16614380

Journal of Materials Processing Technology 162–163 (2005) 551–557

Calculation of the forward tension in drawing processes

E.M. Rubioa,∗, A.M. Camachoa, L. Sevillab, M.A. Sebastiana

a Department of Manufacturing Engineering, National Distance University of Spain (UNED), Juan del Rosal 12, Madrid, Spainb Department of Materials and Manufacturing Engineering, University of Malaga, Plza. EI Ejido, s/n M´alaga, Spain

Abstract

Drawing process is one of the most used metalforming process within the industrial field, particularly, in automotive and electric sectors.Then, different analytical, numerical, empirical and experimental methods have been developed in order to analyse it and to optimise it.However, exact solutions have not achieved yet due to the great number of factors involved in this type of processes and to the mathematicalcomplexity that they present. In this work, the main variants of the drawing process have been studied by different methods. Concretely,wire drawing and plate drawing have been modelled and simulated by means of the slab method (SM) and finite element method (FEM). Inaddition, the results obtained in both cases have been compared with other solutions found in the literature about these themes, particularly,with Wistreich’s solutions in wire case and with Green and Hill and the upper bound technique ones in plate drawing case.© 2005 Elsevier B.V. All rights reserved.

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eywords:Drawing process; Slab method; Finite element method; Coulomb friction

. Introduction

Drawing process is one of the most used metalformingrocess within the industrial field and, particularly, in auto-otive and electric sectors.The process consists of reducing or changing the cross-

ection of pieces such as wires, rods, bars or plates, makingass them through a die by means of a pulling force. Materials

raditionally used in this kind of manufacturing processes areluminium and copper alloys and steels.

The main variables involved in this type of processes are:ie semiangle,α, cross-sectional area reduction,r and frictionoefficient,µ or m, for representing the friction along theie-workpiece interfaces, depending on if Coulomb frictionr partial one is considered[1].

In general, the complexity of these processes and the greatumber of factors involved in them make very difficult toelect the parameter values properly[2–4]. Then, differentnalytical, numerical, empirical and experimental methodsave been developed in order to analyse the best combinationf them[5–11].

Drawing process began to have theoretical modelsciently developed thanks to the contributions of DavisDokos[12] and Hill and Tupper[13], as well as the empircal and experimental one carried out by several investigsuch as Green and Hill[5] and Wistreich[11].

Nowadays, analytical methods still continue being stuand developed in spite of numerical methods allow obtaisolutions with high precision and detail levels in the analof this type of processes.

Among the analytical methods more commonly useanalyse and simulate these processes are, for exampmogeneous deformation (HD), slip lines field (SLF), smethod (SM) and upper bound technique (UBT). Andnumerical method more widely spread in recent years,possible to mention the finite element method (FEM).

In this work, slab method and finite element method hbeen applied to calculate the drawing force necessary toout a wire and a plate drawing process. Slab method hready been used in the analysis of other metalformingcesses such as forging[14–17], rolling [10,18–20], extrusion[17,21]or deep drawing[22,23].

∗ Corresponding author.E-mail address:[email protected] (E.M. Rubio).

Slab method was initially developed by Sachs in 1928[9].It is a method very easy to apply and that does not requirebig calculation resources neither of time. It takes into accountthe homogeneous deformation of the material and the fric-

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924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserveoi:10.1016/j.jmatprotec.2005.02.122

552 E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557

tion between the workpiece material and the die-interfacesto estimate the necessary force (stress, energy or power) tocarry out the process.

Finite element method has been used as well in sev-eral studies about metalforming processes recently[24–29].The main advantages of the FEM are: the capability of ob-taining detailed solutions of the mechanics in a deformingbody, namely, velocities, shapes, strains, stresses, tempera-tures, or contact pressure distributions; and the fact that acomputer code, once written, can be used for a large va-riety of problems by simply changing the point data[30].Its main disadvantages are the bigger calculation resources,the need of having a good knowledge of the used softwareprogramme and the investments in hardware and softwareequipment.

In addition, the results obtained by the both mentionedmethods have been compared with the obtained one by Ru-bio et al.[2–4] using the upper bound technique, by Greenand Hill [5] using empirical methods, and with experimentalworks made by Wistreich[11].

2. Wire drawing

2.1. Modelling

bym zoneh

itiald an-g ef

b d byCc cient

.

Fig. 2. Stress local analysis for wire drawing.

that the friction coefficient will be low. The forces acting onan elemental frustum have been plotted inFig. 2.

