ORIGINAL ARTICLE

Three-dimensional analysis of round-to-angle sectionextrusion through straight converging die

Susanta Kumar Sahoo & B. Sahoo & L. N. Patra &

U. C. Paltasingh & P. R. Samantaray

Received: 22 April 2009 /Accepted: 6 November 2009 /Published online: 26 November 2009# Springer-Verlag London Limited 2009

Abstract The increasing interest in the modeling of metal-forming processes in recent years has brought the develop-ment of different analytical and/or numerical technique.However, due to the complexity nature of the problem,most of the attempts are made with plain strain assump-tions. Among the different techniques used, the upperbound method is a convenient tool for evaluating the rate ofwork in processes involving predominantly plastic defor-mation of rigid/perfectly plastic material. The present studyis an endeavor to remodel and apply the spatial elementaryrigid region technique for analyzing extrusion of angle-section bars from round billets through the linearlyconverging die. Optimized values of the nondimensionalaverage extrusion pressure at various area reductions havebeen computed and compared with experimental results. Itis observed that the proposed technique can be usedeffectively with adequate accuracy to predict the optimaldie geometry which requires a minimal forming stress atdifferent reduction of areas and friction conditions.

Keywords Extrusion . Upper bound . Taper die .

Plastic behavior . Velocity discontinuity

NomenclatureAb, Ae area of the billet and the product cross sectionsAfj area of jth face having friction between dies and

work piecef function representing the equation of a planer

surfaceL length of each side of the approximating polygonm friction factor on the die surfaceM number of sides of the approximating polygonPav average extrusion pressureR billet radiusΔV velocity discontinuityVb, Ve billet and product velocity

Greek letterσo yield stress in uniaxial tension or compressionθ internal angle of a polygon�"ij components of strain rate tensor

1 Introduction

Extrusion is the process by which a block of metal isreduced in cross section by forcing it to flow through a dieorifice under high pressure. The performance of theextrusion process strongly depends on die design becauseit controls material flow. Now, it is becoming essential totake greater attention for the extrusion of section rod fromround stock as this operation offers the promise of aneconomic production route. Theoretical considerations ofthis process have been neglected as simple analyticaltechniques cannot yield valid relationships. Therefore, it

S. K. Sahoo (*) : L. N. PatraDepartment of Mechanical Engineering, N.I.T.,Rourkela 769 008, Orissa, Indiae-mail: sksahoo_rec@rediffmail.com

B. SahooDepartment of Mechanical Engineering, I.G.I.T.,Sarang 759 146, Orissa, India

U. C. PaltasinghDepartment of Mechanical Engineering,Padmanava College of Engineering,Rourkela-2, Orissa, India

P. R. SamantarayDepartment of Mechanical Engineering, BOSE,Cuttack 753007, Orissa, India

Int J Adv Manuf Technol (2010) 49:505–512DOI 10.1007/s00170-009-2427-7

has been very difficult for us to find out the appropriateworking conditions of extrusion and to design the optimumdie shape and dimensions for the required product. For longtime, those matters have been based on empirical knowledge.For extrusion of sections, the flat faced or so-called the squaredie has the disadvantages of limited extrusion speed andrequiring high extrusion force. The converging dies withlubricants is more preferred to flat-faced square dies, as itprovides a gradual change in shape and reduction of areasimultaneously. Despite the advantage of converging dies, afew theoretical approaches to the extrusion or drawingprocesses have been published. As the demand of anglesection bar in the industry is growing, it is important tonumerically model the process so that the extrusion variablescan be predicted with adequate accuracy beforehand.

