Journal of Elasticity

, Volume 102, Issue 2, pp 191–200 | Cite as

Nonlinear Correction to the Euler Buckling Formula for Compressed Cylinders with Guided-Guided End Conditions

  • Riccardo De Pascalis
  • Michel Destrade
  • Alain Goriely
Short Communication

Abstract

Euler’s celebrated buckling formula gives the critical load N for the buckling of a slender cylindrical column with radius B and length L as
$$N/(\pi^3B^2)=(E/4)(B/L)^2,$$
where E is Young’s modulus. Its derivation relies on the assumptions that linear elasticity applies to this problem, and that the slenderness (B/L) is an infinitesimal quantity. Here we ask the following question: What is the first non-linear correction in the right hand-side of this equation when terms up to (B/L)4 are kept? To answer this question, we specialize the exact solution of incremental non-linear elasticity for the homogeneous compression of a thick compressible cylinder with lubricated ends to the theory of third-order elasticity. In particular, we highlight the way second- and third-order constants—including Poisson’s ratio—all appear in the coefficient of (B/L)4.

Keywords

Column buckling Euler formula Non-linear correction Guided end condition 

Mathematics Subject Classification (2000)

74B15 74B10 74B20 74G05 74G15 74G60 

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References

  1. 1.
    Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1961) Google Scholar
  2. 2.
    Goriely, A., Vandiver, R., Destrade, M.: Nonlinear Euler buckling. Proc. R. Soc. A 464, 3003–3019 (2008) MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Wilkes, E.W.: On the stability of a circular tube under end thrust. Q. J. Mech. Appl. Math. 8, 88–100 (1955) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Biot, M.A.: Surface instability of rubber in compression. Appl. Sci. Res. A 12, 168–182 (1963) MATHGoogle Scholar
  5. 5.
    Fosdick, R.A., Shield, R.T.: Small bending of a circular bar superposed on finite extension or compression. Arch. Ration. Mech. Anal. 12, 223–248 (1963) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ogden, R.W.: On isotropic tensors and elastic moduli. Proc. Camb. Philos. Soc. 75, 427–436 (1974) MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Ogden, R.W.: Non-Linear Elastic Deformations. Dover, New York (1984) Google Scholar
  8. 8.
    Dorfmann, A., Haughton, D.M.: Stability and bifurcation of compressed elastic cylindrical tubes. Int. J. Eng. Sci. 44, 1353–1365 (2006) CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Shuvalov, A.L.: A sextic formalism for three-dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials. Proc. R. Soc. A 459, 1611–1639 (2003) MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Pergamon, New York (1986) Google Scholar
  11. 11.
    Murnaghan, F.D.: Finite Deformations of an Elastic Solid. Wiley, New York (1951) Google Scholar
  12. 12.
    Toupin, R.A., Bernstein, B.: Sound waves in deformed perfectly elastic materials. Acoustoelastic effect. J. Acoust. Soc. Am. 33, 216–225 (1961) CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Bland, D.R.: Nonlinear Dynamic Elasticity. Blaisdell, Waltham (1969) MATHGoogle Scholar
  14. 14.
    Eringen, A.C., Suhubi, E.S.: Elastodynamics, vol. 1. Academic Press, New York (1974) MATHGoogle Scholar
  15. 15.
    Norris, A.N.: Finite amplitude waves in solids. In: Hamilton, M.F., Blackstock, D.T. (eds.) Nonlinear Acoustics, pp. 263–277. Academic Press, San Diego (1999) Google Scholar
  16. 16.
    Porubov, A.V.: Amplification of Nonlinear Strain Waves in Solids. World Scientific, Singapore (2003) MATHCrossRefGoogle Scholar
  17. 17.
    Wochner, M.S., Hamilton, M.F., Ilinskii, Y.A., Zabolotskaya, E.A.: Cubic nonlinearity in shear wave beams with different polarizations. J. Acoust. Soc. Am. 123, 2488–2495 (2008) CrossRefADSGoogle Scholar
  18. 18.
    Catheline, S., Gennisson, J.-L., Fink, M.: Measurement of elastic nonlinearity of soft solid with transient elastography. J. Acoust. Soc. Am. 114, 3087–3091 (2003) CrossRefADSGoogle Scholar
  19. 19.
    Destrade, M., Ogden, R.W.: On the third- and fourth-order constants of incompressible isotropic elasticity (submitted) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Riccardo De Pascalis
    • 1
    • 2
    • 3
  • Michel Destrade
    • 4
  • Alain Goriely
    • 5
  1. 1.CNRS, UMR 7190Institut Jean Le Rond d’AlembertParisFrance
  2. 2.UPMC Univ Paris 06, UMR 7190Institut Jean Le Rond d’AlembertParisFrance
  3. 3.Dipartimento di MatematicaUniversità del SalentoLecceItaly
  4. 4.School of Electric, Electronic and Mechanical EngineeringUniversity College DublinDublin 4Ireland
  5. 5.OCCAM, Institute of MathematicsUniversity of OxfordOxfordUK

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