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Preform design in extrusion by the FEM and its experimental confirmation
Journal of Materials Processing Technology, 41 (1994) 237-248 Elsevier
237
Preform design in extrusion by the FEM and its experimental confirmation Beom-Soo K a n g a, B y u n g - M i n Kim b and J a e - C h a n Choi b aDepartment of Aerospace Engineering, Research Institute of Mechanical Technology, Pusan National University, Pusan 609-735, South Korea bDepartment of Mechanical Design Engineering, Research Institute of Mechanical Technology, Pusan National University, Pusan 609-735, South Korea (Received November 16, 1992; accepted in revised form July 3, 1993)
Industrial Summary An undesirable convex configuration at the front of an axisymmetric product occurs in the extrusion of a short bar, that is required to be removed by machining in order that the front of the product be flat. However, a flat front to a product can be obtained without machining by the use of an appropriate design of preform. Here the preform shape to produce a flat front after extrusion is designed using the rigid-plastic finite-element method and its backwardtracing scheme. Experiments into the extrusion process with the same conditions as those of the numerical simulations have been carried out to confirm the results relating to the numerical preform design, the results showing that the numerically designed preform is satisfactory in achieving the design purpose of a fiat front in the axisymmetric extrusion processes. The confirmation of the backward-tracing scheme of the finite-element method is valuable and necessary for wide application of the design scheme to preform design in practical forming processes.
1. Introduction The a p p l i c a t i o n of the finite-element m e t h o d to metal-forming processes has become wide-spread, and is n o w one of the most successful fields in engineering. Especially, the m e t h o d has been utilized for the design of preforms, an i m p o r t a n t aspect of r e s e a r c h into metal-forming processes. A l t h o u g h t h e r e has been c o n s t a n t effort to find b e t t e r preforms, the p r o p e r design of preforms r e m a i n s difficult. Rebelo et al. [1] h a v e proposed a new a p p r o a c h , called the ' b a c k w a r d - t r a c i n g scheme', u s i n g the capabilities of the finite-element m e t h o d for preform design
Correspondence to: Dr. Beom-Soo Kang, Department of Aerospace Engineering, Research Institute of Mechanical Technology, Pusan National University, Pusan 609-735, South Korea. 0924-0136/94]$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0924-0136(93)E0075-R
238
B.-S. Kang et al./ Preform design in extrusion
in metal forming. The concept of the approach is to trace the loading path of a forming process backwards from a given final configuration. The first application of the scheme by the above-mentioned researchers was the design of the preform in shell nosing. Preform design in square and rectangular cup drawing was carried out by Toh et al. [2,3]. The backward-tracing scheme was applied to the design of a disk-forging process where a uniformly deformed disk is required in the presence of friction at the die-workpiece interface [4] and the technique, including temperature calculations into backward tracing, was used also for preform design in shell nosing at elevated temperatures [5]. As more general approaches in controlling the boundary conditions during backward-tracing simulations, the scheme has been used to obtain preforms in blade forging as a two-dimensional plane-strain problem [6] and in the closeddie forging of axisymmetric H-shaped cross-sections [7]. Recently a study of preform design in ring-rolling processes demonstrated the extension of the scheme into three dimensions [8]. The backward-tracing scheme has been also utilized for the design of the process sequence in a four-stage heading process, in which more practical design conditions such as the die load, plastic buckling, and surface cracks, in addition to geometrical filling, have been taken into account [9]. Until now the backward-tracing scheme, however, has seemed to be in lack of confirmation by experiment. Since the rigid-plastic finite-element formulation gives good results in bulk metal forming fields [10], the backward-tracing scheme using the rigid-plastic finite-element formulation should be applicable to the design of preforms for metal-forming processes. The difficulty in confirming numerical results by experiment usually arises because of the high cost and the great effort involved in dealing with real processes and real materials. Proofs of the scheme by performing of experiments are valuable and necessary, even though these may not be numerous, for wide application of the design scheme to practical forming processes. Extrusion processes are usually kinematically steady-state except for at the start and the end of the deformation. A wide variety of theoretical solutions relating to steady-state processes has been obtained by applying the slipline theory and the upper-bound theorems [11,12], and by the finite-element method [13,14]. Shah and Kobayashi [15] analyzed axisymmetric extrusion through frictionless conical dies by the rigid-plastic finite-element method, the steady-state deformation characteristics in extrusion and drawing being obtained as functions of the material properties, die-workpiece interface friction, die angle, and reduction [16]. For bar extrusion, an undesirable convex configuration occurs at the front of the axisymmetric workpiece at the start of deformation, as shown in Fig. 1. A flat front can be obtained, however, by employing a preform of appropriate design. Here the preform shape to produce a flat front after extrusion is designed using the rigid-plastic finite-element method and the backwardtracing scheme, experiments being carried out to confirm the predictions of the analysis.
