a theoretical and experimental study for the load … · a theoretical and experimental study for...

A THEORETICAL AND EXPERIMENTAL STUDY FOR THE LOAD OPTIMIZATION OFGEAR-LIKE PROFILES BY USING FORWARD AND LATERAL EXTRUSION

Tahir Altinbalik and Önder AyerFaculty of Engineering, Department of Mechanical Engineering, Trakya University, Edirne, Turkey

E-mail: [email protected]

Received December 2013, Accepted July 2014No. 13-CSME-202, E.I.C. Accession 3660

ABSTRACTThe main purpose of this research is to investigate the minimum deformation load by selecting a suit-able forming method for manufacturing of gear-like sections and to compare the load estimation methodsbetween Upper Bound Analysis and DEFORM-3D. Forward and lateral extrusion were chosen as two dif-ferent forming methods. The effect of die transition geometry on deformation load was also investigatedby straight tapered and cosine profiles. A newly kinematical admissible velocity field to analyze differentprofiles of extrusion dies was proposed by upper bound analysis. Al 1070 was used as working material.Experiments using five sets of dies with gear-like form were performed, and the measured forming loadresults were compared with the predictions of the theoretical solutions. Experiments were carried out on the150 metric ton hydraulic press.

Keywords: extrusion; aluminum; upper bound; DEFORM-3D.

UNE ÉTUDE THÉORIQUE ET EXPÉRIMENTALE POUR L’OPTIMISATION DE LA CHARGE DES PROFILS DE PIÈCES D’ENGRENAGE EN UTILISANT LES EXTRUSIONS

AVANCÉE ET LATÉRALE

RÉSUMÉLes objectifs principaux de cette étude est la détermination de la méthode adéquate pour la fabricationde pièces comme des engrenages, de faire l’estimation de charge pour un minimum de déformation, etde comparer les méthodes d’estimation de charge entre l’analyse de borne supérieure et DEFORM-3D.On a opté pour les extrusions avancée et latérale comme deux méthodes différentes de formation. L’effetgéométrique de la charge sur la déformation, a aussi été investigué sur des profils droits et cosinus. Afind’analyser les différents profils d’extrusion par la méthode d’analyse de la borne supérieure, on a écritun nouvel espace de vitesse cinématique. Comme matériel d’essai, on a utilisé AL 1070. Les essais sontréalisés avec 5 différents ensembles de moulage au total et les valeurs de force mesurées sont comparéesavec les résultats théoriques. Les essais sont réalisés avec une presse hydraulique d’une capacité de 150tonnes métriques.

Mots-clés : extrusion; aluminium; borne supérieure; DEFORM-3D.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015 53

NOMENCLATURE

r,θ ,z Cylindrical coordinate systemA,B,C,D Die inlet and outlet definition parametersRinlet Inlet profile of the dieRoutlet Outlet profile of the dieRtop Straight tapered die transition profileRcos Cosine curved die transition profileh,H Instantaneous height of billeth0 Initial height of billetVr,Vθ ,Vz Radial, circumferential and axial velocity componentεi j Strain rate components¯εeff Effective strain definitionrr Root circle radiusrp Pitch circle radiusrt Addendum circle radiusα,β ,θ ,φ Angular limits of the material flow regionsWd ,Ws,Wf Work dissipation of plastic deformation, shear and frictional components|∆V | Velocity discontinuityF Deformation loadV0 Press velocityσ ,σm Flow stress of the materialL Length of the diem Friction factorR0 Initial diameter of the billetΩ(θ ,z) Angular velocity definition

1. INTRODUCTION

Gears are important machine element in many types of equipments such as all automotive products, machinetools or general gear boxes. Spur gears are the simplest type of gears. Gears are manufactured by machiningor by a combination of conventional hot forging with machining, which is expensive and requires a lot ofmanufacturing time. Instead of conventional machining, plastic deformation processes can be chosen forthe manufacturing of gear forms because of their advantages. Extrusion is a plastic deformation processin which a block/billet is reduced in cross-section by forcing it trough the die opening of a small cross-sectional area than that of the original billet [1]. Extrusion process can be used for most materials suchas lead, tin, aluminum alloys, copper, titanium, molybdenum, vanadium and steel. As known an optimumdie design will cause sound material flow. Researchers can easily develope techniques to design curved dieprofiles for forward extrusion. Curved dies cover an area of application from simple axisymmetric productsto complicated sections such as gears and splines [2].

