A Successful Attempt at a Synthetic View of Continuum Mechanics on the Eve of WWI: Hellinger’s Article in the German Encyclopaedia of Mathematics

Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 214)

Abstract

This essay analyses the comprehensive nature of a remarkable synthesis published by Hellinger (Die allgemein ansätze der mechanik der kontinua. Springer, Berlin, pp. 602–694, 1914) in a German encyclopaedia. In this contribution Hellinger, a mathematician, succeeds in capturing the progress and subtleties of all what was achieved during the nineteenth century, accounting for most recent works and also pointing at forthcoming developments. On this occasion, the scientific environment of Hellinger is perused and the style of Hellinger and his excellent comprehending of continuum mechanics are evaluated from a document that is a true landmark in the field although often ignored.

Keywords

Continuum mechanics Variational principles Finite strains Oriented media 

References

  1. 1.
    Appell P (1909) Traité de mécanique rationnelle, vol 3. Mécanique des milieux Continus, Gauthier-Villars, ParisGoogle Scholar
  2. 2.
    Boltzmann L (1874) Zur theorie der elastischen Nachwirkung. Sitz K u K Akad Wien 70:275–306Google Scholar
  3. 3.
    Born M (1906) Untersuch, Über die Stabilität der elastischen Linie. Preisschrift, GöttingenGoogle Scholar
  4. 4.
    Born M (1909) Die theorie des starren elektrons in der kinermatik des relativitätsprinzips. Ann der Physik 30:1–56CrossRefMATHGoogle Scholar
  5. 5.
    Brill A (1909) Vorlesungen zur einführung in die mechanik raumerfüllender massen (Lectures on the introduction to mechanics of space-filling—i.e., 3D—masses), LeipzigGoogle Scholar
  6. 6.
    Caratheodory C (1909) Untersuchungen über die Grundlagen der Thermodynamik. Math Annalen 67:355–386CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cosserat E, Cosserat F (1896) Sur la théorie de l’élasticité. Ann Fac Sci Toulouse (1ère série), 10(3-5): I1–I116Google Scholar
  8. 8.
    Cosserat E, Cosserat F (1909). Théorie des corps déformables. Hermann, Paris, 226 pages. (Reprint by Editions Gabay, Paris, 2008; Reprint by Hermann Archives, Paris, 2009 [English translation: N68-15456: Clearinghouse Federal Scientific and Technical Information, Springfield, Virginia NASA, TT F-11561 (February 1968)); another translation by D. Delphenich, 2007] (originally published as a supplement in pp. 953-1173 to: Chwolson OD (1909) Traité de physique (traduit du Russe), Vol. II, Paris, Hermann)Google Scholar
  9. 9.
    D’Alembert J. Le Rond (1743) Traité de dynamique. 1st edn, ParisGoogle Scholar
  10. 10.
    Duhem P (1891) Hydrodynamique, élasticité, acoustique. A. Hermann Editeur, ParisGoogle Scholar
  11. 11.
    Duhem P (1896) Le potentiel thermodynamique et la pression hydrostatique. Ann Ecole Norm Sup, 10(3):187–230Google Scholar
  12. 12.
    Duhem P (1911) Traité d’énergétique ou thermodynamique générale. Gauthier-Villars, Paris Two volumesMATHGoogle Scholar
  13. 13.
    Duvaut G, Lions JL (1972) Inéquations variationnelles en mécanique et en physique. Editons Dunod, ParisGoogle Scholar
  14. 14.
    Duvaut G, Lions JL  (1979) Variational inequations in mechanics and physics. Springer, BerlinGoogle Scholar
  15. 15.
    Eringen AC (1966) Theory of micropolar fluids. J Math Mech 16(1):1–18MathSciNetGoogle Scholar
  16. 16.
    Eringen AC, Maugin GA (1990) Electrodynamics of continua. Springer, New York Two volumesCrossRefGoogle Scholar
  17. 17.
    Flügge S (ed) (1955–1988) Handbuch der physik. Springer, BerlinGoogle Scholar
  18. 18.
    Geiger H, Scheel K (eds) (1926–1933) Handbuch der physik. Julius Springer, BerlinGoogle Scholar
  19. 19.
    Germain P (1973) La méthode des puissances virtuelles en mécanique des milieux continus-I: théorie du second gradient. J de Mécanique 12:235–274 ParisMATHMathSciNetGoogle Scholar
  20. 20.
    Germain P (1986) Mécanique. Editions de l’ecole polytechnique, Palaiseau, Two volumesGoogle Scholar
  21. 21.
    Gibbs JW, Wilson EB (1901) Vector analysis. Yale University Press, New Haven [The first book of its kind, this may however be a wrong attribution since the book in fact is E.B. Wilson’s redaction with an enriched rendering of Gibbs’ lectures in vector analysis at Yale; Wilson was only 22 years old when the book was published (see pp. 228-229 in Crowe MJ (1967) A history of vector analysis, Univ. of Notre Dame Press; reprint Dover, 1985)]Google Scholar
  22. 22.
    Green G (1839) On the laws of reflection and refraction of light at the common surface of two non-crystallized media. Trans Cambridge Phil Soc 7:245–269Google Scholar
  23. 23.
    Hadamard J (1903) Leçons sur la propagation des ondes. Gauthier-Villars, ParisMATHGoogle Scholar
  24. 24.
    Hamel GWK (1908) Über die grundlagen der mechanik. Math Annalen, 66, 350-397 (This was Hamel’s Habilitationsschrift to be further developed in a contribution to the Handbuch der Physik, Die axiome der Mechanik. Geiger and Scheel (eds) Bd. 5. pp.1–42, Springer, Berlin, 1927)Google Scholar
  25. 25.
