Flow stability of a flat plastic ring with free boundaries

Abstract

The problem of two-dimensional unstable flow of an ideally plastic ring acted upon by internal pressure is formulated. The determination of the law of motion for the boundaries and of the time change of pressure is reduced to an ordinary nonlinear differential equation of the second order. For this equation a particular solution of the Cauchy problem is determined; this corresponds to a widening of the ring boundaries with a negative acceleration. For the field of initial velocities an estimate from above is available, expressed in terms of the original parameters. The very particular unstable flow obtained for an ideally plastic ring is also investigated with respect to stability to small harmonic perturbations of the velocity vector, the pressure, or the boundaries of the ring. It is shown that the fundamental flow is stable irrespective of the wave number. This result has been obtained by assuming that the inertial forces in the perturbed flow are small compared to the lasting ones.