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The program is especially effective in cases where a symmetry axis can be defined, because it is able to reduce the computational resources needed for simulation. As typical FE codes, DEFORM-F2 analysis are pided into three stages: pre-processing, simulation and post-processing. As exposed before, four extrusion processes are considered, two direct and two indirect. For each one of these principal groups or types, two subtypes are considered, those related to the geometry of the final product obtained from the initial billet: solid and cup extrusion. Figure 1 shows the two configurations considered for direct extrusion type. In Figure 2, the two extrusion processes for indirect extrusion are represented. Due to the symmetry of the processes only one half of the cross-section of each case is modelled. For all the cases the workpiece is cylindrical with an initial radius of 5 mm and 10 mm in height. The cross-sectional area of the final extrusion is also the same in all cases. Thus, for all the extrusion processes considered, the extrusion ratio is the same, defined as the ratio between the initial and the final cross-sectional area, as indicated in equation (1). Once the geometrical conditions for all the cases have been defined, new parameters must be implemented into the software used. Thereby, all the simulated extrusions correspond with cold forming conditions and the material of the billets is a low carbon steel, concretely AISI 1010. This steel is widely used in extrusion processes as it presents good formability. Not only the dimensions of the workpieces and extrusion tools are enough to define the extrusion process, but also additional information is required by the software. The shape complexity of the mesh that defines the geometry of the work piece must be specified. For all the analysis developed this parameter was considered as moderate. In the same way, the accuracy for the mesh was also defined as moderate. On the other hand, other features for the extrusion process were addressed and introduced into the software. The extrusion semi-angle considered for the simulations of this work is always the same and equal to 90 degrees. A constant ram speed for the punch of 250 mm/s is selected, and in the same way, the total primary die travel is determined, with a value of 7 mm in all the simulations. The friction factor must be established too; Tresca friction law is used in this work, and three different values of the friction factor will be considered for each process in order to compare their influence in the different results. Thus, each one of the four extrusion processes considered run in three different friction conditions; low friction (m=0.08), maximum friction (m=1), and an intermediate situation between both previous values (m=0.5). Once the conditions and components involved in the extrusion process are defined, the simulation analysis based on the Finite Element Method starts. This stage corresponds to the second of the software; the simulation. After finishing the analysis the third stage begins; the post- processing. The software enables the visualization of the results in different ways and allows their exportation for comparative analysis in order to be able to draw conclusions that help the optimization of manufacturing processes. 3. FE model validation Several models have been developed in order to quantify properly the actual true strain and ram forces involved in extrusion processes [19, 20]. With the aim of validating the FEM extrusion model, empirical models based in Johnson studies [21] are employed for estimating the extrusion force in direct and indirect solid extrusion processes. Johnson developed the following equation in order to estimate the extrusion deformation (2): ߝ௫ ൌܾܽή ݎ ௫
(2) The extrusion ratio, rx, is obtained as indicated in (1). Empirical constants a and b use to take the typical values 0.8 and 1.2, respectively. In indirect solid extrusion processes, the extrusion load can be approached by (3) according to [2]: ܨൌ ܣ ή ߪ ത ήߝ௫