Sometimes analytical methods can’t simulate processes exactly due to the complex nature of the deforma-tion processes. However an acceptable solution may be obtained by the computer techniques such as finiteelement method (FEM), upper bound element technique and geometric methods. A number of analyticalstudies have been carried out during the past few decades to estimate the deforming load for the extrusionof metal through several die shapes such as curved or conical etc. In the mentioned studies researchers an-alyzed extrusion processes of different sections using slab method, finite element method and upper boundapproach [3, 4]. The upper bound theorem was first formulated by Prager and Hodge and later modified byDrucker et al to include velocity discontinuities [5]. Many attempts have been made to calculate the loadusing the upper bound method which is a very useful and economical method to obtain knowledge and in-formation on the die designing of metal forming processes. According to Bramley [5] upper bound analysis

54 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

can provide rapid but approximate simulation and preform design when compared with currently availablefinite element based methods. Gunasekera and Hoshino [6] obtained an upper-bound solution for the extru-sion of polygonal sections by converging dies. A kinematically admissible velocity field for a generalizedthree-dimensional extrusion of arbitrarily sections was derived in which all the velocity components wereexpressed in general function forms by Han et al. [7]. Extrusion of clover section from round billets hasbeen chosen as a computational example. The extrusion die assumed to have the cosine profile to extrudecircular billet to circular shape and an upper bound solution done by Narayanasamy [8]. Choi [9] studiedthe two types of forgings of helical gears by using the upper-bound method and experiments have beencarried out by lateral extrusion. Song and Im [10] developed a computer aided design system for manu-facturing a spur gear in cold forward extrusion. Then, in another study Song and Im [11] investigated theapplicability of the developed system in [10] for the cold forward extrusion of solid and hollow spur gearsby carrying out experiments with forward extrusion. A theoretical and experimental study was performedby Çan and Misirli [12] choosing a gear-like section for comparing of the methods of lateral extrusion andclosed-die forging. New kinematically admissible velocity fields for three dimensional extrusion of cloversections were suggested for both to obtain the optimum die length of shaped inlet dies and determinationof the extrusion load by Altinbalik [13]. Recently, Karami and Abrinia [14] developed a new generalizedformulation based on the upper bound method for the three-dimensional analysis of the forward extrusionof shapes such as L-shape, rectangle, hexagon and spur gear form. The method proposed by Karami hasbeen formulated in such a way as to present a more realistic definition of deforming region by eliminatinginternal and velocity discontinuities at the entry and exit surfaces which are defined by three-dimensionalcurved surfaces.

As mentioned before accurate prediction of loads, strains and stresses in complex bodies such as gearscan also be obtained by FEM. Many studies have been performed during the past few decades to determinethe deformation load for the extrusion of metal through a variety of die shapes such as curved or conical byusing FEM. DEFORM-3D is a powerful FEM based programme. It is a practical and efficient programme topredict the material flow in industrial forming operations without the cost and delay of trial-error approach.Chen et al. [15] performed DEFORM-3D finite element code to examine the separately influences of the diesemi-angle, the extrusion ratio and the friction factors on the plastic deformation behaviour of an aluminumbillet during its axisymmetric extrusion through a conical die. In order to explain the non-steady stateextrusion process by means of the extrusion load, He et al [16] also carried out a theoretical investigationbased on DEFORM-3D FEM simulation.

On the other hand, lateral extrusion is a branch of extrusion in which the material, placed in an injectionchamber, is injected into a die cavity in a form which is prescribed by the exit geometry and is charac-terised by combined axial and radial flow of material to fill the die cavity [17]. Less load requirement,comparing the closed-die forging, is the major advantage of lateral extrusion especially for the net-formingof flange-typed parts. Balendra [18, 19] investigated effects of process parameters on metal flow and loadrequirement for suitable geometries. A number of researchers simulated the lateral extrusion of solid billetsfor different geometries by using FEM. The FEM on the combination of lateral and forward or backwardextrusion was also investigated by Lee et al. [20] and Choi et al. [21] with the effect of punch diam-eter and the friction factor on the forming load parameters. The lateral extrusion of segmented flangessuch as splines and spur gear forms are very scarce research areas in the literature. By this time, Çan etal. [22] and Altinbalik [23] used Upper Bound Method on the load requirement of lateral extrusion ofgear-like parts.

