A Slip Line Field for the Hot Rolling Process

Abstract

By using a geometrical representation, recently proposed by Prager (1953)†, of the equations defining the flow in plane strain of a plastic rigid material, it is possible to obtain a solution to the statically undetermined problem of hot rolling. The geometrical method facilitates adjustment, by inspection, of the slip line field until both velocity and stress boundary conditions are satisfied. Accuracy of the method is ensured by continual reference to the stress plane, in which an orthogonal family of identical cycloids defines the slopes of both slip lines and hodograph characteristics at corresponding points. To allow for the effect of rotation of the roll surface it has been necessary to consider the effect of rotation on hodographs in detail. The main assumptions which have to be made are that the yield stress of the material is constant throughout the plastic zone, and that the shear stress along the boundary between roll and material attains the yield shear stress of the material. This latter assumption corresponds with Orowan's condition of ‘sticking’ throughout the complete arc of contact (Orowan 1943), and implies that the slip lines of one family meet the roll surface orthogonally. An interesting feature of the solution is the occurrence of a thin slice of rigid material adjacent to the roll surface at the entry point having a large velocity discontinuity along its boundary.