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Prediction of neutral plane and effects of the process parameters in radial forging using an upper bound solution
Journal of Materials Processing Technology 186 (2007) 147–153
Prediction of neutral plane and effects of the process parameters in radial forging using an upper bound solution M. Sanjari a , A. Karimi Taheri a,∗ , A. Ghaei b a
Department of Materials Science and Engineering, Sharif University of Technology, P.O. Box 11365-9866, Tehran, Iran b Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Received 31 May 2006; accepted 14 December 2006
Abstract Radial forging is an open die forging process used for reducing the diameter of shafts, tubes and axels. In the present research, a new velocity field was developed to analyze the radial forging process by the upper bound method. The forging load was predicted by minimizing the deformation power with regards to the location of neutral plane. The accuracy of the model was verified by comparing the predicted forging loads with experimental results published by the previous researcher. A good agreement was found between the two sets of results. © 2007 Elsevier B.V. All rights reserved. Keywords: Radial forging; Upper bound method; Analysis; Location of neutral plane
1. Introduction Radial forging is a cost-effective and material saving forming process for reducing the cross-section of rods, tubes and shafts made of any metal being suitable for the cold forming process. The advantages of radial forging are smooth surface finish, considerable material or weight savings, preferred fiber structure, minimum notch effect and increased material strength [1]. In this process, two pairs instead of one pair of opposing dies are used to deform the work piece by a number of short-stroke side-pressing operations [2]. Using slab method analysis, Lahoti et al. [3,4] analyzed the mechanics of radial forging process for single and compound angle dies. Ghaei et al. [5] also used slab method analysis to study the effects of circular die shapes on deformation in the radial forging process. Using a modular upper bound technique, Subramanian et al. [6] modeled the metal flow in die cavity in radial forging for rifling of gun barrels under plane strain condition. Yang [7] conducted a study on the radial forging process using the combination of slip-line theory and upper bound method. He used a slip-line field and a hodograph coupled with the use of a non-linear optimization technique to find the field angles and other defining parameters. Moreover, many
∗
Corresponding author. Tel.: +98 21 66165220; fax: +98 21 66005717. E-mail address: [email protected] (A. Karimi Taheri).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.12.029
researchers have used FEM to model the radial forging process [8–13]. One of the important parameters in the radial forging process is the location of neutral plane. Plastic deformation is not feasible if the neutral plane lies outside the zone of deformation, since the material would simply slip away from the radially approaching hammer dies without any deformation taking place [3]. Although, Lahoti and Altan [3] studied the effect of process parameter, such as the die angle, reduction in area and friction coefficient on the location of neutral plane by the slab method analysis, but surprisingly a detailed and complete upper bound analysis does not yet exist in the literature to predict the position of neutral plane and its effect on the forging load. In present work, by using the upper bound theory and minimizing the deformation power with regards to the location of the neutral plane both the forging load and the location of neutral plane are predicted.
2. Analysis In Fig. 1, the general model of the radial forging process for tubes, considered in this study, is illustrated. It is assumed that in the innermost position of the hammer dies, the inlet section of the dies form a perfect conical surface and the die land is perfectly cylindrical. Thus, the small clearances existing between the hammer dies at the end of a forging stroke are neglected. Based on this
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Nomenclature A As1 , As2 dv Fr k L1 L2 L3 m R Rm Rn R0 R1
cross-section area of the element shear surfaces (Fig. 5) the element volume radial force yield shear stress of the part length of the sinking zone length of the forging zone length of the sizing zone friction shear factor outer radius of the element radius of the mandrel outer radius of tube at the neutral plane outer radius of the preform outer radius of the element at sinking–forging zones interface R2 outer radius of the product t0 initial wall thickness of the preform t final wall thickness of the element Vr absolute radial velocity of dies Vsli relative velocity between the part and die in the sinking zone Vx velocity in the X-direction in the sinking zone velocity in the X-direction at the beginning of Vx0 sinking zone Vx1 velocity in the X-direction at the sinking–forging zones interfaces Vz axial velocity in the forging zone Vz1 axial velocity in the sinking–forging zones interface Vz2 axial velocity in the forging–sizing zones interface Vz3 axial velocity at the end of sizing zone Wfd1 rate of friction work at the part–die interface in the sinking zone Wfd2 rate of friction work at the part–die interface in the forging zone Wfd3 rate of friction work at the part–die interface in the sizing zone Wfm2 rate of friction work at the part–mandrel interface in the forging zone Wfm3 rate of friction work at the part–mandrel interface in the sizing zone Wp1 , Wp2 rate of plastic work in the sinking and forging zones, respectively Wp3 rate of plastic work in the sizing zone Ws1 , Ws2 rate of shear work over the surfaces As1 and As2 , respectively Greek symbols α die angle εx strain rate in the X-direction εy strain rate in the Y-direction ε¯˙ effective strain rate σ¯ flow stress of the material in compression
Fig. 1. Schematic representation of radial forging for tubes and variables.
