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.

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