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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on December 1, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2008 61(1):1-24; doi:10.1093/qjmam/hbm021
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Q. Jl Mech. Appl. Math, Vol. 61. No. 1 © The author 2007. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Buckling of an axisymmetric vesicle under compression: the effects of resistance to shear

S. P. Preston, O. E. Jensen and G. Richardson{dagger}

( School of Mathematical Sciences, University Park, University of Nottingham, Nottingham NG7 2RD )

{dagger} < giles.richardson{at}nottingham.ac.uk>

Received 29 May 2007. Revise 23 August 2007. Accepted 24 August 2007.


  Abstract

We consider the axisymmetric deformation of an initially spherical, porous vesicle with incompressible membrane having finite resistance to in-plane shearing, as the vesicle is compressed between parallel plates. We adopt a thin-shell balance-of-forces formulation in which the mechanical properties of the membrane are described by a single dimensionless parameter, C, which is the ratio of the membrane's resistance to shearing to its resistance to bending. This results in a novel free-boundary problem which we solve numerically to obtain vesicle shapes as a function of plate separation, h. For small deformations, the vesicle contacts each plate over a small circular area. At a critical value of plate separation, hTC, there is a transcritical bifurcation from which a new branch of solutions emerges, representing buckled vesicles which contact each plate along a circular curve. For the values of C investigated, we find that the transcritical bifurcation is subcritical and that there is a further saddle-node bifurcation (fold) along the branch of buckled solutions at h = hSN (where hSN > hTC). The resulting bifurcation structure is commensurate with a hysteresis loop in which a sudden transition from an unbuckled solution to a buckled one occurs as h is decreased through hTC and a further sudden transition, this time from a buckled solution to an unbuckled one, occurs as h is increased through hSN. We find that hSN and hTC increase with C, that is, vesicles that resist shear are more prone to buckling.


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