Bending of an elastoplastic strip with isotropic and kinematic hardening

 We propose an exact analysis for finite bending of a compressible elastoplastic strip with combined hardening at a given stretch normal to the bending plane. We apply the self-consistent eulerian rate-type elastoplastic model based on the logarithmic rate, in conjunction with a Tresca-type loading function. Utilizing the maximum circumferential stretch at the outer surface as an independent parameter, we derive exact analytic expressions for the bending angle, the bending moment, the outer and inner radii, the radii of the two elastic-plastic interfaces and the circumferential stretches at these two interfaces, as well as the stress distributions in every current cross section. In particular, we establish an explicit relation between the two circumferential stretches at the two elastic-plastic interfaces, and we show that this relation is universal for all cases of hardening. We show also that the maximum and minimum circumferential stretches at the outer and inner surfaces obey a reciprocal relation in the course of both elastic and elastic-plastic deformations.