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Identification of the Norton-Green Compaction Model for the Prediction of the Ti–6Al–4V Densification During the Spark Plasma Sintering Process

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Identification of the Norton-Green Compaction Model for the Prediction of the Ti–6Al–4V Densification During the Spark Plasma Sintering Process
  DOI:  10.1002/adem.201600348 Identi fi cation of the Norton-Green Compaction Model forthe Prediction of the Ti – 6Al – 4V Densi fi cation During theSpark Plasma Sintering Process** By  Charles Mani   ere, Ugras Kus, Lise Durand, Ronan Mainguy, Julitte Huez, Denis Delagnes and  Claude Estourn  es*  One of the main challenges for the industrialization of the spark plasma sintering (SPS) is to resolveissues linked to the compaction of real parts with complex shapes. The modeling of powder compactionisaninterestingtooltopredicthowthedensi  fi cation  fi eldvariesduringsintering.However,expressingthe behavior law which re  fl ect the powder compaction is often a dif   fi cult and long step in the modelestablishment. In this paper, a simple methodology for the identi  fi cation of the densi  fi cation parametersis proposed. Dense and porous creep tests combined with SPS die compaction tests are employedto determine a complete densi  fi cation law on a Ti  –  6Al  –  4V alloy directly in a SPS machine. Thecompaction model obtained is successfully validated through prediction of the densi  fi cation of newSPSed samples. 1. Introduction The spark plasma sintering process is a breakthroughtechnology based on the simultaneous application of highpressure and high temperature. This process is close to hotpressing (HP) except that the heat is generated, both in thetools and sample if electrically conducting, by a pulsedcurrent applied through the entire column. [1] This processyields high temperatures and pressures, respectively, up to2000  Cand150MPa(withgraphitetools)andarapidheatingrate (up to 1000  Cmin  1 ). This technology can produce awide range of highly dense materials from polymers, metals,and alloys including refractory materials and is a promisingtechnique to obtain better-controlled and homogeneousmicrostructures. [2] Despite all these advantages, it is dif  fi cultto predict and control the temperature and densi fi cation  fi eldduring the experiment. To avoid long and expensive trial-and-error experiments,  fi nite element modeling (FEM) can bea very powerful tool.Two main phenomena are usually modeled: Joule heatingand sintering. Concerning the electro-thermal part (heating),many authors consider the pure resistive model to be a goodapproximation of the heating phenomena. [3 – 6] They point outthe concentration of heat in the punches and the importanteffect of electric and thermal contacts [7 – 12] on the temperature fi eld. On the other hand, the powder compaction phenome-non is often modeled using the visco-plastic approachthrough Olevsky, [13,14] Abouaf, [15] or Camclay [16] models.Thisapproachrequirestheidenti fi cationofcreeplawsonboththe dense and the porous material.In aclassical approach,theidenti fi cation of the creep parameters is particularly long anddif  fi cult and needs compression experiments at differenttemperatures and pressures with dense and porous materialsamples. Instrumented hot isostatic pressing (HIP) tests alsohave to be performed, [17,18] they generally need very longheating times (i.e., several hours) compared to our target SPSprocess(afewminutes).Theimpactoftheheatingtimeonthemicrostructure can be potentially high and cause discrep-ancies in the model. Consequently, an increasingly number of [*]  Dr. C. Estourn  es, Dr. C. Mani  ere, U. KusCIRIMAT, Universit  e de Toulouse, CNRS, INPT, UPS 118route de Narbonne, 31062 Toulouse cedex 9, FranceE-mail: estournes@chimie.ups-tlse.frDr. C. Mani  ere, Dr. L. DurandCEMES, CNRS UPR 8011 and Universit  e de Toulouse, 29 rue Jeanne Marvig, 31055 