B.-S. Kang et al./ Preform design in extrusion
239
convex front
Fig. 1. A convex configuration at the front of an axisymmetric extrusion product.
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Fig. 2. Relationship between the flow stress and the effective strain of SM15C. (1 kgf/mm2 = 9.81 MPa).
2. Material properties and friction factor T h e w o r k p i e c e m a t e r i a l is SM15C steel, the m e c h a n i c a l p r o p e r t i e s of w h i c h w e r e o b t a i n e d by the c o m p r e s s i o n t e s t w i t h a h y d r a u l i c press a n d the r e s u l t a n t w o r k - h a r d e n i n g s t r e s s - s t r a i n r e l a t i o n s h i p is s h o w n in Fig. 2. T h e r e l a t i o n s h i p b e t w e e n the flow stress a n d t h e t o t a l effective s t r a i n of the m a t e r i a l is exp r e s s e d by:
a~Y= (1 + 78.75) 0.1 s
(1)
w h e r e a is the effective stress, e is the effective strain, a n d Y is the initial yield stress.
240
B.-S. Kang et al./ Preform design in extrusion
The friction factor at the die-workpiece interface is critical to the results of FEM simulations. Calcium grease with 10 weight percent of MoS2 powder was used as the lubricant between die and workpiece. A ring compression test was performed using the same lubricant to measure the friction factor at the die-workpiece interface, the resultant friction factor being found to be 0.15.
3. Preform design by FEM An extrusion process with a short circular workpiece is considered for preform design, the shape of the front of the workpiece to be designed such that the front shape of the extruded product is flat.The flattened shape at the front of the product can reduce machining cost and material waste when producing short shafts. Preform design by the finite-element method in the extrusion process requires not only forward-loading simulation but also backward tracing of the loading. The finite-element method used in the present study is based on the rigid-plastic formulation, a general description of the method being available elsewhere [17]. Backward tracing refers to the prediction of the part configuration at any stage in a deformation process, when the final part geometry and process conditions are given. The application of backward tracing is straightforward if the changes of the boundary conditions during a process are known. The boundary conditions for backward tracing are usually derived from the loading simulations of a trial preform. Even if the trial preform does not satisfy the design criteria completely, its boundary conditions during loading simulations could be used in the backward-tracing simulations. In the FEM simulations of the extrusion process, the change of boundary conditions is relatively simple because of the clear transition from contact to detachment between the workpiece and the die. 3.1. Computational conditions
The schematic view of an axisymmetric extrusion process is shown in Fig. 3. The semi-included die angle is 30 ° and the ratio of area reduction is 50%. For the finite-element computations, the extrusion die and the ram are assumed to be rigid. Friction at the interface between the die and the workpiece is expressed by the constant friction factor law, in this problem a friction factor of 0.15 being used, as obtained from the ring-compression test. The material is work-hardening, the relationship between the effective stress and the effective strain obtained in the compression test being used in the finite-element simulations. The diameter of the workpiece is 1.0 and the length of the workpiece is 2.0. The ram speed is assumed to be 1.0, and the interval of the deformation step is 0.01, which varies due to the contact or separation of the workpiece and the die. One hundred and forty nodes and 117 four-node elements are used. The initial mesh configuration of the workpiece is shown in Fig. 4(a).
B.-S. Kang et al./ Preform design in extrusion
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Fig. 3. Schematic view of an axisymmetric extrusion process.
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Fig. 4. Loading simulation of a circular bar extrusion: (a) before extrusion; (b) after extrusion, showing a convex front.
