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An upper-bound elemental technique approach to the process design of asymmetric forgings
Journal of Materials Processing Technology 72 (1997) 141 – 151
An upper-bound elemental technique approach to the process design of asymmetric forgings Jong-Heon Lee a, Young-Ho Kim b,*, Won-Byong Bae c a
b
Department of Vehicle and Machinery, Kyung-Nam Junior College, Pusan, South Korea Department of Mechanical and Production Engineering, Pusan National Uni6ersity, 30, Jangjeon-dong, Kumjeong-gu, Pusan 609 -735, South Korea c Department of Mechanical Design Engineering, Pusan National Uni6ersity, Pusan, South Korea Received 10 June 1996
Abstract An upper-bound elemental technique (UBET) program has been developed to predict forging load, die-cavity filling, preform in asymmetric forging. To analyze the process easily, it is suggested that the deformation is divided into two different parts. Those are axisymmetric part in corner, plane-strain part in lateral. The plane-strain and axisymmetric parts are combined by building block method. The total energy is computed through combination of two deformation parts. A dumbbell-type preform has been obtained from height and volumetric compensations of the billet based on the backward simulation. Experiments have been carried out with pure plasticine at room temperature. Theoretical predictions are in good agreement with experimental results. © 1997 Elsevier Science S.A. Keywords: Upper-bound elemental technique; Asymmetric forging; Process design
1. Introduction There are many components of rib-web type of crosssection in airplane structural components or vehicle components. A lot of research has been carried out on rib-web type forging, most of which involved planestrain or axisymmetric problems, with few studies having been made on asymmetric shapes because of the difficulty in analysis. In analyzing the metal forming process, the finite element method (FEM) [1] is regarded as one of the most effective tools because it can yield more reliable results than other methods. However, it has some problems such as a vast amount of calculation and limitation in its application to actual fields and it is not economic in that it needs a great computer capacity and experts who can deal with it. Nevertheless, the upperbound elemental technique (UBET) has had its merits recognized because it is easy to apply, needs much
* Corresponding author. Fax: [email protected]
+82
51
5147640;
e-mail:
0924-0136/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 9 2 4 - 0 1 3 6 ( 9 7 ) 0 0 1 4 5 - 3
shorter CPU time and the results are comparatively useful. Several studies have been performed on asymmetric shapes. The asymmetric forging problem having side flash was analyzed by Kiuchi [2,3] with UBET. The material was divided into various elements, which was then possible to analyze. In the analysis of asymmetric forging, a UBET simulator (FORMS) [4] was developed by Kiuchi to study the velocity fields of complex shapes integrated by easy forging processes (upsetting, extrusion, etc.). In addition, Kiuchi et al. [5,6] performed a study on developing generalized 3-dimensional kinematically-admissible velocity fields using the upper-bound method for the forward-extrusion problem having various cross-sectional shapes. It was found difficult to formulate the 3-dimensional velocity fields, which were very complicated. Wada et al. [7] applied an upper-bound method, dividing the 3-dimensional shape into rectangular parallel-piped, prismatic, right-angle tetrahedron and right-angle pentahedral elements. However, because of the difficulty in optimizing many variables, it is not adequate for application to complex shapes.
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J.-H. Lee et al. / Journal of Materials Processing Technology 72 (1997) 141–151
In studies by FEM, Argyris et al. [8] anaylzed an airplane turbine blade under frictionless conditions, whilst Pillinger et al. [9] analyzed the forging process of an aluminum connecting rod under constrain frictional conditions. In addition, for the asymmetric problem of a ring, Marques et al. [10] analyzed the forging process, where a circular cross-section was formed from rectangular billet after dividing the entire material into planestrain and axisymmetric parts. Even though there have been several research studies on the preform design of asymmetric components by FEM, comparatively fewer studies have been carried out using UBET. Imai et al. [11] suggested the preform for a cylinder having an asymmetric flange using a UBET simulator (FORMS). Also, Kim et al. [12] suggested the preform of a connecting rod using UBET simulation. In the present study, for the process design of asymmetric forgings of rib-web typed cross-section, the asymmetric shape is divided into plane-strain and axisymmetric parts, then the building block method, which combines the two parts, is used. As a simulation method, UBET is adopted. In the case that the rib height-to-width ratio is 1:1, the kinematically admissible velocity field of each part is determined by minimizing the total energy rate, which latter is the summation of the energy at the plane-strain deformation part, at the axisymmetric part and at the boundary of these two regions. By predicting the die-cavity filling processes and the forging loads for various initial billets, the characteristics of each billet are studied.
Fig. 1. Analytical model of asymmetric forging.
Fig. 2. Flow chart for forward UBET simulation.
If the rib height-to-width ratio is 2:1, the final shape is divided into the plane-strain and axisymmetric deformation parts and then the preform obtained from the backward-tracing scheme [12] in each part is calibrated in the height compensation of the rib. After that, the die-cavity filling is checked by forward simulation. If the cavity of the rib is not filled in the plane-strain or axisymmetric parts, a simple shape of preform of dumb-bell type is designed by performing volumetric compensation of the upper part of the billet.
Fig. 3. Overlap of the flow pattern of the boundary between the plane-strain and axisymmetric parts.
J.-H. Lee et al. / Journal of Materials Processing Technology 72 (1997) 141–151
143
Fig. 5. True stress – true strain curve of plasticine.
effective strain. Fig. 2 shows the flow chart for forward simulation.
2.1. Kinematically admissible 6elocity fields
Fig. 4. Flow chart of backward simulation for preform design.
To check the validity of this theory, a model-material test was performed.
2. UBET analysis The analytical model in this study is shown in Fig. 1. An asymmetric shape of rib-web type is analyzed by dividing the total region into plane-strain and axisymmetric parts and then by combining them again. The plane-strain and axisymmetric parts are divided again into a few simple elements. The velocity field of each divided element should satisfy the velocity boundary condition between adjacent elements and the volume constancy condition. For velocity fields, those suggested by Oudin [13] and Kiuchi [14] are used, whilst for the plane-strain deformation part, rectangular and trapezoid elements, suggested by Oudin [13] and for the axisymmetric part, rectangular and triangular ring elements, suggested by Kiuchi [14], are used. The work hardening of the material is considered under the assumption that the flow stress is dependent only on
The total power consumption rate in a forging process is as follows: J*= % W: i + % W: s + % W: f
(1)
where W: i is the internal power of the plane-strain and axisymmetric parts, W: s is the shear loss at the interface of two adjacent elements which includes the shear loss between plane-strain and axisymmetric parts and W: f means the friction loss.
