Finite amplitude azimuthal shear waves in a compressible hyperelastic solid

Lagrangian equations of motion for finite amplitude azimuthal shear wave pr opagation in a compressible isotropic hyperelastic solid are obtained in co nservation form with a source term. A Godunov-type finite difference proced ure is used along with these equa tions to obtain numerical solutions for w ave propagation emanating from a cylindrical cavity, of fixed radius, whose surface is subjected to the sudden application of a spatially uniform azim uthal shearing stress. Results are presented for waves propagating radially outwards; however, the numerical procedure can also be used to obtain solu tions if waves are reflected radially inwards from a cylindrical outer surf ace of the medium. A class of strain energy functions is considered, which is a compressible generalization of the Mooney-Rivlin strain energy functio n, and it is shown that, for this class, an azimuthal shear wave can not pr opagate without a coupled longitudinal wave. This is in contrast to the pro blem of finite amplitude plane shear wave propagation with the neo-Hookean generalization, for which a shear wave can propagate without a coupled long itudinal wave. The plane problem is discussed briefly for comparison with t he azimuthal problem.