de l Université de recherche Paris Sciences et Lettres PSL Research University

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1 THÈSE DE DOCTORAT de l Univerité de recherche Pari Science et Lettre PSL Reearch Univerity Préparée à MINES PariTech Deformation of teel ingot by punch preing during their olidification. Numerical modelling and experimental validation of induced hot cracking and macroegregation phenomena Ecole doctorale n 364 Spécialité Science et Génie de Matériaux COMPOSITION DU JURY : Pr. Steven LE CORRE Univerité de Nante, France Rapporteur Pr. Menghuai WU Univerité de Leoben, Autriche Rapporteur Dr. Tohiyuki KAJITANI Nippon Steel & Sumitomo Metal Corporation, Japon Examinateur Soutenue par Takao KOSHIKAWA le h h Pr. Wilfried KURZ Ecole Polytechnique Fédérale de Lauanne, Suie Examinateur Dr. Benjamin RIVAUX ArcelorMittal Maizière, France Examinateur Pr. Michel BELLET MINES PariTech, France Directeur de thèe Dr. Charle-André GANDIN CNRS, France Directeur de thèe

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3 Acknowledgement Firt of all, I would like to thank my director, Pr. Michel Bellet and Dr. Charle-André Gandin for their acceptance that I came to their lab, great upport, kind guidance, many advice and encouragement. My two year tay in CEMEF and three year tudy through viio meeting from Japan enriched my knowledge but alo gave me challenge pirit to omething difficultie I have faced. I never forget all. I am incerely grateful to the member of the jury, Pr Steven Le Corre, Pr. Menghuai Wu, Pr. Wilfried Kurz, Dr. Tohiyuki Kajitani and Dr. Benjamin Rivaux for their great work on my thei committee. I would like to thank Olivier Jaouen and Dr Frédéric Cote from TRANSVALOR (Mougin, France), for their kind help and intereting dicuion about thermomechanical modeling uing oftware THERCAST. Without their help, my thei would not be completed. I would like to thank all the CEMEF taff. Special thank are ent to Patrick Coel, Marie-Françoie Guenegan, Carole Torrin and Suzanne Jacomet for their great help and to the director Yvan Chatel and Eliabeth Maoni. I would like to thank all the member of the TMP and SP2 group and the colleague: Gilda, Tommy, Alexi, Ala, Ali, Shijia, Thi-Thuy-My and other for their kindne and intereting talk with me. Many thank to Siham and Ana-Laura for their friendhip and having lunch together lot of time. I would like to thank to Abba for hi friendhip and cheerful talk at our bureau. Thi work i upported by Nippon Steel & Sumitomo Metal Corporation in a collaborative project with ArcelorMittal. I would like to thank the partner of the project and everyone in NSSMC who gave me thi great opportunity to go abroad and work on the project. At lat, I would like to thank my family, epecially my wife for her great upport. Thank you for your coming to France and challenging difficultie with our mall daughter. i

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6 Chapter I Introduction 1 1. Proce preentation 2 2. Typical defect in CC 3 3. Preentation of hot tearing and macroegregation defect Short preentation of defect in the CC proce Hot tearing phenomena Solidification tage Different mode of hot tearing Variou parameter effect on hot tearing Deformation induced egregation Prediction of hot tearing through numerical modeling Prediction of macroegregation by numerical modeling Microegregation modeling for multicomponent teel Numerical modeling Preentation of THERCAST Preentation of R2SOL Large cale experiment 25 Chapter II Computation of Phae Tranformation Path in Steel by a Combination of the Partial- and Para-equilibrium Thermodynamic Approximation 29 Chapter III Study of Hot Tearing During Steel Solidification Through Ingot Punching Tet and It Numerical Simulation 41 Chapter IV Experimental tudy and two-phae numerical modelling of macroegregation induced by olid deformation during punch preing of olidifying teel ingot.. 59 Chapter V Summary of the main reult Summary on microegregation modeling Summary on hot tearing prediction Summary on macroegregation calculation.. 86 iv

7 Chapter VI Perpective and recommendation for future work Perpective on microegregation modeling Perpective on hot tearing prediction Perpective on macroegregation calculation Concluion and future work 95 Reference 97 v

8 Lit of principal ymbol Latin alphabet A B rheological coefficient depending on the olid volume fraction rheological coefficient depending on the olid volume fraction Ct i contant value for element i J mol -1 Ct1 contant value J mol -1 Ct2 contant value J mol -1 c P pecific heat per unit ma depending on temperature J kg -1 K -1 D, diffuion coefficient in the liquid phae for each olute i m 2-1 l i d thickne of olidified hell mm dev deviatoric part of a tenor - F HT criterion function - F train-baed criterion - WYSO HT - - WYSO F HT,e maximum value of the quantitie WYSO,i F HT,e - WYSO,i F HT,e train-baed criterion in each finite element e at a time increment, i g gravity vector m -2 g l volume fraction of liquid phae - g volume fraction of olid phae - crit g coherency olid volume fraction at which the compreibility of the olid phae i extremely high ( B ) H train hardening coefficient depending on temperature Pa h average pecific enthalpy per unit ma depending on temperature - - J kg -1 vi

9 h average enthalpy of the olid-liquid mixture J kg -1 I unit tenor (identity tenor) - K vicoplatic conitency depending on temperature Pa m k A parameter of rheological function A - k parameter of rheological function B - B k i partition coefficient for each olute i - L latent heat J kg -1 l liquid phae - m train rate enitivity depending on temperature - m* parameter of train limit - m i liquidu lope for each olute i K ma% -1 N 0 contant mole mol (k-1) N p j initial phae fraction for phae p j at calculation tep k - k N p j phae fraction for phae p j at calculation tep k - n train enitivity depending on temperature - n* parameter of train limit - n p number of phae - n number of chemical element - P 0 contant preure Pa p preure Pa p j phae with p j = { 1, 2,, l} and j = [1, n p ] - p l liquid phae preure Pa l l p intrinic average liquid phae preure Pa olid phae - deviatoric tre tenor Pa l average deviatoric tre tenor in the liquid phae Pa T temperature K (k-1) T initial temperature at calculation tep k K k T temperature defined, k T = (k-1) T - T K T 0 initial temperature K T END arbitrary temperature for calculation top K T L Liquidu temperature K tr trace of a tenor - vii

10 T S olidu temperature K T urf urface temperature K T ZD zero ductility temperature K k p u j ic average u-fraction of ubtitutional element defined by k p u j ic = k xic p j / ( 1 - k p x j C) intertitial element, where C indicate carbon a v l liquid velocity vector m -1 l l v intrinic average liquid phae velocity m -1 v olid velocity vector m -1 v intrinic average olid phae velocity m -1 w l, i w i (k-1) xi p j k xi p j intrinic average concentration in the liquid phae for each olute i average olute ma concentration of the olid-liquid mixture for each olute i initial compoition for phae p j and element i, i = [1, n ], at calculation tep k compoition for phae p j and element i, i = [1, n ], at - ma% ma% ma% ma% calculation tep k x i0 initial compoition ma% (k-1) x i p j initial average compoition for phae p j and element i, i = [1, n ], at calculation tep k k x i p j average compoition for phae p j and element i, i = [1, n ], at calculation tep k ma% ma% viii

11 Greek alphabet γ acceleration vector m -2 T mall temperature K T BTR extent of the brittle temperature range (BTR) K BTR local cumulated train for a REV in the muhy zone during it cooling within a range of olid fraction - c critical train at which crack form - cumulated platic train - c ˆ train limit - train rate -1 el ε elatic train rate tenor -1 th ε thermal train rate tenor -1 vp ε vicoplatic part of the train rate tenor ε ( v )/ 2-1 ε ( v l ) ymmetric part of the liquid velocity gradient vl -1 ε ( v ) ymmetric part of the olid velocity gradient v -1 generalized train rate -1 ˆ train rate perpendicular to temperature gradient direction -1 i ˆ e train rate perpendicular to temperature gradient direction in each finite element e at a time increment i equivalent train rate -1 poor ductility length mm κ Permeability of the porou medium contituted by the olid phae thermal conductivity depending on temperature W m -1 K -1 2 econdary dendrite arm pacing m average thermal conductivity of the olid-liquid mixture W m -1 K -1 T v -1 m 2 k i p j ( k x i p j ) chemical potential for phae p j and element i with compoition k x i p j J mol -1 ix

12 k p j i ( k p x j i ) average chemical potential for phae p j and element i with average compoition k p x j i -1 J mol l dynamic vicoity of the liquid phae Pa parameter of train limit - average denity depending on temperature kg m -3 ρ L denity at the liquidu temperature kg m -3 ρ S denity at the olidu temperature kg m -3 (T) denity of olid phae depending on temperature kg m -3 average denity of the olid-liquid mixture kg m -3 I firt principal tre Pa equivalent von Mie tre Pa σ average tre tenor Pa Σ i j ˆ e ji_tart t j effective macrocopic tre tenor for the olid phae in the muhy tate cumulated train in each finite element e at a time increment i, i_tart i the time increment at which element j e enter the BTR, t i the time tep at time increment j parameter of train limit - gradient - T local temperature gradient K mm -1 divergence - vector product - Pa - x

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15 Chapter I Introduction 1

16 1. Proce preentation In general, the continuou cating (CC) proce i widely ued in teel indutrie for olidifying molten teel (95.6 % of world production in 2015 [World Steel Aociation]). The chematic of the CC machine i hown in Figure 1. The molten teel i charged in the ladle and it i poured into the mould through the tundih. It i firt cooled in the mould (primary cooling). Spray cooling i then applied in the roller apron; in other word, egment. Thi i called econdary cooling. Thank to cooling, molten teel i olidified at the exit of the machine. The olidified teel i cut by the torch cutter at required length for later proceing. The typical lab ize i 0.25 m thick, 2 m width and 10 m long typically for automotive indutry. For machine length and machine height around 40 m and 10 m, repectively, thi figure correponding to the radiu of bending zone. Looking at the hape of lab in the CC, the bending hape i found firt from the mould and the traight hape i een cloe to the exit of the CC. It indicate that the bending lab i traightened at the bottom of the bending zone, called unbending. Ladle Turret Molten teel Ladle Tundih Mould Roller Apron Dummy bar Torch Cutter Figure 1 Schematic of CC proce An example of recent tandard regarding machine pecification can be found in the literature [Yamaguchi 2012]. The advantage of the CC proce are productivity, yield, energy, labor efficiency and quality aurance [Okumura 1994]. 2

17 2. Typical defect in CC In the CC proce, two type of defect appear: urface defect and internal defect. < Surface defect > - Surface crack due to the deformation of olidified kin in the mould or the deformation of olidified hell in the bending and unbending zone. Thee could be removed after the CC proce by mean of carfing proce up to a certain extent and although thi affect the yield and cot. - Incluion problem due to lag or mould powder which are entrapped in the olidified kin in the mould. Thee defect are reduced thank to optimization of liquid melt flow in the mould with electromagnetic tool. They could be removed a well a urface crack. < Internal defect > - Hot tearing due to the deformation of olidified kin in the mould or the deformation of the olidified hell in the bending and unbending zone. Such crack cannot be removed ince they are located inide the lab. - Macroegregation due to the deformation of olidified hell in the egment cloe to the end of olidification. Thee heterogeneitie in the chemical concentration of alloy element cannot be removed a well. Two critical internal defect are hot tearing (in other word, internal crack) and macro egregation or axial egregation taking place during the econdary cooling proce. Hot tearing lead to defect on a teel coil for automotive indutry or make crucial trouble during hot rolling before the coil i obtained due to fiure of the teel heet. Macroegregation lead to non uniform mechanical propertie of the product and o-called high quality product for line pipe or pecial tanker component require contant propertie, which require minimum macro egregation. 3

18 3. Preentation of hot tearing and macroegregation defect 3.1 Short preentation of defect in the CC proce The phenomena are undertood a follow: during econdary cooling proce, molten teel heat i removed from the urface of the olidified hell. Regarding microtructure, dendritic crytal grow toward liquid core from the olidified hell along the temperature gradient. It mean the dendritic grain are contrained to the direction perpendicular to the lab urface. Tenile deformation along the cating direction perpendicular to which the dendrite are aligned generate due to bulging, bending, unbending and mialignment between roller (ee Figure 2 and Figure 3). The tenile deformation make hot tearing [Lankford 1972, Brimacombe 1977] in between dendrite deep in the muhy zone; typically over 0.9 fraction of olid phae. Since hot tearing take place in the muhy zone, it i eaily imagined that the olidification path baed on microegregation phenomena ha an important role. (Mialignment) Figure 2 Schematic of bulging and mialignment [Lankford 1972] and conequence in term of tenile or compreive tree. 4

19 Figure 3 Schematic of bending proce and imple etimation of train and train rate [Lankford 1972]. Cloe to the olidification end, macroegregation generate due to enriched melt flow becaue of the complex combination among olidified hell deformation, hrinkage, induced fluid flow and olute tranport [Miyazawa 1981], [Kajitani 2001], [Fachinotti 2006] and [Mayer 2010]. Thi defect i alo connected with olidification path becaue it occur in muhy zone, but in the pecific context of CC proce, olidified hell deformation and hrinkage could have an important role for the formation of macroegregation. Concerning olidified hell deformation, it could be ure that bulging take place in between roll cloe to the end of olidification becaue of ferrotatic preure depending on machine height. It then make deformation of muhy zone in which the olid phae i deformed and the liquid phae i circulated due to olid phae movement. Becaue of the flow of the liquid with enriched olute, compoition field become varied, leading to macroegregation. Second, hrinkage make alo liquid flow in muhy zone, reulting in variation of compoition, that i to ay, macroegregation. In the CC proce, it i quite popular that oft reduction i applied to olidifying lab cloe to the end of the melting pool. It mean that lab thickne i gradually reduced due to decreaing roll gap on a certain ditance. The mechanical thickne reduction of the lab can compenate olidification hrinkage, leading to le liquid circulation cloe to the end of olidification, and, a a conequence, le marked egregation. 5

20 3.2 Hot tearing phenomena Solidification tage Solidification i the tranformation from liquid phae to olid phae. Thi tranformation occur continuouly and a mixture of olid and liquid phae coexit in what i called the muhy zone. The range between the olidu and the liquidu temperature i called olidification interval. For a columnar tructure, thi olidification interval i generally divided into three region, chematized in Figure 4. In region 1 and 2, the muhy zone can aborb external tenile tre becaue of the capacity of the olid and the liquid phae to move. Thu, the enitivity to cracking i relatively low. In contrat, when the olidification reache region 3, the enitivity become higher. Indeed, in thi region, liquid film are iolated and the permeability in the muh fall dramatically up to the completion of olidification. Crack that form in thee condition are known a hot tear, they are generally intergranular Different mode of hot tearing Uniaxial tenile tet reult, uing aluminum alloy in muhy zone carried out by Braccini [Braccini 2000] and Ludwig et al. [Ludwig 2004], how that olid fraction ha an important role for the material behavior. The olidification interval i divided into three tage. cohe g g : There i no mechanical reitance becaue of no olid bridge (region 1 in Figure 4(a)). cohe coal g g g : The mechanical reitance appear and it reult in the lo of ductility (region 2 in Figure 4(a)). coal g g : Both the ductility and the mechanical reitance rapidly increae (region 3 in Figure 4(a)). Figure 5 illutrate chematic of the main phenomena occurring in the muhy zone at characteritic olid fraction, above mentioned, explaining the lo of ductility in a given Brittle Temperature Range of the olidification. 6

21 (a) (b) Figure 4 On the top, chematic of the olidification interval of a columnar tructure, on the bottom, relation between the permeability and the olid fraction [Wintz 1994]. Braccini [Braccini 2000] have brought experimental evidence, in the cae of aluminum alloy, of ome pecific cracking mode, depending on the olid fraction. coal In the cae of low olid fraction: g g, hot tearing i due to propagation of rupture in the liquid film at the interface of the grain (Figure 6(a)). Then the urface tenion of the liquid reit the propagation of the rupture. When thi region i ubject to exce tre, liquid film finally tear and peak of liquid remain, a hown in Figure 6(c). In the cae of the high olid fraction: coal g g, platic deformation occur in between olid bridge and lead to ductility (Figure 6(b)). The platic deformation coexit with the previou rupture mechanim. However, thee two type of rupture in thi region can be ditinguihed (appearance between the dendrite of ame grain or grain). 7

22 Figure 5 Schematic of the main phenomena occurring in the muhy zone at characteritic olid fraction, explaining the lo of ductility in a given Brittle Temperature Range [Bellet 2009]. (a) (b) (c) Figure 6 Illutration of face of fracture urface [Braccini 2000]: (a) brittle failure in interdendritic region, (b) ductile rupture of olid bridge, (c) obervation of a peak of tretched liquid Variou parameter effect on hot tearing (1) Influence of olidification interval Chemical compoition determine olidification interval. The larger the olidification interval i, the higher the hot tearing enitivity become. The fragility trongly depend on the olidification path through the kinetic of liquid diappearance near the end of olidification. Conequently, hot tearing depend on the chemical compoition, particularly for element uch a ulfur, phophoru and boron: thee element make olidu temperature lower and hot tearing enitivity increae [Wintz 1994]. The effect of ulfur can be prevented by adding manganee. Indeed, manganee favor 8

23 precipitation of (Fe,Mn)S intermetallic, with higher olidu temperature. The effect of carbon content i not a clear. While it i believed that carbon make the olidification interval widen, it wa oberved that, over 0.3wt% carbon content in Fe-C alloy, the higher the carbon content i, the maller the hot tearing rik become [Pierer 2007]. (2) Influence of microtructure The microtructure i linked to the olidification path. A fine microtructure decreae the enitivity for the appearance of the rupture by the adaptation of the deformation: gliding at grain boundary can aborb deformation [Pierer 2007]. Ductility of the equiaxed tructure i better than with columnar tructure [Ekin 2004]. The microtructure alo influence the mechanical behavior in the muhy zone and the permeability. The metallurgical phae in preence alo influence the occurrence of hot tearing during olidification. Several tudie illutrate thi influence in the cae of tainle teel [Katayama 1985] and [Kotecki 1993]. It i explained that it i more difficult to propagate the rupture in a mixed tructure. Thi influence i verified uing two element, that i, ulfur and phophoru. The reult i that thee two element are eaily oluble in ferrite but not in autenite; thi mean that the kinetic of the diappearance of the liquid near the end of olidification depend on the quantity of the autenite phae. In addition, hrinkage correponding to the tranformation from ferrite to autenite generate the deformation in the muh and modifie the hot tearing enitivity. Grain boundary miorientation angle ha alo an effect on the occurrence of hot tearing during olidification [Wang 2004]. Over a certain critical angle between two metallurgical grain, the undercooling required to overcome the repulive force before the grain bridge i large. A a conequence, and a illutrated in Figure 7, there i an extended region where interdendritic liquid film remain, creating a low trength zone along the future grain boundary. When a tenile train i applied perpendicular to the average dendrite growth direction, hot tearing take place preferably in thi brittle region along the grain boundary. 9

24 Figure 7 Schematic illutration from [Wang 2004] howing the effect of delayed coalecence in cae of ignificant miorientation between grain. Thi lead to the peritence of liquid pocket along uch grain boundarie below the nominal coalecence temperature of the ingle crytal. (3) Influence of olid/liquid wetting angle The wetting angle at the interface of the olid alo influence the enitivity to hot tearing [Braccini 2000]. The wetting angle reult in the capacity of the liquid to pread on the urface of the olid a a function of temperature and the liquid compoition. Thi determine the ditribution of the liquid film at the interface of the grain, that i, film are continued, dicontinued or become pocket, hown a Figure 8. The angle i defined by GB 2 SL co, where and GB are repectively the urface energy of the grain SL boundary and the urface energy of the liquid-olid interface. Figure 8 how typical ditribution of liquid a a function a the angle. Some tenile tet in a SEM to invetigate the relation between the wetting angle and the rupture were carried out by Fredrikon and Lehtinen [Fredrikon 1979]. The author how that the angle i null for Al-Sn alloy and hot tearing appear in the wetting grain. In the cae of Al-Cd alloy, the angle i high and the wetted pocket do not influence the rupture. Finally the author pointed out the role of the grain boundarie wetted by the liquid and oriented in a direction perpendicular to the traction rupture. Thickne of the liquid film alo influence the enitivity to hot tearing. The larger the thickne i, the higher the appearance of the hot tearing become. However, it i neceary to invetigate the characteritic of the liquid film at high temperature [Gerd 1976]. 10

25 Figure 8 Ditribution of liquid film in a urface of a grain [Braccini 2000]. (4) Influence of thermomechanic Thermomechanic ha an important role for the occurrence of hot tearing. Phyical property of alloy i trongly aociated with temperature. Thu, thermal evolution determine the mechanical effect, uch a deformation and hrinkage. It reult in a driving force for the occurrence of the hot tearing. Geometric effect (for intance, bending and unbending material during olidification in continuou cating proce) alo lead to the appearance of hot tear. 3.3 Deformation induced egregation A explained in the previou ection, macroegregation reult from deformation of the olidified hell in the CC proce. In fact, deformation of the muh i the origin of macroegregation. The muh i compoed of olid and liquid phae. The muhy olid i conidered a ponge like material which contain enriched liquid. When the ponge i in compreion, the enriched liquid i expelled outide, leading to negative macroegregation in the deformation area. When the ponge i in traction, the enriched liquid i ucked inide of the ponge, leading to poitive macroegregation. Thi i called deformation induced egregation and the liquid phae i tranported with enriched olute according to the muhy olid deformation. Concerning the intenity of the macroegregation with repect to olute element, olute for intance like P and S which have low partition coefficient make trong egregation. 11

26 4. Prediction of hot tearing through numerical modelling Regarding with modeling of hot tearing, there are everal recent tudie in thi field. They can be divided into two group; micro and macrocopic cale tudie. The lat decade ha een a ignificant development of numerical imulation directly operated at the cale of a Repreentative Elementary Volume (REV) of the muhy material, in view of encompaing mot of the mall-cale phyical phenomena mentioned above. Phillion et al. [Phillion 2008] howed the influence of microtructual feature on tenile deformation of a REV of emi-olid aluminum alloy. Sitaninia et al. [Sitaninia 2011] have developed a numerical imulation uing a dicrete element method in order to account for the train inhomogeneity in the muhy zone due to crack initiation and propagation, in other word, direct imulation of hot tearing. Zaragoci et al. [Zaragoci 2012] conidered a domain of few cubic millimeter deduced from in-itu X-ray tomography of a tenile tet in an Al-Cu ample maintained in the muhy tate. A finite element dicretization of the 3D domain uing the level et method and adaptive meh refinement wa conducted to accurately eparate liquid film and grain. The liquid flow and grain deformation were calculated, intrinic propertie of the phae obeying a vicoplatic contitutive equation. However, in thoe recent work [Phillion 2008], [Sitaninia 2011] and [Zaragoci 2012], only fragment of the different phyical phenomena taking part in hot tearing were accounted for. It can be thought that uch mall-cale model will develop in the future, integrating more and more relevant phyical feature, and thu will become ueful tool for a better fundamental undertanding of hot tearing. But upcaling of uch model to deduce rule that can be applied at the proceing cale, uch a in continuou cating, will require time due to their computational cot and actual approximation of the phyical phenomena. Therefore, it i neceary to focu on more macrocopic analyi method to tudy hot tearing. Thoe contitute the econd group of reearche mentioned previouly, if one i targeting proce cale application. In the literature, lot of tudie have propoed hot tearing criteria baed on thermal conideration [Clyne 1977], olid mechanic [Prokhorov 1962], [Rogberg 1987], [Nagata 1990], [Yamanaka 1991], [Won 2000], [Cerri 2007] and [Bellet 2009] or olid and fluid mechanic [Rappaz 1999]. Cerri et al. [Cerri 2007], [Bellet 2009] have evaluated four of thoe hot tearing criteria for teel [Clyne 1977], [Prokhorov 1962], [Won 2000] and [Rappaz 1999] by mean of contrained hrinkage tet and ingot bending tet. Sytematic numerical imulation of the tet wa conducted with the finite element package THERCAST equipped with the different macrocopic criteria. The author concluded that the only criterion able to 12

27 how a qualitative agreement with the experimental tet wa the train-baed criterion propoed by Won et al. [Won 2000]. They propoed an enriched expreion to reach quantitative agreement [Cerri 2007], [Bellet 2009]. In addition, Pierer et al. [Pierer 2007] evaluated, by ue of ubmerged plit chill tenile tet, a tre-baed criterion [Rogberg 1987], a train-baed criterion [Won 2000], a criterion baed on train rate and including liquid feeding conideration [Rappaz 1999] and a criterion baed on the ole BTR value [Clyne 1977]. The author found that only the tre-baed criterion and the train-baed criterion had good capability for predicting hot tearing. A a conequence, in the framework of the preent tudy, we will eentially focu on hot tearing criteria baed on critical train. 13

