Asymptotic analysis of asymmetric thin sheet rolling

An analytical model for asymmetric rolling is presented, which includes asymmetry in roll friction, roll size and roll speed, for a rigid, perfectly plastic thin sheet deformed with Coulomb friction. This model is solved asymptotically, based on the systematic assumptions that both the roll gap aspect ratio and the friction coefficient are small. While the leading order solution is shown to be consistent with an existing slab model, we are able to derive additional detail by looking to higher orders. We compare our higher order solution and the leading order solution with finite element simulations, and use the results to determine the practical range of validity of our analytical model. Within this region, it gives reasonable quantitative predictions of the force and torque results from finite element simulations and approximates through thickness variation of stress and strain with orders of magnitude shorter computation times. A MATLAB implementation of this solution is included in the supplementary material.