assumption, no material spreading would be allowed between the dies at the end of the forging strike and the deformation could be modeled axi-symmetrically. There is a small gap between the mandrel and the inner surface of the billet. In other words, forging under the conical portion of the dies is accompanied by a certain amount of sinking. This sinking causes a longitudinal back-push applied on the billet at the radial plane where the inner surface of the tube first touches the mandrel. In addition, the hammer dies also contain a cylindrical sizing zone. Thus, nearly all the deformation occurs in the conical inlet section, i.e., in the forging zone shown in Fig. 1. In the sizing zone, most of the tube material is elastically deformed to the yield limit and only a small amount of plastic deformation takes place. Nevertheless, a considerable amount of energy, supplied by the dies, is needed for this operation. As shown in Fig. 1, there are three distinct regions of deformation in radial forging of tubes: (a) the sinking zone, (b) the forging zone and (c) the sizing zone. In the following analysis of the process, all three zones are considered. It is obvious that in some cases, such as in finish forging of tubes, the sinking zone may be very small and its contribution to the overall radial forging load can be excluded in the final evaluation [3]. Depending on the inlet cone angle, the friction at the tube–mandrel and tube–die interfaces, the axial push and pull forces and the length of die land, the deforming material may flow towards the product or towards the preform or both directions simultaneously. Thus, in most general case a neutral plane exists somewhere within the deforming tube. The material being exactly on this plane is only deformed radially and does not flow axially. On both sides of the neutral plane the metal is deformed radially as well as axially, so it flows away from neutral plane in both directions, towards either the preform or the product. Theoretically, the neutral plane could lie in any of the three zones of deformation. However, it is usually located in the forging zone in most of practical conditions [3]. In the upper bound method, the total power required to deform the workpiece is divided into three components: (a) the rate of plastic work or rate of strain energy, (b) the rate of friction work and (c) the rate of shear work which is dissipated when the material flow direction is changed. In the present analysis, the total power required for the process is obtained using the following assumptions: (1) the material is rigid plastic; (2) friction at the die–tube and mandrel–tube interfaces produces a constant friction shear stress;
M. Sanjari et al. / Journal of Materials Processing Technology 186 (2007) 147–153
149
(3) the wall thickness of the tube remains constant throughout the sinking zone; (4) the strain rate is homogenous in each element; (5) there are no front-pull and back-push forces; (6) neutral plane is located in the forging zone. 2.1. Analysis of forging zone As the dies are striking, the outer radius of the billet is reduced and the material flows axially. Strain rates in the radial forging are influenced by the location of neutral plane. Theoretically, the neutral plane could lie in any of the three zones of deformation. However, it is usually located in the forging zone in most of practical conditions [3]. Therefore, it is assumed that the neutral plane is located in the forging zone. Using the condition of volume constancy, the axial velocity for the forging zone is given by: 2 R − R2n Vr Vz = (1) tan α R2 − R2m Because of symmetry, the angular velocity component is taken as zero and it is sufficiently accurate to take the axial strain rate as the effective strain rate. Therefore, the effective strain rate is given by: ε˙ zz = 2Vr R
(R2n − R2m )
(2)
2
(R2 − R2m )
Regarding Fig. 2, the following geometrical relations are derived: dR = −dz tan α dv = π(R
2
(3)
− R2m ) dz
(4)
The rate of plastic work and the rate of friction work at the mandrel–tube and die–tube can be calculated as follows: 2 − R2 ¯ 2n − R2m ) σ(R R πV r m 1 ˙ p2 = σ¯ ε¯˙ dv = (5) ln W tan α R22 − R2m ˙ fm2 = W
(mk) (Vz ) (2πRm ) dz
=
Rn
− R2n −dR (2πR ) m R2 − R2m tan α R2
Vr tan α R1 R 2 Vr R2 − R2n −dR + (2πRm ) (mk) tan α R2 − R2m tan α Rn (mk)
Fig. 2. Schematic representation of forging zone.
(6)
Fig. 3. Schematic representation of sizing zone.
Vz dz (mk) (2πR) cos α cos α Rn 2 −dR Vr R − R2n = (2πR) (mk) sin α R2 − R2m sin α R1 Rn −dR Vr R2 − R2n + (2πR) (mk) sin α R2 − R2m sin α R1
˙ fd2 = W
(7)
2.2. Analysis of sizing zone In the sizing zone, the outer diameter of deforming part has reached to the diameter of final product (Fig. 3). However, due to spring back of tube material and axial feed per stroke the material is deformed almost to the yield point in the last moment of stroke. Moreover, the dissipated energy by friction at the tube–mandrel and tube–die is significant. In the start of sizing zone, it is sufficiently accurate to take the axial velocity equal to that at the exit point of forging zone. As a consequence of volume constancy in the last moment of stoke, the radial velocity induces an axial velocity in the sizing zone: 2Vr R2 L3 R22 − R2m Vz at z = L2 ⇒ Vz = Vz2 Vz at z = L2 + L3 ⇒ Vz = Vz3 + Vz2 Vz3 =
(8)
Assuming that the change of axial velocity is linear from Vz2 to Vz3 for any axial location the following relationship is derived. Vz3 (z − L2 ) + Vz3 Vz = (9) L3 It is sufficiently accurate to take the axial strain rate as the effective strain rate as follows: Vz3 L3 ¯ 22 − R2m ) R21 − R2m πVr σ(R ln = σ¯ ε¯˙ dv = tan α R22 − R2m
ε˙ zz =
(10)
˙ p3 W
(11)
˙ fm3 = (mk) (Vz ) (2πRm ) L3 W
(12)
˙ fm3 = (mk) (Vz ) (2πR2 ) L3 W
(13)
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Fig. 5. Material flow direction and shear surfaces.
