Roll shape design for foil rolling of a four-high mill

In this paper, a mathematical model for a four-high mill is proposed to analyze the elastic deformation of the backup and work rolls, including their respective axial deflection and surface flatness. The contact pressure between the backup and work rolls and that between the work roll and foil are functions of the roll’s position in the axial direction. In this analytical model, the rolls and foil are divided into many small regions and finite difference or matrix methods are used to derive the deflections of the work and backup rolls as a function of the force density in the foil width direction. A camber shape at the surfaces of the backup and work rolls is designed to counteract the rolls’ elastic deformation and to make the rolled foil as flat as possible. The effects of the selected roll materials upon the rolled thickness distribution are also discussed. A thickness variation within 1 μm can be obtained for a foil of 50-μm thick and 40-mm wide under reduction of 32 % using a WC work roll with a camber radius of 150 m.