In the first place, the characteristics of the flow curves for the selected metallic materials were determined in a compression test (Fig. 2), which represented the cold metal forming processes (annealed Al) and warm metal forming processes (reinforced Al).

Such materials were selected so that it would be possible to demonstrate, based on the proposed plastic similarity condition, how the value of matching of the modelling material to the metallic one affects the accuracy of the obtained results. The forward extrusion process was purposefully performed for two types of aluminum, whose flow curves are different in character. The determined coefficients of sensitivity to the deformation speed *m* (in the range of 0.001 to 0.1 s-1) were respectively: 0.03 for reinforced aluminum and 0.04 for annealed aluminum.

Next, for both of these metallic materials characterizing cold and warm bulk metal forming, appropriate mixtures of model materials were then selected to best simulate the behavior of metallic materials.

Based on the macroscopic analysis, large differences in the flow pattern of individual materials were observed. For annealed aluminum, characterized by strong deformation hardening, the material deformation distribution is very uniform in the cross-section of the sample.

This is evidenced by the constant bending of the coordination grid lines perpendicular to the extrusion direction. In the case of hardened aluminum showing a slightly less hardening, the deformation is mainly located in the areas close to the die. The radius of curvature (Rin) of the coordination grid lines perpendicular to the extrusion direction in the central part of the sample for reinforced aluminum (Fig. 3a) is smaller than that for annealed aluminum (Fig. 3b). However, in the case of the radii of curvature of the line at the outer walls (Ro) of the sample, the situation is the opposite for both type of aluminum. For all the materials, in the first phase of the extrusion process, there is a rapid increase in force (Fig. 3c) caused by the upsetting of the material placed in the container. The upsetting is due to the slight difference in dimensions between the container and the material to be squeezed out, necessary for the material to be freely inserted into the container. The main extrusion process then begins with the material filling the die and then flowing out of it; at this stage of the process, the extrusion force increases further. Only at the end of the extrusion process does the extrusion force decrease due to the reduction of the friction surface. The courses obtained for both types of aluminum are very similar, only the extrusion force of the reinforced aluminum is about 7% lower than that for the annealed aluminum. This may be due to the slightly greater sensitivity of the material to the strain rate in the case of annealed aluminum (m=0.04).

The selection of the modelling material for both types of aluminum was made based on the similarity condition of the material in the plastic scope developed by the authors [9]. It assumed a quantitative assessment of the matching degree of the shape of the yield stress-deformation curves for the modelling material and the actual material; two parameters are determined: the scale factor:

and the similarity coefficient:

where:

yield stresses of the real material in point i,

yield stresses of the modelling material in point i,

k – number of points on the reinforcement curves of the modelling material and the real material, for which the similarity coefficient t was determined.

The coefficient t is a non-dimensional quantity, which makes it possible to determine simply and quickly the degree of matching of the reinforcement curve of the model material to the real material.

In the case of a theoretically ideal matching of both curves, this coefficient should equal zero. Fig. 4 shows a graphic interpretation of the quantities present in equations (1) and (2). Of course, both the scale factor and the similarity coefficient t, in the analyzed case, were determined as average values for the three strain rates (0.001 to 0.1 s-1).

On this basis, many mixtures of model materials have been tested and their characteristics were determined in upsetting tests (Fig. 5).

Finally, for reinforced aluminum characterizing in low work hardening, a mixture of filia with 20 % kaolin was ultimately chosen, for which the respective coefficients were t = 0.042 and C = 183.11. For reinforced aluminum, the best match was obtained for filia with 28 % kaolin and 8 % lanolin, where the scale factor C= 130.73 and the similarity coefficient t = 0.069 were determined (Table 1). Fig. 6a shows an examples the courses of the yield stress for the applied mixture as well as reinforced aluminum with the strain rate 0.01 s-1 and Fig. 6b for the second pair of materials.

In the case of annealed aluminum, which has very strong reinforcement a problem with the selection of the modelling material was encountered. Studies of modelling materials have shown that through an increase of the kaolin content, it is possible to obtain mixtures of increasingly high work hardening; however, a kaolin addition in the amount of over 20 % lowers the limit deformations and such a material tends to crack faster than the real material. The strongest work hardening as well as highest limit deformations corresponding to the real materials were obtained for a mixture of 68 % filia 28 % kaolin and 4 % lanolin. The relatively high value of the similarity coefficient proves poor matching of the modelling curve to the real material.

Based on the analysis of the above data, we can notice significant differences in the character of the course of both curves, especially for big deformations, which is also confirmed by a relatively high value of the similarity coefficient t.

**Physical modelling of forward extrusion in flat deformation state **

Based on the presented results for both modelling materials best matched to both types of aluminum, extrusion processes were performed, which were then compared for both compiled pairs of materials. As the lubricant, magnesium stearate was used for both aluminum types, for which *µ* equaled 0.05. In turn, in the physical model, petrolatum was applied, which was placed in specially made grooves on the surface of the samples. For such conditions, and for different modelling material compositions, the obtained friction coefficient *µ* equaled 0.055. In the modelling processes, similarly to the actual ones, the extrusion force in the function of the punch path was recorded. In order to compare both courses, the results obtained from the physical modelling were recalculated into the actual process through their multiplication by the scale factor C. Fig. 7 shows the courses of the extrusion forces as well as comparative macro-analyses of the deformed samples with coordinate grids, and also the distributions of the equivalent deformation on the surface for this pair of materials obtained with the use of a digital image analysis program, ASAME. The character of the force course obtained from the physical model after a proper transformation is similar to the course obtained in the real process (similarity coefficient t=0.042).