Referring to Fig. 4, the sliding velocity or relative velocity between the die and tube in the sinking zone is given by:
Fig. 4. Schematic representation of sinking zone.
2.3. Analysis of sinking zone As the dies are striking, both of the outer and inner surfaces of tube are reduced. The wall thickness of tube is slightly increased, especially when the ratio of thickness to outer diameter of tube is small. Therefore, it is acceptable to neglect changes in the wall thickness of tube, i.e., material flows only in the X-direction in Fig. 4. Referring to this figure and using the condition of volume constancy, the velocity field of material flow in the X-direction can be written as: Vx =
Vx1 (2R1 − t0 cos α) sin α + 2Vr (R − R1 ) cos2 α (2R − t0 cos α) sin α
(14)
The following geometrical relation is derived from Fig. 4: R = R1 + x sin α
Vsli = Vx + Vr sin α
(20)
Thus, the rate of friction work in the sinking zone is: ˙ fd1 = (mk) (Vsli ) (2πR) dx W
(21)
2.4. Rate of shear work Considering the flow pattern depicted in Fig. 5, the material changes its flow direction at the start and end of the sinking zone, i.e., on As1 and As2 surfaces. Therefore, two velocity discontinuities occur on As1 and As2 surfaces. The associated hodographs are shown in Fig. 6. The rates of energy dissipation on these surfaces are given by: ˙ s1 = kAs1 Vz1 W tan α
˙ s2 = kAs2 2Vx0 sin α W 2
(15)
where
Since variation in the wall thickness of tube is disregarded in the sinking zone, the strain rate in the Y-direction is zero. Therefore, the strain rate in the X-direction and effective strain rate in the sinking zone are given by:
As1 =
∂Vx (2Vr ε˙ x = = ∂x −
cos2 α)(2R
+ 2x sin α − t0 cos α) [(2R − t0 cos α) sin α]2 1
(2 sin α)(Vx1 (2R1 −t0 cos α) sin α+2Vr (R−R1 ) cos2 α) [(2R − t0 cos α) sin α]2 (16)
2 ε¯˙ = √ ε˙ x 3 2R dv = πt0 − t0 dx cos α Thus, the rate of plastic work is given by: L1 ˙p = W σ¯ ε¯˙ dv
(17) (18)
(22) (23)
πt0 t0 2Rm + cos α cos α
(24)
πt0 (2R0 − t0 ) cos(α/2)
(25)
and As2 =
Finally, the total power is obtained by adding all rate of works calculated above as a function of position of neutral plane. By minimizing the power with respect to the position of neutral plane, the location of neutral plane and the actual power required for deformation are evaluated. The radial load applied to the dies is obtained by equating the external power with the actual power: ˙ total = (W ˙ p1 + W ˙ fd1 ) + (W ˙ p2 + W ˙ fm2 + W ˙ fd2 ) W ˙ p3 + W ˙ fm3 + W ˙ fd3 ) + (W ˙ s1 + W ˙ s2 ) + (W
(26)
(19)
0
where dv = πt0 (2R − t0 cos α) x. Numerical methods may be used to calculate Eq. (18) easily.
Fig. 6. The hodographs used for the analysis: (a) surface As2 and (b) surface As1 .
M. Sanjari et al. / Journal of Materials Processing Technology 186 (2007) 147–153
Fr =
˙ total W Vr
151
(27)
3. Results and discussion The purpose of this study is to find the location of neutral plane during the radial forging process using a new upper bound solution. For this reason, the dimensions of three zones of deformation were calculated from the tool and work piece geometry and the strain components were calculated from the velocity field. To obtain the maximum force in a stroke, each component of work was described above was considered at the last step. In present research for the radial forging of tubes, the experimental results of forging loads were not available. So, the loads measured by Uhlig [13] for cold swaging of AISI 1015 steel rods and used by Lahoti and Altan [3], were utilized to verify the load predicted by the present analysis. As the analysis has been based on cold radial forging, the work hardening has a considerable effect on the billet flow stress. For this purpose, a relation of the type σ¯ = K¯εn was assumed for the billet material. The K and n values at room temperature were evaluated as [1]: K = 618.14 MPa,
Fig. 7. Rate of work vs. the location of neutral plan.
n = 0.1184.
Since the material was assumed to be rigid-plastic, an average flow stress for the preform and the tube was used in the analysis. The variables used in the analysis were: a = 4.3◦
m = 0.15,
From Eq. (26) if the rate of work is plotted versus the location of neutral plane, the minimum value of curve is the theoretical rate of work. Using Eq. (27), the load relating to each forging tools can be calculated. Fig. 7 shows this operation as an example using the data presented in Table 1. Referring to Table 1, the predicted loads show a good agreement with experiment and are 10–20% greater than the experimental load. It is interesting to note that the results show a good agreement for samples (1) and (2), where the length of forging zone and sizing zone are nearly equal. However, the predicted loads are higher when the forging zone is much smaller than the sizing zone being the case in the finishing forging [6]. According to Subramanian et al. [6], this discrepancy may be attributed to the possibility of a lower friction in the sizing zone in experiment. Table 1 also shows the location of neutral plane relating to the start of the forging zone when the diameter of the billet decreases. To verify the potential of the model for predicting the effects of process parameters, a further theoretical analysis was per-
Fig. 8. The effect of die angle on the radial load.
