| Typically, a slab of very wide material is pulled between two rotating rolls and experiences inhomogeneous compression, produced by the stresses within the gap region. It is the stress normal to the rolls, called the roll pressure, that the rolling analyst would most like to obtain.;In this treatise, after the aforementioned contributions of von Karman and Orowan (of the general rolling problem) are reviewed, the current asymptotic study is then undertaken to take particular advantage of the obvious physical feature of the small one-pass reduction. The variables within the governing equations are nondimensionalized and the velocity and stress fields are expanded in terms of the small reduction parameter. A set of triply coupled equations for the roll pressure, shear stress and the longitudinal deviatoric stress result and they are compared with the classical inhomogeneous deformation theory due to Orowan. Two effects of transverse inhomogeneous deformation arise. One occurs within the shear stress as a nonlinear transversely varying term. The other effect occurs with the flow rule expression and is shown to be a shear deformation contribution to the work hardening.;A relaxation-iteration numerical scheme was implemented upon the set of triply coupled equations. The transverse variation of stresses due to the shear strain-rate (within the flow rule) can alter the front and back tensions significantly (relative to Orowan). The effect of inhomogeneous deformation on the shear stress for this case is virtually undetectable. However, when the material parameters are chosen to exhibit maximal shear effect, the resulting influence upon the other stresses is not as significant.;Although hot rolling is not covered here, conclusions based upon the asymptotic analysis of the cold rolling problem can be drawn about the validity of current hot rolling theories. This is predicated on the fact that the systematic study done here dictates the limitations of the theory and its conclusions, and subsequently the conclusions of past researchers. The asymptotic development results in a well posed mathematical problem in which the Coulomb boundary condition is satisfied throughout the gap region. It is never necessary to switch boundary conditions to one of yielding at the wall as is necessary in slap analysis. (Abstract shortened with permission of author.)... |
