Generalized Upper Bound Formulation Of Some Rigid-Triangle Velocity Fields For Plane-Strain Extrusion And Tube Drawing

In this paper, upper bound solutions for plane-strain extrusion and plug drawing are presented using rigid-triangle velocity fields, assuming both the shear friction (τ=mk) and the Coulomb friction (τ=μp) conditions at the interfaces. For shear friction condition, the solutions are obtained following the classical approach. For this case, the energy dissipation rates of individual triangles are shown to be equal. For Coulomb friction condition, the analysis is carried out by the generalized upper bound formulation proposed by Collins. The energy dissipation rates of the individual triangles, in the deformation zone, for this friction condition, are found to be in geometric progression. Formulae are provided for the calculation of the lengths of contact and the contact pressures at the die/metal and plug/metal interfaces. Results are presented for mean extrusion/drawing stresses, as a function of reduction, for specified friction conditions and die/plug geometry. Values of optimum die/plug angles are determined graphically for given reductions and friction conditions. Finally, results for an alternative velocity field are presented for which the contact pressures on the die and the plug are nearly equal. The theoretical values are compared with some numerical and experimental results available in the literature.