Radial oscillations of thin cylindrical and spherical shells: investigation of Lie point symmetries for arbitrary strain-energy functions

The Lie point symmetry structure of the second order differential equations which describe non-linear radial oscillations of thin-walled hyperelastic cylindrical and spherical shells is investigated. The differential equations depend on the strain-energy function and on the net applied surface pressure. If the net applied surface pressure is time independent, the differential equations admit the Lie point symmetry corresponding to time translational invariance for arbitrary strain-energy functions. Other Lie point symmetries exist for each equation only for special classes of strain-energy function. For the cylindrical shell the special class includes the Mooney-Rivlin strain-energy function and the differential equation reduces to the Ermakov-Pinney equation. A new solution is obtained for a specific time dependent net applied surface pressure. For the spherical shell the special class does not include the Mooney-Rivlin strain-energy function. For free oscillations the differential equation reduces to the Ermakov-Pinney equation but there also exists a special net applied surface pressure and for this pressure the differential equation is more general than the Ermakov-Pinney equation.