Buckling of spherical capsules

Abstract

We investigate buckling of soft elastic capsules under negative pressure or for reduced capsule volume. Based on nonlinear shell theory and the assumption of a hyperelastic capsule membrane, shape equations for axisymmetric and initially spherical capsules are derived and solved numerically. A rich bifurcation behavior is found, which is presented in terms of bifurcation diagrams. The energetically preferred stable configuration is deduced from a least-energy principle both for prescribed volume and prescribed pressure. We find that buckled shapes are energetically favorable already at smaller negative pressures and larger critical volumes than predicted by the classical buckling instability. By preventing self-intersection for strongly reduced volume, we obtain a complete picture of the buckling process and can follow the shape from the initial undeformed state through the buckling instability into the fully collapsed state. Interestingly, the sequences of bifurcations and stable capsule shapes differ for prescribed volume and prescribed pressure. In the buckled state, we find a relation between curvatures at the indentation rim and the bending modulus, which can be used to determine elastic moduli from experimental shape analysis.

Numerical deflation of beach balls with various Poisson’s ratios: From sphere to bowl’s shape

“…The mechanism of buckling in axisymmetric conformations was quite recently investigated in detail by Knoche et al [12], with the study of various metastability branches. At first significative order, their calculations show that for the trivial isotropic ("spherical") deformation, the relative volume variation due to an external overpressure ∆P = P ext − P int expresses:…”

“…A few months ago, stability analysis allowed a detailed study of buckling toward axisymmetric conformations [12]. Experimental [13,3,14,15,16,1,17] and numerical [3,18,17] deflation studies showed shapes holding several dimples, also called "multiple indentation".…”

“…It is expected to happen when the external overpressure ∆P = P ext − P int reaches the critical value [9,12]:…”

“…Careful examination of [21], interrupted grey to [18]. 0.014, pressure difference ∆P presents the type of evolution calculated by [12], i.e. quasiplateauing after buckling (variation of about 15% during the whole deflation, plus some occasional dispersion due to numerical procedure), up to autocontact.…”

The secondary buckling transition: Wrinkling of buckled spherical shells

“…Furthermore, nonlinear shell theory has been used to investigate the postbuckling behaviour [13], which revealed that the buckled shape is unstable with respect to further volume reduction if the pressure is controlled. Numerical analyses of nonlinear shape equations have been used to obtain bifurcation diagrams for the axisymmetric deformation behaviour [14,15]. In this paper, we will approach this classical axisymmetric buckling using two different models for axisymmetric deformations.…”