Fig. 6. Upper and lower dies for the experiments with plasticine.
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metric deformation parts and W: boundary is the shear loss at the interface between the plane-strain and axisymmetric deformation parts. Also, the friction loss, W: f is as follows: W: f = m
s¯ 0
&
DVf dSf (5)
3 s f where m is friction constant and DVf is the velocity discontinuity along the surface between the die and material. The total power consumption rate, J*= J*(p1, p2, p3, …, pn, a1, a2, a3, …, an ) is minimized by the flexible polyhedron search (FPS) method [15] which is a kind of direct search method. Here, pi, ai are pseudo-independent parameters at the plane-strain and axisymmetric parts, respectively. By minimizing J*, the kinematically admissible velocity fields at time t can be determined. The forming load, L, is given as follows: L=
J* VD
(6)
where VD is the die velocity.
2.2. Element regeneration To investigate the die-cavity filling process, the total forming time is divided according to time increment Dt.
Fig. 7. Dimensions and configuration of the initial billet for asymmetric forging (H/B =1:1).
The internal power at each element, W: i is: W: i =
&
6
s¯ to¯; t dV
(2)
where the effective strain-rate, o¯; t = (2/3)oijoij In Eq. (2) s¯ t is the effective stress of each element at time t and it is assumed that s¯ t is the function of effective strain only: s¯ t = Co¯ nt
(3)
W: s represents the shear loss along the velocity discontinuity surface, expressed as follows: W: s =
s¯ 0
3
&
DVs dSs +W: boundary
(4)
s
where, s¯ 0 is the mean value of flow stresses of two adjacent elements, DVs is the velocity discontinuity along the shear surface in the plane-strain and axisym-
Fig. 8. Die-cavity filling of the final step for the various initial billets in Fig. 7.
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it is assumed that the two rounded parts of both ends deform axisymmetrically and that the linear parts between the two rounded parts is a plane-strain problem because their length is much greater then their width. Marques [10] and Kim et al. [12] also based their research on this kind of assumption. As the die advances, the plane-strain and axisymmetric parts have different deformation patterns, which results in the discrepancy of the elemental boundary. The analytical model for calculating the shear loss due to this boundary discrepancy is shown in Fig. 3. The shear loss at the interface of the plane-strain and axisymmetric parts is calculated as follows: W: boundary =
s¯ 0
3
&
DV dS
(8)
and DV =[(Uaxi,R − Uplane,X )2 + (Vaxi,Z − Vplane,y )2]1/2 where dS is the infinitesimal elemental boundary area owned by a plane-strain and an axisymmetric element at the same time, Uaxi,R and Vaxi,Z represent the R- and Z-direction velocity components of the axisymmetric elemental boundary, respectively and Uplane,X and Vplane,Y are the X- and Y-direction velocity components of the plane-strain elemental boundary, respectively. In Fig. 3, the element system is superimposed along the boundary of each deformed part when the height reduction reaches a particular point. As is shown in this figure, when the die advances, there is a boundary Fig. 9. Flow pattern of the final step for multi-layered plasticine for the various initial billets in Fig. 7.
At each step, the coordinate of the deformed shape of each element is determined by the kinematically-admissible velocity fields after minimization. The coordinate of each element after time incremental Dt is as follows: X% =X+Dt · U
(7)
where X% is the position of each element after deformation and X and U represent the position and velocity of an elemental boundary before deformation, respectively. After deformation, sliding occurs between adjacent elements, resulting in the discontinuity of the elemental boundary. Therefore, in order to proceed to the next step, the element regeneration scheme [12] by vertical and horizontal projection is used.
2.3. Building-block method Products of a complex shape such as asymmetric, can be approximated into the combination of simple shapes which can be analyzed easily. Fig. 1 shows a schematic diagram of the final products. To simplify the analysis
Fig. 10. Variation of experimental forging loads for the various initial billets in Fig. 7.
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discrepancy between the plane-strain and axisymmetric parts owing to the difference of the flow pattern in the two parts. In this case the shear loss at each step is calculated as follows. Along the X-axis the node of each element is assigned the coordinate AX(1), AX(2), AX(3), …, AX(N), the computer program perceiving the number of the plane-strain or axisymmetric element corresponding to these assigned values. For element A, both the plane-strain and axisymmetric elements have element number (1, 1). Then the shear loss of element A is obtained from the boundary velocity and the strain-rate value of the element, (1, 1) of both parts. For element B between AX(3) and AX(4), the element number of the plane-strain and axisymmetric parts is (3, 1) and (2, 1) respectively. Therefore for the plane-strain and axisymmetric parts, the data of element number, (3, 1), (2, 1) are used to calculate the shear loss, respectively. For the case of element C, the shear loss is not calculated because only the element of the plane-strain part exists. Each of every other element is included in one of the element types, A, B or C and analyzed likewise.
2.4. Preform design In forming process design, the preform design is one of the most important steps. In closed-die-forging, the preform design includes the dimensions of preform, the number of steps and the design of the configuration. In the present study, the authors have tried to design a comparatively simple preform by calibrating the height and volume, based on the backward-tracing scheme, suggested by Park et al. [16] and applied to UBET by Kim et al. [12] in the preform design. The final products are divided into plane-strain and axisymmetric parts and the surface contacting with die is considered as a free surface. From this surface the nodal points of the prior step are backward traced. As in the forward simulation, there is the elemental boundary discrepancy between the plane-strain and axisymmetric parts in the backward simulation, owing to the different flow pattern of the two parts. Therefore the configuration of the preform by the backward tracing process shows a stepped shape in the upper and outer direction of the preform. In this case, because of the complexity of the preform die, these is the need to simplify the preform configuration. Therefore, in the configuration of final step obtained from the backward tracing process, in order to equalize the height of the ribs in the plane-strain and axisymmetric parts, only the upper direction of the preform is compensated under volume constancy. Then the preform compensating the height of the rib is forward simulated and in constant thickness, the upper direction of the preform containing the web part is compensated by the calculated volume of the cavity between the
Fig. 11. Die-cavity filling process of the element system for Billet 3 in Fig. 7 (H/B= 1:1).
material and the die. Therefore, the authors have tried to design a comparatively simple preform of dumb-bell type. Fig. 4 shows the flow chart for the backward simulation.
