Skip to main content
SpringerLink
Log in

Some Recent Results on Saint-Venant’s Principle

  • Chapter
Nonlinear Analysis and Continuum Mechanics

  • 256 Accesses

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].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Boley, B.A. The determination of temperature, stresses and deflections in two-dimensional thermoelastic problems. J. Aerospace Sci., 23 (1956), 67–75.

    MathSciNet  MATH  Google Scholar 

  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. Boley, B.A. Upper bounds and Saint-Venant’s principle in transient heat conduction. Quart. Appl. Mech., 18 (1960), 205–207.

    MathSciNet  MATH  Google Scholar 

  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. 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. Dou, A. Upper estimate of the potential elastic energy of a cylinder. Comm. Pure Appl. Maths., 19 (1966), 83–93.

    Article  MathSciNet  MATH  Google Scholar 

  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. Edelstein, W.S. A spatial decay estimate for the heat equation. Z. Angew. Math. Phys., 20 (1969), 900–905.

    Article  MathSciNet  MATH  Google Scholar 

  9. Fichera, G. Il principio di Saint-Venant: Intuizione dell’ingegnere e rigore del matematico. Rend. Mat., 10 ( Ser. VI ) (1977), 1–24.

    MathSciNet  MATH  Google Scholar 

  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. 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.

    MathSciNet  MATH  Google Scholar 

  12. Flavin, J.N. and Knops, R.J. Some spatial decay estimates in continuum mechanics. J. Elasticity, 17 (1987), 249–264.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  14. Flavin J.N. and Knops, R.J. Some convexity considerations for a two-dimensional traction problem. ZAMP, 39 (1988), 166–176.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  17. Goodier, J.N. A general proof of Saint-Venant’s principle. Phil. Mag., (7), (1937), 607.

    Google Scholar 

  18. Goodier, J.N. Supplementary note on “A general proof of Saint-Venant’s principle.” Phil. Mag., (7), (1937), 24, 325.

    Google Scholar 

  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. 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. 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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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. Maisonneuve, O. Sur le Principe de Saint-Venant. Thèse d’Etat, Université de Poitiers, March 1971.

    Google Scholar 

  30. Mielke, A. Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity. Arch. Rational Mech. Anal., 102 (1988), 205–229.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Mielke, A. Normal hyperbolicity of center manifolds and Saint-Venant’s principle. Arch. Rational Mech. Anal., 110 (1990), 353–372.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. von Mises, R. On Saint-Venant’s Principle. Bull. Amer. Math. Soc., 51 (1945), 555–562.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  ADS  Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. Payne, L.E. and Weinberger, H.F. Note on a lemma by Finn and Gilbarg. Acta Math., 98 (1957), 297–299.

    Article  MathSciNet  MATH  Google Scholar 

  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. Sternberg, E. On Saint-Venant’s Principle. Quart. Appl. Math., 11 (1954), 393–402.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Synge, J.L. The problem of Saint-Venant for a cylinder with free sides. Quart. Appl. Math., 2 (1945), 307–317.

    MathSciNet  MATH  Google Scholar 

  42. Toupin, R.A. Saint-Venant’s principle. Arch. Rational Mech. Anal., 18 (1965), 83–96.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Zanaboni, O. Dimostrazione generale del Principio del de Saint-Venant. Atti Accad. Naz. Lincei Rend., 25 (1937), 117–120.

    Google Scholar 

Download references

Authors
  1. R. J. Knops
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Lupoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pisa, I-56127, Pisa, Italy

    Giuseppe Buttazzo

  2. Department of Engineering, University of Ferrara, I-44100, Ferrara, Italy

    Giovanni Paolo Galdi

  3. Department of Mathematics, University of Bologna, I-40126, Bologna, Italy

    Ermanno Lanconelli

  4. Department of Mathematics, University of Perugia, I-06123, Perugia, Italy

    Patrizia Pucci

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Knops, R.J., Lupoli, C. (1998). Some Recent Results on Saint-Venant’s Principle. In: Buttazzo, G., Galdi, G.P., Lanconelli, E., Pucci, P. (eds) Nonlinear Analysis and Continuum Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2196-8_6

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2196-8_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7455-1

  • Online ISBN: 978-1-4612-2196-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation