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An asymptotic analysis of the buckling of a highly shear-resistant vesicle. (English) Zbl 1406.74257

Summary: The static compression between two smooth plates of an axisymmetric capsule or vesicle is investigated by means of asymptotic analysis. The governing equations of the vesicle are derived from thin-shell theory and involve a bending stiffness \(B\), a shear modulus \(H\), the unstressed vesicle radius \(a\) and a constant surface-area constraint. The sixth-order free-boundary problem obtained by a balance-of-forces approach is addressed in the limit when the dimensionless parameter \(C = Ha^2/B\) is large and the plate displacements are small. When the plate displacement is of order \(aC^{-1/2}\), the vesicle undergoes a sub-critical buckling instability which is captured by leading-order asymptotics. Asymptotic linear and quadratic force-displacement relations for the pre- and post-buckled solutions are determined. The leading-order post-buckled solution is described by a simple fourth-order problem, exhibiting stress-focusing with stretching and bending confined to a narrow boundary layer. In contrast, in the pre-buckled state, stretching occurs over a larger length scale than bending. The results are in good qualitative agreement with numerical simulations for finite values of \(C\).

MSC:

74G60 Bifurcation and buckling
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74K25 Shells
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