DESIGN OF THE DIE PROFILE FOR THE INCREMENTAL RADIAL FORGING PROCESS *

IJST, Transactions of Mechanical Engineering, Vol. 39, No. M1, pp 89-100 Printed in The Islaic Republic of Iran, 2015 Shira University DESIGN OF THE DIE PROFILE FOR THE INCREMENTAL RADIAL FORGING PROCESS * M. AFRASIAB 1, H. AFRASIAB 2**, M.R. MOVAHHEDY 3 AND G. FARAJI 4 1,4 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, I. R. of Iran 2 Mechanical Engineering Departent, Babol University of Technology, Babol, I. R. of Iran Eail: afrasiab@nit.ac.ir 3 Center of Excellence in Design, Robotics, and Autoation, Departent of Mechanical Engineering, Sharif University of Technology, Aadi Ave., Tehran, I. R. of Iran Abstract In this paper, based on the slab ethod of analysis, a novel and general approach is developed for studying the radial forging process with curved profile dies. The presented approach is not only ore general with respect to previous studies, but it is also easier to understand and use and can be efficiently used for optiiation of the die profile depending on the forging geoetry and conditions. The obtained general equations reduce to those obtained in previous studies for the special case of linear dies. The process is also siulated by the finite eleent ethod to further enhance the results of the study. The obtained results provide useful inforation for the optial design of the radial forging die. Keywords Radial forging, slab ethod, finite eleent ethod, die profile design 1. INTRODUCTION Radial forging is a odern and cost effective process used for precision forging of round and tubular coponents, with or without an internal profile. Coponents produced by radial forging typically have good echanical and etallurgical properties including sooth surface finish, preferred fiber structure, iniu notch effect and increased aterial strength [1-3]. Deforation in radial forging results for a large nuber of short stroke and high speed pressing operations by four haer dies, while work-piece rotates and axially advances between the dies after each blow. Dies are arranged radially around the work-piece, as shown in Fig. 1. Fig. 1. Arrangeent of dies Aong paraeters affecting the deforation pattern and quality of the forged product, the die shape is of prie iportance [1, 4, 5]. Generally, the die profile shape is ade of two sections, the inlet section, which fors a conical surface and the die land, which is cylindrical, as shown in Fig. 2. Received by the editors March 10, 2014; Accepted June 18, 2014. Corresponding author

90 M. Afrasiab et al. Fig. 2. A scheatic side view of the radial forging die Using slab ethod of analysis in [4], Ghaei et al. considered linear and circular (convex, concave and hybrid) profiles for the die inlet one and studied the effects of the die shape on the deforation pattern and quality of radially forged products. In another study, Ghaei and Movahhedy [6] odeled the radial forging process by finite eleent ethod and studied the effect of the die shape in the cross-section area on the work-piece deforation. Using finite eleent analysis and icrohardness test in [1], Sanjari et al. concluded that aong dies with different linear and circular profiles, the die with a convex profile leads to a product with iniu inhoogeneity. In the above-entioned studies, the analyses were restricted to special, naely linear and circular, die profiles. However, here a generalied slab ethod analysis is presented that is capable of odeling the radial forging process with virtually any curved shape die profile. Finite eleent siulations are also perfored to further exaine the effect of the die profile on the process paraeters. This paper is organied as follows. The slab ethod forulation is developed in section 2. The odeling procedure is explained in section 3. The results of different analyses are presented and discussed in section 4. Finally, the concluding rearks are suaried in section 5. 2. THE SLAB METHOD ANALYSIS As shown in Fig. 3, three distinct regions of deforation exist in the radial forging process: (1) sinking one, (2) forging one and (3) siing one. The sinking one is a part of the conical region of the die in which the die contacts the outer surface of the tube, but the inner surface of the tube has not reached the andrel yet. In this region, both the inner and outer diaeter of the tube are reduced under the die pressure. In the forging one, the inner diaeter of the tube has reached the andrel and thus only the outer diaeter of the tube is changed under the die pressure. As a result, the aterial flows in the axial direction and the tube length increases. Most of the axial deforation happens in this region. In the siing one, both the inner and outer diaeters have alost arrived at their final sie and thus the deforation in this region is ostly elastic. The principle function of this region is to iprove the finish quality of the inner surface. Fig. 3. Different deforation ones in the radial forging process The following assuptions are ade for the slab ethod analysis: 1. The tube thickness reains constant throughout the sinking one. 2. Friction at the die-tube interface produces a constant friction shear stress. 3. The noral stress acting on the slab does not vary over the cross-section and it is also a principle stress. 4. The slab is free fro shear stresses. 5. The aterial is rigid-perfectly plastic. IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1 April 2015

