Available online at Fatigue Simulation of Stage I-Short Crack Propagation in Forged Ti6Al4V

Transcription

1 Avalable onlne at Proceda Proceda Engneerng Engneerng 2 (2010) 00 (2009) Proceda Engneerng Fatgue 2010 Smulaton of Stage I-Short Crack Propagaton n Forged T6Al4V Phlpp Köster a,b *, Helge Knobbe b, Claus-Peter Frtzen a, Hans-Jürgen Chrst b, Martn Redler c a Insttut für Mechank und Regelungstechnk - Mechatronk, Unverstät Segen, D Segen, Germany b Insttut für Werkstofftechnk, Unverstät Segen, D Segen, Germany c Böhler Schmedetechnk GmbH & Co KG, A-8605 Kapfenberg, Austra Receved 27 February 2010; revsed 9 March 2010; accepted 15 March 2010 Abstract The propagaton of stage I-cracks s smulated n a mll-annealed mcrostructure of forged T6Al4V, consstng of prmary alpha grans and lamellar alpha/beta colones. The crack growth mechansms are nvestgated expermentally by means of fatgue tests yeldng the followng results: Wthn prmary alpha grans cracks usually grow on the basal plane or a prsmatc plane. In the lamellar colones crack propagaton often occurs parallel to the orentaton of the lamellae. The msorentaton between actve slp bands n neghbourng grans s measured by electron backscatter dffracton. These fndngs have been mplemented nto a two-dmensonal, mechansm-based short-crack model, whch descrbes crack propagaton as a partally rreversble dslocaton glde on a crystallographc slp plane. The model s solved numercally usng dslocaton dpole boundary elements. The nonunform propagaton knetcs of short cracks s consdered by defnng gran boundares as obstacles to plastc slp and crack propagaton. The descrbed model s used to smulate crack propagaton n vrtual mcrostructures, whch are based on Voronodagrams. Statstcal parameters such as gran sze and volume fracton can be adjusted to agree wth the real mcrostructure. c 2010 Publshed by Elsever Ltd. Open access under CC BY-NC-ND lcense. Keywords: short crack propagaton, stage I, vrtual mcrostructure, boundary element method 1. Introducton In aeronautcal applcatons ttanum alloys are often used for hghly stressed components due to ther hgh specfc strength n order to fulfll the requrements n lghtweght desgn. Furthermore, ttanum alloy wth dfferent types of mcrostructures can be obtaned dependng on the thermo-mechancal processng route. Changng the mcrostructure drectly affects the mechancal propertes so that t can be optmzed to a hgh crack growth resstance or to maxmum hgh cycle fatgue strength. In the hgh cycle fatgue regme, the total lfetme of a component s domnated by the phase of crack ntaton and short crack growth, whle the followng long crack growth only covers a small fracton of the lfetme. Thus, t s essental for a lfetme predcton model based on crack propagaton to descrbe the stage of short crack growth correctly. * Correspondng author. Tel.: ; fax: E-mal address: koester@mr.mb.un-segen.de c 2010 Publshed by Elsever Ltd. do: /j.proeng Open access under CC BY-NC-ND lcense.

