Application of cellular automata method to simulate the hot deformation behavior of a dual phase titanium alloy

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1 TECHNISCHE MECHANIK, 30, 1-3, (2010), 1-14 sumitted: Octoer 31, 2009 Appliction of cellulr utomt method to simulte the hot deformtion ehvior of dul phse titnium lloy R. Bstji, N. Boutn, P. Bocher, M. Jhzi A mthemticl model sed on the Eshely theory nd using the cellulr utomt method (CA) is developed to study microstructure evolution during forging of two-phse lloys. Specificlly, the method is pplied to the cse of ner lph titnium lloy, IMI834, minly used in the high-pressure prts of xil compressors of the most recent gs turine engines. Viscoplstic lws re used to descrie the flow ehvior of this lloy for equixed microstructures. The results otined y CA models cn resonly well descrie the mteril ehvior oth in the high temperture et phse nd in the two phse lph + et region. Experimentl dt of hot compression in the pure et field re exploited in order to quntify the thermomechnicl ehvior of this phse, then processing in the lph + et field re used to clculte phse ehvior. Comprisons etween the numericl modelling results nd the experimentl ones indictes tht the CA method hs good cpility for predicting the influence of forging conditions on the locl nd glol mechnicl chrcteristics of IMI834 lloy. 1. Introduction In titnium lloys, locl microstructurl chnges (Germin et l. 2005) cn generte dmge in service with disstrous consequences (Bch nd Evns 1983; Toul et l. 2009; Ut 2009). This prolem ecomes of prmount importnce when it comes to Ti lloys used for erospce pplictions nd results in premture removl of prts from service s precutionry mesure. It ws shown tht these heterogeneities re generted during deformtion (Germin 2007) nd tht proper thermomechnicl processing cn reduce or even cncel their detrimentl effects (Ut 2009). The lrge numer of the process prmeters tht could contriute to control the genertion nd olitertion of these texture heterogeneities, justify etter understnding of the reltionship etween the micro nd mcro ehvior of titnium lloys during processing. In recent yers with the dvent of powerful numericl methods it hs ecome possile to simulte multivrile mnufcturing processes nd predict the ehvior of the component. The finite element method, self consistent models (Semitin et l. 2002) nd recently the cellulr utomt method (CA) (Boutn et l. 2008) re the mjor numericl methods used to crry out the ove mentioned studies. All three methods llow to study oth locl nd glol ehvior of the mteril, n dvntge tht is sent in ll experimentl methods. Among these methods, the CA method hs the gret dvntge of eing more flexile with much shorter clcultion times (Boutn et l. 2008). In the present work, computer code using the CA method sed on Eshely s (1957) pproch is developed to study the mechnicl ehvior of heterogeneous two phse mteril. The flow ehvior prediction is pplied to the hot deformtion of ner-lph titnium lloy nd the effect of phse proportion nd phse morphology is studied. The selected lloy is the IMI 834 which is used in the compressor section of ircrft engines. The forging routes for such prts require severe deformtions in the two-phse field to ttin the trgeted microstructure nd mechnicl properties. The ojectives of the present pper re then threefold: to develop mthemticl model using the cellulr utomt method sed on Eshely s mechnism to simulte the forging process of IMI 834 with equixed microstructure; to generte the nonliner viscoplstic prmeters of the IMI 834 lloy; to pply the CA method to study the influence of processing conditions t the locl level nd its impct on producing microstructurl heterogeneities during the forging process. 1

