Journal of Elasticity

, Volume 102, Issue 2, pp 191–200

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

• Riccardo De Pascalis
• 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

## Preview

### 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)
3. 3.
Wilkes, E.W.: On the stability of a circular tube under end thrust. Q. J. Mech. Appl. Math. 8, 88–100 (1955)
4. 4.
Biot, M.A.: Surface instability of rubber in compression. Appl. Sci. Res. A 12, 168–182 (1963)
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)
6. 6.
Ogden, R.W.: On isotropic tensors and elastic moduli. Proc. Camb. Philos. Soc. 75, 427–436 (1974)
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)
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)
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)
13. 13.
Bland, D.R.: Nonlinear Dynamic Elasticity. Blaisdell, Waltham (1969)
14. 14.
Eringen, A.C., Suhubi, E.S.: Elastodynamics, vol. 1. Academic Press, New York (1974)
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)
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)
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)
19. 19.
Destrade, M., Ogden, R.W.: On the third- and fourth-order constants of incompressible isotropic elasticity (submitted) Google Scholar

## Authors and Affiliations

• Riccardo De Pascalis
• 1
• 2
• 3