Summary: The paper deals with a rigid cone having a rough wall that rotates around its axis of symmetry in an incompressible rigid/plastic solid. It is assumed that the shear yield stress of the solid depends on the equivalent strain and a damage variable. We give an exact analytical solution to this problem. The plastic zone where hardening and softening due to damage occur, develops near the cone. The remainder of the solid is rigid. The stress field is extended into the rigid zone. It is shown that for any hardening law commonly used for metals, the solution does not exist after some amount of deformation, in particular it may not exist at all.