Application of Collins’ upper bound theorem to upsetting processes

Abstract

The standard formulation of the upper bound theorem is incompatible with the Coulomb friction law. However, this law is widely used in metal forming processes analysis. An alternative formulation based on non-kinematically admissible velocity fields can evaluate the load required to deform the material. The present paper adopts this formulation for evaluating the compression force in solid and hollow cylinder compression processes. The trial velocity fields are continuous. The field for the solid cylinder involves no parameter for minimization, and the field for the hollow cylinder involves one parameter. Parametric analyses of the solutions are provided. The compression forces found are compared to available results at small friction coefficients. The solution is also in agreement with physical expectations. In particular, the solution for the hollow cylinder compression predicts that the radial velocity is everywhere positive in the friction coefficient is small enough. For larger friction coefficients, the radius where the radial velocity vanishes propagates from the inner radius as the friction coefficient increases. The final result can be used in conjunction with standard procedures for evaluating the friction coefficient.