The Upper Bound Theorem in forging processes: Model of Triangular Rigid Zones on parts with horizontal symmetry

Abstract

The analysis to determine the necessary forces with which to achieve a plastic deformation in metallic materials, in particular, in forging processes and under conditions of plain strain, has been raised over the years through a double approach; on the one hand, by analytical methods that involve a great complexity in their developments but that allow a direct understanding of the parameters that direct these processes. On the other hand, numerical methods, in which, thanks to the enormous development of computer technology, they provide solutions with a high approximation but, in most cases, do not allow to interpret independently the effect of each one of the parameters that come into play. The development of computers relegated analytical methods to the background. An alternative of great interest to apply these methods comes from the study of the Upper Bound Theorem by means of the Triangular Rigid Zones (TRZ) Model. One of the main limitations in the application of this model come from the fact that it is necessary to define a kinematically admissible velocity field and for complex geometric configurations of parts, this field becomes increasingly complicated. A new approach has delimited, from a theoretical perspective, a modular configuration based on a General Module formed by three TRZ that adapts to any geometry of flat surfaces of the part. Another limitation of the Upper Bound Method is the consideration of the plain strain represented by a flat section with double symmetry. Obviously, this imposition only allows to study a limited number of part configurations, which restricts its application in forging processes since the great majority of forged parts do not present geometrically this double symmetry. The present work releases one of these boundary conditions allowing to expand the possibilities of application of this method.