Some Recent Results on Saint-Venant’s Principle

  • R. J. Knops
  • C. Lupoli
Chapter

Abstract

Saint-Venant’s principle has been accepted by structural engineers, numerical analysts, and others as a reliable guide to how far edge and boundary effects penetrate into a body. Nevertheless, this conjecture by Saint-Venant in 1855, even though based upon plausible intuitive argument, has from the beginning provoked scepticism and the history of the subject has been characterized by attempts to formulate a precise mathematical definition supported by rigorous mathematical proof. Two main approaches may be discerned. One group investigates relevant properties of exact solutions: within this category are authors such as Boussinesq [4], Clebsch [5], Dougall [7], Synge [41], von Mises [32], Sternberg [39], and by slight extension, Mielke [30], [31] who has applied center manifold arguments. The other main area of study considers energy, and early contributors include Zanaboni [43], Goodier [23], [18], and Dou [6], the classic paper being by Toupin [42]. Later writers are, for example, Fichera [9], [10], [11], Oleinik et al. [24]–[28], [36], Payne and Horgan and their respective coworkers. The subject has been comprehensively surveyed by Gurtin [19], Maisonneuve [29], and Horgan and Knowles [20].

Keywords

Transient Heat Conduction Poincare Inequality Classical Linear Elasticity Rigorous Mathematical Proof Numerical Analyst 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Boley, B.A. The determination of temperature, stresses and deflections in two-dimensional thermoelastic problems. J. Aerospace Sci., 23 (1956), 67–75.MathSciNetMATHGoogle Scholar
  2. [2]
    Boley, B.A. Some observations on Saint-Venant’s principle. Proc. 3rd Nat. Congr. Appl. Mech., ASME, (1958), pp. 259–264.Google Scholar
  3. [3]
    Boley, B.A. Upper bounds and Saint-Venant’s principle in transient heat conduction. Quart. Appl. Mech., 18 (1960), 205–207.MathSciNetMATHGoogle Scholar
  4. [4]
    Boussinesq, J. Application des Potentials à l’Étude de l’Equilibre et des Mouvements des Solides Élastiques. Gauthier-Villars, Paris, 1885.Google Scholar
  5. [5]
    Clebsch, A. Théorie de l’Élasticité des Corps Solides. (Translation of Theorie der Elastizität fester Körper by B de Saint-Venant and Flamant.) Dunod, Paris, 1883. Johnson Reprint Cooperation, New York, 1966.Google Scholar
  6. [6]
    Dou, A. Upper estimate of the potential elastic energy of a cylinder. Comm. Pure Appl. Maths., 19 (1966), 83–93.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Dougall, J. An analytical theory of the equilibrium of an isotropic elastic rod of circular section. Trans. Roy. Soc. Edinburgh, 49 (1914), 895–978.Google Scholar
  8. [8]
    Edelstein, W.S. A spatial decay estimate for the heat equation. Z. Angew. Math. Phys., 20 (1969), 900–905.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Fichera, G. Il principio di Saint-Venant: Intuizione dell’ingegnere e rigore del matematico. Rend. Mat., 10 ( Ser. VI ) (1977), 1–24.MathSciNetMATHGoogle Scholar
  10. [10]
    Fichera, G. Remarks on Saint-Venant’s principle. In Complex Analysis and Its Applications. I.N. Vekua Anniversary Volume, pp. 543–557, Nauka, Moscow, 1978.Google Scholar
  11. [11]
    Fichera, G. Sull’esistenza e sul calcolo della soluzioni dei problemi al contorno relativi all’equilibrio di un corpo elastico. Ann. Scuolo Norm. Sup. Pisa, 4 (1950), 35–99.MathSciNetMATHGoogle Scholar
  12. [12]
    Flavin, J.N. and Knops, R.J. Some spatial decay estimates in continuum mechanics. J. Elasticity, 17 (1987), 249–264.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Flavin, J.N. and Knops, R.J. Some decay and other estimates in two-dimensional linear elastostatics. Quart. J. Mech. Appl. Math., 41 (1988), 223–238.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Flavin J.N. and Knops, R.J. Some convexity considerations for a two-dimensional traction problem. ZAMP, 39 (1988), 166–176.MathSciNetADSMATHCrossRefGoogle Scholar
  15. [15]
    Flavin, J.N., Knops, R.J., and Payne, L.E. Decay estimates for the constrained elastic cylinder of variable cross section. Quart. Appl. Math., 47 (1989), 325–350.MathSciNetMATHGoogle Scholar
  16. [16]
    Flavin, J.N., Knops, R.J., and Payne, L.E. Asymptotic behavior of solutions to semi-linear elliptic equations on the half-cylinder. Z. Angew. Math. Phys.,43 (1992), 405–421.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Goodier, J.N. A general proof of Saint-Venant’s principle. Phil. Mag., (7), (1937), 607.Google Scholar
  18. [18]
    Goodier, J.N. Supplementary note on “A general proof of Saint-Venant’s principle.” Phil. Mag., (7), (1937), 24, 325.Google Scholar
  19. [19]
    Gurtin, M.E. The Linear Theory of Elasticity. Handbuch der Physik, Vol. VI a/2, pp. 1–295. Springer-Verlag, New York, 1974.Google Scholar
  20. [20]
