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

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, h(TC), 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 = h(SN) (where h(SN) > h(TC)). 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 h(TC) and a further sudden transition, this time from a buckled solution to an unbuckled one, occurs as h is increased through h(SN). We find that h(SN) and h(TC) increase with C, that is, vesicles that resist shear are more prone to buckling.