Numerical simulation of the thermoforming of multi-layer polymer sheets

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1 Numerical simulation of the thermoforming of multi-layer polymer sheets Marie-Hélène Vantal, Bernard Monasse, Michel Bellet To cite this version: Marie-Hélène Vantal, Bernard Monasse, Michel Bellet. Numerical simulation of the thermoforming of multi-layer polymer sheets. S.F. Shen and P.R. Dawson Editors. Proceedings NUMIFORM 95, 5th Int. Conf. on Numerical Methods in Industrial Forming Processes, Jun 1995, Ithaca, NY, United States. Balkema, pp.pages , <hal > HAL Id: hal Submitted on 14 Mar 2011 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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.

2 Numericalsimulationofthethermoformingofmultilayerpolymersheets MarieHélèneVantal,BernardMonasse,MichelBellet!! "# $%&'()*+(( $* $, -."&/0 11(2)*(2"(& ABSTRACT: This paper presents a 3D finite element model of the thermoforming process. The polymer materialisassumedtoobeytheviscoplasticlawproposedbyg'sellandjonas,thecoefficientofwhichhave beenidentifiedforapolystyrene.theimplementationofthisconstitutivemodel,theheattransfercouplingand theproposedmultilayerapproacharedetailedinthepaper.axisymmetricvalidationtestsandanapplicationto thethermoformingofanindustrialcomponentarealsoreported. 1INTRODUCTION The thermoforming process consists in heating a polymer sheet and shaping it inside a mold. The deformation results from apressurecycleeventually coupled with the use of a moving tool. The forming ofthinproductsmadeofvariouspolymermaterialsis carriedoutbythismethod:rubberyamorphous,solid semicrystalline or multilayer composites. The heating temperature depends upon the selected polymer. At low temperature, the forming is limited bythetoohighrigidityofthesheet.onthecontrary, athightemperature,thesheetdeformsbygravityand the forming is very difficult to control. Hence, the thermoformingisoperatedabovetheglasstransition temperature(t g ),intherubberlikebehaviordomain for amorphous polymers, and close to the melting temperatureforsemicrystallinepolymers. Themainproblemoftheprocessisthethinningin thecornersoftheparts,whichleadstoadecreaseof themechanicalpropertiesoftheshapedcomponents. The optimization of the final thickness profile is generally achieved by trialanderror, changing the design of the component, the polymer material, and the process parameters such as the heating temperature distribution, the mold temperature, the pressurecurve,orusingapunchfordeepparts.the numerical modelling should then result in a more efficientoptimizationoftheprocess. Various numerical simulation models have been proposed(warby& Whiteman 1988, De Lorenzi& Nied 1991, Kouba et al. 1992, Shrivastava& Tang 1993). Regarding the application to three dimensional formings, they are generally based on the membrane mechanical approximation, associated with the finite element method. Those computations use either hyperelastic(de Lorenzi& Nied 1991, Shrivastava&Tang1993)orviscoelastic(Warby& Whiteman 1988, Kouba et al. 1992) constitutive equations, without any heat transfer coupling and a sticking contact hypothesis at polymertools interface. Itshouldbepointedoutthatthislatterassumption hasnoexperimentalevidence.mostofthetime,itis used just because the computation is isothermal and cannot account for the decrease of the polymer temperature and its "freezing" after tool contact. Another thermomechanical coupling is the high self heating source term due to the high strain rates. As the behavior of polymers is known to be highly temperaturedependent, it seems essential to couple heattransferandmechanicalmodels. In the present paper, we describe a 3D finite element model with membrane approximation to predict the deformation. Concerning the contact conditions with punch and mold, either sliding contact with Coulomb's friction law or sticking contact can be considered. The paper is focused on thethermalevolutioneffectsduringtheforming,and ontheapproachtomultilayersheetforming. 2MATERIALBEHAVIORIDENTIFICATION The onedimensional constitutive equation initially proposed by G'Sell and Jonas(1979) has been selected in the context of polymer thermoforming: σ=k p (T)1exp(wε)exphε 2 ε m (1) where T is the temperature, ε the von Mises equivalent strainrate, ε the von Mises equivalent strain.k p,w,handmarematerialparameters. This law accounts for the material behavior of a greatnumberofpolymers,eitheramorphousorsemi crystalline, in a large temperature interval. Duffo et al. (1994) have identified parameters for

