Probing the strain distribution within a single crystal superalloy during high temperature testing

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1 MATEC We of Conferenes 14, (2014) DOI: /mteonf/ Owned y the uthors, pulished y EDP Sienes, 2014 Proing the strin distriution within single rystl superlloy during high temperture testing Alin Jques 1,, Mohmed Biskri 1, Thoms Shenk 1, Jen Philippe Chteu Cornu 1, nd Pierre Bstie 2 1 Institut Jen Lmour (UMR CNRS-UL N 7198), Lex DAMAS, Pr de Surupt, Nny, Frne 2 LiPhy, 140 Avenue de l Physique, BP. 87, Sint-Mrtin-d Hères, Frne Astrt. The omintion of in situ high resolution diffrtion experiments with synhrotron rdition nd mehnil nd diffrtion modelling is used to investigte the mirostruture nd mehnil stte of single rystl superlloys during high temperture tests: very high temperture nneling without externl lod nd reep test. 1. Introdution Single rystl superlloys re oth strtegi industril mterils nd model mterils to understnd nd predit the plsti ehviour of polyphsed mterils. As the onversion effiieny of therml motors suh s turoretors nd gs turines is linked to their operting tempertures, the ility to mnufture the omponents of their hottest prts (high pressure turine ldes nd vnes) nd to predit their ehviour nd life time under vrious onditions from norml use to extreme events is of mjor industril interest. Besides their industril importne, single rystl superlloys hve either uoid mirostruture, or solled rfted mirostruture: they re mde of oherent uoids or of semi oherent pltelets of hrd L1 2 phse perpendiulr to the [001] tensile xis [1, 2] emedded within dutile f mtrix. Differenes etween the mehnil ehviour of the two phses nd lttie mismth of few 10 3 generte internl stresses whih superimpose to the pplied lod. At high temperture, the mtrix deforms y glide of perfet /2<110> dislotions, while the plsti strin of the phse is elieved to our y lim of <100> nd/or <110> dislotions exhnging vnies [3]. In order to uild physilly (i.e. dislotion) sed onstitutive lws not only for the omposite mteril, ut for eh phse, mrosopi dt suh s the pplied stress nd the verge strin rte re insuffiient: we need t lest to know the verge stress (strin) stte of eh phse, or etter the distriution of stresses (strins) within eh phse. We lso need tool to hek the vlidity of suh lws. The im of the present pper is to show tht the omintion of in situ experiments using high resolution synhrotron X-Ry diffrtometry nd diffrtion pek modelling n e suh tool. Corresponding uthor: lin.jques@univ-lorrine.fr 2. Experiments Figure 1 shows the lyout of Three Crystl Diffrtometer [4]. The initil polyhromti X-Ry em first goes through (311) silion monohromtor. The monohromtized em is then diffrted y the smple ((200) diffrting vetor): different res in Brgg onditions with slightly different lttie prmeters stter the em into slightly different diretions. A seond (311) silion rystl lled nlyser diffrts gin one of those ems into n energy sensitive Ge detetor. A rottion of the nlyser proes the distriution of the ems sttered y the speimen, i.e. the distriution of interplnr distnes within the speimen, while rottion of the speimen mesures the distriution of orienttions of the diffrting plnes. As the (311) refletion of silion is very nrrow, the instrumentl width of the devie is lower thn seond of r for 3.65 two thet ngle nd n e negleted. A further redution of the kground n e otined y use of doule rystl diffrtometer. Using the high energy em of the ID 15 emline of the ESRF or the P07 emline of the PETRA III ring t Hsyl (etween 100 kev nd 150keV), it is possile to do experiments in trnsmission nd to mesure lttie prmeters in the ulk of the speimens. Thnks to the high intensity of the synhrotron em, 200 points sn n e reorded in less thn 5 minutes: it is possile to study the response of the mteril t short time intervls following hnge of the pplied lod or temperture. 3. Sttering y strined mteril The shpe of diffrtion pek depends on the distriution of lttie strins within the mteril nd the size of the oherent zones. If we ssume tht the oherene length of X-Rys is lrger thn few mirometres, the mplitude sttered in the viinity of vetor G of the reiprol lttie long vetor q = G+q n e written s: A(q ) = FT{f(x). exp[2iπg.u(x)]}. (1) This is n Open Aess rtile distriuted under the terms of the Cretive Commons Attriution Liense 4.0, whih permits unrestrited use, distriution, nd reprodution in ny medium, provided the originl work is properly ited. Artile ville t or