2 1702 P. Köster et al. / Proceda Engneerng 2 (2010) P. Köster et al./ Proceda Engneerng 00 (2010) Nomenclature a half crack length a left, a rght projected crack length of left and rght half da/dn crack propagaton rate b t, b n dsplacement vector components n t-n-coordnate system c slp band length C materal-specfc constant n crack propagaton law CTOD crack tp openng dsplacement CTSD crack tp slde dsplacement CTSD range of crack tp slde dsplacement G j nfluence functon, j ndex m exponent n crack propagaton law S Schmd factor n, t local coordnates, normal and tangental to crack N number of cycles p number of elements n the crack q number of elements on the gran boundary r number of elements n the plastc zone m von Mses stress nn external normal stress on element nn resolved normal stress on element y yeld stress b mcrostructural yeld stress tn resolved shear stress on element external shear stress on element tn In polshed test specmens cracks often ntate trans- or ntercrystallne after a relatvely low number of cycles at locatons n the mcrostructure subject to elevated stress. Ths stress ncrease can be caused by stress rasers such as nclusons or the elastc ansotropy of the crystals. The followng stage I-crack propagaton s strongly nfluenced by mcrostructural features. Transcrystallne crack growth occurs on sngle slp bands and s controlled by the resolved shear stress on the respectve slp plane [1]. Thus, under push-pull loadng crack planes are usually nclned about 45 to the drecton of loadng. Furthermore, a second mechansm s ntercrystallne crack growth on gran boundares. As the crystal orentaton changes between neghborng grans, stage I-cracks are characterzed by a zgzag-shaped crack path. After crossng several grans addtonal slp bands are actvated at the crack tp and the effects of mcrostructural features on the crack propagaton decrease. The crack contnues to grow n stage II and the drecton changes nto a plane normal to the appled load. Untl now, several models for stage I-cracks have been presented, whch consder the abnormal propagaton behavour of short cracks descrbed above. Very promsng one-dmensonal analytcal models for stage I-crack growth have been developed by Tara [2] and Navarro and de los Ros [3]. These models account for the barrer effect of gran and phase boundares by blockng the extenson of the plastc zone at the next gran boundary. Only f the stress on a dslocaton source behnd the gran boundary reaches a crtcal value, slp s transferred over the barrer resultng n an ncreased crack propagaton rate. In order to take real mcrostructures wth arbtrary gran shapes and orentatons nto account the analytcal approaches have been extended to a two-dmensonal model, whch has been appled successfully to smulate crack growth n a duplex stanless steel [4,5]. The resultng boundary value problem s solved numercally by means of a dscretsaton wth dslocaton dpole boundary elements. Ths model s taken as the bass for the smulaton of short crack growth n forged T6Al4V. Below, the expermental results of fatgue tests are presented.

3 P. Köster et al. / Proceda Engneerng 2 (2010) Stage I-crack growth n forged T6Al4V P. Köster et al./ Proceda Engneerng 00 (2010) The model s based on expermental nvestgatons on the ttanum alloy T6Al4V wth a mll-annealed mcrostructure, whch s obtaned by an annealng treatment after the forgng process (Fg. 1) [6]. It has a prmary alpha volume fracton of 70.9%, the mean dameter of the p -grans and the lamellar colones s 9.9µm and 6.8µm, respectvely. Fg. 1: Forged T6Al4V wth mll-annealed mcrostructure. Durng fatgue testng, two types of crack ntaton stes were observed. Frstly, crack ntaton s observed at the nterface between two lamellae. But because of the Burgers relaton between alpha and beta phase durng phase transformaton, ths nterface plane s parallel to one prsmatc plane n the hcp-crystal of the alpha phase, whch s a possble slp system [7]. Secondly, cracks are found to ntate on slp bands nsde of prmary alpha grans. Once ntated, dfferent growth mechansms were found for the short crack propagaton n stage I. Insde of prmary alpha grans or sngle lamellae, the crack grows on slp bands. By means of EBSD-measurements the cracks are dentfed to be ether on the basal plane ore the prsmatc plane. The actve slp bands are favorably orentated slp systems characterzed by a hgh Schmd factor S. An example of such a crack s shown n Fg. 2a; the crack starts to grow on the basal plane wth a Schmd factor of S = After crossng a gran boundary the crack s slghtly deflected and propagates on a prsmatc plane, once agan characterzed by a hgh Schmd factor S = Thus, the crack propagaton rate s controlled by crack tp slde dsplacement CTSD. The gran boundares act as obstacles to crack propagaton, as they prevent a transmsson of slp nto the neghborng gran. Ths yelds a dslocaton ple up n front of the barrer resultng n a decreased crack growth rate. If the boundary s broken, the stress ntensty wll be releved by slp n the next gran and the crack propagaton rate wll ncrease agan, resultng n an oscllatng crack growth rate. In addton to that, crack growth s also found on gran boundares, as can be seen at the rght crack tp n Fg. 2a. Here, the crack path s n a plane perpendcular to the appled load so that the normal stress, whch causes a crack openng, seems to be responsble for crack propagaton. The crack n Fg. 2b grows on the nterface between two lamellae. As ths plane s equal to a prsmatc plane wth a Schmd factor of S = 0.49 the crack growth mechansm can be explaned by slp as well. However, cracks were also found between two lamellae orented normal to the appled load, where the Schmd factor on the respectve prsmatc plane s small.

