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In the classical 'first approximation' theory of thin-shell structures, the constitutive relations for a generic shell element--i.e. the elastic relations between the bending moments and membrane stresses and the corresponding changes in curvature and strain, respectively--are written as if an element of the shell is flat, although in reality it is curved. In this theory it is believed that discrepancies on account of the use of 'flat' constitutive relations will be negligible provided the ratio shell-radius/thickness is of sufficiently large order. In the study of drawing of narrow, cylindrical 'tethers' from liposomes it has been known for many years that it is necessary to use instead a constitutive law which explicitly describes a curved element in order to make sense of the mechanics; and indeed such tethers are generally of 'thick-walled' proportions. In this paper we show that the proper constitutive relations for a curved element must also be used in the study, by means of shell equations, of the buckling of initially spherical thin-walled giant liposomes under exterior pressure: these involve the inclusion of what we call the 'M[kappa]' terms, which are not present in the standard 'first-approximation' theory. We obtain analytical expressions for both the bifurcation buckling pressure and the slope of the post-buckling path, in terms of the dimensions and elastic constants of the lipid bi-layer, and also the initial state of bending moment in the vesicle. We explain physically how the initial bending moment can affect the bifurcation pressure, whereas it cannot in 'first-approximation' theory. We use these results to map the conditions under which the vesicle buckles into an oblate, as distinct from a prolate ('rugby-ball') shape. Some of our results were obtained long ago by the use of energy methods; but our aim here has been to identify precisely what is lacking in 'first-approximation' theory in relation to liposomes, and so to put the 'shell equations' approach onto a firm footing in mechanics.

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 [R.sub.0] 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. [DOI: 10.1115/1.1500341]