- Email: [email protected]
Physical and mathematical modelling of extrusion processes
Journal of Materials Processing Technology 106 (2000) 2±7
Physical and mathematical modelling of extrusion processes Mogens Arentofta,*, Zbigniew Gronostajskib, Adam Niechajowiczb, Tarras Wanheima a Institute for Manufacturing Engineering, Technical University of Denmark, Building 425, DK-2800 Lyngby, Denmark Institute of Mechanical Engineering and Automation, Technical University of Wroclaw, ul. Lukasiewicza 3/5, 50-371 Wroclaw, Poland
b
Abstract The main objective of the work is to study the extrusion process using physical modelling and to compare the ®ndings of the study with ®nite element predictions. The possibilities and advantages of the simultaneous application of both of these methods for the analysis of metal forming processes are shown. # 2000 Published by Elsevier Science B.V. Keywords: Physical modelling; Mathematical modelling; Extrusion
1. Introduction Currently, for the designing of metal forming processes, mathematical modelling is a very popular method as an assisting system. It is often treated as a universal tool in all problems of designing metal forming processes. Despite incontrovertible advantages, one should also remember that there are limitations in using assisting systems which could cause fault in the designed process; in order to avoid such a situation, veri®cation of the processes is necessary. The best way of veri®cation is testing on the real object. This stage is the most time consuming and expensive part of designing, therefore means which would allow the elimination of or which would limit this stage are being sought. Physical modelling can be one of the methods for this and this is the reason why the method has become so popular. The basic idea in the model-material technique is to substitute the real metal to be plastically formed with a soft model material, which behaves in a way that is analogous to the real metal that is being deformed. This method makes it possible to obtain information about the real process; the directions of metal ¯ow, the strain and strain-rate distributions, the load, the friction conditions, defects, and so on. The advantages of using soft materials for study of the deformation of metals are closely connected with the fact that their yield stresses are of the order of magnitude of 500± 1000 times lesser than that of metals. Consequently, there is the possibility of using cheap tool materials, and simple and cheap presses to observe and register the ¯ow of material and to shorten time of experiments [1,2]. A survey of the *
Corresponding author. Fax: 45-45-93-01-90
0924-0136/00/$ ± see front matter # 2000 Published by Elsevier Science B.V. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 6 2 9 - 4
technique and an extensive literature review is given in papers [2,3]. The correct transformation of model-experiment results to reality is the basic problem. In order that transformation can be possible, the model and the real process must be similar. For metal working processes the following important similarity conditions can be assumed [3,4]: geometrical, plastic, frictional, thermal, elastic and dynamic. The ful®lment of all of these conditions is practically impossible. In most cases, the ®rst four conditions are the most important, although they are also very hard to ful®l. Therefore, it is essential to determine and select which material properties and process parameters are the most important for the purpose of the experiment. For instance, with large plastic deformation, the effect of elastic strain may be taken as negligible. A choice of the model material for the workpiece is practically always the critical step. It is impossible to determine a wax as the real material, which responds to changing temperature and strain rate in the same way as the real material, therefore the processes are assumed to be taking place under isothermal conditions. Then the stress± strain curve of real material can be expressed in the form s Cen e_ m , and it can be accepted that the plastic similarity condition is ful®lled if the n and m values are equal for the real and the model material. This method of choosing the wax material is a simpli®cation and cannot be used at all times. The model-material technique is a cheap and fast method of experiment as compared with the real-metal investigation experiment [5±8]. However, modelling a complex, 3D shapes become costly and labour consuming. Therefore it would be convenient to replace a complex model 3D shape part by plane strain or axisymmetrical models and to transfer
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
the results from these simpli®ed models to the complex part. This is the normal procedure for a large complex structure, which may be divided into parts with a characteristic type of material ¯ow, analysed separately and the results joined together [9,10]. The main aim of the present work is the application of physical modelling to the analysis of cold mild steel extrusion and the comparison of the results obtained with the results from FEM.
