Geometry of principal stress trajectories for a Tresca material under axial symmetry

Abstract

In the case of Tresca’s yield criterion it is usually convenient to subdivide all possible deformations into two subclasses: deformations corresponding to a face of the yield surface and deformation corresponding to an edge of the yield surface. The present paper deals with the second subclass of deformations under conditions of axial symmetry. In this case the stress equations comprising the equilibrium equation and yield criterion can be investigated independently of any flow rule and the final result is also independent of whether elastic strains are included. It is shown that it is always possible to introduce such a principal line coordinate system (i.e. the coordinate system whose coordinate curves coincide with principal stress trajectories) that the product of the scale factors is equal to unity. Using this result the system of equations for mapping between the principal line coordinate and cylindrical coordinate systems is derived and it is shown that this system of equations is hyperbolic.