2 2. Description of mthemticl model 2.1. Cellulr utomton pproch Cellulr Automt re dynmicl systems of cells whose ehviors re defined y locl reltions (Wolfrm (1984)). Schemticlly, CA cn e presented s periodic spce grid constructed nd run in ccordnce with the following steps: Cells nd their relted chrcteristics (sttes) re defined. Reltionships etween neighors re defined. In the present model 2D hexgonl rry is generted nd it is ssumed tht ech cell cn only communicte with its first six neighors. In order to represent schemticlly microstructure, ech cell is ssocited on one-to-one sis with grin, sugrin or homogeneous phse domin nd its respective physicl properties. Cell trnsition rules governing the evolution of the cell from one stte to nother re specified. As the CA is timed system evolution rules re to e defined: In the present work the stte of ll cells chnge simultneously ccording to the trnsition rule (Wolfrm (1984) nd Smith (1994)) Genertion of the initil microstructure The genertion of microstructure in CA permits the derivtion of results for vrious topologies. The cells pertining to ech phse cn e rndomly distriuted, ligned or pcked in clusters nd these clusters cn lso e rndomly distriuted or rrnged ccording to specific order (Briottet nd Montheillet 2005). In the present work, ggregte of grins mde of two distinct phses, nmely α nd β re rndomly generted. Ech grin is ssocited with cell of two-dimensionl cellulr utomt. The numer of neighors to ech cell is ssumed to remin constnt nd equl to six. The nerest neighor reltions re displyed in the form of the hexgonl rry Appliction of the Eshely theory in CA Most of models tht cn give n estimte of the mechnicl ehvior of multiphse mterils t the locl level re sed on Eshely s theory. Eshely (1957) studied n isotropic ellipsoidl inclusion with the elstic constnt different from those of the isotropic infinite mtrix, suject to homogeneous deformtion t infinite e (Figure 1). His pproch consists of determining the tensor of deformtion in the inclusion y sumitting homogeneous medium to specific deformtions (eigenstrins) without knowing the velocity field in the mtrix. The specific deformtions re usully non-elstic deformtions. Figure 1. Schemtic representtion of Eshely pproch Initilly the pproch of Eshely ws relted to elstic mterils, ut ws lter generlised to nonliner viscoplstic components (Montheillet nd Gilormini 1996). By considering the cse of unixil compression, the 2

3 locl strin rte written s I e& pplied to the inclusion I nd resulting from the remote strin rte e& t the infinite cn e e & I = d Ie& (1) c where d I is the strin rte locliztion fctor. The locl strin rtee& ij of given cell C (identified y the i, j indices in the rry) is clculted in the present model using the following eqution e & d c c ij = e& (2) d c where d is the strin rte locliztion fctor in the cell nd c d c the verge strin rte locliztion fctor in the entire rry. For n equixed inclusion, the strin rte locliztion fctor cn e clculted using the following eqution = 1 ( 1 + ) { 1 - tnh[ ( + c)( S c - 1) ]} + + ( d - 1) tnh[ ( + e)( S - 1) ] d c (3) c,, c, d nd e vry with the strin rte sensitivity coefficient m (Briottet nd Montheillet 2005). As in the present cse oth phses hve m vlues close to 0.2, the coefficients will e 0.69, 0.1, 0.863, 7.67 nd , respectively. S c is the consistency rtio defined y the following eqution where prmeter k c is the viscosity of the cell nd k v is given y the following eqution k c  c = (4) kv k v is the verge viscosity of the neighors of the involved cell. The Ê [ 1 ˆ k Á v 2 24r Ë 6 + ( 3-2r + 5( 1- f )( r -1)) + ( 3-2r + 5( 1- f )( r -1)) ] k V = v (5) where f v is the volume frction of the phse in the neighorhood (6 neighors surrounding the cell), nd r is the viscosity rtio etween phse nd phse k / k ). It is importnt to mention tht in the ( present work, the verge strin rte locliztion fctors d c is clculted y the mixture low. After clculting the locl strin rte for ech cell, the locl stress s c is derived from the nonliner viscoplstic lw of ech phse using the following eqution (Oikw et l. 1994) s p m ( K exp ( Q RT ) e& ) V = p (6) where p represents the phse ( or ), is the constnt of the perfect gs, Kelvin, nd K p is constnt relted to the chemicl composition of the phse, R Q p is the ctivtion energy of the phse, T is the deformtion temperture in m p is the strin rte sensitivity of the phse. 3

