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This and the next image are from:
G.W. Jones, S.J. Chapman, and D.J. Allwright, “Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity”, Preprint no. 07/2008, OxMOS: New Frontiers in the Mathematics of Solids Mathematical Institute University of Oxford http://www2.maths.ox.ac.uk/oxmos/ 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.
The next slide shows axisymmetric buckling modes when the loading at infinity is uniaxial compression, that is, axial compression in the vertical direction.
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