Determination of Contact Stresses when Upsetting a Rectangular Strip

Abstract

Closed analytical solutions of a plane strain problem for contact stresses in zones with variable friction forces for upsetting the strip made of ideal rigid-plastic material are derived. It is established that the use of the Mises exact plasticity condition for the slip zone instead of the approximate one written in principal stresses allowing for the Coulomb friction law leads to an essential (by a factor of 2–4 at f > 0,2) increase in the calculated slip zone length. To match the theoretical results with the experimental ones, a new method for calculating the contact stresses is proposed, in which their distributions in the slip and stagnation zones are derived from the solution of the plane strain problem using a parabolic approximation of the plasticity condition. The sizes of slip and stagnation zones are found depending on the characteristics of contact friction and conditions of appearance of the peaks near the strip end in the normal stress diagram are determined. The possibility of occurrence of retardation zones under tangent contact stresses lower than the yield point under simple shear τs are determined. Formulas for determining the upsetting force are refined.