On non-linear radial oscillations of an incompressible, hyperelastic spherical shell

Non-linear radial oscillations of a thin-walled spherical shell of incompressible isotropic hyperelastic material are considered. The oscillations are described by a second order differential equation which depends on the strain-energy function and the net applied pressure at the surfaces. The condition on the strain-energy function for the differential equation to be an Ermakov-Pinney equation is derived. It is shown the condition is not satisfied by a Mooney-Rivlin strain-energy function. The Lie point symmetry structure of the differential equation for a Mooney-Rivlin material is determined. Three approximate solutions are derived for free oscillations of a neo-Hookean material. The approximate solutions have the form of non-linear superpositions similar to the solutions for the non-linear radial oscillations of a thin-walled cylindrical tube.