3
ening that is caused by the diffusion phenomena during forming. The approximated curves shown in Fig. 1 allow an estimation of the suitability of the applied function for the description of the wax stress±strain curves. Of the base stress±strain curves for different materials, the mixture consisting of 84% ®lia, 12% kaolin, 4% lanolina, and 2% FeO was chosen, this material being characterised by n 0:20 and C 0:15 MPa. 3. Experiments
2. Material The material for these tests was chosen in order to create the possibility to investigate the model behaviour of mild steel in the cold extrusion processes; this determines the n value for the wax material. The C value was selected as very low in consideration of the force limit of the transducer that had to be used. The wax material was chosen in order to ful®l all of the above-mentioned requirements. Filia wax was base component. For the change of its properties, the following additions were used: kaolin, lanolin, silicon, M1, harpix. The different characteristic shapes of the stress±strain curves obtained in the experiments are shown in Fig. 1, together with curves from the approximation of the experimental results by the use of a function of the form s Cen . It was possible to change the character of the stress±strain curves over a wide range by a suitable selection of components for the soft model materials; from curves with a high rate of hardening that are typical for cold working materials to curves characteristic for hot working with typical soft-
The experiments performed were mainly based on physical modelling. Mathematical simulation by the DEFORM program was used as a supplement to the physical modelling. The correctness and accuracy of the simulations depend mainly on the assumed boundary conditions, the model of the material and the errors in numerical calculations. One of the most important factors was friction. The constant Coulomb friction law was assumed, the friction coef®cient, determined by ring test, being equal to 0.14. The form s Cen was chosen as the model of material with n 0:2 and C 0:15 MPa. Two kinds of experiments were performed, the ®rst being concerned with plane-strain extrusion which was made as symmetrical, or as non-symmetrical with the die situated only on one side; whilst the second was concerned with axisymmetrical and non-axisymmetrical extrusion, where full bars and tubes were extruded. The dimensions of the tools and the samples were determined by the similarity conditions between these two kinds of experiment.
Fig. 1. The characteristic stress±strain curves.
4
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
Fig. 2. The load±stroke curves.
4. Results The load and stroke for tube extrusion were measured and pictures of deformed grid were made. The obtained load± stroke curves are shown in Fig. 2, and consist of two different ranges. The ®rst range concerns the ®lling of the die by material and is independent of the reduction of area and mandrel diameter; whilst the second range appears when the material ¯ows through the die. In the experiments two types of curves were obtained, these differing in the shape in the second range. This difference is caused by dissimilarity of the friction area. When this area is large, because the material sticks after extrusion to the mandrel and glass, the curve is increasing in second range. For symmetrical tests
with small area reduction when material does not stick to the glass and the mandrel, the curve is almost ¯at in the second range (Fig. 2, curves ax9 and ax10). In order to compare plane strain, axisymmetrical and nonaxisymmetrical extrusion, the maximum average pressure on the punch was determined; this is presented in Fig. 3. The points that present this relationship lie on one curve and their values do not depend on the kind of extrusion. This testi®es that the extrusion load is almost independent of the eccentricity of the mandrel. The next stage of experiments concern plane strain. In order to secure very similar friction on the mandrel surface in the plane-strain test as in tube extrusion, two kinds of tests were made: a symmetrical test for comparison with bars and
Fig. 3. The relationship between the average pressure on the punch and the area reduction.
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
5
Fig. 4. The relationship between the average pressure on the punch and the area reduction, including DEFORM predictions.
a non-symmetrical test for comparison with tube extrusion. In the non-symmetrical plane test the die was at one side, and the metal bar was at the opposite side that was treated as mandrel. The reduction in plane-strain extrusion was chosen to ful®l geometrical or area-reduction similarity to that of tube extrusion. The experiments show that the extrusion force is very similar for symmetrical and non-symmetrical plane-strain extrusion. The following stage of experiments concern a comparison of the loads and of the grids in tube extrusion and planestrain extrusion. A comparison of the average pressure on the punch for the plane-strain test and tube extrusion for
different area reduction is shown in Fig. 3. It appears that the average pressure on the punch for plane strain and tube extrusion is very similar for the same area reduction. For small area reduction the average pressure is a little longer for plane strain than for tube extrusion. It changes with increasing area reduction because in order to obtain a greater area reduction in tube extrusion a larger mandrel is used. The contact area of the mandrel and the samples has very high friction and it increases the average pressure. The simulation by DEFORM gives a lower average pressure for all area reductions, because the coef®cient of friction that is determined by the ring test and which is used in simulation by the
Fig. 5. Comparison of the grid obtained by physical modelling and the DEFORM simulation for axisymmetrical tube extrusion.
6
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
Fig. 6. The method of calculation of the parameter wg.
