanwhile I had revised and expanded the text, and I had learned many things, especially from Cauchy's works. I called my paper back, but the publisher refused to return it. I rewrote it at once from an imperfect copy and my rough notes. In 1952 it appeared, under the title "The Mechanical Foundations of Elasticity and Fluid Dynamics", in the first issue of a journal that had just been founded at Indiana University (7), to which I had moved in 1950. More important than that was the encouragement I received from my students. My examinations then (and for lustra thereafter) included historical questions, and among the papers proposed for study and analysis were always one or two by Cauchy. Jerry Ericksen, who was one of the students in my course on elasticity in the spring of 1951, began to study Cauchy's papers on continuum mechanics, and for some years thereafter we discussed by correspondence particular questions

(6) C. Truesdell, A New Definition of a Fluid, Journal de Mathématiques Pures et Appliquées (9), XXIX (1950), 215-244, and XXX (1951), 111-155, corrigenda 156-158.

(7) Journal of Rational Mechanics and Analysis, I (1952), 125-300, corrigenda and addenda II (1953), 593-616, and III (1954), 801. Corrected reprint : C. Truesdell, Continuum Mechanics, I (New York : Gordon & Breach, 1966).

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treated in them. So far as I know, Ericksen has never written a historical paper, but what I learned with his help was of much value a little later, in the portions of The Classical Field Theories that I composed.

The introduction to that work states : "Our subject is largely the creation of Euler and Cauchy." Indeed, in mechanics Cauchy wrote in much the same style as Euler did, and in that field he should be regarded as the great continuator of the deepest aspects of Euler' s researches. Euler' s publication lasted longer than his life, ending only five years before Cauchy' s death; in contrast with Euler's, Cauchy's contributions to mechanics referred mostly to the foundations, with little attention to special problems.

The Classical Field Theories refers to fourteen papers by Cauchy. The passages of text affected by them are found on eighty-three of its more than 700 pages. Another measure of Cauchy's influence may be seen from the dates of the papers cited, his and others: his all lie within a span of twenty-one years, while the list of references in the treatise runs from 1678 through 1960, and more than half of those came from the last forty-six years. Finally, during the twenty-one years in which Cauchy wrote those fourteen papers, he wrote dozens of others, on nearly every branch of mathematics, mechanics, and physics current in his time — and then there were, from 1836 onward, the 589 notes in the Comptes Rendus.

The references in The Classical Field Theories are far from being mere citations. Details from Cauchy's papers are presented and related to the works of others: predecessors, successors, and contemporaries. The Classical Field Theories is not a history. It is a connected, mathematical treatise on the foundations of continuum mechanics as they appeared in 1960 — a treatise with detailed attributions, which I strove to render as complete and correct as I could. In most instances the demonstrations published in that work are shorter than the original ones.

It is a matter of fact, confirmed by the Citation Index and by the bibliographies of articles on continuum mechanics from 1960 until the present time, that The Classical Field Theories has been and still is being widely consulted. So much for the slender pipe through which Cauchy's discoveries in the mechanics of continua came to be known in the twentieth century.

I turn now to some particular aspects of Cauchy's work that have been developed later by mathematicians. None of the studies

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I shall mention could justly be dismissed as "applied mathematics" ; all concern the foundations, the concepts of mechanics; and all are presented in Cauchy's tradition, employing, as he himself did, recently introduced tools of analysis which led to demonstrations at the contemporary standard of mathematical rigor.

III. — Cauchy's program and first discoveries

Cauchy's first paper on the foundations of continuum mechanics is his "Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non-élastiques", 1823 (8). In Navier's theory of elastic planes, Cauchy writes, there are two kinds of forces: those that lie in the plane and are perpendicular to lines on which they act, and those that are perpendicular to the plane. Cauchy could reduce these two forces to one, but instead he turns at once to three-dimensional elastic solids. This choice reflects his sagacity. For fluids, in the eighteenth century the theories began with one-dimensional hydraulics and then easily proceeded to flows in two and three dimensions. In elasticity, on the contrary, the grand success of the theory of the elastica in the plane was followed by failures with plates and shells, culminating in the debacle of Sophie Germain and, later, of Poisson. Whether Cauchy had read Euler's paper deriving the general equations of motion (1774) for a plane line subject to both tension and shear in its plane (9), I do not know, but they would not have suggested the right course to him. He writes:

"Si dans un corps solide élastique ou non élastique on vient à rendre rigide et invariable un petit élément du volume terminé par des faces quelconques, ce petit élément éprouvera sur ses différentes faces, et en chaque point de chacune d'elles, une pression ou tension déterminée. Cette pression ou tension sera semblable à la pression qu'un fluide exerce contre un élément de l'enveloppe d'un corps solide, avec cette seule dif-

(8) Bulletin des sciences, par la Société Philomatique de Paris, (1823), 9-13. Reprinted in OC (2), II, 300-304.