4 3. Clcultion of the volume frction of nd phses for IMI 834 The viscoplstic prmeters of the phses nd re otined from compression tests which hve een mde in pure nd in + domins. The s-received microstructure of the specimen is composed of 20% of equixed phse within trnsformed mtrix (lmellr phse) (Figure 2). β trnsformed α equixed Figure 2. As-received microstructure with 20% equixed primry in trnsformed mtrix The compression tests re conducted t tempertures of 950 C to 1100 C, with strin rtes of 0.1 s -1 nd 1 s -1. Deformtions re rnging from 0.7 to 1.4. The smples chosen re cylindriclly shped 76 mm in dimeter nd 11.4 mm in length. The chemicl composition of the smples used in this study is given in Tle 1. Tle 1. Chemicl composition of titnium lloy IMI 834 Element Al Sn Zr N Mo Si C Ti Content lnce (%) All smples re heted t the deformtion temperture (T def ) for 15 min, nd deformed. Schemtics of the therml cycles for the tests re shown in Figure 3. Typicl stress-strin curves, corrected for friction nd ditic heting, disply n initil work hrdening period to pek stress t strins less thn 0.2 followed y vried levels of flow softening (Figure 4). Stedy stte stress ws reched rpidly t high tempertures nd is pproximtely 15 % lower thn pek vlues t 1060 C nd ove. Greter degrees of flow softening re oserved t lower tempertures with the mount of softening incresing with decresing temperture (i.e. stress drop of ~30% t 975 C) (Figure 4). Temperture 1.5 C/s WQ 15min Time Figure 3. Schemtic of heting cycle employed for n initil equixed 4

5 Stress (MP) e & = 1 s Strin 975 C 1000 C 1025 C 1060 C 1100 C Figure 4. Stress-strin curves t strin rte of 1 s Thermomechnicl chrcteristion of ech phse 4.1. Stedy stte ehvior of phse during hot deformtion The clcultion of the locl stress in the phse, s is derived from the nonliner viscoplstic lw using Eqution. (6) (Oikw et l. 1994). The description of the prmeter K hs een introduced y Semitin et l. (2002) using pulished experimentl curves for severl titnium lloys. The uthors found tht the K prmeter depends on the equivlent content of Vndium (V eq ) in the phse. Thus, K is given y the following eqution K where = nd = ( ln( Veq) + ) =10 (7) V = 0.27Al + 0Sn+ 2Mo+ 0.3Zr+ Cr+ 0.4N+ 13Si (8) eq eq Al eq is the percentge of equivlent luminum for phse, nd V eq is the percentge of equivlent vndium. In the dul phse region, Aleq chnges s function of temperture s following Vo et l. (2002) [( T - 45) ( ) ( - 45) - 29] Al eq = T (9) Tle 2 shows the vlues of K for different forging tempertures including the two phse region. Tle 2. Clcultion of K vlues ccording to temperture T ( C) K

6 The clcultion of Q nd m re mde from three experimentl dt using the stedy stte stress presented in Figure 4 nd using the liner reltion derived from Eqution (6) ln( K ) + ln( e& ) 1 = ln( s ) m Q - RT ln( s ) (10) Figure 5 shows the plot of Eqution (10). The vlues hve een found to e: Q = J/mol nd m = 0.23 which re in greement with those reported in the literture (Semitin et l. 2002): J/mol nd 0.23, respectively. Figure 5. Grphic clcultion of IMI 834 constnts Q nd m 4.2. Constitutive equtions for the phse during stedy stte hot deformtion Method The clcultion of the constnts governing the mechnicl ehvior of the phse is done in similr mnner s tht of the phse. The locl stress of phse s is derived from the sme type of nonliner viscoplstic lw (Eqution 6). The prmeter K depends on the content of equivlent Aluminum where = nd = Aleq from Vo et l. (2002) ( Al eq + ) K = 10 (11) Al eq = Al + ( 1 / 3Sn + 1/ 6Zr + 10C) (12) 6

7 In contrst to K, K is found to e independent of the temperture nd equl to 0.75 for IMI 834. The phse stedy stte stress s cnnot e clculted directly s there is mixture of phses. As first pproximtion, s is mesured from the experimentl dt nd pplying the following lw of mixture s s = ( - f ) s f where s + is the overll stress of the mixture otined from the experimentl dt, s is the phse stress in the mixture (extrpoltion in the two phses domin from results otined in the pure domin) nd f is the volume frction of phse present during deformtion. (13) Mesurement of the volume frction of ctully present during deformtion Difficulties were met during the mesurement of reel percentge of phse present during the stedy stte deformtion. In fct, equixed phses (primry, ) re esy to identify, some lrge size lmelle were p present t the deformtion temperture s they pper deformed when microstructures is looked t higher mgnifictions. This complictes the estimtion of the volume frction of the phse. Specificlly, t lower temperture (T def = 975 C), microstructures look like mixture of primry prticles in mtrix of trnsformed. Figure 6. Equixed microstructure otined fter deformtion t T def =975 C, e& =0.1s -1, e =1 Figure 7. Equixed microstructure otined fter deformtion t T def =975 C, e& =0.1s -1, e =1 (Arrows pointing out the presence of deformed corse lmellr structures) 7

