Through-Thickness Modelling of Metal Rolling using Multiple-Scale Asymptotics

A new semi-analytic model of the metal rolling processes is presented and validated against finite element simulations. The model generalizes the classical Slab Method of cold rolling, and for the first time is able to predict the through-thickness stress and strain oscillations present in long-aspect-ratio roll gaps. The model is based on the asymptotic Method of Multiple-Scales, with the systematic assumptions of a long thin roll gap and a comparably small Coulomb friction coefficient. The leading-order solution varies only on the long length scale of the roll gap length $\hat{\ell}$, and matches with Slab Theory. The next-order correction varies on both the long length scale and the short length scale of the workpiece thickness $2\hat{h}_0$, and reveals rapid stress and strain oscillation both in the rolling direction and through-thickness. For this initial derivation, the model assumes a rigid perfectly-plastic material behaviour. Despite these strong assumptions, the model is shown to compare well with finite element simulations using realistic elasticity and hardening material models. These assumptions facilitate the simplest possible model to provide a foundational understanding of the complex through-thickness behaviour observed in the finite element simulations, while requiring an order of only seconds to compute. Matlab code for evaluating the model is provided in the supplementary material.