Slip Deformation Analysis Based on Full Constraints Model for -Titanium Alloy at Low Temperature*

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1 Materals Transactons, Vol. 5, No. 8 () pp. 595 to 6 # The Japan Insttute of Lght Metals Slp Deformaton nalyss ased on Full onstrants Model for -Ttanum lloy at Low Temperature* Motoak Morta and Osamu Umezawa Graduate School of Mechancal Engneerng and Materals Scence, Yokohama Natonal Unversty, Yokohama 4-85, Japan Department of Materals Scence and Engneerng, Yokohama Natonal Unversty, Yokohama 4-85, Japan The effects of restrcted slp condtons on both the Taylor factor and plastc work rate under the condton of tensle yeldng have been analyzed n -ttanum alloys at low temperatures, usng the full constrants model. The role of secondary slp systems,.e., the ha basal slp and hc þ a pyramdal slp, was clarfed, when the ha prsmatc slp was domnant. lthough no nfluence of secondary slp condtons on the Taylor factor was detected, the plastc work rate was senstve to the operatng secondary slp systems. When the basal system was chosen as the secondary slp system, the plastc work rate ncreased n all tensle axes, especally around h. In addton, no basal slp operaton decreased the plastc stran energy. The plastc work rate was the hghest along the h tensle axs, and the operaton of the hc þ a pyramdal slp was necessary to acheve plastc deformaton along c axs. Hgh elastc stran energy, therefore, must accumulate to a hgh level around h, because the pyramdal slp s hardly actve owng to ts very hgh crtcal resolved shear stress. [do:.3/matertrans.l-m85] (Receved February 4, ; ccepted prl 3, ; Publshed July 5, ) Keywords: Taylor factor, crtcal resolved shear stress, hexagonal close packed structure, fatgue, prmary slp. Introducton Subsurface crack generaton n the hgh-cycle fatgue of -ttanum alloys s domnant at low stress regmes and temperatures. ) The crack ntaton stes appear at crystallographc facets such as the () transgranular crackng. 5) The dslocaton movement of the alloys s nearly planar and fgh arrays ple up near gran boundares. ) The local stress concentraton near a gran boundary due to heterogeneous slp may lower the hgh-cycle fatgue strength and cause the subsurface crack ntaton. Therefore, t s necessary to consder crystal plastcty n the cyclc deformaton and fatgue fracture mechansm. Here, the fatgue process s dvded nto four stages: () development of a saturated dslocaton structure by cyclcal mcroplastc stran accumulaton, () generaton of localzed slp and/or mcrocrackng to relax the stress concentraton at the vcnty of a boundary, (3) mcrocrack growth and transton to man crack, and (4) crack propagaton. 3) In stages 3 and 4, lnear mechancs can be appled to evaluate the crtcal sze of the subsurface crack (facet) for propagaton 6) and the crack propagaton lfe. 7) However, stages and, whch are based on estmated models, do not exhbt drect evdence of crackng and macroscopc support. Thus, the evaluaton based on crystal plastcty complements the constructon of a sophstcated model of subsurface fatgue crack ntaton. In stage, the fgh domnant slp operates n some grans; ths slp depends on the crystal orentaton. Moreover, both plastc and elastc deformatons coexst n polycrystallne specmens. In stage, regardless of whether the orgn of transgranular crackng s slp-off (slp localzaton) or mcrocrackng on a crystal plane, the plastc deformaton on the facet plane hardly yelds and the elastc feld normal to the facet plane has to accumulate. The actvty *Ths Paper was Orgnally Publshed n Japanese n J. JILM 6 () In Regard to the Paper, the Erratum was Publshed on Page 96, Vol. 6 of J. JILM. of slps, needed for any mcro-yeldng, can be evaluated on the bass of the full constrants model usng Taylor analyss. 