8 However, t higher mgnifiction lrge deformed lmellr cn e clerly distinguished. The fct tht these lmelle re deformed proves tht they re present during the deformtion of the lloy. Figure 8 shows higher mgnifiction view of region where mixture etween equixed grins, lmellr phses (corse nd deformed) nd light stright lmellr phses of trnsformed re visile. The quntifiction of the volume frction of these lrge corse lmellr regions is prticulrly difficult ut essentil in order to ccurtely determine the volume frction of phse present t the deformtion temperture. Trnsformed et Equixed lph Deformed lph lmelle Figure 8. Microstructure deformed t T def = 975 C, e& = 1s -1, e = 1 showing deformed lmellr nd trnsformed phse The volume frction of the deformed lmellr phse ( f C) is evluted from high mgnifiction microgrphs tken from different regions of the smple. Tle 3 shows the percentges of primry equixed phse ( f P), the percentge of the deformed lmellr phse in the rest of the microstructure f C (equixed zone not included) nd the percentge of the lmellr phse (t) present during the deformtion f U, in the entire microstructure for different tempertures clculted from the following eqution f = f ( 1 f P ) (14) U C - Using the ove nlysis, the totl volume frction of the phse present in the microstructure during deformtion is otined y dding the volume frction of lmellr phses f U to the volume frction of equixed phse f P : = fp + fu (15) f Tle 3. Volume frctions for vrious tempertures 975( o C ) 1000( o C ) 1025( o C ) f P (%) f C (%) f U (%)

9 As shown in Tle 4, comprison etween the results otined in this investigtion nd those reported in Flower (1990) indictes reltively good greement with those mesured y Nel et l The difference in vlues etween the three mesurements in the literture could e due to differences in the morphology of smples under investigtion nd the techniques used to quntify the corse nd deformed lmellr phses volume frction f c. Tle 4. Volume frctions for vrious tempertures found in the literture 975( C ) 1000( C ) 1025( C ) Flower (1990) Nel (2001) Nel (1985) present work Thermomechnicl processing prmeters Using Eqution (13) nd the volume frction of the phse clculted ove, the vlues of Q nd m for IMI834 re otined nd re presented in Figure 9. These vlues re close to the vlues reported in the Semitin et l. (2002): J/mol nd 0.20, respectively. 4.3 Summry of the phse prmeters Figure 9. Clcultion curve for Q nd m for IMI 834 To recpitulte the clcultion results, the ctivtion energies nd strin rte sensitivities of nd phse used for the simultions re presented in Tle 5. Tle 5. Vlues of ctivtion energies nd strin rte sensitivities for nd phses Phse Phse Q (J/mol) m

10 5. CA results nd discussion To show the possiilities of CA to study the locl nd glol mechnicl chrcteristics of typicl dul phse forging, n equixed microstructure consisting of phses rndomly dispersed in mtrix is simulted nd nlyzed in detil. Figure 10 shows the sptil distriution of the phses of typicl simulted microstructure (in this cse, 15% ). Three locl regions of interest re selected in the microstructure re illustrted in Figure 10: Zone 1, where n cell is immersed in environment, zone 2 where cell is surrounded y n environment; nd finlly zone 3 where the mximum stress of microstructure is locted. Zone 1 Zone 2 Zone 3 Figure 10. Microstructure studied with three zones 5.1. Prediction of lloy mechnicl properties From glol viewpoint, the CA model vlidtion is mde y compring the stedy stte stress otined experimentlly ( s EXP ) with those otined y CA method ( s CA ) nd those otined y using the lw of mixture ( s ML ) for different forging conditions of IMI 834 (Tle 6). Tle 6. Comprison etween the experimentl results nd results given y cellulr utomt nd mixture lw T ( C ) e& (s -1 ) s (MP) EXP s CA (MP) (Mrgin of error) s ML (MP) (Mrgin of error) In generl, the CA model results re in greement with the experimentl ones nd provide good estimtes for the mechnicl ehvior of the lloy, lthough they hve the tendency to somewht underestimte y few percentges the experimentl dt. On the other hnd, the sme clcultions of stress vlues for forging conditions sed on the mixture lw s ML, overestimte the stedy stte flow stress. The reltively lower flow stress vlues predicted y the CA method re proly due to the fct tht in the CA pproch, not ll cells deform with the sme strin rte. In fct, s indicted in the CA nlysis, not ll cells of 10

