Circularly symmetric deformation of shallow elastic membrane caps

We consider shallow elastic membrane caps that are rotationally symmetric in their undeformed state, and investigate their deformation under small uniform vertical pressure and a given boundary stress or boundary displacement. To do this we use the small-strain theory developed by Bromberg and Stoker, Reissner, and Dickey. We deal with the two-parameter family of membranes whose undeformed configuration is given in cylindrical coordinates as \[ z\left ( x \right ) = C\left ( {1 - {x^\gamma }} \right ), \qquad \left ( 1 \right )\] which includes the spherical cap as a special case ($\gamma = 2$ and $C$ small). We show that if $\gamma > 4/3$ then a circularly symmetric deformation is possible for any positive boundary stress (or any boundary displacement) and any positive pressure, but if $1 < \gamma < 4/3$ then no circularly symmetric deformation is possible if the stress and pressure are positive and small (or for non-positive boundary displacement and small positive pressure).