formed assuming that the material is AISI 1015 and changing the magnitudes of some process parameters. The effect of die angle on the location of neutral plane at the end of stroke is shown in Fig. 8. The limiting position of the neutral plane is at the entrance to or the exit from the hammer die. Plastic deformation is not feasible if the neutral plane lies outside the zone of deformation, since the material would simply slip away from the radially approaching hammer dies without any deformation taking place [3]. Fig. 3 shows that when the die
Table 1 Comparison of predicted loads with the experimental results of Ref. [3] Sample number
Billet diameter (mm)
Product diameter (mm)
Length of forging zone (mm)
Length of sizing zone (mm)
Location of the neural plane (mm)
Predicted maximum load* (kN)
Load* from experiment (kN)
1 2 3 4
15.97 15.97 15.03 13.99
13.18 13.25 13.11 13.03
18.55 18.09 12.77 6.38
18.00 18.00 18.00 18.00
15 14.75 10.35 5.03
192.5 191 152 112.5
172.00 167.00 124.00 74.50
*
Per hammer.
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Fig. 9. The effect of die angle on the location of neutral plane. Fig. 10. The effect of friction factor on the radial load at different die angles.
angle increases the location of neutral plane is approached to the start of the forging zone. Thus, the limitation of increasing the die angle discussed above does not allow to raising this factor so much. The effect of die angle on the radial load at the end of stroke for m = 0.15 is shown in Fig. 9. This figure reveals that as the die angle increases, the radial load decreases because at a constant reduction in area, the length of contact and the friction work are decreased. Moreover, when the die angle increases, the radial component of die normal force reduces and the axial component increases. Therefore, one may conclude that there is a limitation for increasing the die angle, because if the die angle increases excessively, the axial load becomes so large that it may make the preform to buckle or throw the part away from the forging box. Thus, as the axial force is known, one can easily determine whether the part will buckle or not. Also, it is possible to predict the magnitude of the least required back-push force preventing the slipping the part away. This force can also be used to find the maximum die angle within a certain process condition such as reduction in area, length of performs, etc. Fig. 10 shows the effect of friction factor on the location of neutral plane at the end of stroke at various die angles. This figure indicates that when the friction factor increases the location of neutral plane is approached to the start of the forging zone. Fig. 11 shows the effect of friction shear factor on the radial load at the end of stroke at various die angles. It is observed that when m is increased, the radial load is also increased due to the increase in friction works. These changes are increased when the die angle is increased. Fig. 12 shows the effect of reduction in cross-section area on the location of neutral plane at the end of stroke at various friction shear factor. Referring to this figure by increase in reduction in area the work consumed in the sizing zone increases and the location of neutral plane approaches to the sinking zone. Fig. 13 shows the effect of reduction in cross-section area on the radial load at the end of stroke at various friction factors. As can be seen, when the reduction is increased, the radial load is increased. This increase is due to the increase in plastic, friction and shear works. It has been reported that in order to enjoy
Fig. 11. The effect of friction shear factor on the location of neutral plane.
Fig. 12. The effect of reduction in cross-section area on the radial load at the end of stroke.
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(2) when the friction factor increases the location of neutral plane is approached to the start of the forging zone and the radial load is increased; (3) by increasing the reduction in area the location of neutral plane approaches to the sinking zone and radial load is increased. Acknowledgement The authors would like to thank the research board of Sharif University of Technology, Tehran, Iran, for the financial support of this work. References
Fig. 13. The effect of reduction in cross-section area on the location of neutral plane.
the benefits of radial forging process, the reduction must be so large that the plastic deformation penetrates to the core of the material [8]. Thus, it is important to choose a machine or finding the minimum number of passes when it is not possible to forge the tube in one pass. This is possible with calculating the maximum required power for deformation. Obviously, if the calculated power is less than the machine power, the tube can be produced in one pass. Otherwise, the tube should be produced in more than one pass so that the power of each passes to be less than the machine power. 4. Conclusion In this research, by using the upper bound theory and minimizing the deformation power with regards to the position of the neutral plane both the forging load and the position of neutral plane are predicted. For verification of the model, the predicted loads are compared with those of published experimental results. Also, the effect of the various process parameters is assessed by the model and compared with the published results. The following conclusions are achieved from the study: (1) when the die angle increases the location of neutral plane is approached to the start of the forging zone and the radial load is decreased;
[1] T. Altan, S.I. Oh, H. Gegel, Metal Forming Fundamentals and Applications, American Society for Metals, Materials Park, OH, 1983. [2] J.P. Domblesky, R. Shivpuri, B. Painter, Application of the finite-element method to the radial forging of large diameter tubes, J. Mater. Process. Technol. 49 (1995) 57–74. [3] G.D. Lahoti, T. Altan, Analysis of the radial forging process for manufacturing of rods and tubes, J. Eng. Ind. 98 (1976) 265–271. [4] G.D. Lahoti, P.V. Dembowski, T. Altan, Radial forging of tubes and rods with compound-angle dies, in: Proceedings of NAMRC-IV, Columbus, OH, May 17–19, 1976. [5] A. Ghaei, M.R. Movahhedy, A. Karimi Taheri, Study of the effects of die geometry on deformation in the radial forging process, J. Mater. Process. Technol. 170 (2005) 156–163. [6] T.L. Subramanian, R. Venkateshwar, G.D. Lahoti, F.M. Lee, Experimental and computer modeling of die cavity fill in radial forging of rifling, process modelling—fundamentals and applications to metals, Proceedings of Process Modeling Sessions, 1978 and 1979, USA. [7] S. Yang, Research into GFM forging machine, J. Mater. Process. Technol. 28 (1991) 307–319. [8] D.Y. Jang, J.H. Liou, Study of stress development in axi-symmetric products processed by radial forging using a 3-D finite-element method, J. Mater. Process. Technol. 74 (1998) 74–82. [9] A. Ameli, Finite element simulation of the radial forging process, M.Sc. Thesis, Sharif University of Technology, 2004. [10] S.P. Bourkine, N.A. Babailov, Y.N. Loginov, V.V. Shimov, Energy analysis of a through-put radial forging machine, J. Mater. Process. Technol. 86 (1999) 291–299. [11] E.G. Thompson, O. Hamzeh, L.A. Jackman, S.K. Srivatsa, A quasi-steadystate analysis for radial forging, J. Mater. Process. Technol. 34 (1992) 1–8. [12] J.P. Domblesky, R. Shivpuri, M. Mohamdein, FEM simulation of multiple pass radial forging of pyromet, in: Proceeding of the International Symposium, Pittsburg, June 26–29, 1994. [13] A. Uhlig, Investigation of the motions and the forces in radial swaging, Doctorial dissertation, Technical University Hannover, 1964.