In turn, the level of the force in the modelling process is about 10 % lower than that in the actual process. This can result from the slight differences in the shape of the work hardening curves between the modelling material and the real material, as well as the slightly different friction conditions present in both processes resulting from a few times lower pressures, and also a bit smaller dead zone in the die area for the modelling materials in respect of aluminum. Based on the presented results, it can be stated that, in the physical model, the manner of material deformation was very close to the flow manner of the actual material, which is also confirmed by the distribution of the equivalent deformation for this pair of materials obtained by means of the ASAME program.

A similar comparative analysis was performed for the other pair of materials. i.e. annealed aluminum and the mixture of filia with 28 % of kaolin and 4 % of lanolin. In this case, both the force courses and the deformation of the coordinate grid for the physical process significantly deviate from the actual process (Fig. 8). Such results could be expected, as the similarity coefficient had the highest value of all the examined material pairs and it equaled t = 0.069.

Additionally, the modelling material demonstrated cracks in the areas of biggest deformations. Despite intense attempts to select a modelling material for annealed aluminum, it was impossible to find an appropriate wax mixture which would properly reflect the behaviour of the actual material during plastic deformation. Also the distribution of equivalent deformation obtained from the ASAME program for the material pair of annealed aluminum and for the mixture of filia with 28 % kaolin and 4 % lanolin (Fig. 8c) shows the biggest differences in reference to the previous pairs of materials.

Moreover, it should be noted that the biggest discrepancies in the deformation distributions for the particular pairs of materials are located in strongly deformed areas, where very high deformation of the coordinate grid is observed. These differences can result from damages of the grids in these areas, which, in turn, made it difficult to interpret the grids by means of the image analysis system ASAME.

**Numerical modelling of forward extrusion in flat deformation state**

Numerical modelling of the forward extrusion process was performed for all the real materials. This study only presents the test results referring to reinforced aluminum extrusion in the flat deformation state. This material was selected because, in physical modelling according to the new plastic similarity condition, for this material, the similarity coefficient t = 0.042 had an intermediate value among all the other pairs of materials. The numerical thermo-mechanical model of the actual process was constructed by means of the Marc Mentat program. The yield stress-deformation curves for annealed aluminum were run in a numerical form for different values of deformation, deformation rate and temperature. For the discretization of the deformed material, 1280 quadrangular elements type Solid 11 were used (Quad 4). An automatic reconstruction of the deformed grid was applied, with a change of the number and density distribution of the elements, due to the deformation and penetration of the element into the tools. In the numerical model of the reinforced aluminum extrusion process, all the tools. i.e. the die, the punch and the container, were assumed as rigid. The assumed boundary conditions were close to those present in the actual process. The Coulomb friction model was applied as well as the following heat transfer coefficients: for aluminum– specific heat 0.92 kJ/(kgK) and thermal conductivity 203 W/(mK) and for the steel tools – specific heat 0.46 kJ/(kgK) and thermal conductivity 15 W/(mK). Initially, in numerical modelling, a constant value of the coefficient of friction was introduced for the particular tools, equaling 0.05, determined in a ring test. For such deformation conditions, the manner of material flow obtained from FEM (Fig. 9a) only slightly deviated from the actual process. In turn, the level of extrusion force in the function of the punch path (the maximal extrusion force equaled 62 kN) was over 15 % lower than that in the actual process (Fig. 9b). Also, we should note that, in numerical modelling, it was not possible to obtain the formation of a dead zone for the same deformation conditions, which had been observed both in the actual process and, to an even higher degree, in the physical modelling. Probably, the cause of those differences was the assumption of a constant value of the coefficient of friction for the entire tool (punch, recipient and die) in the numerical model. In the actual process, as well as the physical one, probably as a result of poor lubrication in the area of the die impression, a discontinuity of the lubricant layer was observed, and the coefficient of friction reached values over 0.05, as it was assumed in the numerical model.

Next, numerical simulations were performed for higher values of the coefficient of friction. Ultimately, similar manners of material flow as well as course of the extrusion force were obtained in the numerical model and the real process when different coefficients of friction were assumed for the particular tools.

For the punch, the coefficient equaled 0.05, for the container, it was 0.1 and for the die, 0.25. Fig. 10a shows the obtained courses of the force in the function of the punch path for numerical modelling, for new friction conditions, as well as for the actual process. In turn. Fig. 10b demonstrates the analyses of both the similarity of the material flow manner and the deformation distributions by means of the image analysis program ASAME, as well as the results obtained from FEM.

After the introduction of new boundary conditions into the numerical model, the obtained results were very close to those in the actual process. Only the extrusion force in the numerical model was about 5 % lower than that in the real process. This difference can result from the fact that the numerical model referred to the flat deformation state and differed from the actual process in that it did not consider the friction forces between the side wall of the sample and the container.

As it has been demonstrated by the presented investigation results, during the designing of plastic forming processes by means of FEM, it is difficult to perform a correction of the boundary conditions as, for that, the information obtained in the actual process is needed. And so, for the verification of the model, we can use the physical model. However, in such a case, we should consider the fact that this model can be burdened with errors resulting from insufficient matching of the modelling material to the actual one, as well as slightly different other properties of the modelling materials. Nevertheless, the presented results demonstrate good agreement and confirm the possibility of applying physical modelling for the analysis and verification of both numerical modelling and the actual process of forward extrusion.