3 polypropylene by means of tensile tests on sheets, yielding: K p = exp[ /T]MPa.s m w=20,h=0.4,m=0.087 We have studied the material behavior of a polystyrene compound, by uniaxial tensile testing at constantaxialstrainrate.testshavebeencarriedout at four constant temperatures: 110, 120, 130 and 140 C(whichcovertheformingtemperaturedomain of this polymer) and five different strainrates between 10 4 and 10 2 s 1. The associated determination of the coefficients of equation (1) shows the transition from a solidlike behavior to a liquidlikeone:seefigure1.theconsistencyk p of thematerialdecreasessuddenlyabovet g andkeeps lowvalues( 2)intheexperimentaldomain.Theso called viscoelastic coefficient w is found low( 3.7) and independent on temperature. Finally, it has appeared that the basic equation (1) has to be modified, in order to include a temperature dependencyofthestrainratesensitivitycoefficientm andthehardeningcoefficienth. h & m Kp h m temperature ( C) h m Kp 100 Fig.1Coefficientsm,handK p (MPa.s m )ofequ.(1) forpolystyrenevstemperature.(greydotsidentified bytensiletests,blacksquaresissuedfromliterature). 3MECHANICALFORMULATION " #$# According to membrane mechanical assumption, the deformed sheet is considered as a geometric surface, neglecting flexion and transverse shear. A material point is identified by two curvilinear coordinates: θ 1 and θ 2 whichare,inthecaseofan initialsheetinxyplane,theinitialxandycoordinate respectively. At any point of the deformed sheet (vector)thelocaltangentbasisisdefinedby: 1 = Ž Žθ 1 2 = Ž Žθ = (2) 0 The equilibrium of the deformed sheet is expressed by the principle of virtual work(without inertia): * σ ij v * i j ed P 3. * d. * d=0 Ω Ω p Ω c (3) where σ ij are the covariant components of the Cauchystresstensor(planestress:i,j=1,2), j denotes covariantderivationwithrespecttoθ j,eisthesheet thickness,arethefrictionstressesonregions Ω c contacting the mold, P is the inflation pressure appliedtothedomainω p ofthedeformedsurfaceω. "%&'"( StartingwithabalancedconfigurationΩattimet, the problem consists in determining the unknown equilibrated configuration Ω' at t+ t. Variables at t+ taredenoted"prime".byapplicationofeq.(3)at t+ t,wehave,foranyvelocityfield*: σ' ij * i j e'd P'' 3. * d '. * d=0 Ω' Ω' p Ω' c (4) This equation is solved for the incremental displacementfieldbetweenωandω',providedthat ',e'andσ'canbecalculatedfrom.thoserelations areexposedhereunderandtheresolutionin3.3. )* as regard the contact conditionappliedtoω c,itmaybeeithersticking(no relativevelocitywithrespecttothemold)orsliding: in this case, the tangential stress is supposed to be givenbythecoulomb'sfrictionlaw(coefficientµ), inwhichthenormalstressistheinflationpressure: '=µp'(1/ ) (5) &!+,*thenewlocalthicknesse'is deducedfrommaterialincompressibility.denoting themetrictensor(g ij = i. j ),wehave: e'=e det()/det(') (6) ) - $*theonedimensionalconstitutiveequation (1) can be written as a classical viscoplastic power law: σ=kε m (7) Hence, the flow rule derives from a viscoplastic potentialq.underisotropyassumption,ityields: ε= ŽQ Žσ =ŽQ Žσ Žσ Žσ =εžσ Žσ (8) Inconvectivecurvilinearcoordinates,wehave: ε 2 =2/3ε ij ε ij σ 2 =3/2s ij s ij =σ T σ (9) A ijkl =3/2g ik g jl 1/2g ij g kl (10) Theconstitutiveequationcanthenbewritten:

