analysis and die design of flat-die hot extrusion process 2. numerical design of bearing lengths

International Journal of Mechanical Sciences 44 (2002) 935946

Analysis and die design of at-die hot extrusion process2. Numerical design of bearing lengths

Geun-An Leea, Yong-Taek Imb;

aKorea Institute of Industrial Technology, 35-3 Hongchonri, Ibjangmyun, Chonansi 330-825, South KoreabComputer Aided Materials Processing Laboratory, Department of Mechanical Engineering, ME3227, KoreaAdvanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-gu, Taejon 305-701, South Korea

Received 28 February 2001; received in revised form 25 February 2002

Abstract

This paper deals with the assignment of bearing lengths for the control of material ow in the at diehot extrusion. The design process makes the use of the three-dimensional non-steady analysis using thethermo-rigidviscoplastic 5nite element method that includes an automatic remeshing module. The exit velocitydistribution of the workpiece obtained from the analysis results was used to 5nd appropriate values for thefactors used in the proposed bearing length design equation. This equation for designing bearing lengths is afunction of the cross-sectional thickness and distance from the die center of die exit section. A geometric factorwas included in formulation of the design equation to consider the end region of the die exit. The appropriatevalues of factors were determined from three-dimensional analyses of at-die hot extrusion processes withsingle and double channel-sections. The analysis of a at-die hot extrusion process with a L-section was usedto verify the proposed design equation. It was found that the design equation determined bearing lengths thatresulted in a fairly uniform exit velocity distribution throughout the extruded section. From the results of thisstudy, it was found that the proposed design equation can be e7ectively used to estimate appropriate bearinglengths. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Hot extrusion; Material ow; Bearing length; Exit velocity; De ection; Channel-section

1. Introduction

Hot extrusion is a deformation process used to produce long and straight products such as bars,solid sections and tubes. In order to obtain a high-production rate with acceptable quality, manyprocess parameters must be controlled in consideration of the material characteristics and section

Corresponding author. Tel.: +82-42-869-3227; fax: +82-42-869-3210.E-mail address: [email protected] (Yong-Taek Im).

0020-7403/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0020-7403(02)00030-9

936 G.-A. Lee, Y.-T. Im / International Journal of Mechanical Sciences 44 (2002) 935946

Nomenclature

b bearing lengthr radial distance from the die centert section thicknessk geometric factorC1; C2 factorsbr reference bearing lengthbavg average bearing lengthEi value of errors in each sub-sectionEavg average error valueN the number of sub-sections

geometry. In particular, the material should have uniform exit velocity throughout its cross-sectionwhen being extruded to prevent de ection of the 5nal product. Without such a ow control, evensimple shapes cannot be extruded straight. Therefore, die designs must properly consider bearinglengths to maintain uniform velocity of the workpiece. Thus, numerous studies have been madeboth experimentally and numerically for the development of better die design methods consideringmaterial ow at the bearing land.Avitzur [1] predicted the forming load and material ow for wire drawing and extrusion through

conical dies with large cone angles. Keife [2] studied experimentally the material ow to obtain theappropriate bearing lengths and the position of die openings in extrusion through two die openings.Many other studies for the analysis of material ow in hot extrusion can also be found in literature[3,4]. With the studies of material ow during hot extrusion, Zasadzinski et al. [5] and Tashiro etal. [6] investigated the in uence of the exit velocity to minimize distortion of the 5nal product.Numerical analysis has also been used in the design of extrusion dies as many researchers have

investigated the in uence of material ow during hot extrusion [7,8]. In particular, the assignmentof bearing lengths must be carefully considered since it has a signi5cant e7ect on the velocitydistribution of the extruded section. In most cases, the assignment of bearing lengths has relied onexperiments or experience. Akeret et al. [9] studied the in uence of bearing lands experimentally andMiles et al. [10,11] studied the assignment of bearing length using the medial axis transform method.Other researchers conducted studies for the assignment of bearing lengths by controlling metal owduring hot extrusion [12,13]. For an understanding of the metal ow in the bearing section duringextrusion, Kiuchi et al. [14] studied the in uence of process parameters for at die extrusion throughrectangular and angle sections using the three-dimensional 5nite element analysis.In this study, in order to control the material ow, the length distribution of bearing land at

the die outlet was designed in consideration of cross-sectional thickness and distance from the diecenter of die exit sections. Information provided by non-steady three-dimensional 5nite elementsimulations was used in this design process. As mentioned in the companion paper (Part 1) of thisstudy, thermo-rigid-viscoplastic simulations of hot at-die extrusions were carried out to obtain thedeformation behavior of the workpiece and its velocity distribution at the die exit. Accuracy andeKciency are important factors for such non-steady-state simulations of at-die extrusion. That is,

