## 1 Introduction
The mathematical formulation of the upper bound theorem (UBT) presented by [PERSON] and [PERSON] [8] indicates that, taking into account the discontinuity surfaces between the different rigid blocks considered, the actual kinematically admissible velocity field among the possible velocity fields is that one that minimizes the following expression:  
\[\int_{S_{v}}T_{i}\;v_{i}dS_{v}\;\leq\;\int_{S_{D}}k\;[v^{*}]\;dS_{D}^{*}+\; \int_{S_{F}}\;Ti\;v_{i}^{*}\;dS_{F} \tag{1}\]  
where \(T_{i}\): external surface stresses on the workpiece to form; \(v_{i}\): actual velocity field; \(S_{v}\): surfaces where the external forces are applied; \(k\): shear yield stress; \(v^{*}\): velocity discontinuities; \(S^{*}_{D}\): discontinuity surfaces; \(v^{*}_{i}\): kinematically admissible virtual velocity field; \(S_{F}\): external surfaces exposed to external surface stresses. For this, it is useful to divide the deformed part into several zones, with a rigid behavior and which are called triangular rigid zones (TRZ) (Kudo [9; 10; 11; 12; 13]), in each of which the velocity field and its derivatives must be continuous. The application of this method is done by straight lines, considering that only along them there are velocity discontinuities (Rubio [14; 15; 16], Qin [17], and Yang [18]). The rest of the points that make up each block move at the same speed and with the same direction. In a particularly unique way, and with reduced technological changes, the forging processes are fully adapted to the application of this method. Modifications are aimed at achieving a flow behavior of material very similar to that of flat deformation, which is achieved by designing pieces of straight generatrix in the plane perpendicular to the one under study. These TRZ allow the incorporation of different variables present in the plastic forming of metal alloys. In this way, and given that the forge constitutes a nonstationary process, it is possible to determine different natures of the friction (Tresca or Coulomb [19; 20; 21]) acting on the different flat surfaces of contact between part and tool, and, even, assigning different values to each surface. Other parameters, such as the temperature of the process, also have the possibility of being analyzed. One of the main limitations in the classic application of the UBT by means of TRZ comes from the imposition of the double symmetry, although different approaches have been proposed to relative nonsymmetrical parts [22; 23; 24] that prevents us from considering a large number of the geometric configurations present in the industry. In the present work, one of these symmetries has been eliminated, thus significantly increasing the range of application of the method. Consequently, compared with the two options (complex and numerical analytical methods with high computational cost), the analytical method based on the UBT is presented as an interesting alternative to calculate the minimum load to reach the permanent deformation in forging processes, considering a plane strain state. This method offers a reasonable rough value of the necessary load (in terms of pressure) to guarantee the plastic deformation of the part. The application of the UBT, using the model of triangular rigid zones, gives solutions with an extremely low computational cost, since it is possible to develop it immediately by using a simple spreadsheet.