28 5. Prediction of macroegregation by numerical modeling Several reearcher tudied thee phenomena in the context of CC proce with numerical modeling. Miyazawa and Schwerdtfeger [Miyazawa 1981] modelled olidification and macroegregation, conidering bulging between upport roll and the aociated muhy zone deformation. In their model, ma, liquid phae momentum, olute and energy conervation equation are decribed in a fully Eulerian approach. The olid phae momentum equation i not olved o the velocity field of the olid hell and of the olid phae in the central muhy region are arbitrarily given. The numerical imulation how the formation of a center line with poitive macroegregation due to bulging. Kajitani et al. [Kajitani 2001] and Mayer et al. [Mayer 2010] tudied the o-called oft reduction proce uing the ame approach and dicued the quantitative impact of the oft reduction technique to reduce the intenity of the central macroegregation. Fachinotti et al. [Fachinotti 2006] developed a different approach to model the muhy zone in the context of CC, uing an arbitrary Lagrangian-Eulerian approach in which olid and liquid velocity field are concurrently olved for, avoiding then the trong hypothei over the olid velocity field which wa preent in previou work. With thi model, the author imulated continuou cating and imilarly retrieved the effect of bulging on the formation of central macroegregation [Bellet 2007]. More recently, Rivaux [Rivaux 2011] developed an alternative approach to model the deformation of the olid phae and it Darcy-type interaction with the liquid phae in the muhy tate, through a taggered cheme in which the two field are eparately and ucceively olved for at each time increment. However, in the end, it hould be aid that depite lot of effort in developing thoe numerical model, none of them ha been uccefully applied up to the cale of a 3D imulation repreentative of the complexity of the indutrial proce. Thi i certainly due to the coniderable amount of computational power required to run the model encompaing a ignificant part of the econdary cooling ection of an indutrial cater. 14

29 6. Microegregation modeling for multicomponent teel Average alloy propertie in multiphae temperature interval are important for thermomechanical proce imulation. Numerical tool have been developed to follow the tranformation path baed on thermodynamic conideration. Two claical limit are routinely computed: the lever rule (LR) approximation and the Gulliver Scheil (GS) approximation [Gulliver 1913] and [Scheil 1943]. For LR, full equilibrium i aumed, meaning that all chemical element can diffue rapidly. In contrat, the GS approximation neglect diffuion in olid phae where element are completely frozen. In practice, mot indutrial teel include intertitial and ubtitutional element. Intertitial element, uch a carbon, have high diffuivity in the olid phae while ubtitutional element have low diffuivity. Thu, the above LR and GS approximation do not fully apply. To overcome thi problem, the partial equilibrium (PE) approximation [Chen 2002] and [Zhang 2013] ha been developed. PE i intended to take into account perfect diffuion of intertitial element but no diffuion of ubtitutional element. However, the peritectic tranformation often encountered in commercial alloy cannot take place if only PE i conidered. Thi limitation could be circumvented by a LR treatment of the -BCC (or -ferrite)/-fcc (or -autenite) mixture during the peritectic tranformation [Chen 2006], thu permitting the activation of the -BCC to -FCC phae tranformation. Thi will be later refereed a PE+LR olidification path. It i to be mentioned that, while in agreement with obervation, no experimental quantitative data i yet available that could be directly compared with predicted kinetic of the peritectic phae tranformation. The LR treatment of the peritectic tranformation i alo a very crude approximation that doe not fix the problem in coherent manner with repect to the PE approximation. In the preent tudy, a new numerical cheme i preented for the computation of phae tranformation in teel. It i coupled with thermodynamic equilibrium calculation baed on call to the oftware Thermo-Calc [Thermo-Calc 2013] through the TQ-interface uing the TCFE6 databae [Shi 2008]. The o-called para-equilibrium (PA) approximation preented by Hillert [Hillert 1998] ha been introduced to deal with the peritectic reaction during a PE olidification path, later refereed a PE+PA approximation. Thi i implemented in previou development preented by Zhang et al. [Chen 2002] for the computation of LR, GS and PE+LR olidification path. Becaue PA wa initially developed for olid tate tranformation, it i alo ued to imulate phae tranformation during cooling experienced during teel proceing after olidification. The PA treatment allow the activation of the -BCC to -FCC phae 15

30 tranformation during olidification with the conideration of perfect diffuion of intertitial element but no diffuion of ubtitutional element which i conitent with PE. The numerical cheme i applied to teel alloy experiment alo preented in thi contribution. Concerning coupling between microegregation modeling and macroegregation modeling for teel, ome reearcher employed the implet ytem; Fe-C binary ytem [Miyazawa 1981], [Kajitani 2001], [Fachinotti 2006] and [Mayer 2010]. The advantage of thi cae i to implify the implementation into the code. In cae of binary ytem, contant liquidu lope and partition coefficient are aumed and LR or GS could be applied. However, commercial teel are generally multicomponent ytem o that microegregation modeling for multicomponent teel i required for application to the CC imulation or an experiment for teel. Recently, ome reearcher [Carozzani 2012, Saad 2016] have developed a model for multicomponent where enthalpy and olute concentration are tabulated a priori. According to the reult of global reolution uch a enthalpy, momentum and olute conervation equation, the code refer the table and chooe uitable value. With thi model, peritectic tranformation can be taken into account for and more realitic imulation could be performed. In thi preent tudy, imple binary model i extended to multicomponent ytem with ome aumption for coupling the macroegregation and microegregation although peritectic tranformation cannot be conidered. Thi model give acce to analyze the effect of each olute element and the conequent macroegregation compared to the experiment with teel. The detail of the model i explained in the following chapter. 16

31 7. Numerical modeling To undertand and analyze the above two defect, numerical modeling i one of the powerful tool. Potentially, the advantage of the numerical model i to predict the defect. It i alo poible to optimize cating condition according to numerical imulation reult. Therefore, one objective in thi thei i to develop a numerical model which would be able to give a reliable and detailed decription of the main phenomena leading to the defect mentioned above. In term of modeling of CC proce, ANSYS-FLUENT [FLUENT] i quite well known to be applied to melt flow modeling in tundih and mould. Regarding with modeling of lab deformation in econdary cooling proce, THERCAST [Bellet 2004] and [Bellet 2005] developed by Tranvalor trongly upported by CEMEF, MINES Paritech, i one of the olution becaue it i dedicated to the CC proce modeling, epecially econdary cooling proce with deformation. It can be performed in three dimenional modeling with finite element approach. Regarding macroegregation modeling in the CC, R2SOL [Fachinotti 2006] already developed at CEMEF i alo dedicated to econdary cooling proce epecially with muhy zone deformation and olute tranport modeling with finite element approach. Although the modeling i limited to two dimenion, olving deformation of the muh together with olute tranport phenomena i quite intereting to undertand what happen in the lab. A mentioned above, microegregation i key role to hot tearing and macroegregation phenomena. In addition, average alloy propertie in multiphae temperature interval are important for thermomechanical proce imulation. Therefore, microegregation modeling i intereting. In thi tudy, Thermocalc [Thermo-Calc 2013] with thermodynamic conideration i ued. Although boundary condition i limited, like full olute diffuion or not, realitic olidification path and thermo phyical propertie with multicomponent alloy would be obtained. 7.1 Preentation of THERCAST The numerical imulation of the experiment conit of a thermomechanical tre/train analyi. It i conducted uing the 3D finite element code THERCAST. The eential characteritic of the code can be found in Reference [Bellet 2004] and [Bellet 2005]. Non-linear average conervation equation for the total ma, momentum and energy are olved at each time increment on all interacting domain. The olution cover the whole domain occupied by the metal, whatever it tate: liquid, olid, or muhy and the component of the modeling ytem. The emi-olid teel (compoed of 17

32 liquid and olid phae) i imply conidered a a homogenized continuum with averaged propertie and a unique average velocity field v. The thermal problem and the mechanical problem are olved equentially at each time increment. Firt, the average energy conervation i olved in all domain taking into account heat exchange between them. Thi equation can be expreed by: dh T (1) dt where i the average denity, h i the average pecific enthalpy per unit ma, i the average thermal conductivity and T i the temperature. Second, the mechanical problem i conidered. The following average momentum and ma conervation equation are olved concurrently in the framework of a velocity-preure formulation. p g γ 0 vp trε 0 (2) where i the deviatoric tre tenor, p i the preure, g i the gravity vector, γ i the acceleration vector and vp ε i the vicoplatic part of the train rate tenor: T v ε ( v )/ 2 (3) The train rate tenor i expreed a the um of three different contribution: ε el vp th ε ε ε (4) th where ε i the thermal train rate tenor. The alloy i modeled with a hybrid contitutive equation. Detail can be found elewhere [Thoma 2008], only main line being recalled here. Over the olidu temperature, the alloy i conidered a a non-newtonian fluid obeying a temperature-dependent vicoplatic multiplicative law, a follow: m m 3 1 K (5) where i the equivalent von Mie tre, K i the vicoplatic conitency, i the generalized train rate and m i the train rate enitivity. The value of m over the liquidu temperature i equal to 1 (Newtonian fluid). In the frame of the preent tudy, liquid flow in liquid region i ignored and the liquid vicoity i defined a 10 Pa for 18

33 numerical tability reaon. Below the olidu temperature, the alloy obey an elatic-vicoplatic contitutive equation, expreed by: K m m n 1 H 3 (6) in which i the cumulated platic train, H i the train hardening coefficient and n i the train enitivity. Phyical propertie and contitutive parameter are temperature dependent. 7.2 Preentation of R2SOL The macroegregation form due to concurrent deformation of the olid phae in the muhy zone, and from the liquid flow induced by thi deformation. In order to conider eparately - but olve concurrently - for the liquid and olid velocity field in the muhy zone, we apply the o-called "two-phae" model initially developed by Bellet et al. [Fachinotti 2006] and [Bellet 2007] to addre macroegregation in continuou cating. Thi model allow imulating olute tranport phenomena leading to macroegregation induced by deformation, hrinkage and advection. It i performed with the 2D finite element code R2SOL. The main aumption and feature of the model are reminded hereafter. The conervation equation for the olid-liquid mixture are expreed on a repreentative elementary volume (REV) and are obtained uing the patial averaging method [Ni 1991]. The muhy material i conidered a a aturated two-phae medium only made of olid,, and liquid, l (i.e. no poroity and g g 1 where g k denote the volume l fraction of phae k). At the o-called microcopic cale, within the REV, the liquid phae i conidered a an incompreible Newtonian fluid. After patial averaging, the macrocopic behavior of the liquid phae i Newtonian compreible. In the ame way, the olid phae i conidered intrinically a an incompreible non-newtonian fluid, and it macrocopic averaged behavior i the one of a compreible non-newtonian fluid for which inertia effect are neglected. The momentum interaction between olid and liquid phae i formulated with an iotropic Darcy law. 19

34 Solidification hrinkage i taken into account by conidering different denitie for the olid and liquid phae, repectively ρ S and ρ L. Thoe value are aumed contant within the olidification interval. A multi component ytem under lever rule approximation i aumed, meaning that the partition coefficient k i and the liquidu lope m i are contant for each olute element i. Local thermal equilibrium hold within the REV, enuring the uniformity of temperature within the different phae. The previou aumption lead to the following et of averaged conervation equation: Σ g p l g 2 l 1 ( v v ) g g 0 (7) l l S l g p g l l 2 l 1 ( glvl ) l ( vl v) gllg L L( glvl vl ) (8) t S L S g ( glv l ) ( gv) (9) L L t h h v L ( gl g ( vl v)) ( T) 0 (10) t wi ( wi v ) ( glwl, i( vl v)) ( gldl, iwl, i) 0 (11) t in which v and v l are the olid and liquid velocity vector (intrinic average phae velocitie: v v ; l l l v v ), p l i the liquid phae preure ( p l p l l ), μ l i the dynamic vicoity of the liquid phae, and are the average denity and the average heat conductivity of the olid-liquid mixture, h i the average enthalpy of the mixture. w i and l i w, are repectively the average olute ma concentration and the intrinic average concentration in the liquid phae for each olute i. In the momentum equation, the Darcy term introduce the permeability κ of the porou medium contituted by the olid phae. It i derived from the liquid volume fraction and the econdary dendrite arm pacing 2, through the Carman-Kozeny model. 20

35 ) (1 180 l l g g (12) Contitutive model for the liquid and the olid phae in the muhy zone are preented in detail in reference [Fachinotti 2006] and [Bellet 2007]. In ummary, the average deviatoric tre tenor in the liquid phae i expreed by: )) ( tr( 3 1 ) ( 2 ) ( dev 2 l l l l l l l l g g v ε v ε v ε (13) where ) ( l v ε i the ymmetric part of the liquid velocity gradient l v. For olid, it i aumed that the behavior of the fully olid material i vicoplatic, of power law type, a will be introduced later. The train-rate enitivity coefficient of the fully olid material being denoted m, it can be hown that the effective macrocopic tre tenor for the olid phae in the muhy tate, I σ Σ l p, i a degree m homogeneou function with repect to the train rate tenor ) ( v ε ε. Adopting a compreible vicoplatic potential, it expreion write: I ε ε I ε ε Σ m m B A K A B A K tr 9 1 ) dev( 1 ) 3 ( 3 tr ) 3 ( (14) in which K i the vicoplatic conitency of the olid material and m it train rate enitivity, both taken at the olidu temperature. The equivalent train rate i expreed by

36 1 ε A : ε 1 dev( ε A 1 1 (tr ε 9B 3A ) : dev( ε ) 1 ) (tr ε 9B ) (15) In the flow rule (Eq. (14)), A and B are two rheological coefficient depending on the olid volume fraction. In the preent tudy, the expreion introduced in reference [Fachinotti 2006] are ued: B k B g 1 g g crit 3 A (1 k AB) (16) 2 The value of the parameter are taken from the ame reference. Taking the trace of Eq. (14), it can be een that the compreibility of the olid phae i eentially controlled by the value of coefficient B: trσ m K 1 1 ( 3 ) tr ε (17) B A expreed by Eq. (16a), crit g can be een a the coherency olid volume fraction at which the compreibility of the olid phae i extremely high ( B ). Converely, when the olid fraction tend to one (fully olid material), A and B tend toward 3/2 and 0, repectively: the uual power law relating tre and train rate for a dene metal at high temperature i retrieved. In the finite element code, the weak (integral) form of Eq. (7) to (11) i implemented in 2D triangular finite element. At each time increment, the equence of numerical reolution i a follow: - The thermal reolution i operated on the et of the different domain involved: in the preent context, the olidifying ingot, the different component of the mold, and the preing punch. The (non-linear) heat tranfer reolution i carried out on each domain, ucceively, up to convergence (i.e. tabilization of the temperature field). In the ingot, a finite element (FE) reolution of Eq. (10) i done, while in the other domain a more claical ingle phae form of heat diffuion i olved. Heat 22

37 exchange at interface between domain are expreed by the Fourier law, uing heat tranfer coefficient. - The macroegregation reolution i performed in the ingot, through a FE reolution of Eq. (11) for each conidered olute i. In thee reolution, like in the thermal one, the microegregation model (here the lever rule) i conidered a it link the average concentration in the liquid phae w, with the average mixture concentration l i w i, and the averaged enthalpy h with the temperature T. - The mechanical reolution i performed in the ingot, the other domain being aumed perfectly rigid and fixed. A non-linear Newton-Raphon algorithm i ued, providing the nodal value of v, v l and p l through a ingle FE reolution of Eq. (7) to (9). A the mechanical formulation preented above encompae the central muhy zone and the fully olidified region, a pecific treatment applie. In the finite element aembly procedure, element are conidered either muhy or fully olid, according to the temperature at their center. In muhy element, the contribution to the reidue directly reult from the weak form of Eq. (7) to (9). In fully olid element, it can be noted that, by applying thee equation, the liquid velocity i till preent but ha no real phyical meaning: it i then kept cloe to the olid velocity field by mean of the ole Darcy term in the liquid momentum equation, acting a a penalty term. The deviatoric tre tenor i expreed a a function of the velocity field by the vicoplatic Norton-Hoff power law including train hardening: n 2K ( 3 ) m1 dev( ε ) (18) where dev( ε ) denote the deviatoric part of the train rate tenor, K the vicoplatic conitency, m the train rate enitivity coefficient, the generalized train rate and the generalized train. Strain-hardening i aumed null at olidu n temperature and over. Thi i why only K (and not K ) appear in Eq.(14) to expre the behavior model of the olid phae in the muh. The tenor equation (18) yield the one dimenional relation between the von Mie tre and the generalized train-rate : K( 3) m1 n m (19) 23

38 The fully olid metal i then platically incompreible and thi i why Eq. (9) i replaced in the finite element which are found fully olid (temperature at center lower than the olidu temperature) by the following equation which imply account for thermal dilatation term: v 1 d ( T) 0 ( T) dt (20) 24

39 8. Large cale experiment In order to undertand olidification defect; hot tearing and macroegregation, a large cale experiment developed at Nippon Steel & Sumitomo Metal i employed. It view i hown in Figure 9. In the late 1970, uch a large cale ingot punching tet ha been developed by teel indutrie in Japan [Sato 1975], [Miyamura 1976], [Marukawa 1978], [Narita 1978] and [Sugitani 1980] to etimate the CC-proce viability. Such tet ha been alo developed in Europe [Wintz 1994] in They conit in bending, punching, or preing the olidifying hell of an ingot during it olidification by mean of a punch moved perpendicularly to the hell plane (or urface of the ingot), i.e. along the dendritic growth direction when olidification proceed directionally away from the mold wall. The ize of the ingot and the control of the bending/punching condition have to repreent the olicitation undergone by the olid hell during CC when it pae by the upport roll. Note that in CC, the alloy alo endure the metallotatic preure which induce bulging between each roll tand. The concept of the apparatu i imilar to the above reference, the deformation being applied on an ingot which i not fully olidified. The teel ingot thickne i 0.16 m o that olidification time i cloe to real proce, around 15 min. It give imilar olidification microtructure and defect. What i nice in thi named ingot punch preing tet i to control deformation uing a ervo hydraulic cylinder, aying that diplacement and velocity of the punch are eaily controlled. In addition to the control of deformation, olidification progre i monitored uing thermocouple in the ingot. Uing thi monitoring together with numerical modeling, it i then poible to predict the ingot core olidification. At lat, the defect like hot tearing and macroegregation are uccefully created. The detail of the experiment i preented in following chapter. 25

40 Tundih car Servo cylinder Controller Mould Figure 9 Entire view of ingot preing tet developed at Nippon Steel & Sumitomo Metal Corporation. Steel mould i located at the centre of the photo and 450 kg molten metal i prepared in a ladle over the mould and it i introduced into the mould through a tundih. In fact, the ervo cylinder which i in change to the punch movement and o the deformation on an ingot. Thi thei conit of 3 article; Chapter II explain microegregation modeling for multicomponent teel baed on thermodynamical point of view. A new numerical cheme i developed and the imulation reult are dicued with experimental reult. Chapter III preent prediction of hot tearing through 3D thermomechanical modeling with ingot punch preing tet. An excellent correlation in between the numerical reult and oberved crack are een and olidification path effect i then dicued. Chapter IV i linked to prediction of macroegregation by 2D numerical modeling with ingot punch preing tet. The numerical imulation reult how the eential of the driving force of the generation of macroegregation. Chapter V and VI are then ummary and perpective. 26

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43 Chapter II Computation of Phae Tranformation Path in Steel by a Combination of the Partialand Para-equilibrium Thermodynamic Approximation Takao KOSHIKAWA, Charle-André GANDIN, Michel BELLET, Hideaki YAMAMURA and Manuel BOBADILLA DOI: Page: 30 to 40 in thi thei 29

44 ISIJ International, Vol. 54 (2014), No. 6, pp Computation of Phae Tranformation Path in Steel by a Combination of the Partial- and Para-equilibrium Thermodynamic Approximation Takao KOSHIKAWA, 1,2) * Charle-André GANDIN, 1) Michel BELLET, 1) Hideaki YAMAMURA 3) and Manuel BOBADILLA 4) 1) MINES PariTech & CNRS, CEMEF UMR 7635, Sophia Antipoli, France. 2) Nippon Steel & Sumitomo Metal Corporation, Oita Work Equipment Diviion, 1 Oaza-Nihinou, Oita City, Oita Prefecture, Japan. 3) Nippon Steel & Sumitomo Metal Corporation, Steelmaking R&D Diviion, 20-1 Shintomi, Futtu City, Chiba Prefecture, Japan. 4) ArcelorMittal Maizière, Reearch and Development, BP 30320, Maizière-lè-Metz Cedex, France. (Received on Augut 29, 2013; accepted on February 13, 2014) A model combining the partial-equilibrium and para-equilibrium thermodynamic approximation i preented. It account for fat diffuion of intertitial element, uch a carbon, and low diffuion of ubtitutional element in the olid phae, while complete mixing i aumed for all element in the liquid phae. Thee conideration are turned into claical mathematical expreion for the chemical potential and the u-fraction, to which ma conervation equation are added. The combination of the two model permit application to teel, dealing with partial-equilibrium for olidification and para-equilibrium for both the δ -BCC to γ-fcc peritectic tranformation and the γ-fcc to α-bcc olid tate tranformation. The numerical cheme make ue of call to Thermo-Calc and the TQ-interface for calculating thermodynamic equilibrium and acceing data from the TCFE6 databae. Application are given for a commercial teel. The reult are dicued baed on comparion with claical microegregation model and experimental data. KEY WORDS: olidification; microegregation; partial-equilibrium; para-equilibrium; peritectic tranformation. 1. Introduction Average alloy propertie in multiphae temperature interval are important for thermomechanical proce imulation. Numerical tool have been developed to follow the tranformation path baed on thermodynamic conideration. Two claical limit are routinely computed: the lever rule (LR) approximation and the Gulliver-Scheil (GS) approximation. 1,2) For LR, full equilibrium i aumed, meaning that all chemical element can diffue rapidly. In contrat, the GS approximation neglect diffuion in olid phae where element are completely frozen. In practice, mot indutrial teel include intertitial and ubtitutional element. Intertitial element, uch a carbon, have high diffuivity in the olid phae while ubtitutional element have low diffuivity. Thu, the above LR and GS approximation do not fully apply. To overcome thi problem, the partial-equilibrium (PE) approximation 3,4) ha been developed. PE i intended to take into account perfect diffuion of intertitial element but no diffuion of ubtitutional element. However, the peritectic tranformation often encountered in commercial alloy cannot take place if only PE i * Correponding author: takao.kohikawa@mine-paritech.fr DOI: conidered. Thi limitation could be circumvented by a LR treatment of the δ-bcc (or δ-ferrite)/γ-fcc (or γ-autenite) mixture during the peritectic tranformation, 5) thu permitting the activation of the δ-bcc to γ-fcc phae tranformation. Thi will be later refereed a PE + LR olidification path. It i to be mentioned that, while in agreement with obervation, no experimental quantitative data i yet available that could be directly compared with predicted kinetic of the peritectic phae tranformation. The LR treatment of the peritectic tranformation i alo a very crude approximation that doe not fix the problem in coherent manner with repect to the PE approximation. In the preent tudy, a new numerical cheme i preented for the computation of phae tranformation in teel. It i coupled with thermodynamic equilibrium calculation baed on call to the oftware Thermo-Calc 6) through the TQ-interface uing the TCFE6 databae. 7) The o-called para-equilibrium (PA) approximation preented by Hillert 8) ha been introduced to deal with the peritectic reaction during a PE olidification path, later refereed a PE + PA approximation. Thi i implemented in previou development preented by Zhang et al. 4) for the computation of LR, GS and PE + LR olidification path. Becaue PA wa initially developed for olid tate tranformation, it i alo ued to imulate phae tranformation during cooling expe ISIJ 1274