G.-A. Lee, Y.-T. Im / International Journal of Mechanical Sciences 44 (2002) 935946 937

special care must be given in handling the contact algorithm to prevent penetration of contact nodesinto die surfaces since sharp edges are involved in at-die extrusion. Thus, the contact algorithmwas made to determine appropriate normal vectors and local transformation coordinates for variouscontact states to ensure that contact nodes would travel only along die surfaces. Also, non-steady-statesimulations usually require long computing time due to the large number of necessary remeshings.The very simple method of section sweeping which properly considers the pro5le of the extrudedworkpiece was used for eKcient remeshing. This simple remeshing scheme allowed for the entiresimulation procedure of solving, remeshing and transferring of state variables to be carried outautomatically without manual intervention to greatly reduce the required simulation time.The information provided by such numerical simulations was used to determine appropriate values

of factors required in the design of bearing lengths. More speci5cally, in the proposed design equationfor determining suitable bearing lengths, numerical factors were applied to the thickness and distanceparameters, and a geometric factor was used to consider the end regions of the die outlet. Values ofsuch factors were determined from the numerical analysis results of at-die hot extrusion processeswith single and double channel-sections. And 5nally, the e7ectiveness of the design equation wasexamined through 5nite element analysis of a at-die hot extrusion process with a L-section usingthe designed bearing lengths. It was found that the designed bearing lengths resulted in a relativelyuniform distribution of exit velocity.

2. The design of bearing lengths

In order to obtain straight extrusion products without de ection, the die designer must ensurethat the workpiece ows uniformly through the die exit. Generally, the designer can control thematerial ow by constraining the ow in thick sections by increasing the bearing length. In otherwords, variation of the bearing length is the main method of controlling the material ow and feed.There are no absolute rules for calculating appropriate bearing lengths, but the general procedure isto increase the bearing length in thicker sections and to reduce the length with increasing distancefrom the center of the die. Also, the bearing length should be reduced at the end regions of the dieopening to decrease friction levels which are usually high in such regions. Fig. 1 shows an exampleof longer bearing lengths assigned to the thicker cross-section region.The design equation used in the present study for the design of bearing lengths is as follows:

b(r; t) = k{C1t + C2(rmax r)}: (1)Here, b is the bearing length, r is the radial distance from the center of the die, t is the sectionthickness and k is a geometric factor used to consider the end regions. C1 and C2 are factors appliedto the cross-sectional thickness and radial distance parameters in the above equation, respectively.The appropriate values for k, C1 and C2 cannot be easily determined. Thus, non-steady 5nite

element analysis was used in determining these factors. More speci5cally, analysis results of at-diehot extrusion processes with single and double channel-sections using constant bearing lengths wereused. The exit velocity distributions of these two cases are given in Figs. 14 and 15 of Part 1 ofthis study. It should be mentioned that these two channel-section cases are representative of thetwo main classes of channel sections, namely, the class of sections in which the end section is thethickest and the other class in which the end section is the thinnest.


Section AA

A

ABB Section BB

Bearing

Bearing length

Die

Fig. 1. The assignment of bearing lengths in a channel-section.

3. The determination of factors

The determination of C1 and C2 was carried out by the procedure shown in Fig. 2. The input datafor this procedure are the geometry of the die exit and the exit velocity distribution obtained fromthe 5nite element analysis using constant bearing lengths. The geometry of the die exit is dividedinto sub-sections of the uniform cross-sectional thickness and the end regions are also separated assub-sections. This division of sub-sections is the same method that was used in Part 1 of this studyto illustrate the die exit velocity distribution, that is, Figs. 14 and 15 in Part 1. The exit velocitiesobtained from the simulation using constant bearing lengths are averaged in each sub-section andthe reference bearing length is calculated from this average exit velocity for each sub-section. Morespeci5cally, the sub-section with the maximum exit velocity is 5rst given the maximum bearing lengthand the other sub-sections are given bearing lengths according to the ratio of average velocity relativeto the maximum velocity. C1 and C2 are set to be the values in 0.53.0 and 0.0010.5, respectively.After the bearing lengths are calculated by Eq. (1) for each sub-section, these are averaged in eachsub-region. The errors are calculated by Eq. (2) using the reference and this average bearing lengths.Then, the average error is calculated by Eq. (3) using the errors in sub-sections. After the aboveprocedure is carried out within the given ranges, C1 and C2 are determined when the average errorEavg is the minimum value

