j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 1 ( 2 0 0 8 ) 128–132

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Inhomogeneous deformation and residual stress inskin-pass axisymmetric drawing

Heng-Sheng Lin ∗, Yuan-Chuan Hsu, Chia-Chow KehDepartment of Mechanical and Computer-Aided Engineering, National Formosa University, Hu-Wei, Yunlin 632, Taiwan

a r t i c l e i n f o

Keywords:

Skin-pass

Axisymmetric drawing

Inhomogeneous deformation

Residual stress

a b s t r a c t

Skin-pass drawing is usually conducted in the final stage of wire, rod or shaped-section

drawing or shape rolling. It helps improve shape accuracy and reduce residual stress of

the drawn or rolled workpieces. However, its characteristic low reduction induces large

�-parameter, and causes significant inhomogeneous deformation in the workpiece. In

this work, finite-element software DEFORM 2D was utilized to investigate the effect of �-

parameter on inhomogeneous deformation and residual stress of mid-carbon steel through

various combinations of area reduction and die semi-angle. The effect of the friction was

also investigated. The results showed that the inhomogeneity factor of effective strain

decreased with area reduction and increased with die semi-angle and friction. The level

of inhomogeneous deformation could be reduced by drawing with small die semi-angle,

and the influence of the friction was only substantial near the surface of the drawn work-

piece. The axial and circumferential residual stresses would approach to compressive on

the drawn surface in extreme light reductions.

Hc

1. Introduction

Skin-pass drawing is usually conducted in the final stage ofwire, rod or shaped-section drawing or shape rolling. It helpsimprove shape accuracy and reduce residual stress of drawnor rolled workpieces. However, its characteristic low reductioninduces large �-parameter, and causes significant inhomo-geneous deformation in the workpiece. The �-parameter isdefined as

� = h

L(1)

where h is the mean diameter of the workpiece and L is the

contact length between the deforming workpiece and the die.The �-parameter, though not perfectly, is a measure of theinfluence of the deformation geometry upon the drawn work-piece (Hosford and Caddell, 1983). For axisymmetric drawing,

∗ Corresponding author. Tel.: +886 5631 5311; fax: +886 5 631 5310.E-mail address: hslin@nfu.edu.tw (H.-S. Lin).

0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2007.11.126

© 2007 Elsevier B.V. All rights reserved.

� can be expressed as

� = sin ˛

r(1 + √

1 − r)2

(2)

where � is the die semi-angle, r is the reduction of thecross-sectional area. Eq. (2) indicates that � increases when˛ increases or r decreases. However, the small reduction usedin skin-pass drawing intuitively bears a relatively large � valueand impairs its goal of improving shape accuracy.

Backofen (1972) defines a hardness inhomogeneity factorto evaluate the level of inhomogeneous deformation as

IF = Hs − Hc (3)

where Hs and Hc are the Vickers hardness at the surface andcenter, respectively. This definition is suitable for experimen-tal approach, whereas in the FE simulation the distribution

t e c

osoo

I

wtssitaaTdiae

Ebaafefmoaw

�

wtoai

ε

rnrswpb

2

Fsi

which yields the minimum drawing stress is the optimum diesemi-angle of the respective reduction. The optimum anglegradually increases from 4 to 6◦ as the reduction increasesfrom 2 to 10%. Same trends had been reported by Avitzur (1990)

j o u r n a l o f m a t e r i a l s p r o c e s s i n g

f hardness is not available. Since hardness is caused by thetraining of the material, and in order to facilitate the analysisf the data obtained from FE simulation, Backofen’s definitionf inhomogeneity factor is further modified as

Fε = εs − εc

εs(4)

here εs and εc are the effective strain at the surface and cen-er, respectively. The denominator is replaced by the effectivetrain at the surface because the magnitude of the effectivetrain might approach to very small value for skin-pass draw-ng and cause the inhomogeneity factor of strain to approacho a quite large value. A larger IFε represents a larger discrep-ncy between the effective strain of the surface and center, i.e.,drawing condition with more inhomogeneous deformation.hrough the definition of Eq. (4), the level of inhomogeneouseformation incurred by the process parameters can be read-

ly analyzed by extracting the effective strain of the surfacend center out of the FE simulation, without consorting to thexperimental measurement of hardness.