45 ISIJ International, Vol. 54 (2014), No. 6 rienced during teel proceing after olidification. The numerical cheme i applied to teel alloy experiment alo preented in thi contribution. 2. Experiment 2.1. Unidirectional Solidification Experiment To invetigate olidification path in Fe C Mn S Si P Al teel, Bridgman olidification experiment have been carried out. The principle of thi claical method i explained elewhere. 9,10) The ample i prepared with a 1 kg furnace. The nominal compoition of the invetigated alloy S-1 i hown in Table 1. The pecimen wa haped to 5 mm diameter rod. After etting the ample to the apparatu, it wa heated over the liquidu temperature in the hot zone of the Bridgman apparatu and then withdrawn at a contant velocity toward the cold zone, thu achieving olidification. The velocity wa et to m 1. The meaured temperature gradient along the main longitudinal axi of the ample reached 30 Ccm 1 during the experiment. The ample wa finally quenched while till muhy in order to abruptly olidify the remaining liquid. One could thu ditinguih the remaining liquid quenched into a fine microtructure from the olid phae formed before quenching. The ample wa cut along tranvere ection at variou height along the rod axi. The fraction of the olid phae formed before quenching wa then etimated by mean of image analyi at each tranvere ection. The evolution of the olid fraction veru poition along the length of the rod could be deduced. Uing the temperature gradient, the length wa converted into a temperature. A profile of olid fraction veru temperature could then be plotted. Note that even during the quenching proce, olidification of the previouly formed olid could continue. The profile i thu conidered a an overetimate of the olid fraction (underetimate of the remaining liquid fraction) prior to quenching. To etimate the liquidu temperature, differential thermal analyi (DTA) wa ued on heating rate 15 C/min for the teel S-1-DTA given in Table 1. The apparatu ued i a SETARAM Thermobalance, with 10 mm height - 5 mm outer diameter alumina crucible. The ample weight i around 500 mg. Note that other reaction, uch a eutectic, were not detected with ufficient preciion to be ued EPMA Meaurement on a Steel Alloy Sample Electron Probe Micro-Analyi (EPMA) wa carried out on a 3 cm 5cm 1 cm ample extracted from the center of a 450 kg ingot cating (0.5 m width, 0.16 m thickne and 0.75 m height), in a region where no macroegregation wa oberved. It nominal compoition, S-2, i given in Table 1. After polihing, EPMA wa performed for manganee, phophor and ulfur with a 5 μm diameter beam pot. The beam current wa et to 0.5 A, the integration time wa 0.5 Table 1. Nominal compoition (ma%) of teel alloy tudied. Fe C Mn S Si P Al S-1 Bal S-1-DTA Bal S-2 Bal and the meaurement tep wa 50 μm. The analyzed area wa 10 mm 5 mm with a total number of probed point reaching Sorting method propoed in literature were ued, 11) namely the Fleming-Gungor (FG) ort and the Weighted Interval Rank Sort (WIRS). FG i the implet approach, in which all olute are orted independently a acending order of the concentration. Only acending order i employed ince partition coefficient for manganee, phophor and ulfur are le than unity. Each ordered data are aigned with a rank number from 1 to the total number of meaurement point. Finally, the aigned rank i divided by the total number of point, providing with an etimation of the olid fraction. A chemical compoition veru olid fraction curve could thu be drawn. WIRS take into account the probed meaurement data uncertainty in order to weight the compoition. The uncertainty i etimated by the difference between the average compoition meaured by EPMA and the nominal compoition. The weighted compoitional value i averaged at each probed poition and the data are orted according to the average weighted compoitional value in acending order. For a better analyi, the orted data are finally averaged in each 0.01 olid fraction range. 3. Numerical Algorithm Computation of advanced phae tranformation path are baed on thermodynamic equilibrium calculation conducted with variou condition together with ma balance. The four baic approximation found in the literature 1 4) are reminded hereafter ince they erve a the bai for the new combination derived in the preent contribution Lever Rule (LR) Approximation LR aume complete mixing of all phae and thermodynamic equilibrium at interface, meaning that the chemical potential i equal in all phae for each chemical element. A cloe ytem i conidered with contant mole, N 0, compoition, x i0, and preure P 0. It i defined by it n p phae, including olid phae 1, 2, plu a liquid phae l when preent, i.e. p j phae with p j = { 1, 2,, 1} and j = [1, n p ], a well a it total number of chemical element, n, including both intertitial and ubtitutional element. The ytem i at temperature (k 1) T with an initial phae fraction, (k 1) N pj, at compoition, ( k 1 ) p x j, with p j = { 1, 2,, 1}, j = [1, n p ] and i = [1, n i ]. A mall temperature decreae by ΔT i impoed to the ytem in order to compute it thermodynamic equilibrium at a lower temperature k T = (k 1) T ΔT. Mathematical formulation of LR firt relie on equal chemical potential of each element repectively in all phae, LR k pj i LR k pj μ ( xi ). Conidering ma conervation in the ytem, the et of equation are obtained a follow: k LR n pj i k pj LR xi μ ( )= LR Ct i p p = { 1, 2,,}, 1 j = [, 1 n ], i = [, 1 n ]... (1) j k pj p... (2) LR xi = 1 pj = { 1, 2,,}, 1 j = [, 1 n ], i = [, 1 n ] i= ISIJ

46 ISIJ International, Vol. 54 (2014), No. 6 p p n n k pj k pj ( k 1) pj ( k 1) LR i LR = i j= 1 j= 1 x N x N pj = { 1, 2,, 1}, j n p = [, 1 ], i = [, 1 n ].. (3) k p where are phae fraction and are compoition LR N j k p LR i at equilibrium. The number of unknown i n (n p +1)+n p : n n p compoition, n p phae fraction plu n contant LRCt i. The number of equation i the ame: n n p Eq. (1), n p Eq. (2) plu n Eq. (3). In the flow chart given in Fig. 1, thi correpond to the dahed rectangle on the left-hand-ide. It hould be noticed that the temperature tep ΔT i omewhat arbitrary and hould be choen mall enough to enure convergence of the entire tranformation path. A ingle value equal to 0.05 C wa ued hereafter for all calculation. pj 3.2. Gulliver-Scheil (GS) Approximation GS 1,2) aume complete mixing of the liquid and thermodynamic equilibrium at the olid/liquid interface, with no diffuion of the chemical element in the olid phae. The ytem i at temperature (k 1) T with an initial phae fraction, (k 1) N pj, at average compoition, k 1 x, with p j = { 1, 2,, ip j 1}, j = [1, n p ] and i = [1, n ]. A mall temperature decreae by ΔT i impoed to a ubytem reduced to the above liquid, with initial phae fraction (k 1) N 1 and compoition ( k 1) 1 x i, in order to compute a new thermodynamic equilibrium at a lower temperature, k T = (k 1) T ΔT, uing Eq. (1) (3). In order to ditinguih between LR (full equilibrium) and the preent local equilibrium tep for the remaining liquid, the output of the olution of Eq. (1) (3) are denoted with index EQ intead of LR: and with p j = { 1, 2,, 1}, j = [1, n p EQ k N pj k p ] and i = [1, n EQ x j i ]. x j ( ) Fig. 1. Schematic of the numerical cheme accounting for combination of variou thermodynamic approximation: lever rule (LR), Gulliver-Scheil (GS), partial-equilibrium (PE) and para-equilibrium (PA). (Online verion in color.) 2014 ISIJ 1276

47 ISIJ International, Vol. 54 (2014), No. 6 Becaue of the abence of diffuion in the olid phae under the GS approximation, the average olid compoition of the entire ytem at temperature k T i directly computed by adding the fraction of olid formed during cooling of the ubytem to the already exiting olid uing: ( ) ( p ) = = k pj k p k p k p k p GS x 1 ( j j j j i = 1 ) N 1 ( 1 ) x 1 i + EQ N 1 EQ x 1 i ( k 1) j k pj p N + EQ N j [, 1 n ], i [, 1 n ]... (4) The new total fraction of the olid phae for the entire ytem i equal to: k p 1 k 1 p 1 1 GS EQ k p p N j ( ) = N j + N j j = [, 1 n ], i = [, 1 n ]... (5) Liquid fraction and compoition are imply given by the equilibrium olution: k 1 k 1 x = x i =[, 1 n ]... (6) GS i EQ i k GS N = 1 k 1 EQ... (7) 3.3. Partial-equilibrium (PE) Approximation The PE approximation i intended to be ued for ytem with large difference in the diffuivity of chemical element. Thi i typically the cae in teel containing intertitial element with high diffuivity and ubtitutional element with low diffuivity. It i een a an extenion of the GS approximation and wa initially developed for imulation of the olidification path in teel 3,4) where carbon i intertitial and diffue rapidly compared to e.g., ubtitutional chromium. To implify the problem, carbon i the only intertitial element conidered in thi tudy. Mathematical formulation of PE firt relie on equal average chemical potential of carbon in all phae, PE k pj C PE k pj μ ( xi ). It alo conider unchanged value for the average u-fraction of the ubtitutional element defined by PE k p i C PE k p i C PE k p u j = x j /( 1 xc ) j with repect to the value provided by the lat GS tep at temperature k T for each phae participating to equilibrium, GS k p i C GS k p i C GS k p u j = x j /( 1 xc ) j. Thee condition are added the ma conervation of carbon and of all the ubtitutional element over all participating phae conidering the value provided by the lat GS tep at temperature k T. Thu, PE i initialized from a GS calculation a hown in Fig. 1. The above four conideration tranlate into: k PE k PE pj C k pj PE xi μ ( )= p p PE p p = { 1, 2,,}, 1 j = [, 1 n ], i = [, 1 n ]... (8) j Ct j k j p ui C= GSui C pj = { 1, 2,,}, 1 j = [, 1 n ], i = [, 1 n ]... (9) N p p n n k pj k pj k pj k PE C PE = GS C GS j= 1 j= 1 x N x N n n k pj k pj k pj k PE i C PE = GS i C GS i= 1 i= 1 j x N x N... (10) p = {,,, }, j n p = [, 1 ], i = [, 1 n ]... (11) pj pj = { 1, 2,, 1}, j = [ 1, n p ] pj / The number of unknown i (n +1)n p +1: (n n p ) compoition, n p phae fraction plu the contant PECt. The number of equation i the ame: n p Eq. (8), (n 1)n p Eq. (9), (10) plu n p Eq. (11) Para-equilibrium (PA) Approximation On the bai of Hillert idea, 8) the PA approximation wa originally developed for a tranformation taking place between olid phae with local equilibrium at the phae interface and frozen compoition of ubtitutional element. It permit to deal with olid phae tranformation with the ame u-fraction of ubtitutional element and equality of chemical potential for intertitial element. The condition on the intertitial element thu write imilarly a for the PE approximation with equal average chemical potential among olid phae. Thi mean that diffuion of intertitial element i aumed very high, a for PE. For ubtitutional element, condition i given on the ummation of the product of the average chemical potential by the average u-fraction which i deduced from no driving force on the interface. A detailed derivation i given in Ref. 8). A tated before, it i choen to conider a unique average u-fraction for each k p ubtitutional element in all the olid phae,, i = [1, n PEui C ], it value being defined below. Finally, a for previou approximation, ma conervation of intertitial and ubtitutional element over all olid phae mut alo be verified. Conidering only carbon a the intertitial element, the following et of equation mut thu be atified: k PA pj C k pj PA xi μ ( )= PA p p = {,, }, j = [ 1, n ], i = [ 1, n ]... (12) j Ct1 p n k pj k pj k pj PA i C PA μi C PA i j= u ( x )= PACt2 p p = {,, }, j = [ 1, n ], i = [ 1, n ]... (13) j 1 2 k pj k p p PAui C = PEui C pj = 1 2 j= 1 n i= 1 n {,, }, [, ], [, ] p p n n k pj k pj k pj k pj PA xc PA N = PE xc PE N j= 1 j= 1 j = { 1, 2, }, = [ 1, p j n p p n n n n k pj k pj k pj k PA i C PA = PE i C PE j= 1 i= 1 j= 1 i= 1 x N x N j... (14)... (15) p = {,, }, j [, n p 1 2 = 1 ], i = [ 1, n ]... (16) The number of unknown compoition i (n + 1) n p + 2: (n n p ) compoition, n p phae fraction plu the contant PACt1 and PACt2. The number of equation i the ame: n p Eq. (12), n p Eq. (13), n p (n 1) Eq. (14), (15) plu Eq. (16). Comparing the et of equation for PA to PE approximation, the main difference appear through Eq. (13). In the literature, application of the PA approximation i uually limited to 2 olid phae. The above PA et of equation can be olved for a ubytem made of two olid phae p j = { 1, 2} undergoing a peritectic reaction, in which cae thi ubytem i extracted from a previou PE cal- pj p ] ISIJ

48 ISIJ International, Vol. 54 (2014), No. 6 culation, thu explaining the index PE ued in the above Eq. (14) (16). For the u-fraction, it i to be noticed that an k p k p k p average value i defined, u = x / 1 x where k p PE i C PE i C p p n n k pj k pj k PE i C PE / PE j= 1 j= 1 the ubytem of the phae conidered under PA. It i alo worth noticing that when only olid phae coexit, a imple lever rule calculation can be ued for initialiation. Thu, PA provide with a local equilibrium condition at an interface that differ from LR or full equilibrium. PA and LR applied at a olid/olid interface can thu be combined with the PE approximation ued to deal with a olid/liquid interface Numerical Scheme The numerical cheme for a general cooling equence during olidification i hown in Fig. 1. Several combination of the above approximation are poible: LR (i.e., not GS in Fig. 1). It obviouly correpond to full thermodynamic equilibrium tranformation path and i valid from a temperature higher than the liquidu temperature up to room temperature. GS (i.e., not PE in Fig. 1). A explained above, it implie that all intertitial and ubtitutional element in all olid phae are frozen, not permitting the peritectic reaction to take place. Becaue uch approximation i difficult to maintain up to low temperature, a critical fraction of liquid (0.0001%) i choen to top the tranformation, below which the average compoition of all phae i kept contant: GS k pj 1 ( k 1) pj 1 xi = xi and GS k p j 1 ( k 1) p j 1 N = N. PE (i.e., not BCC FCC in Fig. 1). Conidered alone, it conider full equilibrium of intertitial while ubtitutional element are frozen. Thi doe change the olidification path compared to GS, yet till not permitting the peritectic reaction to occur. PE + LR (i.e., not PA in Fig. 1). It keep the PE approximation while handling the peritectic reaction by uing a imple LR approximation among the δ -BCC and γ-fcc olid phae. 5) Note that the γ-fcc to α-bcc olid tate tranformation i conidered in uch ituation. PE + PA (i.e., PA in Fig. 1). Thi configuration provide with the mot advanced configuration where the δ -BCC to γ-fcc peritectic tranformation and the γ-fcc to α-bcc olid tate tranformation are both conidered under PA condition. PE p i C ( ) j x x N N. It i thu computed for = PE C to tranform the olid into liquid, and hence uncertainty in the determination of the liquidu temperature. The other poible reaon i inherent to the preciion of the thermocouple and the accuracy of the thermodynamic databae. Overall, the comparion i thu conidered a very good. Figure 2(a) how calculated olidification path a a function of temperature for compoition S-1. Figure 2(b) and 2(c) preent the ame reult for maller temperature interval. Meaurement data are added in Fig. 2(a) and 2(b). Difference clearly appear between computation uing the variou approximation. All imulation yet lie within the LR and GS curve. Thi wa indeed expected ince, in the abence of undercooling, LR and GS could be regarded a the two extreme thermodynamic limit with repect to olid tate diffuion. Under LR, the total olid Table 2. Liquidu temperature ( C) and onet temperature for the peritectic tranformation ( C) deduced from DTA meaurement and from the TCFE6 databae 11) for alloy S-1 uing variou approximation. Liquidu ± Peritectic ± tranformation DTA LR GS PE PE + LR PE + PA 4. Application to Steel Alloy and Dicuion 4.1. Solidification Path for S-1 The meaured liquidu temperature by mean of DTA and the calculated one are hown in Table 2. The calculated value baed on thermodynamic equilibrium i lightly lower than meaurement. It i yet within the uncertainty of the experimental method. During DTA, the temperature difference between the alloy ample and a reference ample i recorded during heating to reveal the latent heat required during melting. Such a phae tranformation, however, need a driving force leading to overheating of the ample Fig. 2. Solidification path diplayed for (a) the maximum olidification interval deduced from the prediction, (b) the experimental range, and (c) around the peritectic reaction. Meaurement data are added for comparion. Steel S-1 (Table 1), 5 microegregation model. (Online verion in color.) 2014 ISIJ 1278

49 ISIJ International, Vol. 54 (2014), No. 6 fraction increae within the mallet temperature range, i.e. [1 510 C, C], while it i the larget for GS, i.e. [1 510 C, 958 C]. Thi i a common finding for alloy olidification. In the abence of diffuion in the olid phae and with egregation of element in the liquid, olidification proceed up to low temperature and can lead to the formation of additional intermetallic phae. Figure 2(b) provide with a temperature range that permit a better comparion with experimental reult. It i clear from thi graph that even the mot favorable imulation, i.e. LR, ignificantly underetimate the fraction of olid up to more than 90%, i.e. from the liquidu temperature up to C. In the preent ituation, the procedure for collecting the experimental data can certainly be dicued. During olidification in the Bridgman furnace, thermoolutal convection could take place that would lead to a nonuniform alloy compoition in the quenched ample, thu modifying the olidification path. Thi i a well-known phenomenon in upward Bridgman configuration, often reported in the literature for imilar experimental et-up or even during in-itu obervation. 12) Furthermore, a tated before, olidification of the olid phae can continue during quenching, thu overetimating the fraction of olid prior to quenching. Of coure the approximation of the imulation are alo of prime importance, among which the kinetic of the microtructure. A previouly tated for fuion, olidification proceed with diffuion limited kinetic. With undercooling, the microtructure form with a udden burt of the olid fraction jut behind it growing interface, i.e. from it working temperature. Such udden increae i reported by the obervation, the fraction of olid increaing from 0 to more than 80% from the liquidu temperature, ± 5 C, to ± 5 C, i.e. within a 10 C interval. Similar obervation wa reported in the literature. It wa initially explained by conideration of the olute ma balance at the growth front 13) or more phenomenological analye. 14) More ophiticated model were developed for thi purpoe, all taking into account growth undercooling of the microtructure due to limited diffuion in the liquid phae. 15) The analyi wa made available in the preence of combined dendritic, peritectic and eutectic reaction. 16) However, it i till limited when conidering multicomponent alloy. A further magnification i provided in Fig. 2(c) around the peritectic reaction, i.e. the tranformation of δ-bcc to γ-fcc. At around C, all the curve preent a lope change due to the formation of the peritectic γ-fcc phae. Thi i better een in Fig. 3 where the C and Mn compoition of δ-bcc to γ-fcc are drawn for the ame temperature interval a Fig. 2(c). Under LR, the δ-bcc to γ-fcc tranformation tart at C and end at C. In the abence of diffuion in the olid phae (GS), the δ-bcc fraction i frozen at it compoition up to room temperature. The et of condition and equation for PE i hown to alo freeze δ-bcc and it Mn compoition while the C compoition continue to evolve. However, it i not atifying ince peritectic reaction ha to take place. For thi reaon, the PE + LR and PE + PA were computed. The difference between thee two olidification path can only be een in Fig. 2(c). The preent PE + PA tranformation path differ very little from the original PE + LR treatment propoed by Chen et al. 5) One can yet mention the difference oberved in Fig. 3. During the peritectic tranformation under LR, PE + LR and PE + PA, the carbon compoition decreae in both phae. In contrat, the manganee compoition increae in both phae in cae of LR and PE + PA, while it lightly decreae in cae of PE + LR. Fig. 3. Phae compoition within the temperature range of the peritectic tranformation for (left) carbon and (right) manganee in (top) BCC and (bottom) FCC phae. Steel S-1 (Table 1), 5 microegregation model. (Online verion in color.) ISIJ

50 ISIJ International, Vol. 54 (2014), No Tranformation Path for S-2 In order to better analyze the PE + LR and PE + PA approximation, experimental and numerical tudie were conducted on alloy compoition S-2 given in Table 1. Figure 4 how the evolution of BCC and FCC fraction when cooling the alloy from the liquid tate up to room temperature. Thee evolution are alo given for intermetallic MnS, M 3P, and other phae in Fig. 5. Similar obervation are found a for alloy S-1 upon primary δ-bcc olidification and the δ-bcc to γ-fcc peritectic tranformation for the five approximation conidered, i.e. δ-bcc i frozen below the peritectic temperature under GS and PE. Note in Fig. 5(a) and 5(b) the large difference found between the precipitation temperature of the MnS and M 3P phae, repectively. The GS approximation predict the lowet precipitation temperature for thee phae. It alo reveal precipitation of Cementite and Graphite from the melt at around 950 C (Fig. 5(c)). Such large olidification interval i not likely to be achieved and thi i clearly a limitation of the GS approximation. The PE approximation alo predict formation of the MnS and M 3P phae lightly above C. At lower temperature, all phae fraction are locked under GS and PE. A tated before, thi i coherent with olute ma balance analyi of frozen ubtitutional Fig. 4. Ma fraction of phae a a function of temperature for (top) δ-bcc and (bottom) γ-fcc. Steel S-2 (Table 1), 5 microegregation model. (Online verion in color.) Fig. 5. Ma fraction of phae a a function of temperature for (top) MnS, (middle) M 3P and (bottom) other intermetallic. Steel S-2 (Table 1), 5 microegregation model. (Online verion in color.) 2014 ISIJ 1280

51 ISIJ International, Vol. 54 (2014), No. 6 element at phae interface. When conidering LR, PE + LR, and PE + PA, olid-olid interface are activated and phae can continue to evolve up to room temperature. Thee phae tranformation path are obviouly more realitic ince α-bcc (or α-ferrite) i claically found at room temperature. Figure 4 how that the tranformation tart below 800 C. Difference yet clearly appear on the curve, with LR prediction abruptly departing from the PE + LR and PE + PA curve at around 750 C with the tranformation of γ-fcc into α-bcc. Thi behavior of the LR tranformation path i only realitic if ubtitutional element can rapidly equilibrate, leading to a full diappearance of γ-fcc. It alo lead to the formation of Cementite, Graphite, M 5C 2 and M 7C 3 phae at lower temperature a hown in Fig. 5(c). On the contrary, PE + LR and PE + PA how a continuou tranformation of γ-fcc into α-bcc (Fig. 4) and no additional phae other than MnS and M 3P. The latter path are thu more realitic. Figure 6 preent the BCC and FCC compoition in C and Mn predicted by the five approximation. Obviouly, C i only frozen after olidification under GS while Mn i frozen for GS, PE and PE + PA. For LR and PE + LR, Mn compoition continue to change after olidification. Conequently, Mn i fully tranferred from δ-bcc to γ-fcc at room temperature, while autenite i not likely to be preent anymore for thi alloy. In cae of PE + PA, the Mn content i maintained cloe to the nominal compoition (1.47 ma%) in both FCC and BCC phae. For a better aertion of the mot meaningful combination of the thermodynamic approximation, EPMA meaurement conducted on the S-2 compoition i preented in Fig. 7. It preent the predicted average Mn compoition at 20 C (left) and the interfacial Mn compoition obtained during GS tep denoted a in Fig. 1 (right). One p hould EQ k xmn j 1 Fig. 6. Phae compoition a a function of temperature for (left) carbon and (right) manganee in (top) BCC and (bottom) FCC phae. Steel S-2 (Table 1), 5 microegregation model. (Online verion in color.) Fig. 7. Average Mn compoition a a function of the ma fraction of olid with (left) FG and (right) WIRS orting method. Steel S-2 (Table 1), 5 microegregation model. Parameter for the WIRS analyi are given in Table 3. (Online verion in color.) ISIJ

52 ISIJ International, Vol. 54 (2014), No. 6 Table 3. Data for WIRS (ma%). σ i uncertainty, etimated a the difference between nominal compoition and average compoition of EPMA meaurement. Mn S P Min σ notice that the ample analyzed contain MnS, thu explaining the rapid increae of Mn content in the compoition profile for olid fraction above The GS, PE and PE + PA approximation are found to provide better agreement with the meaurement, uing both the FG and WIRS orting method, i.e. when Mn diffuion i frozen. A clear change in the compoition profile i yet predicted under GS, PE and PE + PA, when the olid fraction reache around It i due to the δ-bcc to γ-fcc tranformation. According to previou conideration, the overall bet agreement i thu found for the combination of PE + PA approximation. 5. Concluion (1) Combination of the partial- and para- equilibrium (PE + PA) thermodynamic approximation i propoed for the tudy of phae tranformation in teel, from the liquid tate to room temperature. (2) Calculated olidification path for multi-component alloy are dicued with repect to the peritectic tranformation. It i found that, in cae of PE + PA, the olidification path doe not differ ignificantly from the lever rule (LR) and from the combination of the partial equilibrium plu lever rule (PE + LR) approximation. (3) Difference with experimental reult are dicued baed on analye of a ample proceed by Bridgman olidification, howing the need for more accurate experimental reult and removal of everal modeling hypothee (abence of microtructure kinetic and no treatment of limited diffuion in all phae). (4) Regarding the olid tate phae tranformation, PE + PA i found to provide the mot realitic compoition profile of ubtitutional element and evolution of the phae fraction. PE + PA i therefore recommended a the tandard et of thermodynamic approximation to be ued for the tudy of phae tranformation in multicomponent teel. Acknowledgment Thi work ha been upported by Nippon Steel & Sumitomo Metal Corporation (Tokyo, Japan) in a collaboration project with ArcelorMittal Maizière Reearch (Maizière-lè-Metz, France). REFERENCES 1) G. H. Gulliver: J. Int. Met., 9 (1913), ) E. Scheil: Z. Metallkd., 34 (1942), 70. 3) Q. Chen and B. Sundman: Mater. Tran., 43 (2002), ) H. Zhang, K. Nakajima, C. A. Gandin and J. He: ISIJ Int., 53 (2013), ) Q. Chen, A. Engtrom, X.-G. Lu and B. Sundman: MCWASP-XI, ed. by Ch.-A. Gandin and M. Bellet, TMS, Warrendale, PA, (2006), ) Thermo-Calc TCCS Mnual, Thermo-Calc Sftware AB, Stockholm, Sweden, (2013). 7) P. Shi: TCS teel/fe-alloy Dtabae, V6.0, Thermo-Calc Software AB, Stockholm, Sweden, (2008). 8) M. Hillert: Phae Equilibria, Phae Diagram and Phae Tranformation, Cambridge Univerity, UK, (1998), ) W. Kurz and D. J. Fiher: Fundamental of Solidification, Tran Tech Publication Ltd, Switzerland, (1998), 7. 10) J. A. Dantzig and M. Rappaz: Solidification, EPFL Pre, Lauanne, Switzerland, (2009), ) M. Ganean and P. D. Lee: Mater. Tran. A, 36 (2005), ) A. Bogno, H. Nguyen-Thi, G. Reinhart, B. Billia and J. Baruchel: Acta Mater., 61 (2013), ) J. A. Sarreal and G. J. Abbachian: Mater. Tran. A, 17 (1986), ) B. Giovanola and W. Kurz: Mater. Tran. A, 21 (1990), ) C. Y. Wang and C. Beckermann: Mater. Sci. Eng. A, 171 (1993), ) D. Tourret, Ch.-A. Gandin, T. Volkmann and D. M. Herlach: Acta Mater., 59 (2011), ISIJ 1282

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55 Chapter III Study of Hot Tearing During Steel Solidification Through Ingot Punching Tet and It Numerical Simulation Takao KOSHIKAWA, Michel BELLET, Charle-André GANDIN, Hideaki YAMAMURA and Manuel BOBADILLA DOI: / x Page: 42 to 58 in thi thei 41