Among the various techniques available to modelmathematically metal-forming processes in general andextrusion in particular, the upper bound method providesa good compromise between accuracy for practical pur-poses and speed of calculation. The upper bound techniqueappears to be a useful tool for analyzing three-dimensionalmetal-forming problems when the objective of such ananalysis is limited to prediction of the deformation loadand/or study of metal flow during the process. This is sobecause the classical slip line field solution is not applicableto this class of problems, and the finite element method isconstrained by computational difficulties to achieve accu-racy in these cases. Even though upper bound techniqueswere applied to plane strain and axisymmetric metal-forming problems long ago, three-dimensional problemsattracted attention only in the early 1980s. One of theearliest three-dimensional analyses of extrusion wasreported by Juneja and Prakash [1]. Subsequently, a numberof techniques were reported [2–3] to construct kinematical-ly admissible velocity fields that lead to upper boundsolutions of three-dimensional extrusion problems. Nagpaland Altan [4] introduced the stream function to expressthree-dimensional flow in the die and analyzed the force ofextrusion from round billet to elliptical bars. Basily andSansom [5] made an upper bound analysis on drawing ofsquare sections from round billets by using triangularelements at entry and exit of the die. Yang et al. [6]formulated kinematically admissible velocity fields ofbillets having generalized cross section for the extrusion,where the similarity in the profile of cross section wasassumed to be maintained throughout deformation. Prakashand Khan [7] made an upper bound analysis on extrusionand drawing through dies of polygonal cross sections withstraight stream lines, where the similarity in shape waspresented. Boer et al. [8] made an upper bound approach todrawing of square rod from round bars, by employing amethod of coordinate transformation. Another methodbased on discretization of the deformation zone into rigid

tetrahedral blocks has also been proposed by Gatto andGiarda [9]. Hosino and Gunasekera [10] obtained an upperbound solution for extrusion and drawing of square sectionsfrom round billets through converging dies formed by anenvelope of straight lines. Maity et al. [11] proposed anupper bound analysis, derived from a kinematicallyadmissible velocity fields using a dual-stream functiontechnique, for the extrusion of square sections from squarebillets through curved dies with prescribed profiles. Kim etal. [12] proposed a velocity field, based on upper boundanalysis, for the prediction of the extrusion load in thesquare die forward extrusion of circular-shaped bars fromregular polygonal billets. Chitkara and Celik [13] studiedthe extrusion of nonsymmetric T-shaped sections and theirpositioning to get better qualitative extruded product. Wuand Hsu [14] presented a model based on upper boundtheorem to analyze the extrusion of composite clad rodswith nonaxisymmetric cross section. Kar and Das [15]reported some experimental results of section extrusionfrom square billet through square die. Kar et al. [16]analyzed the extrusion of angle section bars from rectan-gular billets using discontinuous velocity fields based onupper bound method. Ajiboye and Adeyemi [17] extend theupper bound method to investigate the effects of die landlength on the extrusion pressure. Mostafa and Hossein [18]conducted series of experiments to get L sections by directextrusion using two different streamlined dies with andwithout intermediate sections. Noorani et al. [19] used acombined upper bound and slab method to estimate thedeformation load for cold rod extrusion of aluminum andlead in an optimum curved die profile.

Spatial elementary rigid regions (SERR) technique,based on discretization of the deformation zone into rigidtetrahedral blocks, is applicable to the problems where thedie walls are planer in nature. The main objective of thepresent study is to extend the above-mentioned investiga-tion to three-dimensional extrusion and numerically predictthe influences of the semicone angle of the die, theextrusion ratio, and the friction factors. The extrusion ofangle section from round billet (Fig. 1) through a straightconverging die is selected as working example.

2 SERR technique

SERR is based on discretizing the deformation zone intobasic tetrahedral rigid blocks separated from each other byplanes of velocity discontinuity. Each rigid block has itsown internal velocity vector consistent with the boundingconditions. Thus, if there are N rigid blocks, then thenumber of unknown internal velocity vectors is also N(thus, 3N spatial velocity components). The velocity atentry to the deformation zone (the billet velocity) is

506 Int J Adv Manuf Technol (2010) 49:505–512

considered to be prescribed, and the velocity at the exit hasa single component since its direction is known from thephysical description of the problem. Therefore, the totalnumber of unknown velocity components in the globallevel becomes 3N+1. All these unknown velocity compo-nents can be uniquely determined by applying the masscontinuity condition to the bounding faces of all thetetrahedral rigid blocks taken together. It may be noted thatthe set of velocity equations so generated becomesconsistent and determinate if and only if the SERR blocksare tetrahedral in shape, so that, the number of triangularbounding faces automatically becomes 3N+1.

