An Efficient Method of Calculating the Force and Torque in the Upsetting of Cylinders with Rotating Dies

The process of upsetting, with rotating dies, is used to reduce the force required to deform the workpiece and to receive more homogeneous deformation compared to the same process without the rotational motion of the dies. The upper bound method is an efficient tool for a quick estimate of process parameters. The accuracy of upper bound solutions depends on the chosen class of kinematically admissible velocity fields. The present paper provides an efficient method for choosing kinematically admissible velocity fields that satisfy some stress boundary conditions if the associated flow rule is considered. The method applies to the upsetting of cylinders. It is expected that it leads to accurate solutions if friction is high enough. Besides, the kinematically admissible velocity field accounts for a rigid region near the axis of symmetry. Such a region inevitably occurs in exact solutions because the friction stress must vanish at the axis of symmetry. The final expression for the upper bound, on a combination of the force and torque, involves two arbitrary parameters. These parameters are determined using the upper bound theorem. An example is provided to illustrate the method