Prediction of neutral plane and effects of the process parameters in radial forging using an upper bound solution M. Sanjari a , A. Karimi Taheri a,∗ , A. Ghaei b a
Department of Materials Science and Engineering, Sharif University of Technology, P.O. Box 11365-9866, Tehran, Iran b Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Received 31 May 2006; accepted 14 December 2006
Abstract Radial forging is an open die forging process used for reducing the diameter of shafts, tubes and axels. In the present research, a new velocity field was developed to analyze the radial forging process by the upper bound method. The forging load was predicted by minimizing the deformation power with regards to the location of neutral plane. The accuracy of the model was verified by comparing the predicted forging loads with experimental results published by the previous researcher. A good agreement was found between the two sets of results. © 2007 Elsevier B.V. All rights reserved. Keywords: Radial forging; Upper bound method; Analysis; Location of neutral plane
1. Introduction Radial forging is a cost-effective and material saving forming process for reducing the cross-section of rods, tubes and shafts made of any metal being suitable for the cold forming process. The advantages of radial forging are smooth surface finish, considerable material or weight savings, preferred fiber structure, minimum notch effect and increased material strength [1]. In this process, two pairs instead of one pair of opposing dies are used to deform the work piece by a number of short-stroke side-pressing operations [2]. Using slab method analysis, Lahoti et al. [3,4] analyzed the mechanics of radial forging process for single and compound angle dies. Ghaei et al. [5] also used slab method analysis to study the effects of circular die shapes on deformation in the radial forging process. Using a modular upper bound technique, Subramanian et al. [6] modeled the metal flow in die cavity in radial forging for rifling of gun barrels under plane strain condition. Yang [7] conducted a study on the radial forging process using the combination of slip-line theory and upper bound method. He used a slip-line field and a hodograph coupled with the use of a non-linear optimization technique to find the field angles and other defining parameters. Moreover, many
∗
Corresponding author. Tel.: +98 21 66165220; fax: +98 21 66005717. E-mail address: [email protected] (A. Karimi Taheri).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.12.029
researchers have used FEM to model the radial forging process [8–13]. One of the important parameters in the radial forging process is the location of neutral plane. Plastic deformation is not feasible if the neutral plane lies outside the zone of deformation, since the material would simply slip away from the radially approaching hammer dies without any deformation taking place [3]. Although, Lahoti and Altan [3] studied the effect of process parameter, such as the die angle, reduction in area and friction coefficient on the location of neutral plane by the slab method analysis, but surprisingly a detailed and complete upper bound analysis does not yet exist in the literature to predict the position of neutral plane and its effect on the forging load. In present work, by using the upper bound theory and minimizing the deformation power with regards to the location of the neutral plane both the forging load and the location of neutral plane are predicted.
2. Analysis In Fig. 1, the general model of the radial forging process for tubes, considered in this study, is illustrated. It is assumed that in the innermost position of the hammer dies, the inlet section of the dies form a perfect conical surface and the die land is perfectly cylindrical. Thus, the small clearances existing between the hammer dies at the end of a forging stroke are neglected. Based on this
148
M. Sanjari et al. / Journal of Materials Processing Technology 186 (2007) 147–153
Nomenclature A As1 , As2 dv Fr k L1 L2 L3 m R Rm Rn R0 R1
cross-section area of the element shear surfaces (Fig. 5) the element volume radial force yield shear stress of the part length of the sinking zone length of the forging zone length of the sizing zone friction shear factor outer radius of the element radius of the mandrel outer radius of tube at the neutral plane outer radius of the preform outer radius of the element at sinking–forging zones interface R2 outer radius of the product t0 initial wall thickness of the preform t final wall thickness of the element Vr absolute radial velocity of dies Vsli relative velocity between the part and die in the sinking zone Vx velocity in the X-direction in the sinking zone velocity in the X-direction at the beginning of Vx0 sinking zone Vx1 velocity in the X-direction at the sinking–forging zones interfaces Vz axial velocity in the forging zone Vz1 axial velocity in the sinking–forging zones interface Vz2 axial velocity in the forging–sizing zones interface Vz3 axial velocity at the end of sizing zone Wfd1 rate of friction work at the part–die interface in the sinking zone Wfd2 rate of friction work at the part–die interface in the forging zone Wfd3 rate of friction work at the part–die interface in the sizing zone Wfm2 rate of friction work at the part–mandrel interface in the forging zone Wfm3 rate of friction work at the part–mandrel interface in the sizing zone Wp1 , Wp2 rate of plastic work in the sinking and forging zones, respectively Wp3 rate of plastic work in the sizing zone Ws1 , Ws2 rate of shear work over the surfaces As1 and As2 , respectively Greek symbols α die angle εx strain rate in the X-direction εy strain rate in the Y-direction ε¯˙ effective strain rate σ¯ flow stress of the material in compression
Fig. 1. Schematic representation of radial forging for tubes and variables.
