Effects of Die Dimensions for Curvature Extrusion of Curved Rectangular Bars+1

Yoichi Takahashi1,+2, Shigefumi Kihara1, Ken Yamaji2,+3 and Mitsunobu Shiraishi3

1Department of Mechanical Engineering, National Institute of Technology, Kagawa College, Takamatsu 761-8058, Japan2Advanced Course, National Institute of Technology, Kagawa College, Takamatsu 761-8058, Japan3Faculty of Engineering, Kinki University, Higashi-hiroshima 739-2116, Japan

The effects of die dimensions on curvature-extrusion curved rectangular bars were investigated by experiments and numerical simulations.We conducted simulations and experiments of rectangular bar production using various die dimensions. The calculated trend of curvatureshowed good agreement with the experiment results, thereby validating the numerical simulation results. The curvature increases concomitantlywith increasing exit height and exit width. The exit height ratio affects the curvature about 2.1 times more than the exit width ratio does. Theeffect of die thickness on curvature is slight. Regarding the conditions of die dimensions, the exit height is the most important factor that affectsthe curvature. The curvature increases when the difference in exit velocity becomes large. The generation of a dead metal area, along withcontainer configurations, affects the curved rectangular bar curvature. Results show that the curvature is determined by the difference in exitvelocity, in conjunction with the die dimensions. [doi:10.2320/matertrans.P-M2015805]

(Received January 20, 2015; Accepted March 23, 2015; Published May 8, 2015)

Keywords: extrusion, forward extrusion, finite element method (FEM), deformation behavior, manufacturing system

1. Introduction

Extrusion technology enables the easy manufacture of longproducts of a uniform sectional shape. It has been usedextensively as a primary processing method for products suchas bars, wire, tubes, and profiles.1) Generally the industrialcommodity is finished to an end-product shape by secondaryprocessing, such as bending and cutting, after primaryprocessing. Whereas needs for these products are beingdiversified year by year, and smaller production lots of a widevariety are requested, strong demand exists for the reductionof manufacturing costs and process numbers. A formingtechnique for the reduction of manufacturing processnumbers has been proposed, which fabricates a productdirectly by extrusion with cross-section dimension that variesalong the longitudinal axis. Therefore, flexible extrusionattracting attention.2­4) It requires a technique that controlsthe curvature because of material flow near a die exit at thetime of extrusion. The authors have proposed extrusion withinclined dies as a working process for curvature control atextrusion.5) Moreover, they described the effect of theaperture position and the extrusion ratio on curvature duringthe forming of rectangular curved bars and rectangulartubes.6,7) However, because the extrusion ratio is the ratio ofareas, the effects of the aperture height ratio and the aperturewidth ratio are not necessarily equivalent. It is thereforenecessary that each be evaluated independently and that thedie thickness effect also be examined, for detail investigationof the die dimension effects.

In this study, experiments and numerical simulations wereconducted at the time of extrusion of rectangular curved barsusing colored clay as a raw material. The study evaluated theeffects on curvature of the aperture height ratio, the aperturewidth ratio, the die thickness ratio, and the die inclination

angle independently, with further examination of the results.Furthermore, the exit velocity difference was defined as anindex with which the effects of die dimensions can beevaluated systematically. The relation between die dimen-sions and curvature was investigated.

2. Experimental and Simulation Methods

Figure 1 shows a Schematic illustration of extrusionprocess used for this study. Colored clay charged into thecontainer as a raw material is pressurized using a punch; itis squeezed out from the center of the die aperture. Thedimension parameters of a defined die are the die inclination¦, the die aperture height hs, the die aperture width Ws, andthe die thickness t. The experiment and simulation wereperformed on each condition shown in Table 1. Both theexperiment and simulation were conducted at ¦ = 15° and30°, although the simulations were performed only at ¦ = 20°

Fig. 1 Schematic illustration of extrusion process for producing curvedrectangular bar.

Table 1 Dimension of dies.

¦/° hs/mm Ws/mm t/mm

15, 20, 25, 30 13, 15, 17, 20, 25 5.6 3

15, 20, 25, 30 17 5.6, 6.6, 7.6, 14 3

15, 30 17 5.6 3, 5.6, 9

+1This Paper was Originally Published in Japanese in J. JSTP 55 (2014)954­958.