3.2. Preliminary simulations Some knowledge of the metal flow involved in the given extrusion process at the unsteady state of deformation would be helpful in progressing further in designing the next preform. This preliminary simulation contributes to the determination of a trial preform to be simulated before applying the backwardtracing scheme. The workpiece configuration used in the preliminary simulation is a short circular bar having a mesh contained in the die and the ram as shown in Fig. 4(a): the results of extrusion simulations are shown in Fig. 4(b). Since the main concern in the preform design is the front shape of the extruded product, the simulation is stopped when the deformation at the front becomes steady-state, t h a t is, when the front shape no longer changes. The simulation result, Fig. 4(b), shows that at the initial stage the front of the extrudate is not flat. The configuration of the front shape depends on material properties as
242
B.-S. Kang et al./ Preform design in extrusion
well as on the friction factor at the interface between the workpiece and the die. The observations from preliminary simulations are useful items of information for determining the next trial preform. The convex configuration of the front of the extrudate suggests the use of a trial preform with a concave front.
3.3. A modified trial preform A better preform prior to the application of the backward-tracing scheme is desirable since it requires boundary conditions of loading simulation that are close to the desired configuration. To derive a trial preform from the results of the preliminary simulation of a circular bar, an approximate method was applied, as shown in Fig. 5. The volume of the convex part in the final result of the preliminary simulations is calculated and removed from the original circular workpiece, based on the assumption that the convex volume is distributed linearly over the cross-section in the radial direction to make a concave preform. However, it is not expected that a preform derived by this method will satisfy the design criterion of a flat-front configuration in the extruded product, since the configuration of the convex front at the extruded product is not linear in the radial direction. The derived preform is shown in Fig. 6(a). The result of loading simulations using the modified trial preform is shown in Fig. 6(b). The deformation pattern in the front part of the extruded product has an irregular configuration. The modified preform does not improve the flatness of the front in the extruded product. In this state, however, the application of the backward-tracing scheme may be feasible in view of the relatively small changes of the coordinates in the front part required to produce the flat configuration in the final product. Experience and knowledge of the backward-tracing scheme by the finite-element method make it possible to predict that the scheme is easily applicable to this problem, since this
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Fig. 5. An approximate method for deriving a trial preform from the result of preliminary simulation.
B.-S. Kang et al./ Preform design in extrusion
243
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244
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Fig. 7. (a) Preparation for the application of the backward-tracing scheme; and (b) the result of backward-tracing simulation.
Fig. 8. Configuration of the final preform derived from the result of backward-tracing simulation. slight modification of the r e s u l t a n t c o n f i g u r a t i o n of the b a c k w a r d - t r a c i n g simulations. In order to confirm t h a t the preform r e s u l t i n g from b a c k w a r d t r a c i n g satisfies the final design condition, forward loading simulations are performed, the s i m u l a t i o n r e s u l t being shown in Fig. 9. The flatness of the f r o n t of the
245
B.-S. Kang et al./ Preform design in extrusion
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Fig. 9. Loading simulation of the finally designed preform and its distribution of effective strain. extruded product is improved far more than that with the modified trial preform. Even if it is not perfectly flat in the radial direction, the flatness of the front surface is considered to be satisfactory for numerical analysis using meshes.
4. Experiment The extrusion process with a circular bar is relatively simple and shows clearly the front configuration of the product. A schematic view of experimental apparatus for axisymmetric extrusion is shown in Fig. 10. A hydraulic press of 200 ton capacity was used for the experimental extrusion. The dies consist of an upper container, a middle container, and a lower container, all made of SCM4 material, The semi-included angle of the die is 30 deg. and the ratio of area reduction is 0.5, which are the same conditions as in the FEM simulations. The inlet diameter of the die is 30.0 mm and the outlet diameter is 21.1 mm. SM15C steel was used as the workpiece material, the diameter of the workpiece being 29.8 mm. Calcium grease with 10 percent weight of MoS2 powder was used as the lubricant between the die and the workpiece, the friction factor at the interface for this condition being found to be 0.15 by the ring-compression test. Two kinds of specimens were prepared, one specimen being a short circular bar and the other being the preform shape predicted from the FEM simulations and the backward tracing to be such as will produce a flat front in the extruded product. The preform was machined to have the same configuration as the geometry derived from the numerical preform design. The two specimens are shown in Fig. 11. The ram speed was 2 mm/min during the experimental process. The configurations of the extruded products are in Fig. 12, from which it is noted that the two front shapes compare well.
B.-S. Kang et al./ Preform design in extrusion
246
l
Fig. 10. Schematic view of the experimental apparatus for axisymmetric extrusion: 1 punch; 2 dummy block; 3-upper container; 4-die; 5 - d i e holder; 6 - billet; 7 - lower container.
