BIFURCATION TO PEAR-SHAPED EQUILIBRIA OF PRESSURIZED SPHERICAL MEMBRANES

We study the problem of non-spherical, axisymmetric equilibria of an inflated, spherical membrane. We model the membrane as a two-dimensional elastic body characterized by a general class of strain-energy functions, and we consider a general class of loading devices, including (soft) pressure control and (hard) control of the total mass of gas enclosed by the membrane as special cases. Employing tools of modern bifurcation theory, we illuminate the precise necessary and sufficient conditions for bifurcation from the spherical state to an axisymmetric "pear-shaped" state, and we perform a local post-bifurcation analysis. In particular, for a large class of physically reasonable strain-energy functions, we demonstrate the existence of an isola bifurcation (closed loop of non-spherical solutions), which is consistent with experimental observations.