Detailed UB analysis and comparison of Upper Bound solution with solid model programs for gears andgear-like elements which are produced by forward extrusion and lateral extrusion are found to be missing inthe light of the literature survey. In the presented study, a theoretical and experimental investigation of spur


Table 1. Die lengths and solution methods for different die profiles.Deformation Type Die Length or Height Solution Model

Cosine Curved Die for Forward ExtrusionL = 15 mm Upper Bound

DEFORM-3D

L = 20 mm Upper BoundDEFORM-3D

Straight Tapered Die for Forward ExtrusionL = 15 mm Upper Bound

DEFORM-3D

L = 20 mm Upper BoundDEFORM-3D

Lateral Extrusion H = 15 mm Upper BoundDEFORM-3D

Fig. 1. Cylindrical coordinate system of forward extrusion.

gear form is performed in detail. Gear forms of the same dimension were produced by forward extrusionand lateral extrusion. Four different die geometries were chosen for the forward extrusion; including straighttapered transition profile and cosine transition profile, and each of them having two different die lengths.Both the die transition profile and the die length have been changed at the same time different from theinvestigated literature. Die transition profiles and die lengths which are determined according to deformationmethod are given in Table 1. Then a new kinematically velocity field has been proposed for the five differentsituations. Besides, two manufacturing processes were simulated by a commercial FEM programme calledDEFORM-3D in order to predict the extrusion load requirement and compare both the UB results and theexperiments.

2. THEORETICAL ANALYSIS OF FORWARD EXTRUSION

The upper bound analysis predicts a load that is at least equal to or greater than the exact load needed tocause plastic deformation. It considers a kinematically admissible velocity field which satisfies the incom-pressibility condition and the velocity boundary condition for describing the metal flow. There are manykinematically admissible velocity fields for a particular forming process however the best solution is deter-mined by minimizing the total energy rate.

In the presented study outlet profile is obtained by using two different transition profiles called straighttapered and cosine curved profile as seen in Fig. 1. Inlet and outlet profile can be defined mathematically as


in Eq. (1). Four different parameters are used for the purpose of describing arbitrary shapes.

Routlet(θ ,z) = A+Bcos(4θ) ·Rinlet(θ ,z) =C+Dcos(4θ). (1)

Parameters A,B,C and D in the inlet and outlet sections and are varied the shape of the die. Different billetsand product sections can be combined related with these parameters. For this study, inlet profile is chosenas 28 mm diameter and the first equation is defined by parameter C alone because of its circular shape. Theoutlet profile is defined by two parameters where A is 8.7 and B is 3.5 according to the die set outlet profile.

Rinlet(θ ,z) = 14, Routlet(θ ,z) = 8.7+3.5cos(4θ). (2)

Transition profiles of this study are cosine curved and straight tapered. The transition profiles givenin Eq. (3) define the die profile and transition profile so as to produce satisfactory material flow and thedeformation load.

Rcos(θ ,z) =(Rinlet +Routlet)

2+

(Rinlet−Routlet)

2cos

(θzL

),

Rtap(θ ,z) = Routlet +(Rinlet−Routlet)(L− z)

L(3)

2.1. VELOCITY FIELDS of COSINE CURVED and STRAIGHT TAPERED DIESThe kinematically admissible velocity fields of the cosine curved and straight tapered dies are given inTable 2.

2.2. TOTAL WORK COMPONENTS of COSINE CURVED and STRAIGHT TAPERED DIESOnce effective strains are calculated from the velocity fields as given in Table 2, then total work componentswhich are deformation, shear and friction for this study can be calculated by using numerical integrationmethods, given in Table 3.

The total work is the sum of these components.

Wtotal =Wd +Ws +Wf (4)

After the total work is calculated, extrusion load can be calculated as

F =Wtotal

V0where V0 is punch velocity. (5)

3. THEORETICAL ANALYSIS OF LATERAL EXTRUSION

Velocity fields of lateral extrusion given in Table 4 are calculated by using rotational symmetry so profile isdivided into eight portions. Then each portion is also divided into six different regions. General and detailedview of velocity regions are given in Figs. 2a and 2b. On the other hand, the velocity fields for each regionwhich are given in Table 5 are used to calculate the total work components.

Total work definition of the upper bound solution for the lateral extrusion process is the sum of all regions’shear, deformation and friction components.

Wtotal = (Wd +Ws +Wf )∗8 (6)

and total deformation load is calculated by Eq. (5).


Table 2. Velocity fields of cosine curved die.