Fig. 11. Two specimens for the experimental extrusion process: left-hand side a circular bar with a flat inlet; right-hand side the numerically-designed preform.
Fig. 12. Two configurations in the extruded product: left-hand side- the convex front configuration of the circular extrusion; right-hand side the flat-front configuration of the preform extrusion.
B.-S. Kang et al./ Preform design in extrusion
247
5. D i s c u s s i o n and concluding remarks Discussions are focused on the confirmation of the preform design using FEM simulations and backward tracing. The convex front shapes in the numerical simulations and experiment are typical of extrusion of a circular bar. The convex front shape in the numerical simulation of a circular bar, Fig. 4(b), can be compared to the shape shown in the photograph of Fig. ll(a), from which it is noted that the difference between the FEM simulations and the results of experiment are not considerable. The main concern in this study is a comparison between the front shape of the extruded product in the FEM for the numerically designed preform and the shape in experiment. The designed preform for FEM analysis is shown in Fig, 8 and t h a t for experiment on the right-hand side of Fig. 11. The loading simulations of the preform shows an almost flat front except for the central part. As shown on the right-hand side of Fig. 12, the experimental result shows good agreement with the numerical result. A tiny sharp eruption in the central part can be seen in both the FEM result and the experimental result. The extruded product from experiment is fiat over the front part, so that the design criterion of a flat front in the extruded product has been achieved in the experiment using the preform shape derived from FEM simulations and backward tracing. The suitability of the preform design in the extrusion process obtained using forward-loading simulation of FEM and backward-tracing simulation is confirmed by the experiment. This work is valuable and necessary to the wider industrial application of preform design using the backward-tracing scheme of the finite-element method. It is thus concluded that the preform design strategy using the backward-tracing scheme is applicable to industrial processes and that the information obtained from the backward-tracing simulations as well as from the forward-loading simulation is useful for arriving at satisfactory preform design in metal forming processes.
Acknowledgements The first author would like to t h a n k Professor Shiro Kobayashi, Emeritus and FANUC Professor, the University of California at Berkeley, for his encouragement and guidance, which enabled the undertaking of this work. The authors also thank the Korean National Science Foundation for financial support.
References [1] J.J. Park, N. Rebelo and S. Kobayashi, A new approach to preform design in metal forming with the finite element method, Int. J. Mach. Tool. Des. Res., 23 (1983) 111. [2[ C.H. Toh and S. Kobayashi, Deformation analysis and blank design in square cup drawing, Int. J. Mach. Tool Des. Res., 25 (1985) 15. [3] N. Kim and S. Kobayashi, Blank design in rectangular cup drawing by an approximate method, Int. J. Mach. Tool Des. Res., 26 (1986) 125.
248
B.-S. Kang et al./Preform design in extrusion
[4] S.M. Hwang and S. Kobayashi, Preform design in disk forging, Int. J. Mach. Tool Des. Res., 26 (1986) 231. [5] S.M. Hwang and S. Kobayashi, Preform Design in Shell Nosing at Elevated Temperatures, Int. J. Mach. Tool Manuf., 27(1) (1987) 1-14. [6] B.S. Kang, N. Kim and S. Kobayashi, Computer-aided preform design in forging of an airfoil section blade, Int. J. Mach. Tool Manuf., 30 (1990) 43. [7] N. Kim and S. Kobayashi, Preform Design in H-Shaped Cross-Sectional Axisymmetrical Forging by the Finite Element Method, Int. J. Mach. Tool Manuf., 30 (1991) 243. [8] B.S. Kang and S. Kobayashi, Preform Design in Ring Rolling by the Three-dimensional Finite Element Method, Int. J. Mach. Tool Manuf., 31 (1991) 139. [9] B.S. Kang, Process sequence design in a heading process, J. Mater. Process. Technol., 27 (1991) 213. [10] C.R. Boer, P. Gudmundson and N. Rebelo, Comparison of elastic-plastic FEM, rigidplastic FEM and experiments for cylinder upsetting, Proc. Conf. Numerical Methods in Industrial Forming Processes, Pineridge Press, Swansea, 1982, p. 217. [11] W. Johnson, R. Sowerby and R.D. Venter, Plane-Strain Slip Line Fields for Metal Deformation Processes, Pergamon Press, Oxford, 1982. [12] B. Avitzur, Handbook of Metal Forming Processes, Wiley, New York, 1983. [13] K. Iwata, K. Osakada and S. Fujino, Analysis of hydrostatic extrusion by the finite element method, Trans. A S M E , J. Eng. Ind., 94 (1972) 697. [14] C.H. Lee, H. Iwasaki and S. Kobayashi, Calculation of residual stresses in plastic deformation processes, Trans. ASME, J. Eng. Ind., 95 (1973) 283. [15] S.N. Shah and S. Kobayashi, A theory on metal flow in axisymmetric piercing and extrusion, J. Prod. Eng., 1 (1977) 73. [16] C.C. Chen, S.I. Oh and S. Kobayashi, Ductile fracture in axisymmetric extrusion and drawing-Part I, Deformation mechanics of extrusion and drawing, Trans. ASME, J. Eng. Ind., 101 (1979) 23. [17] S. Kobayashi, S.I. Oh and T. Altan, Metal Forming and the Finite-Element Method, Oxford University Press, New York, 1989.