56 Study of Hot Tearing During Steel Solidification Through Ingot Punching Tet and It Numerical Simulation TAKAO KOSHIKAWA, MICHEL BELLET, CHARLES-ANDRE GANDIN, HIDEAKI YAMAMURA, and MANUEL BOBADILLA Experimental and numerical tudie of hot tearing formation in teel are reported. On the one hand, an ingot punching tet i preented. It conit in the application of a deformation at the urface of a olidifying 450 kg teel ingot. The experimental parameter are the diplacement of the preing tool, together with it velocity, leading to variation of a global train rate. On the other hand, three-dimenional finite element thermomechanical modeling of the tet i ued. The time evolution of the train tenor erve to compute an index to evaluate the uceptibility to create hot tear. It i baed on the integration of a hot tearing criterion (HTC) that compare the local accumulation of train with the expreion of a critical value propoed in the literature. The main variable of the criterion i the brittlene temperature range (BTR) that refer to the olidification interval during which train accumulate and create hot crack or tear. Detailed comparion of the imulation reult with the meaurement reveal the importance of the BTR for the prediction a well a excellent capabilitie of the HTC to predict the formation of hot tear. DOI: / x The Mineral, Metal & Material Society and ASM International 2016 I. INTRODUCTION THE preent paper focue on a major defect encountered during econdary cooling in teel Continuou Cating (CC) known a hot tearing. Alo named hot cracking or olidification cracking, thi defect take place deep in the muhy zone in region cloe to the end of olidification, typically exceeding 90 pct of olid. A comprehenive decription of the phyical phenomena involved in hot tearing i found in Reference 1. The liquid phae remaining at the late tage of olidification take the form of thin interdendritic and intergranular film that cannot reit poible mechanical deformation. Thee are tenile tree ariing from external olicitation exerted by the immediate environment, for intance olidification hrinkage, thermal contraction of the neighboring region, or boundary condition TAKAO KOSHIKAWA, Manager, i with Nippon Steel & Sumitomo Metal Corporation Oita Work, Equipment diviion, Oita, Japan, and alo Ph.D. Student at MINES PariTech, Centre de Mie en Forme de Matériaux (CEMEF), UMR CNRS 7635, Sophia Antipoli, France. Contact kohikawa.4eb.takao@jp.nmc. com MICHEL BELLET, Profeor and Group Leader, and CHARLES-ANDRÉ GANDIN, CNRS Reearcher and Group Leader, are with MINES PariTech, CEMEF UMR CNRS HIDEAKI YAMAMURA, formerly Chief Reearcher with Nippon Steel & Sumitomo Metal Corporation, Steelmaking R&D Diviion, i now Secretary General with The Japan Intitute of Metal and Material, , Ichibancho, Aoba-ku, Sendai , Japan. MANUEL BOBADILLA, Group Manager, i with ArcelorMittal Maizière, Reearch and Development, BP 30320, Maizière-lè- Metz, France. Manucript ubmitted January 3, retraining the product. Even low local deformation cannot be compenated by liquid feeding becaue of the extremely low permeability of the olid network at uch high olid fraction. Thi lead to the rupture of the remaining liquid film, thu creating crack. The material appear particularly vulnerable in a critical range of olid fraction, which i known a the Brittlene Temperature Range (BTR). Proceing by CC lead to hot tearing formation during the bending or unbending of the trand, or a a conequence of bulging of the hell between upport roll under the effect of metallotatic preure. Locally and at mall cale, variou parameter influence the occurrence of hot tearing. The firt of them i the chemical compoition of the alloy. The larger the olidification interval i, the higher i it hot tearing enitivity. Indeed, fragility trongly depend on the olidification path through the kinetic of liquid tranformation near the end of olidification. In teel, ome chemical element uch a ulfur, phophoru or boron tend to increae the hot tearing enitivity a they increae the olidification interval and thu decreae the olidu temperature. [2] Second, the microtructure alo influence the hot tearing enitivity. A fine microtructure decreae the enitivity due to a better adaptation to the deformation: gliding at grain boundarie can accommodate mall train at low train rate. [3] Accordingly, tearing reitance of an equiaxed tructure i better than that of a columnar tructure. [4] Note that the microtructure alo determine the permeability and hence the compenation of train by liquid feeding. Thirdly, wetting of the olid phae by the liquid phae depend on both temperature and liquid compoition: a METALLURGICAL AND MATERIALS TRANSACTIONS A

57 low wettability will promote the formation of olid bridge between grain which will help to reit tenion. [5] At lat, the local thermomechanical behavior play an important role in the appearance of hot tearing, itelf being dependent on microtructure. Recently, everal reearche focued on hot tearing phenomena. They can be divided into two group; micro- and macrocopic cale tudie. The lat decade ha een a ignificant development of numerical imulation directly operated at the cale of a Repreentative Elementary Volume (REV) of the muhy material, in view of encompaing mot of the mall-cale phyical phenomena mentioned above. Phillion et al. [6] howed the influence of microtructural feature on tenile deformation of a REV of emi-olid aluminum alloy. Sitaninia et al. [7] have developed a numerical imulation uing a dicrete element method in order to account for the train inhomogeneity in the muhy zone due to crack initiation and propagation, in other word, direct imulation of hot tearing. Zaragoci et al. [8] conidered a domain of few cubic millimeter deduced from in itu X-ray tomography of a tenile tet in an Al-Cu ample maintained in the muhy tate. A finite element dicretization of the 3D domain uing the level et method and adaptive meh refinement wa conducted to accurately eparate liquid film and grain. The liquid flow and grain deformation were calculated, intrinic propertie of the phae obeying a vicoplatic contitutive equation. However, in thoe recent work, [6 8] only fragment of the different phyical phenomena taking part in hot tearing were accounted for. It can be thought that uch mall-cale model will develop in the future, integrating more and more relevant phyical feature, and thu will become ueful tool for a better fundamental undertanding of hot tearing. But upcaling of uch model to deduce rule that can be applied at the proceing cale, uch a in continuou cating, will require time due to their computational cot and actual approximation of the phyical phenomena. Therefore, it i neceary to focu on more macrocopic analyi method to tudy hot tearing. Thoe contitute the econd group of reearche mentioned previouly, if one i targeting proce cale application. In the literature, lot of tudie have propoed hot tearing criteria baed on thermal conideration, [9] olid mechanic [10 15] or olid and fluid mechanic. [16] Bellet et al. [15] have evaluated four of thoe hot tearing criteria for teel [9,10,14,16] by mean of contrained hrinkage tet and ingot bending tet. Sytematic numerical imulation of the tet wa conducted with the finite element package THERCAST equipped with the different macrocopic criteria. The author concluded that the only criterion able to how a qualitative agreement with the experimental tet wa the trainbaed criterion propoed by Won et al. [14] They propoed an enriched expreion to reach quantitative agreement. [15] In addition, Pierer et al. [3] evaluated, by ue of ubmerged plit chill tenile tet, a tre-baed criterion, [11] a train-baed criterion, [14] a criterion baed on train rate and including liquid feeding conideration, [16] and a criterion baed on the ole BTR value. [9] The author found that only the tre-baed criterion and the train-baed criterion had good capability for predicting hot tearing. A a conequence, in the framework of the preent tudy, we will eentially focu on hot tearing criteria baed on critical train. Section II of thi paper will conit of a review of macrocopic train-baed criteria and of their characterization through ingot bending or punching tet, a it will be hown that uch tet are well uited to reproduce the context of continuou cating. Section III will preent the ingot punching tet developed in the framework of the preent tudy. In Section IV, the thermomechanical finite element model will be preented including the effective implementation of the elected hot tearing criterion in the computation code THERCAST. Reult of the numerical imulation of the punching tet and their comparion to experimental meaurement will be preented and dicued in Section V. II. A REVIEW OF STRAIN-BASED CRITERIA AND ASSOCIATED CHARACTERIZATION THROUGH INGOT BENDING OR PRESSING TESTS A initially uggeted by Yamanaka et al., [13] the baic expreion to quantify the rik for the occurrence of hot tearing due to accumulation of train i written a follow: F HT ¼ e BTR e c ; ½1 where e BTR i the local cumulated train for a REV in the muhy zone during it cooling within a range of olid fraction. The econd train, e c, i the critical train at which crack form. Thee quantitie will be detailed and dicued hereafter. From Eq. [1], a poitive value of the criterion function F HT interpret a the onet of crack a the cumulated train locally exceed the critical train. Note that if one aume a direct relationhip between temperature and fraction of olid, the range of olid fraction tranform into a temperature interval called the brittlene temperature range (BTR). For a given alloy, the critical train value ha to be etablihed by experimental teting. The latter hould be repreentative of the condition of the olidification proce for which the criterion i ued. With thi repect, the ingot bending or punching tet developed ince the late 70 by teelmaker to etimate the CC-proce viability are good candidate. Such tet have been developed by everal author. [2,17 21] They conit in bending, punching, or preing the olidifying hell of an ingot during it olidification by mean of a punch moved perpendicularly to the hell plane (or urface of the ingot), i.e., along the dendritic growth direction when olidification proceed directionally away from the mold wall. The ize of the ingot and the control of the bending/punching condition have to repreent the olicitation undergone by the olid hell during CC when it pae through the upport roll. Note that in METALLURGICAL AND MATERIALS TRANSACTIONS A

58 CC, the alloy alo endure the metallotatic preure which induce bulging between each roll tand. Narita et al. [20] performed ingot punching tet on 72 kg teel ingot. Thi wa done by moving a punch through an opening window deigned in the mold. The diplacement and the velocity of the punch were recorded during the tet, but the punching force wa not meaured. Another example i provided by Wintz et al. [2] who developed a three-point bending tet performed on 300 kg teel ingot ( mm 3 ) during their olidification in a thin 6-mm-thick mold made of mild teel. The whole et ingot plu mold wa bent by a cylindrical punch (maximum force 450 kn), the velocity of which wa maintained contant in the range from 0.1 to 5.5 mm 1 during a hort time. The urface temperature of the mold wa meaured by a pyrometer. The knowledge of the olidified thickne at the bending intant wa obtained by a ulfur print made on a cro ection of a tet-ingot in which the remaining liquid pool wa on purpoe and intantly enriched in ulfur at the intant of bending. In thi type of experiment, author evaluated the applied deformation cloe to the olidification front by ome imple bending formulae. The applied train rate wa directly deduced by dividing the total diplacement impoed to the punch by the duration of it travel. It i worth noting that thoe etimation do not take into account neither the three-dimenional character of the tet nor the fact that highly non-uniform thermal gradient prevail through the hell thickne, inducing a gradient in the material trength. Therefore, the quantitative reult deduced from thee tet hould be conidered cautiouly. However, from a qualitative point of view, reult clearly demontrate that the firt-order variable for crack occurrence i train. Strain rate appear a a econd-order parameter, a the critical train depend on train rate (the higher the train rate, the maller the critical train). Nagata et al. [12] ummarized lot of experimental reult conducted on teel and draw the concluion that the critical train i related to the train rate, _e, bya power law. The lope of the log-form of thi power law i a contant whatever the teel grade. The correponding relationhip thu i written a follow: e c ¼ A_e m ; where m* i a contant while A depend on teel grade (chemical compoition). In addition, the author introduced the concept of poor ductility zone in the direction of the ingot thickne, the extent of which i denoted g. Poor ductility i conidered by the author between the olidu temperature T S and a lower temperature called Zero Ductility Temperature T ZD. Thi hould not be een a being contradictory to the definition of the BTR given above. Indeed, thi may expre the effect of intergranular coalecence. A demontrated by Rappaz et al. [22] and illutrated by Wang et al., [23] table liquid film could till cover grain boundarie to a lower temperature than the olidu temperature. Thu, in their paper, Nagata et al. ½2 propoed an expreion of T ZD, or equivalently of the coalecence undercooling, a a function of the carbon, phophoru, and ulfur content. [12] Taking into account the non-linear temperature profile through the ingot hell (non-contant temperature gradient), the expreion propoed for g i " g ¼ d 1 T # ZD T 1:291 urf ; ½3 T S T urf where d i the thickne of olidified hell and T urf i the urface temperature. It i worth noticing that g can equivalently be expreed by the following formula: g ¼ T S T ZD ; krtk g ½4 which involve the local temperature gradient krtk g within the poor ductility zone. Nagata et al. noticed that the critical train alo obey the following expreion [12] : e c ¼ Cg n ; where C and n* are contant with poitive value. Interpretation i that the maller the poor ductility zone i, the larger i the critical train required to form crack (that i the maller i the rik of hot tearing). Finally, in line with Eq. [2] and [5], and on the bai of a compilation of everal ingot bending experiment, the expreion of the critical train, a uggeted by Nagata et al., i written a follow [12] : e c ¼ / g n _e m where m* = and parameter / and n* are a follow: If 1 mm<g<3 mm; then / ¼ 0:0602 and n ¼ 2:13 If 3 mm<g; then / ¼ 0:0077 and n ¼ 0:258 Won et al. [14] propoed a imilar expreion. But rather than uing the poor ductility extent g, which require the knowledge of both the urface temperature T urf and the olidified hell thickne d, aineq.[3] and [6], they preferred to ue directly the BTR value. The propoed expreion for the critical train i then u e c ¼ ; ½8 _e m ðdt BTR Þ n where DT BTR i the extent of the BTR in Kelvin and u, m*, and n* are parameter. Through the analyi of many experimental data, [18,20,24,25] the author identified contant value for thoe three parameter: u = , m* = , n* = They alo mention that DT BTR hould be calculated with a non-equilibrium microegregation analyi, taking into account the teel compoition and the cooling rate. Bellet et al. [15] conducted a imilar dicuion, howing the trong ½5 ½6 ½7 METALLURGICAL AND MATERIALS TRANSACTIONS A

59 Table I. Compoition and Experimental Condition Compoition (Ma Pct) C Si Mn P S Al Mold Removal () Punching Start () Diplacement (mm) Velocity (mm 1 ) Target N N N N N Fig. 1 Schematic of half of the hot tearing experiment etup developed at Nippon Steel & Sumitomo Metal Corporation howing configuration (a) prior to the filling tage and (b) at the onet of the punching tage when filling, partial olidification with the mold configuration hown in (a), removal of the top-right mold and diplacement of the cylindrical punching tool to come into contact with the ingot. The coordinate ytem i defined in (a). The Y = 0 m plane i defined by the top urface of the bottom mold part, alo defining the lowet ingot urface in (b). The Y-axi, oppoite to the gravity vector, i directed toward the top of the ingot at mid-width and mid-thickne. The X-axi i directed toward the fixed left-mold urface. Thermocouple, TC-1 and TC-2, are located at (X, Y, Z) poition (0, 0.43, 0.06) m and (0.02, 0.43, 0.16) m, repectively. A chematized, pyrometer meaurement (PM) i performed around the punched urface of the ingot, before and after punching. dependence of the effective value of the critical train, e c, with repect to the microegregation model ued for the determination of DT BTR. Accordingly, and in line with Won et al., [14] they propoed a new expreion for the critical train, baed on a BTR value calculated with a microegregation model in equilibrium condition, but with correction term expreing the effective extent of the brittlene domain epecially due to ulfur and phophoru. Thi model wa validated againt the experimental reult of Wintz et al. [2] In the preent tudy, a train-baed criterion hereafter named WYSO from the initial of it author will be conidered with the hape of Eq. [8], the BTR extent being calculated by mean of different microegregation model preented in the literature. [28] The latter point will be detailed and dicued in Section V. III. INGOT PUNCHING TEST The preent punch pre tet ha been deigned and operated by Nippon Steel and Sumitomo Metal Corporation in the R&D Centre of Futtu, Japan. Steel grade for the preent tudy i hown in Table I. A chematic of the tet i hown in Figure 1. It ha been deigned to mimic the thermomechanical olicitation taking place during CC proceing of teel, for intance, with repect to the evolution of the cooling rate and deformation. The procedure of the experiment wa a follow: the molten metal wa prepared in an electric ladle furnace and the temperature wa maintained at 1913 K (1640 C), i.e., 129 K (129 C) above the liquidu temperature. The molten metal wa poured into the mold from the top through a tundih. The filling duration wa about METALLURGICAL AND MATERIALS TRANSACTIONS A

60 70 econd during which the temperature of the molten metal tarted decreaing due to heat exchange with the tundih and with the mold. After filling, powder wa added at the top of the ingot to limit heat exchange with the air. The ize of the ingot wa 0.75 m height, 0.5 m width, and 0.16 m thick. It ma wa approximately 450 kg. In practice, a thermal inulator whoe thickne wa 5 mm covered the bottom part of the inner mold wall from the bae of the ingot and up to 0.2 m height. Thi wa required to avoid molten metal ticking and to make demolding eaier. At a precie time after filling, the upper part of one of the two larget ingot urface (top-right mold in Figure 1) wa moved upward. The lateral urface of the ingot wa then in direct contact with the air. Note in Figure 1 that thi ide of the mold i made of two part poitioned one on top of the other. The bottom part could thu be kept at the ame place when the top part wa removed. A punching tool wa then put into contact with the lateral vertical free urface of the ingot. It conit of a horizontal 0.12-m-diameter and 0.3-m-long cylinder with length centered on the main vertical face of the ingot and with it longitudinal axi aligned with the Z direction at height 0.45 m from the bottom of the ingot. The tool velocity and diplacement are controlled by mean of a hydraulic ytem and recorded during the tet. The reaction force i alo meaured uing the time evolution of the hydraulic ytem preure. The tet condition referred to a N-2 to N-6 are ummarized in Table I, together with meaured nominal compoition in the ladle before filling. Note that a unique alloy compoition wa targeted for the experimental reult reported hereafter. To know temperature evolution during the tet, type-b thermocouple labeled TC-1 and TC-2 in Figure 1 were poitioned in the ingot. For that purpoe, two horizontal hole were created in the fixed large face of the mold (at left in Figure 1). Thermocouple were inerted into thee hole from outide. They were poitioned in the mold cavity at deired location. The hole diameter wa 5.5 mm. For ealing purpoe, thermal inulator wa put into the hole after etting thermocouple. In addition, type-k thermocouple were placed in the fixed large face of the mold in trial N-4. The ingot urface temperature wa alo meaured with a pyrometer, labeled PM in Figure 1, in trial N-4 before and after deformation when the top-right mold wa removed. After completion of cating and cooling to room temperature, the ingot wa cut and appropriate etching wa conducted to invetigate hot tearing occurrence conidering it intenity and location in micrograph. The cut wa performed in the ection Z =0m a chematized in Figure 1(b), i.e., a tranvere ection that reveal the entire ingot thickne and height and i located at mid-width. Figure 2(a) how a typical ection for trial N-4. The punching tool come from the right-hand ide in the figure. In front of the punching tool, a columnar dendritic microtructure grew from the ingot urface toward the ingot center. The columnar dendrite are lightly oriented upward a hown in Figure 2(b). Thi could be due to a downward liquid flow in the melt in front of the growing dendrite, a expected from previou analye. [29] Around the center of the ingot, the morphology i changed to equiaxed due to the decreae of the temperature gradient in the liquid ahead of the growing front. [30] At the top, primary hrinkage i oberved with a large void located lightly below the top olid hell. In order to avoid thermal exchange at the top, a thermal inulator powder wa put on the free liquid/air interface jut after filling. However, thermal exchange wa not completely topped and a thin olid hell formed. Upon further cooling, hrinkage flow lead to a typical ingot top-urface depreion hape and a large void underneath. Conidering the bottom of the ingot, the morphology i almot equiaxed and void are alo found. Thee obervation are attributed to hrinkage upon olidification of an iolated hot liquid pocket. In fact, olidification i delayed in the bottom part of the ingot compared to the area located in front the punching tool becaue of the thermal inulator material depoited at the bottom urface of the mold to avoid metal ticking. The temperature gradient i alo lowered at the bottom part, facilitating the formation of an equiaxed zone and large hrinkage poroity. Yet, void formation i not tudied hereafter, a the main interet for thi tudy i hot tearing in the ection of the ingot located in front of the punching tool. Detailed analyi reveal the formation of the hot tear, a hown in Figure 2(b). The crack are aligned with the columnar dendrite, a expected from previou obervation in the literature. [31] In thi micrograph, it eem topped by the equiaxed zone. It length reache 20 mm in the cro ection, although hot tear are obviouly three-dimenional object and thi value hould be een a an underetimation. Alo worth noticing i the abence of hot tear on the ide of the ingot oppoite to the punching tool (e.g., in region X > 0 according to Figure 1). The total number and length of crack oberved in the tranvere metallographic cro ection were meaured to etimate the intenity of hot tearing a a function of the experimental condition. IV. THERMOMECHANICAL FINITE ELEMENT MODELING A. Finite Element Modeling with THERCAST The numerical imulation of the punching tet conit of a thermomechanical tre/train analyi. It i conducted uing the 3D finite element code THERCAST. The eential characteritic of the code are found in Reference 32 and 33. Non-linear average conervation equation for the total ma, momentum, and energy are olved at each time increment on all interacting domain a hown in Figure 1: the ingot, the different component of the mold, and the preing punch. Yet, a regard the mechanical problem, only the ingot i conidered a deformable; the mold component and the punch are aumed rigid. In the ingot, the olution cover the whole domain occupied by the metal, whatever it tate: liquid, olid, or muhy. The emiolid teel (compoed of liquid and olid phae) i imply conidered a a homogenized continuum with METALLURGICAL AND MATERIALS TRANSACTIONS A

61 Fig. 2 Micrograph for trial N-4 with (a) global view of the full ingot ection in plane Z = 0 m (Fig. 1) and (b) magnified view in the rectangular red window hown in (a) revealing a hot tear circulated with a red dahed contour. Punching wa performed from the right-hand-ide urface of the ingot at the window height (Color figure online). averaged propertie and a unique average velocity field v. The thermal problem and the mechanical problem are olved equentially at each time increment. Firt, the average energy conervation i olved in all domain taking into account heat exchange between them. Thi equation can be expreed by q dh dt ¼rðkrTÞ; ½9 where q i the average denity, h i the average pecific enthalpy per unit ma, k i the average thermal conductivity and T i the temperature. Second, the mechanical problem i conidered. The following average momentum and ma conervation equation are olved concurrently in the framework of a velocity-preure formulation. r rp þ qg qc ¼ 0 tr _e vp ¼ 0 ½10 where i the deviatoric tre tenor, p i the preure, g i the gravity vector, c i the acceleration vector and _e vp i the vicoplatic part of the train rate tenor _e ¼ðrv þr T vþ=2. The alloy i modeled with a hybrid contitutive equation. Detail can be found elewhere, [34] only main line being recalled here. Over the olidu temperature, METALLURGICAL AND MATERIALS TRANSACTIONS A

62 the alloy i conidered a a non-newtonian fluid obeying a temperature-dependent vicoplatic multiplicative law, a follow: p r ¼ K ffiffi mþ1_ 3 e m ½11 where r i the equivalent von Mie tre, K i the vicoplatic conitency, _e i the generalized train rate and m i the train rate enitivity. The value of m over the liquidu temperature i equal to 1 (Newtonian fluid). In the frame of the preent tudy, liquid flow in liquid region i ignored and the liquid vicoity i defined a 10 Pa for numerical tability reaon. Below the olidu temperature, the alloy obey an elatic-vicoplatic contitutive equation, which i expreed a p r ¼ K ffiffi mþ1_ 3 e m þ He n ; ½12 Table II. Extent of the Brittlene Temperature Range (BTR) Calculated with (LR) the Lever Rule Approximation, (PE+PA) the Partial- Plu Para-Equilibrium Approximation and (EX) Deduced from Meaurement LR PE+PA EX BTR extent (K) The correponding olidification path and exact alloy compoition are reported in Fig. 3. in which e i the cumulated platic train, H i the train hardening coefficient and n i the train enitivity. Phyical propertie and contitutive parameter are temperature dependent. B. Implementation of the Hot Tearing Criterion In thi tudy, the WYSO hot tearing criterion propoed by Won et al. [14] i evaluated. Being interpreted like in Reference 15, thi criterion i baed on the cumulated train in the plane perpendicular to the dendritic growth direction, thi direction being aumed a the temperature gradient direction. The correponding train rate i denoted _^e. Following Eq. [1] and [8], the train-baed criterion F WYSO HT i expreed a follow: Z F WYSO u HT ¼ _^edt ^e c with ^e c ¼ _^e m ðdt BTR Þ n BTR ½13 where ^e c denote the train limit which depend itelf on the train rate perpendicular to the thermal gradient and on the BTR extent. The value of parameter u, m*, and n* have already been reported in Section II. [14] The value of the characteritic olid fraction bounding the BTR will be dicued further. A regard _^e, it i calculated a follow. The train rate tenor _e i firt calculated in a reference frame attached to the temperature gradient direction, the two complementary direction defining the perpendicular plane. From the four component of the tenor _e in thi plane, it i poible to calculate two eigen value. Conidering that hot tearing take place under tenion, _^e i defined a the larget poitive eigen value. In THERCAST, given the linear interpolation of the velocity field in tetrahedral element, the velocity gradient i uniform in each element, and o i the value of _^e. Hence, the criterion F WYSO HT ha been implemented a a contant field in each finite element, e. It i een from Eq. [13] that the definition of the train rate ued to define the critical train ^e c can give rie to interpretation Fig. 3 Three olidification path for targeted alloy compoition (Table I) correponding to (blue curve, LR) the lever rule approximation, (red curve, PE+PA) a combination of the partial- and para-equilibrium approximation [26,27] and (green mark with error bar, EX) experimental invetigation [28] (Color figure online). and dicuion. The elected implementation procedure i detailed here. At each time increment i, the following quantity i calculated: F WYSO;i HT;e ¼ Xi j¼i tart _^e j u e Dtj m ½14 _^e i e ðdt BTR Þ n where i_tart i the time increment at which element e enter the BTR, and Dt j i the time tep at time increment j. The time increment at which element e exit the BTR i denoted i_end. During the croing of the BTR, the maximum value of the quantitie F WYSO;i HT;e i retained to form the hot tearing criterion. Finally, in each element e, the criterion value i defined a F WYSO HT;e ¼ Max i¼i tart;i end FWYSO;i HT;e : ½15 C. Simulation of Ingot Punching Tet A illutrated in Figure 1, half of the etup i imulated, becaue of the ymmetry with repect to the central tranvere vertical ection plane. The imulation encompae 7 domain: the ingot, 5 mold component, and the punching tool. In the ingot, the initial meh ize METALLURGICAL AND MATERIALS TRANSACTIONS A

