zbMATH — the first resource for mathematics

Nonlinear correction to the Euler buckling formula for compressed cylinders with guided-guided end conditions. (English) Zbl 1273.74104
Summary: 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 \((B/L)^{4}\).

74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text: DOI
[1] Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1961)
[2] Goriely, A., Vandiver, R., Destrade, M.: Nonlinear Euler buckling. Proc. R. Soc. A 464, 3003–3019 (2008) · Zbl 1152.74342 · doi:10.1098/rspa.2008.0184
[3] Wilkes, E.W.: On the stability of a circular tube under end thrust. Q. J. Mech. Appl. Math. 8, 88–100 (1955) · Zbl 0064.18602 · doi:10.1093/qjmam/8.1.88
[4] Biot, M.A.: Surface instability of rubber in compression. Appl. Sci. Res. A 12, 168–182 (1963) · Zbl 0121.19004
[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) · Zbl 0108.37102 · doi:10.1007/BF00281227
[6] Ogden, R.W.: On isotropic tensors and elastic moduli. Proc. Camb. Philos. Soc. 75, 427–436 (1974) · Zbl 0327.73007 · doi:10.1017/S0305004100048635
[7] Ogden, R.W.: Non-Linear Elastic Deformations. Dover, New York (1984) · Zbl 0541.73044
[8] Dorfmann, A., Haughton, D.M.: Stability and bifurcation of compressed elastic cylindrical tubes. Int. J. Eng. Sci. 44, 1353–1365 (2006) · Zbl 1213.74145 · doi:10.1016/j.ijengsci.2006.06.014
[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) · Zbl 1058.74044 · doi:10.1098/rspa.2002.1075
[10] Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Pergamon, New York (1986) · Zbl 0178.28704
[11] Murnaghan, F.D.: Finite Deformations of an Elastic Solid. Wiley, New York (1951) · Zbl 0045.26504
[12] Toupin, R.A., Bernstein, B.: Sound waves in deformed perfectly elastic materials. Acoustoelastic effect. J. Acoust. Soc. Am. 33, 216–225 (1961) · doi:10.1121/1.1908623
[13] Bland, D.R.: Nonlinear Dynamic Elasticity. Blaisdell, Waltham (1969) · Zbl 0236.73035
[14] Eringen, A.C., Suhubi, E.S.: Elastodynamics, vol. 1. Academic Press, New York (1974) · Zbl 0291.73018
[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)
[16] Porubov, A.V.: Amplification of Nonlinear Strain Waves in Solids. World Scientific, Singapore (2003) · Zbl 1058.74005
[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) · doi:10.1121/1.2890739
[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) · doi:10.1121/1.1610457
[19] Destrade, M., Ogden, R.W.: On the third- and fourth-order constants of incompressible isotropic elasticity (submitted)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.