Ei = |{(br)i (bavg)i}=(br)i|: (2)Eavg =

(Ei)=N: (3)

Here, br and bavg are the reference bearing lengths in each sub-section and the average bearinglengths, respectively. Ei is the value of errors in each sub-section, Eavg the average error value andN the number of sub-sections.From these procedures, the values of C1 and C2 were determined for single and double channel-

sections, respectively. For the single-channel-section case, C1 and C2 were determined as the valuesof 1.0 and 0.27, respectively. For the double-channel-section case, C1 and C2 were determined asthe values of 1.0 and 0.04, respectively.


Fig. 2. Flow chart for determining the values of C1 and C2 using the exit velocity obtained from the 5nite elementanalysis.

Also, as mentioned previously, k is the geometric factor used to consider the end regions. Sincethe exit velocity was the smallest in the end regions of the exit due to the high friction in theseareas as shown in Figs. 14 and 15 of Part 1 of this study, k is given the values of 0.30.5 to reducethe friction in the end regions.

4. Results and discussion

Using C1 and C2 obtained from the procedure of Fig. 2, the bearing lengths were designed usingEq. (1) for the single channel-section extrusion example. The geometry of the single channel-sectioncase is depicted in Fig. 3(a). Comparison of bearing lengths between the design and actual datafrom industry is shown in Fig. 3(b). As shown in this 5gure, although the designed bearing lengths


t1= 7.1t2= 5.2

t1

t2CL

40.6

25.4

Dim.: mm

(a)

(b)

30

40

50

60

70

80

90

100

(E)(F)(D)

(C)

(B)

(A)

(D) Exit velocity distribution resulting from (A)(E) Exit velocity distribution resulting from (B)(F) Exit velocity distribution resulting from (C)

Exit

velo

city

(mm/

sec)

-4

0

4

8

12

16

20

Bea

ring

leng

th (m

m)

(A) Constant bearing length of 5 mm(B) Designed bearing lengths(C) Actual bearing lengths from industry

Fig. 3. (a) The double-channel-section geometry; and (b) comparison of exit velocity distributions resulting from bearinglengths assumed to be constant, obtained from the current design, and obtained from industry for the single-channel-sectionextrusion case.

deviate from the actual industry data, the resulting exit velocity distribution is quite uniform andcompares well with the industry bearing length results. The improvement compared to the casewith constant bearing lengths can be clearly seen. Fig. 4 compares the deformed workpiece shapesbetween the cases using constant bearing lengths of 5 mm and designed bearing lengths at the presentinvestigation.The geometry of the double channel-section case is depicted in Fig. 5(a) and in Fig. 5(b), the

exit velocity distribution with a constant bearing length of 5 mm is compared to the results obtainedfrom the designed bearing lengths. It can be seen that the exit velocity distribution of curve (D) ismore uniform than the result of curve (C). However, the gap between the maximum and minimumvelocities in curve (D) is about 10 mm=s, which is not a small value. This was mainly due tothe fact that the sub-sections of and in Fig. 5(a) were too large. That is, these sub-sectionsshould be divided further for the bearing land in this region to be able to accommodate the velocity


Fig. 4. The deformed shapes of the workpiece using: (a) constant bearing lengths of 5 mm; and (b) the designed bearinglengths for the single channel-section at punch stroke of 1:32 mm.

distribution. Thus, these sub-sections were divided into more sub-sections as depicted in Fig. 6(a).As shown in Fig. 6(b), introduction of more sub-sections improves the uniformity of the resultingvelocity distribution compared to the previous results of Fig. 5(b). In this case, di7erence betweenthe maximum and minimum velocities was found to be reduced to about 4 mm=s.Fig. 7 shows the improvement of using the designed bearing lengths compared to the case using

constant bearing lengths. These results show that the current approach gives a reasonable design ofbearing lengths for extrusion processes.Next, the validity of the design equation was investigated by designing bearing lengths for a