assumption, no material spreading would be allowed between the dies at the end of the forging strike and the deformation could be modeled axi-symmetrically. There is a small gap between the mandrel and the inner surface of the billet. In other words, forging under the conical portion of the dies is accompanied by a certain amount of sinking. This sinking causes a longitudinal back-push applied on the billet at the radial plane where the inner surface of the tube first touches the mandrel. In addition, the hammer dies also contain a cylindrical sizing zone. Thus, nearly all the deformation occurs in the conical inlet section, i.e., in the forging zone shown in Fig. 1. In the sizing zone, most of the tube material is elastically deformed to the yield limit and only a small amount of plastic deformation takes place. Nevertheless, a considerable amount of energy, supplied by the dies, is needed for this operation. As shown in Fig. 1, there are three distinct regions of deformation in radial forging of tubes: (a) the sinking zone, (b) the forging zone and (c) the sizing zone. In the following analysis of the process, all three zones are considered. It is obvious that in some cases, such as in finish forging of tubes, the sinking zone may be very small and its contribution to the overall radial forging load can be excluded in the final evaluation [3]. Depending on the inlet cone angle, the friction at the tube–mandrel and tube–die interfaces, the axial push and pull forces and the length of die land, the deforming material may flow towards the product or towards the preform or both directions simultaneously. Thus, in most general case a neutral plane exists somewhere within the deforming tube. The material being exactly on this plane is only deformed radially and does not flow axially. On both sides of the neutral plane the metal is deformed radially as well as axially, so it flows away from neutral plane in both directions, towards either the preform or the product. Theoretically, the neutral plane could lie in any of the three zones of deformation. However, it is usually located in the forging zone in most of practical conditions [3]. In the upper bound method, the total power required to deform the workpiece is divided into three components: (a) the rate of plastic work or rate of strain energy, (b) the rate of friction work and (c) the rate of shear work which is dissipated when the material flow direction is changed. In the present analysis, the total power required for the process is obtained using the following assumptions: (1) the material is rigid plastic; (2) friction at the die–tube and mandrel–tube interfaces produces a constant friction shear stress;
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(3) the wall thickness of the tube remains constant throughout the sinking zone; (4) the strain rate is homogenous in each element; (5) there are no front-pull and back-push forces; (6) neutral plane is located in the forging zone. 2.1. Analysis of forging zone As the dies are striking, the outer radius of the billet is reduced and the material flows axially. Strain rates in the radial forging are influenced by the location of neutral plane. Theoretically, the neutral plane could lie in any of the three zones of deformation. However, it is usually located in the forging zone in most of practical conditions [3]. Therefore, it is assumed that the neutral plane is located in the forging zone. Using the condition of volume constancy, the axial velocity for the forging zone is given by: 2 R − R2n Vr Vz = (1) tan α R2 − R2m Because of symmetry, the angular velocity component is taken as zero and it is sufficiently accurate to take the axial strain rate as the effective strain rate. Therefore, the effective strain rate is given by: ε˙ zz = 2Vr R
(R2n − R2m )
(2)
2
(R2 − R2m )
Regarding Fig. 2, the following geometrical relations are derived: dR = −dz tan α dv = π(R
2
(3)
− R2m ) dz
(4)
The rate of plastic work and the rate of friction work at the mandrel–tube and die–tube can be calculated as follows: 2 − R2 ¯ 2n − R2m ) σ(R R πV r m 1 ˙ p2 = σ¯ ε¯˙ dv = (5) ln W tan α R22 − R2m ˙ fm2 = W
(mk) (Vz ) (2πRm ) dz
=
Rn
− R2n −dR (2πR ) m R2 − R2m tan α R2
Vr tan α R1 R 2 Vr R2 − R2n −dR + (2πRm ) (mk) tan α R2 − R2m tan α Rn (mk)
Fig. 2. Schematic representation of forging zone.
(6)
Fig. 3. Schematic representation of sizing zone.
