When a circular shaft is twisted the shearing stresses are greatest in the surface of the shaft.
If the limit of proportionality of the material in shear is at a stress , then this stress is first attained in the surface of the shaft at a torque
where J is the polar second moment of area, and a is the radius of the cross-section.
The torque sustained by the elastic core is
where subscripts 1 refer to the elastic core. The torque sustained by the outer plastic zone is
where L is the length of the shaft. We assume that the outer plastic region suffers the same angle of twist; this is tantamount to assuming that radii remain straight during plastic torsion of the shaft