To illustrate the application of the above principles, letthe ith bounding face in the assembly of tetrahedrons be

f x; y; zð Þ � a1ixþa2iyþa3izþ1 ¼ 0 ð1ÞThe coofficiants a1i, a2i, and a3i in Eq. 1 above can bedetermined by specifying the coordinates of the threevertices of this triangular face. Then, the unit normal vectorto this face is

bn ¼ rf

rfj j ð2Þ

If V1 and V2 are the velocity vectors on both sides of the ithplane, the condition for continuity is

bni �V1 ¼ bni �V2 ð3ÞA determinate set of velocity equations is generated byapplying Eq. 3 to all the bounding faces in the assembly oftetrahedrons. The bounding conditions on the velocity fieldare also enforced through this equation. For example, if aface lies on a plane of symmetry then the right hand side ofEq. 3 is to be made zero to admit the condition that no massflow occurs normal to these faces. On the other hand, if aface is an entry plane, then the right hand side of Eq. 3should be bni � Vb where Vb is the billet velocity.

3 Present problem

For the sake of the present analysis, it is assumed that thecentroid of the die aperture lies on the billet axis, and nodead metal zones are formed on the sides of the die orifice.As mentioned earlier, the SERR technique can be appliedwhere there are plane boundaries. Hence, the curvedsurface is to be replaced by planer surfaces so as toaccommodate the SERR analysis. To approximate thecircular cross section of the billet into a regular polygon,the cross-sectional areas of the billet and the approximatingpolygon must be maintained equal. This condition iswritten as:

pR2 ¼ M1

4L2Cot

q2

� �ð4Þ

For our analysis, the round billet has been approximatedto a regular polygon of 24 sides (as there is a negligiblechange of final computed value by further increasing thesides, shown in Fig. 2). Since the angle section has one foldof symmetry, one half of the deformation zone (domain ofinterest) can be considered for the analysis. The subzonesof deformation can be delineated in the domain of interestby taking suitably located floating points as proposed byKar and Das [15]. Figure 3 shows one half of thedeformation zone (M=24) with one floating on the planeof symmetry (1-2-16-15), and all the corner points of thedie orifice are joined to them. The resulting pyramid, prism,and tetrahedrons are the ultimate deformation subzone forthis SERR formulation. This single-point formulation givesrise to three pyramids (2-3-16-17-19, 8-9-17-18-19 and 14-1-18-15-19) and ten tetrahedrons (3-4-17-19, 4-5-17-19, 5-6-17-19, 6-7-17-19, 7-8-17-19, 9-10-18-19, 10-11-18-19,11-12-18-19, 12-13-18-19 and 13-14-18-19). The double-point formulation (Fig. 4), one point on the plane ofsymmetry and other one at an arbitrary position in the

DIE

Z

X

Y

BILLET

a) Schematic diagram b) Half sectional view

BILLET

CONTAINER

DIERAM

PRODUCT Z

Fig. 1 Direct extrusion of anglesection from round billet. aSchematic diagram. b Half-sectional view

Int J Adv Manuf Technol (2010) 49:505–512 507

deformation zone, gives two prism (2-3-16-17-19-20 and 1-14-15-18-19-20), one pyramid (8-9-17-18-20), and tentetrahedrons (3-4-17-20, 4-5-17-20, 5-6-17-20, 6-7-17-20,7-8-17-20, 9-10-18-20, 10-11-18-20, 11-12-18-20, 12-13-18-20 and 13-14-18-20).

Single-point formulation results in 16 tetrahedrons(SERR blocks), and the total number of alternative waysit can be obtained are eight (2� 2� 2). All these subzones

are interconnected and have common triangular faces.When continuity condition is applied to all the 49 boundingfaces of these 16 SERR blocks, equal number of velocityequations can be obtained.

Double-point formulation gives rise to 18 tetrahedrons,and the total number of alternative schemes is 72(6� 6� 2). These 18 SERR blocks have 55 boundingfaces, and when continuity condition is applied to all thefaces, equal number of velocity equations can be obtained.Though more number of subzones can be generated addingfurther floating points, double points is considered here asthere is no further improvement of final computed value byincreasing the number of floating points. The discretizationdetails are summarized in Table 1.

Here, it is to be noted that the length of the die is takenas per the equivalent semicone angle. The equivalentsemicone angle is defined as the semicone angle of aconical die where the reduction area is the same as that ofpolygonal sections. The assumptions made in this analysisare the length of deformation zone is same as that of die, nodead metal zone is formed inside the die or container, andthe billet material is perfectly rigid plastic.