11 given phse deform similrly which is logicl s the strin rtes in the phse re somewht smller thn those in the phse. Moreover, the ehvior of the grin is function of its neighorhoods Mechnicl properties t the locl level Simultions re presented for strin rte of 1 s -1 nd for cse with 15 % phse. As mentioned efore, the histogrm in Figure 11 shows tht the strin rtes in the phse re glolly smller thn those of the phse. On the other hnd, the histogrm of stress vlues shows tht the stress vlues in the phse re smller thn those in the phse (Figure 12). Figure 11. Histogrm of stin rte distriution in the microstructure for e& = 1 s -1 Figure 12. Histogrm of stress distriution in the microstructure for e& = 1 s -1 Similr results re found t locl level in zone 1 (Figure 13). The locl stress of the cell is higher y 14% thn those of its neighors. A stress of n grin surrounded y pure environment is lower y 2% compred with grin surrounded y ( + ) neighorhood. On the other hnd, the nlysis of strin rtes shows tht the strin rte is miniml in the cell, while its neighorhood deforms fster thn the verge (out 1.66%). cells, since they hve viscosity coefficient higher thn tht of cells, they deform less rpidly thn their neighors. This increses loclly the strin rte of their neighorhood. 11

12 Stress (MP) Strin rte (s -1 ) Figure 13. Zoom of zone 1 where cell is surrounded y six neighors: Representtion of stress nd strin e& = s -1 rte distriution in the microstructure during forging in the stedy stte for 1 In the second zone cell surrounded y six neighors cn e seen (Figure 14). The CA nlysis shows tht the stress in the cell increses y 1.55% compred to the glol stress. It lso deforms fster thn its neighors y 13%. This is due to the influence of the neighorhood ( grins) which tend to trnsfer some pressure on the cell y incresing the rigidity of the neighorhood. Stress (MP) Strin rte (s -1 ) Figure 14. Zoom of zone 2 where cell is surrounded y six neighors: Representtion of the distriution of stress nd strin rte in the microstructure during the forging in the stedy stte for e& = 1 s -1 Zone 3 (see Figure 15) loctes the mximum stress in the microstructure. The vlue of the mximum stress is locted in n cell whose neighorhood is mde up of severl cells. The stress of this cell reches vlue of 78 MP, i.e., n increse of 2% compred with n grin imedded in pure environment, nd 12 % higher thn the verge stedy stte flow stress of 68 MP. In fct, the neighorhood of the grin influences the stress level tht it reches when undergoing deformtion. It ws noted tht the greter the numer of neighors, the greter is the stress of the grin. Moreover the mximum strin rte is locted on cell while the minimum stress is locted in grins immersed in region. Future development tking into ccount the heterogeneous nture of the elstic properties of the phse will provide more comprehensive results nd mye n explntion for the occurrence of lrge microstructure heterogeneities in forged prts. 12

13 Stress (MP) Strin rte (s -1 ) Figure 15. Expnsion of zone 3 where there is mximum stress: Representtion of stress distriution in the microstructure during the forging in the stedy stte fore& = 1 s CONCLUSIONS A mthemticl model using CA nd sed on Eshely s inclusion is developed to etter understnd the processing of forged titnium lloy IMI 834 forging. The following conclusions cn e drwn from this study: Glolly the CA model results re in greement with experimentl results. Moreover, s the CA gives the opportunity to look t the locl informtion, it is found tht the locl stin rtes re quite close to the vlue of the remote strin rte with the locl strin rtes of grins eing lower thn tht of the mtrix. On the other hnd, the strin rte of the phses is higher thn the remote strin rte. It is oserved tht the stress of the grin is significntly higher thn tht of the surrounding neighorhood. Within the microstructure, the mximum stress is locted t the grins. It is noticed tht the locl stress level increses with the percentge of the grins in the neighorhood. By contrst, the stress in the grins increses s the percentge of phse increses in the neighorhood. The present work shows how CA could e used to investigte the deformtion of multiphse lloys. Interesting trends hve een shown, however, more fetures should e considered to etter explin the heterogeneous deformtion of imodl ner lph titnium lloys. Acknowledgement The uthor wishes to grtefully cknowledge the Nturl Sciences nd Engineering Reserch Council of Cnd for the finncil support of this work nd Phoung Vo for mny fruitful discussions nd ccess to relevnt dt. References Bch, M.R.; Evns, W.J.: Sensitive ftigue response of titnium lloys for power plnts ppliction. J. of Engineering for Gs Turines nd Power, Vol. 125, (2003) Boutn, N.; Bocher, P.; Jhzi, M.; Piot, D.; Montheillet, F.: Microstructurl modeling of cold creep/ftigue in ner lph titnium lloys using cellulr Automt Method. Trnsctions of the Cndin Society for Mechnicl Engineer, 32, 2, (2008), Eshely, J.D.: The determintion of the elstic field of n ellipsoidl inclusion nd relted prolems. Proc.Roy.Soc, A 241, (1957),

