# What Happened on September 30, 1822, and What Were its Implications for the Future of Continuum Mechanics?

## Abstract

This contribution offers a discussion about the notion of stress in a general continuum as initially proposed in a magisterial paper by Cauchy in 1822 (but published only in 1828) without using arguments involving molecules. This is here presented in its historical context. Cauchy’s view is the currently accepted view among mechanicians and engineers although attempts (including by Navier and Cauchy himself) to start from a molecular description in the manner of Newton and Laplace were constantly offered in both nineteenth and twentieth centuries. The discussion introduces other secondary stress definitions such as those by Piola, Kirchhoff, and more recently Eshelby. The question naturally arises of what happens with the possibility to introduce other internal forces such as hyperstresses (in so-called gradient theories) and couple stresses (e.g., in Cosserat continua), and whether some introduced stresses have associated with them a meaningful boundary condition. Also pondered is the question whether one can identify a stress concept in physical approaches still considering interactions between point particles (lattice dynamics, kinetic theory, nonlocal theory, statistical-mechanics approach). The chapter is concluded by a more in depth discussion of the notion of stress-energy-momentum, culminating in that of pseudo-tensor of energy-momentum in gravitation theory.

## Keywords

Cauchy stress Piola-Kirchhoff stress Eshelby stress Energy-momentum tensor Natural boundary conditions## Notes

### Acknowledgments

Heartful thanks go to Dr Martine Rousseau in Paris and Professor James Casey in Berkeley for their critical careful reading of this contribution that led to much improvement and readability. Mme Florence Greffe (“Conservateur du Patrimoine”) from the Archives Library of the Paris Academy of Science is to be thanked for her definite help in providing the “birth certificate” of continuum mechanics.