4 Three-dimensional modeling of metal flowusing reformulated SERR technique

In Fig. 3, 1-2-3…-14 is the billet section, and 15-16-17-18is the die orifice (for illustration, one half of thedeformation zone for single-point formulation with M=24

24

7

8

9 1

10

19

16

17 15

18

3 5

6

11121314

20

Fig. 4 One half of the deformation zone for double point formulation(M=24). Subzones are 1 prism 2-3-16-17-19-20, 2 prism 1-14-15-18-19-20, 3 pyramid 8-9-17-18-20, 4 tetrahedron 3-4-17-20, 5 tetrahe-dron 4-5-17-20, 6 tetrahedron 5-6-17-20, 7 tetrahedron 6-7-17-20,8 tetrahedron 7-8-17-20, 9 tetrahedron 9-10-18-20, 10 tetrahedron 10-11-18-20, 11 tetrahedron 11-12-18-20, 12 tetrahedron 12-13-18-20, 13tetrahedron 13-14-18-20

18

24

7

8

9 1

10

19

16

17 15

3 5

6

11121314

Fig. 3 One half of the deformation zone for single-point formulation(M=24). Subzones are 1 pyramid 2-3-16-17-19, 2 pyramid 8-9-17-18-19, 3 pyramid 14-1-18-15-19, 4 tetrahedron 3-4-17-19, 5 tetrahedron4-5-17-19, 6 tetrahedron 5-6-17-19, 7 tetrahedron 6-7-17-19, 8 tetra-hedron 7-8-17-19, 9 tetrahedron 9-10-18-19, 10 tetrahedron 10-11-18-19, 11 tetrahedron 11-12-18-19, 12 tetrahedron 12-13-18-19, 13tetrahedron 13-14-18-19

0 10 20 30 40Number of sides of the approximating polygon

0

2

4

6

8N

on-d

imen

sion

al e

xtru

sion

pre

ssur

e,

Area reduction = 90%

= 80%

= 70%= 60%= 50%= 40%

Equivalent semi-cone angle=25 degreeFriction factor=0.00Double point formulatin

Pav σ o

Fig. 2 Variation of extrusion pressure with number of sides ofapproximated regular polygon

508 Int J Adv Manuf Technol (2010) 49:505–512

is taken into account). Surfaces like 2-3-16-17, 3-4-17, andthe eleven other similar planes are the friction (die metalcontact) surfaces. The deformation zone is obtained byjoining points 1, 2, 3, ––14 and 15, 16, 17, 18 to thefloating point “19” on the symmetry plane (1-2-16-15). Ascan be noticed, there are three pyramidal and ten tetrahedralsubzones of deformation. Just as the elementary rigidregions are planar in two-dimensional problems, the three-dimensional (or spatial) elementary rigid regions aretetrahedral, as proposed by Gatto and Giarda [9]. Apyramidal subzone can be split into two tetrahedral blocksby a diagonal of the quadrilateral base. In case of morecomplicated problems, like extrusion of I- and T-sections,the number of floating points may be more, and in suchcases, prismatic subzones may result. In a similar manner, aprismatic subzone can be discretized into three tetrahedronsin six different ways. Thus, the deformation zone ultimatelybecomes an assembly of tetrahedrons separated from eachother by planes of velocity discontinuity. As the metal(rigid) push to the deformation zone, it passes throughdifferent elementalized deformation zone (tetrahedralblocks), its velocity changes (termed as velocity disconti-nuities), and ultimately comes out through the orifice. Bycalculating and optimizing the energy requirements forthese velocity changes and at friction surfaces, we can findthe extrusion load requirements (as per SERR technique).

For easy visualization of the metal flow and the relationexisting among the various internal velocity vectors, theprismatic subzone 2-3-16-17-19-20 is isolated from Fig. 4 andredrawn in Fig. 5a. The characteristics of the facesconstituting this subzone are explained in the figure itself.The discretization of this subzone into three tetrahedralblocks according to a typical scheme is shown in Fig. 5b. Intetrahedron 2-20-16-19, as the face 2-16-19 lies on a plane ofsymmetry, hence the internal velocity vector for this SERRblock is parallel to the plane. Secondly, the face 2-19-20 isone of the planes through which the billet enters thedeformation zone. So the internal velocity vector is relatedto the billet velocity by applying Eq. 3 to this face. Thirdly,the plane 19-20-16 is the plane through which the extrudedproduct goes out from the deformation zone. Finally, theplane 2-20-16 is an internal plane separating the first andsecond tetrahedrons of the subzone under consideration.