Toulouse, FranceU. Kus, R. Mainguy, Dr. J. HuezCIRIMAT, Universit  e de Toulouse, CNRS, INPT, UPS 4 all  eeEmile Monso, 31030 Toulouse cedex 4, FranceDr. D. DelagnesUniversit  edeToulouse,CNRS,MinesAlbi,INSA,UPS,ISAE,ICA (Institut Cl  ement Ader), Campus Jarlard, 81013 Albi,France [**]  The support of the Plateforme Nationale CNRS de FrittageFlash (PNF2/CNRS) is gratefully appreciated. C. M. and C. E.thank the French National Research Agency (ANR) for  fi nancial support of this study within project ANR09 MAPR-007 Impuls  e. DOI: 10.1002/adem.201600348 © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  wileyonlinelibrary.com  1 ADVANCED ENGINEERING MATERIALS  2016 , F  UL  L  P AP E R  authors use inverse analysis to determine the powdercompaction parameters directly on the SPS tests. [14 – 20] OtherauthorssubstituteHIPtestsbyanothertypeofmechanicaltestsuch as the determination of the radial strain rate in acylindrical porous creep test, [21] or a die compaction test. [22,23] In the present work, a methodology is proposed to determinethe powder compaction parameters for the Norton-Greenmodel [24] also called the Abouaf model. The methoddeveloped includes creep and SPS die compaction testssimilar to that reported by Geindreau et al. [23] The method isapplied to SPS machine using different con fi gurations. Thesecreep and die compaction con fi gurations and operationalconditions were improved in order to obtain a good stabilityof the temperature and accurate results.The main contribution of this work is then to adapt theclassical methodology [15 – 23] to the SPS conditions (highheating rate and pressure) in the aim of being very close tothe target application of the model. The proposedmethodology is then perfectly suited to determine densi fi -cation properties for potential applications using  fi nemicrostructures or for a preserved speci fi c phase ormicrostructures.Theidenti fi cationprocessisdividedintotwosteps:i)creepexperiments are performed by direct/indirect heating of cylindrical samples of dense material at different temper-atures (current assisted or current insulated). ii) the determi-nation of the compaction law is performed using the resultsobtained from creep tests with cylindrical samples of porousmaterial and SPS die compaction tests.In this paper, the densi fi cation of the Ti – 6Al – 4V alloy alsocall TA6V is investigated. This alloy is widely used inaerospace applications, [25,26] due to its low density and itsgood mechanical properties at high temperature (up to400  C). [27 – 29] Moreover, this alloy is also used for medicalapplications due to its bio-compatibility. [2,30] 2. Experimental Section All the experiments were performed on the SPS machine(Dr. Sinter 2080, SPS Syntex Inc, Japan) of the  “ PlateformeNationale CNRS de Frittage Flash ”  located at the Universit  eToulouse III-Paul Sabatier. For this study, the densi fi cation behavior of a Ti – 6Al – 4V alloy was investigated with densesamples produced by die forging (for more details see aprevious study [31,32] ), and a prealloyed powder kindlyprovided by Aubert and Duval. This powder is made of spherical granules ranging in size from 40 to 400 m m. An SEMimage and the composition of the powder are given inFigure 1 and Table 1, respectively. In the following, when werefertothecastmaterialwewillspeakaboutdenseTi – 6Al – 4V,otherwise we are referring to the powder material.For the identi fi cation of the densi fi cation model differentexperimental con fi gurations reported in Figure 2 were used.The determination of creep behavior on dense samples wasperformed using the con fi gurations reported in Figure 2a and b. The determination of the powder compaction behaviorrequires two types of experiment: i) creep experiments onporous samples as reported in Figure 2c and powder SPS diecompaction experiments using the con fi guration reported inFigure 2d.The creep experiments were performed on denseTi – 6Al – 4V cylinders 8mm in diameter and 