8,9) Subsequently, the deformaton behavor of the secondary slps n a locally yelded gran descrbes the elastc (stress) feld n stages and. The full constrants model s based on the assumpton of homogeneous deformaton n polycrystallne specmens and has been manly appled to understand the homogeneous and large-scale deformaton n face-centered cubc (f.c.c.) metals. 8,9) Occasonally, the full constrants model has been appled to slp analyss n hexagonal closed packed (h.c.p.) metals, because twnnng has a great nfluence on ther homogeneous and large-scale deformaton. ) In the present study, we adopted the full constrants model n order to evaluate the crtcal resolved shear stresses (RSSs) n the prmary slp system of - ttanum at low temperatures. In addton, we calculated the external (nternal) plastc work rate and Taylor factor to dscuss the effect of the prmary slp system on slp behavor and that of the crystal orentaton on the stress feld.. Procedure. pplcaton of full constrants model gran n the tensle stress feld that develops by cyclc deformaton s constraned by ts neghbors n a polycrystallne specmen. The full constrants model, n whch all the grans n the feld exhbt the same stran, was adopted to sustan the stran compatblty between neghborng grans. That s, to satsfy the full constrants, multple slps were ntroduced n a gran. In the tenson mode, the compatblty n a polycrystallne specmen can be acheved by operatng fve ndependent slp systems. 8,9) When an unaxal tensle stran s parallel to the z-axs n the specmen s coordnate system, XYZ, gran deformaton takes place under axal symmetry at a fxed volume (d" xx þ d" yy þ d" zz ¼ ): ) d" xx ¼ d" yy ¼ d" zz ðþ d xy ¼ d yz ¼ d zx ¼ ðþ

2 596 M. Morta and O. Umezawa where d" xx, d" yy and d" zz are the plastc stran rate, and d xy, d yz and d zx are the plastc shear stran rate n a gran. The nternal plastc work rate dw s the ncrement of work per volume and t s the sum of the work of fve ndependent slp systems n a gran: dw ¼ X d ¼ X jd j ð ¼ ; ; ; 5Þ ð3þ where s the RSS and d s the slp rate n the -th slp system. There are a number of combnatons of operatng slp systems that satsfy the external work constrant, but only one combnaton s to be chosen. Each tme deal plastc deformaton take places, dw equals the external plastc work rate as n eq. (4): dw ¼ d ¼ zz d" zz ð4þ The Taylor factor M s calculated usng eq. (4) as zz ¼ d ¼ M ð5þ d" zz M represents the total of the slp rates n fve ndependent slp systems for the straned gran, and depends on the relatonshp between the tensle axs (z-axs) and gran orentaton.. Evoluton of full constrants model appled to lowtemperature slp deformaton n -ttanum.. oordnate transformaton n hexagonal lattce The stran rate tensor d" j n the specmen space s equal to the stran rate tensor P k Ek j, whch s the sum of the stran rate tensors n fve ndependent slp systems: d" xx d xy d xz d" j d yx d" yy d yz ¼ X E k ð; j ¼ ; ; 3Þ j ðk ¼ ; ; ; 5Þ d zx d zy d" k zz ð6þ In the specmen coordnate system, the Ej k are descrbed by the slp drecton vector b and the normal vector drecton to the slp plane n, where the coordnate transformaton from the Mller-ravas