Elastic-plastic torsion of circular ring sectors

The solution is presented to the problem of uniform torsion of circular ring sectors with circular cross section under the assumption of perfect plasticity. The elastic-plastic problem is solved by a semi-inverse method. When the entire ring is plastic a discontinuity of stress appears which may be regarded as the limiting case of an elastic core; such discontinuities have recently been discussed in the literature [1]. The solution for the discontinuous fully plastic stress distribution is exact, that for the elastic-plastic case approximate in the sense that it is found exactly for cross sections differing slightly from the circular. This difference is negligible for ratios of ring radius $R$ to cross section radius $\rho$ occurring in practical applications to helical springs of small pitch.