Abstract Here we consider axially symmetric shear waves propagating from a cylindrical cavity in an incompressible hyperelastic solid, whose strain energy function is expressible as a truncated power series in terms of the basic strain invariants. The solid is assumed to be unbounded. A continuous pulse is initiated at the boundary of the cavity and can break in finite time. We determine what shock waves can subsequently occur using an approximate solution obtained by Whitham's nonlinearization technique. We find that under mild restrictions on the material parameters, a shock wave develops near the front or back of the pulse, and propagates indefinitely. In addition, a transient shock can occur and exist for a finite length of time. The set of shock paths will be referred to as the shock pattern. We show how the material parameters influence the shock pattern. As well, the analysis presented here provides accurate estimates of the breaking distance and time, and the location of the shock path, for any shock waves that occur. Results of the analysis are illustrated with numerical solutions obtained using a relaxation scheme for systems of conservation laws.