ndex n the crystal to the orthogonal coordnate system n the specmen was done under the coordnates shown n Fg.. Frst, b mb [jkh] gven by the Mller-ravas ndex (axs: a, a, a 3, c) was transformed nto b norh [(-k) (-j) (c=a) h] n the nonorthogonal coordnate system (axs:,,, ¼ ), and then b orh n the orthogonal one (axs: x, y, z )as b orh ¼ Q b norh ð7þ Q s the coordnate transformaton tensor whch represents the relatonshp between the unt vector of axes n the nonorthogonal coordnate system ð ; ; Þ and that n the orthogonal one ðx ; y ; z Þ. Second, the normal vector to the slp plane n orh was estmated from the exteror product of the two slp drectons b orh and b orh on a slp plane as n orh ¼ b orh b orh ð8þ Subsequently, the n orh and b orh, represented n the orthogonal coordnate system n the crystal, were transformed to the n and b n the orthogonal coordnate system n the specmen as eq. (9): a 3 α, a y x z,, c β x cos sn ðn or b Þ¼@ cos sn cos cos sn sn sn sn cos cos ðn orh α or b orh Þ β z β y y,, a Fg. Defnton of and on the standard stereographc projecton of () plane wth Mller-ravas ndex ða ; a ; a 3 ; cþ, orthogonal coordnates system ðx ; y ; z Þ, and nonorthogonal coordnates system ð ; ; Þ n crystal and orthogonal coordnates system ðx; y; zþ n specmen. ð ¼ ; ; 3Þ ð9þ where angular parameters and represent the transformaton from the orthogonal coordnates n the crystal to those n the specmen, xyz, (Fg. ). The rotaton angle ( <<9 ) along the x-axs n the orthogonal coordnates, xy z, gves the transform nto the orthogonal coordnates, xyz. The angle ( <<9 ) along z -axs n the orthogonal coordnates, x y z, gves the transform nto the xy z... Influence of crtcal resolved shear stress Only the prncpal slp systems,.e., fgh, fgh, and fgh3 were taken nto account for the deformaton mode n -ttanum alloy. 3 6) In addton, no deformaton twnnng and "-martenste were consdered. ecause the RSSs n the slp systems are dfferent from each other, especally at low temperatures,, 4,6) eqs. (3) and (5) are modfed to permt the multple slps as gven by eqs. () and () under the full constrants model: dw ¼ X jd j ð ¼ ; ; ; 5Þ ðþ X jd j ¼ M ð ¼ ; ; ; 5Þ ðþ d" zz where M s the newly defned sum of the slp rates n the operatng slp systems wthn a gran per unt stran, because t cannot be drectly found by the yeld stress. In a randomly orented polycrystal under unaxal tenson parallel to the z-axs, the deformaton s denoted as

3 Slp Deformaton nalyss ased on Full onstrants Model for -Ttanum lloy at Low Temperature 597 Table Ratos n crtcal resolved shear stresses of major slp systems. (reference: fgh at 3 K) Type Rato of RSS fgh fgh fgh [] =.79 ;(α, β)=(57, 3) M mn =.75 ;(α, β)=(9, 8) [] [] Fg. 3 Dependence of Taylor factor on tensle axs under the condton of homogeneous prmary slps. 77K [] K, 3) M mn =.75 ;(α, β)=(9, 8) = 3.8 ; (α, β)=(33, 3) -4) -5) [] [] Fg. Relatonshp between crtcal resolved shear stress (RSS) and temperature for ha prsmatc, ha basal, and hc þ a pyramdal slps n - T. Dashed curved lnes ndcate scales wth 5,, and 5 tmes as large as the RSS of ha prsmatc slp. The dashed-dotted lne ndcates the athermal component of ha basal slp n pure T. 5) X jd j¼ zz d" zz ð ¼ ; ; ; 5Þ ðþ To evaluate the effect of the RSSs n the prncpal slp systems on M and dw, sx types of models were examned (Table ). In Type, the RSSs of the three actve slp systems are the same. In Types 6, fgh s fxed as the prmary slp system. Types 4 smply represent the act of