4 ε=(ε/σ)σ=(1/k)ε (1m) σ (11) A semiimplicit time integration scheme is used overtheincrement.theincrementalstraintensor ε, the covariant components of which depend on the displacementaccordingto(12)iswrittenas(13): ε ij =1/2(u i j +u j i +u m i u j m ) (12) ε= t(1η)ε+ηε' (13) Equations(1113) clearly permit to deduce the newlocalstresstensorσ',knowingthedisplacement field. Practically, a fully implicit scheme(η=1) is used. "*)!.)"""/ using a Galerkin 1D f.e.m.(figure 2) and a semi implicittimeintegrationscheme(vantal1995). The coupling between the mechanical and the thermal resolution is carried out at each time incrementasexplainedinfigure3. 5MULTILAYERFORMULATION The multilayer approach proposed here is based onthefollowingassumptions: eachlayerissubmittedtothesamedeformation as the mean surface of the sheet. This is consistent with Finally, injection of (56) and (1113) in the equilibrium equation (4) leads to a nonlinear equation for the displacement field. Its spatial discretization by f.e.m. (linear triangles or quadrangles) is detailed by Bellet(1988, 1990). At everytimestepanonlinearsystemforthevector ofthenodaldisplacementsissolvedbythenewton Raphsonmethod,withaconsistenttangentmatrix. 4HEATTRANSFERRESOLUTION membrane finiteelement mesh Considering the thinness of polymer sheets, the short processing times, and the low diffusivity of polymers, it can be assumed that heat transfer is essentially one dimensional across the thickness of the sheet(vantal 1995). Consequently, s being the coordinateinthethicknessdirection(s=θ 3 ),the1d heattransferequationcanbeexpressed: ρc dt dt =Ž Žs λžt Žs +σ:ε (14) whereρisthespecificmass,ctheheatcapacityand λ the heat conductivity. The following boundary conditionsareaccountedfor.atsheet/airinterface: λ ŽT Žs sgn()=h conv(tt air ) (15) whereh conv isthecoefficientforheatexchangeby convection,t airtheairtemperatureandsgn()is±1 depending on the orientation of the outward normal unitvector.atsheet/toolinterface,duetothemuch higher diffusivity of metals, we willassumethatthe surface temperature of the polymer sheet is prescribedtotheinterfacetemperaturegivenby: T inter=(b tool T tool +b sheet T sheet )/(b tool +b sheet ) (16) wherebisthethermaleffusivity λρc and T sheet is theaveragetemperatureofthesheetinthethickness. The initial temperature profile is assumed to be knownatthebeginningoftheprocess. Equation(14) is discretized in space and time, at each integration point of membrane finite elements, 1Dfiniteelementmesh Fig.21Dapproachforthermalcoupling

5 A/Thermalresolution forallmembraneelements forallintegrationpointsoftheelement performa1df.e.computationofthenew temperatureprofilet'acrossthethickness. Thesourcetermisgivenbythe mechanical resolutionof previoustimestep. forallintegrationpointsipthof1dmesh updatethetemperaturedependent coefficients:k' p (T'),m'(T')andh'(T'). B/Mechanicalresolution B1)computationoftheresidualvector( ) forallmembraneelements forallintegrationpointsoftheelement forallintegrationpointsipthof1dmesh solvetheconstitutiveequationsforσ'(ipth) and σ'/ ε(ipth),usingtheupdated valuesofmaterialcoefficients:k' p (ipth), m'(ipth)andh'(ipth) computethicknessaveragedvalues<σ'> and< σ'/ ε>andsuminresidualvector. B2)iterativeNewtonRaphsonproceduretosolve ( )= Fig.3Thermomechanicalcouplingalgorithm. the membrane approach in which transverse shear andflexionareneglected.hence,thethicknessratio of the different layers remains constant during the process. accordingly,themechanicalandthermalcontact are assumed perfect(no sliding, no thermal contact resistance)atinterfacesbetweenlayers. The algorithm is identical to the one for heat transfer coupling, except that all thermal and mechanical parameters used at integration points in thickness(ipth)willnowdependuponthematerialin which they are located (see fig. 4). Each layer material has an identified temperature dependent behaviorlaw. Suchaformulationisexpectedtobemoreprecise than the reduction of the multilayer to a single "equivalent" material, especially when steep temperature gradients appear in the sheet thickness whenonesideofthemultilayercontactsthetooling. nlayers 0" *1!), Formings were carried out on a small machine equipedwithcontactsensorsandpressuregauges. POLYMER SHEET CONTACT SENSORS MOLD PressurizedAir PRESSURE GAUGE PRESSURE GAUGE Fig.5Experimentalthermoformingsystem(the optionalpunchsystemisnotshownhere). Y X Z Fig.6Moldandpunchview. The mold geometry is axisymmetric(diameter 140mm,depth60mm),includingacentralinsertin thebottom(seefig.6)whichmakesitverysensitive both to cooling and friction effects. The forming parameters are the followings: aluminium mold, 20 C, air temperature = 20 C, linear pressure vs time:0.6mpa.s 1,initialsheettemperature130 C. The onelayer polystyrene sheet is initially 1.1 mm thick. Its orientation due to extrusion is relatively high, and it involves an evolution of the thickness during infrared heating as it is clamped. Consequently, the measured final thickness profile depends upon the measurement direction(extrusion ortransversedirection,seefig.7). 1Dmesh Fig.4Multilayerformulation(schematic). 6TESTOFTHEMODEL