3. Experimental For the forging experiments of the asymmetric products of rib-web type, pure plasticine was used. To obtain the flow stress-and-strain characteristics, a cylindrical billet, of which the height and diameter are equal, is used. In the experiments, using vaseline lubricant, (m¥0.1) which show comparatively little bulging, the cylinder is compressed up to 50% of its initial height at constant die velocity (0.5 mm s − 1) at room temperature. Fig. 5 shows the material characteristics from the experiments. The relationship of the flow stress and strain of the material is as follows: s¯ =0.178o¯ 0.082 (MPa)
(9)
To obtain information about the material flow in the workpiece, plasticine was used as a model material. The billets are laminated with two colors (white and black), after being kneaded to remove air from inside the material. The initially rectangular billets are made into a rounded shape at the ends.
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The material of the die is carbon steel. To make it easy to release the plasticine product from the die, the upper dies are divided two parts, joined with a pin and bolt. The experiments are performed with a material testing system (MTS) at a constant die velocity (0.5 mm s − 1). The lubricant is talc powder (m ¥0.4). Using the X– Y plotter attached to MTS, the load curve is obtained. To keep the temperature of billets constantly during experiments, the die and billets were put into an over for 24 h and the room temperature is kept constant using a thermo-hygrostat. Fig. 6 shows configuration of the forging die.
4. Results and discussion
4.1. Initial billets (H/B = 1:1) Fig. 7 shows three billets used having a ratio of height-to-width of rib 1:1. Figs. 8 – 10 show the numerical and experimental results of the billets in Fig. 7, these being: Final die-cavity filling by forward simulation; final flow pattern; and forging load by experiment, respectively. Here, when billet 1 is used, at the planestrain part the die cavity is filled and flash forms adequately, but at the axisymmetric part the material Fig. 13. Material flow pattern of the multi-layered plasticine for billet 3 in Fig. 7 (H/B=1:1).
Fig. 12. Die-cavity filling process of the grid distortion pattern for billet 3 in Fig. 7 (H/B= 1:1).
filling around the rib part is not completed. To enhance the die-cavity filling in the axisymmetric rib part, the dumb-bell type billet 2, which is made by calibrating the volume to the outward direction of the axisymmetric part is used, which effects the filling of the die-cavity of the plane-strain and axisymmetric parts and also the amount of flash is adequate. However, it is difficult to use the dumb-bell type billet 2 as a initial billet because it requires another process. Accordingly, a billet that can ensure die-cavity filling at the axisymmetric part and is of comparatively simple shape, is needed. Thus the outside boundary dimension of axisymmetric part of billet 2 is extended up to the plane-strain part, the extended billet being billet 3. In the case of billet 3 even if quite a greater amount of flash is produced at the plane-strain part than that for billet 2, the simplicity of the configuration is recognized. As is indicated in Fig. 10, which shows the forging loads by experiment, the loads that increase up to a reduction in height of 40% are similar, because billets 1 and 2 have the same dimensions of the plane-strain part. In the case of billet 2, for the reduction in height
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of 40%, the flash is produced at the plane-strain and axisymmetric parts simultaneously. On the other hand, billet 1 has flash only at the plane-strain part, so for a reduction in height of around 40%, billet 2 has higher loads than is the case for billet 1. At the reduction in height of around 17%, billet 3 starts to have flash at the plane-strain part and for a reduction in height of 55%, flash is produced at the axisymmetric part. Figs. 11 and 12 show the die-cavity filling and grid distortion processes of the plane-strain and axisymmetric parts of billet 3 from the initial to the final step. Fig. 13 shows the die-cavity filling process at several steps in plasticine experiments by cutting away one-fourth of the billets. From Figs. 11 – 13, it is noted that the boundary of the plane-strain and axisymmetric parts is out of agreement because as the process proceeds the material flow in the upper and outward directions becomes different: This can be verified by the results of experiment shown in Fig. 13. Thus to analyze this boundary, the building-block method of Section 2.3 is used. Further, the flash at the plane-strain part is produced first and at the moment when the die-cavity of the axisymmetric part is filled completely, the planestrain part already has a significant amount of flash. Fig. 14 shows the forging-load variation for the case of billet 3. In this diagram that the results of numerical analysis are a little higher than those of experiments is due to the use of upper-bound analysis. However, on the whole, the results are acceptable. Fig. 15. Backward process from final configuration in asymmetric forging (H/B =2:1).
4.2. Preform design
Fig. 14. Comparison between theoretical and experimental forging loads for billet 3 in Fig. 7 (H/B= 1:1).