(9) L. Euler, De gemina methodo tam aequilibrium quam motum corpora flexibilium determinandi et utriusque egregio consensu (1774), Novi commentarii academiae Scientiarum Petropolitanae, XX (1775), 1776, 180-193. Reprinted in OO (2), XI, 180-193.

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férence, que la pression exercée par un fluide en repos contre la surface d'un corps solide, est dirigée perpendiculairement à cette surface de dehors en dedans, et indépendante en chaque point de l'inclinaison de la surface par rapport aux plans coordonnés, tandis que la pression ou tension exercée en un point donné d'un corps solide contre un très petit élément de surface passant par ce point, peut être dirigée perpendiculairement ou obliquement à cette surface, tantôt de dehors en dedans, s'il y a condensation, tantôt de dedans en dehors, s'il y a dilatation, et peut dépendre de l'inclinaison de la surface par rapport aux plans dont il s'agit. De plus, la pression ou tension exercée contre un plan quelconque se déduit très facilement, tant en grandeur qu'en direction, des pressions ou tensions exercées contre trois plans rectangulaires donnés [...].

"Du théorème énoncé plus haut, il résulte que la pression ou tension en chaque point est équivalente à l'unité divisée par le rayon vecteur d'un ellipsoïde. Aux trois axes de cet ellipsoïde correspondent trois pressions ou tensions que nous nommerons principales, et l'on peut démontrer que chacune d'elles est perpendiculaire au plan contre lequel elle s'exerce. Parmi ces pressions ou tensions principales se trouvent la pression ou tension maximum, et la pression ou tension minimum. Les autres pressions ou tensions sont distribuées symétriquement autour des trois axes..."

Thus Cauchy has proved the existence of a stress tensor and some of its properties. In terms of it, the pressures and tensions on planes through a given point are determined. Cauchy continues:

"Cela posé, si l'on considère un corps solide variable de forme et soumis à des forces accélératrices quelconques, pour établir les équations d'équilibre de ce corps solide, il suffira d'écrire qu'il y a équilibre entre les forces motrices qui sollicitent un élément infiniment petit dans le sens des axes coordonnés, et les composantes orthogonales des pressions ou tensions extérieures qui agissent contre les faces de cet élément. On obtiendra ainsi trois équations d'équilibre qui comprennent, comme cas particulier, celles de l'équilibre des fluides. Mais, dans le cas général, ces équations renferment six fonctions inconnues des coordonnées [...]. Il reste à déterminer les valeurs de ces six inconnues; mais la solution de ce dernier problème varie suivant la nature du corps et son élasticité plus ou moins parfaite."

That is, the stress tensor, having only six possibly independent components, is symmetric, and Cauchy has separated the general dynamics of a continuous body from the nature of the material composing it. That had been done earlier in narrower contexts by James Bernoulli and Euler but was hot given much attention during the period around the turn of the century. We now call specification of "the nature of the body and its elasticity, more

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or less perfect", a constitutive equation. Cauchy's examples are, first, a material now called isotropic and Cauchy elastic, and, second, "un corps entièrement dépourvu d'élasticité", later called "un corps mou", which he does not specify completely enough to obtain a theory of plasticity. He does indicate that he has found the distribution of condensations and dilatations at a point and has discovered what are now called the principal axes of deformation and the principal extensions.

During the nineteenth and early twentieth centuries there were scattered attempts to determine constitutive relations of various kinds. V. Mises' lecture of 1930 concerned them, and the rise of modern continuum mechanics from the 1940s onward made them a primary subject of study. In 1958 Walter Noll constructed a fairly general theory of constitutive relations (10). He was co-author of The Non-linear Field Theories of Mechanics (11). While that book refers to five works by Cauchy, it rests largely on ideas conceived in the twentieth century, of course not without precursors in special instances.