6 1,10 0,90 0,70 0,50 0,30 thickness(mm) EXPERIMENTS : MODEL : extrusion direction transversal direction R(mm) 0,10 0,0 18,0 36,0 54,0 72,0 90,0 Fig.7Comparisonbetweenmeasured andcomputedthicknessdistribution. 0"%)! Due to axisymmetry, only a narrow sector has been meshed with 160 triangles. We have used the rheological data of fig.1 andthefollowingvaluesof thermalparameters: ρc= J.m 3.K 1, λ=.2 W.m 1.K 1,h=33W.m 2.K 1.ACoulombfriction hasbeenaccountedfor(µ=0.4). The predicted final thickness profile, using the coupled thermomechanical model, is in good agreementwiththemeasurements(figure7). Regardingkinematics,itshouldbenoticedthatthe measured differential pressure was found very low. The use of these values as a prescribed boundary condition(p(t) in equation(3)) has lead to much slower forming rates than the actual ones. However, ifthemeasuredupperpressureisusedinsteadofthe differential pressure, then computed forming times areinagreementwithexperimentalones,asshownin table 1. This unexpected result needs more investigation. contacttime(s)atsensor# experiment computation 0"())!, 0.095s 0.400s 0.510s 0.102s 0.311s 0.390s Table1 coefficient of the material for the contact zones, as the. strainrate sensitivity coefficient decreases. The evolution of these coefficients localize the deformation in the warmer zones: the strainrate values are almost zero in the"frozen" zones. The nonisothermalmodelismuchmorerepresentativeof thelocalphenomenainthermoformingandisableto account independently for interface tribology and heattransfer. thickness(mm) 1,10 0,90 0,70 0,50 0,30 ANISOTHERMAL Coulomb s contact µ=0,4 ISOTHERMAL sticking contact ISOTHERMAL Coulomb s contact µ=0,4 R(mm) 0,10 0,0 16,0 32,0 48,0 64,0 80,0 Fig.8Computedthicknessforthreemodels. 0" 21 Wehavesimulatedtheformingofabilayer:80% polystyrene (PS), 20% polypropylene (PP), total thickness 1mm. The rheological parameters for both polymers are those given in section 2. The sheet is formedsuccessivelywithapunch(z=30mmat0.15 s) and pressure(linear increase of 0.8 MPa between 0.15sand0.65s).Theinitialtemperaturesare20 C fortoolsand150 Cforthesheet. It is shown in table 2 and figure 9 that the deformationofthebilayerasimpleadditionor averageofthedeformationofeachlayercomponent PPonly PP_PS PS_PP PSonly PPontop PSontop 0.59s 0.50s 0.39s 0.45s Table2Predictedformingtimes thickness(mm) On figure 8, three different computed thickness profiles have been plotted, using three different computationaloptions. Asalreadysaid,theassumptionofstickingcontact has no clear experimental evidence. However, as shown on figure 8, it is the only means to get reasonableresultsifanisothermalmodelisused.for example the combination(isothermal;highfriction) yieldscompletelyerroneousthicknesses(fartoolow ontheinsert,despitethehighfrictioncoefficient)! On another hand, the use of the present non isothermal formulation permits to decouple clearly the frictional and the thermal effects: the quick cooling of the polymer when contacting the mold increases the consistency and the hardening.91 PS.71 PS_PP.52 PP PP_PS.32 R(mm) Fig.9Computedthicknessprofiles.

