The axisymmetric branching behavior of complete spherical shells

The purpose of this paper is to describe the axisymmetric branching behavior of complete spherical shells subjected to external pressure. By means of an asymptotic integration technique (based on the smallness of the ratio of the shell thickness to the shell radius) applied directly to a differential equation formulation, we are able to continue the solution branches from the immediate vicinity of the bifurcation points, where the solution has the functional form predicted by the linear buckling theory, to the region where the solution consists of either one or two “dimples” with the remainder of the shell remaining nearly spherical. The analysis deals with a novel aspect of bifurcation theory involving “closely spaced” eigenvalues.