    Hamel GWK (1912) Elementare Mechanik. Teubner, BerlinMATHGoogle Scholar
  26. 26.
    Hellinger E (1914) Die allgemein ansätze der mechanik der kontinua. In: Klein F, Müller CH (eds) Enz math wiss, vol 4, part 4, Article 30, Springer, Berlin, pp 602–694Google Scholar
  27. 27.
    Hellinger E, Toeplitz O (1927) Integralgleichungen und gleichungen mit unenlichvielen unbekannten (originally intended for the Enz Math Wiss, reprinted by Chelsea, New York, 1953)Google Scholar
  28. 28.
    Herglotz G (1911) Über die mechanik des deformierbaren körpers vom standpunke der relativitätstheorie. Ann der Physik, 36(4):493–415Google Scholar
  29. 29.
    Heun K (1904) Ansätze und allgemeine methoden der systemmechanik. In: Klein F, Müller CH (eds) Enz math wiss, vol 4/2, Article 11, Springer, Berlin, pp 359–504Google Scholar
  30. 30.
    Kármán Th von (1914) Physikalische grundlagen der festigkeitslehre. In: Klein F, Müller CH (eds) Enz math wiss, vol 4/4, Article 31, Springer, BerlinGoogle Scholar
  31. 31.
    Klein F, Mueller CH (eds) (1907–1914) Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (in brief: Enz Math Wiss), B.G. Teubner (Verlag), Leipzig (vol 4, Part 4 on Mechanics is available on the site of the University of Toronto-Gerstein Science Information Centre as archive.org/details/p4encyclopedie04-akaduof )Google Scholar
  32. 32.
    Lagrange JL (1788) Mécanique analytique, First edition. Vve Courcier, ParisGoogle Scholar
  33. 33.
    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 28:523–579MATHMathSciNetGoogle Scholar
  34. 34.
    MacCullagh J (1839) An essay towards a dynamical theory of crystalline reflection and refraction. Trans Roy Irish Acad Sci 21:17–50Google Scholar
  35. 35.
    Maugin GA (1980) The method of virtual power in continuum mechanics: Application to coupled fields. Acta Mech 35(1):1–70CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Maugin GA (1992) The thermo-mechanics of plasticity and fracture. Cambridge University Press, UKCrossRefGoogle Scholar
  37. 37.
    Maugin GA (2012) The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. Cont Mech and Thermodynam 25:127–146 (Special issue in the honor of G. Del Piero)Google Scholar
  38. 38.
    Maugin GA (2013) Continuum mechanics through the twentieth century—a concise historical perspective. Springer, DordrechtCrossRefMATHGoogle Scholar
  39. 39.
    Maugin GA (2013) What happened on September 30, 1822, and what were its implications for the future of continuum mechanics? (Preprint UPMC, also Chapter 3 in this book)Google Scholar
  40. 40.
    Maugin GA (2013) About the Cosserats’ book of 1909 (Preprint UPMC, also Chapter 8 in this book)Google Scholar
  41. 41.
    Maugin GA (2013) A course of continuum mechanics at the dawn of the twentieth century (Volume III of Appell’s Treatise on Rational Mechanics) (Preprint UPMC, also Chapter 11 in the present book)Google Scholar
  42. 42.
    Minkowski H (1908) Die grundgleichungen für die electrodynamischen Vorgänge in bewegten Körpern. Nach Ges d W Gottingen, math-phys KL. p 53Google Scholar
  43. 43.
    Molk J, Appell P (eds) (1904–1916) Encyclopédie des sciences mathématiques pures et appliquées, twenty two volumes, [Facsimile reprint by Gabay, Paris, 1991-1995] (French translation from the German of: F. Klein and C.H. Mueller (eds), Enz Math Wiss, B.G. Teubner (Verlag), Leipzig)Google Scholar
  44. 44.
    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
  45. 45.
    Reissner E (1953) On a variational theorem for finite elastic deformation. J Math Phys 32:129–135 MIT, CambridgeMATHMathSciNetGoogle Scholar
  46. 46.
    Sedov LI (1968) Variational methods of constructing models of continuous media. In: Parkus H, Sedov LI (eds) Irreversible aspects of continuum mechanics (IUTAM Symposium, Vienna, 1966), Springer, Berlin, pp 346–358Google Scholar
  47. 47.
    Timoshenko SP (1953) History of strength of materials. McGraw Hill, New York (Reprint by Dover, New York, 1985)Google Scholar
  48. 48.
    Truesdell CA, Noll W (1965) The nonlinear field theories of mechanics In: S. Flügge (ed) Handbuch der Physik, vol. III/3. Springer-Verlag, BerlinGoogle Scholar
  49. 49.
    Truesdell CA, Toupin RA (1960) The classical theories of fields. In: Flügge S (ed) Handbuch der Physik,vol III/1. Springer-Verlag, BerlinGoogle Scholar
  50. 50.
    Voigt W (1887) Teoretische studien über elasticitätverhältinsse der krystalle, I.II. Abh K Ges Wissen Göttingen 34(3-52):53–100Google Scholar
  51. 51.
    Voigt W (1895/1896) Kompendium der theoretischen physik (Two volumes). Teubner, LeipzigGoogle Scholar
  52. 52.
    Voss A (1904) Principles of Rational mechanics (in German). In: Klein F, Mueller CH (eds) Enz Math Wiss, vol 4 Article 1, B.G. Teubner (Verlag), Leipzig (translated into French by the Cosserats in Molk and Appell (1904–1916))Google Scholar
  53. 53.
    Washizu K (1968) Variational methods in elasticity and plasticity. Pergamon Press, OxfordMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut Jean Le Rond d’AlembertUniversité Pierre et Marie CurieParis Cedex 05France

Personalised recommendations