Analysis of the non-linear behaviour of circular membranes with an arbitrary radial profile

Equations are presented and a numerical-analytical algorithm is constructed for analysing the non-linear behaviour of a circular membrane with an arbitrary meridional profile. The branch points are determined and the supercritical behaviour is investigated in the case of a specific membrane. In the case of membranes with a relatively high initial flexure due to the choice of the shape of the profile, it is established that it is possible, instead of snapping (popping) the membrane, to obtain a smooth stable transition from one axisymmetric state to a subsequent non-contiguous state through a non-axisymmetric shape deformation step proportional to the cosine of the angular coordinate.