Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

Buckling, barrelling, and surface instabilities of a finite, transversely isotropic circular cylinder


Author: K. T. Chau
Journal: Quart. Appl. Math. 53 (1995), 225-244
MSC: Primary 73Hxx; Secondary 73B40, 73K99
DOI: https://doi.org/10.1090/qam/1330650
MathSciNet review: MR1330650
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper investigates all possible geometric instabilities, such as buckling, barrelling, surface wrinkling, and orange-peel mode instabilities, for a finite length transversely isotropic circular cylinder under axisymmetric tension or compression. The constitutive responses of the cylinder can, in general, be transversely isotropic, compressible, and elastic-plastic; and no existence of rate potential is assumed. A general procedure is given to calculate the smallest critical ratio of radius to length, $ {\left( a/L) \right)_{cr}}$, of a circular cylinder that separates the buckling (antisymmetric) and the barrelling (axisymmetric) instabilities. Numerical results show that $ {\left( a/L \right)_{cr}}$ is very sensitive to the constitutive model used. The stress at the maximum load point is obtained as the first term of the axisymmetric long wavelength limit; the Euler buckling formula is obtained as the first term of the antisymmetric long wavelength limit. Three different types of surface instabilities are considered. The eigenstress for the longitudinal short wavelength limit corresponds to that for the wrinkling instability of a halfspace under plane deformations; the circumferential short wavelength limit is always possible; and the longitudinal and circumferential short wavelength limit, which corresponds to the surface instability with an ``orange-peel'' appearance, is available along the tensile elliptic-parabolic boundary. Numerical results show that eigenmodes of $ n \ge 2$ (where $ n$ is the circumferential wave number) can be the first bifurcation encountered under tension depending on the geometric ratio, $ a/L$. However, such eigenmodes will not be the first possible bifurcation under compression unless the ratio of the incremental transverse shear modulus to the incremental longitudinal shear modulus drops below, approximately, 0.25.


References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. A. Stegun (eds.). Handbook of Mathematical Functions, Dover, New York, 1965
  • [2] J. L. Bassani, D. Durban, and J. W. Hutchinson, Bifurcations at a spherical hole in an infinite elastoplastic medium, Math. Proc. Camb. Phil. Soc. 87, 339-356 (1980)
  • [3] M. F. Beatty, Elastic stability of rubber bodies in compression, Finite Elasticity, 27, ASME, AMD, 1977, pp. 125-150 (R. S. Rivlin, ed.)
  • [4] M. F. Beatty and P. Dadras, Some experiments on the elastic stability of some highly elastic bodies, Internat. J. Engrg. Sci. 14, 233-238 (1976)
  • [5] M. F. Beatty and D. E. Hook, Some experiments on the stability of circular rubber bars under end thrust, Internat. J. Solids Struct. 4, 623-635 (1968)
  • [6] M. A. Biot, Mechanics of Incremental Deformations, Wiley, New York, 1965
  • [7] K. T. Chau, Non-normality and bifurcation in a compressible pressure-sensitive circular cylinder under axisymmetric tension and compression, Internat. J. Solids Struct. 29, 801-824 (1992)
  • [8] K. T. Chau, Anti-symmetric bifurcations in a compressible pressure-sensitive circular cylinder under axisymmetric tension and compression, ASME J. Appl. Mech. 60, 282-289 (1993)
  • [9] K. T. Chau and J. W. Rudnicki, Bifurcations of compressible pressure-sensitive materials in plane strain tension and compression, J. Mech. Phys. Solids 38, 875-898 (1990)
  • [10] P. J. Davies, Buckling and barrelling instabilities in finite elasticity, J. Elast. 21, 147-192 (1989)
  • [11] P. J. Davies, Buckling and barrelling instabilities of nonlinearly elastic columns, Quart. Appl. Math. 49, 407-426 (1991)
  • [12] R. Hill, A general theory of uniqueness and stability in elastic-plastic solids, J. Mech. Phys. Solids 6, 236-249 (1958)
  • [13] R. Hill, Bifurcation and uniqueness in non-linear mechanics of continua, Problems of Continuum Mechanics, contributions in Honor of the Seventieth Birthday of Academician N. I. Muskhelishvili (Ed. by M. A. Lavrent'ev et al.; English edition edited by J. R. M. Radok), Society for Industrial and Applied Mathematics, Philadelphia, PA, 1961, pp. 155-164
  • [14] R. Hill, Aspects of invariance in solid mechanics, Adv. Appl. Mech. 18, 1-75 (1978) (C. S. Yih, ed.)
  • [15] M. Levinson, Stability of a compressed neo-Hookean rectangular parallelepiped, J. Mech. Phys. Solids 16, 403-415 (1968)
  • [16] W. Prager, Introduction to the Mechanics of Continua, Dover, New York, 1973
  • [17] R. Rice, The localization of plastic deformation, Proc. 14th Internat. Congr. Theoretical and Appl. Mech. (W. T. Koiter, ed.), Delft, North-Holland, Amsterdam, Vol. 1, 1976, pp. 207-220
  • [18] D. Rittel, The influence of microstructure on the macroscopic patterns of surface instabilities in metals, Scripta Metallurgica et Materialia 24, 1759-1764 (1990)
  • [19] D. Rittel, R. Aharonov, G. Feigin, and I. Roman, Experimental investigation of surface instabilities in cylindrical tensile metallic specimens, Acta Metall. Mater. 39, 719-724 (1991)
  • [20] J. W. Rudnicki, The effect of stress-induced anisotropy on a model of brittle rock failure as localization of deformation, Energy Resources and Excavation Technology, Proc. 18th U. S. Symposium on Rock Mechanics, Keystone, Colorado, June 22-24, 1977, pp. 3B4-1-3B4-8
  • [21] J. W. Rudnicki and J. R. Rice, Conditions for the localization of pressure-sensitive dilatant materials, J. Mech. Phys. Solids 23, 371-394 (1975)
  • [22] F. J. Santarelli and E. T. Brown, Failure of three sedimentary rocks in triaxial and hollow cylinder compression tests, Internat. J. Rock Mech. Min. Sci. & Geomech. Abstr. 26, 401-413 (1989)
  • [23] J. A. Shimer, This Sculptured Earth: The Landscape of America, Columbia University Press, New York, 1960
  • [24] H. C. Simpson and S. J. Spector, On barrelling instabilities in finite elasticity, J. Elast. 14, 103-125 (1984)
  • [25] H. C. Simpson and S. J. Spector, On barrelling for a special material in finite elasticity, Quart. Appl. Math. 42, 99-111 (1984)
  • [26] H. C. Simpson and S. J. Spector, On the positivity of the second variation in finite elasticity, Arch. Rat. Mech. Anal. 98, 1-30 (1987)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73Hxx, 73B40, 73K99

Retrieve articles in all journals with MSC: 73Hxx, 73B40, 73K99


Additional Information

DOI: https://doi.org/10.1090/qam/1330650
Article copyright: © Copyright 1995 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website