10mm inheight. Two different con fi gurations, reported inFigure 2a and b, were used to carry out these experiments:Con fi guration 1, direct heating creep tests (Figure 2a): thecurrent is allowed to pass through the sample.Con fi guration 2, indirect heating creep tests (Figure 2b):the current passes only through the die that is added to heatthe sample. For this con fi guration the sample is electricallyinsulated by a boron nitride spray deposited on the upperand lower punch faces in contact with the sample. Forthe porous creep experiments, a similar con fi gurationwas used (Figure 2c) except the sample was a porouscylinder obtained by interrupted SPS compaction. The lastcon fi guration (Figure 2d) is SPS die compaction performedon the powder inserted into an 8mm innerdiametergraphitedie.The creep tests (Figure 2a – c) were performed at differentholding temperatures between 750 and 900  C at a heatingrate of 100Kmin  1 . Except for the contact due to the thermalexpansion of the cylinder, only a minimum load was appliedduring the heating ramp. Then only a few MPa are applied before the isothermal stage. Once the temperature setpoint isreached and the thermal expansion is stabilized, differentlevels of pressure were applied and the dwell time for eachwas maintained until stabilization of the strain rate. The SPSdie compaction experiment (Figure 2d) was performed at agiventemperature( T  ¼ 820  C)withdifferentlevelsofappliedpressure until complete densi fi cation of the powder. The Fig. 1. Ti  –  6Al  –  4V powder under the Scanning Electron Microscope (SEM).Table 1. Chemical composition of the Ti  –  6Al  –  4V powder. Element Al V Fe C N O Ti[%wt] 6.12 4.06 0.19 0.014 0.003 0.15 Base C. Mani  ere et al./Identification of the Norton-Green Compaction Model . . . 2  http://www.aem-journal.com © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/adem.201600348ADVANCED ENGINEERING MATERIALS  2016 ,      F     U     L     L     P     A     P     E     R  temperature measurements were performed by a K-typethermocouple positioned in contact to the surface of thesample for the experiments reported in Figure 2a – c. A 1mmdeep hole is previously made at the surface of the sample tomaintain the thermocouple during the shrinkage. For theexperiment corresponding to Figure 2d, the temperature isobtained by a K-type thermocouple placed directly in thepowder bed. Another experiment was performed to ensurethat the thermocouple does not disturb the measurement of the shrinkage curve.To check the model (creep and densi fi cation), two experi-ments (con fi guration displayed in Figure 2d) were performedup to stabilization of the relative density at a constant appliedpressure of 25MPa and with two temperature ramps: i) 50and ii) 100Kmin  1 . These compaction tests were repeatedtwice to determine the temperature of the sample (by anotherthermocouple placed in the powder bed) and the displace-ment of the graphite thermal expansion in the anisothermalpart. 3. Theory and Calculation3.1. Compressive Dense Creep Formulation The Norton-Green model is based on a creep power lawformula de fi ned for visco-plastic materials. _ e eq  ¼  A s  n eq  ð 1 Þ where  _ e eq  is the equivalent strain rate (s  1 ),  s  eq  the equivalentstress (Pa) and  A  a temperature dependent parameter(s  1 Pa  n ):  A ¼  A 0  exp   QRT     ð 2 Þ with,  A 0  the pre-exponential factor (s  1 Pa  n ),  R  the universalgas constant (Jmol  1 K  1 ),  Q  the creep activation energy(Jmol  1 ) and  T   the absolute temperature.The  A  and  n  Norton parameters are identi fi ed usingcompressive creep tests (Figure 2a and b) with the followinglinear regression equation:ln  _ e eq   ¼ ln  A ð Þþ n ln  s  eq    ð 3 Þ Theexperimentalstrainratecanbecalculatedwith the true strain expression: _ e ¼ d  ln  hh 0    dt  ð 4 Þ with,  h 0  and  h , respectively, the initial andat the given time heights of the cylindricalsample.The experimental stress can be calculatedwith the measured