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

Abstract

SummaryThe 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.

Syntactic foam under compressive stress: Comparison of modeling predictions and experimental measurements

“…Fok and Allwright 16 and Jones et al. 17 are among the few groups who considered a spherical shell actually embedded in an unbounded uniform elastic medium. De Pascalis et al.…”

Predicting the pressure–volume curve of an elastic microsphere composite

“…Initial work into the buckling pressure of a spherical shell embedded in an unbounded uniform elastic medium has been carried out by Fok and Allwright (2001) and Jones et al (2008). We shall discuss these models and their assumptions later on in the paper, particularly that of Fok and Allwright (2001) which is the model that we shall adopt for buckling here.…”

Asymptotic Analysis of a Buckling Problem for an Embedded Spherical Shell

“…Restricting the shell buckling patterns to be axisymmetric, in [12] the lowest critical stress for buckling was found, together with the associated buckling pattern, which was confined to a band around the shell's equator. One of the drawbacks of the calculation method was that it was seminumerical, in that the buckling patterns were found as an infinite series whose coefficients had to be found by solving an infinite system of linear equations.…”

“…Of major importance to many applications is the study of how attachment to a substrate affects the buckling behavior of a thin structure. A recent article of ours [12] examined the buckling problem of a spherical shell embedded in a linearly elastic medium, which is subjected to a uniaxial compressive stress. This problem is intended to model the compression of a class of anechoic tiles for underwater structures, whose microstructure consists of a distribution of microspheres (spherical shells whose radii are typically ∼ 20 μm) embedded in a rubber matrix.…”

“…Following a review of the setup and results of [12] in section 2, we consider in section 3 the paradigm problem of a beam attached to a substrate of variable stiffness. This has a similar buckling pattern to the shell problem and is thus an ideal choice to develop our method.…”