DEFORM program is smaller than that in the physical modelling. Load estimation is often one of many tasks of physical modelling. Frequently, it is necessary to measure and calculate the distributions of velocity, strain rate, strain, pressure on the tools, etc. This can be done, but it is labour consuming, therefore measurements are avoided so far as there are other methods to gain indispensable results. Mathematical modelling is one of the methods that allow the required results to be obtained in a quicker and cheaper way, but these results have to be veri®ed. If the results from physical modelling, such as load and the picture of the deformed grid, and the same results obtained by mathematical modelling are convergent, then it can be assumed that other results such as velocity, strain rate, and strain computed by using both methods will be similar. In this way
strain and stress distributions can be calculated by mathematical modelling methods and accepted as correct results so far as physical modelling con®rms the correctness of the calculations. In the present work this way was assumed. For the plain-strain extrusion, axisymmetrical extrusion of bars and tubes mathematical modelling by DEFORM was performed. The calculated loads for these processes and their similarity to the results obtained by physical modelling are shown in Fig. 4. A comparison of distorted grids for extrusion in the planestrain and axisymmetrical state obtained by physical and mathematical modelling is shown in Fig. 5. The grids obtained by the two used methods are very similar, thus it can be testi®ed that mathematical models describe the extrusion processes in the correct way. A quantitative comparison of distorted grid is very dif®cult, therefore for
Fig. 7. The relationship between the parameter wg and the area reduction.
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
quantitative comparison only a very simple parameter wg could be taken. This parameter expresses the bending of the perpendicular line to the axis of sample divided by the width of the samples (Fig. 6). Fig. 7 presents the relationship between these parameters and the area-reduction results calculated from geometrical similarity for plane strain and axisymmetrical extrusion. The relationship is very similar for both of these cases. Taking into consideration this similarity and the similarity of the average pressure on the punch for plane-strain and axisymmetrical extrusion, it can be stated that for this same reduction of area axisymmetrical bars and tubes extrusion can be modelled by the planestrain test with the correct selection of process parameters such as the process scheme, the friction conditions, etc. 5. Conclusions 1. Physical modelling is a very ef®cient way to analyse complex metal forming processes and can be used for the veri®cation of designed processes. 2. Physical modelling gives a lot of detailed information about the way material ¯ows, and die ®lling. Quantitative determination of the strain and stress by physical modelling is less precise because wax material is very sensitive to temperature and strain rate and adequate determination of the friction coef®cient is very dif®cult in physical modelling. 3. The investigation has shown that physical and mathematical modelling give very similar results. 4. There are possibilities to replace a complex 3D structure by a simpler plane-strain model by retaining particular conditions.
7
5. The correct material model and friction conditions are critical for the correctness of the mathematical and physical modelling of metal forming processes. References [1] T. Wanheim, Physical Modelling of Metalprocessing, Procesteknisk Institut, Laboratories for Mekaniske Materialeprocesser, Danmarks Teknisk Hùjskole, Denmark, 1988. [2] S.K. Ghosh, CAD/CAM & FEM in Metal Working, North Staffordshire Poliytechnic, Beaconside, Stafford, UK, Pergamon Press, Oxford, 1988, p. 1. [3] T. Wanheim, Trends in physical simulation of metal working processes, Paper presented at Fourth Cairo University Conference on Mechanical Design and Production, Cairo University, Egypt, December 27±29, 1988. È hnlichkeitstheorie der Unformtechnik, [4] O. Pawelski, Beitrag zur A Archiv fuÈr das EisenhuÈttenwesen 35 (1) (1964) 27±36. [5] A. Azushima, H. Kudo, Physical simulation for metal forming with strain rate sensitive model material, Advanced Technology of Plasticity, Vol. II, Stuttgart, 1987, pp. 1221±1227. [6] H. Tsukamoto, T. Egawa, J. Ibushi, S. Oomori, K. Yagishita, Simulative model test on metal forming using plasticine as a model material, Technical Paper MF74-179, USA, 1974. [7] T. Wanheim, Trends in physical simulation of metal working processes, Paper presented at Fourth Cairo University Conference on Mechanical Design and Production, Cairo University, Egypt, December 27±29, 1988. [8] H. Tsukamoto, T. Egawa, J. Ibushi, S. Oomori, K. Yagishita, Simulative model test on metal forming using plasticine as a model material, Technical Paper MF74-179, USA, 1974. [9] A.P. Green, On unsymmetrical extrusion in plane strain, J. Mech. Phys. Solids 8 (1955) 189±196. [10] H. Shin, D. Kim, N. Kim, A simpli®ed three-dimensional ®niteelement analysis of the non-axisymmetrical extrusion processes, J. Mater. Process. Technol. 38 (1993) 567±587.