IV. — Cauchy's general theory of stress

Cauchy based his development of stress upon the principles of statics: equilibrium of forces and equilibrium of torques. In his paper of 1823, "De la pression ou tension dans un corps solide (12)", he assumed that the resultant applied force on any part P of a body was the sum of two forces:

/ = /в + /с»

the former of which is the resultant of the exterior forces acting upon the interior of P and is usually called the body force, while the latter is the contact force, which is the resultant force arising

(10) W. Noll, A mathematical theory of the mechanical behavior of continuous media, Archive for Rational Mechanics and Analysis, II (1958-1959), 197-226.

(11) C. Truesdell and W. Noll, The Non-linear Field Theories of Mechanics, in Fliigge's Encyclopedia of Physics, vol. III/3 (Berlin : Springer-Verlag, 1965), v-vm and 1-662. (Second edition, corrected, to appear in 1992.)

(12) Exercices de Mathématique, t. 2 (1823), 42-56. Reprinted in ОС (2), VII, 60-78.

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from a vector field tdP acting upon the surface ЭР and called the traction:

fc = I tdPdA, А = area. Jap

In hydrostatics tdP = — pn, in which p is the pressure and n is the outer unit normal of ЭР. For Cauchy the direction and magnitude of tdP was to be subject only to the requirement that, for all ЭР having a common tangent plane at the place x, the vector tdP would be the same:

tap = f(x> n).

This assumption is called the Cauchy postulate. Some people say that Cauchy's reasoning, which is not very clear on this point, does not really require this postulate. Be that as it may, the assumption is standard in careful later presentations.

I shall now recall Cauchy's main arguments in this memoir, merely outlining them roughly and not reproducing details. Cauchy first proved his fundamental lemma:

/(*, - ri) = -f{x,n).

To do so, he introduced "the pillbox argument". A disk S contains the place x and lies upon the tangent plane to a surface within the body at x. We describe a prism perpendicular to that plane, of height e above the plane and e below it, and we suppose that prism to be in equilibrium subject to the body force and contact force acting upon it. In the limit as e — 0 the resultant body force vanishes because the volume vanishes, but the contact forces on the top and bottom of the prism are not null; they must cancel each other. Letting the area of S vanish yields the fundamental lemma.

Cauchy went on to set up what is now called "the tetrahedron argument". An arbitrary force is applied to the inside of the face normal to n, and then the forces on the insides of the faces on the co-ordinate planes needed to keep the tetrahedron in equilibrium are calculated. As the tetrahedron is made to shrink down to the origin, its volume approaches zero faster than does its surface area. The product traction x area on the inside of the slanted

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t(n)

t(-n)

Fig. 1. — Sketch to illustrate "the pillbox argument'

Fig. 2. — Sketch to illustrate "the tetrahedron argument"

face becomes equal to the sum of the three corresponding products on the co-ordinate planes. Application of the fundamental

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lemma delivers the product traction x area for the outside of the slanted face. In modern notation,

t№ = f(x, n) = T(x)n,

in which T denotes a linear operator, the stress tensor. Thus the traction on ЭР at x results from operating by T(x) on the normal n to ЭР at the point x. This statement is Cauchy *s fundamental theorem on existence of the stress tensor.

Cauchy in the great paper we have been considering dealt only with bodies in equilibrium. He neglected to mention that his conclusions here remain valid in dynamics because the inertial forces, like the body forces, are expressed as integrals over volumes.

Cauchy's fundamental theorem is what makes continuum mechanics possible. If we did not have it, we could scarcely set up, let alone solve, any particular problem or demonstrate any valuable theorem.

Constitutive relations usually specify the stress tensor in terms of the deformations: T is determined by the deformations undergone or the histories of those deformations. Without T, we should not know how to approach mathematical spécification of the mechanical response of a material.

Cauchy's march of ideas is splendid. His arguments are easily made mathematically acceptable today by rather straightforward technical assumptions: use of the mean-value theorem, etc. The great masters proved their theorems in the course of discovering them. The moderns, more delicate, when they see the strokes of broad chisels driven by giants' malls, discern rough spots, missing elements, and sometimes even an armature ready to collapse and hence in need of replacement. The aesthetic criteria as well as the sharpened faculty of criticism of modern mathematics rarely leave unimproved any old piece of mathematics, especially if it is a major theorem. A mathematician today easily sees places where the assumptions made to prove Cauchy's fundamental theorem are unnecessarily strong. I list them in order of increasing importance.