For the case one rib height-to-width ratio (H/B= 1:1), it is verified that the dumb-bell type billet, billet 2, is adequate regarding the amount of flash and the forging load, the die-cavity filling process. For the case two rib height-to-width ratio (H/B= 2:1), it can be predicted that the dumb-bell type billet is adequate as a preform to be forged into the final product. If backward-traced from the asymmetric final product, as the results at the final step of Fig. 15, the height and width of the rib become different because of different flow pattern of the plane-strain and axisymmetric parts. When this shape is adopted as a preform, the die of preform becomes very complex, resulting in difficulty in numerical analysis and die manufacturing. Thus the configuration of preform must be simplified. Accordingly, first of all, the heights of the rib, y2, y3 which are obtained from backward tracing, can be equalized into the calibrated height, yn according to the volume-constancy condition. With this billet, forward simulation is performed and the amount of unfilled die-cavity is checked, then after performing the volume calibration of the total upper part of billet including the
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web shown in Fig. 16(b), the preform of dumb-bell type shape shown in Fig. 16 is obtained. In Fig. 16(b) it can be noted that the calibration volume is more than the uncalibrated volume, which results from the situation that at the beginning of the backward tracing, the flash length is backward traced from the end point of the flash land. When the actual asymmetric forging process where quite a large amount of flash occurs is considered, it is known that the volume of the preform backward trace from the end point of flash land is relatively small. In addition, the volume is calibrated only in the upper direction, the outward direction of billet being confined, otherwise when the volume calibration is performed in the outward direction, the flow to the flash part becomes large and the flow to the rib part becomes small. In this case Fig. 17. Dimensions and configuration of the preform in asymmetric forging (H/B =2:1).
it is difficult to find a preform that fills the die-cavity well. Fig. 18 shows the die-cavity filling process when the preform of Fig. 17 is forward simulated. Fig. 19 shows the shape of the initial billet, the flow pattern of the preform and final product by cutting away 1/4 of the total part, for experiments performed with plasticine. In Figs. 18 and 19, it is noted that at the plane-strain and
Fig. 16. Height-and volumetric-compensations based on the final billet in Fig. 15.
Fig. 18. Die-cavity filling process of the element system from the preform in asymmetric forging (H/B =2:1).
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J.-H. Lee et al. / Journal of Materials Processing Technology 72 (1997) 141–151
6. Nomenclature H/B m n pi, ai U X X% W: boundary DVf DVs
rib height-to-width ratios friction factor at the die–workpiece interface work-hardening exponent pseudo-independent parameter for planestrain and axisymmetric parts velocity of the boundary between deforming elements the position of an element before deformation the position of an element after the deformation shear loss of the boundary between plane-strain and axisymmetric parts magnitude of velocity discontinuity at the die–workplace interface magnitude of velocity discontinuity at the shear surface between elements
Greek letters o¯ effective strain o¯; effective strain-rate of an element at time t s¯ effective stress effective stress of an element at time t s¯ t s¯ 0 average effective stress between adjacent elements Fig. 19. Initial billet, preform and final product using plasticine in asymmetric forging (H/B=2:1).
axisymmetric parts, the die-cavity is filled completely. There also is relatively good agreement, between experiment and simulation in the results for preform and final shape. 5. Conclusions In this study, to predict the initial billet and preform of an asymmetric forging product, the upper-bound elemental technique was used. In the analysis, the material is divided into plane-strain and axisymmetric parts. The total power consumption rate including the shear loss at the boundary of both parts is optimized to approach the asymmetric problem easily. The excellence of the dumb-bell type billet is verified by plasticine experiments and numerical analysis. By height and volume calibration of the backward-traced shape of the plane-strain and axisymmetric part, a relatively simplified preform is obtained. There is good agreement between simulation and experiment in respect of forging load and material flow. However, to design the preform of asymmetric forgings, a more developed design method is needed.
References [1] W.T. Wu, S.I. Oh, ALPIDT: A general purpose FEM code for simulation of non-isothermal forming processes, in: Proc. NAMRC-XIII, Berkeley, CA, 1984, p. 449. [2] M. Kiuchi, A. Karato, Application of UBET to non-axisymmetric forging, Adv. Technol. Plast. 11 (1984) 967. [3] M. Kiuchi, A. Karato, Simulation of flow behaviors of billet to fill side-flush cavity of die-application of UBER to non-axisymmetric forging I, J. JSTP 26 (290) (1985) 307. [4] M. Kiuchi, T. Muramatsu, T. Imai, Analysis on non-axisymmetric complex forging — Development of simulation method for forging process I, J. JSTP 30 (342) (1989) 997. [5] M. Kiuchi, H. Kishi, M. Ishikawa, Upper bound analysis of extrusion and/or drawing of square, rectangular, hexagonal and other asymmetric bars and wires — Study on non-symmetric extrusion and drawing I, J. JSTP 24 (266) (1983) 290. [6] M. Kiuchi, M. Ishikawa, Upper bound analysis of extrusion and/or drawing of L-, T- and H-sections — Study on non-symmetric extrusion and drawing II, J. JSTP 24 (270) (1983) 722. [7] T. Wada, T. Nanba, Numerical method for plastic working processes in three dimensions, Adv. Technol. Plast. 11 (1987) 1005. [8] J.H. Argyris, J. St Doltsinis, J. Luginsland, Three-dimensional thermomechanical finite element calculation of three-dimensional analysis of forming processes, in: Proc. Int. Workshop Simulation of Metal Forming Processes by the Finite-Element Method (SIMON-I), Stuttgart, 1985, Springer-Verlag, Berlin, 1986, p. 125.
J.-H. Lee et al. / Journal of Materials Processing Technology 72 (1997) 141–151 [9] I. Pillinger, P. Hartley, C.E.N. Sturgess, G.W. Rowe, A three-dimensional analysis of the cold forging of a model aluminium connecting rod, Proc. Inst. Mech. Eng. 199 (1995) 319. [10] M.J.M. Barata Marques, P.A.F. Martins, A rigid-plastic finite element analysis of ring forging, Adv. Technol. Plast. 1 (1990) 13. [11] T. Imai, M. Kiuchi, T. Muramatsu, Analysis of non-axisymmetric upsetting — Development of simulation method for forging process IV, J. JSTP 33 (374) (1992) 253. [12] H.Y. Kim, D.W. Kim, Computer-aided preform design in the closed-die forging process, J. Mater. Process Technol. 41 (1994) 83.
.
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[13] J. Oudin, Y. Ravalard, An upper bound method for computing loads and flow patterns in plane-strain forging, Int. J. Mach. Tool Des. Res. 21 (1981) 237. [14] M. Kiuchi, Y. Murata, Study on application of UBET (Upper Bound Elemental Technique) 1st Report — Simulation of axisymmetric metal forming process, J. JSTP 22 (244) (1981) 495. [15] D.M. Himmelblau, Applied Nonlinear Programming, McGrawHill, New York, 1972, p. 152. [16] J.J. Park, N. Rebelo, S. Kobayashi, A new approach to preform design in metal forming with the finite element method, Int. J. Mach. Tool Des. Res. 23 (1) (1983) 71.
