Some approximate solutions of the dynamic problem of axisymmetric shock deformation of a previously unstressed incompressible elastic medium

Approximate theoretical solutions are presented for the boundary value problem of the shock load on the boundary of a circular cylindrical cavity in the space occupied by an incompressible elastic medium undeformed previously. We assume that the shock load causes the movement of the medium particles along helical trajectories. Data on the types and velocities of shock waves are based on the analysis of the dynamic conditions for the compatibility of discontinuities, supplemented by relations along the characteristic directions. We show that the leading front of the dynamic process in a medium undeformed previously, which is a plane-polarized shock wave, is simultaneously included in one of the families of characteristics. This property results in the constancy of shear direction on the plane-polarized shock wave. This condition makes it possible to significantly simplify the obtaining of approximate theoretical solutions for the near-front neighborhood of the shock wave. The two approximate solutions of the problem are presented. One of them bases on solving a system of evolutionary equations that was obtained using the matched asymptotic expansions method; and the second solution, on a version of the ray method.