Thus, the internal velocity vector of the first tetrahedron isrelated to that of the second by applying Eq. 3 to theconnecting face 2-20-16. Similarly, the internal velocityvector for the second tetrahedron (2-3-16-20) is related tothat of the third tetrahedron (3-20-16-17) by applying Eq. 3to the connecting plane 3-20-16. The face 2-3-16 and 3-16-17 are extrusion die metal faces admitting no mass flow indirection normal to itself. Hence, the internal velocity vectoris also parallel to this face. These conditions are enforced bytaking the right-hand side equal to zero when Eq. 3 isapplied to these faces. Continuing in this way, the internalvelocity vector for the first tetrahedron of the secondsubzone (tetrahedron 3-4-17-20) is related to that of thethird tetrahedron (3-20-16-17) of the first subzone byapplying Eq. 3 to the connecting face 20-17-3. In thismanner, all internal velocity vectors are interrelated throughapplication of Eq. 3 to different faces of all the SERR blocksin the global system. The exit velocity is related to theinternal velocity vector of a tetrahedron if an exit planecontains one of the faces of this tetrahedron.

Table 1 Summary of discretization schemes for one half of the deformation zone (M=24)

Item Single-point formulation Double-point formulation

Type of subzones 3 pyramids and 10 tetrahedrons 2 prisms, 1 pyramids, and 10 tetrahedrons

No. of SERR blocks 3� 2þ 10� 1 ¼ 16 2� 3þ 1� 2þ 10� 1 ¼ 18

No. of discretization schemes 2� 2� 2 ¼ 8 6� 6� 2 ¼ 72

No. of triangular bounding faces 49 55

No. of velocity components 16×3=48 for 16 SERR+1 at exit, total=49 18×3=54 for 18 SERR+1 at exit, total=55

(a)

(b)

2

17

16

19

3

20

2

17

16

19

3

20

Fig. 5 Interconnection of SERRblocks. a Subzone prism 2-3-16-17-19-20. b Discretization ofsubzone (tetrahedron 2-20-16-19, 2-3-16-20 and 3-20-16-17)

Int J Adv Manuf Technol (2010) 49:505–512 509

5 Application of the upper bound theorem

In the upper bound method, the problem is posed as theminimization of the functional for the rate of energy. In ourcase, the functional J is defined by

J¼J1 þ J2 þ J3 ð5Þ

where

J1 ¼ 2soffiffiffi3

pZv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2�"ij

�"ij

rdv

¼ work dissipated for internal deformation

ð6Þ

J2 ¼ soffiffiffi3

pZAi

ΔVij jdAi

¼ work dissipated at surfaces of velocity discontinuity

ð7Þ

J3 ¼ m soffiffiffi3

pAfj

jΔVj j dAfj

¼ work dissipated due to friction at die� work

piece interface jth faceð Þ

ð8Þ

In the present formulations with a discontinuous velocityfield, the strain rate components �

"ijare all zero inside therigid blocks. This leads to

J1 ¼ 0 ð9Þ

Since, velocity discontinuity |ΔVi| and |ΔVj| are constantover all the faces, it can be written as:

J ¼ soffiffiffi3

pX

ΔVij jAi þ m ΔVj

�� ��Afj

� � ð10Þ

The nondimensional average extrusion pressure is then canbe written as

Pav

so¼ J

AbVbsoð11Þ

6 Solution process

A comprehensive computational model is developed to makean upper bound analysis for the extrusion of angle section.The solution process consists of the following steps:

1. Generation of data for each triangular face in theassembly of tetrahedrons

2. Determination of the coefficients of equations repre-senting the bounding faces of the tetrahedral blocks

3. Determination of the coefficients of the velocityequations applying the mass continuity condition tothe faces

4. Computation of the deformation work by Eq. 10 andthe normalized extrusion pressure by Eq. 11

5. Optimization of the normalized extrusion pressureusing a multivariate unconstrained optimization rou-tine. The unknown coordinates of the floating pointsserve as the optimization parameters to minimize theextrusion pressure.