.
An upper-bound elemental technique approach to the process design of asymmetric forgings Jong-Heon Lee a, Young-Ho Kim b,*, Won-Byong Bae c a
b
Department of Vehicle and Machinery, Kyung-Nam Junior College, Pusan, South Korea Department of Mechanical and Production Engineering, Pusan National Uni6ersity, 30, Jangjeon-dong, Kumjeong-gu, Pusan 609 -735, South Korea c Department of Mechanical Design Engineering, Pusan National Uni6ersity, Pusan, South Korea Received 10 June 1996
Abstract An upper-bound elemental technique (UBET) program has been developed to predict forging load, die-cavity filling, preform in asymmetric forging. To analyze the process easily, it is suggested that the deformation is divided into two different parts. Those are axisymmetric part in corner, plane-strain part in lateral. The plane-strain and axisymmetric parts are combined by building block method. The total energy is computed through combination of two deformation parts. A dumbbell-type preform has been obtained from height and volumetric compensations of the billet based on the backward simulation. Experiments have been carried out with pure plasticine at room temperature. Theoretical predictions are in good agreement with experimental results. © 1997 Elsevier Science S.A. Keywords: Upper-bound elemental technique; Asymmetric forging; Process design
1. Introduction There are many components of rib-web type of crosssection in airplane structural components or vehicle components. A lot of research has been carried out on rib-web type forging, most of which involved planestrain or axisymmetric problems, with few studies having been made on asymmetric shapes because of the difficulty in analysis. In analyzing the metal forming process, the finite element method (FEM) [1] is regarded as one of the most effective tools because it can yield more reliable results than other methods. However, it has some problems such as a vast amount of calculation and limitation in its application to actual fields and it is not economic in that it needs a great computer capacity and experts who can deal with it. Nevertheless, the upperbound elemental technique (UBET) has had its merits recognized because it is easy to apply, needs much
* Corresponding author. Fax: [email protected]
+82
51
5147640;
e-mail:
0924-0136/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 9 2 4 - 0 1 3 6 ( 9 7 ) 0 0 1 4 5 - 3
shorter CPU time and the results are comparatively useful. Several studies have been performed on asymmetric shapes. The asymmetric forging problem having side flash was analyzed by Kiuchi [2,3] with UBET. The material was divided into various elements, which was then possible to analyze. In the analysis of asymmetric forging, a UBET simulator (FORMS) [4] was developed by Kiuchi to study the velocity fields of complex shapes integrated by easy forging processes (upsetting, extrusion, etc.). In addition, Kiuchi et al. [5,6] performed a study on developing generalized 3-dimensional kinematically-admissible velocity fields using the upper-bound method for the forward-extrusion problem having various cross-sectional shapes. It was found difficult to formulate the 3-dimensional velocity fields, which were very complicated. Wada et al. [7] applied an upper-bound method, dividing the 3-dimensional shape into rectangular parallel-piped, prismatic, right-angle tetrahedron and right-angle pentahedral elements. However, because of the difficulty in optimizing many variables, it is not adequate for application to complex shapes.
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J.-H. Lee et al. / Journal of Materials Processing Technology 72 (1997) 141–151
In studies by FEM, Argyris et al. [8] anaylzed an airplane turbine blade under frictionless conditions, whilst Pillinger et al. [9] analyzed the forging process of an aluminum connecting rod under constrain frictional conditions. In addition, for the asymmetric problem of a ring, Marques et al. [10] analyzed the forging process, where a circular cross-section was formed from rectangular billet after dividing the entire material into planestrain and axisymmetric parts. Even though there have been several research studies on the preform design of asymmetric components by FEM, comparatively fewer studies have been carried out using UBET. Imai et al. [11] suggested the preform for a cylinder having an asymmetric flange using a UBET simulator (FORMS). Also, Kim et al. [12] suggested the preform of a connecting rod using UBET simulation. In the present study, for the process design of asymmetric forgings of rib-web typed cross-section, the asymmetric shape is divided into plane-strain and axisymmetric parts, then the building block method, which combines the two parts, is used. As a simulation method, UBET is adopted. In the case that the rib height-to-width ratio is 1:1, the kinematically admissible velocity field of each part is determined by minimizing the total energy rate, which latter is the summation of the energy at the plane-strain deformation part, at the axisymmetric part and at the boundary of these two regions. By predicting the die-cavity filling processes and the forging loads for various initial billets, the characteristics of each billet are studied.
Fig. 1. Analytical model of asymmetric forging.
Fig. 2. Flow chart for forward UBET simulation.
If the rib height-to-width ratio is 2:1, the final shape is divided into the plane-strain and axisymmetric deformation parts and then the preform obtained from the backward-tracing scheme [12] in each part is calibrated in the height compensation of the rib. After that, the die-cavity filling is checked by forward simulation. If the cavity of the rib is not filled in the plane-strain or axisymmetric parts, a simple shape of preform of dumb-bell type is designed by performing volumetric compensation of the upper part of the billet.
Fig. 3. Overlap of the flow pattern of the boundary between the plane-strain and axisymmetric parts.
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Fig. 5. True stress – true strain curve of plasticine.
effective strain. Fig. 2 shows the flow chart for forward simulation.
2.1. Kinematically admissible 6elocity fields
Fig. 4. Flow chart of backward simulation for preform design.
To check the validity of this theory, a model-material test was performed.
2. UBET analysis The analytical model in this study is shown in Fig. 1. An asymmetric shape of rib-web type is analyzed by dividing the total region into plane-strain and axisymmetric parts and then by combining them again. The plane-strain and axisymmetric parts are divided again into a few simple elements. The velocity field of each divided element should satisfy the velocity boundary condition between adjacent elements and the volume constancy condition. For velocity fields, those suggested by Oudin [13] and Kiuchi [14] are used, whilst for the plane-strain deformation part, rectangular and trapezoid elements, suggested by Oudin [13] and for the axisymmetric part, rectangular and triangular ring elements, suggested by Kiuchi [14], are used. The work hardening of the material is considered under the assumption that the flow stress is dependent only on
The total power consumption rate in a forging process is as follows: J*= % W: i + % W: s + % W: f
(1)
where W: i is the internal power of the plane-strain and axisymmetric parts, W: s is the shear loss at the interface of two adjacent elements which includes the shear loss between plane-strain and axisymmetric parts and W: f means the friction loss.
