Finite element modelling of cold drawing for high-precision tubes

(1)

HAL Id: hal-01822637

https://hal-mines-albi.archives-ouvertes.fr/hal-01822637

Submitted on 25 Jun 2018

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

high-precision tubes

Florian Boutenel, Myriam Delhomme, Vincent Velay, Romain Boman

To cite this version:

Florian Boutenel, Myriam Delhomme, Vincent Velay, Romain Boman. Finite element modelling of

cold drawing for high-precision tubes. Comptes Rendus Mécanique, Elsevier Masson, 2018, 346 (8),

pp.665-677. �10.1016/j.crme.2018.06.005�. �hal-01822637�

(2)

Computational modeling of material forming processes / Simulation numérique des procédés de mise en forme

Finite

element

modelling

of

cold

drawing

for

high-precision

tubes

Florian Boutenel

a

,

∗

,

Myriam Delhomme

b

,

Vincent Velay

a

,

Romain Boman

c

a_ICA_(Institut_{Clément-Ader),}_Université_de_Toulouse,_CNRS,_IMT_Mines_Albi,_INSA,_UPS,_{ISAE–SUPAERO,}_Campus_Jarlard,

81013 Albi CT cedex 09, France

b_Minitubes_ZAC_Technisud,_21,_rue_{Jean-Vaujany,}_BP_2529,₃₈₀₃₅_Grenoble_cedex_02,_France

c_Department_of_Aerospace_and_Mechanical_Engineering,_University_of_Liège,_9,_allée_de_la_Découverte,_B-4000_Liège,_Belgium

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received29August2017 Accepted9January2018 Availableonlinexxxx Keywords:

Precisionmetalforming Coldtubedrawing Tubesinking Mandreldrawing Finiteelementmethod Largedeformation

Coldtubedrawingisametalformingprocessthatallowsmanufacturerstoproduce high-precision tubes.Thedimensionsofthe tubeare reducedbypullingit throughaconical convergingdiewithorwithoutinnertool.Inthisstudy,ﬁniteelementmodellinghasbeen usedtogiveabetterunderstandingoftheprocess.

Thispaperpresentsamodelthatpredictsthefinaldimensionsofthetubewithveryhigh accuracy.Itisvalidatedthankstoexperimentaltests.Moreover,fivestudiesareperformed withthismodel,suchasinvestigatingtheinfluenceofthedieangleonthedrawingforce ortheinfluenceofrelativethicknessontubedeformation.

1. Introduction

Medicaldevices,likestents,cardiacvalves,andimplants,aremanufacturedwiththin-walledtubesofsmalldiameters [1]. Asapplicationsinthebiomedicalﬁeld,thesetubesrequireaveryhighprecisionindimensionsandsurfaceﬁnish.Also,these propertiesarestronglylinkedtothequalityofthemetalformingprocess.

Coldtube drawing enablesmanufacturers to producetubes withcontrolled dimensions, good surfaceﬁnish,andhigh mechanicalproperties [2].Thismetal formingprocess givesabettertubequality comparedtohot forming.Tubedrawing consistsinreducingthetubedimensionsbypullingitthroughaconicalconvergingdiewithorwithoutinnertool.Different drawingmethodsexist [3].Inthispaper,twotechniquesarestudied:tubesinkingandmandreldrawing(Fig.1).

Forbothofthem,thediecalibratesthetube’souterdiameter.Tubesinkingistheonlymethodthatdoesnotuseaninner tool.Theinnerdiameterisreducedbecauseofthefreedeformationinsidethetube.Inconsequence,theinnersurfaceﬁnish is degraded.In mandrel drawing,the inner tool,named mandrel, moveswiththetube andcalibrates itsinner diameter. Themaindrawbackofthistechniqueisrelatedtotheendofthedrawingoperationwherethetubeisclampedaroundthe mandrel.Thus,areelingoperationisrequiredtoremovethetool.

Themetalformingindustrywantstoperpetuallyimproveproductivityandproductquality.Inordertoreachthispurpose, a betterunderstanding ofthe processesis necessary. Ontheone hand,a large seriesofexperimental testscan be done.

*

Correspondingauthor.

E-mailaddresses:ﬂorian.boutenel@mines-albi.fr(F. Boutenel),m.delhomme@minitubes.com(M. Delhomme),vincent.velay@mines-albi.fr(V. Velay),

r.boman@ulg.ac.be(R. Boman).

URLs:http://www.institut-clement-ader.org/(F.BoutenelandV.Velay),http://www.minitubes.com/(M. Delhomme),http://www.ltas-mnl.ulg.ac.be/

(R. Boman).

https://doi.org/10.1016/j.crme.2018.06.005

(3)

Fig. 1. Sketch of the two drawing methods studied in this paper (adapted from [4]).

However, thistype of approach maybe time and moneyconsuming. On the other hand,lots of testscan be performed virtually thankstoﬁniteelement(FE)modelling.Italsogivesaccesstophysicalvalues,suchasstressesandstrains,which arenotmeasurableduringtheprocess.Thus,FEmodellingseemstobeahelpfultool.