Physical and mathematical modelling of extrusion processes Mogens Arentofta,*, Zbigniew Gronostajskib, Adam Niechajowiczb, Tarras Wanheima a Institute for Manufacturing Engineering, Technical University of Denmark, Building 425, DK-2800 Lyngby, Denmark Institute of Mechanical Engineering and Automation, Technical University of Wroclaw, ul. Lukasiewicza 3/5, 50-371 Wroclaw, Poland
b
Abstract The main objective of the work is to study the extrusion process using physical modelling and to compare the ®ndings of the study with ®nite element predictions. The possibilities and advantages of the simultaneous application of both of these methods for the analysis of metal forming processes are shown. # 2000 Published by Elsevier Science B.V. Keywords: Physical modelling; Mathematical modelling; Extrusion
1. Introduction Currently, for the designing of metal forming processes, mathematical modelling is a very popular method as an assisting system. It is often treated as a universal tool in all problems of designing metal forming processes. Despite incontrovertible advantages, one should also remember that there are limitations in using assisting systems which could cause fault in the designed process; in order to avoid such a situation, veri®cation of the processes is necessary. The best way of veri®cation is testing on the real object. This stage is the most time consuming and expensive part of designing, therefore means which would allow the elimination of or which would limit this stage are being sought. Physical modelling can be one of the methods for this and this is the reason why the method has become so popular. The basic idea in the model-material technique is to substitute the real metal to be plastically formed with a soft model material, which behaves in a way that is analogous to the real metal that is being deformed. This method makes it possible to obtain information about the real process; the directions of metal ¯ow, the strain and strain-rate distributions, the load, the friction conditions, defects, and so on. The advantages of using soft materials for study of the deformation of metals are closely connected with the fact that their yield stresses are of the order of magnitude of 500± 1000 times lesser than that of metals. Consequently, there is the possibility of using cheap tool materials, and simple and cheap presses to observe and register the ¯ow of material and to shorten time of experiments [1,2]. A survey of the *
Corresponding author. Fax: 45-45-93-01-90
0924-0136/00/$ ± see front matter # 2000 Published by Elsevier Science B.V. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 6 2 9 - 4
technique and an extensive literature review is given in papers [2,3]. The correct transformation of model-experiment results to reality is the basic problem. In order that transformation can be possible, the model and the real process must be similar. For metal working processes the following important similarity conditions can be assumed [3,4]: geometrical, plastic, frictional, thermal, elastic and dynamic. The ful®lment of all of these conditions is practically impossible. In most cases, the ®rst four conditions are the most important, although they are also very hard to ful®l. Therefore, it is essential to determine and select which material properties and process parameters are the most important for the purpose of the experiment. For instance, with large plastic deformation, the effect of elastic strain may be taken as negligible. A choice of the model material for the workpiece is practically always the critical step. It is impossible to determine a wax as the real material, which responds to changing temperature and strain rate in the same way as the real material, therefore the processes are assumed to be taking place under isothermal conditions. Then the stress± strain curve of real material can be expressed in the form s Cen e_ m , and it can be accepted that the plastic similarity condition is ful®lled if the n and m values are equal for the real and the model material. This method of choosing the wax material is a simpli®cation and cannot be used at all times. The model-material technique is a cheap and fast method of experiment as compared with the real-metal investigation experiment [5±8]. However, modelling a complex, 3D shapes become costly and labour consuming. Therefore it would be convenient to replace a complex model 3D shape part by plane strain or axisymmetrical models and to transfer
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
the results from these simpli®ed models to the complex part. This is the normal procedure for a large complex structure, which may be divided into parts with a characteristic type of material ¯ow, analysed separately and the results joined together [9,10]. The main aim of the present work is the application of physical modelling to the analysis of cold mild steel extrusion and the comparison of the results obtained with the results from FEM.