+2Corresponding author, E-mail: ytaka@t.kagawa-nct.ac.jp+3Undergraduate Student, National Institute of Technology, KagawaCollege

Materials Transactions, Vol. 56, No. 6 (2015) pp. 844 to 849©2015 The Japan Society for Technology of Plasticity

and 25°. The height hb and width Wb of a billet were,respectively, constant at 32mm and 17mm. A 300 kNuniversal testing machine (UH-F300KNI; Shimadzu Corp.)was used for the experiment: the punch velocity was 0.5mm/s and the extrusion length was about 30mm, so that thecurvature was unaffected by the weight of the extruded clay.Curvature was measured using a material extruded at a steadystate, i.e., squeezed out at a constant curvature. Soapy waterwas used as a lubricant.

Figure 2 shows the simulation model used for this study.One half of the analysis region was modeled considering thebilateral symmetry of the model. Figures 2(a)­2(c) respec-tively portray the analytic space, with particles representingthe raw material and the container. Forging analysis softwaredeveloped by one of the present authors8) was employed forthe simulation. It is a characteristic of this analysis that theanalysis regions consist of regions in which the material isand is not present. Letting the former region, that where thematerial is present before extrusion, be a material region andletting the latter region, where the material is expected to bepresent after extrusion, be a material-front region then thematerial can be defined using a group of particles. Thecontainer inwalls were defined with rigid plane elements. Theanalytic space, consisting of the material region and thematerial-front region, were defined with 3,500­4,500 ofhexahedral elements. The velocity, stress, strain, strain rate,and work per unit volume of analysis regions were computedbased on the rigid plastic finite element method consideringcompressibility.9) The self-weight of the material was notconsidered. A reference value10) given by eq. (1) was adoptedas the deformation resistance of the colored clay. Container-material and die-material friction were assumed to obeyCoulomb friction. The friction coefficient was set as 0.1based on a comparison between the calculation andexperimentally obtained results.

· ¼ 0:28¾0:2 MPa ðRoom temperature 24�CÞ ð1Þ

3. Results and Discussions

3.1 Effect of die dimensions on curvatureFigure 3 shows the result of an extrusion experiment with

the aperture height ratio H as a parameter at die inclinationsof ¦ = 15° and 30°, and the analysis result only at ¦ = 30°, asexamples. The aperture height ratio H was defined as the ratioof the billet height hb to the die aperture hs. Other conditionsinclude the die aperture width Ws = 5.6mm and the diethickness t = 3mm, both constant. Downward curvature wasobserved both in the experiment and simulation, as the figureshows. Curvature became greater as the die inclination andaperture height ratio increased, so that varying the dieinclination and the die height ratio changed the flow state of amaterial at extrusion, thereby affecting curvature. Downwardcurvature was observed in every condition in this study.

Figure 4 portrays the relation between the die dimensionsand curvature acquired from the experimental and simulationresults. Figure 4(a) shows the curvature µ¹1 when varyingthe aperture height ratio H to 1.3, 1.6, 1.9, 2.1, and 2.5.The experimentally obtained result for the die inclination of¦ = 15° reveals that the curvature at H = 2.5 increased toµ¹1 = 0.038mm¹1, whereas that at H = 1.3 was µ¹1 =0.018mm¹1. Similarly, the experimentally obtained resultfor ¦ = 30° shows a trend by which µ¹1 increased as Hbecame greater; µ¹1 became larger than the result for ¦ = 15°.The simulation result also showed a trend by which µ¹1

increased for greater H, and the error was 18% or less. Thiserror is presumed to be attributable to the fact that frictionconditions might vary according to sites in experiment,whereas the friction coefficient was assumed to be constant inthe simulation. Figure 4(b) shows the curvature µ¹1 whenvarying the aperture width ratio W to 1.2, 2.2, 2.6, and 3.0.The aperture width ratio W was defined as the ratio of thebillet width Wb to the die aperture width Ws. The die height hsand thickness t were, respectively, constant at hs = 17mmand t = 3mm. µ¹1 was 0.018­0.032mm¹1 when varying Wto 1.2­3.0 in experiment for ¦ = 15°, whereas 0.033­0.048mm¹1 for ¦ = 30°. µ¹1 increased as W became largerat any ¦ similarly to the variation of the aperture height ratio.Figure 4(c) shows curvature µ¹1 when varying the diethickness ratio T to 1.0, 1.9, and 3.0. The die thickness ratioT was defined as the die thickness normalized by t = 3mm.

Fig. 2 Simulation model of extrusion process for producing curved bar.

Inclination angle of die ζ

15 / ° 30 / ° 30 / °

Experiment Simulation

Exi

t hei

ght r

atio

s H 2.

51.

91.

3

Fig. 3 Particle flow patterns and product profiles of rectangular bar atvarious exit height ratios.