Analytical solutions havebeen developedby several authorsto studythe effectsof process parameters. Um etal. [5] obtained anupper boundsolution toﬁxed plug drawing thatcan be simpliﬁedin thecaseoftube sinking.This method hasbeenadaptedtomandreldrawingby Alexandrova [6] inthecaseofmandreldrawing.Later,theHill’sgeneralmethod of analysis for metalworking processes and a fracture criterion have been added to this analytical model to study the workabilityofmandreldrawing [7].Zhaoetal. [8] haveproposedananalyticalsolutiontotubesinkingusinganintegration methodofstrainratevectorinnerproduct.

Numerical studies of tube drawing using FE can also be found in the literature. A ﬁnite element analysis has been conductedby Sawamiphakdiet al. [9] todetermine theinitial tubesizes that give theappropriate mechanicalproperties after drawing.Linardon etal. [10] combineda conicalmandrel tubedrawing test withaFE modellingtoselecta failure criterion. Thepotentialmaking ofhigh-qualitythintubeswithshape-memoryalloysthankstomandrel drawinghasbeen studied by Yoshida etal. [3]. Karnezis etal. [11] used a FEmodel to investigatethe possibilityof reducing the number of drawing passes. Palengatet al. [12] underlined the importance ofthe propertiesof the interface (tube withtools) on the drawinglimits.Severalstudiesdealingwithtoolsdesignhavebeenachievedinordertoimprovetheprocess.Sheuet al. [13],Leeet al. [14] and Bélandet al. [15] focused on thedie geometry,while Kim etal. [16] had an interest forthe mandrelone.

Inthispaper,aFEmodelisdesignedtopreciselypredicttheﬁnaldimensionsofthetube.Thisaxisymmetricsteady-state modeltakesintoaccountdifferentissues,includingtoolgeometry,tubeelastoviscoplasticbehaviour,toolelasticbehaviour, contactsandfriction.Moreover,thenumericalconvergenceisbuilttoreacha1-micronaccuracy.Inconsequence,thismodel canbeusedtoanalysethetubedrawingprocess,andthus,permitsabetterunderstanding.

This modelling was performedon Metafor [17–19], an in-house nonlinear ﬁnite element code of the Department of AerospaceandMechanicalEngineeringoftheUniversityofLiège,Belgium.

This paperis organised asfollows.A detailedmodelformulation isgiven inSection 2.Then, Section 3 compares the numericalresultswithdrawingexperimentsinordertovalidatethemodel.Furthermore,anumericalanalysisoftheprocess ispresentedinSection4.Finally,Section5reportsontheconclusionsofthiswork.

2. Formulationoftheﬁniteelementmodel

ThissectiondescribespreciselytheFEmodelling.Tubesinkingandmandreldrawingaretreatedtogetherbecauseoftheir manysimilarities.Formandreldrawing,thereelingoperationisnotmodelled.Itimpliesthat,attheendofthesimulation, thetuberemainsclampedaroundthemandrel.

(4)

Fig. 2. Geometry of the model (mandrel drawing case).

Table 1

Chemicalcompositionofthe316LVMstainlesssteel(ASTM-F138 [20]).

Element Fe Cr Ni Mo Mn Si Cu N C P S

Mass % Balance 17–19 13–15 2.25–3 <2 <0.75 <0.5 <0.1 <0.03 <0.025 <0.01

2.1. Geometry

Byassumption,thetubeandthetoolsaregeometricallyperfect.Theseelementsarealsosupposedtobecoaxial. More-over,thegeometryandtheloadingconditionshaveanaxialsymmetry.Thus,theprocessisaxisymmetricandcanbesolved in2D.

First,thetubeisdescribedbyitsouterandinnerdiameters(respectivelyDout andDin).Then,thediegeometryincludes variousdimensions:the diameterDdie,the bearinglength Lb,theentrydieangle

α

.Inthecaseofmandreldrawing,the innertoolisrepresentedbyitsdiameterDm.

ThegeometryofthemodelisgiveninFig.2.Inpractice,thebeginningofthetubeisinsertedinthediebeforestarting thedrawingoperation.Theresultsofthisareaarenotrelevant.Besides,evenifthedieseemstobesymmetric,thetoolis notreversible.Infact,thedieholderensuresthatthedieisusedinthecorrectway.

2.2. Materials

Inthismodel,thetubeandthetoolsarebothrepresentedbydeformablesolids.

Thetubeismadeof316LVMstainlesssteel.Thismaterialiscommonlyusedforbiomedicalapplicationsbecauseofits excellentresistancetovarioustypesofcorrosion.ItschemicalcompositionisgiveninTable1.

Palengatetal. [12] observedanisotropicbehaviourofthismaterial.Then,Fréchardetal. [21] showedthat:

•

atambienttemperature(temperatureoftheprocess),nitrogen-alloyedausteniticstainlesssteelsarestrain-ratesensitive;

•

theviscoplasticmodelofJohnson–Cookisthemostappropriatetodescribethebehaviourofthesematerialsatdifferent strainrates.