The definition of the inhomogeneity factor of hardness,q. (3), only takes into account the discrepancy of hardnessetween the surface and center of the drawn workpiece. It alsossumes that the maximum hardness occurs on the surfacend the minimum hardness occurs at the center. However,rom the FE simulation of this work, it was observed that theffective strain does not necessarily increase monotonicallyrom the center to the surface. The maximum effective strain

ight occur somewhere near the surface. Therefore a morebjective definition, the mean variation of effective strain, waslso applied to assess the level of inhomogeneous deformationhich is defined as

ε̄ =∑n

i=1|εi − ε̄|r2i

− r2i−1

r2f

(5)

here ri is the radius of the ith point, and i = 0 corresponds tohe center and i = n corresponds to the surface; rf is the radiusf the drawn workpiece. εi is the effective strain at point i,nd ε̄ is the mean effective strain of the cross-section whichs defined as

¯ =∑n

i=1εi(r2i

− r2i−1)

r2f

(6)

Up to now, there has been abundant work reported forod or wire drawing of medium reductions. However limitedumber of work was devoted to the drawing process of lighteductions. This work focuses on investigating the relation-hip of the inhomogeneous deformation of effective strain asell as residual stress with the area reduction of the work-iece, the die semi-angle and workpiece/die friction conditiony finite-element simulation.

. Finite-element simulation

inite-element software DEFORM was used in simulating thekin-pass drawing process. Since the process is axisymmetricn nature, the 2D Version 8.0 axisymmetric module was cho-

h n o l o g y 2 0 1 ( 2 0 0 8 ) 128–132 129

sen. Mid-carbon steel AISI-1045 was used as the workpiecematerial, and the built-in stress–strain relationship of elasto-plasticity at 20 ◦C with yield strength 640 MPa was selected.Heat transfer effect was not considered. There were 5000 ele-ments used in meshing the workpiece, and finer meshes wereconstructed close to the surface in order to better scope the“skin-pass” process. Billet length was 20 mm and the outletdiameter of the drawing die was 10 mm. Reduction of cross-section selected was from 2 to 10%. Converging die was usedand assumed to be rigid. Values of the die semi-angle are from2 to 20◦. Die bearing length was 4 mm and the radius of the fil-let joining the bearing and the converging zone was 0.2 mm.The drawing speed was fixed at 67 mm/s. The frictional law ofconstant shear strength was used and friction factors 0.1, 0.3and 0.5 were selected. Usually drawing with a small frictionfactor corresponds to a forming condition of better lubrica-tion at the workpiece/die interface and vice versa. Drawingwith the billet is unsteady at the beginning and near the endof the process. Steady values appear at the middle of the billetand satisfactory data acquisition of the drawing load, cross-sectional distributions of the effective strain and the residualstress can be obtained.

3. Results and discussion

3.1. Optimum die semi-angle

Though minimizing drawing stress is not the primary consid-eration of skin-pass drawing, the evaluation of the optimumdie semi-angle is still essential in understanding the char-acteristics of the process. Fig. 1 shows the variation of thenormalized drawing stress with die semi-angle with areareductions ranging from 2 to 10%. The normalized drawingstress was obtained by dividing the drawing load with theoutlet area of the drawn workpiece and normalized with theyield strength of workpiece. The friction factor was 0.1. Fora selected reduction, the drawing stress first decreases andthen increases with die semi-angle, as a consequence that thefrictional work decreases and the redundant work increaseswith die semi-angle (Hosford and Caddell, 1983). The angle

Fig. 1 – Variation of drawing stress with die semi-angle forvarious reductions.

130 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 1 ( 2 0 0 8 ) 128–132

Fig. 4 – Distributions of effective strain for variousreductions and friction factors.

Fig. 2 – The influence of friction factor on the optimumangle.

with larger reductions and incurring with larger optimum diesemi-angles.