63 Fig. 5 Temperature-dependent average propertie for thermomechanical imulation including (a) pecific heat per unit ma c P and denity q while auming full equilibrium [38,39] and (b) thermal conductivity k. [40] Fig. 4 Strain-tre (e-r) curve etablihed by mean of (mark) high-temperature tenile tet [36,37] and (curve) calculation baed on Eq. [12] for (a) 3 temperature from 1273 K to 1573 K (1000 C to 1300 C) at contant train rate, , (b) at a higher temperature, 1673 K (1400 C), for 3 train rate, a) , b) , and c) , and (c) obtained contitutive parameter a a function of temperature. i 8 mm and the number of element i approximately 600,000. For better accuracy in the deformed region, remehing i performed to decreae the meh ize to 4 mm jut before punching, leading to 1,000,000 element. The filling tage i not taken into account o that the thermomechanical imulation tart at the end of the filling tage (70 econd after filling tart). The correponding initial temperature of the ingot at that time, 1823 K (1550 C), i etimated thank to the meaurement. It i et to 293 K (20 C) for all other domain. The thermal boundary condition in between ingot and mold i defined through a heat tranfer coefficient, the value of which depend on the air gap which may form at the interface becaue of hrinkage and thermal contraction. When the ingot i into contact with the mold, the heat tranfer coefficient i calibrated by comparion with meaurement. It value i et to 850 W m 2 K 1. When air gap form, it width i conidered and a heat tranfer coefficient i computed taking into account of radiation and heat conduction through the air gap. [34] Regarding the ingot top, an adiabatic condition i ued, thu modeling the ue of the powder thermal inulator. After partial mold removal, a free urface heat exchange condition i applied to thi region of the ingot urface by accounting for radiation and convection with the air at 293 K (20 C), with the value for emiivity and convective heat tranfer coefficient taken a 0.8 and 15 W m 2 K 1, repectively. Three different olidification path are given in Figure 3. The firt one i baed on the lever rule (LR) METALLURGICAL AND MATERIALS TRANSACTIONS A

64 Fig. 6 Cooling curve (dahed) meaured during trial N-4 and (plain) computed. Poition of thermocouple TC-1 and TC-2 and pyrometer PM are reported in Fig. 1. approximation, meaning that full thermodynamic equilibrium i aumed in all exiting phae at any temperature. The econd one i a combination of the partial equilibrium (PE) approximation together with the para-equilibrium approximation (PA). It account for the fact that C i an intertitial element with large diffuivity compared to ubtitutional element that are frozen in the olid phae. In addition, the PE+PA approximation account for the peritectic reaction taking place in the teel grade invetigated in thi tudy. The detail of thee approximation and the olidification path for imilar alloy compoition are given elewhere. [28,35] The lat olidification path (EX) i extracted from meaurement conducted on a ample uing interrupted unidirectional olidification. Finally, the BTR choen in the preent tudy i limited by olid fraction 0.9 and Calculated value are lited in Table II for the three olidification path given in Figure 3. Graph in Figure 4(a) and (b) how a comparion of experimental tre-train plot with calculated curve baed on Eq. [13]. The comparion indicate a good agreement. The temperature dependence of the correponding contitutive coefficient in Eq. [12] and [13] i given in Figure 4(c). Thermophyical propertie hown in Figure 5(a) are deduced from the TCFE6 databae auming full equilibrium olidification path. [38,39] Latent heat i etimated to 260,000 J kg 1. Tabulation of the thermal conductivity a taken from Reference 36 i alo given in Figure 5(b). V. RESULTS AND DISCUSSION Reult for experiment N-4 are firt conidered, a global view of the tructure oberved in a tranvere metallographic cro ection being provided in Figure 2. Analye of the temperature hitory and punch force evolution permit to introduce meaurement and their comparion with numerical imulation. Thi i done conidering the 3 olidification path hown in Figure 3, correponding to the BTR value reported in Table II, a parameter for the imulation. The et of data detailed for N-4 are then compared with thoe for N-5 and N-6. A can be deduced from Table I, the punching time i doubled in N-5 a the total diplacement of the punching tool i the ame a for N-4 while the velocity of the punching tool i twice maller. For trial N-6, the punching time i the ame a for N-4 ince the total diplacement ha been reduced by a factor 2 with the ame velocity a for N-5. Finally, reult for all trial are reported in the ame graph for a global analyi and dicuion. A. Thermomechanical Validation Figure 6 how the cooling curve recorded in the N-4 trial ingot by thermocouple TC-1 and TC-2. TC-1 (blue curve in Figure 6) i located at the center of the ingot thickne and at 190 mm from the narrow face (or inner wall of the ide mold). Comparatively, TC-2 (red curve in Figure 6) i cloer to the narrow face (90 mm from the ide mold) but till cloe to mid-thickne (only 20 mm off). Both thermocouple are at the ame height, i.e., only 20 mm below the impact poition of the punching tool with the ingot urface. Surface temperature at the height of the punching tool uing the pyrometer are alo reported (circle mark in Figure 6). The calculated temperature evolution at the location of the thermocouple i in good agreement with meaured data from about 300 econd. Before thi time, imulated cooling curve reveal an overetimation of the temperature compared to meaurement. A erie of additional imulation were performed by progreively METALLURGICAL AND MATERIALS TRANSACTIONS A

65 Fig. 7 Time evolution of the reaction force due to the diplacement of the punching tool during trial (a) N-4, (b) N-5, and (c) N-6. Curve correpond to (black mark) meaurement thank to hydraulic ytem ued to impoe the movement of the cylindrical tool and (curve) imulation. Reult are preented a a function of the 3 olidification path preented in Fig. 3. decreaing the liquid vicoity during the firt 300 econd. With more realitic propertie of the melt, the predicted temperature came cloer to the meaurement. It hould alo be pointed out that the filling tage of the mold by the molten metal wa not modeled. It i expected that it would further homogenize the temperature. Temperature evolution during the punching tet i yet well predicted. After 300 econd, the top-right mold located in front of the punching tool i removed (Table I). Simulation how that the urface temperature of the ingot then increae (green curve in Figure 6). Thi i expected a the heat flux decreae in the abence of the mold. At 484 econd, identified by a dahed vertical orange line, the punching tool i in contact with the ingot urface. Preing tart 15 econd later, at 499 econd, and lat for 12 econd (Table I). The tart/end of the preing equence i identified by the 2 vertical orange line in Figure 6. While no thermal ignature of the punching tet i made viible with the thermocouple located in the core of the ingot, it can be een on the imulated curve extracted at the ingot urface in front of the punching tool. A large temperature decreae precede the punching equence, with a minimum temperature reached immediately after punching. While the punching tool goe backward with a velocity of 1.0 mm/ after punching, the temperature increae up to approximately the ame level a before punching. Although it cannot be meaured during preing, the calculated temperature evolution i in good agreement with the pyrometer meaurement (green ymbol in Figure 6). In order to further validate the thermomechanical imulation, the time evolution of the reaction force acting againt the impoed diplacement of the punching tool i plotted in Figure 7(a) for trial N-4. It i to be compared with the recorded force on the punching tool. Globally, the agreement of the imulation with meaurement i good. When comparing the prediction obtained with the different olidification path, it can be een that an excellent agreement with the meaurement i reached for the imulation uing the PE+PA approximation. No large variation can be found when uing EX or LR. However, the fact that the prediction uing LR lead to a larger reaction force reveal the preence of a tiffer olid hell when preing i applied. Indeed, according to LR, a higher quantity of olid i indeed preent a expected from Figure 3 when conidering a given temperature ditribution. Overall, the comparion of temperature and force evolution demontrate that thermomechanical evolution are well imulated from about 300 econd and up to the completion of olidification, thu demontrating the ue of atifying thermophyical and rheological propertie, a well a boundary condition. Similar obervation could be done for the other trial a hown for N-5 and N-6 in Figure 7(b) and (c). Thee reult confirm that quantitative agreement i found for the thermomechanical load applying on the ingot urface. METALLURGICAL AND MATERIALS TRANSACTIONS A

66 Fig. 8 Predicted reult for trial N-4 (Table I) with the PE+PA olidification path (Fig. 3, Table II) howing (a) the temperature field, (b) the region with poitive value of the firt principal tre, i.e., under tenion, the gray region within the ingot being under compreion, and (c) the hot tearing criterion F WYSO HT. The top and bottom line how the different field jut before and after punching, repectively. Black vertical line indicate io-liquid fraction contour from 0 to 100 pct with 10 pct tep. Thi provide a ound bai in the ue of the model for further interpretation of the ditribution of tree and train, a well a their exploitation to evaluate the elected hot tearing criterion. B. Stre Ditribution and Hot Tearing Criterion Figure 8 how the numerical imulation reult jut after punching for trial N-4 with the olidification path labeled PE+PA in Figure 3 and Table II. The variou part of the mold and the punching tool appear with a uniform gray color, while the ingot ection and urface are colored according to the ditribution of (a) the temperature, (b) the firt principal tre component r I, and (c) the hot tearing criterion F WYSO HT. A thi i a magnified view through the variou domain of the imulation hown in Figure 1, only the left, bottom-right and ide part of the mold are partially made viible. The top-right part of the mold ha already been removed and the bottom part i inviible in thi field of view. The cylindrical punching tool i alo viible. From thi view, one can better ee the analogy made with the interaction with roll in the econdary cooling zone of continuou cating machine. At the end of punching, for trial N-4, it can firt be een that a region of fully molten liquid teel till exit at the center of the ingot where the temperature i higher than the liquidu temperature of the alloy, 1784 K (1511 C), while the urface temperature remain over 1273 K (1000 C). Iotherm alo reveal that the average temperature of the right-hand-ide domain of the ingot i hotter than the left-hand ide. Thi i compatible with the previou decription of the predicted cooling curve hown in Figure 6 a cooling wa found to be delayed in the abence of the top-right mold. Figure 8(b) preent the field of the firt principal tre component, jut before and jut after punching. For a better repreentation, only the tenile tate i colored, region in compreion (with negative value of thi firt principal tre component) being repreented with the ame gray color a the mold part and tool. Firt, it can be een that a mall region below the ingot urface, in front of the tool, i compreed. Second, and a expected from the bending created by the punch movement, tenile tree generate and their value increae when going deeper into the ingot. However, the maximum value (around 6 MPa) i not found in the muhy zone, but in the olid, jut beide the muhy zone. Thi i becaue the tiffne of the material i continuouly decreaing with increaing temperature, with an METALLURGICAL AND MATERIALS TRANSACTIONS A

67 Fig. 9 Magnified micrograph for trial N-4, N-5, and N-6 on the top. In all cae, hot tear are oberved and circled with dahed red contour. Note that the top circle crack in N-4 i the ame a the one hown in Fig. 2(b). Alo, the view being limited, not all crack found are hown. Profile along the ingot thickne for (black curve, right-hand-ide legend) the number of hot tear and (colored curve, left-hand-ide legend) the hot tearing criterion F WYSO HT on the bottom. Reult are preented a a function of the 3 olidification path preented in Fig. 3 for trial N-4, N-5, and N-6 (Color figure online). METALLURGICAL AND MATERIALS TRANSACTIONS A

68 abrupt decay at the olid/muhy tranition. Tenile tree alo form on the left-hand ide but with a much lower intenity than on the punch ide. The ditribution of the value of the hot tearing criterion F WYSO HT i diplayed in Figure 8(c). It i reminded here that thoe value reult from an incremental build-up when croing the BTR, a expreed by Eq. [14] and [15]. Like for the tenile tre, only repreentation of region with poitive value are hown, correponding to higher rik to form tear defect. Before punching and outide of the punching zone, there i no trace of hot tearing rik. The negative value of the criterion expre that the natural olidification of the ingot i globally ound. Converely, after bending, poitive value appear, the higher one being found on the punching ide, at the height of the punching tool, cloe to the end of olidification, within the BTR. Thi i clear in Figure 8(c): after punching the maximum value are within the contour defined for the fraction of olid 0.9 and 1. At the end of a cating imulation, when the ingot i fully olid, the final F WYSO HT map i made by retaining thoe value and reveal a high hot tearing enitivity only in the ingot region facing the punching tool. C. Hot Tearing Prediction v Obervation Figure 9 how a magnified view of the micrograph for trial N-4. It i extracted from the cut chematized in Figure 1(b) and hown in Figure 2(a). In fact, the top hot tear circled in Figure 9 i the ame a the one hown in Figure 2(b). Crack were only found on the punching ide of the ingot. Their number and poition were meaured. Thi wa done by counting the number of crack along the height of the ingot a a function of the ditance from the punching ide, i.e., the right-hand-ide limit of the image. Reult are alo preented in Figure 9 for trial N-4. The black tepwie line i thu a direct reading of the intenity of defect found. In fact, the range over which crack are preent depend on the crack length and the crack denity i directly proportional to the number of crack. A total of maximum 3 tear were counted 50 mm away from the ingot urface where preing wa applied. Crack are preent between 40 to 60 mm. Note that thi range i cloe to the length of the circled crack in Figure 2(b), thu the longet in thi ample. Profile are alo provided for the criterion F WYSO HT in Figure 9. Poitive value normally correpond to the occurrence of crack. Conidering the PE+PA olidification path (red curve), crack are predicted to occur over a ditance ranging from 36 to 50 mm. Thi i not far from the prediction uing the experimental olidification path (EX, green curve). However, it alo become clear from Figure 9 that prediction uing the LR approximation (blue curve) i not atifying a it almot never lead to poitive value for F WYSO HT. The reaon i linked to the mall extent of the BTR (Table II) in the cae of LR. Cumulated train computed within thi mall temperature interval doe not permit to reach the critical train thu poitive value of F WYSO HT over a atifying ditance. Thi alo corroborate the finding decribed for the imulated reaction force that LR doe not lead to reult a good a the PE+PA approximation. However, while difference between PE+PA and LR when predicting the reaction force were not large, the effect on F WYSO HT become pectacular. The effect of the olidification path i thu evidenced, revealing the crucial role of the microegregation model for the prediction of hot tearing. D. Comparion of Experiment and Dicuion Figure 7 and 9 alo preent reult of analye for trial N-5 and N-6. Similar obervation can be given for the reaction force prediction and it comparion with meaurement through Figure 7. In Figure 9, however, a large number of crack are viible for trial N-5 while very few and hort one are found for trial N-6. Again, a more quantitative illutration i given in the bottom line of Figure 9. Thi obervation i retrieved in the ditribution with a maximum number of crack count reaching 10 at around 50 mm reported in Figure 9 for trial N-5, while only a maximum of 2 crack are reported in Figure 9 for trial N-6. Similar trend are predicted with the ditribution of the F WYSO HT value along the profile. The maximum F WYSO HT value i reached for trial N-5 while it only lightly overpae the zero-limit from which defect are expected to form for trial N-6. In Figure 9, trial N-5 correpond to twice the time for the reaction time and a lower reaction force for trial N-6 ince the total diplacement ha been reduced by a factor 2 compared to trial N-4. Such reult confirm the pertinence of the elected hot tearing criterion. Similar analye a for N-4 again reveal the critical role played by the olidification path to determine the BTR extent. In an effort to analyze the cumulated train criterion and to compare reult with literature, Figure 10 preent limit curve predicted by the WYSO criterion when baed on the three olidification path. The continuou curve correpond to the train limit ^e c ð _ e Þ that contitute the econd term of the WYSO criterion in Eq. [13]. Fig. 10 Computed cumulated train R _^e BTR dt a a function of the train rate _^e perpendicular to the temperature gradient calculated when paing through the BTR. Value are taken from the F WYSO HT profile at the punching height, an example being hown in Fig. 9. The three continuou curve indicate the train limit ^e c (Eq. [13]). METALLURGICAL AND MATERIALS TRANSACTIONS A

69 Note that in the expreion of ^e c ð _ e Þ, the BTR extent i known a priori from the olidification path (Table II). Thu, for each curve, the half pace located above the curve correpond to F WYSO HT >0, expreing a higher rik to form hot tearing defect. According to it maller BTR extent, the LR limit curve i hifted toward the top-right region of the graph while the PE+PA limit curve i located in a lower (left-bottom) region, the EX limit falling in between. Calculated value are uperimpoed on the figure for trial N-2 to N-6. To thi end, for each tet, the profile of F WYSO HT along the thickne of the ingot at the punching tool height i exploited. The maximum value for F WYSO HT i taken from the profile a well a the correponding value at the ame poition for the perpendicular cumulated train: R _^e BTR dt. The correponding perpendicular train rate i then recomputed with Eq. [13], uing the obtained value for the maximum F WYSO HT and the cumulated train. The ditribution of the reult in the graph i linked to the applied diplacement and velocity of the punching tool a hown in Table I. It i found that the microegregation model doe not play a major role on the perpendicular cumulated train and the perpendicular train rate in the BTR ince the three point for each trial are very cloe. Note that, while metallographic invetigation of all thee trial have revealed the preence of hot tear, only the PE+PA and EX curve give a good repone. The imulation uing LR doe not work a N-6 i not located in the crack rik region and wa yet hown to preent defect. Conidering the reult preented above, it appear that the expreion of F WYSO HT not only provide u with a criterion to predict the formation of hot tear but alo ha potential to predict their magnitude. In order to reveal thi behavior, a hot tearing index i built by conidering a imple addition of the crack length meaured in each of the experiment. Figure 11 preent the value of cumulated hot tearing length, expreed in millimeter, a a function of the maximum value for Fig. 11 Cumulated hot tearing length meaured a a function of the calculated value of the maximum hot tearing criterion F WYSO HT uing the PE+PA olidification path. obtained from it profile at punching height baed on the PE+PA olidification path for each experiment. A poitive lope i found that how a ignificant correlation between the calculated value of the criterion function and the effective crack intenity a deduced from cumulated meaured crack length. Thi indicate that the train-baed criterion can alo be ued to predict the hot tearing magnitude. F WYSO HT VI. CONCLUSIONS A hot tearing tudy ha been performed during teel olidification. Analye include 1. punching tet uing a tool to pre the emi-olid ingot while it i olidifying, thu mimicking deformation under the roll of a continuou cating machine, 2. everal trial conducted for variou value of the total deformation and velocity of the preing tool, 3. meaurement including thermal hitory through the ue of thermocouple and a pyrometer a well a the reaction force acting againt the impoed diplacement of the preing tool, 4. thermomechanical imulation of each trial uing finite element imulation, 5. metallographic invetigation of a tranvere cro ection of the ingot and ytematic count of all crack found to evaluate the intenity of defect formation, 6. integration of a cumulated train-baed hot tearing criterion (HTC), 7. comparion of prediction with obervation. Main finding are a follow: 1. the HTC i able to predict the rik of crack formation, provided that the brittlene temperature range (BTR) i adequately reproduced, 2. prediction in a cumulated train veru train rate perpendicular to the temperature gradient, a well a the reaction force, do not evidence large variation with the BTR, nor did the thermal hitory, 3. the role of the BTR i evidenced when drawing the HTC profile along the ingot thickne, thu revealing the critical effect of the elected olidification path, 4. excellent correlation i found between meaurement and imulation concerning the poition and intenity of the hot tear. A a perpective to the preent work, conidering the trong enitivity of the hot tearing prediction to the olidification path, coupling with macroegregation ha to be modeled. Indeed, deformation-induced compoition variation are expected to generate variation in olidification path. [37] Thi i particularly true when conidering alloying element uch a C, S, or P for which low concentration variation can generate large deviation of the olidu temperature. The numerical modeling of thi coupling with macroegregation require in turn the calculation in the muhy zone of the concurrent deformation of the olid phae and liquid flow. Some work ha METALLURGICAL AND MATERIALS TRANSACTIONS A

70 been initiated by the author of the preent paper [41 43] by ue of a two-dimenional two-phae thermomechanical model. [42,43] However, a general 3D framework implementing concurrently macroegregation and hot tearing i till miing in the literature. Further development in thi direction are foreeen. ACKNOWLEDGMENTS Thi work i upported by Nippon Steel & Sumitomo Metal Corporation (NSSMC) in a collaborative project with ArcelorMittal (AM). The author are deeply grateful to Dr Olivier Jaouen and Dr Fre de ric Cote from TRANSVALOR (Mougin, France), for their kind help and dicuion about thermomechanical modeling uing oftware THERCAST. REFERENCES 1. J. Campbell: Cating, Butterworth-Heinemann, Oxford, M. Wintz, M. Bobadilla, and M. Jolivet: La Revue de Me tallurgie, 1994, vol. 4, pp R. Pierer, C. Bernhard, and C. Chimani: La Revue de Me tallurgie, 2007, vol. 2, pp D.G. Ekin, Suyitno, and L. Katgerman: Prog. Mater. Sci., 2004, vol. 49, pp M. 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Won, T.J. Yeo, D.J. Seol, and K.H. Oh: Metall. Mater. Tran. B, 2000, vol. 31A, pp M. Bellet, O. Cerri, M. Bobadilla, and Y. Chatel: Metall. Mater. Tran. A, 2009, vol. 40A, pp M. Rappaz, J.M. Drezet, and M. Gremaud: Metall. Mater. Tran. A, 1999, vol. 30A, pp H. Sato, T. Kitagawa, K. Murakami, and T. Kawawa: Tetu to Hagane, 1975, vol. 61, p. S K. Miyamura, A. Ochi, K. Kanamaru, and N. Kaneko: Tetu to Hagane, 1976, vol. 62, p. S K. Marukawa, M. Kawaaki, T. Kimura, and S. Ihimura: Tetu to Hagane, 1978, vol. 64, p. S K. Narita, T. Mori, K. Ayata, J. Miyazaki, and M. Fujimaki: Tetu to Hagane, 1978, vol. 64, p. S Y. Sugitani, M. Nakamura, H. Kawahima, and M. Kawaaki: Tetu to Hagane, 1980, vol. 66, p. S M. Rappaz, A. Jacot, and W.J. Boettinger: Metall. Mater. Tran. A, 2003, vol. 34A, pp N. Wang, S. Mokadem, M. Rappaz, and W. Kurz: Acta Mater., 2004, vol. 52, pp H. Fujii, T. Ohahi, and T. Hiromoto: Tetu to Hagane, 1976, vol. 62, p. S T. Matumiya, M. Ito, H. Kajioka, S. Yamaguchi, and Y. Nakamura: ISIJ Int., 1986, vol. 26, pp Q. Chen and B. Sundman: Mater. Tran., 2002, vol. 43, pp M. Hillert: Phae Equilibria, Phae Diagram and Phae Tranformation, Cambridge Univerity, Cambridge, T. Kohikawa, C.A. Gandin, M. Bellet, H. Yamamura, and M. Bobadilla: ISIJ Int., 2014, vol. 54, pp T. Carozzani, C.A. Gandin, H. Digonnet, M. Bellet, K. Zaidat, and Y. Fautrelle: Metall. Mater. Tran. A, 2013, vol. 44A, pp C.A. Gandin: Acta Mater., 2000, vol. 48, pp H. Fujii, T. Ohahi, and T. Hiromoto: Tetu to Hagane, 1976, vol. 62, pp M. Bellet and V.D. Fachinotti: Comput. Method Appl. Mech. Eng., 2004, vol. 193, pp M. Bellet, O. Jaouen, and I. Poitrault: Int J. Num. Method Heat Fluid Flow, 2005, vol. 15, pp B.G. Thoma and M. Bellet: ASM Handbook Volume 15: Cating, Diviion 4: Modeling and Analyi of Cating Procee, American Society of Metal, 2008, pp T. Kohikawa, M. Bellet, H. Yamamura and M. Bobadilla: SteelSim2013, 5th Int. Conf. on Modelling and Simulation of Metallurgical Procee in Steelmaking, Otrava, Czech Republic, September 10 12, 2013, DVD publihed by the Czech Metallurgical Society. 36. Iron and Steel Intitute of Japan: Mechanical behavior in Continuou Cating, ISIJ, 1985, NCID:BN (Citation Information by National intitute of informatic: CiNii). 37. A. Palmaer, A. Etienne, and J. Mignon: Stahl und Eien, 1979, vol. 99, pp P. Shi: TCS teel/fe-alloy databae V6.0, Thermo-Calc Software AB, Stockholm, Thermo-Calc: TCCS Manual, Thermo-Calc oftware AB, Stockholm, Britih Iron and Steel Reearch Aociation: Phyical Contant of Some Commercial Steel at Elevated Temperature (Baed on Meaurement Made at the National Phyical Laboratory, Butterworth Scientific Publication, Teddington), G. Leoult, C.A. Gandin, and N.T. Niane: Acta. Mater., 2003, vol. 51, pp V.D. Fachinotti, S. Le Corre, N. Triolet, M. Bobadilla, and M. Bellet: Int. J. Num. Method Eng., 2006, vol. 67, pp T. Kohikawa, M. Bellet, C.A. Gandin, H. Yamamura and M. Bobadilla: Proc. MCWASP XIV, 14th Int. Conf. on Modeling of Cating, Welding and Advanced Solidification Procee, Awaji Iland, Japan, June 2015, H. Yauda, ed., IOP Conference Serie: Material Science and Engineering METALLURGICAL AND MATERIALS TRANSACTIONS A