Design of the die profile for the 91 a) The sinking one Figure 4 shows the stresses acting on an eleent of aterial in the sinking one. In order for the eleent to be in equilibriu, we ust have: ( d )( A da) A pa sin A cos 0 (1) 1 1 1 1 1 1 where σ 1 is the noral stress parallel to the die surface, A 1 and A 2 are the cross sectional and lateral area of the slab, p and τ are the radial pressure and shear stress due to friction, and α is the slab angle defined in Fig. 4. Fig. 4. Stresses acting on an eleent in the sinking one The following geoetrical relations can be deduced fro Fig. 4: A [ R( ) ( )] da[2 RR ( ) ( d ) 2 ( ) ( d ) ] 1 1 d A2 2 R( ) cos The ʹ sybol in this equation represents the first derivative with respect to, ρ is the slab inner radius and R is its outer radius. Substituting Eq. (2) into Eq. (1) yields: 12 [ R( ) R ( ) d ( ) ( ) d] d 1 [ R ( ) ( )] d d p[2 R( ) ]sin [2 R( ) ]cos 0 cos cos which can be siplified to: 2 [ R( R ) ( d ) ( ) ( d ) ] d [ R( ) ( )] 1 1 2 pr( )tand 2 R( ) d 0 Also fro the geoetry in Fig. 4, we can write: tan R( ) (5) This reduces Eq. (4) to: d1 2 1[ RR ( ) ( ) ( ) ( )] 2 prr ( ) ( ) 2 R ( ) (6) d [ R ( ) ( )] Next, we consider equilibriu of the eleent in the slab radial direction to obtain (see Fig. 5): P R( ) ( ) (7) R ( ) 3 (2) (3) (4) April 2015 IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1

92 M. Afrasiab et al. Fig. 5. Free body diagra of an eleent in the sinking one Where σ 3 is the hoop stress on the slab as shown in Fig. 5. In the sinking region, the radial pressure on the die is sall copared with the principal longitudinal and circuferential stresses, and therefore the approxiate yield condition is given by: 13 (8) where is the yield stress of the workpiece aterial. Solving Eqs. (7) and (8) for p yields: R( ) ( ) p ( 1) (9) R ( ) Substituting p fro Eq. (9) into Eq. (6) gives: d1 2 1 ()[ () R()] 2 R()[ R() ()] 2 1R() d [ R ( ) ( )] [ R ( ) ( )] [ R ( ) ( )] (10) Using the reasonable assuption that the inner profile of the tube ρ() is parallel to its outer profile R(), we will have: R() () (11) Substituting Eq. (11) into Eq. (10) gives: d1 2 R()[ R() ()] 2 R() d R R [ ( ) ( )] [ ( ) ( )] (12) In the analysis of the radial forging process, it is norally assued that the frictional shear stress varies according to the sticking condition [4, 6, 7]: (13) 3 where is the constant friction factor. Applying this assuption to Eq. (12) leads to: d1 2 R( ) R( )[ R( ) ( )] ( ) (14) d [ R ( ) ( )] R( ) 3 This equation gives the variation of the axial stress σ 1 in the sinking one with respect to the axial distance. Using the derivative chain rule, we can write: d1 1 d1 (15) dr R ( ) d IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1 April 2015

