A hamiltonian approach to the bifurcating mechanics of elastocapillary shells held by needles

An increasing number of research problems addresses the mechanical response of bubbles onto which a thin polymeric layer has been grown in a manner that their properties are governed by a complex interplay of interfacial tension and elasticity. This “elastocapillary” interplay gives rise to long-term stability or intriguing shape changing capacities and is often quantified by analyzing the inflation and deflation of a bubble held by a needle in a fluid. Even though the response of such objects to an excess pressure can be predicted using modern numerical tools, it remains important to tackle the problem via a theoretical route to reveal more clearly the underlying physics.

The mechanical response of purely elastic shells held by a needle is known to be a notoriously difficult theoretical problem, solved by Föppl and von Karmán after they had realized that the problem is intrinsically non-linear even at modest deformations. In view of that, it is interesting to quantify to what extent this unusual behavior is impacted by an interface combining an interfacial tension ? and an interfacial elasticity Gh, where G is the elastic shear modulus of the polymeric layer and h its thickness. We rationalized this problem in the framework of a Hamiltonian theoretical approach and show that it can be fully analyzed for small deformations. We show that a curious bifurcation occurs when the elastocapillary coefficient Gh/? reaches a certain value which depends on the size of needle to which the initially spherical cap is attached. We discuss also the experimental implications of these results.