Buckling patterns of complete spherical shells filled with an elastic   medium under external pressure
  The critical buckling characteristics of hydrostatically pressurized complete spherical shells filled with an elastic medium are demonstrated. A model based on small deflection thin shell theory, the equations of which are solved in conjunction with variational principles, is presented. In the exact formulation, axisymmetric and inextensional assumptions are not used initially and the elastic medium is modelled as a Winkler foundation, i.e., using uncoupled radial springs with a constant foundation modulus that is independent of wave numbers of shell buckling modes. Simplified approximations based on a Rayleigh-Ritz approach are also introduced for critical buckling pressure and mode number with a considerable degree of accuracy. Characteristic modal shapes are demonstrated for a wide range of material and geometric parameters. A phase diagram is established to obtain the requisite thickness to radius, and stiffness ratios for a desired mode profile. The present exact formulation can be readily extended to apply to more general cases of non-axisymmetric buckling problems. 
Motohiro Sato, M. Ahmer Wadee, Kohtaroh Iiboshi, Takafumi Sekizawa and   Hiroyuki Shima
cond-mat.soft cond-mat.mes-hall