The buckling of elastic spherical caps (Q1179713)

The author studies the elastic buckling of a spherical cap simply- supported at its edge and subjected to a constant uniform external pressure under the assumption, concerning the deformation of the cap, that there is a plane of symmetry. John's shell equations are used and the problem is formulated as a single operator equation on an appropriate Hilbert space. First it is shown that contrary to the high multiplicity of the critical eigenvalues of the problem of buckling of a full sphere the multiplicity of the critical eigenvalue for the cap is either one or two. Hence the Lyapunov-Schmidt reduction can be well used to solve the problem. For a shallow cap it is shown that solutions possessing circular, pear-shaped, elliptical, triangular, square-shaped, pentagonal and a variety of other symmetries branch from the unbuckled state of the shell. The stability of these solutions is discussed and some numerical results are presented.