7 Results and discussion

Computations are carried out for all the eight globaldiscretization schemes of single-point formulation, and thescheme giving the least upper bound is identified. Thediscretized deformation zone corresponding to the leastupper bound is named here as the optimum configuration.The optimum configuration in case of double-point formu-lation is also similarly determined testing all 72 schemes.Table 2 gives a comparison of the computed results of thetwo formulations for their respective optimum configura-tions. It is obvious from this table that the double-pointformulation gives the best results. Therefore, this formula-tion only is used for further computation.

For the single-point formulation, the floating point lieson the symmetry plane (1-2-16-15). Thus, it has twoundetermined coordinate (z-coordinate and x-coordinate).These two undetermined quantities serve as the optimiza-tion parameters to minimize the extrusion pressure for thisformulation. In case of the double-point formulation, thereare two floating points, one on the symmetry plane(z-coordinate and x-coordinate undetermined), and thesecond floating point arbitrarily located in the deformationcavity has three undetermined coordinates (x-, y-, and z-

Table 2 Comparison of results for the different formulations (M=24,m=0.25)

Area reduction, % Single-pointformulation

Double-pointformulation

60 2.481 2.265

65 2.646 2.501

70 3.178 2.819

75 3.562 3.259

80 4.021 3.891

85 5.316 4.860

90 6.820 6.450

510 Int J Adv Manuf Technol (2010) 49:505–512

coordinates). These five undetermined coordinates serve asthe five optimization parameters for this formulation.

The optimum configuration (corresponding to the leastupper bound) is utilized for computation of normalizedextrusion pressure variation with equivalent semicone anglein degrees and percentage of area reduction at differentfriction factor. Figure 6 shows that the nondimensionalextrusion pressure increases with area reduction and alsowith friction. It is obvious from the fact that, with theincrease of area reduction and friction, the redundant workincreases. Figure 7 indicate that the extrusion pressuredecreases with increase of equivalent semicone anglebefore reaching a minimum value and moves up withfurther increase of equivalent semicone angle. It is alsoclear from the figure that the optimal semicone angle(requires minimal extrusion pressure) increases with theincrease of friction. This is probably due to the fact that theredundant work is minimal at a particular combination ofequivalent semicone angle and friction factor for the sectionconsidered. The present findings can be used to predict theforming stress and optimal die shape for designing thesectioned die, assessing the frictional condition eitherthrough an empirical way or a simulation test.

Comparison of the present solution is also made with theanalytical results of Kar and Das [15] and experimentalresults of Kar et al. [16] at different area reductions (Fig. 8).In those experiments, square billets of commercial lead areextruded through the rough square dies. The approximationof the curved surface to planer surfaces, assumed velocityfield, and the three-dimensional nature of the presentproblem give higher value than the experimental results.This higher value obtained by this study is due to the fact

that the computed results give upper bounds rather than theexact.

The accuracy of the predicted results by the proposedtechnique is about 20% which is adequate for a productionprocess. And the most important is that the computationaltime is very minimal (about 60 s), which is much less thanfinite element method (FEM). This ensure that, where thestress field is needed, an upper bound solution gives thecomplete solution relatively more accurately and uses lessCPU time. Thus, the usefulness of the upper bound analysishas not diminished with more and more application of FEM.

0 10 20 30 40Equivalent semi-cone angle, degree

0

2

4

6

8

Non

-dim

ensi

onal

ext

rusi

on p

ress

ure,

Area reduction = 60 %

m=0.1

m=0.0

m=0.2

m=0.4m=0.3

Pav σ o

Fig. 7 Variation of extrusion pressure with die angle

40 60 80 100Percentage area reduction

0

2

4

6

8

Non

-dim

ensi

onal

ext

rusi

on p

ress

ure,

Equivalent semi-cone angle=25 degree

m=0.0m=0.1m=0.2

m=0.3m=0.4

Pav σ o

Fig. 6 Variation of extrusion pressure with reduction of area

40 60 80 100Percentage area reduction

0

2

4

6

8

Non

-dim

ensi

onal

ext

rusi

on p

ress

ure, Present solution, m=0.3

Theoretical results (Kar and Das,1998)Experimental results (Kar et al., 2002)

Pav σ o

Fig. 8 Comparison of the present solution

Int J Adv Manuf Technol (2010) 49:505–512 511

8 Conclusions

From the present investigation, the following conclusionscan be drawn:

1. Among the two SERR formulations carried out in thisstudy, the double-point formulation gives the lowestupper bound to the extrusion pressure for extrudingangle-section bars.