Vz dz (mk) (2πR) cos α cos α Rn 2 −dR Vr R − R2n = (2πR) (mk) sin α R2 − R2m sin α R1 Rn −dR Vr R2 − R2n + (2πR) (mk) sin α R2 − R2m sin α R1
˙ fd2 = W
(7)
2.2. Analysis of sizing zone In the sizing zone, the outer diameter of deforming part has reached to the diameter of final product (Fig. 3). However, due to spring back of tube material and axial feed per stroke the material is deformed almost to the yield point in the last moment of stroke. Moreover, the dissipated energy by friction at the tube–mandrel and tube–die is significant. In the start of sizing zone, it is sufficiently accurate to take the axial velocity equal to that at the exit point of forging zone. As a consequence of volume constancy in the last moment of stoke, the radial velocity induces an axial velocity in the sizing zone: 2Vr R2 L3 R22 − R2m Vz at z = L2 ⇒ Vz = Vz2 Vz at z = L2 + L3 ⇒ Vz = Vz3 + Vz2 Vz3 =
(8)
Assuming that the change of axial velocity is linear from Vz2 to Vz3 for any axial location the following relationship is derived. Vz3 (z − L2 ) + Vz3 Vz = (9) L3 It is sufficiently accurate to take the axial strain rate as the effective strain rate as follows: Vz3 L3 ¯ 22 − R2m ) R21 − R2m πVr σ(R ln = σ¯ ε¯˙ dv = tan α R22 − R2m
ε˙ zz =
(10)
˙ p3 W
(11)
˙ fm3 = (mk) (Vz ) (2πRm ) L3 W
(12)
˙ fm3 = (mk) (Vz ) (2πR2 ) L3 W
(13)
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Fig. 5. Material flow direction and shear surfaces.
Referring to Fig. 4, the sliding velocity or relative velocity between the die and tube in the sinking zone is given by:
Fig. 4. Schematic representation of sinking zone.
2.3. Analysis of sinking zone As the dies are striking, both of the outer and inner surfaces of tube are reduced. The wall thickness of tube is slightly increased, especially when the ratio of thickness to outer diameter of tube is small. Therefore, it is acceptable to neglect changes in the wall thickness of tube, i.e., material flows only in the X-direction in Fig. 4. Referring to this figure and using the condition of volume constancy, the velocity field of material flow in the X-direction can be written as: Vx =
Vx1 (2R1 − t0 cos α) sin α + 2Vr (R − R1 ) cos2 α (2R − t0 cos α) sin α
(14)
The following geometrical relation is derived from Fig. 4: R = R1 + x sin α
Vsli = Vx + Vr sin α
(20)
Thus, the rate of friction work in the sinking zone is: ˙ fd1 = (mk) (Vsli ) (2πR) dx W
(21)
2.4. Rate of shear work Considering the flow pattern depicted in Fig. 5, the material changes its flow direction at the start and end of the sinking zone, i.e., on As1 and As2 surfaces. Therefore, two velocity discontinuities occur on As1 and As2 surfaces. The associated hodographs are shown in Fig. 6. The rates of energy dissipation on these surfaces are given by: ˙ s1 = kAs1 Vz1 W tan α
˙ s2 = kAs2 2Vx0 sin α W 2
(15)
where
Since variation in the wall thickness of tube is disregarded in the sinking zone, the strain rate in the Y-direction is zero. Therefore, the strain rate in the X-direction and effective strain rate in the sinking zone are given by:
As1 =
∂Vx (2Vr ε˙ x = = ∂x −
cos2 α)(2R
+ 2x sin α − t0 cos α) [(2R − t0 cos α) sin α]2 1
(2 sin α)(Vx1 (2R1 −t0 cos α) sin α+2Vr (R−R1 ) cos2 α) [(2R − t0 cos α) sin α]2 (16)
2 ε¯˙ = √ ε˙ x 3 2R dv = πt0 − t0 dx cos α Thus, the rate of plastic work is given by: L1 ˙p = W σ¯ ε¯˙ dv
(17) (18)
(22) (23)
πt0 t0 2Rm + cos α cos α
(24)
πt0 (2R0 − t0 ) cos(α/2)
(25)
and As2 =
Finally, the total power is obtained by adding all rate of works calculated above as a function of position of neutral plane. By minimizing the power with respect to the position of neutral plane, the location of neutral plane and the actual power required for deformation are evaluated. The radial load applied to the dies is obtained by equating the external power with the actual power: ˙ total = (W ˙ p1 + W ˙ fd1 ) + (W ˙ p2 + W ˙ fm2 + W ˙ fd2 ) W ˙ p3 + W ˙ fm3 + W ˙ fd3 ) + (W ˙ s1 + W ˙ s2 ) + (W
(26)
(19)
0
where dv = πt0 (2R − t0 cos α) x. Numerical methods may be used to calculate Eq. (18) easily.
Fig. 6. The hodographs used for the analysis: (a) surface As2 and (b) surface As1 .
M. Sanjari et al. / Journal of Materials Processing Technology 186 (2007) 147–153
Fr =
˙ total W Vr
151
(27)
3. Results and discussion The purpose of this study is to find the location of neutral plane during the radial forging process using a new upper bound solution. For this reason, the dimensions of three zones of deformation were calculated from the tool and work piece geometry and the strain components were calculated from the velocity field. To obtain the maximum force in a stroke, each component of work was described above was considered at the last step. In present research for the radial forging of tubes, the experimental results of forging loads were not available. So, the loads measured by Uhlig [13] for cold swaging of AISI 1015 steel rods and used by Lahoti and Altan [3], were utilized to verify the load predicted by the present analysis. As the analysis has been based on cold radial forging, the work hardening has a considerable effect on the billet flow stress. For this purpose, a relation of the type σ¯ = K¯εn was assumed for the billet material. The K and n values at room temperature were evaluated as [1]: K = 618.14 MPa,
Fig. 7. Rate of work vs. the location of neutral plan.
n = 0.1184.
