(3) For direct solid extrusion processes, where the friction at the container-workpiece interface has an important effect, the load can be estimated by (3): ܨൌ ܣ଴ ή ߪ ത௙  ήቀߝ௫ ൅ ଶ௅஽బቁ           
(4) In Figure 3 a comparison of the ram forces obtained by FEM with those calculated by (3) and (4), for indirect and direct solid extrusion, respectively, is shown. The figure shows that the empirical model for direct solid extrusion (4) provides an upper limit for the ram force, and it is very close to the force obtained by FEA and the maximum friction factor (m = 1); whereas the empirical model for indirect solid extrusion (3) gives a lower limit of the applied force. In both cases, but especially in the case of direct  extrusion, results from FEA are in  good agreement with the values calculated by both empirical methods; the relative difference is less than 9% for indirect extrusion, and less than 3% for direct extrusion, so the FE models can be considered validated. 4. Results and discussion Diagrams relating the extrusion force and the punch stroke are developed for each one of the cases of the study. Then, the different curves obtained are compared in a same diagram. 4.1. Solid extrusion Figure 4 shows the results obtained for direct and indirect solid extrusion. Three curves are represented in each diagram, one for each friction factor considered in the simulations. The curves obtained from the values given by the code present oscillations. During the simulation process mesh nodes are constantly coming into contact or losing contact with  the die/container surface, and this means a constant updating of the mesh system. Due to this,  it is recommended the use of  trend  lines  for the analysis of the results, being the polynomial type the most appropriate [18].  Fig. 4. Direct solid extrusion for different friction factors (left); Indirect solid extrusion for different friction factors (right). In Figure 4  it is observed that in direct extrusion processes the  required punch loads are higher than in indirect extrusion processes. The higher the friction factor, the higher the difference found. For the maximum coefficient of friction,  the difference between punch loads in both processes, direct and indirect,  is more pronounced. This is typical of direct extrusion processes, where friction at the container-billet interface makes the biggest difference in the required load. Thus, for  m=1  (sticking conditions), one can see how the load-stroke curves have a clear descending  trend, while this phenomenon is less important for lower values of the friction factor.
The fact that the required load decreases with a constant slope is because the friction  load against the billet movement decreases at constant speed when the extrusion is taking place, as the remaining billet length decreases.  In indirect extrusions, the workpiece displacement within the container does not occur, and therefore there is no friction between both, consequently, the loads required in the process are significantly lower. However, there is a contribution to the friction load that has to be considered in both direct and indirect extrusion: this is the friction at the die-billet interface located at the end of the container. As a general trend, in both cases, the higher the friction factor, the higher the forces needed. 4.2. Cup extrusion On the other hand, Figure 5 compares the ram forces needed in cup extrusion processes. Figure 5 (left) presents results for direct cup extrusion process and Figure 5 (right) presents the ones in indirect cup extrusion process. In both cases, again, three different friction values are considered (m=0.08, m=0.5 and m=1).   Fig. 5. Direct cup extrusion for different friction factors (left); Indirect cup extrusion for different friction factors (right). As in solid extrusion, friction has a great influence in the value of the necessary loads. The higher the value of the friction factor, the higher the force required. This situation can be seen in both variants, direct and indirect extrusion processes. As in solid extrusion, for the highest values of friction, a descendent  trend of the curve associated with the direct extrusion processes can be identified; while in the indirect extrusion cases, the curves are more horizontal, the loads remain more constant throughout the process at stationary conditions. It is observed that the necessary loads for the simulated direct extrusion processes are greater than the required for indirect extrusion under the same friction conditions. This is as well due to the contact of the forming material with the container before being extruded through the extrusion die. In direct extrusion, the material is displaced by the punch and it is affected by the friction with the container walls. In the indirect extrusion this does not occur, as it can be observed in Figure 2; therefore, in comparison with direct extrusion, this component of the friction load does not exist so the necessary force to run the process is lower. As a final comment, it is observed that forces required in  cup extrusion are significantly higher than in solid extrusion for the same extrusion ratio. 4.3. Friction load contributions Due to the relevance of friction in these processes, the contribution to the friction load is analyzed in more detail. Total load required in every extrusion process can be expressed as indicated in (5). ܨ ௘௫ ൌܨ௛ௗ ൅ܨ ௙ ൅ܨ ௦             (5)  In direct extrusion processes, the component Ff can be pided as well as follows (6): ܨ ௙ ൌܨ ௖ ൅ܨௗ௜௘             (5) To quantify these contributions to the friction load, limit situations are considered: the highest friction value m=1 (corresponding to sticking conditions) and the ideal situation of a frictionless process m=0. As the software does not allow introducing  the value of zero for the coefficient of friction, a value of 0.00001  is introduced  as the closest value to the desired theoretical conditions. Thus, comparing  the necessary forces required in both limit situations, the contributions to the friction load can be obtained and discussed.  In Figures 6 and 7, the contributions  to the friction load are  represented by calculating the difference between the required load under sticking conditions and perfectly sliding conditions, for direct and indirect solid extrusion.   Fig. 6. Friction load for direct solid extrusion.  Fig. 7. Friction load for indirect solid extrusion. When the initial billet is displaced along  the container in direct extrusion processes, the final friction load  is related to two different friction phenomena as explained above (Fc + Fdie). In indirect extrusion processes this does not happen. In both cases, direct and indirect, there is a friction load  contribution  consequence of the contact between the material and the die, Fdie. Only in direct extrusion the friction load at the container walls  (Fc) will appear, so the maximum contribution of the component Fdie, that is common to both kinds  of processes, can be directly obtained from the friction load contribution in indirect extrusion process (green curve in Figure 7), and can be estimated around 20 kN. Figures 8 and 9 present the friction contribution to the load for the cup extrusion both in direct and indirect processes. As in the previous case, both graphs can be compared to evaluate the force associated to each type of friction condition, thanks to the different situations that take place in direct and indirect extrusion processes. Comparing the results for cup extrusion processes, and following the same methodology, the friction load at the die, Fdie, can be directly obtained from the indirect cup extrusion process (green curve, Figure 9). In this case, the maximum friction load contribution due to the die-billet contact is around 100 kN, that is much higher than in the case of solid extrusion. On the contrary, the maximum friction load contribution due to the container wall is much higher in the case of solid extrusion than in cup extrusion. As one example of the FE simulations, strain effective diagrams are shown in Figure 10 for a friction factor of m=0.08 and the same punch stroke during the deformation process. In this work, solid direct and indirect extrusion processes as well as cup direct and indirect extrusion processes of low carbon steel (AISI-1010) have been compared. DEFORM  F2  has been used to simulate  different extrusion processes in order to analyze the friction load contributions to the final load required in this kind of processes. 有限元挤压工艺英文文献和中文翻译(3):http://www.751com.cn/fanyi/lunwen_34305.html