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Mathematics and Mechanics of Solids
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On Non-Linear Radial Oscillations of an Incompressible, Hyperelastic Spherical Shell

N. Roussos

Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

D. P. Mason

Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

D. L. Hill

Department of Mathematics and Statistics, University of Western Australia, Nedlands, Western Australia 6907, Australia

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.

Mathematics and Mechanics of Solids, Vol. 7, No. 1, 67-85 (2002)
DOI: 10.1177/1081286502007001228
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