3
ening that is caused by the diffusion phenomena during forming. The approximated curves shown in Fig. 1 allow an estimation of the suitability of the applied function for the description of the wax stress±strain curves. Of the base stress±strain curves for different materials, the mixture consisting of 84% ®lia, 12% kaolin, 4% lanolina, and 2% FeO was chosen, this material being characterised by n 0:20 and C 0:15 MPa. 3. Experiments
2. Material The material for these tests was chosen in order to create the possibility to investigate the model behaviour of mild steel in the cold extrusion processes; this determines the n value for the wax material. The C value was selected as very low in consideration of the force limit of the transducer that had to be used. The wax material was chosen in order to ful®l all of the above-mentioned requirements. Filia wax was base component. For the change of its properties, the following additions were used: kaolin, lanolin, silicon, M1, harpix. The different characteristic shapes of the stress±strain curves obtained in the experiments are shown in Fig. 1, together with curves from the approximation of the experimental results by the use of a function of the form s Cen . It was possible to change the character of the stress±strain curves over a wide range by a suitable selection of components for the soft model materials; from curves with a high rate of hardening that are typical for cold working materials to curves characteristic for hot working with typical soft-
The experiments performed were mainly based on physical modelling. Mathematical simulation by the DEFORM program was used as a supplement to the physical modelling. The correctness and accuracy of the simulations depend mainly on the assumed boundary conditions, the model of the material and the errors in numerical calculations. One of the most important factors was friction. The constant Coulomb friction law was assumed, the friction coef®cient, determined by ring test, being equal to 0.14. The form s Cen was chosen as the model of material with n 0:2 and C 0:15 MPa. Two kinds of experiments were performed, the ®rst being concerned with plane-strain extrusion which was made as symmetrical, or as non-symmetrical with the die situated only on one side; whilst the second was concerned with axisymmetrical and non-axisymmetrical extrusion, where full bars and tubes were extruded. The dimensions of the tools and the samples were determined by the similarity conditions between these two kinds of experiment.
Fig. 1. The characteristic stress±strain curves.
4
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
Fig. 2. The load±stroke curves.
4. Results The load and stroke for tube extrusion were measured and pictures of deformed grid were made. The obtained load± stroke curves are shown in Fig. 2, and consist of two different ranges. The ®rst range concerns the ®lling of the die by material and is independent of the reduction of area and mandrel diameter; whilst the second range appears when the material ¯ows through the die. In the experiments two types of curves were obtained, these differing in the shape in the second range. This difference is caused by dissimilarity of the friction area. When this area is large, because the material sticks after extrusion to the mandrel and glass, the curve is increasing in second range. For symmetrical tests
with small area reduction when material does not stick to the glass and the mandrel, the curve is almost ¯at in the second range (Fig. 2, curves ax9 and ax10). In order to compare plane strain, axisymmetrical and nonaxisymmetrical extrusion, the maximum average pressure on the punch was determined; this is presented in Fig. 3. The points that present this relationship lie on one curve and their values do not depend on the kind of extrusion. This testi®es that the extrusion load is almost independent of the eccentricity of the mandrel. The next stage of experiments concern plane strain. In order to secure very similar friction on the mandrel surface in the plane-strain test as in tube extrusion, two kinds of tests were made: a symmetrical test for comparison with bars and
Fig. 3. The relationship between the average pressure on the punch and the area reduction.
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
5
Fig. 4. The relationship between the average pressure on the punch and the area reduction, including DEFORM predictions.
a non-symmetrical test for comparison with tube extrusion. In the non-symmetrical plane test the die was at one side, and the metal bar was at the opposite side that was treated as mandrel. The reduction in plane-strain extrusion was chosen to ful®l geometrical or area-reduction similarity to that of tube extrusion. The experiments show that the extrusion force is very similar for symmetrical and non-symmetrical plane-strain extrusion. The following stage of experiments concern a comparison of the loads and of the grids in tube extrusion and planestrain extrusion. A comparison of the average pressure on the punch for the plane-strain test and tube extrusion for
different area reduction is shown in Fig. 3. It appears that the average pressure on the punch for plane strain and tube extrusion is very similar for the same area reduction. For small area reduction the average pressure is a little longer for plane strain than for tube extrusion. It changes with increasing area reduction because in order to obtain a greater area reduction in tube extrusion a larger mandrel is used. The contact area of the mandrel and the samples has very high friction and it increases the average pressure. The simulation by DEFORM gives a lower average pressure for all area reductions, because the coef®cient of friction that is determined by the ring test and which is used in simulation by the
Fig. 5. Comparison of the grid obtained by physical modelling and the DEFORM simulation for axisymmetrical tube extrusion.
6
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
Fig. 6. The method of calculation of the parameter wg.