1. The Cauchy postulate should be a consequence of the nature of contact forces, not an assumption.

2. The tetrahedron argument delivers a fourth vector from three linearly indépendant vectors, but any four vectors are linearly

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dependent. Thus there seems to be one vector too many in the argument.

3. The assumption that body forces and contact forces exist should be replaced by mathematical proof that the systems of forces appropriate to continuum mechanics necessarily are sums of forces of these two, different kinds.

In a long sequence of studies beginning with Walter Noll's communications to two symposia on the axiomatic method, one at Berkeley in 1957 and the other at Paris in 1959 (13), and continuing with papers by him and by M. E. Gurtin, W. O. Williams, V. Mizel, M. Šilhavý, W. Ziemer, R. L. Fosdick, E. Virga, and others (14), all these lacunae have been filled, and new directions have been opened and pursued. In what follows I will list only those studies that concern Cauchy's line of argument directly.

First, in 1957 Noll proved the "Cauchy Postulate" to be superfluous ; it follows as a theorem from Cauchy's fundamental lemma and geometrical considerations. Stephen M. Winters in 1976 found imperfect arguments to the same effect in works by Hamel (1909-1927) (15). Perhaps something of the same kind is included in Cauchy's own presentation, and so a historical question remains here.

(13) These two papers are reprinted as Nos. 9 and 21 in W. Noll, The Foundations of Mechanics and Thermodynamics (New York : Springer- Verlag, 1974).

(14) Most of these papers, fairly numerous, but not all of them, appeared in Archive for Rational Mechanics and Analysis, from 1967 to the present time. The earliest ones are developments of Noll's сше lectures (1965), The Foundations of Mechanics in Nonlinear Continuum Mechanics (Rome : Cremonese, 1966), and a corrected version in a report of the Mathematics Department, Carnegie-Mellon University, written later but issued in 1965. The later literature is most easily traced by consulting the indices for the first 100 volumes of the Archive, at the ends of volumes 50 (1973) and 100 (1988). Works published in journals other than the Archive may be located by consulting the lists of references of those that did appear there.

(15) I rely here on the unpublished M. S. thesis of Stephen M. Winters, Johns Hopkins University, 1976. The work of Hamel on the foundations of mechanics began with: Uber die Grundlagen der Mechanik, Mathematische Annalen, LXVI (1908), 350-397, which was an attempt to solve a part of Hubert's sixth problem. Hamel in his article, Die Axiome der Mechanik, in H. Geiger and К. Scheel (eds.), Handbuch der Physik, V (Berlin : Springer- Verlag, 1927), explicitly adopted Hilbert's basic idea: The concepts used in a mathematical theory are the primitive quantities and the axioms laid down for them. While Hamel's analysis is crude and insufficient, he must be regarded as the pioneer in twentieth-century axiomatic studies of mechanics.

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In 1967 Walter Noll constructed an ingenious proof, using no co-ordinates, of Cauchy's fundamental theorem based on use of three arbitrary, linearly independent vectors (16).

V. — Substantial universes and systems OF FORCES IN CONTINUUM MECHANICS

In continuum mechanics Euler and Cauchy always considered only resultant forces upon bodies, and the later tradition did the same. That made reasoning on the foundation of continuum mechanics basically different from that which is used in the analytical dynamics of mass-points, which always considers systems of forces in which each point attracts or repels other points. That difference forced Cauchy to introduce contact forces and body forces a priori. Walter Noll in his lectures of 1965 (17) made mechanics in general rest upon mutual forces between pairs of bodies. In 1959 and 1965 he had introduced the concept of "material universe" (18), which is a Boolean algebra, the elements of which are bodies or the shapes of bodies. In recent work of Noll & Virga (19), those shapes are "fit sets" in Euclidean space, namely, regularly open, bounded sets of finite perimeter and negligible boundary.

On a substantial universe a mapping of pairs of separate bodies, Ct, С into a vector-space may be interpreted as delivering the force f(GL, Q) that С exerts upon G. The resultant force upon (B is then /((B, (B*), in which (Be is the exterior of (B. If /((B, (Be) = 0 for every (B, the system of forces is balanced. For three-dimensional continua the value of /(Ct, 6) is a vector in the translation space of Euclidean 3-space. The forces in the system are subject to rather natural axioms. Noll proved that if the system of forces (including inertial force) is balanced, then the system of forces is pairwise equilibrated:

да,е)= -де,а).