force and the samplesection  S . The section can be determined eachtime increment by the height  h  and the initialsection  S 0  of the sample assuming conservation of thecylindrical shape. S ¼ S 0 h 0 h  ð 5 Þ 3.2. Norton-Green Model Description The strain rate tensor of the Norton-Green model forporous solids is as follows [17] : _ e  ¼  A s  eq n  1  32 cs þ  fI  1  i    ð 6 Þ with,  s  the deviatoric stress tensor,  I  1  the stress tensor  fi rstinvariant (the trace of the stress tensor),  i  the identity tensor,  c and  f   functions of the relative density  r . [24] The equivalent stress is de fi ned by the followingequation: s  eq  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 cJ  2 þ  fI  21 q   ð 7 Þ with,  J  2  the deviatoric stress tensor second invariant.Developing the expression of the deviatoric stress tensor s  in (6) gives the stress tensor expression as follows: s   ¼  A  1 n _ e eq 1 n  1  23 c _ e þ  19  f     29 c   tr  _ e ð Þ  i    ð 8 Þ This expression is close to the expression used in theOlevsky model in a pure visco-plastic approach [13] :The equivalent strain rate is de fi ned by the followingequation: _ e eq  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23 c _ e  : _ e þ  19  f     29 c   tr  _ e ð Þ 2 s   ð 9 Þ To link the variation of the relative density  r  and the localvolume change, mass conservation is de fi ned as follows: _ rr ¼ tr  _ e ð Þ ð 10 Þ Fig. 2. Creep and compaction test con  fi  gurations: (a) direct heating creep of dense samples (currentassisted), (b) indirect heating creep of dense samples (current insulated), (c) indirect heating creep of poroussamples, (d) classical SPS test (SPS die compaction test). C. Mani  ere et al./Identification of the Norton-Green Compaction Model . . . DOI: 10.1002/adem.201600348 © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim http://www.aem-journal.com  3 ADVANCED ENGINEERING MATERIALS  2016 , F  UL  L  P AP E R  3.3. Compressive Porous Creep Formulation The macroscopic stress tensor in compressive creep with aporous sample (Figure 2c) can be reduced to: s    0 0 00 0 00 0  s   z 0B@1CA  ð 11 Þ with,  z  the revolution axis of the cylindrical sample.Theexpressionoftheequivalentstress(7)canbereducedto: s  eq  ¼  s   z j j  ffiffiffiffiffiffiffiffiffiffi c þ  f  q   ð 12 Þ Combining (12), (11), and (6) we obtain the  fi rst  c  and  f  identi fi cationexpressionfortheporouscreeptests(Figure2c): _ e  z j j¼  A f   þ c ð Þ n þ 12 s   z j j n ð 13 Þ 3.4. SPS Powder Die Compaction The SPS case can be assimilated to uniaxial compaction,along the z-axis, of the powder in the die. Then themacroscopic strain rate tensor can be reduced to: _ e   0 0 00 0 00 0  _ e  z 0B@1CA ð 14 Þ The equivalent strain rate (9) is then given by the relation: _ e eq  ¼  _ e  z j j 13  ffiffiffiffiffiffiffiffiffiffiffi 4 c þ 1  f  s   ð 15 Þ Combining (15), (14), and (8) we obtain the second  c  and  f  identi fi cation expression for the SPS die compaction tests(Figure 2d): s   z j j¼  A  1 n _ e  z j j 1 n 13   1  nn 4 c þ 1  f    n þ 12 n 19  ð 16 Þ 3.5. c and f Identi  fi cation Equations The  c  and  f   functions can then be determined by theresolution of the following system of two equations (17).  c þ  f  was obtained from porous creep tests (with con fi gurationFigure 2c) and 4/ c þ 1/  f   from SPS die compaction tests (withcon fi guration Figure 2d): c þ  f   ¼  _ e  z j j  A  s   z j j n   2 n þ 14 c þ 1  f   ¼  9  A 1 n  s   z j j  _ e  z j j  1 n  13    1  nn 0B@1CA 2 nn þ 1 8>>>>>>>>>><>>>>>>>>>>: ð 17 Þ To summarize,  fi ve parameters have to be determined inorder to model powder compaction. The parameters  A 0 ,  Q ,and  n  for the power law creep were determined from creeptests on dense samples, and the  c  and  f   functions weredetermined from both creep and SPS die compaction tests onporous samples. 