Asymmetric dynamic instability of axisymmetric polar dimpling of thin shallow spherical shells

Abstract

If the parameter ɛ², which measures the thickness-to-rise of the shell, is small, the axisymmetric polar dimpling of shallow spherical shell due to quadratic pressure distribution is dynamic instability, i.e., a small perturbation can change it to an asymmetric polar dimple mode. In two cases, the problem can be reduced to an eigenvalue problem Tw_n=c_n w_n, where T can approximately be reduced to a Sturm-Liouville operator if ɛ²≪1. The existence of at least one real eigenvalue of T, which means that the axisymmetric polar dimpling is dynamically unstable, is proved by spectral theorem or Hilbert theorem. Furthermore, an eigenfunction, which represents one of the asymmetric modes of the unstable dimple shell, belonging to an eigenvalue of T is found.