a generalized spherical velocity field for bi-metallic tube extrusion through dies of any shape

in this paper, an upper-bound approach is used to analyze the extrusion process of bi-metallic tubes through dies of any shape with moving cylindrical shaped mandrel. a generalized kinematically admissible velocity field using spherical coordinate system is developed to evaluate the internal power and the power dissipated on frictional, velocity discontinuity surfaces and the total power. the extrusion process is also simulated using the finite element code, abaqus. analytical results are compared with the results given by experiments of other researchers and also by the finite element method. these comparisons show a good agreement.