The workpiece material can be considered as a rigid-perfectly plastic material then, its yield stress,Y, will be aconstant.

In finite element method case, the drawing process mod-elling has been carried out by means of the finite elementcode called ABAQUS[31].

The workpiece has been meshed with the CGAX4R el-ement. This type of elements belongs to the ABAQUS el-ement library. It is a four-node bilinear, reduced integrationand hourglass control element. An example of mesh with thistype of element is shown inFig. 3. This is exactly the meshused in this work for wire drawing analysis.

2.2. Simulations algorithms

The adimensional forward tension has been chosen as out-put variable. The main reason for making this election is thisvariable can be assimilated to the necessary energy to carryout the process (of course in no dimensional terms as well).

This fact is very useful, especially, in the selection of ma-chines and equipment to obtain a certain type of parts sincesimply multiplicand by the volume piece it is possible to

In order to make a stress local analysis of the problemeans of the slab method the plastically deformationas been modelled like it is shown inFig. 1.

There, it is possible to see a circular section wire of iniameterDi which is drawn through a conical die of semile α, to reduce its diameter toDf at the exit of the die. Th

orward tension isσz and the normal die pressure isp.The used die has a constant semiangle,α. The friction

etween the workpiece and the die interfaces is modelleoulomb’s coefficient, which is represented byµ and, in thisase, is constant. The lubrication is considered so effi

Fig. 1. Modelling of the plastically deforming zone for wire drawing
Fig. 3. FEM model by CGAX4R elements.

E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557 553

Table 1Parameters values range

Die semiangle,α 2.5–16◦Coulomb friction coefficient,µ 0–0.2Cross-sectional area reduction,r 0.20–0.40

know the necessary power to carry out the process. In thiswork, it will be very useful, as well, in order to compare theresults obtained using other different methods.

In the slab method case, making the forces balance (Fig. 2)alongz- andx-directions, neglecting differential quantitiesof second order and simplifying adequately, it is possible towrite [9]:

σzf

Y= 1 + B

B[1 − (1 − r)B] (1)

where σzf is the forward tension,Y, the yield stress,B=µ cotgα and r the cross-sectional area reduction givenby:

r =π4 (D2

i − D2f )

π4D2

i

= 1 −(

Df

Di

)2

(2)

In finite element method case, the value of the expressionσzf /Y, can be calculated by:

σzf

Y= Ff

YAf=∑

NFORC

YAf(3)

whereFf is the drawing force;Af the cross-sectional area atthe die exit;Y the yield stress of the material and; NFORC anodal variable obtained from the ABAQUS code by extrap-o

2

latedb erv

f them pre-p

ichp char-a

incea tressY

TP

DYPYP

Fig. 4. Comparison between slab method and finite element method solu-tions forr = 0.20-0.30-0.40: (a)µ = 0.10 and (b)µ = 0.20.

2.4. Results and comparison with other previoussolutions

Fig. 4shows the obtained results by both described meth-ods for cross-sectional area reductions of 0.20; 0.30 and 0.40and a Coulomb friction value equal to 0.10 and 0.20. In thatfigure, it is possible to see that FEM solutions are bigger thanSM ones. This is due to slab method only takes into accountthe terms of homogenous deformation and the friction onewhile the FEM takes into account other terms.

In Fig. 5, SM, FEM and Wistreich solutions have beenpresented forµ = 0.03 Coulomb friction coefficient and twocross-sectional area reduction. Concretely,r = 0.30 and 0.40.The main reason to select these values is that the experimentalresults found in the classical literature about these themes usethis coefficient for them[11]. In the same way that inFig. 4,FEM and Wistreich’s experimental solutions are bigger thanSM ones, and, for this smallµ value, lower and closed amongthem.

3. Plate drawing

3.1. Modelling

onef likei

lation of the stress values at the integration points.

.3. Applications

The adimensional drawing stress value has been calcuy the expressions(1) and (3) for the range of parametalues showed inTable 1.

In the finite element method case, some properties oaterial used in the process must be introduced in therocessing stage.

Particularly, the used material is an aluminium whresents a rigid-perfectly plastic behaviour and the nextcteristics collected inTable 2.