L-section and examining the resulting velocity distribution from the 5nite element analysis. Thedesign of bearing lengths was carried out by using the factors determined in the two channel-sectioncases. The input data for this procedure is the geometry of the die exit. The geometry of the dieexit is divided into sub-sections of uniform cross-sectional thickness and the end regions are alsoseparated as sub-sections.The distances from the die center are calculated along the boundary of the cross-section. Then, C1

and C2 are applied by checking whether end regions are thicker or not than the other sub-sections.The value of k is applied by checking whether the current sub-section is an end region or not. For thequadruple channel-section case, C1 and C2 were 1.0 and 0.04, respectively. k was given the value of0.5 at end regions. Using the factors, the bearing lengths are calculated by Eq. (1). After followingthe above procedure, the bearing lengths are determined in sub-sections of the cross-section.The L-section geometry and sub-sections are depicted in detail in Fig. 8(a). The designed bearing

lengths are shown in curve (A) of Fig. 8(b). Three-dimensional 5nite element analysis was carriedout for this quadruple L-section using the designed bearing lengths. The analysis conditions aregiven in Fig. 9. Due to symmetry of the problem, only a quarter section with appropriate boundary


CLt1= 6.0t2= 7.0

t1

t2

Dim.: mm

43.0

19.0

(a)

(b)

10

20

30

40

50

60

70

80

Bea

ring

leng

th (m

)

Exit

velo

city

(mm/

sec)

0

4

8

12

16

20

24(A) Constant bearing length of 5 mm(B) Designed bearing lengths(C) Exit velocity distribution resulting from (A)(D) Exit velocity distribution resulting from (B)

(A)

(B)

(C)

(D)

Fig. 5. (a) The double-channel-section geometry; and (b) the exit velocity distribution using the designed bearing lengthsand constant bearing lengths of 5 mm.

conditions was used. The initial workpiece was taken as a cylindrical billet with diameter of 200 mmand height of 50 mm, and the initial temperatures of the workpiece, die and atmosphere were set as420C, 400C and 18C, respectively. The friction constant was assumed to be 0.3 and the punchvelocity was set as 1 mm=s. The ow stress obtained from the compression test of Al6061-T6 wasused in simulations as follows:

O = 91:0 O0:09

(MPa): (4)

Here, O and O are the e7ective stress and the e7ective strain rate, respectively.The exit velocity distribution and deformed shape of the workpiece were obtained from the sim-

ulation. The exit velocity distribution is shown in curve (B) of Fig. 8(b). It was found that the exitvelocity distribution was fairly uniform with velocity deviations less than 1:0 mm=s. Also, Fig. 10shows the deformed shapes of the workpiece at various strokes for the quadruple L-section extrusionprocess. It can be con5rmed that the workpiece exits the die without having much de ection.


CLt1= 6.0t2= 7.0

t1

t2

Dim.: mm

43.0

19.0

(a)

(b)

10

20

30

40

50

60

70

80

Bea

ring

leng

th (m

m)

Exit

velo

city

(mm/

sec)

0

4

8

12

16

20

24(A) Constant bearing lengths of 5mm(B) Designed bearing lengths(C) Exit velocity distribution resulting from (A)(D) Exit velocity distribution resulting from (B)

(A)

(B)

(C)(D)

11

12

13

11 12 13

Fig. 6. (a) The double-channel-section geometry divided into more sub-sections; and (b) the resulting exit velocity distri-butions using the designed bearing lengths and constant bearing lengths of 5 mm.

Fig. 7. The deformed shapes of the workpiece using: (a) constant bearing lengths of 5 mm; and (b) the designed bearinglengths for the double channel-section at punch stroke of 1:5 mm.


CL

30.0

30.0

6.0

Dim.: mm

x

y

(a)

(b)

17

18

19

20

21

22

23

Bea

ring

leng

th (m

m)

Exit velocity distribution

Exit

velo

city

(mm/

sec)

0

4

8

12

16

20 Designed bearing length

11 12 13

11

12

13

Fig. 8. (a) The L-section geometry; and (b) the designed bearing lengths and resulting exit velocity distribution.

This shows that the proposed equation for designing bearing lengths can be applied for similarchannel-section type extrusion processes.