DEFORM program is smaller than that in the physical modelling. Load estimation is often one of many tasks of physical modelling. Frequently, it is necessary to measure and calculate the distributions of velocity, strain rate, strain, pressure on the tools, etc. This can be done, but it is labour consuming, therefore measurements are avoided so far as there are other methods to gain indispensable results. Mathematical modelling is one of the methods that allow the required results to be obtained in a quicker and cheaper way, but these results have to be veri®ed. If the results from physical modelling, such as load and the picture of the deformed grid, and the same results obtained by mathematical modelling are convergent, then it can be assumed that other results such as velocity, strain rate, and strain computed by using both methods will be similar. In this way
strain and stress distributions can be calculated by mathematical modelling methods and accepted as correct results so far as physical modelling con®rms the correctness of the calculations. In the present work this way was assumed. For the plain-strain extrusion, axisymmetrical extrusion of bars and tubes mathematical modelling by DEFORM was performed. The calculated loads for these processes and their similarity to the results obtained by physical modelling are shown in Fig. 4. A comparison of distorted grids for extrusion in the planestrain and axisymmetrical state obtained by physical and mathematical modelling is shown in Fig. 5. The grids obtained by the two used methods are very similar, thus it can be testi®ed that mathematical models describe the extrusion processes in the correct way. A quantitative comparison of distorted grid is very dif®cult, therefore for
Fig. 7. The relationship between the parameter wg and the area reduction.
M. Arentoft et al. / Journal of Materials Processing Technology 106 (2000) 2±7
quantitative comparison only a very simple parameter wg could be taken. This parameter expresses the bending of the perpendicular line to the axis of sample divided by the width of the samples (Fig. 6). Fig. 7 presents the relationship between these parameters and the area-reduction results calculated from geometrical similarity for plane strain and axisymmetrical extrusion. The relationship is very similar for both of these cases. Taking into consideration this similarity and the similarity of the average pressure on the punch for plane-strain and axisymmetrical extrusion, it can be stated that for this same reduction of area axisymmetrical bars and tubes extrusion can be modelled by the planestrain test with the correct selection of process parameters such as the process scheme, the friction conditions, etc. 5. Conclusions 1. Physical modelling is a very ef®cient way to analyse complex metal forming processes and can be used for the veri®cation of designed processes. 2. Physical modelling gives a lot of detailed information about the way material ¯ows, and die ®lling. Quantitative determination of the strain and stress by physical modelling is less precise because wax material is very sensitive to temperature and strain rate and adequate determination of the friction coef®cient is very dif®cult in physical modelling. 3. The investigation has shown that physical and mathematical modelling give very similar results. 4. There are possibilities to replace a complex 3D structure by a simpler plane-strain model by retaining particular conditions.
7
5. The correct material model and friction conditions are critical for the correctness of the mathematical and physical modelling of metal forming processes. References [1] T. Wanheim, Physical Modelling of Metalprocessing, Procesteknisk Institut, Laboratories for Mekaniske Materialeprocesser, Danmarks Teknisk Hùjskole, Denmark, 1988. [2] S.K. Ghosh, CAD/CAM & FEM in Metal Working, North Staffordshire Poliytechnic, Beaconside, Stafford, UK, Pergamon Press, Oxford, 1988, p. 1. [3] T. Wanheim, Trends in physical simulation of metal working processes, Paper presented at Fourth Cairo University Conference on Mechanical Design and Production, Cairo University, Egypt, December 27±29, 1988. È hnlichkeitstheorie der Unformtechnik, [4] O. Pawelski, Beitrag zur A Archiv fuÈr das EisenhuÈttenwesen 35 (1) (1964) 27±36. [5] A. Azushima, H. Kudo, Physical simulation for metal forming with strain rate sensitive model material, Advanced Technology of Plasticity, Vol. II, Stuttgart, 1987, pp. 1221±1227. [6] H. Tsukamoto, T. Egawa, J. Ibushi, S. Oomori, K. Yagishita, Simulative model test on metal forming using plasticine as a model material, Technical Paper MF74-179, USA, 1974. [7] T. Wanheim, Trends in physical simulation of metal working processes, Paper presented at Fourth Cairo University Conference on Mechanical Design and Production, Cairo University, Egypt, December 27±29, 1988. [8] H. Tsukamoto, T. Egawa, J. Ibushi, S. Oomori, K. Yagishita, Simulative model test on metal forming using plasticine as a model material, Technical Paper MF74-179, USA, 1974. [9] A.P. Green, On unsymmetrical extrusion in plane strain, J. Mech. Phys. Solids 8 (1955) 189±196. [10] H. Shin, D. Kim, N. Kim, A simpli®ed three-dimensional ®niteelement analysis of the non-axisymmetrical extrusion processes, J. Mater. Process. Technol. 38 (1993) 567±587.