4. Results and Discussion Sections 4.1 and 4.2 are devoted to the determination of thecreep power law parameters  A 0 ,  Q , and  n  with the direct(Figure 2a) and indirect (Figure 2b) heating con fi gurations.The identi fi cation of   c  and  f   is developed in Section 4.3.Section 4.4 is devoted to the validation of the powdercompaction model with independent SPS tests. 4.1. Power Law Creep Identi  fi cation in Direct HeatingCon  fi  guration The experimental curves of strain rate for increasingapplied load obtained at 750  C are reported in Figure 3a. Anexternal camera is added for this experiment to verify theaccuracy of the displacement curves provided by the SPSmachine and to control that the eventual disturbance of thethermal expansion due to thermal non-equilibrium remainssuf  fi ciently low. The strain rate curves (camera vs. SPS)reported in Figure 3a in isothermal regime are very close. TheSPSdataarethenacceptablemeaningthethermalequilibriumis quickly reached or the temperature is suf  fi ciently stable toallow displacement measurements. The experimental datapointsofthestrainrateasafunctionofthestressrateobtainedfor temperatures of 750, 800, 850, and 900  C are reported in alog-log graph (Figure 3b).Considering Equation 3 for each temperature, the experi-mentalpointsshouldnaturallyalignonastraightlinegivingaslope that allows the n exponent to be determined. RegardingFigure3,thereisahighlevelofscatteringinallofthesecurves,giving  n  values between 1 and 3 with an average value of 2.3. According to the Norton-Green model (Eq. 3), theexperimental curves are expected to be roughly linear andpositioned from the right (high pressure) for the lowtemperatures to the left (low pressure) for the high temper-atures. These results showan inappropriate positioningofthecurves, for instance the curve at 800  C is expected to be onthe left of the curves obtained at 750  C. Moreover, therepeatability in the curves positioning at a single temperatureof 750  C is not satisfactory. This problem of positioning can be explained by the pictures of the sample, reported inFigure 3c, taken during a creep experiment at the beginningand the end of the isothermal dwell time. A strong thermalgradient was observed between the center of the cylindricalsample (hot) and the edge in contact with the punches (cool).This thermal gradient implies a higher deformation of thesample at his center and creates a non-cylindrical deformedsample at the end of the dwell. This phenomenon of thermalgradient can be explained by the output heat  fl ux on thesample/punch interface that is strongly in fl uenced by the C. Mani  ere et al./Identification of the Norton-Green Compaction Model . . . 4  http://www.aem-journal.com © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/adem.201600348ADVANCED ENGINEERING MATERIALS  2016 ,      F     U     L     L     P     A     P     E     R  thermal contact resistance (TCR) of this interface. The less thepressure is, more is this TCR. Consequently, for a lowerpressure, the TCR is higher and the output heat  fl ux at theinterface is lower, the overall temperature of the sample isthen higher. In practical terms, this could explain why one of the curves at 750  C was positioned on the left (higher overalltemperature) for lower pressure as compared to the others.Based on our previous work [11] on the Joule heatingsimulation of the spark plasma sintering, the maximummagnitude of the thermal gradient of the sample in directcon fi guration can be investigated numerically. The tempera-ture dependent parameters of the SPS column elements aretaken from ref. [11] except for the properties of Ti – 6Al – 4VwhicharereportedinTable2.Atthespecimen/punchcontact,the thermal effect of the papyex on the specimen thermalgradient is a key parameter. As previously explained, thethermal contact resistance decreases the heat  fl ow across theinterface and then decreases the specimen thermal gradient.Inordertoevaluatethemaximummagnitudeofthespecimenthermal gradient,the papyex effect ismodeled by an interfacecondition that takes into account the 