4 1704 P. Köster et al. / Proceda Engneerng 2 (2010) P. Köster et al./ Proceda Engneerng 00 (2010) Fg. 2: (a) Stage I-crack n prmary alpha on slp plane and (b) crack growth on lamellae nterface. 3. Short crack model The two-dmensonal short crack model consders the crack and the plastc zone as yeld strps. In the cracked area an openng and a tangental dsplacement of the crack flanks s possble. In the plastc zone n front of the crack tp a dfferentaton between two cases s made, whch results n dfferent crack growth mechansms. In the frst case the crack propagates on a slp band and plastc slp occurs on ths slp plane, f the resolved shear stress reaches the crtcal value for dslocaton moton b. In the second case the crack grows on a gran boundary, whch does not represent a slp system of the crystal. Therefore, the only way to reduce the stress ntensty n front of the crack tp s a plastc deformaton nsde the neghborng grans leadng to a crack tp openng dsplacement. As the crystals at each sde of the gran boundary have dfferent orentatons, n general t s lkely that more than one slp system s actvated n each gran n order to ensure a compatble plastc deformaton (Fg. 3a). In the model ths plastc deformaton s projected onto the yeld strp on the gran boundary where both a tangental and a normal relatve dsplacement s possble. As the plastc deformaton occurs on multple slp systems n the neghborng grans, the von Mses stress m s used as the crteron for plastc deformaton (Fg. 3b). Fg. 3: Crack on gran boundary: a) plastc zone around the crack tp and b) yeld strp model of the plastc zone. As gran boundares act as obstacles to plastc deformaton, the extenson of the plastc zone s blocked by the next gran boundary. Thus, the crack propagaton rate, whch s controlled by the plastc deformaton at the crack tp, decreases. Only f a crtcal stress ntensty s reached on a dslocaton source beyond the barrer n the neghbourng gran, the respectve slp band s actvated and the plastc zone extends nto the new gran. The dslocaton ple up n front of the gran boundary s released resultng n a sgnfcant ncrease of the crack tp slde dsplacement and the crack propagaton rate. Ths mechansm yelds an oscllatng crack growth rate that s characterstc of mcrostructurally short fatgue cracks.

5 P. Köster et al. / Proceda Engneerng 2 (2010) P. Köster et al./ Proceda Engneerng 00 (2010) The crack problem s solved numercally by means of a dscretsaton wth dslocaton dpole boundary elements. Such a dpole element conssts of a negatve and a postve dslocaton and represents a constant relatve dsplacement over the element. Wthn a crack element two dpoles wth a Burgers vector normal and tangental to the crack are combned to allow for an openng and a tangental dsplacement. On a slp band only a slde deformaton s possble, so the respectve boundary elements only consst of a dpole representng a tangental dsplacement. The gran boundary elements are dentcal to the crack elements as they also allow for a normal and a tangental dsplacement. To montor the stress state behnd the gran boundary sensor elements wth a constant length are postoned at the ntersecton pont between actve slp band and gran boundary. The sensor orentatons match those of possble slp planes so that t s possble to calculate, n whch drecton a slp band s actvated. Fg. 4: Stage I-crack model wth boundary element dscretsaton. Two arbtrary elements are connected to each other by an nfluence functon G j, whch descrbes the stress n an element due to a dsplacement n an element j. G j s derved from the analytcal soluton of the stress feld around a dslocaton n an nfnte plate [8]. The total stress nn and tn s obtaned by summaton over all elements plus the external normal and shear stress and : nn pq j1 G j nn j pqr j j1 nn j tn nn, nbn Gnn, tbt (1) tn pq j1 j tn j pqr j j1 j tn G, nbn Gtn, tbt (2) The consderaton of the boundary condtons yelds the followng system of nequaltes: nn 0 0 tn b 1...p 1... p p q 1... p q r (3) (4) m 2 2 y nn 3 p 1... p q (5) tn b p (6) n