The influence of friction on the optimum angle is shown inFig. 2 with friction factors of 0.1, 0.3 and 0.5 and for area reduc-tions of 2 and 10%, respectively. A significant increase of theoptimum angle from 6 to 8 and 10◦ is observed in drawing withthe higher reduction of 10%, whereas no significant increaseof optimum angles is observed for the smaller reduction of2%. In drawing with relatively low reduction, the decreaseof the frictional work with die semi-angle becomes less sig-nificant because the contact length between the deformingworkpiece and the die becomes relatively small. Therefore,the result suggests, on the basis of minimizing drawing stress,using a larger die semi-angle when the lubrication conditionis poor, especially in drawing with larger reductions. However,the optimum angle remains unchanged with the lubricationcondition in skin-pass drawing.

3.2. Inhomogeneous deformation

3.2.1. Distribution of effective strainFig. 3 illustrates the distribution of effective strain for 2%reduction. Friction factor was fixed at 0.1. The distributionshows that the minimum effective strain occurs at the cen-ter and its value is close to its nominal cross-sectional areareduction. The maximum effective strain occurs near the sur-face of the drawn workpiece for the selected die semi-angle.

The larger the maximum effective strain, the higher level ofinhomogeneous deformation is produced by the drawing pro-cess. The difference in effective strain among the respective

Fig. 3 – Distribution of effective strain for 2% reduction.

Fig. 5 – Variation of inhomogeneity factor with�-parameter for various die semi-angles.

die semi-angles is not significant near the center whereas nearthe surface the effective strain increases dramatically withlarger die semi-angles. Such trend becomes less substantialwhen drawing with larger reductions. However, in drawingwith smaller die semi-angles, e.g. 2 or 4◦, the increase in effec-tive strain is insignificant and the strain distribution becomesmore uniform. The level of inhomogeneous deformation ishence reduced.

The influence of friction on inhomogeneous deformationis illustrated in Fig. 4. It shows the distributions of effec-tive strain in drawing with 6 and 2% reductions, respectivelyand 8◦ die semi-angle with various friction factors. The dis-tributions indicate that the effective strain at the center isnot affected by the chosen friction factor, i.e., the lubricationcondition between the workpiece and the die. The influencebecomes significant near the surface and the effective strainis the greatest with friction factor of 0.5, followed by that of 0.3and 0.1. The slight drop in the effective strain on the surfacecan reduce the discrepancy of the effective strain between thesurface and center, and hence reduce the inhomogeneity ofdeformation.

3.2.2. Correlating inhomogeneity factor with �-parameterFig. 5 shows the inhomogeneity factor of effective strain plot-ted against �-parameter for various die semi-angles. The

definitions of the IFε and � are given by Eqs. (4) and (2),respectively. The plot shows a monotonic increase of the inho-mogeneity factor of effective strain with �-parameter for eachdie semi-angle selected. At large values of �-parameter, all the

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 1 ( 2 0 0 8 ) 128–132 131

Fig. 6 – Variation of mean variation of effective strain with�

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Fig. 7 – Distribution of axial residual stress of various diesemi-angles.

Fig. 8 – Residual stress becomes compressive on the

-parameter for various reductions.

urves converge, indicating the level of inhomogeneous defor-ation is determined solely by the �-parameter. However, at

mall values of �-parameter, the curves diverge and the devi-tion becomes more obvious especially for drawing with 2◦

ie semi-angle. This indicates an identical �-parameter wouldear different levels of inhomogeneous deformation at small.

Moreover, from the observation of the strain distributionf Fig. 3, using the discrepancy of effective strain between theurface and the center was not an objective way in determin-ng the inhomogeneous deformation because the maximumffective strain might occur somewhere near the surfacenstead of at the surface. Therefore a statistical definition, the

ean variation of effective strain, Eq. (5), is applied to assesshe level of inhomogeneous deformation. Fig. 6 shows the

ean variation of effective strain plotted against �-parameteror various reductions. The plot shows that the statistical def-nition on the contrary leads to an even obvious deviation ofnhomogeneity among each reduction selected, indicating theecessity of modifying the �-parameter in correlating defor-ation inhomogeneity with the process parameters such as

rea reductions, die semi-angles and friction factor altogether.

