New models for cold rolling: generalized slab theory and slip lines for fast predictions without finite elements

In this work, a new mathematical model for cold rolling processes is presented. Starting from the governing equations and assuming only a narrow roll gap aspect ratio (in effect, large rolls on a thin strip), we find a solution by introducing two length scales inherent to the problem. The solution consists of a large scale, along with small (next order) correction at a small scale. The leading-order solution depends on the large length scale and matches with slab theory. The next-order correction depends on both the large and small length scales, and reveals rapid stress and strain oscillation. These oscillations are also seen in preliminary FE simulations. The oscillations resemble the slip-line fields, and the FE simulations suggest a strong connection between these oscillations and the residual stress in the resulting strip. The modelling approach used here has potential applications for modelling many metal forming processes, just as the slip-line theory itself did, but with the distinct advantage of simplicity and quick computation.