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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on July 24, 2008
The Quarterly Journal of Mechanics and Applied Mathematics 2008 61(4):475-495; doi:10.1093/qjmam/hbn018
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© The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity

G. W. Jones{dagger}, S. J. Chapman and D. J. Allwright

( Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford OX1 3LB )

{dagger} <jonesg{at}maths.ox.ac.uk>

Received 20 August 2007. Revise 21 September 2007. Revise 13 June 2008.
  Abstract

The problem of a thin spherical linearly elastic shell perfectly bonded to an infinite linearly elastic medium is considered. A constant axisymmetric stress field is applied at infinity in the matrix, and the displacement and stress fields in the shell and matrix are evaluated by means of harmonic potential functions. In order to examine the stability of this solution, the buckling problem of a shell which experiences this deformation is considered. Using Koiter's nonlinear shallow shell theory, restricting buckling patterns to those which are axisymmetric and using the Rayleigh–Ritz method by expanding the buckling patterns in an infinite series of Legendre functions, an eigenvalue problem for the coefficients in the infinite series is determined. This system is truncated and solved numerically in order to analyse the behaviour of the shell as it undergoes buckling and to identify the critical buckling stress in two cases, namely, where the shell is hollow and the stress at infinity is either uniaxial or radial.


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