Geometry of principal stress trajectories for piece-wise linear yield criteria under axial symmetry

Rigid- and elastic- plastic solids obeying quite a general yield criterion represented by linear functions of the principal stresses are considered. A general axisymmetric state of stress satisfying the hypothesis of Haar and von Karman is analyzed in quasi-static flow. The circumferential velocity vanishes. A superimposed restriction on the yield criterion is that the system of stress equations is hyperbolic. The primary objective of this study is to select optimal coordinates and unknowns for deriving the integrable differential relations in their most convenient form for numerical treatments. It is shown that there exists a simple relation between the scale factors of the principal lines coordinate system (the coordinate curves of this coordinate system coincide with the trajectories of the principal stresses). Another simple relation exists between the scale factors and the principal stresses. The mapping between the principal lines and cylindrical coordinates is determined by a system of hyperbolic differential equations.