14 Germin, L.; Gey, N.; Humert, M.; Bocher, P.; Jhzi, M.: Anlysis of shrp microtexture heterogeneities in imodl IMI 834 illet. Act Mterili, 53, 13, (2005), Germin, L.; Vo P.; Gey N.; Jhzi M.; Humert M.; Bocher P.: Development of shrp locl lph microtextures during lph/et thermo-mechnicl processing of TIMETAL 834, Ti-2007 (Kyoto) Science nd Technology, Jpn Institute of Metls, 2, (2007), Flower, H.M.: Microstructurl development in reltion to hot working of titnium lloys. Mteril Science nd Technology, 6, (1990), Smith, M.A.: Cellulr Automt Methods in Mthemticl Physics. Ph.D. Thesis, Msschusetts Institute of Technology, (1994), Briottet L.; Montheillet F.: Prévision des hétérogénéités de déformtion dns un grégt de deux phses viscoplstiques, L Revue des Métllurgie-CIT/Science et génie des mtériux, (1997), 94:1212 Montheillet, F.; Gilormini, P.: Predicting the mechnicl ehvior of two phse mterils with cellulr utomt. Int. J. Plsticity, 12, (1996), Nel, D.F.: Mterils Design Approches nd Experiences. In: J.C. Zho; M. Fhrmnn; Pollock, T.M, eds., TMS, Wrrendle, (2001), Nel, D.F.: Titnium Science nd Technology, Deutsche Gesellschft fur Metllkunde. In: G. Lütjering; U. Zwicker; W. Bunk, eds., Oerursel, (1985), Oikw, H.: Metllurgy nd technology of prcticl titnium lloys. In: S. Fujishiro; D. Eylon; T. Kishi, eds., TMS, Wrrendle, (1994), Semitin, S. L.; Montheillet F.; Shen, G.; Jons, J. J.: "Self-Consistent Modeling of the Flow Behvior of Wrought Alph/Bet Titnium Alloys under Isotherml nd Nonisotherml Hot-Working conditions," Metllurgicl nd Mterils Trnsction, 33A, (2002), Toul, L.; Bocher, P.; Moreu, A.: Dwell ftigue life dispersion of ner lph titnium. Alloy. Interntionl Journl of Ftigue 31, (2009), Ut, E.: Etude des hétérogénéités de texture et de microstructure u sein de disques forgés en IMI834 : Influence sur les propriétés en ftigue-dwell. Ph.D. thesis, Université de Metz (2009). Vo, P.; Jhzi, M.; Yue, S.; Bocher, P.: Modelling flow stress of ner- titnium lloy, Mterils Science nd Engineering. 447, (2007), Wolfrm, S.: Cellulr utomt s models of complexity. Nture, 311, 5985, (1984) Addresses: Redoune Bstji, Nil Boutn, Philippe Bocher, École de technologie supérieure, 1100 rue Notre-Dme Ouest, Montrél (QC) Cnd H3C 1K3 ; Mohmmed Jhzi, Centre des Technologies de Friction en Aérosptile (CTFA), 5145, venue Decelles, Montrél, Cnd H3T 2B2. emil: Philippe.Bocher@etsmtl.c; Mohmmd.Jhzi@cnrc-nrc.gc.c. 14