Finite Amplitude Azimuthal Shear Waves in a Compressible Hyperelastic Solid

Lagrangian equations of motion for finite amplitude azimuthal shear wave propagation in a compressible isotropic hyperelastic solid are obtained in conservation form with a source term. A Godunov-type finite difference procedure is used along with these equations to obtain numerical solutions for wave propagation emanating from a cylindrical cavity, of fixed radius, whose surface is subjected to the sudden application of a spatially uniform azimuthal shearing stress. Results are presented for waves propagating radially outwards; however, the numerical procedure can also be used to obtain solutions if waves are reflected radially inwards from a cylindrical outer surface of the medium. A class of strain energy functions is considered, which is a compressible generalization of the Mooney-Rivlin strain energy function, and it is shown that, for this class, an azimuthal shear wave can not propagate without a coupled longitudinal wave. This is in contrast to the problem of finite amplitude plane shear wave propagation with the neo-Hookean generalization, for which a shear wave can propagate without a coupled longitudinal wave. The plane problem is discussed briefly for comparison with the azimuthal problem.