237
Preform design in extrusion by the FEM and its experimental confirmation Beom-Soo K a n g a, B y u n g - M i n Kim b and J a e - C h a n Choi b aDepartment of Aerospace Engineering, Research Institute of Mechanical Technology, Pusan National University, Pusan 609-735, South Korea bDepartment of Mechanical Design Engineering, Research Institute of Mechanical Technology, Pusan National University, Pusan 609-735, South Korea (Received November 16, 1992; accepted in revised form July 3, 1993)
Industrial Summary An undesirable convex configuration at the front of an axisymmetric product occurs in the extrusion of a short bar, that is required to be removed by machining in order that the front of the product be flat. However, a flat front to a product can be obtained without machining by the use of an appropriate design of preform. Here the preform shape to produce a flat front after extrusion is designed using the rigid-plastic finite-element method and its backwardtracing scheme. Experiments into the extrusion process with the same conditions as those of the numerical simulations have been carried out to confirm the results relating to the numerical preform design, the results showing that the numerically designed preform is satisfactory in achieving the design purpose of a fiat front in the axisymmetric extrusion processes. The confirmation of the backward-tracing scheme of the finite-element method is valuable and necessary for wide application of the design scheme to preform design in practical forming processes.
1. Introduction The a p p l i c a t i o n of the finite-element m e t h o d to metal-forming processes has become wide-spread, and is n o w one of the most successful fields in engineering. Especially, the m e t h o d has been utilized for the design of preforms, an i m p o r t a n t aspect of r e s e a r c h into metal-forming processes. A l t h o u g h t h e r e has been c o n s t a n t effort to find b e t t e r preforms, the p r o p e r design of preforms r e m a i n s difficult. Rebelo et al. [1] h a v e proposed a new a p p r o a c h , called the ' b a c k w a r d - t r a c i n g scheme', u s i n g the capabilities of the finite-element m e t h o d for preform design
Correspondence to: Dr. Beom-Soo Kang, Department of Aerospace Engineering, Research Institute of Mechanical Technology, Pusan National University, Pusan 609-735, South Korea. 0924-0136/94]$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0924-0136(93)E0075-R
238
B.-S. Kang et al./ Preform design in extrusion
in metal forming. The concept of the approach is to trace the loading path of a forming process backwards from a given final configuration. The first application of the scheme by the above-mentioned researchers was the design of the preform in shell nosing. Preform design in square and rectangular cup drawing was carried out by Toh et al. [2,3]. The backward-tracing scheme was applied to the design of a disk-forging process where a uniformly deformed disk is required in the presence of friction at the die-workpiece interface [4] and the technique, including temperature calculations into backward tracing, was used also for preform design in shell nosing at elevated temperatures [5]. As more general approaches in controlling the boundary conditions during backward-tracing simulations, the scheme has been used to obtain preforms in blade forging as a two-dimensional plane-strain problem [6] and in the closeddie forging of axisymmetric H-shaped cross-sections [7]. Recently a study of preform design in ring-rolling processes demonstrated the extension of the scheme into three dimensions [8]. The backward-tracing scheme has been also utilized for the design of the process sequence in a four-stage heading process, in which more practical design conditions such as the die load, plastic buckling, and surface cracks, in addition to geometrical filling, have been taken into account [9]. Until now the backward-tracing scheme, however, has seemed to be in lack of confirmation by experiment. Since the rigid-plastic finite-element formulation gives good results in bulk metal forming fields [10], the backward-tracing scheme using the rigid-plastic finite-element formulation should be applicable to the design of preforms for metal-forming processes. The difficulty in confirming numerical results by experiment usually arises because of the high cost and the great effort involved in dealing with real processes and real materials. Proofs of the scheme by performing of experiments are valuable and necessary, even though these may not be numerous, for wide application of the design scheme to practical forming processes. Extrusion processes are usually kinematically steady-state except for at the start and the end of the deformation. A wide variety of theoretical solutions relating to steady-state processes has been obtained by applying the slipline theory and the upper-bound theorems [11,12], and by the finite-element method [13,14]. Shah and Kobayashi [15] analyzed axisymmetric extrusion through frictionless conical dies by the rigid-plastic finite-element method, the steady-state deformation characteristics in extrusion and drawing being obtained as functions of the material properties, die-workpiece interface friction, die angle, and reduction [16]. For bar extrusion, an undesirable convex configuration occurs at the front of the axisymmetric workpiece at the start of deformation, as shown in Fig. 1. A flat front can be obtained, however, by employing a preform of appropriate design. Here the preform shape to produce a flat front after extrusion is designed using the rigid-plastic finite-element method and the backwardtracing scheme, experiments being carried out to confirm the predictions of the analysis.