7 with the same thickness. In addition, the results clearlydependonwhichmaterialisontop. This is due to the fact that the cooling effect is different for PS or PP: the consistency of PS decreasessuddenlyabovet g whereasitfollowsan ArrheniuslawforPP.Alsomandharetemperature dependentforpswhereasconstantforpp. length 7APPLICATION:INDUSTRIALFORMING Inordertotesttherobustnessandtheresultsofthe code,theformingofaonelayershallowcomponent forfoodpackaginghasbeenstudied(fig.10). diag2 diag1 diag3 width Fig.11Finaldeformedfiniteelementmeshand selecteddirectionsforthicknesscomparison. Fig.10Selectedpolypropylenetestpart. Thematerialispolypropylene,initially0.475mm thick. The industrial forming conditions have not been accurately measured, but the following figures can be considered realistic: aluminum mold(5 C), linear pressure reaching 0.7MPa at 0.5s, air temperature20 C,initialsheettemperature150 C. Onlyaquarterofthesymmetrizedactualforming has been computed, using 7591 and 6463 triangles for the sheet and mold meshes respectively. A Coulomb's friction law(^=0.4) is assumed at the mold surface, except near the edge where sticking contactisprescribed.asshownonfig.12,thecentral bulgingofthesheetisveryfast,reachingthebottom at 0.04s. The forming is then slowed down until the end.

8 experimentaldispersionhasbeenfoundhigh(upto 18%ofthe Z(mm) EXPERIMENTAL THICKNESS (mm).calculated THICKNESS (mm) length (mm) Fig. 13 Example of comparison between measured and computed thickness ("length" on figure12). average value). The average of relative errors between computed and average experimental thicknessisgood:10%atthebottom,18%atwall. 8CONCLUSION Thermomechanical coupling and a multilayer approach have been implementedinafiniteelement membrane model. They are shown to be very efficient tools to improve the predictive character of finiteelementsimulationsofthermoforming. 9ACKNOWLEDGEMENT Theexperimentalpartofthisworkhasbeendone at ElfAtochem(Cerdato) which has also supported thisstudy. 10REFERENCES Fig.12Computeddeformedsheet Figure11showsthedeformedfiniteelementmesh at the end of the process. Isovalues of temperature (notshownhere)indicatethatduringformingthefree regions are the warmer (close to the initial temperature), and that in contacting regions, the longerthecontactduration,thecooleristhesheet. The final thickness of actual parts has been measured along the five directions mentioned on figure 11. An example of comparison with the computedvaluesisshowninfigure13.experimental pointsareissuedfrommeasurementsontwodifferent parts, yielding four values per point. The Bellet M Modélisation numérique du formage superplastiquedetôles.doctoratethesis(infrench),ecole desminesdeparis. Bellet M., Massoni E. and Chenot J.L Numerical simulation of thin sheet forming processes by the finite elementmethod.eng.comp.,7,p.21. DeLorenziH.G.&NiedH.F.1991.Finiteelementsimulation of thermoforming and blow molding. Modelling of Polymer Processing, A.I.Isayev (ed.), Hanser Verlag, chap.5,pp Duffo P., Monasse B., Haudin J.M., G'Sell C.& Dahoun A Rheology of polypropylene in the solid state. J. Mater.Sci.,29. G'Sell C. & Jonas J.J Determination of the plastic behavior of solid polymers at constant true strain rate. J. Mat.Sci.,14,pp Kouba K., Bartos O. & Vlachopoulos J Computer simulation of thermoforming in complex shapes. Polym. Eng.Sci.,32,pp Shrivastava S. & Tang J Large deformation finite elementanalysisofviscoelasticmembraneswithreference tothermoforming.j.strainanal.,28,pp Vantal MH Etude numérique et expérimentale du thermoformage des polymères. Doctorate thesis (in french),ecoledesminesdeparis.

9 Warby M.K.& Whiteman J.R Finite element model of viscoelastic membrane deformation. Comput. Struct., 68, pp.3354.