4. DEFORM-3D FINITE ELEMENT METHOD (FEM) SIMULATION PROCEDURE

In the presented study, commercial FEM based software DEFORM-3D v10.2 was used for extrusion processsimulations. DEFORM-3D developed by Scientific Forming Technology Corporation (SFTC)® in the US.In the simulations, aluminium billets were 28 mm in diameter and 45 mm in length. Die components wasassumed as rigid bodies and workpiece was chosen as plastic material obeying Von-Mises flow rule and itssolid mesh is approximately 6000 nodes and 26000 tetrahedral elements. The friction factor was definedas 0.4 for all contact surfaces between material and die components. Punch velocity for the simulations


Velocity Components

Cosine Curved

Die

L

zBACBAC

CVVz cos4cos4cos2

0

2

0

cos4cos4cos2

sin4cos

L

zBACBAC

L

L

zBACCV

Vr

L

zC

L

zB

L

zAC

L

zA

L

zACCBA

L

zACBACCBA

L

zCBACBA

L

zCBACABL

L

zCBACAB

L

zCBABAC

L

zCBABAC

L

zCVr

V

cos1cos1

cos21cos

cos2

cos22

cos

cos

cos

cos

cos

sin47.39

22

2

2

1

2222

222222

2

1

02

Str. Tap.

L

zLBACBA

CVVz

4cos4cos

0

2

0

4cos4cos

4cos

L

zLBACBAL

rBACCVVr

222

2

122222222

2

1

02

2

222

278.6

BACAzACLLzC

CBAzACzLzCACLzLC

CBAzCLCBAzCL

CBACLzLCVr

V

Table 3. Work components of cosine curved and straight tapered dies.

was chosen as 5 mm/s and stroke was 30 mm to represent the experiment conditions realistically. Duringthe course of the simulations, frequent remeshing was developed by a result of the large and localizeddeformations.

5. EXPERIMENTAL STUDY

A series of experiments for gear-like extrusion by two different manufacturing methods were performedin order to obtain the optimum die surface design and compare the measured experimental loads valueswith those calculated from UB analysis and FEM. Firstly an extrusion container with internal diameterof 28.2 mm having an outer diameter of 55 mm and a punch of 28.2 mm were machined. The die inletgeometries for forward extrusion were chosen cosine curved and straight tapered and the length of the dieswere L = 15 mm and L = 20 mm according to the literature [13]. Schematic representation of the forwardextrusion dies are given in Figs. 3a and 3b and the die assembly for lateral extrusion are given in Fig. 3c.The dies have been designed in flexible manner and both the forward extrusion experiments and also thelateral extrusion experiments have been carried out with similar arrangement. For the lateral extrusionexperiments the exit geometry given in Eq. (1) have been machined on a die having a height of 15 mm.Dies were machined at CNC due to geometrical complexity. Dies, containers and the punches were madefrom 1.2344 DIN hot worked tool steel and hardened to 54 HRC. Photographical view of the dies and theexperimental set-up are shown in Figs. 4a and 4b, respectively. Al 1070 used as experimental material forinvestigation. For the forward extrusion aluminium specimens were cut from the bar and machined to 28 mmdiameter and 45 mm in length. For the lateral extrusion the initial diameter of specimens was 10 mm, whichis assumed to be the dedendum circle of the gear. Simple compression test was performed to obtain the


Fig. 2. Detailed view of lateral extrusion zones. (a) General view of the velocity regions. (b) Detailed view and regionsof primarily deformation zones.

stress-strain relationship of the material. Its equation is

σ = 144ε0.162 (MPa). (7)

Experiments were carried out on a 150 metric ton hydraulic press with constant ram speed of 5 mm/sec.A PLC interfaced pressure transducer supplied a database load vs. displacement values . The upper plateof the press reached the previously determined punch stroke, the experiment was stopped by means of thesoftware. Data files about load versus stroke were stored. Photographs of the forward and lateral extrudedparts appears in Fig. 4c.