Fig. 6. Upper and lower dies for the experiments with plasticine.
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metric deformation parts and W: boundary is the shear loss at the interface between the plane-strain and axisymmetric deformation parts. Also, the friction loss, W: f is as follows: W: f = m
s¯ 0
&
DVf dSf (5)
3 s f where m is friction constant and DVf is the velocity discontinuity along the surface between the die and material. The total power consumption rate, J*= J*(p1, p2, p3, …, pn, a1, a2, a3, …, an ) is minimized by the flexible polyhedron search (FPS) method [15] which is a kind of direct search method. Here, pi, ai are pseudo-independent parameters at the plane-strain and axisymmetric parts, respectively. By minimizing J*, the kinematically admissible velocity fields at time t can be determined. The forming load, L, is given as follows: L=
J* VD
(6)
where VD is the die velocity.
2.2. Element regeneration To investigate the die-cavity filling process, the total forming time is divided according to time increment Dt.
Fig. 7. Dimensions and configuration of the initial billet for asymmetric forging (H/B =1:1).
The internal power at each element, W: i is: W: i =
&
6
s¯ to¯; t dV
(2)
where the effective strain-rate, o¯; t = (2/3)oijoij In Eq. (2) s¯ t is the effective stress of each element at time t and it is assumed that s¯ t is the function of effective strain only: s¯ t = Co¯ nt
(3)
W: s represents the shear loss along the velocity discontinuity surface, expressed as follows: W: s =
s¯ 0
3
&
DVs dSs +W: boundary
(4)
s
where, s¯ 0 is the mean value of flow stresses of two adjacent elements, DVs is the velocity discontinuity along the shear surface in the plane-strain and axisym-
Fig. 8. Die-cavity filling of the final step for the various initial billets in Fig. 7.
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it is assumed that the two rounded parts of both ends deform axisymmetrically and that the linear parts between the two rounded parts is a plane-strain problem because their length is much greater then their width. Marques [10] and Kim et al. [12] also based their research on this kind of assumption. As the die advances, the plane-strain and axisymmetric parts have different deformation patterns, which results in the discrepancy of the elemental boundary. The analytical model for calculating the shear loss due to this boundary discrepancy is shown in Fig. 3. The shear loss at the interface of the plane-strain and axisymmetric parts is calculated as follows: W: boundary =
s¯ 0
3
&
DV dS
(8)
and DV =[(Uaxi,R − Uplane,X )2 + (Vaxi,Z − Vplane,y )2]1/2 where dS is the infinitesimal elemental boundary area owned by a plane-strain and an axisymmetric element at the same time, Uaxi,R and Vaxi,Z represent the R- and Z-direction velocity components of the axisymmetric elemental boundary, respectively and Uplane,X and Vplane,Y are the X- and Y-direction velocity components of the plane-strain elemental boundary, respectively. In Fig. 3, the element system is superimposed along the boundary of each deformed part when the height reduction reaches a particular point. As is shown in this figure, when the die advances, there is a boundary Fig. 9. Flow pattern of the final step for multi-layered plasticine for the various initial billets in Fig. 7.
At each step, the coordinate of the deformed shape of each element is determined by the kinematically-admissible velocity fields after minimization. The coordinate of each element after time incremental Dt is as follows: X% =X+Dt · U
(7)
where X% is the position of each element after deformation and X and U represent the position and velocity of an elemental boundary before deformation, respectively. After deformation, sliding occurs between adjacent elements, resulting in the discontinuity of the elemental boundary. Therefore, in order to proceed to the next step, the element regeneration scheme [12] by vertical and horizontal projection is used.
2.3. Building-block method Products of a complex shape such as asymmetric, can be approximated into the combination of simple shapes which can be analyzed easily. Fig. 1 shows a schematic diagram of the final products. To simplify the analysis
Fig. 10. Variation of experimental forging loads for the various initial billets in Fig. 7.
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discrepancy between the plane-strain and axisymmetric parts owing to the difference of the flow pattern in the two parts. In this case the shear loss at each step is calculated as follows. Along the X-axis the node of each element is assigned the coordinate AX(1), AX(2), AX(3), …, AX(N), the computer program perceiving the number of the plane-strain or axisymmetric element corresponding to these assigned values. For element A, both the plane-strain and axisymmetric elements have element number (1, 1). Then the shear loss of element A is obtained from the boundary velocity and the strain-rate value of the element, (1, 1) of both parts. For element B between AX(3) and AX(4), the element number of the plane-strain and axisymmetric parts is (3, 1) and (2, 1) respectively. Therefore for the plane-strain and axisymmetric parts, the data of element number, (3, 1), (2, 1) are used to calculate the shear loss, respectively. For the case of element C, the shear loss is not calculated because only the element of the plane-strain part exists. Each of every other element is included in one of the element types, A, B or C and analyzed likewise.
2.4. Preform design In forming process design, the preform design is one of the most important steps. In closed-die-forging, the preform design includes the dimensions of preform, the number of steps and the design of the configuration. In the present study, the authors have tried to design a comparatively simple preform by calibrating the height and volume, based on the backward-tracing scheme, suggested by Park et al. [16] and applied to UBET by Kim et al. [12] in the preform design. The final products are divided into plane-strain and axisymmetric parts and the surface contacting with die is considered as a free surface. From this surface the nodal points of the prior step are backward traced. As in the forward simulation, there is the elemental boundary discrepancy between the plane-strain and axisymmetric parts in the backward simulation, owing to the different flow pattern of the two parts. Therefore the configuration of the preform by the backward tracing process shows a stepped shape in the upper and outer direction of the preform. In this case, because of the complexity of the preform die, these is the need to simplify the preform configuration. Therefore, in the configuration of final step obtained from the backward tracing process, in order to equalize the height of the ribs in the plane-strain and axisymmetric parts, only the upper direction of the preform is compensated under volume constancy. Then the preform compensating the height of the rib is forward simulated and in constant thickness, the upper direction of the preform containing the web part is compensated by the calculated volume of the cavity between the
Fig. 11. Die-cavity filling process of the element system for Billet 3 in Fig. 7 (H/B= 1:1).
material and the die. Therefore, the authors have tried to design a comparatively simple preform of dumb-bell type. Fig. 4 shows the flow chart for the backward simulation.
