It is well-known that a “tether” may be drawn out from a pressurized liposome by means of a suitably applied radial-outward force applied locally to the lipid bilayer. The tether is a narrow, uniform cylindrical tube, which joins the main vesicle in a short “transition region.” A first-order energy analysis establishes the broad relationship between the force F needed to draw the tether, the radius $R0$ of the tether, the bending-stiffness constant B for the lipid bilayer and the membrane tension T in the pressurized liposome. The aim of the present paper is to study in detail the “transition region” between the tether and the main vesicle, by means of a careful application of the engineering theory of axisymmetric shell structures. It turns out that the well-known textbook “thin-shell” theory is inadequate for this purpose, because the tether is evidently an example of a thick-walled shell; and a novel ingredient of the present study is the introduction of elastic constitutive relations that are appropriate to the thick-shell situation. The governing equations are set up in dimensionless form, and are solved by means of a “shooting” technique, starting with a single disposable parameter at a point on the meridian in the tether, which can be adjusted until the boundary conditions at the far “equator” of the main vessel are satisfied. It turns out that the “transition region” between the tether and the main vessel is well characterized by only a few parameters, while the tether and main vessel themselves are described by very simple equations. Introduction of the thick-shell constitutive relation makes little difference to the conformation of, and stress-resultants in, the main vessel; but it makes a great deal of difference in the tether itself. Indeed, a kind of phase-change appears to take place in the “transition region” between these two zones of the liposome.

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