6. RESULTS AND DISCUSSION

Comparison of forming loads versus the punch stroke between experimental and theoretical results whichobtained from DEFORM-3D and UB solution for different die profiles are shown in Figs. 5–8. As expectedthe forming load increases with increasing the punch stroke due to the increase in frictional surfaces. Theforming load decreases until the last stage and then increases again sharply as shown in figures becauseof the upper and lower dead zones are facing each other. So, the curves emphasize the three stages ofextrusion called: compression, steady and unsteady stages. The theoretical predictions are always higherfor UB solution and somewhat higher for DEFORM-3D than experimental ones. In Fig. 5 theoreticallycalculated load values and experimental measured are shown for straight tapered profiled die having 15 mmdie length to obtain the gear-like section. As known that the prediction of the maximum extrusion loadis important so as to design all forming parameters such as die design and tool material selection. In thediagram an acceptable agreement can be seen between theoretical results and the experiments. Upper boundsolution is found to be higher than the experimental results for whole the process except the final stage.Because at the last stage of the process assumed kinematically admissible velocity fields do not valid. The


Table 4. Velocity fields of lateral extrusion.

value of the maximum load calculated by the UB, which is significant for determining the press capacity,deviates from the value obtained experimentally only by 9% and this discrepancy is acceptable. On theother hand, DEFORM-3D modelling agrees with experimental data also. As seen the diagram calculatedFEM results deviate only 5.8% from the experimental results for the maximum extrusion load. What leadsto this difference is that the assumptions made in the program don’t entirely match the actual materialproperties. The effect of die length on deformation load was also investigated and for this purpose thenumber of experiments were repeated for each die sets for L = 20 mm die length. Loads against punchstroke measured as a result of the experiments are given together with the analytical solutions for straighttapered die for L = 20 mm die length in Fig. 6. An increase in die length from 15 mm to 20 mm cause a17% increase in deformation load. While load caused friction forces increases about 36% load exerted dueto friction in total load increases from 14 to 18%. As seen in the load-stroke diagram given in Fig. 6 the valueof the maximum load calculated from the UB deviates from the value obtained experimentally only by 8%.On the other hand according to the Fig. 6 the difference between measured and obtained from DEFORM-3D values at the maximum is about 5%. As Figs. 5 and 6 are examined together it may be said that thecurve profiles of the theoretical results are quite close to each other and it agrees with the experimentalresults. In order to investigate effects on die transition profile on the deformation characteristics of thegear-like form experiments were repeated for the cosine transition profiled die L = 15 mm and L = 20 mm.Similar diagrams were obtained and at L = 15 are given in Figs. 7 and 8. As seen in Fig. 7, the UB


Vr Vt Vz

Reg.1 0 0 -Vo

Reg.2

h

rVrV

0

2

2

1102

h

rVV h

zVzV 02

Reg.3

h

BArVrV

4cos03

4

4sin

80

4cos5

2

4

4sin

4

4sin

20

3

BBA

BA

BA

h

rVV 0

Reg.4

h

rVrV

204

2

104 h

rVV h

zVzV 04

Reg.5

h

BArVh

rV

Vr 4cos0

0

5

BA

BA

BA

h

rVV

4

4sin

80

4cos5

2

4

4sin

4

4sin

222

2

05

0

Reg.6

h

BArV

h

rVrV

4cos006

B

BAA

h

rVV

4

4sin

80

4cos5

4

4sin

4

2

44

06

0

Table 5. Total work components of each region for lateral extrusion.


Fig. 3. Schematic illustration of the die assembly for extrusion processes (a) straight tapered die (b) cosine profileddie (c) die assembly of lateral extrusion.

solution gives about 7.5% higher and the DEFORM-3D solution gives 6.5% lower extrusion load than thatexperimentally recorded for the L = 15 mm die length condition of cosine transition profiled die. Theextrusion load increases with increasing die length as clearly seen in Fig. 8. With the die length extendedfrom L = 15 mm to L = 20 mm the extrusion load has risen about 18%. As similar to straight tapereddie, load required to overcome friction increased 34% with increasing die length and the load exerted dueto friction in total load increases from 14 to 17%. Theoretical solutions for the cosine profiled die having20 mm die length are compatible with experimental results. The value of the maximum load which theUB calculated, which is highly effective for determining the press capacity, differs from the recorded onlywith 7%. On the other hand, FEM analysis gives again lower results than the experimental recorded formaximum load point but regarding with Figs. 5–8, curves obtained from the DEFORM-3D agree well withthe experimental data for whole processes. DEFORM-3D solutions, similar to UB solution, gives highervalues compared to measured values during the first half of the process. So both the FEM analysis and theUpper Bound solutions are assumed to be valid. As the diagrams are examined together the extrusion loadfor cosine profiled die having 15 mm die length gives a little lower load than the other die sets. According toFigs. 5–8 that the die transition geometry does not have a significant effect on load for the same die lengthin forward extrusion. The difference of the measured load values between the cosine profiled die and thestraight tapered die in the experiments that have been carried out with die having a length of L = 15 mm is


Fig. 4. Photographic view of the dies with each part (a) dies (b) experimental set-up (c) forward and lateral extrudedparts.