Theconstitutiveequationofthismodel [22] is:

σ

eq

=

A

+

B

ε

_eqn

1

+

C ln

ε

˙

eq

˙

ε

0 eq

1

−

T

−

Tref Tm

−

Tref

m

(1) where

σ

eq representstheﬂowstress,

ε

eq theequivalentplasticstrain, A theyieldstress, B thepre-exponentialfactor,

ε

˙

eq the plasticstrain rate,

ε

˙

0

eq the referenceplastic strain rate, C thestrain ratesensitivity coefficient,n the work-hardening coefficient, T thetemperatureofthematerial,Tm themeltingtemperatureofthematerial, Tref thereferencetemperature, andm thethermalsofteningexponent.Inotherwords,thefirstfactorisrelatedtoworkhardening,thesecondonetostrain ratedependence,andthelastonetothermalsoftening.

However,thethermalsofteningtermcanbeneglectedifthestrainrateoftheprocessdoesnotallowthematerialtobe affectedbythetemperaturerise [23].Ononeside,thestrainrateincolddrawingprocesscanreach40 s−1.Ontheother one, theexperimental measures showedthat the tubetemperaturedoes not exceed130◦C [12], i.e.lessthan one tenth ofthematerial’smeltingpoint(1400◦C).Thus,the thermalsoftening termcan besafelyneglectedin (1).Finally,thevon Misesyieldcriterionisusedtomodelplasticity.

ThemechanicalpropertiesandtheJohnson–Cookparametersof316LVMarerespectivelylistedinTables2and3.A rep-resentationoftheJohnson–CooklawisalsogiveninFig.3.

The die is madeof tungsten carbide,while the mandrel is madeof steel. Both materials are considered as isotropic. Duringtheprocess,noirreversibledeformationisobservedinthetools,sothebehaviourisassumedtobeperfectlyelastic duringdrawing.ThemechanicalpropertiesofthematerialsofwhichthetoolsaremadeofarelistedinTable4.

(5)

Table 2

Mechanicalpropertiesofthematerialofwhichthetubeiscomposed. Density Young’s modulus Poisson’s ratio 7 900 kg·m−3 _{192 GPa} _0.29

Table 3

Johnson–Cookparametersof316LVMstainlesssteel [24].

A B C n ε˙0

eq

287 MPa 1265 MPa 0.021 0.664 0.03 s−1

Fig. 3. Representation of the Johnson–Cook law for 316LVM stainless steel for three plastic strain rates (with parameters of Table3).

Table 4

Mechanicalpropertiesofthematerialsofwhichthetoolsarecomposed. Tungstene carbide Steel Density (kg·m−3₎ _{15 000} _{7 900}

Young’s modulus (GPa) 650 210

Poisson’s ratio 0.3 0.3

Fig. 4. Boundary conditions of the model (mandrel drawing case).

2.3. Boundaryconditions

Twoboundaryconditionsareprescribedinthismodel(Fig.4).

The right-hand extremityof the die is ﬁxed along the drawing directionin order to modelthe contact withthe die holder.Then, adisplacementcondition isappliedtothetubeandthemandrel,ifitexists.Thisdisplacementisprescribed atconstantspeed.Itcorrespondstothesteadyspeedofthedrawingtrolley.

2.4. Contact

(6)

Fig. 5. Drawing force evolution as a function of the friction coeﬃcientμtube/die(tube sinking case).

Table 5

Frictioncoeﬃcientsidentiﬁedbyinverse analysis.

μtube/die μtube/mandrel

0.043 0.139

Intheindustrialprocess,lubricationisusedtoreducefrictionbetweeneachpart.Byassumption,itishomogeneousand constant.

Frictiondependsonseveralparameters:materialsincontact,relativevelocity,surfaceroughness,thicknessoftheoilfilm, normalpressure,etc.Inthisstudy,thecontactsare describedby aCoulombfriction model.Thus, twofrictioncoefficients aredefined:

μ

tube/die and

μ

tube/mandrel.However, thesecoeﬃcientsarediﬃculttomeasureinpractice.Inconsequence,an inverseanalysisisusedtoidentifythem.

Foreachcase,representativetestswereperformedexperimentallyandthen,simulatedwithvariousfrictioncoeﬃcients. Anearlylineardependencewasobservedbetweenthedrawingforceandthefrictioncoeﬃcient(Fig.5).

Thecoeﬃcient, whichpermitsustomatchthesimulatedforcewiththemeasured one,was keptforthewhole study.

μ

tube/die was firstly identified thanksto a tubesinking test. Thiscoefficient is assumedtobe the samein both drawing methods. Then, a mandrel drawing test allowed usto determine

μ

tube/mandrel.The values of the friction coeﬃcientsare listedinTable5.

These values ﬁt in the range of those of the literature concerning cold tube drawing. In fact, usual coeﬃcients are includedbetween0.03 [3] and0.2 [25].

