Stress Amplification in a Ring Caused by Dynamic Instability

Abstract

An analysis is presented for stress amplification in a ring caused by the dynamic instability of symmetric in-and-out breathing oscillations, which results in energy transfer to flexural modes. Stress amplification is shown to depend on a ring stability parameter p that is proportional to the unperturbed hoop strain and inversely proportional to the thickness-to-radius ratio. The analysis is a generalization of earlier work by Goodier and McIvor. They showed that, with p tacity assumed small, stress is amplified by approximately the factor 6, independent of the wave number of the flexural mode into which the breathing energy is transferred. The present analysis shows that for large p (increased instability), higher-order nonlinear terms must be included in the differential equations in order to give bounded solutions. With these terms included, stress amplification departs from 6 as p is increased, the peak compressive stress becoming larger and the peak tensile stress becoming smaller. With damping also included, the amplification approaches unity (no amplification) as p approaches zero, and passes through a maximum as p is increased. An experimental example of flexural fracture caused by such amplification is given.