  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.