Design of the die profile for the Therefore, the σ 1 variation along the radial distance R is as follows: d1 2 R R[ R ] ( ) dr R R R [ ] 3 93 (16) Equation (16) holds for a die that has a general curved shape profile. In the special case of the linear die with the inlet angle of α, we have: R tan (17) Since the tube thickness t 0 is assued constant in the sinking one, for a linear die we can write: R t0 cos (18) R t cos (2Rt cos ) 0 0 Substituting Eqs. (17) and (18) into Eq. (16) gives: d1 2 R t0 ( ) dr t (2R t cos ) R 3sin 0 0 which is the equation obtained previously by Lahoti and Altan in [7]. The corresponding radial pressure in the sinking one is obtained by using Eq. (9). b) The forging one According to Fig. 6, the force equilibriu of an eleent of aterial in the forging one requires that: ( d )( A da ) A p A sin A cos A 0 (20) 1 r 1 r 2 where σ is the axial stress, A and A r are the cross sectional and lateral area of the slab, p 1 and τ 1 are the radial pressure and shear stress due to friction on the die-workpiece interface and τ 2 is the shear stress due to friction on the andrel-workpiece interface. This can be siplified to: where: da d A p A sin A cos A 0 (21) 1 r 1 r 2 A R R da R R d [ ( ) ] 2 ( ) ( ) (19) d Ar 2 R( ) cos A 2 R d (22) Fig. 6. Stresses acting on an eleent in the forging one in which R is the andrel radius. Substituting Eq. (22) into Eq. (21) gives: April 2015 IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1

94 M. Afrasiab et al. 2 RR ( ) ( d ) d [ R( ) R ] d d p1[2 R( ) ]sin 1[2 R( ) ]cos 2[2 Rd] 0 cos cos (23) which is siplified to: 2 RR ( ) ( d ) d[ R( ) R ] 2 pr( )tand2 R( ) d2 R d0 1 1 2 (24) Also, fro the geoetry shown in Fig. 6, it follows that: tan R( ) (25) Substituting this relation into Eq. (24) yields: d 2 R( R ) ( ) 2 prr ( ) ( ) 2 R ( ) 2 R d [ R ( ) R ] 1 1 2 (26) Now, we consider the equilibriu of the eleent in the radial direction. According to Fig. 7, the following relationship is obtained: [ R ( ) R] R ( ) R (27) r1 r2 Where σ r1 and σ r2 are radial stresses at the die and andrel surfaces, respectively, and σ θ is the circuferential stress. Fig. 7. Free body diagra of an eleent in the forging one By neglecting the radial stress variation through the eleent, we can assue that: Applying Eq. (28) to Eq. (27) results in: (28) r1 r2 r Therefore, the yield condition in the forging one becoes: [ R ( ) R] [ R ( ) R] (29) r r (30) r Considering the stress boundary condition at the tube-die interface yields: IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1 April 2015

Design of the die profile for the 95 p1 r 1 R( ) (31) Substituting σ r fro Eq. (30) into Eq. (31), the following relationship is obtained: Using this equation, Eq. (26) changes to: p ( ) R( ) (32) 1 1 d 1 2 { 2 R( R ) ( ) 2 1R ( )[1 R ( )] 2 2R} d [ R ( ) R ] Assuing constant friction factor at the tube-die and tube-andrel interfaces gives: 1 2 1, 2 (34) 3 3 Applying this relation to Eq. (33), after soe siplification, results in: d 2 R ( ) 1 2 R { ( ) [1 ( )] 2 R R } (35) d [ R ( ) R ] 3 3 R( ) This equation gives the variation of axial stress σ in the forging one with respect to axial distance. For obtaining the variation of axial stress σ along the radial distance R, the chain rule is used according to Eq. (15). This way, we will have: d 2 R 1 2 R { [1 ] 2 R R } (36) dr R[ R R ] 3 3 R This equation holds for a die with an arbitrary curved shape profile. For the special case of linear dies, we use Eq. (17) to obtain: d 2 R 1 2 R {tan [1 tan ] 2 } dr tan [ R R ] 3 (37) 3 R Siplifying this equation, we get the result obtained previously by Lahoti and Altan in [7]: (33) d [1 tan ] 2R 2R {(1 ) } dr tan [ R R ] [ R R ] 2 1 2 3 3tan (38) The radial pressure on the die surface is equal to radial stress σ r : c) The siing one pforging r (39) Only a sall portion of the aterial is plastically defored in the siing one. However, all the aterial is elastically defored to the yield point. Thus, it will be assued that the yield condition given by Eq. (30) is also satisfied in the siing one. Considering the equilibriu of forces acting on an eleent in the siing one gives (see Fig. 8): d 2 ( R R ) d R R t 0, L L 1 2 2 3 3 ( 2 ) 1 (40) April 2015 IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1

