On the plastic flow of metals under conditions of axial symmetry

Abstract This paper is concerned with the axially symmetric plastic flow of a rigid-plastic nonhardening material which obeys the Tresca yield criterion of constant maximum shearing stress and the associated flow rule. A general discussion of the basic equations is given. The discussion shows that the hypothesis of Haar and von Kármán is likely to be of great importance in the solution of axially symmetric problems. This conclusion is substantiated by the remainder of the work which considers problems in which the hypothesis is satisfied, i.e. problems in which the circumferential stress is equal to one of the principal stresses in the meridional planes. Possible plastic velocity fields in a circular cylindrical bar stressed to yielding in compression or tension are obtained in §3. Section 4 examines plastic stress fields in the neighbourhood of stress-free conical surfaces. In the final sections of the paper, the plastic stress field and a permissible deformation mode for the problem of the indentation of the plane surface of a semi-infinite body by a circular flat-ended rigid punch are obtained. It is shown that the plastic stress field near the punch can be extended into the rigid region without violating the yield criterion.