Non-linear radial oscillations of a transversely isotropic hyperelastic incompressible tube

The constitutive equation for a transversely isotropic incompressible hyperelastic material is written in a covariant form for arbitrary orientation of the anisotropic director. Three non-linear differential equations are derived for radial oscillations in radial, tangential and longitudinal transversely isotropic thin-walled cylindrical tubes of generalised Mooney–Rivlin material. A Lie point symmetry analysis is performed. The conditions on the strain-energy function and on the net applied surface pressure for Lie point symmetries to exist are determined. For radial and tangential transversely isotropic tubes the differential equations are reduced to Abel equations of the second kind. Radial oscillations in a longitudinal transversely isotropic tube and in an isotropic tube are described by the Ermakov–Pinney equation.