Recent Advances in Rational Mechanics (1956)

  • C. Truesdell


I begin by answering the question most of you asked upon reading the title: What is rational mechanics? It is difficult to define rational mechanics, but no more so than to define chemistry or physics or mathematics. However, a chemist coming before you would not be expected to begin by defining chemistry. The reason most of you wish me to define rational mechanics is that in the United States, at least, it is not a recognized science. Indeed, there are some who disbelieve in its existence1.


Phase Average Vortex Line Vortex Tube Rational Mechanic Bibliographical Note 
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  1. 3.
    I. Newton, Philosophiae Naturalis Principia Mathematica ,London, 1687; 2nd ed., Cambridge, 1713; 3rd ed., London, 1726. The only available English translation is unreliable, especially in the allegedly modernized reprint. Some of the relevant passages, newly translated, are quoted in Essay III, above. A new edition of the Principia ,collating the three original ones, and presumably also to be provided with a correct translation, has been undertaken by I. B. Cohen and the late A. Koyre ; its appearance is eagerly awaited.Google Scholar
  2. 4.
    D. Hilbert, “Mathematische Probleme,” Nachr. Ges. Wiss. Göttingen 1900, 253–297. An English translation of an amplified version is printed in Bull. Am. Math. Soc. (2) 8, 437–479 (1902). Note added for the reprinting. Only two significant attempts to solve the part of Hilbert’s sixth problem that concerns mechanics have been published: that of Hamel, Über die Grundlagen der Mechanik, Math. Annalen 66, 350–397 (1908), and that of Noll, mentioned in Essay VII, above, and presented in full in the notes of his lectures at the C.I.M.E. Course on Recent Developments in the Mechanics of Continua at Bressanone in 1965, Edizioni Cremonese, Rome, 1966, and issued also as a report of the Mathematics Department of the Carnegie Institute of Technology, 1966.Google Scholar
  3. 5.
    J. W. Gibbs, Elementary Principles in Statistical Mechanics, 1902, reprinted in Volume 2 of Gibbs’ Collected Papers.MATHGoogle Scholar
  4. 6.
    H. v. Helmholtz, “Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen,”J. Reine Angew. Math. 55, 25–55 (1858)= Ges. Abb. 1, 101–134. English translation, Phil. Mag. (4) 33, 485–512 (1867).CrossRefMATHGoogle Scholar
  5. 7.
    W. Thomson, “On vortex motion,” Trans. Roy. Soc. Edinburgh 25, 217–260 (1869) = Collected Papers 4, 13–66.Google Scholar
  6. 8.
    H. Ertel, “ Ein Theorem über asynchron-periodische Wirbelbewegungen kompressibler Flüssigkeiten,” Misc. Acad. Berol. 1, 62–68 (1950).MathSciNetGoogle Scholar
  7. 9.
    E. Reissner, “On a variational theorem in elasticity,”J. Math. Phys. 29, 90–95 (1950); “On a variational theorem for finite elastic deformations,” J. Math. Phys. 32, 129–135 (1953). Note added for the reprinting. In fact, the variational principle seems to be due to M. Born, who gave it for two-dimensional problems, with boundary conditions neglected, in § IV of the Anhang to his Habilitationsschrift, Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen ,Göttingen, 1906. In three dimensions, again without boundary conditions, it was given by E. Hellinger in § 7e of his article, “Die allgemeinen Ansätze der Mechanik der Kontinua,” Enz. Math. Wiss. 44, 602–694 (1914). The new contribution of Reissner lies in the formulation of a boundary integral such as to yield appropriate boundary conditions for all the standard problems of linear elasticity, and in applications to cases.MathSciNetMATHGoogle Scholar
  8. 10.
    A. J. C. B. de St. Venant, “Memoire sur la torsion des prismes ...,” Mem. Divers Savants Acad. Sci. Paris 14, 233–560 (1855); also issued separately, “De la Torsion des Prismes ...,” Paris, 1855.Google Scholar
  9. 11.
    J. Boussinesq, Application des Potentiels à l ’Étude de l ’Équilibre et du Mouvement des Solides É lastiques ,Paris, Gauthier-Villars, 1885.Google Scholar
  10. 12.
    R. V. Mises, “On Saint-Venant’s Principle,” Bull. Am. Math. Soc. 51, 555–562 (1945).CrossRefMATHGoogle Scholar
  11. 13.
    E. Sternberg, “On Saint-Venant’s Principle,”Quart. Appl. Math. 11, 393–402 (1954).MathSciNetMATHGoogle Scholar
  12. 16.
    P. G. Bordoni, “ Sopra le trasformazioni termoelastiche finite di certi solidi omogenei ed isotropi,” Rend. mat. e delle sue Applic. (Roma) (5) 12, 237–266 (1953). See §5. Note added for the reprinting. The thermodynamic basis of this work does not now seem clear to me.MathSciNetGoogle Scholar
  13. 18.
    M.Reiner, “A mathematical theory of dilatancy,” Am. J. Math. 67, 350–362 (1945). Reprinted in Rational Mechanics of Materials•, Gordon &Breach, 1965. Cf. also W. Prager, “ Strain hardening under combined stresses,” J. Appl. Phys. 16, 837–840 (1945), where the same algebraic theorem is inferred in a different context.MathSciNetADSCrossRefMATHGoogle Scholar
  14. 19.
    R. S. Rivlin, “The hydrodynamics of Non-Newtonian fluids, I,” Proc. R. Soc. London 193, 260–281 (1948); reprinted in Rational Mechanics of Materials , Gordon &Breach, 1965.MathSciNetADSCrossRefMATHGoogle Scholar
  15. 23.
    A.-L. Cauchy, Sur l’équilibre et le mouvement intérieur des corps considérés comme des masses continues, Ex. Math. 4, 293–319 (1829)= (Euvres (2) 9, 342–369.Google Scholar
  16. 24.
    S. Zaremba, “ Sur une forme perfectionnée de la théorie de la relaxation,” Bull, inter. Acad. Sci. Cracovie 1903, 594–614.Google Scholar
  17. 25.
    J. G. Oldroyd, “On the formulation of rheological equations of state,” Proc. R. Soc. London A 200, 523–541 (1950). Reprinted in Rational Mechanics of Materials ,Gordon &Breach, 1965.MathSciNetADSGoogle Scholar
  18. 26.
    W. Noll, “On the continuity of the solid and fluid states,” J. Rational Mech. Anal. 4, 3–81 (1955). Reprinted in Rational Mechanics of Materials ,Gordon &Breach, 1965.MathSciNetMATHGoogle Scholar
  19. 28.
    J. C. Maxwell, “On the dynamical theory of gases,” Phil. Trans. R. Soc. London A157, 49_88 (1867)= Phil. Mag. (4) 35, 129–145, 187–217 (1868)= Papers 2, 26–78.Google Scholar
  20. 32.
    J. Irving &J. G. Kirkwood, The statistical mechanical theory of transport processes, IV, The equations of hydrodynamics, J. Chem. Phys. 18, 817–829 (1950).MathSciNetADSCrossRefGoogle Scholar
  21. 33.
    W. Noll, “ Die Herleitung der Grundgleichungen der Thermomechanik aus der statistischen Mechanik,” J. Rational Mech. Anal. 4, 627–646 (1955).MathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 1968

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  • C. Truesdell

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