choosng the secondary slp system from fgh and fgh3. Each RSS of the secondary slp systems s from fve to ffteen tmes as hgh as that of fgh n the low-temperature regme (Fg. ). Subsequently, t was assumed that the RSS of the secondary slp systems was ten tmes that of the prmary or more. What the actve of two secondary slp systems was dstngushed, the rato of ether system was chosen as ffteen. In Types 5 and 6, the rato of the RSSs n the three systems was gven by the estmated values at 3 and 77 K, respectvely. The RSSs of fgh3 and fgh were derved from the expermental data for sngle crystal -ttanum,,5) although t s hard to evaluate the RSSs of fgh and fgh3 at low temperatures due to deformaton twnnng. 4) Then the RSS of fgh below K was estmated by the basal ð¼ þ Þ, where the thermal Fg. 4 Dependence of Taylor factor on tensle axs under the condtons of Types, 3, and 4. and athermal component were gven by the thermal actvated process. The RSS of fgh3 was also gven by the data of T-6 mol%l. ) 3. Results 3. Influence of slp systems on Taylor factor When the RSSs of the actve slp systems are equvalent (Type ), s.79 at the tensle axs wth ð; Þ ¼ ð57; 3Þ and M mn s.75 wth ð; Þ ¼ð9; 8Þ as shown n Fg. 3. The M value s :473:79 n the regon of ¼ 456 and there s no bg dfference between and M mn. Thus, the factor M s almost sotropc n any crystallographc orentaton n Type. When the prmary slp system s gven as fgh and regardless of Types 4, no dfference of M and ts dependence on the tensle axs s present (Fg. 4). s 3.8 at the tensle axs wth ð; Þ ¼ð33; 3Þ and M mn s.75 wth ð; Þ ¼ð9; 8Þ. In the tensle axes wth <9 and >8, ther M factors are the same as that n Type. In Types 4, however, the M factor s remarkably hgher than that n Type around the tensle axes wth ð; Þ ¼ ð9; 8Þ and ð; Þ ¼ð7; Þ. hgh gradent n M, therefore, s produced at or near ¼ 9 and ¼ 8. In the smple analyss by the Sachs model where only the prmary slp system operated, the M factor was ) When

4 598 M. Morta and O. Umezawa (a) Type [] [] dw mn =.8 ;(α, β)=(79,) =.88 ;(α, β)=(7, ) dw max = 4.67 ;(α, β)=(,) M mn =.75 ;(α, β)=(9, 8) M mn [] [] [] [] (b) Type 3 [] Fg. 6 Dependence of Taylor factor on tensle axs under the condtons of Types 5 and 6. dw mn = 5.4 ;(α, β)=(55,) dw max = 37. ;(α, β)=(,) M mn [] [] (c) Type 4 dw mn =.9 ;(α, β)=(79, ) dw max = 7.79 ;(α, β)=(43, 3) M mn the homogeneous slp deformaton s deally avalable n a gran as n ths study, the average M, M ave, s about half the M factor of the Sachs model on all tensle axes and types (Type : M ave ¼ :4, Types 4: M ave ¼ :63, Types 56: M ave ¼ :46). 3. Influence of secondary slp systems on plastc work rate When fgh was chosen as the prmary slp system, dw hghly depended on the operatng secondary slp systems. When fgh and fgh3 were operatonally equvalent to secondary slp systems (Type ), the tensle axes of the plastc work rate maxmum, dw max and the mnmum, dw mn were dfferent wth those of M mn and, respectvely (Fg. 5(a)): the dw max s 4.67 at ð; Þ ¼ð; Þ, and the dw mn s.8 at ð; Þ ¼ð79; Þ. dw contnuously decreases as ncreases from to 8. [] [] [] Fg. 5 Dependence of plastc work rate on tensle axs drecton under the condtons of Types, 3, and 4. When fgh was chosen as the secondary slp system and the operaton of fgh3 was prevented (Type 3), dw was hgher than that of Type n all of the tensle axes (Fg. 5(b)). dw max s 37. at the same axs wth ð; Þ ¼ð; Þ as n Type, and dw mn s 5.4 at ð; Þ ¼ ð55; Þ. The tensle axes of dw