B.-S. Kang et al./ Preform design in extrusion
239
convex front
Fig. 1. A convex configuration at the front of an axisymmetric extrusion product.
E E
8O
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;/ ~
6O
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~ ~o w
20
I
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0.2
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|
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Effective
|
1,2
I
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Strain ( E )
Fig. 2. Relationship between the flow stress and the effective strain of SM15C. (1 kgf/mm2 = 9.81 MPa).
2. Material properties and friction factor T h e w o r k p i e c e m a t e r i a l is SM15C steel, the m e c h a n i c a l p r o p e r t i e s of w h i c h w e r e o b t a i n e d by the c o m p r e s s i o n t e s t w i t h a h y d r a u l i c press a n d the r e s u l t a n t w o r k - h a r d e n i n g s t r e s s - s t r a i n r e l a t i o n s h i p is s h o w n in Fig. 2. T h e r e l a t i o n s h i p b e t w e e n the flow stress a n d t h e t o t a l effective s t r a i n of the m a t e r i a l is exp r e s s e d by:
a~Y= (1 + 78.75) 0.1 s
(1)
w h e r e a is the effective stress, e is the effective strain, a n d Y is the initial yield stress.
240
B.-S. Kang et al./ Preform design in extrusion
The friction factor at the die-workpiece interface is critical to the results of FEM simulations. Calcium grease with 10 weight percent of MoS2 powder was used as the lubricant between die and workpiece. A ring compression test was performed using the same lubricant to measure the friction factor at the die-workpiece interface, the resultant friction factor being found to be 0.15.
3. Preform design by FEM An extrusion process with a short circular workpiece is considered for preform design, the shape of the front of the workpiece to be designed such that the front shape of the extruded product is flat.The flattened shape at the front of the product can reduce machining cost and material waste when producing short shafts. Preform design by the finite-element method in the extrusion process requires not only forward-loading simulation but also backward tracing of the loading. The finite-element method used in the present study is based on the rigid-plastic formulation, a general description of the method being available elsewhere [17]. Backward tracing refers to the prediction of the part configuration at any stage in a deformation process, when the final part geometry and process conditions are given. The application of backward tracing is straightforward if the changes of the boundary conditions during a process are known. The boundary conditions for backward tracing are usually derived from the loading simulations of a trial preform. Even if the trial preform does not satisfy the design criteria completely, its boundary conditions during loading simulations could be used in the backward-tracing simulations. In the FEM simulations of the extrusion process, the change of boundary conditions is relatively simple because of the clear transition from contact to detachment between the workpiece and the die. 3.1. Computational conditions
The schematic view of an axisymmetric extrusion process is shown in Fig. 3. The semi-included die angle is 30 ° and the ratio of area reduction is 50%. For the finite-element computations, the extrusion die and the ram are assumed to be rigid. Friction at the interface between the die and the workpiece is expressed by the constant friction factor law, in this problem a friction factor of 0.15 being used, as obtained from the ring-compression test. The material is work-hardening, the relationship between the effective stress and the effective strain obtained in the compression test being used in the finite-element simulations. The diameter of the workpiece is 1.0 and the length of the workpiece is 2.0. The ram speed is assumed to be 1.0, and the interval of the deformation step is 0.01, which varies due to the contact or separation of the workpiece and the die. One hundred and forty nodes and 117 four-node elements are used. The initial mesh configuration of the workpiece is shown in Fig. 4(a).