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73 Chapter IV Experimental tudy and two-phae numerical modelling of macroegregation induced by olid deformation during punch preing of olidifying teel ingot Takao KOSHIKAWA, Michel BELLET, Charle-André GANDIN, Hideaki YAMAMURA and Manuel BOBADILLA DOI: /j.actamat Page: 60 to 76 in thi thei 59

74 Acta Materialia 124 (2017) 513e527 Content lit available at ScienceDirect Acta Materialia journal homepage: Full length article Experimental tudy and two-phae numerical modeling of macroegregation induced by olid deformation during punch preing of olidifying teel ingot Takao Kohikawa a, b, Michel Bellet a, Charle-Andre Gandin a, *, Hideaki Yamamura c, Manuel Bobadilla d a MINES PariTech, PSL Reearch Univerity, CEMEF, CNRS UMR 7635, Sophia Antipoli, France b Nippon Steel & Sumitomo Metal Corporation, Oita Work Equipment Diviion, 1 Oaza-Nihinou, Oita City , Japan c The Japan Intitute of Metal and Material, , Ichibancho, Aoba-ku, Sendai , Japan d ArcelorMittal Maiziere, Reearch and Development, BP 30320, Maiziere-le-Metz, France article info abtract Article hitory: Received 28 April 2016 Received in revied form 7 November 2016 Accepted 8 November 2016 Keyword: Solidification Steel Segregation Modeling Deformation Ingot punching tet are performed on the already formed olid hell of a 450 kg teel ingot during olidification. Such tet i deigned in order to be repreentative of the thermomechanical condition that give rie to macroegregation during econdary cooling in teel continuou cating. In order to undertand the different phyical phenomena, a numerical model of the tet ha been developed, coniting of a two-dimenional planar finite element imulation in the median ection of the ingot. A twophae formulation ha been implemented, in which the velocitie of the liquid and olid phae are concurrently olved for. The imulation how how olute are reditributed through the central muhy zone of the ingot under the effect of the compreion of the olid phae and the induced fluid flow reulting from the punching of the olid hell. By comparion with meaurement of macroegregation operated on two ingot of ame initial compoition but punched under different condition, the imulation prove it capacity to reproduce the main experimental trend. However the predicted intenity of macroegregation i lower than the one meaured. Through dicuion and analyi of different numerical enitivity tet, critical material parameter and model improvement are identified in view of attaining better quantitative prediction in the future Acta Materialia Inc. Publihed by Elevier Ltd. All right reerved. 1. Introduction The paper focue on a major defect encountered during econdary cooling in teel Continuou Cating (CC): macroegregation. Thi critical defect i oberved at the center of continuouly cat emi-product, uch a lab and billet. Like in any olidification proce, macroegregation reult from the concurrent effect of microegregation on the one hand, and the relative movement of the liquid phae with repect to the olid phae on the other. Microegregation conit of the enrichment or depletion of the liquid phae in olute at the cale of the dendritic microtructure. A analyzed in the pioneer work by Fleming and Nereo [1], the differential liquid motion with repect to the olid may have * Correponding author. addre: charle-andre.gandin@mine-paritech.fr (C.-A. Gandin). different caue: i) the olidification hrinkage and the thermal contraction of the olid phae, ii) the convection flow of liquid ariing from it local variation of denity with temperature and chemical compoition, iii) the movement of equiaxed olid grain tranported by the liquid flow, and iv) the deformation of the olid phae within the muhy zone. In teel ingot cating, all coupled phenomena are preent and play a part in the formation of macroegregation, but the lat one (iv) can often be neglected. Regarding numerical modeling, one can refer to Wu and Ludwig, Li et al. [2,3] and Zaloznik and Combeau [4] who developed the more advanced numerical model at the preent time, in the context of the finite volume method. The latter author have alo worked with two author of the preent paper to demontrate the feaibility of finite element (FE) formulation [5]. Converely to ingot cating, the deformation of the olid phae within the muhy zone play a major role in continuou cating of teel. Thi deformation i created by the bulging of the olid hell / 2016 Acta Materialia Inc. Publihed by Elevier Ltd. All right reerved.

75 514 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e527 between upport roll of the CC machine, which i due to the metallotatic preure, the poible bending and unbending of the trand, and by the poible o-called oft reduction coniting of a mechanically applied thickne reduction of the trand operated cloe to the end of olidification. Such a thickne reduction of the cat product generate deformation of the underlying olid phae within the muhy zone and, a a conequence, macroegregation. Several reearcher tudied thee phenomena in the context of CC proce with numerical modeling. Miyazawa and Schwerdtfeger [6] modelled olidification and macroegregation, conidering bulging between upport roll and the aociated muhy zone deformation. In their model, ma, liquid phae momentum, olute and energy conervation equation are decribed in a fully Eulerian approach. The olid phae momentum equation i not olved o the velocity field of the olid hell and of the olid phae in the central muhy region are arbitrarily given. The numerical imulation how the formation of a center line with poitive macroegregation due to bulging. Kajitani et al. [7] tudied the o-called oft reduction proce uing the ame approach and dicued the quantitative impact of the oft reduction technique to reduce the intenity of the central macroegregation. In a more recent contribution, Domitner et al. [8] developed a two-dimenional (2D) model covering the whole length (25 m) of the product in the cater, but till with a predefined motion of the olid phae, like in Ref. [6,7]. Fachinotti et al. [10] developed a different approach to model the muhy zone in the context of CC, uing an arbitrary Lagrangian-Eulerian approach in which olid and liquid velocity field are concurrently olved for, avoiding then the trong hypothei over the olid velocity field which wa preent in previou work. With thi model, the author imulated continuou cating and imilarly retrieved the effect of bulging on the formation of central macroegregation [11]. More recently, Rivaux [12] developed an alternative approach to model the deformation of the olid phae and it Darcy-type interaction with the liquid phae in the muhy tate, through a taggered cheme in which the two field are eparately and ucceively olved for at each time increment. However, in the end and depite lot of effort made in developing thoe numerical model, none of them ha been uccefully applied up to the cale of a three-dimenional (3D) imulation repreentative of the complexity of the indutrial proce. Thi i certainly due to the coniderable amount of computational power required to run the model encompaing a ignificant part of the econdary cooling ection of an indutrial cater. Thi i one of the reaon why, in order to tudy and better undertand the origin of the defect, experiment uch a ingot bending or ingot punch preing, have been developed by everal reearcher for many year [13e18]. The objective of thee experiment i to analyze macroegregation phenomena when they arie eentially from the deformation of the olid phae, and to do thi in a configuration a priori impler than continuou cating: ingot cating. Thi i obtained by deforming the external olid hell of an ingot during it olidification. The preent tudy report on an ingot punch preing tet which ha been deigned and operated by Nippon Steel & Sumitomo Metal Corporation (Tokyo, Japan). In order to analyze the tet, epecially from the macroegregation point of view, a FE numerical imulation ha been developed uing the reearch code R2SOL previouly developed at MINES PariTech CEMEF [10,11]. The paper preent the tet and it application to two teel ingot of imilar initial compoition but punched at different time (that i for different degree of olidification progre). Then, FE modeling of thermomechanic and olute tranport i introduced. Finally, reult are preented and the predicted occurrence, location, and intenity of macroegregation are explained and dicued in the light of experimental meaurement. 2. Ingot punch preing tet The chematic of the ingot punch preing tet, a developed at Nippon Steel & Sumitomo Metal Corporation, i hown in Fig. 1. It ha been deigned to mimic the thermomechanical load taking place during CC proce; for intance, with repect to the evolution of the cooling rate and deformation. The procedure of the experiment i a follow: the molten metal i prepared in an electric ladle furnace and the temperature i maintained at 1640 C before pouring. The molten metal i poured into the mold from the top through a tundih. The filling duration i about 70. After filling, powder i added at the top of the ingot to limit heat exchange with air. The ma of metal i approximately 450 kg. The ize of the ingot i 0.16 m in thickne, 0.5 m in width and 0.75 m in height. In practice, a fibrou thermal inulator of thickne 5 mm cover the inner wall of the lowet part of the mold up to 0.2 m from the bottom of the ingot in order to avoid molten metal ticking. At a precie time after filling, the upper part of one of the two large face of the mold i removed, a illutrated in the right part of Fig. 1(b). The ingot i then deformed during olidification by puhing a horizontal cylindrical tool perpendicularly to the urface of the ingot, it longitudinal axi being located 0.45 m from the bottom of the ingot. The tool velocity and diplacement are controlled by mean of a hydraulic ytem and meaured during the tet. The reaction force i alo meaured uing the time evolution of the hydraulic ytem preure. For the alloy targeted in the preent work, a total of 9 experiment, labelled N-1 to N-9 were performed. They belong to a erie that aim at tudying both macroegregation and hot tearing enitivity a thee defect are frequently found to form concomitantly [19]. However, due to very demanding effort to analyze and preent the trial with repect to macroegregation, only the tet condition referred to a N-1 and N-9 are conidered in the preent work. Parameter are ummarized in Table 1, together with meaured nominal compoition in the ladle before filling. Note that a unique alloy compoition wa targeted for the experimental reult reported hereafter. The two tet differ firt by the time at which the punching tart. For N-1 punching tart earlier on a central muhy zone richer in liquid. Converely, for N-9, olidification i more advanced at the beginning of punching. Tet how alo difference in punch diplacement, the maximum punch force being limited to 450 kn for the protection of the bending equipment. In cae N-9, thi limiting punch force i obtained for a diplacement of 9 mm, after which the punch top. In cae N-1, which i a priori le reitant, due to a le advanced olidification, the limiting force i obtained for a larger diplacement of 13 mm. The temperature evolution during the tet i recorded by a B-type thermocouple poitioned in the ingot cavity, a hown in Fig. 1(a). For the thermocouple etting, a hole along horizontal direction, of diameter 5 mm, i pierced through the fixed large face of the mold. A thermocouple i inerted into the hole from outide and i placed at the deired poition. For ealing purpoe, a thermal inulator i put into the hole after etting of the thermocouple. After the experiment, ingot were cut along their central tranvere ymmetry plane and micrograph were carried out after appropriate etching. Fig. 2 how micrograph for both trial N-1 and N-9. The punching tool move from the right to the left and the urface deformation due to punching appear clearly. Concerning microtructure, one can ee a dendritic columnar microtructure which ha grown from the ingot urface toward the center. Note that the primary trunk of the dendritic tructure i lightly oriented upward. Thi i expected when conidering liquid melt thermal convection oriented downward in front of the growing dendrite. Becaue the direction of dendrite trunk and arm correpond to 100 crytallographic direction in the preent ytem, a texture of

76 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e Fig. 1. Schematic of the ingot punching tet developed at Nippon Steel & Sumitomo Metal Corporation howing (a) initial tate and (b) punching tage after the top right mold wa removed. The configuration hown i half of the etup, conidering ymmetry with repect to the central tranvere ection through the whole ytem. A thermocouple (TC-1) i placed at coordinate (0, 0.43, 0.06), with reference frame centered at the top urface of the bottom mold plate. the columnar zone i thu defined. Further recirculation of the liquid in the muhy zone will no longer modify thi texture ettled during primary olidification. Around the center of the ingot, the morphology i changed to equiaxed due to the decreae of the temperature gradient cloe to the center of the ingot. At the top, primary hrinkage hape i oberved and a macro void i alo found below the very thin top olid hell. In order to avoid thermal exchange at the top, powder i put after filling. However, the thermal exchange i not completely prevented. So, a thin olid hell i formed at the top and the macro void i then generated below, due to hrinkage. Looking at the bottom of the micrograph, the morphology i almot of equiaxed type due to the ame reaon. Thi i aociated here with the preence of the thermal inulator along the bottom urface to avoid ticking; it preence being made viible by the hape of the ingot tranvere ection at bottom and up to approximately 200 mm height. Becaue of the thermal inulation, olidification i delayed, compared to the top area. Cloe to the end of ingot olidification the liquid melt cannot be provided from the upper central region due to earlier completion of olidification there. In the end, a macro void necearily form becaue of hrinkage a hown in Fig. 2 at about 150 mm height from the bottom of the ingot. The vertical white band at the ingot center reveal negative egregation. It appear very clearly o that the olute variation could be large, epecially for tet N-1. For quantitative analyi, Electron Probe Micro Analyzer (EPMA) meaurement were performed on both ingot, for pecie Mn and P. Meaurement condition were a follow: beam diameter 50 mm, beam current 1 ma, integration time 50 m, meaurement tep 50 mm. Line can meaurement were carried out along a horizontal line at punch height. For comparion to numerical imulation reult, the meaurement value were averaged over 2 mm where 40 meaurement point are contained and the tandard deviation were calculated at each average point. Reult will be preented further (Section 4), together with the predicted macroegregation profile, for comparion purpoe. 3. Thermomechanical and macroegregation FE modeling 3.1. Two-phae thermomechanical and macroegregation modeling The macroegregation formed in uch a punch preing tet clearly reult from the concurrent deformation of the olid phae in the muhy zone, and from the liquid flow induced by thi deformation. In order to conider eparately - but olve concurrently - for the liquid and olid velocity field in the muhy zone, we apply the o-called two-phae model initially developed by Bellet et al. [10,11] to addre macroegregation in continuou cating. Thi model i alo named ponge-like model by reference to the title of the paper of Leoult et al. [9] in which the balance equation governing the deformation of an alloyed muhy zone were expreed. Thi model allow imulating olute tranport phenomena leading to macroegregation induced by deformation, hrinkage and advection. It i performed with the 2D FE code R2SOL. The main aumption and feature of the model are reminded hereafter. The conervation equation for the olid-liquid mixture are expreed on a repreentative elementary volume (REV) and are obtained uing the patial averaging method [20]. The muhy material i conidered a a aturated two-phae medium only made of olid,, and liquid, l (i.e. g þ g l ¼ 1 where g k denote the volume fraction of phae k). Table 1 Meaured nominal compoition and experimental punching condition. Nominal compoition (wt.pct) C Si Mn P S Al Mold removal [] Punching tart [] Diplacement [mm] Velocity [mm 1 ] N N

77 516 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e527 Local thermal equilibrium hold within the REV, enuring the uniformity of temperature within the different phae. The previou aumption lead to the following et of averaged conervation equation: V$S g Vp l þ g 2 l m lk 1 ðv l v Þþg r S g ¼ 0 (1) D V$ le g l Vp l gl 2m lk 1 ðv l v Þþg l r L g vðg ¼ r l v l Þ L þ r vt L V$ðg l v l v l Þ (2) V$ðg l v l Þþ r S r L V$ðg v Þ¼ rl r S r L vg vt (3) r v h þ V h $ rv þ LV$ðg vt l g r ðv l v ÞÞV$ð l VTÞ ¼0 (4) vhw i i þ V$ðhw vt i iv ÞþV$ðg l w l;i ðv l v ÞÞV$ g l D l Vw l;i ¼ 0 (5) in which v and v l are the olid and liquid velocity vector (intrinic average phae velocitie: v l hv l i l ; v hv i ), p l i the liquid phae preure ðp l hp l i l Þ, m l i the dynamic vicoity of the liquid phae, r and l are the average denity and the average heat conductivity of the olid-liquid mixture, h i the average enthalpy of the mixture. hw i i and w l,i are repectively the average olute ma concentration and the intrinic average concentration in the liquid phae for each olute i. In the momentum equation, the Darcy term introduce the permeability k of the porou medium contituted by the olid phae. It i derived from the liquid volume fraction and the econdary dendrite arm pacing l 2, through the Carman-Kozeny model. l 2 2 g3 l k ¼ 180ð1 g l Þ 2 (6) Fig. 2. Micrograph at central tranvere ection of (a) N-1 and (b) N-9 ingot. Macroegreation i qualitatively revealed by the variation of the grey level. The print og the punching tool i viible on the right hand ide in thee image. At the o-called microcopic cale, within the REV, the liquid phae i conidered a an incompreible Newtonian fluid. After patial averaging, the macrocopic behavior of the liquid phae i Newtonian compreible. In the ame way, the olid phae i conidered intrinically a an incompreible non-newtonian fluid, and it macrocopic averaged behavior i the one of a compreible non-newtonian fluid for which inertia effect are neglected. The momentum interaction between olid and liquid phae i formulated with an iotropic Darcy law. Solidification hrinkage i taken into account by conidering different denitie for the olid and liquid phae, repectively r S and r L. Thoe value are aumed contant within the olidification interval uing the value at the liquidu temperature for r L and the value at the olidu temperature for r S, the latter being the volume-average of the denity of all olid phae. A multi component ytem under lever rule approximation i aumed, meaning that the partition coefficient k i and the liquidu lope m i are contant for each olute element i. Contitutive model for the liquid and the olid phae in the muhy zone are preented in detail in Ref. [10,11]. In ummary, the average deviatoric tre tenor in the liquid phae i expreed by: D le ¼ 2m l g l devð_εðv l ÞÞ ¼ 2m l g l _εðv l Þ 1 3 trð_εðv lþþ (7) where _εðv l Þ i the ymmetric part of the liquid velocity gradient Vv l. For olid, it i aumed that the behavior of the fully olid material i vicoplatic, of power law type, a will be introduced later. The train-rate enitivity coefficient of the fully olid material being denoted m, it can be hown that the effective macrocopic tre tenor for the olid phae in the muhy tate, S ¼ þ p l I,ia degree m homogeneou function with repect to the train rate tenor _ε ¼ _εðv Þ. Adopting a compreible vicoplatic potential, it expreion write: p ffiffiffi S ¼ 3K 3 _ε m1 1 A _ε þ p ffiffiffi ¼ 3K 3 _ε m B 1 tr _ε I 3A A devð _ε Þþ 1 9B tr _ε I in which K i the vicoplatic conitency of the olid material and m it train rate enitivity, both taken at the olidu temperature. The equivalent train rate _ε i expreed by (8)

78 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e _ε 1 ¼ ¼ 1 9B 1 3A ðtr _ε Þ A _ε : _ε þ 1 A devð _ε Þ : devð _ε Þþ 1 9B ðtr _ε Þ (9) In the flow rule (Eq. (8)), A and B are two coefficient depending on the olid volume fraction. In the preent tudy, the expreion introduced in Ref. [10] are ued: A ¼ 3 2 ð1 þ k ABÞ ; B ¼ k B 1 g g g crit (10) The value of the parameter are taken from the ame reference. Taking the trace of Eq. (8), it can be een that the compreibility of the olid phae i eentially controlled by the value of coefficient B: p ffiffiffi trs ¼ K 3 _ε m11 B tr _ε (11) A expreed by Eq. (10), g crit can be een a the coherency temperature at which the compreibility of the olid phae i extremely high (B/ ). Converely, when the olid fraction tend to one (fully olid material), A and B tend toward 3/2 and 0, repectively: the uual power law relating tre and train rate for a dene metal at high temperature i retrieved. In the FE code, the weak (integral) form of Eq (1)e(5) i implemented in 2D triangular finite element. At each time increment, the equence of numerical reolution i a follow: - The thermal reolution i operated on the et of the different domain involved: in the preent context, the olidifying ingot, the different component of the mold, and the preing punch. The (non-linear) heat tranfer reolution i carried out on each domain, ucceively, up to convergence (i.e. tabilization of the temperature field). In the ingot, a FE reolution of Eq. (4) i done, while in the other domain a more claical ingle phae form of heat diffuion i olved. Heat exchange at interface between domain are expreed by the Fourier law, uing heat tranfer coefficient. - The macroegregation reolution i performed in the ingot, through a FE reolution of Eq. (5) for each conidered olute i. In thee reolution, like in the thermal one, the microegregation model (here the lever rule) i conidered a it link the average concentration in the liquid phae w l,i with the average mixture concentration hw i i, and the averaged enthalpy h with the temperature T. - The mechanical reolution i performed in the ingot, the other domain being aumed perfectly rigid and fixed. A non-linear Newton-Raphon algorithm i ued, providing the nodal value of v, v l and p l through a ingle FE reolution of Eq (1)e(3). A the mechanical formulation preented above encompae the central muhy zone but alo the fully olidified region, a pecific treatment applie. In the FE aembly procedure, element are conidered either muhy or fully olid, according to the temperature at their center. In muhy element, the contribution to the reidue directly reult from the weak form of Eq (1)e(3). In fully olid element, it can be noted that, by applying thee equation, the liquid velocity i till preent but ha no real phyical meaning: it i then kept cloe to the olid velocity field by mean of the ole Darcy term in the liquid momentum equation, acting a a penalty term. The deviatoric tre tenor i expreed a a function of the velocity field by the vicoplatic Norton-Hoff power law including train hardening: ¼ 2Kε n p ffiffiffi m1devð_εþ 3ε _ (12) where devð_εþ denote the deviatoric part of the train rate tenor, K the vicoplatic conitency, m the train rate enitivity coefficient, _ε the generalized train rate and ε the generalized train. Strainhardening i aumed null at olidu temperature. Thi i why only K (and not Kε n ) appear in Eq. (8) to expre the behavior model of the olid phae in the muh. The tenor Eq. (12) yield the one dimenional relation between the von Mie tre and the generalized train-rate _ε: pffiffiffi mþ1ε n ¼ K 3 _ε m (13) The fully olid metal i then platically incompreible and thi i why, in the finite element which are found fully olid (temperature at center lower than the olidu temperature), Eq. (3) i replaced by the following equation which imply account for thermal dilatation term: V$v þ 1 r ðtþ dr ðtþ ¼ 0 (14) dt 3.2. Specific modeling of ingot punch preing tet The numerical imulation i two dimenional, in plane train condition, along the tranvere mid-ection of the cating ytem. Such a 2D approach eem a priori reaonable to analyze the reult produced in thi ymmetry plane, the width of the ingot (500 mm) being more than 3 time greater than it thickne (160 mm). A illutrated in Fig. 1 and 7 domain are involved: the ingot, 5 mold component, and the punch. The bae meh ize i 8 mm but the punching region of the ingot i refined up to 2 mm and the number of element in the ingot i approximately 40,000. The filling tage i not taken into account o that the imulation tart immediately after the end of filling, that i 70 after it tart. The temperature of the liquid teel at that time i uppoed to be homogeneou in the ingot domain. According to meaurement, it i etimated to 1550 C. Temperature i uppoed uniform and equal to 20 C for all other domain. The thermal boundary condition between ingot and Fig. 3. Solidification path with nominal teel compoition N-1 given in Table 1, by ue of Thermo-Calc with the TCFE6 databae [23,24] and auming the lever rule approximation (Al i omitted in the calculation).

79 518 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e527 Table 2 Partition coefficient and liquidu lope for olute [23,24]. C Si Mn P S Partition coefficient k i Liquidu lope m i [K (wt.pct) 1 ] mold component i defined a a contant heat tranfer coefficient which i calibrated with meaurement and et to 550 W m 2 K 1. Regarding the ingot top, an adiabatic condition i ued, modeling the powder thermal inulator. After partial mold removal, the free urface heat exchange take into account radiation and convection, with value for emiivity and convective heat tranfer coefficient taken a 0.8 and 15 W m 2 K 1, repectively. It ha been underlined above that the two-phae olid-liquid FE model can be extended to the fully olidified metal. However, the extenion to the fully liquid tate i not poible. Thi i the reaon why the imulation i actually decompoed in two ucceive tep, a decribed hereafter. -Inafirt imulation tep, tarted at time 70, a tandard thermomechanical calculation i conducted up to the moment of punching. In thi firt calculation, olute tranport reolution i not carried out becaue the muhy zone i conidered a a homogenized continuum in which olid and liquid velocitie are not ditinguihed. Intead, the muh i uppoed to behave like a homogenized non-newtonian fluid with a vicoplatic conitency and a train-rate enitivity varying with the ole olid fraction. The nature of the FE modeling operated i not detailed here and can be found in Ref. [21,22]. During thi imulation tep, olidification i governed by a unique olidification path which i defined a priori by the lever rule, baed on the nominal teel compoition. Thi olidification path i hown in Fig. 3 (ee detail hereunder), the liquidu temperature being 1511 C and the olidu 1412 C. -A econd imulation tep i then chained to the previou one, through an automatic retart procedure including the tranmiion of the patial ditribution of all required variable. In thi econd tep the two-phae model preented above (Section 3.1) now applie. At thi moment, jut before punching, the olidification of the ingot i ignificantly advanced and the central region of the ingot i fully in the muhy tate, without any region containing liquid phae only. In the ingot domain, the imulation cover then the central muhy zone and the Fig. 4. Rheological behavior a train-tre (ε-) relation (a) for 3 temperature from 1000 C to 1300 C at contant train rate, and (b) at a higher temperature, 1400 C, for 3 different train rate etablihed by mean of (mark) high temperature tenile tet [25,26] and (curve) calculation baed on Eq. (13). The temperature dependence of contitutive parameter for Eq. (13), auming full equilibrium olidification, are given in (c).