max and dw mn are also not consstent wth those of M mn and. dw max s.5 tmes as hgh as that n Type and dw s.3 tmes at ¼ 7.t around ¼ 55, dw s almost the same wth that n Type, therefore, the restrcton of the fgh3 operaton s not responsble for the plastc work rate. When fgh3 was chosen as the secondary slp system and the operaton of fgh was prevented (Type 4), dw ncreased n the regon of ¼ 7 where the axs wth ð; Þ ¼ð43; 3Þ gave dw max.3 tmes as hgh as that n Type (Fg. 5(c)). Near the axes of ¼ and ¼ 9, dw s almost the same wth dw mn and that n Type. 3.3 Slp deformaton behavor of -ttanum alloys at low temperature The dependence of the tensle axs drecton on M and dw was evaluated at 3 K. The 3 K RSS of each slp system was adopted, as lsted n Table (Type 5). s equal to.88 at ð; Þ ¼ð7; Þ, as shown n Fg. 6, and dw max s equal to 54.9 at ð; Þ ¼ð; Þ. dw contnuously ncreases as the angle ncreases and equals 7. at ð; Þ ¼ð55; Þ. t 77 K (Type 6), M s the same as that of Type 5 (Fg. 7(b)). oth dw max (74.3) and dw mn (.89) also have the same tensle axes as those n Type 5. Type 3 gave a smlar rato of RSS wth Types 5 and 6. The results at 3 K and 77 K show almost the same dependence of dw on tensle axs wth Type 3, although ther tensle axs of s dfferent from that of Type Dscusson 4. Dependence of domnant slp system on tensle axs The M factor around the tensle axes wth ð; Þ ¼ð33; 3Þ and ð; Þ ¼ð7; Þ s hgher n Types 4, where fgh s the prmary slp system (Fg. 4), than that n Type where the RSSs of three prmary slp systems are

5 Slp Deformaton nalyss ased on Full onstrants Model for -Ttanum lloy at Low Temperature 599 (a) Type 5 [] (a) Type, prsmatc [] 3 dw mn = 7. ;(α, β)=(55, ) dw max = 54.9 ;(α, β)=(, ) M mn [] [] [] [] (b) Type 6 [] (b) Types -4, prsmatc [] 3 dw mn =.89 ;(α, β)=(55, ) dw max = 74.3 ;(α, β)=(, ) M mn [] [] Fg. 7 Dependence of plastc work rate on tensle axs under the condtons of Types 5 (a) and 6 (b). : (α, β)=(33, 3) : (α, β)=(33, ) : (α, β)=(7, ) [] [] Fg. 8 Number of actve prsmatc slp systems under the condtons of Type (a) and Types, 3, and 4 (b). Table ctve fve ndependent slp systems and ther slp rates for tensle drectons,, and shown n Fg. 8(b). - ðþ½š ðþ½š ðþ½š ðþ½š ðþ½3š M ¼ 3: ðþ½š ðþ½š ðþ½š ðþ½š ðþ½3š M ¼ 3: ðþ½š ðþ½š ðþ½š ðþ½3š ðþ½3š M ¼ : ðþ½š ðþ½š ðþ½š ðþ½3š ðþ½3š M ¼ : the same (Fg. 3). It s suggested that the regon of the tensle axs operatng at fgh should be enlarged to ncrease ts slp rates. In ths secton, the relatonshp between the operatng slp system and slp rate s dscussed. The number of actve slps of fgh n Types 4 s ncreased from to at the tensle axs around ¼ 33 relatve to Type (Fg. 8). ecause fgh has two ndependent slps, t seems to domnate the deformaton around ¼ 33 n the tensle axs. However, the ncrease of M depends on the angles as well (Fg. 4), and n that case the number of actve slps n the prmary system s not suffcent to dscuss the domnant slp system durng deformaton. Here, the tensle axes, -ndcated as ponts of : ð; Þ ¼ð33; 3Þ, :ð; Þ ¼ð33; Þ, and : ð; Þ ¼ ð7; Þ n Fg. 8(b) were chosen to represent the axes that ncreased M n Types 4. ll fve actve slp systems and ther slp rates n Types 4 were evaluated (Table ). Pont gves the axs of n Types 4. shows an ncrease of.94 relatve to M at the same axs n Type. There are two sets of fve ndependent slp systems for mnmum work at the axs lsted n Table as - and -. Each set nvolves two knds of fgh,.e., ðþ½š and ðþ½š, and ether ðþ½š or ðþ½š, revealng a large slp rate. However, ther Schmd factors are the same and as low as S ¼ :8 6,8) so that the fgh s geometrcally hard to operate. In Types 4, therefore, the restrctons of other slp systems result n the operaton of fgh because of ther hgher RSSs. The operaton of fgh n tenson along axs should be accompaned by deformaton along the c-axs. s a result, the slp rate of fgh3 may be hgh. t tensle axs, the Schmd factors of the two slp systems, ðþ½š and ðþ½š are the same