In slab method case, this information is not required sll features of the material are represented in the yield s, in this case constant.

able 2roperties of the aluminium used in the process

ensity,ρ 2700 kg/m3

oung’s modulus,E 7× 1010 Paoisson’s ratio,ν 0.33ield stress in simple tension,Y 2.8× 107 Palane strain yield stress,S(=2k) 1.155Y

In the plate drawing case, the plastically deformation zor the analysis with the slab method has been modelledt is shown inFig. 6.


Fig. 5. Comparison among slab method, finite element method and Wistre-ich solutions forµ = 0.03: (a)r = 0.30 and (b)r = 0.40.

There, it is possible to see a rectangular cross-sectionalarea plate with an initial height ofhi which is drawn throughan edged die of semiangleα, to reduce its height tohf atthe exit of the die. The drawing process is carried out underplane strain conditions then, the workpiece width,w (theplate dimension along they-axis), can be considered constantduring the whole process. The forward tension is given byσz and the normal die pressure isp, that is constant on itssurfaces.

Equal to the wire case, the used die has a constant semi-angle,α. The friction between the workpiece and the die

.

Fig. 7. Stress local analysis for plate drawing.

interfaces is of Coulomb kind,µ, and constant. The lubrica-tion is considered so efficient that the friction coefficient willbe low. The forces acting on a differential elemental of theworkpiece have been plotted inFig. 7.

The workpiece material can be considered as a rigid-perfectly plastic material then, its yield stress,Y, will be aconstant.

In finite element method case, the drawing process simu-lation has been carried out by ABAQUS[31] as in the wirecase.

The workpiece has been meshed with the CPE4R element.This is a four-node bilinear, plane strain, quadrilateral, re-duced integration and hourglass control element. An exampleof mesh with this type of element is shown inFig. 8. This isexactly the mesh used in this work.

3.2. Simulations algorithms

In plate case, the adimensional forward tension has beenchosen as output variable by the same reasons explained inthe wire one.

UsingFig. 7 for making the forces balance it is possibleto write the equations of the slab method for the drawingprocess as[9]:( ( )B

)

T ionala -s

Fig. 6. Modelling of the plastically deforming zone for plate drawing

σzf

S= 1 + B

B1 − hf

hi(4)

aking into account that, under plane strain, cross-sectrea reduction can be given byr = 1−hf /hi , then the expresion(4) will be:

σzf

S= 1 + B

B(1 − (1 − r)B) (5)

Fig. 8. FEM model by CPE4R elements.


Applying FEM in the plate drawing case, the expression,σzf /2k, can be calculated by:

σzf

2k= Ff

2kAf=∑

NFORC

2kAf(6)

whereFf is the drawing force;Af the cross-sectional area atthe die exit;k, the shear yield stress and; NFORC a nodalvariable obtained from the ABAQUS code by extrapolationof the drawing stress values at the integration points.

3.3. Applications

The adimensional forward tension value has been calcu-lated by the expressions(5)and(6) for the range of parametervalues showed inTable 1. In the finite element method case,the properties of the material collected inTable 2have beenintroduced in the pre-processing stage like in wire case. Withthe slab method, this information is not required since all fea-tures of the material are represented in the shear yield stress,k.

3.4. Results and comparison with other previoussolutions

Fig. 9shows the obtained results by both described meth-o 0.40a thatfi thanS

andC hod

F solu-t

and the finite element method have been compared with theobtained by Green and Hill (G&H) using empirical methods[5]; since most of the experimental values available at the mo-ment are for axisymmetric case. In addition, all results werecompared with the obtained ones in previous works applyingthe upper bound technique (UBT)[2–4].

The G&H results were obtained empirically by Green andHill for different cross-sectional area reductions, die semian-gles from 5◦ to 15◦ and Coulomb friction coefficients smallerthan 0.10[5].

They gave the next expression in order to fit the results toa theoretical model:σzf

2k= p

2k

r

1 − r[(1 + µ ctgα) − µ(0.2 + 0.08r ctg2α)] (7)

Outside of the intervals described above, the expression(7)is not correct and has not to be used. Then, the application inthis case, has been only made for the values mentioned.

To obtain the results by means of the upper bound tech-nique, first of all, the plastically deformation zone was mod-elled by an only triangular rigid zone as it is shown inFig. 10.This model fits better than others[3] to the results found onthe classical references about these themes[1,5,11,13].