3. Experimental For the forging experiments of the asymmetric products of rib-web type, pure plasticine was used. To obtain the flow stress-and-strain characteristics, a cylindrical billet, of which the height and diameter are equal, is used. In the experiments, using vaseline lubricant, (m¥0.1) which show comparatively little bulging, the cylinder is compressed up to 50% of its initial height at constant die velocity (0.5 mm s − 1) at room temperature. Fig. 5 shows the material characteristics from the experiments. The relationship of the flow stress and strain of the material is as follows: s¯ =0.178o¯ 0.082 (MPa)
(9)
To obtain information about the material flow in the workpiece, plasticine was used as a model material. The billets are laminated with two colors (white and black), after being kneaded to remove air from inside the material. The initially rectangular billets are made into a rounded shape at the ends.
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The material of the die is carbon steel. To make it easy to release the plasticine product from the die, the upper dies are divided two parts, joined with a pin and bolt. The experiments are performed with a material testing system (MTS) at a constant die velocity (0.5 mm s − 1). The lubricant is talc powder (m ¥0.4). Using the X– Y plotter attached to MTS, the load curve is obtained. To keep the temperature of billets constantly during experiments, the die and billets were put into an over for 24 h and the room temperature is kept constant using a thermo-hygrostat. Fig. 6 shows configuration of the forging die.
4. Results and discussion
4.1. Initial billets (H/B = 1:1) Fig. 7 shows three billets used having a ratio of height-to-width of rib 1:1. Figs. 8 – 10 show the numerical and experimental results of the billets in Fig. 7, these being: Final die-cavity filling by forward simulation; final flow pattern; and forging load by experiment, respectively. Here, when billet 1 is used, at the planestrain part the die cavity is filled and flash forms adequately, but at the axisymmetric part the material Fig. 13. Material flow pattern of the multi-layered plasticine for billet 3 in Fig. 7 (H/B=1:1).
Fig. 12. Die-cavity filling process of the grid distortion pattern for billet 3 in Fig. 7 (H/B= 1:1).
filling around the rib part is not completed. To enhance the die-cavity filling in the axisymmetric rib part, the dumb-bell type billet 2, which is made by calibrating the volume to the outward direction of the axisymmetric part is used, which effects the filling of the die-cavity of the plane-strain and axisymmetric parts and also the amount of flash is adequate. However, it is difficult to use the dumb-bell type billet 2 as a initial billet because it requires another process. Accordingly, a billet that can ensure die-cavity filling at the axisymmetric part and is of comparatively simple shape, is needed. Thus the outside boundary dimension of axisymmetric part of billet 2 is extended up to the plane-strain part, the extended billet being billet 3. In the case of billet 3 even if quite a greater amount of flash is produced at the plane-strain part than that for billet 2, the simplicity of the configuration is recognized. As is indicated in Fig. 10, which shows the forging loads by experiment, the loads that increase up to a reduction in height of 40% are similar, because billets 1 and 2 have the same dimensions of the plane-strain part. In the case of billet 2, for the reduction in height
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of 40%, the flash is produced at the plane-strain and axisymmetric parts simultaneously. On the other hand, billet 1 has flash only at the plane-strain part, so for a reduction in height of around 40%, billet 2 has higher loads than is the case for billet 1. At the reduction in height of around 17%, billet 3 starts to have flash at the plane-strain part and for a reduction in height of 55%, flash is produced at the axisymmetric part. Figs. 11 and 12 show the die-cavity filling and grid distortion processes of the plane-strain and axisymmetric parts of billet 3 from the initial to the final step. Fig. 13 shows the die-cavity filling process at several steps in plasticine experiments by cutting away one-fourth of the billets. From Figs. 11 – 13, it is noted that the boundary of the plane-strain and axisymmetric parts is out of agreement because as the process proceeds the material flow in the upper and outward directions becomes different: This can be verified by the results of experiment shown in Fig. 13. Thus to analyze this boundary, the building-block method of Section 2.3 is used. Further, the flash at the plane-strain part is produced first and at the moment when the die-cavity of the axisymmetric part is filled completely, the planestrain part already has a significant amount of flash. Fig. 14 shows the forging-load variation for the case of billet 3. In this diagram that the results of numerical analysis are a little higher than those of experiments is due to the use of upper-bound analysis. However, on the whole, the results are acceptable. Fig. 15. Backward process from final configuration in asymmetric forging (H/B =2:1).
4.2. Preform design
Fig. 14. Comparison between theoretical and experimental forging loads for billet 3 in Fig. 7 (H/B= 1:1).
For the case one rib height-to-width ratio (H/B= 1:1), it is verified that the dumb-bell type billet, billet 2, is adequate regarding the amount of flash and the forging load, the die-cavity filling process. For the case two rib height-to-width ratio (H/B= 2:1), it can be predicted that the dumb-bell type billet is adequate as a preform to be forged into the final product. If backward-traced from the asymmetric final product, as the results at the final step of Fig. 15, the height and width of the rib become different because of different flow pattern of the plane-strain and axisymmetric parts. When this shape is adopted as a preform, the die of preform becomes very complex, resulting in difficulty in numerical analysis and die manufacturing. Thus the configuration of preform must be simplified. Accordingly, first of all, the heights of the rib, y2, y3 which are obtained from backward tracing, can be equalized into the calibrated height, yn according to the volume-constancy condition. With this billet, forward simulation is performed and the amount of unfilled die-cavity is checked, then after performing the volume calibration of the total upper part of billet including the
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web shown in Fig. 16(b), the preform of dumb-bell type shape shown in Fig. 16 is obtained. In Fig. 16(b) it can be noted that the calibration volume is more than the uncalibrated volume, which results from the situation that at the beginning of the backward tracing, the flash length is backward traced from the end point of the flash land. When the actual asymmetric forging process where quite a large amount of flash occurs is considered, it is known that the volume of the preform backward trace from the end point of flash land is relatively small. In addition, the volume is calibrated only in the upper direction, the outward direction of billet being confined, otherwise when the volume calibration is performed in the outward direction, the flow to the flash part becomes large and the flow to the rib part becomes small. In this case Fig. 17. Dimensions and configuration of the preform in asymmetric forging (H/B =2:1).