6 6 M. Morta and O. Umezawa (a) Types 5 and 6, prsmatc [] 3 (b) Types 5 and 6, pyramdal [] 3 [] [] : (α, β)=(33, 3) : (α, β)=(33, ) : (α, β)=(7, ) [] [] Fg. 9 Number of actve slp systems under the condtons of Types 5 and 6 for prsmatc (a) and pyramdal (b). Table 3 ctve fve ndependent slp systems and ther slp rates for tensle drectons,, and shown n Fg ðþ½š ðþ½š ðþ½3š ðþ½3š ðþ½3š M ¼ : ðþ½š ðþ½š ðþ½3š ðþ½3š ðþ½3š M ¼ : ðþ½š ðþ½š ðþ½š ðþ½3š ðþ½3š M ¼ : ðþ½š ðþ½š ðþ½š ðþ½3š ðþ½3š M ¼ : (S ¼ :8) and equal to the ones at axs. Furthermore, ther slp rates are small. Thus, fgh s less conducve to deformaton n total even though t becomes actve. t the tensle axs, both ðþ½š and ðþ½š can easly operate, because ther Schmd factors are hgh (S ¼ :38) and fgh operates more easly when the secondary slp systems are suppressed. In Types 5 and 6, the slp systems and ther slp rates were also evaluated (Fg. 9 and Table 3). ecause every actve slp system and ts slp rates n Types 5 and 6 are the same as those n Types 4 at the tensle axes and, the deformaton behavor of fgh n Types 5 and 6 s smlar to those n Types 4. t the tensle axs, however, M s not hgher than that n Type. No fgh system s chosen, but three systems n fgh3 are actve (Table ). Therefore, fgh and fgh3 are domnant at the tensle axs. ased on the above, the domnant deformaton mode of fgh and the role of the secondary slp systems n the homogeneous deformaton s dscussed n sectons 4. and Domnant deformaton modes n prmary slp systems When all the RSSs of the prmary slp systems are the same (Type ), fgh can hardly operate at the tensle axes around and where M shows a steep nclne. When the operaton of fgh s predomnant (Types 4), both fgh and fgh3 are hardly actve. Then, fgh must operate (Fg. 8(b)), and ts slp rates has to ncrease. Therefore, fgh s the domnant mode. The same mode appeared n Types 5 and 6, but the secondary systems of fgh and fgh3 are also domnant at the tensle axs. t the tensle axs n Types 6, the two slp systems of fgh are chosen (Fgs. 8(b) and 9(a)) but also ther slp rates are small (Tables and 3). Therefore, t s not fgh, but fgh and fgh3 that are the domnant modes at the axs. The tensle axs where the prmary slp system fgh s domnant s evaluated. In Types 4, the steep nclne of M at the tensle axes around ¼ 8 and 77 (Fg. 4) must orgnate n the extenson of the regon where fgh s domnant. The ncrease of M from the condton of Type s smaller at the tensle axes wth ¼ 477, although the Schmd factor s hgher. 6,8) In ths regme, M s comparatvely hgh regardless of the secondary slp systems operaton, and the number of actve fgh s mostly the same (Fg. 8). dw s also not so senstve on the restrctons of the secondary slp systems (Fg. 5). t a tensle axs wth ¼ 477, the secondary slp systems have less of an effect on the deformaton, therefore, fgh s domnant. Hence, the prmary slp system fgh s the domnant mode at the tensle axes around, and for ¼ 477, when the operaton of the secondary slp systems s suppressed (Types 6).

