On the theory of rolling

Abstract

A comprehensive solution to von Karman's basic equation is developed, with the aid of a digital computer, the linear first-order differential equation being integrated by a fourth-order Runge-Kutta process. An analytical solution for the special case of constant interfacial shear stress $\tau $, equal to the yield shear stress k of the material being rolled, is derived and used to test the convergence of the numerical solution. The mixed boundary condition $\tau =\mu _{s}$ or k whichever is smaller is incorporated, and back and front tensions and both elastic arcs of contact are included in all the proposed solutions, as well as variation of the yield stress k through the arc of contact, so that the theory can be applied to both hot and cold rolling situations. Comparisons are made with successively more approximate solutions and with well-known existing theories and it is shown that none of these earlier theories are capable of predicting roll torque with any degree of precision.