80 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e Fig. 5. Temperature dependent average propertie for thermomechanical imulation including denity while auming full equilibrium [23,24] and thermal conductivity [27]. olidified hell, and i run up to complete olidification of the ingot, in order to compare macroegregation profile with EPMA meaurement. During thi econd imulation tep, the olidification path i no more defined a priori, ince olute concentration vary due to liquid flow and olid deformation. Given a multicomponent ytem involving N S olute, the lever rule approximation i ued to model the microegregation phenomena. Auming contant partition coefficient k i and liquidu lope m i for each olute element i, one obtain the following relation: T ¼ T m þ XNS i¼1 m i w l;i (15) hw i i¼ðg l þð1 g l Þk i Þw l;i (16) h ¼ c p T þ g l L (17) where T m i the melting temperature of pure iron, c p i the heat capacity (aumed contant in the preent tudy), and L i the latent heat (alo aumed contant). The number of equation i N S þ2. Fig. 7. Calculated hape of the ingot and ditribution of the liquid fraction jut before punching (a) at time 792 (correponding to trial N-1) and (b) at time 881 (correponding to trial N-9). The average quantitie h and hw i i are known ince thee variable are obtained thank to the global energy and olute conervation reolution. The number of unknown i then N S þ2: T, g l and w l,i. Thu, the et of equation i well poed and can be olved iteratively at each node of the meh. The required characteritic olutal coefficient k i and m i are obtained uing Thermo-Calc with TCFE6 databae [23,24]: their value are given in Table 2, for each of the 5 olutal element which are conidered here: C, Si, Mn, P and S. The melting temperature for pure iron i defined a 1538 C. The contant value of the pecific heat c p and of the latent heat L are 815 J kg 1 K 1 and 260,000 J kg 1, repectively. With the nominal compoition, i.e. in the abence of macroegregation, one can obtain the olidification path hown in Fig. 3. When proceeding to the initialization of nodal variable at the beginning of thi econd imulation tep, at each node the average concentration hw i i are et to the nominal one w i0. Then, knowing h from the firt imulation tep, a firt calculation of T, g l and w l,i i carried out by olving Eq (15)e(17). Regarding now the mechanical data, Fig. 4 how a comparion between experimental tre-train plot, obtained by tenile tet performed at different high temperature and variou train rate, and calculated curve baed on Eq. (13) for optimized value of K, n and m coefficient identified by regreion analyi. The comparion indicate a good agreement over the whole et of condition teted. The temperature dependence of the contitutive parameter i hown in Fig. 4(c). A regard the liquid vicoity, it i defined arbitrarily a Pa [7,8]. In the Darcy term the permeability of the Carman-Kozeny model conider a contant predefined value of 300 mm for the econdary dendrite arm pacing l 2, on the bai of meaurement carried out on micrograph in cae N-1. Finally, the heat conductivity and the denity, a ued in Eq (3) and (14) for the latter, are hown in Fig. 5. Denitie were obtained by ue of Thermo-Calc oftware and TCFE6 databae auming full Table 3 Comparion of calculated liquid fraction at ingot center when punching tart. N-1 N-9 Fig. 6. Meaured cooling hitory recorded at thermocouple TC-1 (Fig. 1) during experiment N-1 and N-9, and calculated with the finite element imulation at the nominal location of the thermocouple. Calculated liquid fraction Calculated temperature, at TC-1 nominal location [ C] Temperature meaured by TC-1 [ C] 1487 ± ± 4

81 520 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e527 Fig. 8. Simulated and oberved ingot hape howing (a) a global view of it deformation at time (0.4 after punch tart) for trial N-1 and (b) a zoom in the top region with hrinkage pipe to be compared with (c) obervation. The imulation alo how (color map) the ditribution of the liquid fraction and (black contour) iofraction of liquid, a well a the (arrow) the relative velocity of the liquid phae with repect to the olid one, v l - v. The darker region at the center of the micrograph i expelled liquid in the primary hrinkage cavity a a reult of punching. (For interpretation of the reference to colour in thi figure legend, the reader i referred to the web verion of thi article.) Fig. 9. Central region in the imulation of ingot N-1, 6 after punch tart, howing (a) the ditribution of the horizontal component of the velocity of the olid phae v, (b) V$v with evidence of negative value in the muhy zone at the height of the punch tool and (c) the vertical component of the relative velocity of the liquid phae with repect to the olid one, v l - v. The imulation alo how (black contour) iofraction of liquid, from 0 to 0.3, and (arrow) the vector field v l - v. equilibrium olidification path [23,24], while the temperature dependence of thermal conductivity wa taken from the literature [27]. 4. Reult and dicuion 4.1. Micrograph for the two trial Coming back to Fig. 2, howing micrograph for the two ingot, we note two important difference between N-1 and N-9. Firt of all, the clear vertical white band oberved in cae of N-1 i not oberved in N-9. Secondly, the microtructure in the top region i found different. In cae of N-1, a darker microtructure i oberved, well delimited and located at the bottom of the main hrinkage pipe. It eem to partially fill the initially formed pipe cavity. Thi i not een in the hrinkage cavity of N-9. Thoe difference are interpreted in the following ection Numerical imulation of trial N-1 and N-9: heat tranfer and mechanical analyi Fig. 6 how a comparion of temperature evolution cloe to the ingot center. It i een that the two cooling hitorie meaured by thermocouple for trial N-1 and N-9 are quite cloe, indicating a good reproducibility obtained through an appropriate control of the procedure of mold preparation and cating. A mall deviation between the two tet can be een after 800, but thi occur after the punching tage and hould not affect the quantitative comparion of macroegregation. The calculated curve hown in the figure ha been obtained by the tandard thermomechanical imulation preented above and i in good agreement with the meaurement, epecially during the

82 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e punching tage (tarting at 792 for N-1, and at 881 for N-9). Thi good agreement i maintained up to full olidification, which i obtained after 900, according to the numerical imulation. Thi expree a good quality of the et of thermophyical data ued in the imulation, a well a a good calibration of heat exchange at interface. The tate of the ingot jut before punching, a imulated by the tandard thermomechanical imulation in the firt imulation tep defined above, can be viualized in Fig. 7. It can be een that at that time, olidification i quite advanced. Table 3 contain quantitative information on temperature and the maximum of liquid fraction at the core of the ingot. In cae of N-1, we note that the calculated liquid fraction i higher (0.39) than for N-9 (0.12). Thi i of coure conitent with the fact that punching i activated earlier in cae N-1 (792 ) than for N-9 (881 ) a reported in Table 1. Let' conider now the deformation of the olid phae and the aociated liquid flow, a calculated in the two-phae approach. Thi i illutrated in Fig. 8, for the imulation of trial N-1. The deformation of the olid hell by the punch induce a compreion of the olid phae in the muhy zone (not viible in Fig. 8), which in turn induce liquid flow. For evident reaon (global quai incompreibility of the muhy material, cloed muhy pool at the bottom of the ingot), liquid flow i necearily oriented upward. In cae N-1, olidification i le advanced. The alloy i till in the muhy tate in it top central region. Thi make poible uch an upward liquid flow, with liquid expulion at the top urface into the primary hrinkage cavity: the expelled melt cover the already formed olid hell. The imulated liquid flow can be een in the central part of Fig. 8. Experimental evidence of uch an expulion of liquid flow i provided by the micrograph on the right part of the figure, in which a reidue of enriched (darker) expelled liquid i preent at the bottom of the wedge-haped hrinkage cavity. It i intereting to compare the hape of the hrinkage cavity between imulation and micrograph. Of coure the real ingot how a complex form, epecially with a reidual thin kin at the extreme top, which reult Fig. 11. Chill effect during punching hown 4 after punching tart with liquid fraction ioline. from a complex thermomechanical interaction with the cover powder. Thi i not imulated. However, the ection of the wedgehape at the lower urface of the cavity i well reproduced by the thermo-mechanical imulation. Fig. 9 reveal in more detail what happen in the muhy zone at the height of the punch tool. Due to the motion of the punching tool, the olid hell in it vicinity i diplaced a hown by the horizontal component of the olid velocity (Fig. 9(a)). The reulting compreion of the olid phae within the muhy zone i expreed by negative value of the divergence of the olid velocity field, V$v, diplayed in Fig. 9(b). A relative liquid flow i thu created a hown in Fig. 9(c). The compreion of the olid phae primarily affect the region of the muh for which the liquid fraction i higher than 0.1. Indeed, when liquid fraction decreae, the value of coefficient B, which control the compreibility of the olid phae (cf. Eq. (11)), decreae a well, tending toward zero which i the limit of the B- value aociated with a fully dene olid material. Thi continuou decreae of coefficient B hinder the compreion of the olid phae Fig. 10. Ditribution of the divergence of the olid phae velocity field, together with the ioline for the liquid fraction, and a vector repreentation of the relative velocity field of the liquid phae with repect to the olid one: v l e v. Information i given for trial N-1 during punching (a) 1 and (b) 4 after punch tart. The horizontal dot line indicate the height of the punching tool.

83 522 at the periphery of the muhy zone, where olidification i more advanced. Conidering the ma balance equation, Eq. (3), and taking into account that advancement of olidification decribed by the right hand ide term i much lower than V$ðg v Þ, the compreion of the olid phae (V$ðg v Þ < 0) create a liquid flow. Becaue of the preence of the olid hell, thi liquid flow i everely contrained along the horizontal direction. Therefore it i found eentially vertical but howing a poitive velocity gradient along the vertical direction, thu enuring V$ðg l v l Þ > 0. Accordingly, the flow of the liquid phae i accelerated vertically. Thi reulting flow can be een in Fig. 9(c). Note that there i no neutral point in front of the punch inide the muhy zone, which would eparate an upper region with upward v l from a lower region with a downward v l.a mentioned earlier, thi i becaue of the cloure of the muhy pool at the bottom of the ingot, where the olid hell cannot move, being blocked by the mold. For thi reaon, the liquid velocity field i uniformly oriented upward throughout the whole muhy zone, but with a vertical gradient in order to comply with the compreion of the liquid keleton through the ma balance equation (Eq. (3)). Looking now at two different intant during punching, like in Fig. 10 at 1 and 4 after punch tart, it can be een that the ditribution of liquid fraction ha changed ignificantly, howing a general decreae and a hift toward the oppoite ide to the punch (here, a hift to the left). Different phenomena contribute to the evolution of the liquid fraction. - Firt, the global olidification of the ingot explain a general decreae of the liquid fraction, which hould be found more or le ymmetric. Looking at the left ide of the muhy zone, thi phenomenon i found negligible on the time interval conidered: the ioline 0.1 and 0.2 for the liquid fraction are tationary. - Second, the chill effect due to the contact with the punch. The cooling aociated with the heat extraction through the punch can be een in Fig. 11. Thi effect i non-ymmetric but ha no ignificant influence on the core of the ingot. Indeed, a quick etimation of the affected ditance (through the expreion T. Kohikawa et al. / Acta Materialia 124 (2017) 513e527 pffiffiffiffiffiffiffiffi adt where a i the thermal diffuivity of the material and Dt ¼ 3 ) lead to a ditance of about 4 mm: the chill effect only affect the kin of the ingot, a hown in Fig Third, the motion of the olid phae, which i compreed and puhed toward the left direction a a conequence of the diplacement of the olid hell induced by the motion of the punch. The punch velocity being 2 mm/, the 6 mm diplacement of the olid hell to the left can be een through the diplacement of the ioline 0 and 0.1 for the liquid fraction on the right ide of the muhy zone. Clearly thi effect i dominant. It alo induce the expulion of the liquid phae, becaue of ma conervation (Eq. (3)). At time 4 after punch tart (Fig. 10(b)), the liquid velocity i more or le maintained, which i normal ince the punch keep moving (total duration of the motion: 6.5 in cae N-1). However, the liquid flow appear to be more uniform through the thickne of the muhy zone than at time 1 (Fig. 10(a)). To undertand thi, one ha to conider again the expreion of coefficient B. It i maximum in the center of the muhy zone (where g i minimum) and continuouly decreae up to the boundary of the muhy zone, where it take the value zero. Therefore, upon combined forced movement of the olid phae and olidification, the maximum value in the center continuouly decreae, and o the module of the gradient of B through the muhy zone. Thi expree that the compreibility of the olid phae become more and more uniform throughout the thickne of the muhy zone. A a conequence, the amplitude of the value of V$v decreae, and o the amplitude of V$v l. Finally, that explain a decreae in the liquid velocity gradient through the thickne of the muh, a hown in Fig. 10(b). Over the punching region, the compreion of the olid phae decreae rapidly with height a it can be een in Fig. 9(b). A a conequence, in thi region, the liquid flow coming from the lower region of the muh i ditributed in the muh according to the permeability, a hown in Fig. 12. That explain higher velocitie in the center, where permeability i the highet, while liquid flow i found much lower at the periphery of the muhy zone, where permeability i lower. In the cae of trial N-9, punching i triggered later in the ingot olidification. A indicated in Table 3, the maximum liquid fraction at ingot center i here limited to Conequently, both the compreibility and the permeability of the olid phae are much lower than in cae N-1. Therefore, relative velocity of the liquid flow with repect to the olid i very low. Thi i illutrated in Fig. 13. The muhy material tend to behave like a homogenized material in which the two phae have approximately the ame velocity field. Thi velocity field i determined by the global incompreibility of the muh and by the diplacement/deformation induced by the punch motion. In cae of N-1 at 4 after punch tart, the maximum relative velocity at the punch height i around 0.02 m 1, while in cae of N-9, it i about m 1 which i the ame order of magnitude of punch velocity. Concerning the divergence of olid velocity, it value in cae of N-4 i and it i in cae of N Macroegregation analyi Fig. 12. Relative velocity field along vertical direction with the ioline of liquid fraction 4 after the tart of punching. According to the numerical imulation decribed in the previou ection, the remaining liquid in the muhy zone, which i enriched with olute, i tranported from the punching tool height to the ingot top. Thi i combined with the compreion of the olid phae, and the local increae of olid fraction at punch height. From thi, a negative macroegregation i expected to take place at the ingot center, a it i effectively hown by the micrograph (ee Fig. 2(a)). Let' analyze the numerical imulation of the formation of

84 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e Fig. 13. Simulation of trial N-9. The punching i triggered at a later tage of olidification. Comparion can be done with Fig. 10 for trial N-1. Same caption a for Fig. 10. macroegregation in more detail before making quantitative comparion. Fig. 14 how the progreion of the macroegregation of manganee upon punching. To undertand thi formation, let' come back to the nature of the governing equation (Eq. (5)) and to it numerical treatment. Firt, in the imulation, the FE node are updated according to the velocity of the olid phae, what i expreed by v mh ¼ v where v mh i the meh velocity. Conequently, the time partial derivative of the olute concentration, which i the firt term in Eq. (5), can be written a (for the ake of Fig. 14. Formation of a negative macroegregation zone for Mn at the core of the ingot for trial N-1 (a) 1, (b) 3 and (c) 6 after punching tart. The ditribution of the relative difference in ma concentration with repect to the nominal one, ð w w 0 Þ=w 0, i repreented, a well a the black iocontour for the liquid fraction.

85 524 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e527 Fig. 15. Comparion of calculated and meaured final egregation profile in cae of trial (1) N-1 and (2) N-9 howing (a) micrograph at punch height, and the relative variation in concentration ðhw i iw i0 Þ=w i0 where w i0 tand for the nominal initial compoition in olute i for (b) Mn and (c) P. Profile are plotted through the thickne of the ingot, at punch height, the mid-ection after punching being around 75 mm from the ingot urface oppoite to the punching ide. EPMA meaurement are indicated by (b) black quare and (c) triangle, while the continuou curve are extracted from the finite element imulation. Error bar indicate tandard deviation. implicity, the olute index i i omitted here): v w vt ¼ v mh w V w $v (18) vt where v mh /vt denote the time derivation with repect to the meh, that i when following grid node (or, equivalently, following the olid phae). After injection in Eq. (5) we get: v mh w ¼ V$ðg vt l D l Vw l Þ fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} diffuion Vw l $ðg l ðv l v ÞÞ fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} advection ðw l V$ðg l ðv l v ÞÞþ w V$v Þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} compreion of olid phae (19) The different contribution to the formation of macroegregation can then be clearly identified: diffuion in the liquid phae, tranport (advection) due to the relative velocitie of liquid and olid phae, compreion of the olid phae. Owing to the very low diffuion coefficient D l, of the order of 10 9 m 2 1, the contribution of diffuion i marginal. The econd contribution i alo quite limited in the preent ituation becaue the gradient of Fig. 16. Relative variation of average olute concentration in the olid phae a a function of the liquid fraction. The dependence i plotted for olute Mn and P, and for two microegregation model: (LR) lever rule and (GS) Gulliver-Scheil. The relative variation of the concentration in the olid phae i calculated a (w,i w i0 )/w i0 where w i0 i the concentration of olute i at the beginning of olidification.

86 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e the liquid concentration i eentially horizontal wherea the relative velocity v l v i eentially vertical. Therefore their dot product remain mall. Finally the only ignificant driving force for macroegregation lie in the lat term aociated with the compreion of the olid phae. Comparing Fig. 14(c) and Fig. 9(a), thi i clearly revealed, a macroegregation principally affect region where liquid fraction i higher than 0.1 and where the maximum value of jv$v j can be found. That alo explain that macroegregation principally form in the punching region and ha a limited extenion apart from thi area. Thi can be een in Fig. 14 where the time evolution of the vertical extent of the egregated zone i limited. Fig. 15 illutrate from a qualitative and a quantitative point of view the final macroegregation in both cae N-1 and N-9. A already commented with Fig. 2, the two micrograph hown in the top line of Fig. 15 indicate a marked egregation in cae N-1 and no clear trend of egregation for trial N-9. Thi i confirmed by the meaurement profile obtained by EPMA, for pecie Mn and P along horizontal line on the tranvere urface hown in Fig. 2, around the punching height. In the cae N-1, a ignificant depletion in Mn and P i meaured in the central region of the ingot: the relative variation in concentration are 15% for Mn and 60% for P. A expected, the egregation i more marked for P, for which the partition coefficient i maller (cf Table 2). Indeed, a lower partition coefficient lead to a higher ratio of liquid v olid concentration within the muh, o that the magnitude of the macroegregation i increaed. In cae N-9, EPMA profile do not how a clear evidence of a central macroegregation, confirming it abence on the micrograph. Regarding the profile iued from the numerical imulation, in cae N-1, they how a central depletion with relative variation of 8% for Mn and 28% for P. In cae N-9, no ignificant macroegregation i calculated. Thu, the numerical imulation reult how the ame trend a the meaurement: more intene central macroegregation for P than for Mn; no ignificant macroegregation in cae N-9. One hould add that a numerical tet wa performed with no punching, i.e. leaving the ingot olidifying with no other external mechanical force than thoe created by hrinkage. No macroegreation wa found, the profile of the olute concentration remaining flat, equal to the nominal alloy compoition, in front of the punching tool. Depite thi good qualitative agreement with meaurement, the intenity of the calculated egregation i found much lower than in reality, roughly by a factor 2. There are everal poible caue for thee difference, which will be dicued in the following ection Dicuion on the amplitude of macroegregation Experimental obervation and two-phae numerical imulation reveal the phenomena taking place in the muhy zone: a central negative macroegregation form under the concurrent effect of compreion of the olid phae, and induced flow of the liquid phae from the punch region toward the top of the ingot. The caue that could explain the too weak intenity of the computed central macroegregation can be claified in two main categorie. The firt one comprie the caue for a poible too low enrichment of the liquid phae. The econd one comprie the caue for inappropriate olid deformation and induced liquid flow. Thee two error ource are dicued hereunder. Enrichment of the liquid phae. Influence of the microegregation model The olute partition between the liquid phae and the olid phae i crucial ince both phae are tranported in thi tudy. The olute enrichment of the liquid phae, or equivalently, the olute impoverihment of the olid phae, i directly linked to the partition coefficient, which are key variable in any microegregation model. Beide the lever rule model, which i ued in the imulation, we conider here the Gulliver-Scheil (GS) model, in which complete mixing i aumed in the liquid phae but no diffuion i aumed in the olid phae. In thi latter cae, and auming a cloed ytem approach (in which the average concentration remain contant all along the olidification for a conidered material point), we have for any olute i [28]: w l;i ¼ hw i ig k i1 l for GS (20) In the cae of the lever rule (LR), we get, under the ame cloed ytem aumption (ee alo Eq. (16)): hw w l;i ¼ i i for LR (21) g l þð1g l Þk i The evolution of the variation of the average concentration in the olid phae a a function of the liquid fraction can be derived eaily from the total ma balance of olute pecie i hw i i¼w l;i g l þð1 g l Þw ;i and Eq (20) and (21). The repective expreion are a follow: w ;i ¼ hw i i 1 gk i l for GS (22) 1 g l k i w ;i ¼hw i i for LR (23) g l þð1g l Þk i Fig. 16 how the evolution of the relative variation of the average olute concentration in the olid phae for Mn and P, (w,i w i0 )/w i0, according to the two microegregation model with w i0 the nominal alloy compoition replacing the average olute content hw i i. The effect of the partition coefficient can be een by comparing the evolution for Mn and P. Additionally, for each pecie, the effect of the microegregation model can alo be een. A expected, the plot for P are found below thoe for Mn. A regard the liquid fraction effect, the lower the liquid fraction i, the higher the relative concentration in the olid phae, and o the lower the amplitude of the relative variation. In the cae N-1, the maximum liquid fraction calculated at punch tart i 0.39, a preented in Table 3. From Fig. 16, the relative variation of the concentration in the olid phae for Mn and P i etimated in the cae of LR a 0.14 and 0.50, repectively. Thi lead to a ratio between thee variation of about 3.6, at punch tart. Looking now at the imulation reult, it i found that the maximum relative variation of concentration, after punching, are and for Mn and P, repectively (ee Fig. 15). Thu, the ratio of thee variation i around 3.7, very cloe to the initial ratio before punching. Thi i fully conitent, a during the 6 of the punching tage, the enriched liquid phae i imply tranported through the muh: the ratio P/Mn i kept unchanged. Beide, it i worth noting that thi ratio i found around 4 for the EPMA meaurement (0.15 and 0.60 for Mn and P, repectively). Thi good agreement on the ratio between the amplitude of egregation for Mn and P indicate that imulation capture well the relative egregation for each pecie. Another iue i the concentration in the olid phae itelf. At punch tart, according to Table 3, the liquid fraction i g l,tart ¼ The average Mn concentration in the olid phae i w,tart ¼ 0.86w 0. Thi latter value i computed from Eq. (23) or deduced from Fig. 16 following the LR approximation. Similarly, the average Mn

87 526 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e527 Table 4 Summary of enitivity tet. N-1 i the reference imulation while S-1, S-2, and S3 are three enitivity tet. The underline value for tet S-1, S-2 and S-3 identifie the modified parameter with repect to N-1. N-1 S-1 S-2 S-3 Multiplier applied to the vicoplatic conitency K over the olidu temperature Multiplier applied to the compreibility coefficient B Secondary dendrite arm pacing l 2 [mm] Maximum relative variation in concentration for Mn Relative variation with repect to reference e þ21% þ18% þ8% concentration in the liquid phae can be deduced uing Eq. (21), which yield w l,tart ¼ 1.22w 0. From the imulation, the liquid fraction after punching wa computed: g l,end ¼ Note that it correpond to an increae by 36% in olid fraction, i.e. (1e0.17)/ (1e0.39) ¼ The imulation alo how that thi compaction of the olid phae i aociated with an almot pure vertical tranport of the fluid that can till circulate in the muhy zone. Evaluation of the Mn egregation index after punching with repect to the nominal compoition w 0, ð w w 0 Þ=w 0, i then poible conidering the total ma balance of olute. Auming the compoition of the two phae unchanged during the hort punching time (6.5 ), we have, at the end of punching: w end ¼ g l;end w l;tart þ 1 g l;end w ;tart (24) Thi lead to a egregation index after punching equal to 0.08 for Mn. Thi value i very cloe to the minimum read for the computed relative Mn concentration profile which i diplayed in Fig. 15(b1), howing that the etimation provided by the preent imple analyi are found quite cloe to the more ophiticated numerical imulation. A tated before, thi reult yet underetimate macroegregation, the minimum exhibited in Fig. 15(b1) for the meaured egregation profile for Mn being It hould be noted here that it wa checked that numerical imulation how almot no evolution of the egregation profile between the end of punching and complete olidification. Actually, after punching the permeability i low enough to prevent any ignificant fluid flow. Therefore, the imple etimation calculated above after punching can be reaonably compared to the meaurement done on fully olidified ingot. Let' now examine what would be the ituation conidering GS intead of LR. Auming the ame initial liquid fraction at punch tart, g l,tart ¼ 0.39, and applying Eq (20) and (22), the Mn concentration in the liquid and olid phae would be w l,tart ¼ 1.32w 0 and w,tart ¼ 0.79w 0, repectively. Further auming the ame increae of olid fraction e through compaction e by 36% in the muhy zone, leading to g l,end ¼ 0.17, the average Mn concentration obtained by Eq. (24) yield now w end ¼ 0:88w 0, that i a egregation index after punching equal to Thi value come cloer to obervation. A a concluion, it can be aid that the choice of the microegregation model ha a ignificant influence on the intenity of the macroegregation (increaing it by 45% in the cae of Mn). Thu, it might be intereting to conider more ophiticated microegregation model, like thoe taking into account partial equilibrium with peritectic tranformation [29]. In addition, undercooling effect could be taken into account, providing a more accurate olidification path and initial liquid fraction before punching. Influence of olid compreion and aociated liquid flow The compreion of the olid phae and the liquid flow obviouly play a prominent role in olute ditribution, a already underlined when commenting Eq. (19) governing the evolution of average olute concentration. Conequently, a few parameter are expected to have an impact on the amplitude of egregation. A firt et of parameter conit of the contitutive parameter of the olid phae, and among them particularly the vicoplatic conitency K and the compreibility parameter B. Indeed a lower value of the conitency, or a greater value of B, would favor the compreion of the olid phae and then increae the amplitude of macroegregation. A econd et of parameter conit of thoe governing the permeability of the olid network and o the ditribution of liquid flow through it. Auming Eq. (6) for the iotropic permeability, the only parameter i in fact the econdary dendrite arm pacing l 2. In order to tet the influence of thoe parameter, a enitivity analyi i conidered. It conit of three tet for which, all other thing being equal, the value of one of the parameter K, B and l 2 ha been changed. The conidered change for each of the coefficient are indicated in Table 4. They are all expected to induce a more marked negative egregation in the ingot center: vicoplatic conitency divided by two; compreibility coefficient doubled; econdary dendrite arm pacing multiplied by three. The lat line of the Table contain the value of the maximum (in abolute value) relative variation of average Mn concentration obtained in each imulation. A expected, the three tet imulation how a more evere central negative macroegregation, which again how a good qualitative repone of the two-phae FE calculation. In comparion with the reference imulation N-1, the magnitude of Mn egregation index i increaed by 21% in imulation S-1 (influence of K), 18% in imulation S-2 (influence of B) and by 8% in imulation S-3 (influence of l 2 ). Conidering the large variation teted for thoe parameter, it appear that their influence i effective, but probably lower than the impact of the choice of the microegregation model, a dicued above. 5. Concluion An ingot punch preing tet ha been developed to better undertand the phenomena affecting the olid hell and the central muhy zone of teel emi-product when croing the econdary cooling ection during continuou cating. The tet conit of the punching of one ide of the olid hell of a partially olidified teel ingot by an external tool. The preent tudy i focued on the formation of macroegregation induced by thi punching operation. The deformation of the olid hell and of the central muhy zone are imulated numerically. The mechanic of the muhy zone i addreed by a two-phae approach, in which the compreible deformation of the olid phae and the liquid flow are olved concurrently, a they are in cloe interaction. Coupling with the numerical olution of olute ma tranfer induced by liquid flow i achieved, thu imulating the occurrence of macroegregation. A 2D FE method i ued to olve all conervation equation in a ingle model. Two experimental tet are elected, coniting of two ingot of the ame teel grade, but punched at different intant, correponding to different progreion of their olidification prior to

