An upper bound solution for extrusion and drawing through conical dies considering Coulomb friction

The upper bound theorem is a primary method for analyzing metal forming processes. A disadvantage of almost all upper bound solutions is that the constant friction law (Prandtl's law) is adopted. The present paper employs an alternative formulation of the theorem that enables the inclusion of Coulomb friction in the boundary conditions of a boundary value problem. The general theorem is specialized to extrusion and drawing processes through conical dies. The general trial velocity field generalizes a typical kinematically admissible velocity field used in solutions based on the conventional formulation of the theorem. This trial velocity field is simplified, again using typical assumptions used in solutions based on the conventional formulation of the theorem. The calculation of the extrusion and drawing forces reduces to evaluating an ordinary integral numerically. The optimal die angle that minimizes the extrusion or drawing force is determined. The restrictions on the range of the validity of the solution are discussed. Comparison with a solution available in the literature is made.