2.5. Mesh

Thetubeandthetoolsaremeshedwith4-node-quadrangularelementsusingselectivereducedintegration(SRI), mean-ingthatthepressureincrementoveranelementisconstantandcomputedatthecentreoftheelement.Whilethedeviatoric partofthestress tensorisevaluated at4Gausspoints. Fig.6 showsa view oftheinitialmesh.In practice,only the up-per halfofthemodel(abovethesymmetricaxis)ismeshed andsolvedby theﬁniteelementmethod.Moreover,it could be mentioned that thereis nocontact betweenthetube andthe tool onthe right-hand part(corresponding tothe tool areaaftertheoutputplane inFig.6).However,inordertosimplifythemeshprocedureandnottoaddanothernumerical parameterintothestudy(Section2.6.2),theright-handpartofthetoolismeshedinthesamewayastheleft-handpartis.

2.6. Analysisofnumericalfactors

This ﬁniteelement model includes various numericalmethods using differentparameters. These methodsare related tothe contactalgorithm, themesh,theintegrationtime, andthe requireddurationofthesimulation toreachthe steady state.Also, theparametersarechosen toassureareliabilityofthemodel.Infact,allthefollowingnumbersresultfroma numericalstudyandensurethebestcompromisebetweenCPUtimeandthedesired1-micronaccuracy(seeSection2.6.5).

2.6.1. Contact

Both contactpairs(tube/dieandtube/mandrel)are numericallymanaged bythe samemethods usingidenticalsetsof parameters.

(7)

Fig. 6. Initial mesh of the model (mandrel drawing case).

Apenaltycontactalgorithm [18] isused.Thepenaltycoeﬃcientalongthenormaldirectionisequalto107 _MPa

_·

_mm−1_. Alongthetangentialdirection,thecoefficientisdefinedasthenormalonemultipliedbythefrictioncoefficient.Thismeans thatthegeometricalerrorisidenticalalongbothdirectionswhenthesurfacesslideovereachother.

Finally,thecontactismodelledasanode-to-surfacecontactusingasingle-pass method [19]. Thisimpliesthat master andslave surfaceshavetobe assigned.Onone side,themastersurface corresponds tothemorerigidone. Thus,ineach case,thesurfaceofthetoolisconsideredasthemastersurface.Ontheotherone,theslavesurfaceisassignedtothetube. Moreover,thedepth,atwhichthedetectionoccursisequaltohalfatubethickness.

2.6.2. Mesh

The tubeismeshed with8square elements inits thickness. Thisnumberiscoherentwithother studies.Forinstance, Palengatetal. [12] haveusedatleast8elementsinthethicknesstostudytheforminglimit.

Inorderto consumelessCPUtime, thetoolsare meshedwithlargerelements.Infact, theglobalsizeoftheelements has been chosen to be tentimes bigger in thedie corethan in the tube.This factoris equalto thirty for themandrel. However, the meshof thetoolsis reﬁnednear thecontactzonesinorder toavoid contactbetweensegments withvery differentlengths.Aspreviously,asizefactorisdeﬁned:0.5forthedieand2forthemandrel.

2.6.3. Timeintegration

A Chung–Hulbert scheme [26] is used fortime integration. This implicitdynamic integration scheme belongsto the generalised

α

-schemefamily.

Then, thetimeincrementislimitedbyauser-defined maximumvalue inordertoensurethatthetubedeformation is progressiveandremainssmallduringeachincrement.Inthisway,thisvalueisdefinedsothatanodecoversthelengthof atubeelementin25steps.Althoughtheresultingsizeofthetimeincrementisrathersmall,itissignificantlylargerthan the one expectedwithanexplicit approach.Moreover,using theproposed implicitmethod,comparedto an explicitone, preventsusfrommanagingadditionalnumericalparameterssuchasmassandloadscalingfactorsandhourglasscontrol.

Finally,theequilibriumtolerancewithineachincrementissetto10−4 toassurea convergencewithouttoomuchCPU timeconsumption.

2.6.4. Steadystate

Thedurationofthesimulatedprocessissetsothatthewholetubeisdrawnandleavethediesothatthespringbackis modelled.

Also, the modelled tube length should be suﬃcient to reach the steady-state conditions far from tube ends during drawing.Inthisway,aminimumlengthof20 mmshouldbeconsidered.

2.6.5. Analysismethod

Allthepreviousparametersweresetinordertoguaranteethereliabilityofthemodel.Theapproachthathasbeenused isexplainedinthisparagraph.

Aparameterischosen:forinstance,thepenaltycoeﬃcientalongthenormaldirection.Thiscoeﬃcientexpresses, numer-ically,thestiffnessofthecontact.Thismeansthatthegreateritis,betterthecontactisrepresented.

Next,thevalueofthisparameterischanged,whiletheothersareﬁxed.Table6givesanexampleofresults.