88 T. Kohikawa et al. / Acta Materialia 124 (2017) 513e punching. Simulation with the two-phae FE model demontrate the ability to capture the eential phenomena that drive macroegregation in the ingot punch preing tet: compreion by mechanical deformation of the porou olid phae combined with liquid flow through the permeable medium. The compreion and it reulting permeability decreae are more marked for the ingot where olidification wa le advanced at the onet of punching. It lead to the formation of intene negative central macroegregation in the upper half part of thi ingot. For the econd ingot punch preing tet, deformation wa alo imulated but it did not generate ignificant macroegregation a olidification wa more advanced at the onet of preing, thu hindering olute tranport by liquid flow in the muhy zone. Quantitative comparion between imulated and meaured compoition profile how that the ponge-like model [9] for the mechanical behavior of the muhy zone and the propoed two-phae numerical reolution [10,11] contitute a tep forward on modeling deformation-induced macroegregation during teel cating procee by combining coupled phenomena. Thi work contitute the firt validation of modeling effort tarted in the ixtie [1] to decribe the role of thermomechanical deformation on the formation of macroegregation. The variou exiting model developed for teel [6e9,12e18] lack the effective coupling between olid deformation and fluid flow demontrated in the preent contribution. However, the intenity of the predicted macroegregation remain lower than the one meaured in reality. Through the dicuion of the reult and in the light of enitivity tet, it appear that the oberved difference are mainly due to the type of microegregation model and to the contitutive model and parameter controlling the compreion of the olid phae in the muhy zone. Regarding the firt point, the ole lever rule ha been ued in the preent tudy. The ue of the Gulliver-Scheil or alternative microegregation model could increae the intenity of calculated macroegregation. A for the mechanic of the muhy zone, the identification of compreibility parameter by eparate elementary rheological tet appear to be eential a they alo influence ignificantly the intenity of macroegregation. Macroegregation remain the main defect to be controlled during CC. The preent contribution report effort to reach quantitative prediction. Regarding applicability to teel continuou cating, it wa already hown to be effective in 2D and for a binary alloy, allowing the analyi of oft reduction operation carried out on teel lab at the end of the econdary cooling zone [11]. In addition to the implementation of better microegregation model, extenion to 3D analyi i neceary to addre the continuou cating of long product, uch a billet or bloom, or to capture with better accuracy edge effect, e.g. cloe to the two lateral narrow face in the cae of lab. The achievement of uch model i of paramount importance for the undertanding and control of the formation of central macroegregation. It i expected that enhanced modeling capabilitie will benefit optimization of current indutrial practice by increaing yield through reducing macroegregation defect. Achieving more quantitative modeling of macroegregation and other defect (e.g., hot tearing [19]), ha alo potential for exploring numerically route to cat high-alloyed teel with variou grade and to reduce the time to production of new alloy. Reference (1967) 1449e1461. [2] M. Wu, A. Ludwig, A three-phae model for mixed columnar-equiaxed olidification, Metall. Mater. Tran. A 37 (2006) 1613e1631. [3] J. Li, M. Wu, A. Ludwig, A. Kharicha, Simulation of macroegregation in a ton teel ingot uing a three-phae mixed columnar-equiaxed model, Int. J. Heat Ma Tranf. 72 (2014) 668e679. [4] M. Zaloznik, H. Combeau, An operator plitting cheme for coupling macrocopic tranport and grain growth in a two-phae multicale olidification model: Part I e model and olution cheme, Comput. Mater. Sci. 48 (2010) 1e10. [5] T.-T.-M. Nguyen, H. Combeau, M. Zaloznik, M. Bellet, Ch.-A. Gandin, Multicale finite element modelling of olidification tructure by a plitting method taking into account the tranport of equiaxed grain, in: H. Yauda (Ed.), Proc. MCWASP XIV, 14th Int. Conf. on Modeling of Cating, Welding and Advanced Solidification Procee, Awaji Iland, Japan, June 2015, IOP Conference Serie: Material Science and Engineering, vol. 84, 2015 article , 10 page. [6] K. Miyazawa, K. Schwerdtfeger, Macroegregation in continuouly cat teel lab: preliminary theoretical invetigation on the effect of teady tate bulging, Arch. Eienhuttenwe 52 (1981) 415e422. [7] T. Kajitani, J.M. Drezet, M. Rappaz, Numerical imulation of deformation induced egregation in continuou cating of teel, Metall. Mater. Tran. A 32 (2001) 1479e1491. [8] J. Domitner, M. Wu, A. Kharicha, A. Ludwig, B. Kaufmann, J. Reiter, T. Schaden, Modeling the effect of trand urface bulging and mechanical oft reduction on the macroegregation formation in teel continuou cating, Metall. Mater. Tran. A 45 (2014) 1415e1434. [9] G. Leoult, Ch.-A. Gandin, N.T. Niane, Segregation during olidification with pongy deformation of the muhy zone, Acta Mater. 51 (2003) 5263e5283. [10] V.D. Fachinotti, S. Le Corre, N. Triolet, M. Bobadilla, M. Bellet, Two-phae thermo-mechanical and macroegregation modelling of binary alloy olidification with emphai on the econdary cooling tage of teel lab continuou cating procee, Int. J. Num. Method. Eng. 67 (2006) 1341e1384. [11] M. Bellet, A two-phae 2D finite element model for macroegregation in teel continuou cating, in: H. Jone (Ed.), Proceeding of the 5th Decennial Int. Conf. on Solidification Proceing, The Univerity of Sheffield, Sheffield, United Kingdom, 2007, pp. 424e427. [12] B. Rivaux, Modeliation multi - echelle de tructure de grain et de egregation dan le alliage metallique, Ph.D. Thei, MINES PariTech, Pari, France, [13] H. Sato, T. Kitagawa, K. Murakami, T. Kawawa, Tetu Hagane 61 (1975) S471. [14] K. Miyamura, A. Ochi, K. Kanamaru, N. Kaneko, Tetu Hagane 62 (1976) S482. [15] K. Marukawa, M. Kawaaki, T. Kimura, S. Ihimura, Tetu Hagane 64 (1978) S661. [16] K. Narita, T. Mori, K. Ayata, J. Miyazaki, M. Fujimaki, Tetu Hagane 64 (1978) S152. [17] Y. Sugitani, M. Nakamura, H. Kawahima, M. Kawaaki, Tetu Hagane 66 (1980) S193. [18] M. Wintz, M. Bobadilla, M. Jolivet, Hot cracking during olidification of teel - effect of carbon, ulphur and phophoru, La Rev. Metall. 4 (1994) 105e114. [19] T. Kohikawa, M. Bellet, Ch.-A. Gandin, H. Yamamura, M. Bobadilla, Metall. Mater. Tran. 47A (2016) 4053e4067. [20] J. Ni, C. Beckermann, A volume-averaged two-phae model for tranport phenomena during olidification, Metall. Mater. Tran. B 22 (1991) 349e361. [21] M. Bellet, V.D. Fachinotti, ALE method for olidification modelling, Comput. Method. Appl. Mech. Eng. 193 (2004) 4355e4381. [22] M. Bellet, O. Jaouen, I. Poitrault, An ALE-FEM approach to the thermomechanic of olidification procee with application to the prediction of pipe hrinkage, Int. J. Num. Method. Heat. Fluid Flow 15 (2005) 120e142. [23] P. Shi, TCS Steel/Fe-alloy Databae V6.0, Thermo-Calc Software AB, Stockholm, Sweden, [24] Thermo-Calc, TCCS Manual, Thermo-Calc oftware AB, Stockholm, Sweden, [25] Iron and Steel Intitute of Japan, Mechanical Behavior in Continuou Cating, ISIJ, NCID: BN (Citation Information by National intitute of informatic: CiNii). [26] A. Palmaer, A. Etienne, J. Mignon, Calculation of the mechanical and thermal tree in continuouly cat trand, Stahl Eien 99 (1979) 1039e1050. [27] Britih Iron and Steel Reearch Aociation, Phyical Contant of Some Commercial Steel at Elevated Temperature (Baed on Meaurement Made at the National Phyical Laboratory, Teddington, United Kingdom), Butterworth Scientific Publication, London, United Kingdom, [28] J.A. Dantzig, M. Rappaz, Solidification, EPFL Pre, [29] T. Kohikawa, C.A. Gandin, M. Bellet, H. Yamamura, M. Bobadilla, Computation of phae tranformation path in teel by a combination of the partial- and para-equilibrium thermodynamic approximation, ISIJ Int. 54 (2014) 1274e1282. [1] M. Fleming, G. Nereo, Macroegregation: Part I, Tran. Metall. Soc. AIME 239

89 75

90 76

91 Chapter V Summary of the main reult 77

92 1. Summary on microegregation modelling A preented in Chapter II, a numerical cheme concerning microegregation model for a general cooling equence during olidification ha been developed. Several combination of the above approximation are poible: - Lever Rule (LR). It obviouly correpond to full thermodynamic equilibrium tranformation path and may be applied from a temperature higher than the liquidu temperature up to room temperature. - Gulliver Scheil (GS). All intertitial and ubtitutional element in all olid phae are frozen, not permitting the peritectic reaction to take place. Becaue uch approximation i difficult to maintain up to low temperature, a critical fraction of liquid (10-6 ) i choen to top the tranformation, below which the average compoition of all phae i kept contant. - Partial Equilibrium (PE). Conidered alone, it conider full equilibrium for intertitial while ubtitutional element are frozen. Thi doe change the olidification path compared to GS, yet till not permitting the peritectic reaction to occur. - Partial Equilibrium plu Lever Rule (PE+LR). It keep the PE approximation while handling the peritectic reaction by uing a imple LR approximation among the δ-bcc and γ-fcc olid phae [Chen 2006]. Note that the γ-fcc to α-bcc olid tate tranformation i alo conidered in uch ituation. - Partial Equilibrium plu Para Equilibrium (PE+PA). Thi i the mot advanced configuration where the δ-bcc to γ-fcc peritectic tranformation and the γ-fcc to α-bcc olid tate tranformation are both conidered. It i applied to multicomponent teel. Analye include - unidirectional olidification experiment conducted for Fe-C-Mn-S-Si-P-Al teel, - microtructure invetigation with image analye at certain ection perpendicular to withdrawal direction in which one can ditinguih the remaining liquid quenched into a fine microtructure from the olid phae formed before quenching, - Electron Probe Micro-Analyi (EPMA) on a teel ample from the centre of a 450 kg ingot cating (0.5 m width, 0.16 m thickne and 0.75 m height), - a numerical cheme for Lever Rule (LR), Gulliver-Scheil (GS), Partial Equilibrium (PE), Para Equilibrium (PA) and combination of PE+LR and PE+PA with Thermocalc and the databae TCFE6, - Computation of tranformation path for two experiment. Main finding are a follow: 1- A new combination of the Partial- and Para- Equilibrium (PE+PA) thermodynamic approximation i propoed for the tudy of phae tranformation in teel, from the liquid tate to room temperature. 2- Calculated olidification path for multi-component alloy are dicued with repect to the peritectic tranformation. It i found that, in cae of PE+PA, the olidification path doe not differ ignificantly from PE+LR approximation. 78

93 Temperature ( C) LR PE+PA EX Solid fraction Figure 1 Three olidification path for targeted alloy compoition, hown in Table 1 correponding to (blue curve, LR) the Lever Rule approximation, (red curve, PE+PA) a combination of Partial Equilibrium and Para Equilibrium approximation and (green mark with error bar, EX) Experimental invetigation. 3- Difference with experimental reult are dicued baed on analye of a ample proceed by Bridgman olidification, howing the need for more accurate experimental reult and removal of everal modeling hypothee (abence of microtructure kinetic and no treatment of limited diffuion in all phae). 4- Regarding the olid tate phae tranformation, PE+PA i found to provide the mot realitic compoition profile of ubtitutional element and evolution of the phae fraction. PE+PA i therefore recommended a the tandard et of thermodynamic approximation to be ued for the tudy of phae tranformation in multicomponent teel. The obtained olidification path i hown in Figure 1. It how how the olidification interval i modified by the variou thermodynamic approximation. A a conequence, the Brittle Temperature Range (BTR), defined where fraction of olid increae from 0.9 to 0.99, i ignificantly changed a given in Table 2. A the BTR alo refer to the olidification interval during which train accumulate up to form the hot crack criterion, it i ued for hot tearing modeling. 79

94 Table 1. Target compoition and experimental condition. Article I. Compoition (ma%) Diplace-m ent Velocity C Si Mn P S Al (mm) (mm -1 ) Target to to 2.0 Table 2. Extent of the brittlene temperature range (BTR) calculated with (LR) the Lever Rule approximation, (PE+PA) the Partial- plu Para-Equilibrium approximation and (EX) deduced from meaurement. The correponding olidification path and exact alloy compoition are reported in Figure 1. LR PE+PA EX BTR extent (K)

95 2. Summary on hot tearing prediction A preented in Chapter III, a hot tearing tudy i performed during teel olidification with ingot punch preing tet. A chematic of the tet i hown in Figure 2. It ha been deigned to mimic the thermomechanical olicitation taking place during CC proceing of teel; for intance, with repect to the evolution of the cooling rate and deformation. The ize of the ingot i 0.75 m height, 0.5 m width and 0.16 m thick. It ma i approximately 450 kg. At a precie time after filling, the upper part of one mold part that define the two larget ingot urface (a top right mold part in Figure 2) i moved upward. The lateral urface of the ingot i then in direct contact with the air. Note in Figure 2 that thi ide of the mold i made of two part poitioned one on top of the other. The bottom part could thu be kept at the ame place while the top part i removed. A punching tool i then put into contact with the lateral vertical free urface of the ingot. It conit of a horizontal 0.12 m-diameter 0.3 m-long cylinder with length centered on the main vertical face of the ingot, it longitudinal axi aligned with the Z direction at height 0.45 m from the bottom of the ingot. The tool velocity and diplacement are controlled by mean of a hydraulic ytem and recorded during the tet. The reaction force i alo meaured uing the time evolution of the hydraulic ytem preure. The nominal compoition i meaured in the ladle before filling and i reported in Table 1. 81

96 Figure 2. Schematic of half of the hot tearing experiment etup developed at Nippon Steel & Sumitomo Metal Corporation howing configuration (a) prior to the filling tage and (b) at the onet of the punching tage after filling, partial olidification with the mold configuration hown in (a), removal of the top-right mold and horizontal diplacement of the cylindrical punching tool to come into contact with the ingot. The coordinate ytem i defined in (a). The Y = 0 m plane i defined by the top urface of the bottom mold part, alo defining the lowet ingot urface in (b). The Y-axi, oppoite to the gravity vector, i directed toward the top of the ingot at mid-width and mid-thickne. The X-axi i directed toward the fixed left-mold urface. Thermocouple, TC-1 and TC-2 are located at (X, Y, Z) poition (0, 0.43, 0.06) m and (0.02, 0.43, 0.16) m, repectively. A chematized, meaurement i alo performed with a pyrometer (PM) around the punched urface of the ingot, before and after punching. 82

97 Time () (a) T (K) (b) I (MPa) K 1823 MPa 10 (c) WYSO F HT K 1823 MPa Figure 3. Predicted reult for one of the tet with the PE+PA olidification path (Figure 1) howing (a) the temperature field, (b) the region with poitive value of the firt principal tre, i.e. under tenion, the grey region within the ingot being under compreion, and (c) the hot tearing criterion F. The top and bottom line how the different field jut before (499 ) and after (511 ) WYSO HT punching, repectively. Black vertical line indicate io-liquid fraction contour from 0 to 1 with 0.1 tep. Main finding are a follow: - the HTC i able to predict the rik of crack formation, provided that the brittlene temperature range (BTR) i adequately reproduced, - excellent correlation i found between meaurement and imulation concerning the poition hown in Figure 4 which i baed on 3D thermomechanical imulation hown in Figure 3, - the role of the BTR i evidenced when drawing the HTC profile along the ingot thickne hown in Figure 4, thu revealing the critical effect of the elected olidification path, - prediction in a cumulated train veru train rate perpendicular to the temperature gradient, a well a the reaction force, doe not evidence large variation with the BTR hown in Figure 5, nor doe the thermal hitory, 83

98 Number of hot tear (-) LR PE+PA EX Crack F WYSO (-) HT Ditance from the ingot urface of the punch ide (m) Figure 4. Magnified micrograph for one of the tet on the top. In all cae, hot tear are oberved and circled with dahed red contour. Alo, the view being limited, not all crack found are hown. Profile along the ingot thickne for (black curve, right-hand-ide legend) the WYSO number of hot tear and (colored curve, left-hand-ide legend) the hot tearing criterion F HT on the bottom. Reult are preented a a function of the 3 olidification path preented in Figure 1. 84

99 Cumulated train (-) 0.03 EX LR Higher crack rik N-2 N-5 N PE+PA N-3 N LR PE+PA EX Le crack rik E-5 1E-4 1E Figure 5 Computed cumulated train Perpendicular train rate ( -1 ) BTR ˆ dt a a function of the train rate ˆ perpendicular to the temperature gradient calculated when paing through the BTR. WYSO Value for the different tet are taken from the F HT profile at the punching height, an example being hown in Figure 4. The three continuou curve indicate the train limit c ˆ (Chapter III Eq. (13)). BTR effect i clearly een a the three train limit curve, wherea the obtained cumulated train a a function of the train rate i not o different in each other with the ame tet cae. 85

100 3. Summary on macroegregation calculation A preented in Chapter IV, a macroegregation calculation ha been carried out during teel olidification with ingot punch preing tet. The detail of the tet i explained in the previou ection. The ingot punch preing tet ha been developed to better undertand the phenomena influencing the olid hell and the central muhy zone of teel emi-product when croing the econdary cooling ection during continuou cating. The tet conit of the punching of one ide of the olid hell of a partially olidified ingot, by an external tool. The tudy i now focued on the macroegregation which i induced by thi punching operation. The deformation of the olid hell and of the central muhy zone are imulated numerically, uing a 2D finite element method. The mechanic of the muhy zone i addreed by a "two-phae" approach, in which the compreible deformation of the olid phae and the liquid flow are olved concurrently, a they are in cloe interaction. The thermo mechanical imulation reult (Figure 6) how the deformation of the olid phae and the aociated liquid flow, a calculated in the "two-phae" approach for the imulation of one of thee tet. The deformation of the olid hell by the punch induce a compreion of the olid phae in the muhy zone (not viible in Figure 6), which in turn induce liquid flow. For evident reaon (global quai incompreibility of the muhy material, cloed muhy pool at the bottom of the ingot), liquid flow i necearily oriented upward. In thi precie tet, olidification i le advanced: the alloy i till in the muhy tate in it top central region. Thi make poible uch an upward liquid flow, with liquid expulion at the top urface into the primary hrinkage cavity: the expelled melt cover the already formed olid hell. The imulated liquid flow can be een in the central part of Figure 6. Experimental evidence of uch an expulion of liquid flow i provided by the micrograph on the right part of the figure, in which a reidue of enriched (darker) expelled liquid i preent at the bottom of the V-hape hrinkage cavity. The imilarity of the hape of the primary hrinkage cavity can be noted between the imulation and the micrograph, expreing the good repreentativity of the thermo-mechanical imulation. Figure 7 reveal in more detail what happen in the muhy zone at the height of the punch tool. Due to the motion of the punching tool, the olid hell in it vicinity i diplaced a hown by the horizontal component of the olid velocity (Figure 7(a)). The reulting compreion of the olid phae within the muhy zone i expreed by negative value of the divergence of the olid velocity field, v, diplayed in Figure 7(b). A relative liquid flow i thu created a hown in Figure 7(c). The compreion of the olid phae primarily affect the region of the muh for which the liquid fraction i higher than 0.1. Indeed, when liquid fraction decreae, the compreibility of the olid phae decreae a well, tending toward a fully dene olid material. Thi hinder the compreion of the olid phae at the periphery of the muhy zone, where olidification i more advanced. 86

101 (a) 0.5 g l 0.0 (b) (c) Figure 6. Simulated and oberved ingot hape howing (a) a global view of the deformation and ditribution of olidification in the 0.16 m width ingot ection at time (0.4 after punch tart) for one of the trial tudied and (b) a zoom in the top region with hrinkage pipe to be compared with (c) obervation. The imulation alo how (color map) the ditribution of the liquid fraction and (black contour) iofraction of liquid, a well a (arrow) the relative velocity of the liquid phae with repect to the olid one, v l - v. The darker region at the center of the micrograph i due to expellation of liquid in the primary hrinkage cavity a a reult of punching. Figure 7. Central region of ingot, 6 after punch tart, howing (a) the ditribution of the horizontal component of the velocity of the olid phae v, (b) v with evidence of negative value in the muhy zone at the height of the punch tool and (c) the vertical component of the relative velocity of the liquid phae with repect to the olid one, v l - v. The imulation alo how (black contour) iofraction of liquid, from 0 to 0.3, and (arrow) the vector field v l - v. Two experimental tet are conidered, coniting of two ingot of the ame teel grade, but punched at different intant, that i for two different advancement of their olidification. A hown in Figure 8, the two-phae finite element imulation i able to account for the eential driving force for induced macroegregation. The general trend are reproduced: formation of a negative macroegregation in one ingot only, which only affect the uperior part of the central region of the ingot. Thi illutrate the 87

102 relevance of the "ponge-like" model for the mechanical behavior of the muhy zone and of the propoed "two-phae" numerical reolution. However, the intenity of the predicted macroegregation remain ignificantly lower than the one meaured in reality. Through the dicuion of the reult and in the light of ome enitivity tet, it appear that the oberved difference are mainly due to the type of microegregation model and to the contitutive model and parameter controlling the compreion of the olid phae in the muhy zone. Regarding the firt point, the ole lever rule ha been ued in the preent tudy. The ue of the Gulliver-Scheil or alternative microegregation model could increae the intenity of calculated macroegregation. A for the mechanic of the muhy zone, the identification of compreibility parameter by eparate elementary rheological tet appear to be eential a they alo influence ignificantly the intenity of macroegregation. 88

103 P relative concentration P relative concentration Mn relative concentration Mn relative concentration Trial N-1 Trial N-9 10mm 10mm Meaured Calc Meaured Calc Meaured Calc Ditance from the oppoite ingot urface to the punching tool (m) Meaured Calc Ditance from the oppoite ingot urface to the punching tool (m) Figure 8 Comparion of calculated and meaured egregation profile in cae N-1 and N-9 (left and right part of the figure, repectively). The top line how two micrograph at punch height. The two bottom line how the relative variation in concentration for Mn and P, repectively: ( wi wi 0) wi 0, where w i0 tand for the nominal initial compoition in olute i (Mn or P). Profile are plotted through the thickne of the ingot, at punch height, the mid-ection after punching being around 75 mm from the ingot urface oppoite to the punching ide. EPMA meaurement are indicated by black quare and triangle, while continuou curve are for finite element calculation. Error bar indicate tandard deviation. 89