An analysis for three-dimensional upset forging of elliptical disks and rings based on the upper-bound method.

• A general form of kinematically admissible velocity field has been proposed for three-dimensional upset forging of elliptical disks and rings. • Three control functions with free parameters have been introduced to take consideration into sidewise spread, lateral bulging and movement of neutral surface in the upset forging of elliptical rings. • By changing the proposed three control functions, different forms of the assumed general kinematically admissible velocity fields can be derived for the upset forging of various shapes of disks and rings. • The forging load and deformed profiles of the billet can be calculated by minimizing the total plastic dissipation power using the upper-bound method and simplex algorithm. An analysis is carried out for three-dimensional upset forging of elliptical disks and rings based on the upper-bound method. A general form of kinematically admissible velocity field has been proposed by introducing three control functions, each of which contains a free parameter, thus the sidewise spread, lateral bulging along thickness direction as well as movement of neutral surface which occurs in the upset forging of elliptical rings can all be taken into consideration. It is shown that the kinematically admissible velocity fields proposed in previous references by other researchers can be obtained by choosing different control functions in the proposed general form. By optimizing the free parameters using the simplex algorithm in the upper-bound method, the forging load and deformed profiles of the billet can be calculated at certain stages of deformation. Calculation for different billet shapes and friction conditions have been conducted, the obtained theoretical results are in good agreement with the experimental results in previous studies and results obtained from finite element analysis carried out in this paper. Image, graphical abstract [ABSTRACT FROM AUTHOR]