Effects of Die Dimensions for Curvature Extrusion of Curved Rectangular Bars 845

The die height hs and width Ws were, respectively, constantat hs = 17mm and Ws = 5.6mm. µ¹1 declined as T increasedboth in experiment and simulation. The experimentallyobtained result when varying T from 1.0 to 3.0 showed thatµ¹1 was 0.032­0.025mm¹1 at ¦ = 15°, and 0.048­0.036mm¹1 at ¦ = 30°, so that the variation was around 25% atmost. Consequently, the effect of the die thickness ratio onµ¹1 was small compared with that of the aperture height ratio

and aperture width ratio shown respectively in Figs. 4(a)and 4(b). The linear approximation of the experimentallyobtained result for ¦ = 15° portrayed in Fig. 4(c) isextrapolated to µ¹1 = 0 around T = 10: it is inferred thatcurvature was suppressed by the material flow near theaperture because of the effect of the die aperture section at adie thickness ratio over a certain value, so that the materialwas pushed out straight.

A dead metal region was determined from the simulationresult to examine the relation between the die dimensions andcurvature in detail. Figure 5 exhibits a dead metal region ineach condition at a die inclination of ¦ = 15°: Fig. 5(a) showsthe results at aperture height ratios of H = 1.3 and 2.5;Fig. 5(b) shows those at aperture width ratios of W = 1.2 and3.0; and Fig. 5(c) shows those at die thickness ratios ofT = 1.0 and 3.0. A dead metal region represents a region of amaterial not squeezed out and left in the container, which isdefined as a region where the travel distance of a particle isnot more than 0.5mm per unit time in this simulation. Denserdots in the figures express a smaller travel distance per unittime. Regarding the effect of the aperture height ratio H inFig. 5(a), the volume difference in dead metal existing in theupper and lower parts of the exit is 298mm3 at H = 2.5 and59mm3 at H = 1.3. Therefore, the volume difference isgreater for a higher H. A similar trend is observed also for theaperture width ratio W in Fig. 5(b). Cases of large volumedifference in dead metal regions in the upper and lower partsof the exit correspond to the result of larger curvature,whereas cases of small volume differences correspond to thatof smaller curvature. Previous studies have revealed that thematerial exit velocity in the upper and lower parts of the exitaffect curvature.6) This study also verified by simulation thatthe material was squeezed out along dead metal regions, andthat the material exit velocity increased in the upper part ofthe exit where the dead metal volume is large, as depicted inFig. 6. Consequently, the material exit velocity in the upperpart exits faster than that in the lower part, generatingdownward curvature in the extruded material, in agreementwith the experimentally obtained result. However, thedistribution of dead metal regions in Fig. 5(c) differs fromthose in Figs. 5(a) and 5(b). Because little difference in theupper and lower parts of the exit both at T = 1.0 and 3.0, thecurvature change attributable to the die thickness ratio isslight, so that the effect of dead metal regions on curvature issmall.

The effects of die dimensions on curvature are summarizedas follows, based on the discussion above. Figure 7 showsthe effect of die dimensions on curvature revealed in thisstudy. The curvature shown in the figure represents variationas the ratio of each factor increases by 1.0. Because thecurvature varies with the die inclination, its averagedvariation is computed. Curvature µ¹1 is affected stronglywhen the exit height ratio H > the exit width ratio W > thedie thickness ratio T, as portrayed in Fig. 7. Consequently, adie dimension with great effect on the curvature is a case inwhich the die aperture height hs > the die aperture widthWs > the die thickness t. Results show that µ¹1 increases byabout 0.021, 0.010, and 0.005mm¹1, respectively, when H,W, and T increase by 1.0. Therefore, the aperture height ratioaffects curvature about 2.1 times more than the aperture

(a)

(b)

(c)

Fig. 4 Relationship between dimension of dies and curvature; relationshipbetween curvature of rectangular bar and (a) exit height ratios, (b) exitwidth ratios, (c) dies thickness.

Y. Takahashi, S. Kihara, K. Yamaji and M. Shiraishi846

width ratio. Conventionally, curvature has been evaluatedusing the extrusion ratio, which is a ratio of area. However,the effects of the aperture height ratio and the aperture widthratio are not equivalent, but differ about 2.1 times. Thisdifference cannot be disregarded. For that reason, that it isnecessary to evaluate their effects independently. The causeof the difference in the effects of the exit height ratio H andthe exit width ratio W is regarded as follows. Figure 8 showsthe die restraining surface and the extrusion direction. Thedifference of both is a restraining area on the die surface. It isanticipated that friction work on material deformation withinthe restraining area is large because the die surface is loadedthe most. Because the material is curved upward ordownward to the extruding direction as presented in Fig. 8,the distance to the exit from the right and left ends of therestraining surface more strongly affects the results than thedistance from the upper and lower ends. This difference in

Fig. 7 Effects of dimension of dies for curvature.