Accordingtotheprevious deﬁnitionofthiscoeﬃcient, thelargestvalue (108 MPa

·

mm−1) istakenasthereference. An errorof39.3 μmisobservedontheouterdiameterifthecoeﬃcientissetto104 _MPa

_·

_mm−1_._This_error_is_bigger_than_the micron accuracy which isexpected. Thus, a coeﬃcient of106 _or₁₀7 _MPa

_·

_mm−1 _allows _us_to _verify_this_accuracy _while consuminglesstimethanthereference.Finally,thevalueof107 _MPa

_·

_mm−1 _was_chosen_for_the_rest_of_the_study.

(8)

Table 7

Toolsdimensions.

Test Die Mandrel

Ddie Lb α Dm

(mm) (mm) (deg) (mm) 1 7.012 2.010 14.859 – 2 6.64 2.4 22.0 5.80

3. Modelvalidationwithdrawingexperiments

Variousdrawingtestswereperformedonindustrialbenchesinordertovalidatethemodel.Twoofthemarepresented inthissection.

3.1. Experiments

Test 1, performed for thispaper, deals with tube sinking. A tube (Dout

=

9.88 mm and Din

=

8.07mm) is pulled througha dieof7.01 mmofdiameter.Thiscorrespondsto anarea reductionof28%.Test 2,extractedfromanotherwork [4], concernsmandrel drawing.The section areaof atube(8.16 mm

×

7.05 mm)is reducedby38% usinga die(Ddie

=

6.64 mm)andamandrel(Dm

=

5.80 mm).

3.1.1. Measurements

Thetubedimensionswerepreciselymeasuredbeforeandafterthedrawingoperation.Theouterdiameterwasgivenby alasermeasure(threepointsontwosectionsofthetube).AnHeidenhaintouchprobewasusedtomeasurethethickness oneightpointsofonetubesection.Theinnerdiameterwasthendeduced.

ThedimensionsofthedieweremeasuredthankstoaMitutoyocoordinatemeasuringmachine.

Astheouterdiameterofthetube,themandreldiameterwasmeasuredthankstoalaser.Inthecaseofmandreldrawing, giventhefactthatthereelingoperationisnotsimulated,thetubeisclampedaroundthemandrelattheendoftheprocess. As a consequence,the measure of the tube outer diameter was done with the mandrel inside. Also, the thickness was deducedby assumingthatthe tube’sinnerdiameterisequaltothemandrel’s diameter.Toolsmeasurementsare listedin Table7.

The tests were performed at ambient temperature (22◦C) and at constant speed (9.37 m

·

min−1 _for _Test ₁ _and 11.4 m

·

min−1 forTest2).

3.1.2. Results

Duringthetests,thedrawingforcewasmeasuredbyaloadcelllocatedbetweenthedieandthedieholder.Fig.7gives theevolutionoftheforceduringdrawing.

Threestagescanbediscerned.Atthebeginningofthedrawing(att

=

0 s),theforceincreasesquickly.Then,amechanical steadystateisdeﬁnedbyaconstantforce.Att

=

12

.

3 s,thetubegetsoutofthedie(endofdrawing),andtheforcebecomes zero.Suchevolutioncanbeobservedforalldrawings.

ThedrawingtestsresultsaresummarisedinTable8.

3.2. ComparisonwiththeFEmodel

BothtestsweresimulatedwiththeﬁniteelementmodeldescribedinSection2.Thisallowedustocalculatethedrawing force,theouterdiameter,andtheﬁnalthicknessofthetubes.

Inordertobeconsistentwiththeexperimentalmeasurements,thenumericaldrawingforceiscalculatedasthereaction force ontheﬁxedsideofthedie(see Fig.4).Thediameterandthethicknessare calculatedonatubesectionthat isnot submissivetosideeffects.

(9)

Fig. 7. Evolution of the experimental drawing force (Test 1).

Table 8

Resultsoftheexperimentaldrawingtests.

Test Initial tubes (mm) Final tubes (mm) Drawing force (daN) Outer∅ Thickness Inner∅ Outer∅ Thickness Inner∅

1 9.875 0.901 8.073 7.019 0.959 5.101 867

±0.002 ±0.002 ±0.0005 ±0.003 ±11 2 8.16 0.555 7.05 6.6726 5.80 0.4363 739

±0.002 ±0.002 ±0.001 ±23

Table 9

Comparisonbetweenexperimental(EXP)andFEresults.

Test Drawing force (daN) Outer diameter (mm) Thickness (mm)

1 EXP 867 7.019 0.959 FE 866 7.0172 0.9821 Difference 1 −0.0018 +0.0231 0.1% 0.03% 2.41% 2 EXP 739 6.6726 0.4363 FE 739 6.6689 0.4423 Difference 0 −0.0037 +0.006 <0.1% 0.06% 1.38% Table 10 CPUstatistics.

Test Real drawing time (s) CPU time Number of steps 1 0.285 53 min 48 s 9994

2 0.208 4 h 26 min 18511

Asexpected, thecalculateddrawingforceisquiteequaltothemeasuredone.Infact,thefrictioncoeﬃcients,identiﬁed inTable5,allowustoobtainthisresult.