(16) The first publication of Noll's proof is found on pages 48-49 of the article by M. E. Gurtin, The Linear Theory of Elasticity in volume VIa/2 (edited by C. Truesdell) of Flugge's Encyclopedia of Physics (Heidelberg : Springer- Verlag, 1972).

(17) Cited above in footnote 14. (18) The former work is cited in footnote 13 ; the latter (сше lectures), in footnote 14. (19) W. Noll and E. Virga, Fit Regions and Functions of Bounded Variation, Archive

for Rational Mechanics and Analysis, CII (1988), 1-21.

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20 С. Truesdell

A statement of this kind is usually called "Newton's Third Law" and is taken as an axiom. Noll's theorem makes it a demonstrated property in the context of a mathematically defined and developed system of forces.

In such a system it is easy to state an axiom delimiting the quality natural to forces exerted by one continuous body upon another:

Let GL and Q be separate bodies, the area of contact of whose shapes is sufficiently small, and let the mass of 6L be sufficiently small. Then

I/(Q, e)| < kao*x(G) n д*х(еу> + кем(а>.

Here К is a positive constant; Ke is a positive, bounded function of С such that Ke — 0 as M(C) — 0; the symbol M denotes "mass of ", x denotes "shape of", and d* denotes "reduced boundary of". The axiom states, roughly, that the magnitude of the force exerted by С on Oř is bounded both by the area of their contact and by the mass of d. Researches of Noll, Gurtin, Williams, Ziemer, and Virga stretching over the past fifteen (20) years demonstrate that from this axiom alone Cauchy's Fundamental Theorem follows. Here are the main intermediate steps (21): the system of forces / is proved to be the sum of two systems, /B and /c, the former bounded by area, the latter, bounded by mass:

second, the same bounds apply to the subbodies of Cfc and C; third, the contact force is delivered as the integral of a bounded traction vector ; fourth, both the systems fB and /c are pairwise equilibrated (though neither, generally, is an equilibrated system).

The final step proves the existence of Cauchy's stress tensor.

VI. — Cauchy's dynamics of continua

In the volume in which Cauchy demonstrates the existence and properties of the stress tensor is a note (22), "Sur les relations

(20) Cf. footnotes 14 and 19. (21) An outline of this development, provided with some of the proofs, is given in

chapter III of C. Truesdell, A First Course in Rational Continuum Mechanics, vol. I, second edition (Boston : Academic Press, 1991).

(22) ОС (2), VII, 141-145.

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qui existent dans l'état d'équilibre d'un corps solide ou fluide entre les pressions ou tensions et les forces accélératrices", in which he obtains the partial-differential equation of equilibrium of a continuous medium. The derivation follows Euler's for the equations of hydrodynamics. Though Cauchy does not use a diagram, we can see one in his equations: infinitesimal forces act on the faces of an infinitesimal parallelepiped. We now regard such proofs as being crude, early arguments toward the divergence theorem, mixed with some application of it, often ingenious. Some books on engineering still dazzle beginners with arguments referring to pictures of this kind. A mathematician today does not consider them wrong; rather, they are ugly, unnecessarily long, repetitious, and boring.

In volume HI (p. 195-226), in the memoir "Sur les équations qui expriment les conditions d'équilibre ou les lois du mouvement intérieur d'un corps solide, élastique ou non élastique (23)", Cauchy adds the force of inertia to the density of the body force.

In concept Cauchy' s dynamics rests upon Euler's Laws of Motion (24), namely, for every part (P of any body,

resultant applied force f = m = rate of change of linear momentum,

resultant applied torque F" = M = rate of change of rotational momentum.

In these statements the balance of linear momentum and the balance of rotational momentum are placed on a par as basic, independent principles of general mechanics:

pxďV =

x0)ApxdV = Fa((?)Xo,

in which x means "the shape of", x is the acceleration, and x0 is an arbitrary place in an inertial frame. When specialized to continuum mechanics these laws become

(23) ОС (2), VIII, 195-226. (24) L. Euler, §27-29 of Nova methodus motum corporum rigidorum determinandi,

Novi Commentarii Academiae Scientiarum Petropolitanae, XX (1775), 1776, 208-238. Reprinted in OO (2), IX, 112-113.