2. Using the proposed technique, the optimal die geom-etry (equivalent semicone angle) which requires aminimal forming stress can be obtained for differentreduction of areas and friction conditions.

3. The extrusion pressure decreases with increase ofequivalent semicone angle and reaches a minimumvalue. Further increase of equivalent semicone anglemakes the extrusion pressure to move upward.

4. Comparison made with existing experimental resultsshows that the present solution can predict reasonableupper bound extrusion pressure.

5. The present method can be extended to obtain thesolution of generalized problems of nonaxisymmetricextrusion or drawing through converging dies.

References

1. Juneja BL, Prakash R (1975) An analysis for drawing andextrusion of polygonal sections. Int J MTDR 15:1–30

2. Chitkara NR, Adeyemi MB (1977) Working pressure and deforma-tion modes in forward extrusion of I and T shaped sections fromsquare sludge. Proc. 8th IMTDR Conf., London: 39–46

3. Nagpal V, Billhardt CF, Altan T (1979) Lubricated extrusion of"T" section from aluminium, titanium and steel using computeraided techniques. Trans ASME Series B J Eng Ind 101:319–324

4. Nagpal V, Altan T (1975) Analysis of the three-dimensional metalflow in extrusion of shapes with the use of dual stream function. Proc.Third N. Am. Met. Res. Conf., Pittsburgh, Pennsylvenia: 26–40

5. Basily BB, Sansome DH (1979) Some theoretical considerationsfor the direct drawing of section rod from round bar. Int J MechSci 18:201–209

6. Yang DY, Lee CH (1978) Analysis of three-dimensional extrusionof sections through curved dies by conformal transformations. Int.J. Mech Sci 20:541–552

7. Prakash R, Khan OH (1979) An analysis of plastic flow throughpolygonal converging dies with generalised boundaries of thezone of plastic deformation. Int J MTDR 19:1–19

8. Boer CB, Schneider WR, Avitzur B (1980) An upper boundapproach for the direct drawing of square section rod from roundbar. Proc 20th Int MTDR Conf, Birmingham: 149–156

9. Gatto F, Giarda A (1981) The characteristics of three-dimensionalanalysis of plastic deformation according to the SERR method. IntJ Mech Sci 23:129–148

10. Gunasekera JS, Hosino S (1982) Analysis of extrusion or drawingof polygonal sections through straightly conversing dies. TransASME Series B J Eng Ind 104:38–45

11. Maity KP, Kar PK, Das NS (1996) A class of upper-boundsolutions for the extrusion of square shapes from square billetsthrough curved dies. J Mater Process Technol 62(1–3):185–190

12. Kim DK, Cho JR, Bae WB, Kim YH, Bramley AN (1997) Anupper bound analysis of the square-die extrusion of non-axisymmetric sections. J Mater Process Technol 71(3):477–486

13. Chitkara NR, Celik KFA (2001) Extrusion of non-symmetric T-shaped sections on analysis and some experiments. Int J Mech Sci43:2961–2987

14. Wu CW, Hsu RO (2000) Theoretical analysis of extrusion ofrectangular, hexagonal and octagonal composite clad rods. Int JMech Sci 42:473–486

15. Kar PK, Das NS (1997) Upper bound analysis of extrusion of I-section bars from square and rectangular billets through squaredies. Int J Mech Sci 39(8):925–934

16. Kar PK, Sahoo SK, Das NS (2002) Section extrusion fromrectangular billet: experimental verification of the SERR analysis.Indian J Eng Mater Sci 9:35–40

17. Ajiboye JS, Adeyemi MB (2007) Upper bound analysis forextrusion at various die land lengths and shaped profiles. Int JMech Sci 49(3):335–351

18. Ketabchi M, Seyedrezai H (2007) Energy analysis of L-sectionextrusion using two different streamlined dies. J Mater ProcessTechnol 189(1–3):242–246

19. Noorani-Azad M, Bakhshi-Jooybari M, Hosseinipour SJ, Gorji A(2005) Experimental and numerical study of optimal die profile incold forward rod extrusion of aluminum. J Mater Process Technol164–165:1572–1577

512 Int J Adv Manuf Technol (2010) 49:505–512