7 Slp Deformaton nalyss ased on Full onstrants Model for -Ttanum lloy at Low Temperature 6 (a) Types -4, pyramdal, = [] 5 [] (b) Types -4, basal [] [] [] Role of secondary slp systems n homogeneous deformaton The tensle deformaton n whch fgh s predomnant s dscussed. In Type 3 where the fgh3 operaton s suppressed (Fg. 5(b)), dw s hgher than that n Type at all tensle axes (Fg. 5(a)). ecause the operatons of fgh and fgh3 are equvalent as the secondary slp system n Type, the deformaton n arbtrary shape needs an operatng system of fgh3 at least (Fg. (a)). Furthermore, fgh3 s the domnant mode at the tensle axs at ¼ 3, because dw s remarkably hgher there. Especally at the [] tensle axs, M s relatvely lower than the rest (Fg. 4), and the operatng fve slps are covered wth only the fgh3 system at the tensle axs at ¼ (Fg. (a)). Thus, an operaton of fgh3 s necessary to deform homogeneously at the tensle axs around []. However, t s dffcult to observe fgh3 dslocatons n tenson along [] at low temperature. ) ecause the RSS of fgh3 s ncreased at lower temperature,,) dw s hgher at the tensle axs of [] (Fg. 7(b)). In the tensle axes at ¼ 779, dw s not senstve to the actvty of fgh3 as shown n Fg. 5, although a few fgh3 systems are operatve there (Fg. (a)). Namely, the fgh3 slp system s not domnant n the tensle axes at ¼ 779. In the tensle axes at ¼ 4565, other slp systems are domnant, because the actvty of fgh3 s not responsble for dw (Fg. 5). 4 3 [] Fg. Number of actve slp systems under the condtons of Types, 3, and 4 for pyramdal (a) and basal (b). 3 prsmatc + pyramdal (Types -4) or basal + pyramdal (Types 5-6) pyramdal [] [] basal + pyramdal prsmatc + basal prsmatc [] Fg. Schematc llustraton of domnant slp systems for tenson n - ttanum at low temperature. The slp systems of fgh and fgh are manly chosen n the tensle axes at ¼ 456 (Fg. ). When the operaton of fgh s suppressed (Type 4), dw n the tensle axes at ¼ 365 s hgher than that n Type where the secondary slp systems of fgh and fgh3 are operatve (Fg. 5). dw max s also gven at the tensle axs wth ð; Þ ¼ð43; 3Þ n Type 4. onsequently, the fgh operaton s domnant n the tensle axes at ¼ 365, but t s not actve n all tensle axes (Fg. (b)). Thus, the fgh operaton results n lowerng dw and s not necessary for arbtrary deformaton. In tensle axes at ¼ 779, the secondary slp systems are not domnant. The prmary slp system fgh that s domnant agrees wth the hgh Schmd factor. 7,8) Fgure shows the domnant slp systems for arbtrary deformaton, when fgh s the prmary slp system. When the operaton of the secondary slp systems s suppressed, plastc deformaton hardly proceeded n the tensle axes at ¼ Slp deformaton n low-temperature hgh-cycle fatgue of -ttanum alloy ccordng to the analyss usng the RSSs of -ttanum alloys at 77 and 3 K (Types 5 and 6), the operaton of fgh and fgh s preferred n the tensle axes at ¼ 465, as shown n Fg. where dw s low. However, the operaton needs a large slp rate to produce unt stran n a gran, because ther M values are hgh. The slp rate may result from the ncreasng dslocaton densty so that the pled-up dslocatons on fgh and fgh should readly occur. The analyss agrees wth the expermental result n whch pled-up dslocatons of fgh were observed n the fatgued samples at 77 K. ) The operaton of fgh3 s domnant at tensle axs wth ð; Þ ¼ð; Þ where dw max s gven. However, the operaton of fgh3 s hard, because ts RSS s hgh at low temperature. In fact, the operaton of fgh3 n the fatgued samples can be hardly observed by transmsson electron mcroscopy. ) Therefore, a large elastc stran feld may accumulate under ths condton. Facet or facets of {} were detected n the subsurface fatgue crack ntaton ste of -ttanum alloys. 5) It supports the development of openng stress (mode I) normal to {}. It s dffcult to understand the transgranular crackng on 3

