This paper outlines the salient features of the “generalized upper bound” technique and uses it to derive upper bounds for plane strain compression and extrusion of metals in the presence of Coulomb friction (τ = µp). For compression the velocity field is established by reformulating the same proposed earlier by Lee and Altan so as to be compatible with the interface Coulomb friction condition. For extrusion, the upper bound velocity field is derived from a stream function for flow through a smooth die of wedge angle (α+λ) where, α is the semi-wedge angle of the rough die, and λ is the friction angle (tanλ = µ). Analysis for this case is also presented assuming the flow to be radial. The analytical results are compared with those obtained from, the slip-line field and finite element analyses and from the rigid triangle velocity fields. These are also validated by some e experimental results available in literature.

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 ([Formula: see text]) and the Coulomb friction ([Formula: see text]) 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.