ASYMMETRIC DYNAMIC INSTABILITY OF AXISYMMETRIC POLAR DIMPLING OF THIN SHALLOW SPHERICAL SHELLS

If the parameterε~2 ,which measures the thickness-to-rise of the shell,is small,theaxisymmetric polar dimpling of shallow spherical shell due to quadratic pressuredistribution is dynamic instability,i.e.,a small perturbation can change it to an asymmetricpolar dimple mode.In two cases,the problem can he reduced to an eigenvalue problemTw=cw,where T can approximately be reduced to a Sturm-Liouville operator ifε~2《1.The existence of at least one real eigenvalue of T,which means that the axisymmetricpolar dimpling is dynamically unstable,is proved by spectral theorem or Hilbert theorem.Furthermore,an eigenfunction,which represents one of the asymmetric modes of theunstable dimple shell,belonging to an eigenvalue of T,is found.