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Cauchy and the modern mechanics of continua - article ; n°1 ; vol.45, pg 5-24


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Revue d'histoire des sciences - Année 1992 - Volume 45 - Numéro 1 - Pages 5-24
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.
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Source : Persée ; Ministère de la jeunesse, de l’éducation nationale et de la recherche, Direction de l’enseignement supérieur, Sous-direction des bibliothèques et de la documentation.


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Cauchy and the modern mechanics of continua
In: Revue d'histoire des sciences. 1992, Tome 45 n°1. pp. 5-24.
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.
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_4229and the modern mechanics Cauchy
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 rigou
reuse 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 mechan
ics 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 mathem
atical theories of large deformation of material bodies. Several
limitations follow.
a) Those who created modern continuum mechanics were not his
torians. 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. Histo
rians 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 6 С. Truesdell
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 mathemat
icians 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 unpub
lished, 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 nota
tions were needed to present them. The hoipolloi of mathemat
ics 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 and the modern mechanics of continua 7 Cauchy
formal manipulations, which Cauchy attacked, and rejected all
that unnecessary rigor he attempted to promote. Modern cont
inuum 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 Ency
clopedia 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 suc
cessful creator of theories of bodies modelled as assemblies of small
masses, often called molecules, especially when taken in arrange
ments 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 socio
logy 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 indi
cating 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. 8 С. Truesdell
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 cont
inuum mechanics and on the statistical mechanics of systems of
mass-points. I feel neither commitment nor guilt.
II. — Background, composition
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 suffi
cient 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. and the modern mechanics of continua 9 Cauchy
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 physic
ists 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 approxi
mations and ugly formulae, opaque and tedious, short on funda
mental 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 sim
plicity; 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 oppos
ite. I searched for the foundations of continuum theories of all
kinds, in kinematics, in dynamics, in variety and nature of re
sponse 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 geo
metry 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 intro
duces 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 deter
mines 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 Mater
ials from Galilei to Lord Kelvin, edited and completed... by Karl Pearson, 2 vol. (Camb
ridge : Cambridge Univ. Press, I : 1886 and II : 1893).
(5) A.-L. Cauchy, ОС (2), XII, 343-377. 10 С. Truesdell
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 ment
ioned 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 stu
dents 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, and
addenda II (1953), 593-616, and III (1954), 801. Corrected reprint : C. Truesdell, Cont
inuum Mechanics, I (New York : Gordon & Breach, 1966). and the modern mechanics of continua 1 1 Cauchy
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 refe
rences 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 mathemati
cs, 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 cont
inuum 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 12 С. Truesdell
I shall mention could justly be dismissed as "applied mathemat
ics" ; 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 demonstrat
ions at the contemporary standard of mathematical rigor.
III. — Cauchy's program and first discoveries
Cauchy's first paper on the foundations of continuum mechan
ics 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 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 theo
ries began with one-dimensional hydraulics and then easily pro
ceeded 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. and the modern mechanics of continua 13 Cauchy
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 condens
ation, 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 ten
sions 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 pres
sions ou tensions que nous nommerons principales, et l'on peut démont
rer que chacune d'elles est perpendiculaire au plan contre lequel elle
s'exerce. Parmi ces pressions ou tensions principales se trouvent la pres
sion ou tension maximum, et la pression ou tension minimum. Les autres
pressions ou tensions sont distribuées symétriquement autour des trois
Cauchy has proved the existence of a stress tensor and some Thus
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