A method of solution for the cyclic bending problem

A new approach to modeling processes of elastoplastic cyclic bending on roller levellers for beam straightening, levelling of long-dimensional profiles with arbitrary cross-sections, and some other applications of a similar kind, is proposed. Standard techniques for modeling the above-mentioned processes do not account for longitudinal motion, and, therefore, are not completely adequate. The stress-strain state (SSS) at every point of the material is determined not only by its position but also by its strain history. In modeling the processes of cyclic bending, use is made of the Euler-Lagrange approach to formulating the governing equations, together with traditional assumptions such as the hypothesis that plane cross-sections remain plane, stationarity of the processes, and the theory of anisotropic hardening. These allow one to take into account the strain history of moving material particles. The method developed makes possible the determination of not only the longitudinal components of the strain and stress tensors, but also the transverse components, as well as the beam profile's curvature, level and distribution of residual stresses.