Inbothtests,theouterdiameterandthethicknessarewellpredicted.

ThesesimulationshaveusedanIntelCorei73.20 GHzprocessor.TheCPUstatisticsarelistedinTable10.

In conclusion,thiscomparison withdrawingexperiments showsthat thepresented modelis ableto predictthe ﬁnal dimensionsofthetubewithagoodaccuracy.

4. Numericalanalysisoftheprocess

The validatedmodelcanbeusedtogive abetterunderstandingofthecoldtubedrawing process.Inthispurpose,ﬁve studiesarepresentedinthissection.

(10)

Fig. 8. Drawing force as a function of the die angle (tube sinking case). 4.1. Inﬂuenceofthedieangleonthedrawingforce

Therequiredpullingforcetodrawatubeisacriticalvalueinthedesignoftheindustrialprocess,becauseitcalibrates thecapacityofthebench.Thus,itisinterestingtoknowtheinﬂuenceofprocessparametersontheforce.

Inparticular,theentrydieangleisstronglylinkedtoit.Inthisway,atubesinkingsimulationwasrepeatedwithvarious anglevalues(from10◦ to30◦ bystepof1◦).Theotherparameters arethoseofTest1(seeTable7).Thevariations ofthe drawingforceareshowninFig.8.Also,theequivalentplasticstrainisgiveninFig.9forthreevaluesofthedieangle.

Béland etal. [15] revealed that an optimum value of the die angle minimises the drawing force. This result can be observedherefor

α

=

17◦.

Whenthedieangledeviatesfromtheoptimumvalue,theforce increases.However,adissymmetryisobservedbecause the increase, on either side of the optimum value, doesnot have the same origin. When the die’s angle is below the optimumvalue, thefrictionpredominatesover thedeformation.Iftheangledecreases,thefriction betweenthetubeand thedieincreasesandsotheforceincreases.Whenthedie’sangleisabove theoptimumvalue,deformationpredominates overfriction.Iftheangleincreases,thetubemustdeformmoreinordertokeepaconstantmaterialﬂow.Asaconsequence, agreaterforceisneeded.

Finally,Fig.9showsthattheequivalent plasticstrain ofthetubeincreaseswiththedieangle.Furthermore,themodel predicts that, inthe presentedcases oftube sinking,no contactoccursbetweenthe tube andthedie along thebearing length.Duetoascaleeffect,thisobservationismorevisibleinFig.9(c).

4.2. Inﬂuenceoftheinitialrelativethicknessonthegeometryofthetube

Various tubethicknesses could be drawn through agiven die.As a consequence,a dimensionless quantity isused to comparethe differentkindsofdrawings. Therelative thicknessofthetube isdeﬁnedastheratiobetweenthethickness andtheouterradius.

Duringdrawing,thetubedeformation isdirectlyinﬂuenced byitsinitial relativethickness. Fig.10showsthevariation rateofthicknessasafunctionoftherelativethickness,fortubesinking.Thisgraphisobtainedbyrepeatingthesimulation ofTest1withvariousvaluesoftheinitialinnerdiameterofthetubeinordertoreachawiderangeofrelativethicknesses. Whenthe initialrelative thicknessincreases,the variationrateofthethicknessdecreases. Also,inthe presentedcase, thethicknessisnotmodiﬁedbythedrawingoperationifthedimensionlessquantityisequalto40%.So,twobehaviourscan bedistinguished. Iftherelativethicknessisbelow40%,thevariationrateispositive.Inother words,thetubegets thicker afterdrawing.Iftherelativethicknessisabove 40%,thevariationrateisnegative, andso,duetodrawing,tube thickness decreases.Thisbehaviouriscommonlyobservedinpracticeintheindustry.

4.3. Inﬂuenceoftheinitialrelativethicknessontheelongation

As the drawing process impliesa diameter reduction, the tubeis stretched. In thisstudy, the inﬂuenceof the initial relative thickness onthe elongation is studied for tubesinking. A representationof the elongation evolution is givenin Fig.11.ThisgraphisactuallyobtainedwiththesamesimulationsasinSection4.2.

Theelongationincreaseswiththerelativethicknessinaquitelinearway.Infact,thegreatertherelativethickness,the greaterthearea reduction.Thus,duetovolumeconservation,thetubeismorestretched.Thisﬁgurealsorevealsthatthe outerandinnersurfaceshavethesameelongation,eveniftheouteroneisconstrainedbythedieandtheinnerisfreeto deform.

(11)

Fig. 9. Equivalent plastic strain of the tube for three values of the die angle, at the steady state (tube sinking case).

Fig. 10. Inﬂuence of the relative thickness on the ﬁnal tube thickness (tube sinking case).

4.4. Distributionofresidualstressesinthetubethickness

The tubeis subjected to strain hardening during cold tube drawing. An analysisof the residual stresses through the thickness allows usto understand the tube state at the end of the drawing. Fig. 12 gives an example for tube sinking (Test 1).