Based on this model the simulation algorithm can be writ-ten as:

ws e

ds for cross-sectional area reductions of 0.20; 0.30 andnd a Coulomb friction value equal to 0.10 and 0.20. Ingure, it is possible to see that FEM solutions are biggerM ones by the same reasons that in the wire case.The results for plate drawing under plane strain

oulomb friction conditions obtained using the slab met

ig. 9. Comparison between slab method and finite element methodions forr = 0.20-0.30-0.40: (a)µ = 0.10 and (b)µ = 0.20.

σzf

2k= AB�ν12 + BC�ν23

hf

[ν3 − µ

sinα+µ cosαν2

] (8)

here:AB andBC are the discontinuity lines,vij, the relativepeed between thei andj zones, andvi, the speed of the zon

Fig. 10. Triangular rigid model in the UBT case.


i. The rest of the variables within the expression(8)have beenpreviously described.

4. Discussion

The adimensional forward tension has been calculated bymeans of the slab method and the finite element method. ingeneral, it is possible to affirm that using both SM and FEMthe necessary forces to carry out the process will be bigger ifthe cross-sectional area reductions increases.

FEM solutions are always bigger than SM ones. This is dueto SM only takes into account the terms of homogenous de-formation and the friction one while FEM takes into accountother terms. Then, SM always underestimates the necessarydrawing stress to carry out the process.

SM/FEM adimensional forward tension curves are de-creasing and quite similar for die semiangles smaller than10◦. Over this value, the slope of the FEM curves are alsoconstant and the SM ones continue decreasing.

In wire drawing case, a more specific analysis can be madeseeingFig. 4. It shows that the solutions obtained by bothmethods are quite close especially for high cross-sectionalarea reductions, low Coulomb coefficient and low die semi-angles.

Particularly, solutions are comparable for die semianglesb en,f rr forb c-t

thee tt emi-a eens

forr om4

bovec and

F upperb(

SM solutions. Only, it could be added that solutions are lightlymore next now.

The results have been compared with the empirical onescarried out by Green and Hill[5], and the analytical onescalculated using the UBT[2–4]. In this case, experimentalvalues were not available. This is because the investigationworks are generally centered in the wire drawing; mainly dueto the importance that they acquire in industrial sectors suchas the electric one.

SeeingFig. 11it can be said that for die semiangles from10◦ to 14◦ the four solutions are very next and this is espe-cially true for values from 10◦ to 12◦.

5. Conclusions

Drawing process has been studied by means of differentmethods in recent years because of its great importance inthe industrial sector.

In this work, the main types of the drawing process hasbeen analysed by means of the slab method and the finiteelement method.

The obtained solutions have been tested with other onesfound in the literature about this theme. Concretely, in thewire drawing case, with the experimental results given byW m-po

hodm withi n inF chc ut int

nd ag

ula-t y not

etween 6◦ and 10◦ and they diverge for bigger ones. Evor die semiangles near 14◦, FEM solutions obtained fo= 0.20 and 0.30 also coincide with SM ones obtainedigger reductions, concretely, forr = 0.30 and 0.40 respe

ively.In addition, both solutions have been compared with

xperimental ones obtained by Wistreich.Fig. 5confirms thahe three solutions are very near especially for small sngles and low friction coefficients. The differences betwolutions decrease as the reduction increases.

Besides, FEM and Wistreich solutions coincide= 0.40,µ = 0.03 in the interval of the die semiangles fr◦ to 10◦.

In plate drawing case, most of the made comments aan be repeated now for the comparison between FEM

ig. 11. Comparison among slab method, finite element method,ound technique and Green and Hill solutions forr = 0.30: (a)µ = 0.03 andb) µ = 0.10.

istreich[11] and, in the plate drawing case, with the eirical ones proposed by Green and Hill[5] and with thebtained applying by the UBT.

The solutions comparison confirm that FEM is a metore accurate than SM because the obtained results

t are nearer to the real results. Besides, as it is showig. 12, FEM provides very intuitive simulations in whian be seen the forward tension not only at the die exit bhe deformation zone as well.

However, FEM needs more calculation resources aood code knowledge by the users.

SM is easy to apply and does not require big calcion resources neither the time but solutions are usuall

Fig. 12. Drawing process simulations by finite element method.


acceptable. SM underestimates the drawing stress value tocarry out a certain process, since it only takes into accountthe homogeneous deformation of the material and the fric-tion between the workpiece material and the die-interfaces.SM forward tension curves only are similar to experimentaland FEM ones for low reductions carried out in dies with lowsemiangles and under low friction conditions.

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16614380