## References

- 1.Belhoste B (1991) Augustin-Louis Cauchy: a biography. Springer, New York (English trans: French original “Cauchy, 1789–1857”, Editions Belin, Paris)Google Scholar
- 2.Casal P (1963) Capillarité interne en mécanique. CR Acad Sci Paris 256:3820–3822zbMATHGoogle Scholar
- 3.Cauchy AL (1823) Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques. Bull Soc Filomat Paris 9–13Google Scholar
- 4.Cauchy AL (1828) Sur les équations qui expriment l’équilibre ou les lois du mouvement intérieur d’un corps solide élastique ou non élastique. Exercices de mathématiques, vol 2, pp 160–187, Sept 1828 (This presents in print the ideas originally submitted to the Paris Academy of Sciences on 30 Sept 1822)Google Scholar
- 5.Cauchy AL (1828) In: Exercices de Mathématiques, vol 3, Sept 1828, pp 188–212, and «De la pression ou tension dans un système de points matériels» Oct 1828, pp 213–236Google Scholar
- 6.Cosserat E, Cosserat F (1909) Théorie des corps déformables. Hermann, Paris (reprint, Gabay, Paris, 2008)Google Scholar
- 7.Costa de Beauregard O (1944) Sur les équations fondamentales, classiques, puis relativistes, de la dynamique des milieux continus. J Math Pures Appl 23:211–217zbMATHMathSciNetGoogle Scholar
- 8.Costa de Beauregard O (1945) Définition covariante de la force. CR Acad Sci Paris 221:743–747Google Scholar
- 9.Costa de Beauregard O (1946) Sur la théorie des forces élastiques. CR Acad Sci Paris 222:477–479zbMATHGoogle Scholar
- 10.Dahan-Dalmenico A (1984–1985) La mathématisation de la théorie de l’élasticité par A.L. Cauchy et les débats dans la physique mathématique française (1800–1840). Sciences et techniques en perspective 9:1–100Google Scholar
- 11.Dell’Isola F, Seppecher P (1995) The relationship between edge contact forces, double forces and intersticial working allowed by the principle of virtual power. CR Acad Sci Paris IIb 321:303–308zbMATHGoogle Scholar
- 12.Dell’Isola F, Seppecher P, Madeo A (2012) How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach
*à la*d’Alembert. Zeit angew Math und Physik 63(6):1119–1141Google Scholar - 13.Dirac PAM (1975) General theory of relativity. Wiley, New York (reprint: Princeton University Press, Princeton, 1996)Google Scholar
- 14.Duhem P (1903) L’évolution de la mécanique (published in seven parts in: Revue générale des sciences, Paris; as a book: A. Joanin, Paris, 1906) (English trans: The evolution of mechanics, Sijthoff and Noordhoff, 1980)Google Scholar
- 15.Eckart CH (1940) The thermodynamics of irreversible processes III: relativistic theory of the simple fluid. Phys Rev 58:919–924CrossRefGoogle Scholar
- 16.Einstein A (1916) Das Hamiltonisches Prinzip und allgemein Relativitätstheorie. Sitzungsber preuss Akad Wiss 2:1111–1115 (also, ibid, 1 (1918): 448–459)Google Scholar
- 17.Eringen AC (2002) Nonlocal continuum field theories. Springer, New YorkzbMATHGoogle Scholar
- 18.Eringen AC, Maugin GA (1990) Electrodynamics of continua, (Two volumes), Springer, New YorkCrossRefGoogle Scholar
- 19.Eshelby JD (1951) Force on an elastic singularity. Phil Trans Roy Soc Lond A244:87–111CrossRefMathSciNetGoogle Scholar
- 20.Germain P (1973) La méthode des puissances virtuelles en mécanique des milieux continus, Première partie: théorie du second gradient. J Mécanique (Paris) 12:235–274zbMATHMathSciNetGoogle Scholar
- 21.Green G (1828) An essay on the mathematical analysis of electricity and magnetism, privately printed, Nottingham (also in: mathematical papers of George Green, Ferrers NM (ed) Macmillan, London, 1871; reprinted by Chelsea, New York, 1970)Google Scholar
- 22.Grattan-Guinness I (1993) The ingénieur-savant (1800–1830): a neglected figure in the history of French mathematics and science. Sci Context 6(2):405–433CrossRefMathSciNetGoogle Scholar
- 23.Gurtin ME (1999) Configurational forces as basic concepts of continuum physics. Springer, New YorkGoogle Scholar
- 24.Korteweg DJ (1901) Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation de la densité. Arch Néer Sci Exactes et Nat Sér. II 6:1–24Google Scholar
- 25.Kunin IA (1982) Elastic media with microstructure I and II. Springer, Berlin (trans: 1975 Russian edition Kröner E, ed)Google Scholar
- 26.Landau LD, Lifshitz EM (1971) The classical theory of fields, vol 2 of course on theoretical physics, 3rd edn (trans: Russian). Pergamon Press, OxfordGoogle Scholar
- 27.Le Corre Y (1956) La dissymétrie du tenseur des efforts et ses conséquences. J Phys Radium 17:934–939CrossRefzbMATHMathSciNetGoogle Scholar
- 28.Le Roux J (1911) Etude géométrique de la torsion et de la flexion, dans les déformations infinitésimales d’un milieu continu. Ann Ecole Norm Sup 28:523–579zbMATHMathSciNetGoogle Scholar
- 29.Mandel J (1971) Plasticité classique et visco-plasticité. CISM Lecture Notes, Udine, ItalyGoogle Scholar
- 30.Maugin GA (1975) On the formulation of constitutive laws in the relativistic mechanics of continua (in French) (Main document of) Thèse de Doctorat ès Sciences Mathématiques, Université de Paris-VI, p 164, ParisGoogle Scholar
- 31.Maugin GA (1980) The method of virtual power in continuum mechanics: application to coupled fields. Acta Mech 35:1–70CrossRefzbMATHMathSciNetGoogle Scholar
- 32.Maugin GA (1988) Continuum mechanics of electromagnetic solids. North-Holland, AmsterdamzbMATHGoogle Scholar
- 33.Maugin GA (2010) Generalized continuum mechanics: what do we understand by that? In: Maugin GA, Metrikine AV (eds) Mechanics of generalized continua: one hundred years after the Cosserats. Springer, New York, pp 3–13CrossRefGoogle Scholar
- 34.Maugin GA (2011) Configurational forces. CRC/ Chapman & Hall/Taylor and Francis, Boca RatonzbMATHGoogle Scholar
- 35.Maxwell JC (1873) Treatise on electricity and magnetism. Clarendon Press, OxfordGoogle Scholar
- 36.Murdoch IA (2012) Physical foundations of continuum mechanics. Cambridge University Press, UKCrossRefzbMATHGoogle Scholar
- 37.Neuenschwander DE (2011) Emmy noether’s wonderful theorem. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
- 38.Noether EA (1918) Invariante variationsprobleme. Nach Akad Wiss Gïttingen, Math-Phys, Kl. II: 235–257 (English trans: Tavel M, Transp Theory Stat Phys I: 186–207, 1971)Google Scholar
- 39.Noll W (1974) The foundations of classical mechanics in the light of recent advances in continuum mechanics. Reprinted in W. Noll, The foundations of mechanics and thermodynamics (selected papers of W. Noll). Springer, Berlin, pp 32–47Google Scholar
- 40.Noll W, Virga E (1990) On edge interactions and surface tension. Arch Rat Mech Anal 111:1–31CrossRefzbMATHMathSciNetGoogle Scholar
- 41.Norton JD (1993) General covariance and the foundations of general relativity: eight decades of dispute. Rep Prog Phys 56:791–858CrossRefMathSciNetGoogle Scholar
- 42.Piola G (1836) Nuova analisi per tutte le questioni della meccanica moleculare. Mem Mat Fis Soc Ita. Modena 21(1835):155–321Google Scholar
- 43.Piola G (1848) Intorno alle equazioni fondamentali del movimento di corpi qualsivoglioni considerati secondo la naturale loro forma e costituva. Mem Mat Fiz Soc Ital Modena 24(1):1–186Google Scholar
- 44.Timoshenko SP (1953) History of the strength of materials. McGraw Hill, New York (Dover reprint, New York, 1983) (review of this book by Truesdell reprinted in Truesdell 1984, pp 251–253)Google Scholar
- 45.Truesdell CA (1968) Essays in the history of mechanics. Springer, New YorkCrossRefzbMATHGoogle Scholar
- 46.Truesdell CA (1984) An idiot’s fugitive essays on science. Springer, New YorkCrossRefzbMATHGoogle Scholar
- 47.Truesdell CA, Toupin RA (1960) The classical theory of fields. Handbuch der Physik, Ed. S.Flügge, Bd III/1, Springer, BerlinGoogle Scholar
- 48.Truesdell CA, Noll W (1965) The Nonlinear field theory of mechanics. Handbuch der Physik, Ed. S. Flügge, Bd III/3, Springer, BerlinGoogle Scholar
- 49.Van Dantzig D (1939) Stress tensor and particle density in special relativity. Nature (London) 143:855CrossRefzbMATHGoogle Scholar