    Horgan, C.O. and Knowles, J.K. Recent developments concerning Saint-Venant’s principle. In Advances in Applied Mechanics (J.W. Hutchinson, ed.). vol. 23,pp. 179–269. Academic Press, New York, 1983. See also: C.O. Horgan. Recent developments concerning Saint-Venant’s principle: An update. Appl. Mech. Rev.,42 (11), (1989), 295–302.Google Scholar
  21. [21]
    Horgan, C.O., Payne, L.E., and Wheeler, L.T. Spatial decay estimates in transient heat conduction. Quart. Appl. Math., 42 (1) (1984), 119–127.MathSciNetMATHGoogle Scholar
  22. [22]
    Horgan, C.O. and Wheeler, L.T. Spatial decay estimates for the heat equation via the maximum principle. Z. Angew. Math. Phys., 27 (1976), 371–376.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Knops, R.J., Rionero, S., and Payne, L.E. Saint-Venant’s principle on unbounded regions. Proc. Roy. Soc. Edinburgh, 115A (1990), 319–336.MathSciNetMATHGoogle Scholar
  24. [24]
    Kondratiev, V.A., Kopachek, I., and Oleinik, O.A. On the behaviour of weak solutions of second-order elliptic equations and the elasticity system in a neighbourhood of a boundary point. Trudy Sem. Petrovsky. 8 (1982), 135–152.Google Scholar
  25. [25]
    Kondratiev, V.A. and Oleinik, O A On the asymptotics at infinity of solutions of elliptic systems with constant coefficients. Uspekhi Mat. Nauk, 40 (5), (1985), 233.Google Scholar
  26. [26]
    Kondratiev, V.A. and Oleinik, O.A. On the asymptotic behaviour of solutions of systems of differential equations. Uspekhi Mat. Nauk., 40 (5), (1985), 306.Google Scholar
  27. [27]
    Kondratiev, V.A. and Oleinik, O.A. On the behaviour at infinity of solutions of elliptic systems with finite energy integral. Arch. Rational Mech. Anal., 99 (1987), 75–89.MathSciNetADSMATHCrossRefGoogle Scholar
  28. [28]
    Kondratiev, V.A. and Oleinik, O.A. Boundary value problems for the system of elasticity theory in unbounded domains. Korn’s inequalities. Uspekhi Mat. Nauk., 43 (1988), 55–98. English translation in Russian Math. Surveys, 43 (1988), 65–119.Google Scholar
  29. [29]
    Maisonneuve, O. Sur le Principe de Saint-Venant. Thèse d’Etat, Université de Poitiers, March 1971.Google Scholar
  30. [30]
    Mielke, A. Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity. Arch. Rational Mech. Anal., 102 (1988), 205–229.MathSciNetADSMATHCrossRefGoogle Scholar
  31. [31]
    Mielke, A. Normal hyperbolicity of center manifolds and Saint-Venant’s principle. Arch. Rational Mech. Anal., 110 (1990), 353–372.MathSciNetADSMATHCrossRefGoogle Scholar
  32. [32]
    von Mises, R. On Saint-Venant’s Principle. Bull. Amer. Math. Soc., 51 (1945), 555–562.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Neapolitan, R.E. and Edelstein, W.S. Further study of Saint-Venant’s principle in linear viscoelasticity. Z. Angew. Math. Phys., 24 (1973), 283–837.MathSciNetCrossRefGoogle Scholar
  34. [34]
    Neapolitan R.E. and Edelstein, W.S. A priori bounds for the secondary boundary value problem in linear viscoelasticity. SIAM J. Appl. Math., 28 (1975), 559–564.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    Oleinik, O.A. and Yosifian, G.A. An analogue of Saint-Venant’s principle and the uniqueness of solutions of boundary-value problems for parabolic equations in unbounded domains. Russian Math. Surveys, 31 (1976), 153–178.MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    Oleinik, O.A. and Yosifian, G.A. On the asymptotic behaviour at infinity of solutions in linear elasticity. Arch. Rational Mech. Anal., 78 (1982), 29–53.MathSciNetADSMATHCrossRefGoogle Scholar
  37. [37]
    Payne, L.E. and Weinberger, H.F. Note on a lemma by Finn and Gilbarg. Acta Math., 98 (1957), 297–299.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    Barré de Saint-Venant, A.-J.-C. Mémoire sur la tension des prismes, avec des considérations sur leur flexion. Mém. Divers Savants., 14 (1855), 233–560.Google Scholar
  39. [39]
    Sternberg, E. On Saint-Venant’s Principle. Quart. Appl. Math., 11 (1954), 393–402.MathSciNetMATHGoogle Scholar
  40. [40]
    Sternberg, E. and Al Khozaie S.A. On Green’s function and Saint-Venant’s principle in the linear theory of viscoelasticity. Arch. Rational Mech. Anal., 15 (1964), 112–146.MathSciNetADSMATHCrossRefGoogle Scholar
  41. [41]
    Synge, J.L. The problem of Saint-Venant for a cylinder with free sides. Quart. Appl. Math., 2 (1945), 307–317.MathSciNetMATHGoogle Scholar
  42. [42]
    Toupin, R.A. Saint-Venant’s principle. Arch. Rational Mech. Anal., 18 (1965), 83–96.MathSciNetADSMATHCrossRefGoogle Scholar
  43. [43]
    Zanaboni, O. Dimostrazione generale del Principio del de Saint-Venant. Atti Accad. Naz. Lincei Rend., 25 (1937), 117–120.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1998

Authors and Affiliations

  • R. J. Knops
  • C. Lupoli

There are no affiliations available

Personalised recommendations