Since the material was assumed to be rigid-plastic, an average flow stress for the preform and the tube was used in the analysis. The variables used in the analysis were: a = 4.3◦
m = 0.15,
From Eq. (26) if the rate of work is plotted versus the location of neutral plane, the minimum value of curve is the theoretical rate of work. Using Eq. (27), the load relating to each forging tools can be calculated. Fig. 7 shows this operation as an example using the data presented in Table 1. Referring to Table 1, the predicted loads show a good agreement with experiment and are 10–20% greater than the experimental load. It is interesting to note that the results show a good agreement for samples (1) and (2), where the length of forging zone and sizing zone are nearly equal. However, the predicted loads are higher when the forging zone is much smaller than the sizing zone being the case in the finishing forging [6]. According to Subramanian et al. [6], this discrepancy may be attributed to the possibility of a lower friction in the sizing zone in experiment. Table 1 also shows the location of neutral plane relating to the start of the forging zone when the diameter of the billet decreases. To verify the potential of the model for predicting the effects of process parameters, a further theoretical analysis was per-
Fig. 8. The effect of die angle on the radial load.
formed assuming that the material is AISI 1015 and changing the magnitudes of some process parameters. The effect of die angle on the location of neutral plane at the end of stroke is shown in Fig. 8. The limiting position of the neutral plane is at the entrance to or the exit from the hammer die. Plastic deformation is not feasible if the neutral plane lies outside the zone of deformation, since the material would simply slip away from the radially approaching hammer dies without any deformation taking place [3]. Fig. 3 shows that when the die
Table 1 Comparison of predicted loads with the experimental results of Ref. [3] Sample number
Billet diameter (mm)
Product diameter (mm)
Length of forging zone (mm)
Length of sizing zone (mm)
Location of the neural plane (mm)
Predicted maximum load* (kN)
Load* from experiment (kN)
1 2 3 4
15.97 15.97 15.03 13.99
13.18 13.25 13.11 13.03
18.55 18.09 12.77 6.38
18.00 18.00 18.00 18.00
15 14.75 10.35 5.03
192.5 191 152 112.5
172.00 167.00 124.00 74.50
*
Per hammer.
152
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Fig. 9. The effect of die angle on the location of neutral plane. Fig. 10. The effect of friction factor on the radial load at different die angles.
angle increases the location of neutral plane is approached to the start of the forging zone. Thus, the limitation of increasing the die angle discussed above does not allow to raising this factor so much. The effect of die angle on the radial load at the end of stroke for m = 0.15 is shown in Fig. 9. This figure reveals that as the die angle increases, the radial load decreases because at a constant reduction in area, the length of contact and the friction work are decreased. Moreover, when the die angle increases, the radial component of die normal force reduces and the axial component increases. Therefore, one may conclude that there is a limitation for increasing the die angle, because if the die angle increases excessively, the axial load becomes so large that it may make the preform to buckle or throw the part away from the forging box. Thus, as the axial force is known, one can easily determine whether the part will buckle or not. Also, it is possible to predict the magnitude of the least required back-push force preventing the slipping the part away. This force can also be used to find the maximum die angle within a certain process condition such as reduction in area, length of performs, etc. Fig. 10 shows the effect of friction factor on the location of neutral plane at the end of stroke at various die angles. This figure indicates that when the friction factor increases the location of neutral plane is approached to the start of the forging zone. Fig. 11 shows the effect of friction shear factor on the radial load at the end of stroke at various die angles. It is observed that when m is increased, the radial load is also increased due to the increase in friction works. These changes are increased when the die angle is increased. Fig. 12 shows the effect of reduction in cross-section area on the location of neutral plane at the end of stroke at various friction shear factor. Referring to this figure by increase in reduction in area the work consumed in the sizing zone increases and the location of neutral plane approaches to the sinking zone. Fig. 13 shows the effect of reduction in cross-section area on the radial load at the end of stroke at various friction factors. As can be seen, when the reduction is increased, the radial load is increased. This increase is due to the increase in plastic, friction and shear works. It has been reported that in order to enjoy
Fig. 11. The effect of friction shear factor on the location of neutral plane.
Fig. 12. The effect of reduction in cross-section area on the radial load at the end of stroke.
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(2) when the friction factor increases the location of neutral plane is approached to the start of the forging zone and the radial load is increased; (3) by increasing the reduction in area the location of neutral plane approaches to the sinking zone and radial load is increased. Acknowledgement The authors would like to thank the research board of Sharif University of Technology, Tehran, Iran, for the financial support of this work. References
Fig. 13. The effect of reduction in cross-section area on the location of neutral plane.
the benefits of radial forging process, the reduction must be so large that the plastic deformation penetrates to the core of the material [8]. Thus, it is important to choose a machine or finding the minimum number of passes when it is not possible to forge the tube in one pass. This is possible with calculating the maximum required power for deformation. Obviously, if the calculated power is less than the machine power, the tube can be produced in one pass. Otherwise, the tube should be produced in more than one pass so that the power of each passes to be less than the machine power. 4. Conclusion In this research, by using the upper bound theory and minimizing the deformation power with regards to the position of the neutral plane both the forging load and the position of neutral plane are predicted. For verification of the model, the predicted loads are compared with those of published experimental results. Also, the effect of the various process parameters is assessed by the model and compared with the published results. The following conclusions are achieved from the study: (1) when the die angle increases the location of neutral plane is approached to the start of the forging zone and the radial load is decreased;
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