2 MATEC We of Conferenes x (µm) Figure 1. Three Crystl Diffrtometer. Where FT indites the Fourier Trnsform, f(x) is the lol sttering ftor (depending on the lol hemistry nd lol struture mplitude), nd u(x) is the displement field within the mteril [5]. The diffrted intensity n then e otined y tking I = A.A*, where A* is the omplex onjugte of A. If the displement field n e lulted, it is thus possile to generte theoretil diffrtion pek whih my e ompred to n experimentl one. Vrious methods n e used: Diret lultion of the superposition of the displement fields of individul uoidl inlusions. Finite element methods. Fourier trnsform methods [6]. While the first two methods re omputer intensive nd, in the first se, require nlytil formuls, Fst Fourier Trnsform (FFT) methods re quite effiient for lrge numers of voxels. (The sme lultion for volume tkes 23 hours on desktop omputer with the first method, nd 12 minutes with the third.) Furthermore, FFT lultions n e extended esily for inhomogeneous mterils with nisotropi elsti properties [7]. Figure 2 shows first exmple of the displement field U x lulted in suh wy for 2 µm ue of superlloy ontining 0.7 volume frtion of preipittes (size pprox µm. See dotted lines) nd lttie mismth. The orresponding pek, nd top view of its enlrged til re given in Figs. 2 nd 2, while Fig. 2d shows the TCD profiles (vertilly offset) expeted for the sme uoids mirostruture, ut with ues hving evelled edges nd orners. While for perfet ues, low nd wide pek is expeted for strin, nrrower ut tller pek is otined t for 0.1 µm evel. Assuming onstnt volume frtion 1-f of mtrix, lrge evel results in more phse t the edges (irleinfig.2) nd orners, nd less in the orridors (retngle) whih eome thinner: the intensity sttered y the orridors is lower nd distriuted on wider pek. The min pek lso eomes slightly thinner s the highly stressed elsti singulrities t the orners of the ues dispper. Lst, the distriution of uoid sizes is wider in rel mirostruture thn in the present model. 4. Spontneous loss of oherene during high temperture nneling 4.1. Experimentl results. Figure 3 shows the evolution of the high resolution (200) diffrtion pek of speimen with uoid Ux (nm) (200) / Figure 2. yx plot of the displement field in the x diretion in superlloy nd profile long the stright vertil line (), nd (200) diffrtion peks (,,d). See text for detils. mirostruture during high temperture nneling t 1130 C. The top of the pek (Fig. 3) ws reorded with 1 seond ounting time per step, while 5 seonds reording nd verging were used for the tils (Logrithmi sle, Fig. 3. All diffrtion profiles were normlized to onstnt unit re for omprison. The pek intensity y d p.2

3 EUROSUPERALLOYS 2014 ' + strin pek deresed. Another wider pek ppered t strin The height of the left til (Logrithmi sle) ws unhnged, while it slightly deresed on the right side. After ooling, polishing nd hemil ething (66% HCl et 33% HNO 3 ), the speimens were oserved y SEM. Lines orresponding to dislotion tres n e seen t the surfe of the uoids, with n verge distne of out 0.02 µm. During the high temperture nneling, the volume frtion ws lower thn t room temperture, nd the orridors wider. The mgnitude of the lttie mismth ws lrger thn 0.003, nd the oherene stresses eme lrger thn the Orown stress, llowing dislotions to glide into the orridors nd prtly relese the oherene stresses. As this plsti relxtion llowed the lttie prmeter of the orridors prllel to (200) to inrese in the [100] diretion, the orresponding pek shifted to the right. The intensity sttered t the position of the pek thus deresed. The high lttie prmeter of the orridors perpendiulr to (200) ws due to the oherene stresses in the (100) plne: s these stresses deresed, the position of the pek shifted to the left Pek modelling Figure 3. High resolution diffrtion pek of superlloy t 1130 C in liner () nd logrithmi () sle. SEM mirogrph (1 µm mrk) fter hemil ething. t the top is 500 (ritrry units) nd the tils level t 0.01: the pek to kground rtio is lose to the 10 5 rnge. The min pek orresponds to the uoids with ontriution of the orridors prllel to the (200) diffrting vetor. The til on the lrge prmeters side (Fig. 3, right) is due to the orridors perpendiulr to (200) [8]. The long rnge tils nerly follow n ε 4 lw: this is proly Hung sttering resulting from the highly strined zones on the viinity of point defets. At intermedite strins, n interesting feture of the peks is their exponentil til (i.e. their liner slope when seen with logrithmi sle). The inverses of these slopes re defined s the W nd W prmeters whih will e disussed elow. During nneling, the width of the min pek nd its til deresed progressively, nd the height of the Figure 4 shows simultion of the diffrtion pek (thin lk line) for omprison with the experimentl pek reorded t 100 minutes (Fig. 3). The displement field ws lulted in 2 µm 3 representtive volume ontining 64 uoids with rndom size nd position (0.42 volume frtion of phse), nd ssuming lttie mismth etween oth phses. By ritrrily giving 0 nd 1 sttering mplitude to the (lue) nd the (red) phses, it is possile to lulte their ontriutions to the peks. The pek ontriutes to the min pek only, while the intensity is distriuted etween the min pek (oherent orridors prllel to the g vetor), the highly strined orridors perpendiulr to the edges (retngles in Fig. 2) nd the ue edges (irles). A evel t the ues orners nd edges ws ssumed: this is proly n underestimted vlue, s the re of the intermedite pek is lower thn the experimentl one, nd the right pek higher. Lst, the width of the lulted intermedite pek is lower thn the experimentl one. Figure 4 ompres the (green) experimentl pek reorded t 480 minutes nd peks simulted ssuming tht the lttie mismth hd een hlf relxed y infinitesiml dislotions. Both nd peks re thinner thn in the unrelxed mirostruture (their width is pproximtely divided y two) nd their positions hve een reltively shifted y , i.e. the plsti strin. However, if the lulted peks hve orret positions, they re thinner thn the experimentl ones: n ingredient is missing in the lultion. We elieve this ft results from the hoie to use infinitesiml dislotions insted of perfet dislotions of the mtrix: s the lttie plnes remin ontinuous etween interfe dislotions, there is no jump of the lttie prmeters nd some intensity should e diffrted etween the two peks of Fig p.3

4 MATEC We of Conferenes strin w ( ) (10 s ) t (10 3 s) W (10-3 ) Figure 4. Simulted high resolution diffrtion pek of superlloy with n unrelxed () nd hlf relxed () mirostruture. 5. Pek widening during reep 5.1. Experimentl results Figure 5 shows the evolution of the W nd W prmeters of the diffrtion peks of rfted superlloy during tensile reep test under vrile lod t 1080 C[9]. The pplied lod σ nd the strin rtes of eh phse ( :thin line nd : thik line) re given for omprison. The W prmeter of the pek remins onstnt during the eginning of the test, nd only egins to inrese fter jump of the pplied lod up to 200 MP. After prtil unloding of the speimen t t = s, it returns to its initil vlue. The W prmeter is initilly It egins to inrese t s, under 150 MP pplied lod, when the strin rte of the phse is in the 10 8 s 1 rnge. It inreses up to Figure 5. ) Plot of the vritions with time (1000 s) of t the mximum lod, then dereses to the pplied lod (top), the W (empty dimonds) nd W (full fter unloding. dimonds) fit prmeters (10 In the W vs. W plot of Fig. 5, the experimentl points 3 strin units, ottom left hnd sle) nd the strin rte of eh phse (ottom right hnd sle). initilly move from A to B, then move to the top nd ) Plot of W vs. W. ) Post mortem TEM mirogrph tken fter downwrds long the stright BC line. The slope of BC vries with temperture in the 980 C to 1130 ooling of the speimen under high stress. C rnge. Under high stresses, the evolutions of W nd W re thus orrelted, nd re elieved to e relted to the dislotion 5.2. Lol stresses nd long rnge stresses density within the phse. This my e heked in Fig. 5, whih shows m 2 dislotion density Long rnge stresses in TEM mirogrph of the sme speimen fter ooling under high lod, s the (W, W ) point ws t C. The sme ehviour ws oserved on ll speimens Figure 6 shemtizes the distriution of stresses within strined t tempertures higher thn 980 C. the rfted mirostruture t the eginning of the seond p.4

5 EUROSUPERALLOYS 2014 stge of reep urve. During stge I, dislotions moved within the (ler) orridors etween (drk) rfts, nd left dislotion segments t the / interfes. If given time, these segments my ret to form new dislotion segments with their Burgers vetor perpendiulr to the [001] tensile xis, nd form more or less relxed dislotion wll. The stress stte of mtter within the mirostruture is then the sum of the pplied lod σ zz = σ, nd of oherene stresses long the x nd y diretions due to the lttie mismth δ nd the stress field of the interfil dislotion wlls. It hs een shown in [9] tht the verge resolved sher stress for the plstiity of the orridors is equl to the verge Orown stress σ O. We my dmit this remins true for eh orridor: new dislotions my enter orridor s soon s the lol Von Mises stress σ VM = σ zz σ xx is equl to the Orown stress for this orridor. These dislotions will stop entering s soon s the inresed stress field of the dislotion wlls t the interfes etween the sme orridor nd its neighouring rfts hs eome lrge enough to repel them. We n thus expet eh orridor to hve on its oth interfes dislotion wlls with the sme dislotion density. On the ontrry, different orridors with different thiknesses nd different Orown stresses will hve different interfil dislotion densities. From this, we n expet heterogeneous distriution of internl stresses in the phse (i.e. wide pek), nd homogenous distriution of stresses in the rfts, euse the stress fields of the interfe dislotions of the orridor etween two neighouring rfts will nel. When the pplied lod is high enough, the rfts egin to deform plstilly. As in the se of orridors, new dislotions will e dded t the interfes on oth sides of rft, or will nel with existing ones. If the plsti strin is the sme in ll rfts, the sme numer of dislotions will dispper on oth sides of ll interfes, nd the distriution of stresses will remin the sme. Conversely, if some rfts deform more or less thn others (Fig. 6), the density of dislotions on oth sides of orridor will eome different, nd the stress fields of oth interfes will no longer nel: the internl stresses within the phse will eome heterogeneous, nd the width of the pek will inrese Dislotions nd reversile hnges in the shpes of the diffrtion peks While the ontriution of dislotions hs not yet een tken into ount in our modelling of diffrtion peks, it is ovious from Fig. 4 tht it is importnt. We shll here first rule out the effet of interfe dislotions, then onsider the effet of dislotions moving within the rfts on the distriution of elsti strins within the mteril. (As dislotions re more moile in the orridors thn in the rfts, their densities should e muh lower). The highly distorted res in the viinity of the ores of interfe dislotions should strongly ontriute to the pek tils, t distnes d lower thn hlf the distne etween these dislotions, i.e. d < m. The orresponding strin is /2πd = , more thn five times lrger thn the distne etween the nd peks. At sles intermedite etween d nd the Figure 6. Homogeneity nd heterogeneity of the interfe dislotion ontriution to the σ xx omponent of the stress tensor, efore nd fter the onset of plstiity within the rfts. orridor thikness, we my expet flututions of the elsti strin whih derese s the distne d i to the interfe inreses. The derese is exponentil for n infinite periodi dislotion wll [10]. If we onsider disordered wlls, the vrine of the strin flututions my derese s (d/di) 1 or (d/di) 3 depending on the type of disorder [11,12]. As the rfts re thiker thn the orridors, the ontriution of these flututions is proly lrger in the se of the pek: the slope of the W vs. W plot Fig. 3 should e smller thn one. The flututions