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3.2. Preliminary simulations Some knowledge of the metal flow involved in the given extrusion process at the unsteady state of deformation would be helpful in progressing further in designing the next preform. This preliminary simulation contributes to the determination of a trial preform to be simulated before applying the backwardtracing scheme. The workpiece configuration used in the preliminary simulation is a short circular bar having a mesh contained in the die and the ram as shown in Fig. 4(a): the results of extrusion simulations are shown in Fig. 4(b). Since the main concern in the preform design is the front shape of the extruded product, the simulation is stopped when the deformation at the front becomes steady-state, t h a t is, when the front shape no longer changes. The simulation result, Fig. 4(b), shows that at the initial stage the front of the extrudate is not flat. The configuration of the front shape depends on material properties as
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well as on the friction factor at the interface between the workpiece and the die. The observations from preliminary simulations are useful items of information for determining the next trial preform. The convex configuration of the front of the extrudate suggests the use of a trial preform with a concave front.
3.3. A modified trial preform A better preform prior to the application of the backward-tracing scheme is desirable since it requires boundary conditions of loading simulation that are close to the desired configuration. To derive a trial preform from the results of the preliminary simulation of a circular bar, an approximate method was applied, as shown in Fig. 5. The volume of the convex part in the final result of the preliminary simulations is calculated and removed from the original circular workpiece, based on the assumption that the convex volume is distributed linearly over the cross-section in the radial direction to make a concave preform. However, it is not expected that a preform derived by this method will satisfy the design criterion of a flat-front configuration in the extruded product, since the configuration of the convex front at the extruded product is not linear in the radial direction. The derived preform is shown in Fig. 6(a). The result of loading simulations using the modified trial preform is shown in Fig. 6(b). The deformation pattern in the front part of the extruded product has an irregular configuration. The modified preform does not improve the flatness of the front in the extruded product. In this state, however, the application of the backward-tracing scheme may be feasible in view of the relatively small changes of the coordinates in the front part required to produce the flat configuration in the final product. Experience and knowledge of the backward-tracing scheme by the finite-element method make it possible to predict that the scheme is easily applicable to this problem, since this
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Fig. 8. Configuration of the final preform derived from the result of backward-tracing simulation. slight modification of the r e s u l t a n t c o n f i g u r a t i o n of the b a c k w a r d - t r a c i n g simulations. In order to confirm t h a t the preform r e s u l t i n g from b a c k w a r d t r a c i n g satisfies the final design condition, forward loading simulations are performed, the s i m u l a t i o n r e s u l t being shown in Fig. 9. The flatness of the f r o n t of the
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4. Experiment The extrusion process with a circular bar is relatively simple and shows clearly the front configuration of the product. A schematic view of experimental apparatus for axisymmetric extrusion is shown in Fig. 10. A hydraulic press of 200 ton capacity was used for the experimental extrusion. The dies consist of an upper container, a middle container, and a lower container, all made of SCM4 material, The semi-included angle of the die is 30 deg. and the ratio of area reduction is 0.5, which are the same conditions as in the FEM simulations. The inlet diameter of the die is 30.0 mm and the outlet diameter is 21.1 mm. SM15C steel was used as the workpiece material, the diameter of the workpiece being 29.8 mm. Calcium grease with 10 percent weight of MoS2 powder was used as the lubricant between the die and the workpiece, the friction factor at the interface for this condition being found to be 0.15 by the ring-compression test. Two kinds of specimens were prepared, one specimen being a short circular bar and the other being the preform shape predicted from the FEM simulations and the backward tracing to be such as will produce a flat front in the extruded product. The preform was machined to have the same configuration as the geometry derived from the numerical preform design. The two specimens are shown in Fig. 11. The ram speed was 2 mm/min during the experimental process. The configurations of the extruded products are in Fig. 12, from which it is noted that the two front shapes compare well.
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Fig. 11. Two specimens for the experimental extrusion process: left-hand side a circular bar with a flat inlet; right-hand side the numerically-designed preform.
Fig. 12. Two configurations in the extruded product: left-hand side- the convex front configuration of the circular extrusion; right-hand side the flat-front configuration of the preform extrusion.