96 M. Afrasiab et al. Fig. 8. Stresses acting on an eleent in the siing one Integrating Eq. (40) yields: 2 ( R R ) ( ), L L 1 e e 3 ( R2 R ) t1 2 3 (41) where σ =σ e at = e. When e =L 3 we have σ e =σ f, where σ f is the front pull per unit area, and when e =L 2, σ e is the axial stress obtained fro the analysis of the forging one. Using Eq. (30), the radial pressure in the siing one is: p (42) Siing 3. FINITE ELEMENT MODELING PROCEDURE If we ignore the sall gap between dies at the end of their stroke, the tube loadings and boundary conditions ay be a function of r and only (and not a function of θ). This eans that the proble can be odeled as axisyetric to reduce the coputational tie. The finite eleent analyses were perfored using a verified FEM code previously used in [1, 4, 6, 8]. Since the deflection of the die and andrel is very sall copared to that of the deforing tube, it is accurate enough to assue that they are rigid bodies. The aterial was assued to be perfectly plastic with a yield point of 120MPa [4]. To odel the friction in the contact surfaces, the penalty forulation was used. The liiting shear stress was obtained by 3, where = 0.15 is the constant friction factor coonly used for cold forging conditions [7]. The process paraeters, including tube, die and andrel geoetry are presented in Table 1. Table 1. Tube, die and andrel geoetry () Outer radius of the prefor 100 Thickness of the prefor 7 Outer radius of the product 90 Thickness of the product 4 Length of the die inlet 71 Length of the die land 10 Radius of the andrel 86 IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1 April 2015

Design of the die profile for the 97 4. RESULTS a) Coparison of finite eleent and slab ethod results The slab ethod equations obtained in Sec. 2 were solved nuerically using an algorith based on the Runge-Kutta schee. Figure 9 copares the radial pressure distribution obtained by the slab ethod and the finite eleent analysis for a linear die with inlet angle of 8. Fig. 9. Coparison of the results of slab ethod and FEM The difference between pressures in the sinking one stes fro the bending of the tube in the die inlet that is not considered in the slab ethod [4]. In practice, during the dies ipact, the tube is bent a little and the aterial is entered to the sinking one. However, in the slab ethod analysis, it was assued that the aterial flows sharply to the sinking one. This bending causes an easier aterial flow to the sinking one and less radial pressure at this one as predicted by the finite eleent analysis. Another source of discrepancy between FEM and slab ethod results can be attributed to the changes in the radial pressure along the thickness of the tube that was assued to be constant in the slab ethod analysis [4]. b) Effect of the die profile shape on the product quality In order to study the effect of the die profile shape on the product quality, three different profiles were considered for the die, naely linear, quadratic and cubic order profiles, as shown in Figs. 10, 11 and 12, respectively. Fig. 10. Die with linear profile Fig. 11. Die with quadratic (2 nd order) profile Fig. 12. Die with cubic (3 rd order) profile April 2015 IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1

98 M. Afrasiab et al. In the case of curved shape dies, to achieve a ore hoogenous structure in the product, it is recoended to use a profile in which the slope gradually changes to ero (equal to the slope of the siing one) at the end of forging one [4]. Therefore, the two eployed curved profiles are chosen to satisfy this recoendation. The residual stress distribution was selected as the easure of the product quality since it is an indication of the degree of the deforation hoogeneity. In other words, the ore hoogenous the deforation, the saller the residual stress [6]. Furtherore, tensile residual stresses ay lead to the opening of cracks which accelerates the failing of the tube while copressive stresses close the cracks and increase the tube life. Moreover, these stresses have a fundaental effect on the diensional stability, wear resistance and fatigue life of the tube and are treated as one of the ost iportant paraeters of the surface quality. Figure 13 copares the finite eleent solution for the residual Von-Mises stress distribution, i.e. the stress distribution after copletion of the entire radial forging process [9] in the inner surface of the tube for different die profiles. As this figure shows, the tube tensile residual stress is iniu for the die with the quadratic profile while it is axiu for the die with the linear profile. Furtherore, the tube residual stress distribution is ore unifor for dies with curved shape profiles. In these cases, the gradual change of the profile slope leads to ore steady and hoogenous deforation and flow of the tube between the die and andrel, producing ore favorable residual stress distribution in the tube. Fig. 13. Residual stress distribution for different die profiles b) Effect of the die profile shape on the radial pressure distribution Figure 14 shows the radial pressure distribution along the forging axis for the three different die profile shapes. Fig. 14. Effect of the die profile shape on the radial pressure distribution IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1 April 2015