.3. Residual stress

esidual stress affects the geometrical precision of the drawnorkpiece. Usually a residual compressive stress is desirablen the surface because it helps increase workpiece’s ability inesisting crack propagation and consequently increase work-iece’s service life. The distribution of residual stress is alsoffected by the selection of the process parameters like reduc-ion and die semi-angle and friction as well. Fig. 7 showshe distributions of axial residual stress of various die semi-ngles at 6% reduction. The residual stress is normalized withield strength, and the simulation also showed axial stressas more dominant than the circumferential stress. The axial

esidual stress exceeds its yield strength near the center andhe surface of the drawn workpiece because of the work hard-

ning effect. Residual stress becomes compressive near theenter and tensile near the surface, and the level of resid-al stress on the surface is minimum in drawing with 2◦ dieemi-angle.

surface at light reductions.

Kuntani and Asakawa (2000) reported that axial residualstress was maximum in drawing with medium die semi-angleor medium reduction, which are the ordinary operation con-ditions in drawing. The axial residual stress would becomecompressive when the reduction is small. Fig. 8 shows theaxial residual stress on the surface plotted against the �-parameter of various friction factors in drawing with diesemi-angle 8◦. The figure shows that the axial residualstress becomes compressive at reduction less than 0.4%, andthe simulation also showed that the circumferential stressbecame compressive at reduction less than 0.3%. However,the relation between residual stress and friction factor washardly to observe. The conditions of drawing with large diesemi-angles were not simulated because of little relevancy inskin-pass drawing.

4. Conclusion

Several points can be drawn from the results of the simulationwith DEFORM 2D as follows:

(1) The evaluation of the optimum die semi-angle suggested

using a large die semi-angle at high reductions and usinga smaller angle at low reductions. When the lubricationcondition was poor, the optimum die semi-angle wouldincrease especially in drawing with larger reductions.

n g t

r

132 j o u r n a l o f m a t e r i a l s p r o c e s s i

However, the optimum angle remained unchanged withthe lubrication condition in skin-pass drawing.

(2) The minimum effective strain occurred at the center andwas close to its nominal cross-sectional area reductionof the drawn workpiece. The maximum effective strainoccurred near the surface. The differences in effectivestrain in drawing with various die semi-angles were notsubstantial near the center whereas near the surface theeffective strain increased dramatically in drawing withlarger die semi-angles. In drawing with very small diesemi-angles, the increase in effective strain was insignifi-cant and the strain distribution became more uniform. Thelevel of inhomogeneous deformation was hence reduced.The effective strain at the center was not affected by thefriction but increased significantly with the friction nearthe surface. The larger the friction, the more inhomoge-neous deformation was incurred on the drawn workpiece.

(3) The plot of inhomogeneity factor of effective strainagainst �-parameter showed all the curves converged atlarge values of �-parameter, indicating the level of inho-mogeneous deformation was determined solely by the�-parameter. However, the curves diverged at small val-ues of �-parameter, indicating an identical �-parameterwould bear different levels of inhomogeneous deforma-tion at small �. The deviation became more obvious when

the mean variation of effective strain was plotted against�-parameter, indicating the necessity of modifying the�-parameters in correlating deformation inhomogeneitywith the process parameters.

e c h n o l o g y 2 0 1 ( 2 0 0 8 ) 128–132

(4) Residual stress was compressive near the center andbecame tensile near the surface of the drawn workpiece,and the level of residual stress on the surface was min-imum in drawing with 2◦ die semi-angle. The residualaxial stress was more dominant than the circumferentialstress. The axial residual stress became compressive onthe drawn surface at reduction less than 0.4%, and thecircumferential stress became compressive at reductionless than 0.3%. The relation between residual stress andfriction factor was hardly to observe.

Acknowledgement

The authors would like to thank the financial support of theNational Science Council of the R.O.C. under the grant of NSC93-2212-E-218-006.

e f e r e n c e s

Avitzur, B., 1990. The use of the personal computer for simulationof the process of wire drawing and extrusion in an interactive,user-friendly mode. Wire J. Int. 23, 48–60.

Backofen, W.A., 1972. Deformation Processing. Addison-Wesley,

Reading, Mass, p. 140.

Hosford, W.F., Caddell, R.M., 1983. Metal Forming: Mechanics andMetallurgy. Prentice-Hall.

Kuntani, N., Asakawa, M., 2000. Analysis of the residual stressinduced by bar and wire drawing. SEAISI Q. (July), 35–41.