B.-S. Kang et al./ Preform design in extrusion
247
5. D i s c u s s i o n and concluding remarks Discussions are focused on the confirmation of the preform design using FEM simulations and backward tracing. The convex front shapes in the numerical simulations and experiment are typical of extrusion of a circular bar. The convex front shape in the numerical simulation of a circular bar, Fig. 4(b), can be compared to the shape shown in the photograph of Fig. ll(a), from which it is noted that the difference between the FEM simulations and the results of experiment are not considerable. The main concern in this study is a comparison between the front shape of the extruded product in the FEM for the numerically designed preform and the shape in experiment. The designed preform for FEM analysis is shown in Fig, 8 and t h a t for experiment on the right-hand side of Fig. 11. The loading simulations of the preform shows an almost flat front except for the central part. As shown on the right-hand side of Fig. 12, the experimental result shows good agreement with the numerical result. A tiny sharp eruption in the central part can be seen in both the FEM result and the experimental result. The extruded product from experiment is fiat over the front part, so that the design criterion of a flat front in the extruded product has been achieved in the experiment using the preform shape derived from FEM simulations and backward tracing. The suitability of the preform design in the extrusion process obtained using forward-loading simulation of FEM and backward-tracing simulation is confirmed by the experiment. This work is valuable and necessary to the wider industrial application of preform design using the backward-tracing scheme of the finite-element method. It is thus concluded that the preform design strategy using the backward-tracing scheme is applicable to industrial processes and that the information obtained from the backward-tracing simulations as well as from the forward-loading simulation is useful for arriving at satisfactory preform design in metal forming processes.
Acknowledgements The first author would like to t h a n k Professor Shiro Kobayashi, Emeritus and FANUC Professor, the University of California at Berkeley, for his encouragement and guidance, which enabled the undertaking of this work. The authors also thank the Korean National Science Foundation for financial support.
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[4] S.M. Hwang and S. Kobayashi, Preform design in disk forging, Int. J. Mach. Tool Des. Res., 26 (1986) 231. [5] S.M. Hwang and S. Kobayashi, Preform Design in Shell Nosing at Elevated Temperatures, Int. J. Mach. Tool Manuf., 27(1) (1987) 1-14. [6] B.S. Kang, N. Kim and S. Kobayashi, Computer-aided preform design in forging of an airfoil section blade, Int. J. Mach. Tool Manuf., 30 (1990) 43. [7] N. Kim and S. Kobayashi, Preform Design in H-Shaped Cross-Sectional Axisymmetrical Forging by the Finite Element Method, Int. J. Mach. Tool Manuf., 30 (1991) 243. [8] B.S. Kang and S. Kobayashi, Preform Design in Ring Rolling by the Three-dimensional Finite Element Method, Int. J. Mach. Tool Manuf., 31 (1991) 139. [9] B.S. Kang, Process sequence design in a heading process, J. Mater. Process. Technol., 27 (1991) 213. [10] C.R. Boer, P. Gudmundson and N. Rebelo, Comparison of elastic-plastic FEM, rigidplastic FEM and experiments for cylinder upsetting, Proc. Conf. Numerical Methods in Industrial Forming Processes, Pineridge Press, Swansea, 1982, p. 217. [11] W. Johnson, R. Sowerby and R.D. Venter, Plane-Strain Slip Line Fields for Metal Deformation Processes, Pergamon Press, Oxford, 1982. [12] B. Avitzur, Handbook of Metal Forming Processes, Wiley, New York, 1983. [13] K. Iwata, K. Osakada and S. Fujino, Analysis of hydrostatic extrusion by the finite element method, Trans. A S M E , J. Eng. Ind., 94 (1972) 697. [14] C.H. Lee, H. Iwasaki and S. Kobayashi, Calculation of residual stresses in plastic deformation processes, Trans. ASME, J. Eng. Ind., 95 (1973) 283. [15] S.N. Shah and S. Kobayashi, A theory on metal flow in axisymmetric piercing and extrusion, J. Prod. Eng., 1 (1977) 73. [16] C.C. Chen, S.I. Oh and S. Kobayashi, Ductile fracture in axisymmetric extrusion and drawing-Part I, Deformation mechanics of extrusion and drawing, Trans. ASME, J. Eng. Ind., 101 (1979) 23. [17] S. Kobayashi, S.I. Oh and T. Altan, Metal Forming and the Finite-Element Method, Oxford University Press, New York, 1989.