Volume of dead metal, 413 mm3

115 mm3

Velocity of particle, 9.2 mm/s

5.4 mm/s

Fig. 6 Dead metal and particle flow (¦ = 15°, H = 2.5).Single view

Single view

Exit height ratio H=2.5

Exit height ratio H=1.3

413 mm3

474 mm3

115 mm3

415 mm3

Side view

Side view

Volume of dead metal

Exit width ratio W=3.0

Exit width ratio W=1.2

463 mm3

393 mm3

123 mm3

522 mm3

Side view

Single view Side view

Single view

Side view

Die thickness ratio T=3.0

Die thickness ratio T=1.0

223 mm3

285 mm3

Single view Side view

Single view

115 mm3

135 mm3

(a)

(b)

(c)

Fig. 5 Distribution of dead metal area (¦ = 15°); (a) effect of exit heightratios, (b) effect of exit width ratios, (c) effect of dies thickness ratios.

Fig. 8 State of extrusion.

Effects of Die Dimensions for Curvature Extrusion of Curved Rectangular Bars 847

this restraining surface engenders differences in the deadmetal regions and in the exit height ratio, the differencebetween the upper and lower ends of the exit, the curvaturebeing affected more than the exit width ratio, and thedifference between the right and left ends.

3.2 Relation between exit velocity difference and curva-ture

Factors that affect the curvature by extrusion with inclineddies include the die hole shape, die inclination, and materialproperties. It is important when designing a die to establishan index that evaluates these factors systematically. Theresults described above revealed that the curvature wasaffected by the difference in material flow states between ofthe upper and lower parts near the exit. Accordingly, the exitvelocity was computed using the simulation results, from thetravel distance of 16 particles a1­a8 and b1­b8 positioned atan equal interval as shown in Fig. 9. A streamline in thefigure represents the travel distance of a particle in 2.5 s.A longer line signifies a higher velocity. The average ofa1­a8 and that of b1­b8 were defined respectively as thevelocity at the upper and lower parts of the exit. Thedifference between them was defined as the exit velocitydifference as eq. (2).

�v ¼X8

i¼1

ai8�

X8

i¼1

bi8

ð2Þ

Figure 10 shows the velocity distribution computed fromthe travel distance of particles for 2.5 s shown in Fig. 9. Thevertical axis denotes y, the distance from the bottom of acontainer of ¦ = 30°. The horizontal axis expresses v, thevelocity at a position distant by 8mm in parallel from the endface of the container. The velocity distributions are shown asexamples for aperture height ratios of H = 1.3, 1.6, 1.9, 2.1,and 2.5 at a die inclination ¦ = 30°. The maximum velocitywas enhanced and the form of velocity distribution wassharpened as H was greater, so that a rectangular bar wasformed with a larger curvature. Simulations were conductedthe same way in all the conditions shown in Table 1. The exitvelocity difference was computed from the acquired velocitydistribution. The relation between the exit velocity differenceand curvature is presented in Fig. 11, which shows that thecurvature increased linearly as the exit velocity increased.Consequently, it was verified that the curvature wasdetermined by velocity difference between the upper and

lower parts of an exit irrespective of extrusion conditions,and obtained by eq. (3):

µ�1 ¼ 3:11� 10�2 ��v ð3Þwhere µ¹1mm¹1 denotes the curvature and "v mm·s¹1

represents the exit velocity difference.

4. Conclusion

(1) Curvature increases as the exit height ratio and the exitwidth ratio become larger, irrespective of the dieinclination. The exit height ratio affects the curvatureabout 2.1 times more than the exit width ratio does. Thecurvature decreases as the die thickness ratio increases,but it is less affected than the exit height ratio and thewidth ratio.

(2) Dead metal occurs in the corner section near the exit.The material flow state attributable to the velocitydifference between the upper and lower parts of the exitaffects the curvature.

(3) The velocity difference between the upper and lowerparts of the exit is defined as an index by whichthe effect of die dimensions on curvature can beevaluated systematically. The relation with curvaturewas elucidated.

Fig. 10 Velocity distribution by the difference in exit height ratios(¦ = 30°).

Fig. 11 Relationship between curvature and difference of exit velocity.

Fig. 9 Method of measurement of particle’s movement.

Y. Takahashi, S. Kihara, K. Yamaji and M. Shiraishi848

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Effects of Die Dimensions for Curvature Extrusion of Curved Rectangular Bars 849