5. Conclusions

In this study, using the exit velocity distribution obtained from 5nite element simulation results,bearing lengths for channel-section-type die geometries were designed in consideration of the thick-ness and distance from the die center of die exit sections and the end region with high frictionwas considered. The designed bearing lengths resulted in improvement of the uniformity of exitvelocity distributions. Also, it was found that the proposed design equation worked reasonably wellfor a L-section die exit geometry under the present investigation and that the current approach o7ersuseful information to the die designer of shape extrusion processes.


Fig. 9. (a) Full analysis model; (b) top view of die container; and (c) simulation conditions for analysis of extrusionthrough the quadruple L-section.

Fig. 10. The deformed shapes of the workpiece for the quadruple L-section die at various strokes: (a) 0:28 mm; (b)0:56 mm; (c) 0:84 mm; and (d) 1:16 mm.


Acknowledgements

The authors wish to acknowledge the support from the BK (Brain Korea) 21 project.

References

[1] Avitzur B. Analysis of wire drawing and extrusion through conical dies of large cone angle. Transactions of ASME,Journal of Engineering for Industries 1964;86:30516.

[2] Keife H. Extrusion through two die openings: A 2D upper-bound analysis checked by plasticine experiments. Journalof Materials Processing Technology 1993;37:189202.

[3] Takuda H, Hatta N. A simple approach to plane strain extrusion with dead metal zone using upper-bound theorem.Metals and Materials 1998;4(4):73741.

[4] Kar PK, Das NS. Upper bound analysis of extrusion of I-section bar from square=rectangular billets through squaredies. International Journal of Mechanical Sciences 1997;39(8):92534.

[5] Zasadzinski J, Richard J, Libura W, Misiolek W, Krakow. Maximization of the exit velocity of aluminum alloysduring hot extrusion. Aluminium 1984;60(1):148.

[6] Tashiro Y, Yamasaki H, Ohneda N, Nakanishi K. Extrusion conditions and metal ow to minimize both distortionand variance of cross sectional shapes. Proceedings of the Fifth International Aluminium Extrusion TechnologySeminar, vol. 2, Chicago, USA, 1992. p. 191205.

[7] Valberg H. A modi5ed classi5cation system for metal ow adapted to unlubricated hot extrusion of aluminum andaluminum alloys. Proceedings of the Sixth International Aluminium Extrusion Technology Seminar, vol. 2, Chicago,USA, 1996. p. 11324.

[8] Kinoshita H, Nakanishi K, Kamitani S, Yoshida T. Metal ow characteristics and deformation analysis inhot extrusion of 6063, 5052 and 5083 Al-alloys. Journal of Japanese Society for Technology of Plasticity1996;37(428):9338.

[9] Akeret R, Strehmel W. Control of metal ow in extrusion. Proceedings of the Fourth International AluminiumExtrusion Technology Seminar, vol. 2, Chicago, USA, 1988. p. 35767.

[10] Miles N, Evans G, Middleditch A. Automatic bearing length assignment using the medial axis transform. Proceedingsof the Sixth International Aluminium Extrusion Technology Seminar, vol. 2, Chicago, USA, 1996. p. 1617.

[11] Miles N, Evans G, Middleditch A. Bearing length for extrusion dies: rational, current practice and requirements forautomation. Journal of Materials Processing Technology 1997;72:16276.

[12] Hardouin JP. Bearing length calculation by control of metal ow pressure. Proceedings of the Fifth InternationalAluminium Extrusion Technology Seminar, vol. 1, Chicago, USA, 1992. p. 291303.

[13] Rodriguez P, Rodriguez A. System to calculate chambers and feeds to obtain a minimum single bearing. Proceedingsof Fifth International Aluminium Extrusion Technology Seminar, vol. 1, Chicago, USA, 1992. p. 3958.

[14] Kiuchi M, Yanagimoto J, Mendoza V. Finite-element analysis of metal ow in bearing section during extrusionprocess of rectangular and angle sections. Journal of Japanese Society for Technology of Plasticity 1997;39(446):2115.

Analysis and die design of flat-die hot extrusion process2. Numerical design of bearing lengthsIntroductionThe design of bearing lengthsThe determination of factorsResults and discussionConclusionsAcknowledgementsReferences

analysis and die design of flat-die hot extrusion process 2. numerical design of bearing lengths

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