TheevolutionofthevonMisesstressthroughthethicknessrevealsthatthetubeismoreconstrainedonitsinnersurface thanonitsouterone.Then,shearandradialstressesarenotsigniﬁcant.Moreover,thecircumferentialstressisnegative in theinner partofthetube,andpositiveintheouterone.Finally,asfarasthelongitudinalstressisconcerned,thetubeis incompressiononitsinnersurfaceandintractiononitsouterone.

(12)

Fig. 11. Elongation as a function of the initial relative thickness of the tube (tube sinking case).

Fig. 12. Distribution of the residual stresses in the thickness for tube sinking.

Table 11

MandreldrawingsofTubesAandBbeforethereelingoperation–extractedfrom[4].

Tube Die Mandrel Speed

Ddie Lb α Dm

(mm) (mm) (deg) (mm) (m·min−1₎

A 7.48 0.9 22.7 6.50 11.4

B 8.13 0.3 33.0 7.00 11.4

Tube Initial tubes (mm) Final tubes (mm)

Outer∅ Thickness Inner∅ Outer∅ Thickness Inner∅

A 9.05 0.64 7.77 7.52 0.51 6.50 B 10.50 0.75 9.00 8.16 0.58 7.00

4.5. Predictionofthereelingoperation

Inthecaseofmandreldrawing,thetubeisclampedaroundthemandrelattheendoftheprocess.Areelingoperation isneededtoremovethemandrelfromthetube.Itconsistsinrelaxingthetubestressesthankstoarollersystem.

Inpractical,thisoperationisdiﬃculttoperformonsometubes.Evenifthereelingoperationisnotsimulatedwiththis model,aqualitativestudyisproposedinthissection,topredictthediﬃcultyofthereelingoperationoftwotubes.

TheexperimentshaveshowedthatthereelingoperationofTubeBismorediﬃcultthantheoneofTubeA.The charac-teristicsofthedrawingsprecedingthereelingoperationarepresentedinTable11.

(13)

Fig. 13. Repartition of the radial stress before the reeling operation.

Consideringthattheclampingonthemandrelisduetotheresidualradialstress,theideaistocomparethedistribution ofthisstressthroughtheﬁnalsectionafterdrawingforbothtubes.Fig.13representstherepartitionofradialstressinthe thicknessforTubesAandB.

For both tubes, the radial stress is negative, except on the outer skin. This compression is coherent with the tube clampingobservedinpractice.ItcanalsobeobservedthatthecompressionismoreimportantinTubeBthaninTubeA(by afactorof1.55ontheinnerskin).Asaconsequence,agreaterrelaxationisnecessaryforTubeB.Thussimulationconﬁrms thatremovingthemandrelfromTubeAbyareelingoperationiseasierthandoingthesamefromTubeB.

Ofcourse,theresultsofthisqualitativestudyshould beideallyconﬁrmedbyafull simulationofthe removalprocess. Thesimulationofthiscompletepost-operationwillbeinvestigatedinfuturemodels.

5. Conclusions

Aﬁniteelementmodelofcoldtubedrawinghasbeenpresentedinthispaper.Also,thenumericalparametershavebeen chosentoassurethereliabilityofthemodel.

Then,drawingexperimentshavebeenachievedandcomparedtosimulationsinordertovalidatethemodel.Thisshowed thattheﬁnaldimensionsofthetubecanbepredictedbythemodelwithagoodaccuracy.

Furthermore, a complete analysis of the process has been presented. It has been shown that the die angle has an inﬂuenceonthedrawingforce.Also, therelative thicknesshasanimpactonthemechanicalbehaviour ofthetubeduring thedrawingoperation.Infact,itcanbecomethickerorthinner,dependingonwhetherrelativethicknessisbeloworabove atransitionvalue(whichis40%inthepresentedcase).Moreover,thetubeelongation,identicaloninnerandoutersurfaces, increaseswithrelativethickness.Finally,ananalysisoftheresidualstressesallowedustounderstandthetubestateatthe endofdrawing.Inparticular,itcanqualifythediﬃcultyofthereelingoperationinthecaseofmandreldrawing.

As part of the optimisation of the process parameters, this ﬁrst work gives uspromising results. Other studies will be considered infuture works: thermal effects,anisotropy of the tube material andchaining severalsuccessive drawing operations.

Acknowledgements

WesincerelywishtothanktheFrenchcompanyMinitubes,whichsupportedandcontributedtothiswork.

References

[1]P.Poncin,D.Ferrier,A.Loshakove,J.Proft,C.Meyer-Kobbe,Acomparisonbetweentwomanufacturingmethods,in:A.P.S.Russell(Ed.),Proceedingsof theInternationalConferenceonShapeMemoryandSuperelasticTechnologies,Asilomar,California,May2000,2000,p. 477.

[2] U.Amborn,S.Ghosh,I.Leadbetter,Modernside-shaftsforpassengercars:manufacturingprocessesII–Monobloctubeshafts,J.Mater.Process.Technol. 63 (1)(1997)225–232,https://doi.org/10.1016/S0924-0136(96)02746-X.