Design of the die profile for the 99 As illustrated in this figure, the die with quadratic order profile creates the largest radial pressure and consequently requires the largest forging force and energy. Furtherore, the position of the neutral plane oves towards the prefor in this case. Therefore, either saller axial force feed or saller reduction in area should be considered in order to prevent overloading of achine. On the other hand, the cubic order curve creates a saller radial pressure on the die while the position of the neutral plane does not change significantly in this case copared with the linear die. Since the product quality (the residual pressure distribution shown in Fig. 13) is also desirable for the cubic order profile, it ay be an optial option for the radial forging die. It is to be noted that while finding the ost optial die profile is not the concern of this paper, the relations obtained in Section 2 can be easily used for further optiiation of the die profile shape depending on the forging geoetry and conditions. 5. CONCLUSION The slab ethod of analysis was used to investigate the radial forging of tubes with curved profile dies. The presented approach is not only ore general with respect to previous studies, but it is also easier to understand and use and can be efficiently used for optiiation of the die profile depending on the forging geoetry and conditions. The obtained general equations reduce to those obtained in previous studies for the special case of linear dies. Good agreeent was observed between the results of the slab ethod and finite eleent analysis. It was shown that, the die with quadratic order profile provides the best residual stress distribution copared with the linear and cubic order dies, but it requires the largest forging force and energy. The cubic order die, on the other hand, provides a good residual stress distribution in the product and requires a saller force and energy. REFERENCES 1. Sanjari, M., Saidi, P., Karii Taheri, A. & Hossein-Zadeh, M. (2012). Deterination of strain field and heterogeneity in radial forging of tube using finite eleent ethod and icrohardness test. Materials and Design, Vol. 38, pp. 147 153. 2. Sahoo, A. K., Tiwari, M. K. & Mileha, A. R. (2008). Six siga based approach to optiie radial forging operation variables. Journal of Materials Processing Technology, Vol. 202, pp. 125 136. 3. Sanjari, M., Karii Taheri, A. & Ghaei, A. (2007). Prediction of neutral plane and effects of the process paraeters in radial forging using an upper bound solution. Journal of Materials Processing Technology, Vol. 186, pp. 147 153. 4. Ghaei, A., Movahhedy, M. R. & Karii Taheri, A. (2005). Study of the effects of die geoetry on deforation in the radial forging process. Journal of Materials Processing Technology, Vol. 170, pp. 156 163. 5. Khaleed, H. M. T., Saad, Z., Mujeebu, M. A., Badaruddin, A., Badruddin, I. A., Abdullah, A. B. & Salan Ahed, N. J. (2012). FEM siulation and experiental validation of flash-less cold forging for producing AUV propeller blade. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, Vol. 36, No. M1, pp. 1-12. 6. Ghaei, A. & Movahhedy, M. R. (2007). Die design for the radial forging process using 3D FEM. Journal of Materials Processing Technology, Vol. 182, pp. 534 539. 7. Lahoti, G. D. & Altan, T. (1976). Analysis of the radial forging process for anufacturing of rods and tubes. Journal of Engineering for Industry, Vol. 98, pp. 265-271. April 2015 IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1

100 M. Afrasiab et al. 8. Ghaei, A., Movahhedy, M. R. & Karii Taheri, A. (2008). Finite eleent odelling siulation of radial forging of tubes without andrel. Materials and Design, Vol. 29, pp. 867 872. 9. Jang, D. Y. & Liou, J. H. (1998). Study of stress developent in axi-syetric products processed by radial forging using a 3-D non-linear finite-eleent ethod. Journal of Materials Processing Technology, Vol. 74, pp. 74 82. IJST, Transactions of Mechanical Engineering, Volue 39, Nuber M1 April 2015