6 1706 P. Köster et al. / Proceda Engneerng 2 (2010) P. Köster et al./ Proceda Engneerng 00 (2010) Here, p s the number of crack elements and q and r are the number of plastc zone elements on a gran boundary and a slp band, respectvely. Roughness nduced crack closure s consdered by preventng a penetraton of the crack flanks (Eq. 6). The crack propagaton on a slp band s calculated from the range of the crack tp slde dsplacement CTSD by a power law functon (Eq. 7), whch s analog to the model of [3]. A physcal explanaton for the mechansm of crack propagaton s gven n [9]. Durng the loadng and the unloadng cycle dslocatons of opposte sgn are created n a dslocaton source n front of the crack tp and move towards the crack. Thus, vacances are generated, whch lead to a crack advance. For crack propagaton on a gran boundary the crack growth s related to the crack tp openng dsplacement CTOD. da dn da dn C CTSD CTSD C CTOD CTOD m CTSD m CTOD (7) (8) The constants C CTSD, C CTOD are materal-specfc and m CTSD, m CTOD are exponents usually very close to one. The cyclc yeld stress y = 725MPa has been determned expermentally from a cyclc stress-stran curve and the mcrostructural yeld stress of b = 112.5MPa s derved usng the analytcal approach of [3]. 4. Vrtual mcrostructures The smulaton model can be used for a vrtual optmzaton of the mcrostructure for maxmum hgh-cyclefatgue strength. The am s to carry out crack growth smulatons n vrtual mcrostructures wth dfferent propertes to fnd out, how the stage I-crack growth resstance can be mproved. Therefore, an algorthm for the generaton of vrtual mcrostructures has been developed, whch s based on the Vorono technque. Intally, such a Vorono dagram has a normal dstrbuton of the gran dameter and s sngle-phased. By assgnng dfferent phases to ndvdual grans, a mcrostructure consstng of prmary alpha grans and colones of alpha lamellae s created. Furthermore, the average gran sze and gran sze dstrbuton s modfed by dvdng selected grans. In a frst step a vrtual mcrostructure has been generated, whch represents the real mcrostructure shown n Fg. 1. The vrtual mcrostructure s shown n Fg. 5 and has a prmary alpha volume fracton of 70.5%, compared to 70.9% of the real one. A good agreement s found n the mean dameter of the p -grans and lamellar grans, whch are 10.0µm and 6.8µm for the vrtual and 9.9µm and 6.8µm for the real mcrostructure. Even the gran sze dstrbuton shows a reasonably good agreement, as can bee seen n Fg. 6 and 7. Here, especally the large grans wth a large effectve slp length are mportant, as they are preferred stes for crack ntaton. Fgure 5: Vrtual mcrostructure of forged T6Al4V.