[3] K.Yoshida,H.Furuya,Mandreldrawingandplugdrawingofshape-memory-alloyﬁnetubesusedincathetersandstents,J.Mater.Process.Technol. 153(2004)145–150,https://doi.org/10.1016/j.jmatprotec.2004.04.182.

[4]M.Palengat,Modélisationdescouplagesmultiphysiquesmatériaux–produits–procédéslorsdel’étiragedetubes :applicationauxalliagesmétalliques usuels,Ph.D.thesis,UniversitédeGrenoble,France,2009(inFrench).

[5] K.-K.Um,D.N.Lee,Anupperboundsolutionoftubedrawing,J.Mater.Process.Technol.63 (1)(1997)43–48,https://doi.org/10.1016/S0924-0136(96) 02597-6.

[6] N.Alexandrova,Analyticaltreatmentoftubedrawingwithamandrel,Proc.Inst.Mech.Eng.,CJ.Mech.Eng.Sci.215 (5)(2001)581–589,https:// doi.org/10.1243/0954406011520968.

(14)

elementanalysis,Int.J.Mech.Sci.70(2013)69–78,https://doi.org/10.1016/j.ijmecsci.2013.02.003.

[13] J.-J.Sheu,S.-Y.Lin,C.-H.Yu,Optimumdiedesignforsinglepasssteeltubedrawingwithlargestraindeformation,Proc.Eng.81(2014)688–693,

https://doi.org/10.1016/j.proeng.2014.10.061.

[14] S.-K.Lee,M.-S.Jeong,B.-M.Kim,S.-K.Lee,S.-B.Lee,DieshapedesignoftubedrawingprocessusingFeanalysisandoptimizationmethod,Int.J.Adv. Manuf.Technol.66 (1)(2013)381–392,https://doi.org/10.1007/s00170-012-4332-8.

[15] J.-F.Béland,M.Fafard,A.Rahem,G.D’Amours,T.Côté,Optimizationonthecolddrawingprocessof6063aluminiumtubes,Appl.Math.Model.35 (11) (2011)5302–5313,https://doi.org/10.1016/j.apm.2011.04.025.

[16] S.Kim,Y.Kwon,Y.Lee,J.Lee,Designofmandrelintubedrawingprocessforautomotivesteeringinputshaft,J.Mater.Process.Technol.187(2007) 182–186,https://doi.org/10.1016/j.jmatprotec.2006.11.134.

[17] METAFORwebsite,UniversityofLiège,Belgium,http://metafor.ltas.ulg.ac.be/.

[18] L.Noels,L.Stainier,J.-P.Ponthot,Simulationofcrashworthinessproblemswithimprovedcontactalgorithmsforimplicittimeintegration,Int.J.Impact Eng.32 (5)(2006)799–825,https://doi.org/10.1016/j.ijimpeng.2005.04.010.

[19] L.Adam,J.-P.Ponthot,Numericalsimulationofviscoplasticand frictionalheatingduringﬁnitedeformationofmetal.PartI: theory,J.Eng.Mech. 128 (11)(2002)1215–1221,https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1215).

[20] ASTMF138-00:StandardSpeciﬁcationforWrought18Chromium–14Nickel–2.5MolybdenumStainlessSteelBarandWireforSurgicalImplants(UNS S31673),https://doi.org/10.1520/F0138-00.

[21]S.Fréchard,A.Redjaïma,G.Metauer,E.Lach,A.Lichtenberger,Comportementdynamiqueetévolutionmicrostructuraled’unacierinoxydable austéni-tiquealliéàl’azote,in:Matériaux2002,Tours,France,2002(inFrench).

[22]G.Johnson,W.Cook,Aconstitutivemodelanddataformetalssubjectedtolargestrains,highstrainrates,andhightemperatures,in:Proceedingsof the7thInternationalSymposiumonBallistics,TheHague,Netherlands,19–21April1983,1983,pp. 541–547.

[23]P.Joyot,Modélisationnumériqueetexpérimentaledel’enlèvementdematière :applicationàlacoupeorthogonale,Ph.D.thesis,UniversitéBordeaux-1, Bordeaux,France,1994(inFrench).

[24]C.Linardon,PrecisionTubeDrawingforBiomedicalApplications:Theoritical,NumericalandExperimentalStudy,Ph.D.thesis,UniversitédeGrenoble, France,2013.

[25] N.Sandru,G.Camenschi,Amathematicalmodelofthetheoryoftubedrawingwithﬂoatingplug,Int.J.Eng.Sci.26 (6)(1988)569–585,https:// doi.org/10.1016/0020-7225(88)90055-9.

[26] J.Chung,G.Hulbert,Atimeintegrationalgorithmforstructuraldynamicswithimprovednumericaldissipation:thegeneralized-αmethod,J.Appl. Mech.60 (2)(1993)371–375,https://doi.org/10.1115/1.2900803.