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22 С. Truesdell

pxdV = tdmdA + pbdV, J *(($>) J àxm J

- Xo) Л /Ш/V = (X - X0) Л ^((P) J az((P)

+ (* - лг0) Л pbďV,

for all parts (P of all bodies in question. Here p is the mass density, b is the density of external force. Putting Tn for tdjc((f) in accordance with Cauchy's Fundamental Theorem and then using the divergence theorem, we readily obtain Cauchy's First Law of Motion for continua:

p'x = div T + pb,

which is a necessary and sufficient condition that linear momentum be balanced for all subbodies in the interior of a region where p'x, pb, T and div T are continuous.

In this line of argument Euler's Second Law is not used, and so the stress tensor T need not be symmetric, but in Cauchy's derivation it is symmetric from the start. That happens because in his paper of 1827, which contains his proof that in equilibrium a stress tensor exists, he proved also that it is symmetric (his Theorem II). He did so by invoking the static balance of moments. Again a lacuna results when Cauchy comes to dynamics. He may well have seen that as the dynamic accelerating forces are given by integrals with respect to volume, the local properties of the stress tensor, because they derive from integrals with respect to area, carry over from statics to dynamics, but he does not say so.

Later authors have preferred a more straightforward argument based directly on Euler's Second Law when applied to continua (above). From that argument we easily obtain what we call Cauchy's Second Law of Motion:

On the assumption that linear momentum is balanced, rotational momentum is balanced if and only if:

тт = T

The superscript T denotes "transpose". This conclusion requires that the stress tensor be symmetric. (This formulation appears in

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Love's Treatise on the Mathematical Theory of Elasticity, 1892, and he writes there: "The theorem expressed by these equations is due to Cauchy.") In the second edition, 1906, Love is more verbose and less clear ; to prove the symmetry of the stress tensor he resorts to an infinitesimal parallelepiped decorated with arrows ; as he wrote in his preface, his first aim was "to make the book more accessible to engineers". Nevertheless he did not delete the divergence theorem entirely.

The essence of Cauchy' s argument can be applied in domains other than elasticity and has been. We begin from an equation or inequality of integral balance:

in which 3) is a domain of integration while В and С are tensor fields, the former being a density with respect to volume, the latter being a density with respect to surface. I omit here mathematical conditions that 3), В and С need satisfy. It is a problem of analysis to determine from the inequality a differential relation valid locally when the arguments В and С are sufficiently smooth. In our book (25) Chao-Cheng Wang and I, making use of analyses by Noll, Gurtin, Mizel, and Williams, treated this problem with all the generality then available. We proved general counterparts of Cauchy's fundamental lemma, fundamental theorem, and equations of motion, all under rather weak assumptions. The local equivalent of the integral inequality is:

В ^ div C

in regions where В and С are sufficiently smooth. On surfaces of discontinuity, if sufficiently regular, the integral inequality delivers a relation connecting the jumps of В and C.

After these conclusions became somewhat known in mathematical circles outside continuum mechanics, analysts began to see that equations of integral balance expressed concepts worthy of abstraction and extension. Among those who developed recently the general theory of what are called "Cauchy fluxes" are Ziemer and Šilhavý.

(25) C.-C. Wang and C. Truesdell, Introduction to Rational Elasticity (Leyden : Noord- hoff, 1973).

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VII. — Conclusion

I have told you about some recent work growing from Cauchy's innovations in the general theory of continuum mechanics. For this lecture I have drawn on only five of Cauchy's papers. Many others retain value for mathematicians today. The article on Cauchy by the late Professor Freudenthal in the Dictionary of Scientific Biography is the earliest fair, scientific account of Cauchy's work that is plausible and comes near to doing justice to his magnificent creations, concepts, and analysis. Now we have Mr Belhoste's splendid book, Augustin-Louis Cauchy : A Biography (Springer- Verlag, 1991), in which we learn at least outlines and often details of much of Cauchy's work. We get a panorama of his scene as it changed in time, and we may understand the firm positions he took and his sacrifices in defending them. Freudenthal wrote of Cauchy's "greatest achievement, [which] would suffice to assure him a place among the greatest scientists: the founding of elasticity theory." Here I have addressed only a part of that achievement, namely, Cauchy's work on the foundations of theories of large strain and the forces that can effect it.

Those who are interested in the modern developments growing from what Cauchy did do and from what he did not do may consult my book, Introduction à la mécanique rationnelle des milieux continus (26) or, better, the second edition in English (27).

Not every mathematician, however great, has established a tradition like Cauchy's, still burning more than 130 years after his death.

Clifford A. Truesdell.

(26) Paris : Masson, 1974. (27) Cf. footnote 21.