8 6 M. Morta and O. Umezawa {} due to the openng stress only. few models of crackng mechansm have been proposed. One s transgranular crackng on {} due to the combnaton of nternal stress concentraton and appled stress. 9) The other s the slp-off on {} and ts growth due to localzed slp deformaton. ) There s a strong possblty that slp-off occurs, because the development of dslocaton ple-ups on {} s preferred. However, no drect evdence of the crackng has been found. In hgh-cycle fatgue fracture of -ttanum alloys, the facet of fg ) or fg,) has been also detected at the crack ntaton ste. It was ponted out that there was a possblty to form the fg facet due to the stress concentraton wth dslocaton ple-ups of fgh and fgh as well as the {} facet. Elastc stress can be accumulated normal to fg at low temperature, because dw s comparatvely hgh at the tensle axs at ¼ 9, as shown n Fg. 7. The crackng of fg twn s dfferent from that of {} and fg crackng, and may occur by another mechansm. n advanced deformaton model that also consders twnnng needs to be developed. 5. onclusons We evaluated the role of prmary slp systems n tensle yeldng and stress accumulaton n -ttanum alloys at low temperatures by usng the Taylor model. The major conclusons are as follows: () The prmary slp system fgh was domnant n the tensle axes near ð; Þ ¼ð33; 3Þ and ¼ 49. () The operaton of fgh decreased the plastc work rate n the tensle axes at ¼ 365 and was responsble for arbtrary deformaton. (3) The secondary slp system fgh3 was domnant n the tensle axes at ¼ 3. The operaton of fgh3 was related to the deformaton n all tensle axes and was necessary for arbtrary deformaton. (4) The plastc work rate was hgh at ð; Þ ¼ð; Þ, because RSS of fgh3 was hgh at low temperatures. (5) The dslocaton ple-ups of fgh and fgh may be the largest n the tensle axes at ¼ 465. (6) The hghest elastc stress may be developed n the tensle axs at ð; Þ ¼ð; Þ and may be responsble for () transgranular crackng. REFERENES ) O. Umezawa and K. Naga: ISIJ Int. 37 (997) ) H. Yokoyama, O. Umezawa, K. Naga, T. Suzuk and K. Kokubo: Metall. Mater. Trans. 3 () ) M. R. ache, W. J. Evans and H. M. Daves: J. Mater. Sc. 3 (997) ). Sarrazn, R. hron, S. Lesterln and J. Pett: Fatgue Fracture Eng. Mater. Struct. 7 (994) ) V. Snha, M. J. Mlls and J.. Wllams: Metall. Mater. Trans. 37 (6) ) O. Umezawa, K. Naga and K. Ishkawa: Tetsu-to-Hagane 76 (99) ) M. Hamada and O. Umezawa: ISIJ Int. 49 (9) ) W. F. Hosford: The Mechancs of rystals and Textured Polycrystals, (Oxford Unv. Press, 993) pp ) G. Y. hn and W. L. Mammel: Trans. Met. Soc. IME 39 (967) ) S. Hanada: Tetsu-to-Hagane 4 (99) ) Notaton for dfferentaton can express as both Lebnz s notaton (dw, d", and d) and Newton s notaton ( _W, _", and _). Lebnz s notaton (dw, d", and d) s used n ths paper. ) H. Numakura: Matera Japan 37 (998) ) H. onrad, M. Doner and. Meester: Ttanum Scence and Technology, Vol., (Plenum, New York, 973) pp ) N. E. Paton and W.. ackofen: Metall. Trans. (97) ) E. D. Levne: Trans. Met. Soc. IME 36 (966) ) F. rder, P. Vllechase and J. Mendez: cta Mater. 53 (5) ) R. rmstrong, I. odd, R. M. Douthwate and N. J. Petch: Phl. Mag. 7 (96) ) F. rder, P. Vllechase and J. Mendez: cta Meter. 56 (8) ) M. R. ache: Int. J. Fatgue 5 (3) ) E. E. Sackett, L. German and M. R. ache: Int. J. Fatgue 9 (7) 5. ). J. eevers and M. D. Hallday: Metall. Sc. 3 (969) ) Y. Ono, M. Demura, T. Yur, T. Ogata, S. Matsuoka and S. Hor: Trans. Jpn. Soc. Mech. Eng. 74 (8)

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