7 P. Köster et al. / Proceda Engneerng 2 (2010) P. Köster et al./ Proceda Engneerng 00 (2010) Fgure 6: Gran sze dstrbuton of p-grans n (a) real and (b) vrtual mcrostructure. Fgure 7: Gran sze dstrbuton of lamellar colones n (a) real and (b) vrtual mcrostructure. The slp systems n the model are chosen based on the expermental observatons durng the fatgue tests. As cracks were found both on the basal plane and on prsmatc planes n p -grans, the respectve systems are consdered n the model. It would be also possble to allow pyramdal slp, but no crack has been found on these planes so far. In realty a lamellar colony conssts of parallel alpha plates n a beta matrx and the orentaton relaton between both phases s defned as the Burgers relatonshp. Thus, there are two parallel slp systems n both phases, (110)[11 1] (0002)[11 2 0] and (1 12)[111] (1100)[11 2 0], and two have only a small msorentaton, (110)[11 2] (0002)[1 2 10] and (11 2)[1 11] (101 0)[1 2 10] [7]. Ths means that slp can be easly transferred over the nterface between alpha plate and beta matrx so that the effectve slp length s equal to the colony dameter. As the effectve slp length s the crucal parameter for crack ntaton and crack growth, a lamellar colony s modeled as one gran n the vrtual mcrostructure wthout resolvng each sngle lamella. Nevertheless, only the slp systems mentoned above, whch stretch over the whole colony, are used n the short crack model. 5. Smulaton results Frst, the model s verfed by comparng smulaton results to real fatgue crack propagaton behavor. For ths purpose the geometry of the crack shown n Fg. 8b s gven as an nput nto the smulaton model. Then, the smulaton s started from the ntal crack length untl the crack tp reaches the end of the slp band. In Fg. 8a the projected crack length of the left and the rght half s plotted over the number of cycles. It can be seen that the crack propagates fast on the left hand sde on a slp band favorably orented for slp. However, when the crack tp approaches the gran boundary the propagaton rate decreases. On the rght hand sde the crack grows sgnfcantly slower on a gran boundary. The comparson of smulaton and expermental data shows satsfyng agreement. Fnally, a crack growth smulaton s carred out n a vrtual mcrostructure (Fg. 9). The smulaton s started wth an ntal crack on a slp band n an p -gran. Then the crack grows autonomously through the mcrostructure. On the left hand sde the crack propagates on a slp band, whereas ntercrystallne crack growth on gran boundares occurs on the rght hand sde.

8 1708 P. Köster et al. / Proceda Engneerng 2 (2010) P. Köster et al./ Proceda Engneerng 00 (2010) Fg. 8: (a) Comparson between smulaton and expermentally observed fatgue crack and (b) crack geometry. Fg. 9: Crack growth smulaton through several grans n vrtual mcrostructure. 6. Conclusons In ths paper, a two-dmensonal model for the smulaton of stage I-crack growth s presented. The model s based on expermental nvestgaton on T6Al4V and reproduces the growth mechansms, whch have been observed on real fatgue cracks. It allows for both transcrystallne crack propagaton on ndvdual slp bands and ntercrystallne crack growth along gran boundares. Gran boundares are consdered as obstacles to crack propagaton and prevent an extenson of the plastc zone untl a crtcal stress ntensty s reached to actvate a slp system n the next gran. For the purpose of lfetme predctons and mcrostructure optmzaton, vrtual mcrostructures have been developed, whch concde well wth the real mcrostructure. By means of crack smulatons n these mcrostructures, the effect of dfferent parameters, such as gran sze, on the short crack growth resstance can be studed. Acknowledgements The authors gratefully acknowledge the fnancal support of Böhler Schmedetechnk GmbH & Co KG.

9 P. Köster et al. / Proceda Engneerng 2 (2010) P. Köster et al./ Proceda Engneerng 00 (2010) References [1] Krupp U. Fatgue crack propagaton n metals and alloys. Wenhem New York: Wley-VCH; [2] Tara S, Tanaka K, Naka Y. A model of crack-tp slp band blocked by gran boundary. Mechancs Research Communcatons 1978;5: [3] Navarro A, de los Ros ER. Short and long fatgue crack growth: A unfed model. Phl. Mag. A 1988;57: [4] Künkler B, Düber O, Köster P, Krupp U, Frtzen CP, Chrst HJ. Modellng of short crack propagaton Transton from stage I to stage II. Engng. Fract. Mech. 2008; 75: [5] Düber O, Künkler B, Krupp U, Chrst HJ, Frtzen CP. Expermental characterzaton and two-dmensonal smulaton of short-crack propagaton n an austentc ferrtc duplex steel. Int. J. of Fatgue 2006;28: [6] Knobbe H, Köster P, Chrst HJ, Frtzen CP, Redler M. Intaton and propagaton of short fatgue cracks n forged T6Al4V. Proceda Engneerng, submtted. [7] Lütjerng G, Wllams JC,. Ttanum. 2nd ed. Berln Hedelberg New York: Sprnger; [8] Hlls DA, Kelly PA, Da DN, Korsunsky AM. Soluton of crack problems. London: Kluwer Academc Publshers; [9] Wlknson AJ, Roberts SG, Hrsch PB. Modellng the threshold condtons for propagaton of stage I fatgue cracks. Acta Mater. 1998;46: