Abstract The slip-line field for frictionless non-steady state extrusion can be constructed by a suitable combination of the slip-line fields for plane strain indentation by smooth and rough dies. The stress state at the vertex of the rigid regions within the slip-line field solution is checked in accordance with Hill's criterion. A statically admissible stress field is constructed by the use of Bishop's method in which some lines of stress discontinuity are introduced. A series method for determining the line of stress discontinuity is proposed by combining Ewing's series method and the method of undetermined coefficients in a power series solution of the differential equation. Some examples of the complete solution are given.