should lso e lrger when the disorder is high, when new dislotions re dded rndomly t the interfes, i.e. when the strin rte of the orridors is high. They might then smooth out when the strin rte dereses gin, nd interfe dislotions my rerrnge into more orderly, low energy wy. The evidene for suh trnsient pek width inrese is sre in Fig. 5. Severl rguments support link etween dislotions within the phse nd the reversile widening of the peks: Both peks thiken when the strin rte of the rfts is high. Dislotion densities oserved y TEM (Fig. 5) re ner m 2 when the experiment is stopped t high strin rtes for lrge vlues of W nd W, nd m 2 or lower when W nd W were llowed to relx under low stress. The slope of the W vs. W plots is higher thn one: s the lrgest flututions our ner the dislotion ores, the orridors re shielded. An ttempt to model the evolution of the distriution of elsti strins with the dislotion density using 2D simultion is shown in Fig. 7. A 2D superell 20 2 µm 2 wide ontining stk of 40 - periods with lognorml size distriution nd 0.6 frtion of ws uilt. Different densities ρ d of edge dislotions with p.5

6 MATEC We of Conferenes Log 10 (I) (u) superlloy thn just pek positions. The peks re very sensitive to the detils of the mirostruture (evelled ues) nd the distriution of elsti strins within the mirostruture. Thus, in situ diffrtion n eome very powerful tool to test the results of models of the plsti ehviour of superlloys. However, full nlysis of those peks will require extensive mehnil nd physil modelling of the mteril, tking mny effets into ount: elsti nisotropy nd inhomogeneity, point defets, nd most of ll dislotions: further work is needed. ε 3 *I Log 10 (w) The uthors would like to thnk the ESRF (ID15A emline) nd DESY (P07 emline, CALIPSO: EU Support of Aess to Synhrotrons/FELs in Europe ) for emtime nd help for the experiments, nd Snem (SAFRAN) for providing the AM1 speimens. ε ρ d Figure 7. 2D modelling of the distriution of elsti strins due to dislotion within the rfts. ) Model; ) strin distriution in oth phses; ) plot of ε 3.I(ε) vs.ε; d)evolutionofthew prmeters with the dislotion density within the rfts. [100] Burgers vetors t rndom positions (ut the sme numer of positive nd negtive signs in eh rft) were dded in the rfts. The distriution of strins ws lulted for n isotropi nd homogenous mteril using periodi onditions, nd is given for oth phses in Fig. 7 for ρ d = As expeted, the pek is wider nd hs lrger til: the plot of ε 3.I(ε) in ) shows tht the tils of the distriution indeed vries s ε 3 [13,14] inthe phse for strins lrger thn 0.01, nd dereses fster in the phse. The doule logrithmi plot in Fig. 7d shows tht W nd W vry s power lws of ρ d with exponents 0.49 (W) nd 0.64 (W ). From sttistil rguments (the vrine of W nd W should vry s the dislotion density), 0.5 exponent is expeted. Lst, the shpe of diffrtion pek does not depend only on the distriution of strin, ut lso on the size of oherent zones, whih nnot e modelled in the present pproximtion. 6. Conlusion From the present exmples, it n e seen tht high resolution diffrtion peks n give muh more dt on d Referenes [1] P. Cron nd T. Khn, Mterils Siene nd Engineering 61, (1983) 173 [2] T.M. Pollok nd A.S. Argon, At Metll. Mter 40, (1992) 1 [3] F. Mompiou nd D. Cillrd, Mter. Si. Eng. A 483, (2008) 143 [4] K. D. Liss et l., J. Synhrotron Rd. 5, (1998) 82 [5] N. Vxelire et l, New Journl of Physis 12, (2010) [6] T. Mur, Miromehnis of defets in solids 2 nd edition, Mrtinus Nijhof editions (1987), ISBN X [7] H. Mouline, P. Suquet, Comp. Meth. Appl. Meh. Engng. 157, (1998) [8] H. Mughri, H. Biermnn nd T. Ungr, Journl of Mteril Engineering nd Performne 2(4), 557 (1993) [9] L. Dirnd et l., Phil. Mg. 93 (2013) 1384 [10] J.P. Hirth nd J. Lothe, Theory of dislotions, 2nd New edition of Revised edition, Krieger Pulishing Compny (1991) ISBN-13: [11] G. Sd nd E. Bouhud, At Metll. 41, (1993) 2173 [12] G. Sd nd D. Sornette, At Metll. 43, (1995) 313 [13] T. Ungár, nd A. Borély, Applied Physis Letters 69, (21), (1996) 3173 [14] I. Grom, nd A. Borély, A. Diff. Anlysis of the Mirost. of Mterils 68 (2004) p.6