Fig. 5. Comparison of theoretical and experimental extrusion load against punch stroke for straight tapered die,L = 15 mm.


Fig. 6. Comparison of theoretical and experimental extrusion load against punch stroke for straight tapered die,L = 20 mm.

Fig. 7. Comparison of theoretical and experimental extrusion load against punch stroke for cosine curved die, L =15 mm.


Fig. 8. Comparison of theoretical and experimental extrusion load against punch stroke for cosine curved die, L =20 mm.

Fig. 9. Comparison of theoretical and experimental lateral extrusion load versus punch stroke.

only about 2%. Similar situation is valid for the dies having lengths of L = 20 mm. For 20 mm die lengththe load difference between the cosine profiled and the straight tapered die is 2.5%. Appreciable differencebetween the measured load values has not been observed because the differences of the die lengths in thepresented work are small.


Fig. 10. Comparison of the deformation loads between the forward and lateral extrusion.

Four toothed gear-like parts were also forged by lateral extrusion in order to make analysis of manufac-turing method. Figure 9 illustrates theoretical extrusion loads calculated from UB and DEFORM-3D andexperimentally measured against the punch movement for lateral extruded parts. Simple lateral extrusion,the tooth formation and the corner filling stages can be seen in the diagram. The tooth formation startsat 5 mm of the punch stroke and continues until the front wall of the gap. The corner filling stage startsat 23 mm of the punch stroke and finishes end of the process. Lateral extrusion requires less deformationload than that of the forward extrusion process for manufacturing the same geometry. As it is expected, UBsolution gives higher values than measured from experiments for every stage of the process. At maximumload point, UB solution is 4.5% higher than the measured value. According to diagram DEFORM-3D solu-tion gives more suitable results comparing the forward extrusion solutions. The value of the maximum loadestimated by the DEFORM-3D is about 9% higher from the value obtained experimentally at the end of theprocess and this deviation is better than the results of the forward extrusion.

In order to compare two different manufacturing processes by means of deformation load, Fig. 10 wasdrawn. Load values obtained by two methods, which are lateral extrusion and forward extrusion in whichminimum load values were obtained, are given in Fig. 10. As seen, maximum deformation load for lateralextrusion of the gear-like form was measured as 101 kN and this value is 3.9 times less than the forwardextrusion.

7. CONCLUSION

In the presented study detailed experimental and theoretical investigations have been carried out to obtainoptimum forming process in terms of lowest load requirements of gear-like form. This has been performedfor the two forming process: lateral extrusion and forward extrusion. Besides, in addition to the experimentalwork, the process has also been analyzed by upper bound method and simulated by using DEFORM-3D andthey were compared to each other. As stated above, it is expected that upper bound method gives slightly


higher result compared to experimental results. So, predicted velocity fields and upper bound method canbe reliable in this specific work. However, results of DEFORM-3D are also be used to obtain the loadrequirements rapidly if upper bound analysis can not be done because of the lack of knowledge.

With regard to detailed analysis and experimental results, the following observations are obtained:

1. Load requirement for cosine curved transition die in forward extrusion is about only 2% less than thatof straight tapered transition die for L = 15 mm die length and only 2.5% for L = 20 mm die length.So, there is no appreciable difference between the load values obtained by forward extrusion in termsof die transition geometry of these die lengths.

2. An increase in die length from 15 to 20 mm causes 17% increase in deformation load for straighttapered die. When the die length is extended from L = 15 mm to L = 20 mm the extrusion load risesabout 18% for cosine curved die.

3. Comparing the forward extrusion with the lateral extrusion by means of the load requirements, lateralextrusion required 74% less deformation load than the forward extrusion.

4. The theoretical predictions are always higher for UB solution. The differences between the experi-mental results and the ones obtained from UB are about 8% for the forward extrusion and 4.5% forthe lateral extrusion.

5. DEFORM-3D solution gives slightly lower results for the forward extrusion and slightly higher resultsfor the lateral extrusion. The differences between the experimental results and the ones obtained fromFEM values are about 6% for the forward extrusion and 5% for the lateral extrusion.

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