it is difficult to find a preform that fills the die-cavity well. Fig. 18 shows the die-cavity filling process when the preform of Fig. 17 is forward simulated. Fig. 19 shows the shape of the initial billet, the flow pattern of the preform and final product by cutting away 1/4 of the total part, for experiments performed with plasticine. In Figs. 18 and 19, it is noted that at the plane-strain and
Fig. 16. Height-and volumetric-compensations based on the final billet in Fig. 15.
Fig. 18. Die-cavity filling process of the element system from the preform in asymmetric forging (H/B =2:1).
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6. Nomenclature H/B m n pi, ai U X X% W: boundary DVf DVs
rib height-to-width ratios friction factor at the die–workpiece interface work-hardening exponent pseudo-independent parameter for planestrain and axisymmetric parts velocity of the boundary between deforming elements the position of an element before deformation the position of an element after the deformation shear loss of the boundary between plane-strain and axisymmetric parts magnitude of velocity discontinuity at the die–workplace interface magnitude of velocity discontinuity at the shear surface between elements
Greek letters o¯ effective strain o¯; effective strain-rate of an element at time t s¯ effective stress effective stress of an element at time t s¯ t s¯ 0 average effective stress between adjacent elements Fig. 19. Initial billet, preform and final product using plasticine in asymmetric forging (H/B=2:1).
axisymmetric parts, the die-cavity is filled completely. There also is relatively good agreement, between experiment and simulation in the results for preform and final shape. 5. Conclusions In this study, to predict the initial billet and preform of an asymmetric forging product, the upper-bound elemental technique was used. In the analysis, the material is divided into plane-strain and axisymmetric parts. The total power consumption rate including the shear loss at the boundary of both parts is optimized to approach the asymmetric problem easily. The excellence of the dumb-bell type billet is verified by plasticine experiments and numerical analysis. By height and volume calibration of the backward-traced shape of the plane-strain and axisymmetric part, a relatively simplified preform is obtained. There is good agreement between simulation and experiment in respect of forging load and material flow. However, to design the preform of asymmetric forgings, a more developed design method is needed.
References [1] W.T. Wu, S.I. Oh, ALPIDT: A general purpose FEM code for simulation of non-isothermal forming processes, in: Proc. NAMRC-XIII, Berkeley, CA, 1984, p. 449. [2] M. Kiuchi, A. Karato, Application of UBET to non-axisymmetric forging, Adv. Technol. Plast. 11 (1984) 967. [3] M. Kiuchi, A. Karato, Simulation of flow behaviors of billet to fill side-flush cavity of die-application of UBER to non-axisymmetric forging I, J. JSTP 26 (290) (1985) 307. [4] M. Kiuchi, T. Muramatsu, T. Imai, Analysis on non-axisymmetric complex forging — Development of simulation method for forging process I, J. JSTP 30 (342) (1989) 997. [5] M. Kiuchi, H. Kishi, M. Ishikawa, Upper bound analysis of extrusion and/or drawing of square, rectangular, hexagonal and other asymmetric bars and wires — Study on non-symmetric extrusion and drawing I, J. JSTP 24 (266) (1983) 290. [6] M. Kiuchi, M. Ishikawa, Upper bound analysis of extrusion and/or drawing of L-, T- and H-sections — Study on non-symmetric extrusion and drawing II, J. JSTP 24 (270) (1983) 722. [7] T. Wada, T. Nanba, Numerical method for plastic working processes in three dimensions, Adv. Technol. Plast. 11 (1987) 1005. [8] J.H. Argyris, J. St Doltsinis, J. Luginsland, Three-dimensional thermomechanical finite element calculation of three-dimensional analysis of forming processes, in: Proc. Int. Workshop Simulation of Metal Forming Processes by the Finite-Element Method (SIMON-I), Stuttgart, 1985, Springer-Verlag, Berlin, 1986, p. 125.
J.-H. Lee et al. / Journal of Materials Processing Technology 72 (1997) 141–151 [9] I. Pillinger, P. Hartley, C.E.N. Sturgess, G.W. Rowe, A three-dimensional analysis of the cold forging of a model aluminium connecting rod, Proc. Inst. Mech. Eng. 199 (1995) 319. [10] M.J.M. Barata Marques, P.A.F. Martins, A rigid-plastic finite element analysis of ring forging, Adv. Technol. Plast. 1 (1990) 13. [11] T. Imai, M. Kiuchi, T. Muramatsu, Analysis of non-axisymmetric upsetting — Development of simulation method for forging process IV, J. JSTP 33 (374) (1992) 253. [12] H.Y. Kim, D.W. Kim, Computer-aided preform design in the closed-die forging process, J. Mater. Process Technol. 41 (1994) 83.
.
151
[13] J. Oudin, Y. Ravalard, An upper bound method for computing loads and flow patterns in plane-strain forging, Int. J. Mach. Tool Des. Res. 21 (1981) 237. [14] M. Kiuchi, Y. Murata, Study on application of UBET (Upper Bound Elemental Technique) 1st Report — Simulation of axisymmetric metal forming process, J. JSTP 22 (244) (1981) 495. [15] D.M. Himmelblau, Applied Nonlinear Programming, McGrawHill, New York, 1972, p. 152. [16] J.J. Park, N. Rebelo, S. Kobayashi, A new approach to preform design in metal forming with the finite element method, Int. J. Mach. Tool Des. Res. 23 (1) (1983) 71.
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