4W (mK)  1 thermalconductivity of the 0.2mm papyex. The graphite/papyex andpapyex/Ti – 6Al – 4V TCR, which decrease with the pressureand temperature, are thus taken equal zero. Considering aperfect TCR is then close to the case of high pressure andtemperature and provides us a reasonable estimation of thespecimen maximum temperature difference in that type of con fi guration. In accordance with the experimental picture(Figure 3c), the simulation picture reported inFigure 3d at the beginning of the dwell show amaximum temperature at the middle height of the specimen and an important decrease of thetemperature in the area closed to the punch.The temperature difference is about 200K. Asimulation similar to the one presented inFigure 3d was also performed for a 5mm heightspecimen and in that case the temperaturedifference is reduced to about 100K. Thetemperature gradient is then signi fi cantly re-duced for a smaller specimen height, butremains too high considering our tests. More-over, considering the 1.8mm of the thermocou-ple diameter, this con fi guration is too small toperform multi-step pressure creep tests thatneed more important shrinkage distances com-pare to the classical approaches.The lack of control of the output heat  fl uxprevents any identi fi cation of the creep param-eters using this con fi guration. To solve this problem, in thenext creep tests, an indirect heating con fi guration wasdeveloped where the heat is not generated in the sample but transferred from external elements in contact. 4.2. Power Law Creep Identi  fi cation in Indirect HeatingCon  fi  guration With the indirect heating con fi guration (Figure 2b),according to the picture taken at the beginning of theisothermal stage (Figure 4a), the temperature distribution inthe sample appears to be more homogeneous. The experi-mental data points (Figure 4b) con fi rm this tendency becauseofthecorrectrelativepositionofeachcurvewithrespecttotheexperimental temperature and the regulargap between them.Pictures of the samples after the creep tests are reported inFigure 5. The deformed sample shape is closer to a cylinderwith indirect heating than with a direct heating con fi guration(Figure 3). The barrel effect is minimized. Linear regressioncan then be performed and gives:  A 0 ¼ 30.6s  1 Pa  n ;  Q ¼ 416kJmol  1 and  n ¼ 2. _ e eq  ¼ 30 : 6  exp   4 : 16  10 5 RT    s  2eq  ð 18 Þ The value of   n  thus obtained is  fi nally close to thatdetermined using direct heating and is in good agreementwiththe tendencyofthe valuesreported intheliterature [29 – 32] that decrease with temperature (Figure 4c). 4.3. Identi  fi cation of c and f  As the creep power law of the dense sample is nowdetermined, the  c  and  f   functions can be identi fi ed. Thesystem of Equations 17 was solved using creep testsperformed on porous samples and SPS die compaction teststhat gave values of the right hand side of both equations. Theresults of these tests are reported in Figure 6 with the upper Fig.3. Directheatingcreeptests:(a) evolutionof thestrain ratecurvesfor increasingappliedloadsobtain at750  C by the SPS device and with an external optical camera, (b) isobaric isotherm experimental points at750, 800, 850, 900  C; (c) images of a sample at the beginning and after dwell for the creep test (750  C),(d) temperature pro  fi le simulated at the beginning of the dwell.Table 2. Temperature-dependent properties of Ti  –  6Al  –  4V. [33] Electrical resistivity ( Ω m) 1.35E-6 þ 1.17E-9T-4.06E-13T 2 Thermal conductivity (w(mK)  1 ) 8.11-0.0149T þ 4.47E-5T 2 -2.27E-8T 3 Specific heat (J(kgK)  1 ) 383 þ 0.671T-5.35E-4T 2 þ 1.64E-7T 3 Density (kgm  3 ) 4467-0.119T-1.28E-5T 2 C. Mani  ere et al./Identification of the Norton-Green Compaction Model . . . DOI: 10.1002/adem.201600348 © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim http://www.aem-journal.com  5 ADVANCED ENGINEERING MATERIALS  2016 , F  UL  L  P AP E R
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