Through-thickness modelling of metal rolling using multiple-scales asymptotics

A new semi-analytic model of the metal rolling process is introduced, which, for the first time, is able to predict the through-thickness stress and strain oscillations present in long thin roll-gaps. The model is based on multiple-scales asymptotics, assuming a long thin roll-gap and a comparably small Coulomb friction coefficient. The leading-order solution varies only on a long lengthscale corresponding to the roll-gap length and matches with slab models. The next-order correction varies on both this long lengthscale and a short lengthscale associated with the workpiece thickness, and reveals rapid stress and strain oscillation both in the rolling direction and through the thickness. For this initial derivation, the model assumes a rigid perfectly-plastic material behaviour. Despite these strong assumptions, this model compares well with finite element simulations that employ more realistic material behaviour (including elasticity and strain hardening). 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. This model can form the foundation of further improved models with more complicated mechanics in the future. Matlab code for evaluating the model is provided in the supplementary material. • We create a semi-analytic model capable of quickly predicting through-thickness stress and velocity during rolling. • Our results compare well with finite element simulations that employ realistic elasticity and strain hardening, despite our modelling assumptions of a rigid perfect-plastic material. • Our model is 16,000 times faster than the comparable finite element simulation in Abaqus. • Our leading-order solution recovers the slab theory result, while being derived using the more systematic assumptions of an asymptotically small roll-gap length-to-height ratio and Coulomb friction coefficient. • Our next-order correction reveals an oscillatory pattern of through-thickness variations, which may have consequences for future studies investigating microstructure development and residual stress from rolling.