Rate Sensitive Analysis of Texture Evolution in FCC Metals

Transcription

1 METALS AND MATERIALS, Vol. 3, No. 4 (1997), pp Rate Senstve Analyss of Texture Evoluton n FCC Metals Sh-Hoon CHOI and Kyu Hwan OH School of Materals Scence and Engneerng, Seoul Natonal Unversty Research Insttute of Advanced Materals San 56-1 Shnrm-dong, Kwanak-ku, Seoul , Korea The rate senstve analyss was used to nvestgate the texture evoluton durng plane stran compresson and combned deformaton (plane stran compresson and smple shear). The rate senstvty of the rate senstve model affects the orentaton densty dstrbutons n Euler space and change the major orentatons. As the rate senstvty of the rate senstve models ncreases, lattce rotaton vectors around major components decreases. 1. INTRODUCTION The texture evoluton durng rollng has been studed by many researchers durng last three decades [1-5]. The nhomogenety of texture through the thckness drecton s attrbuted to the formaton of shear texture at the surface of sheet metals. The rollng condtons such as frcton, roll geometry etc., and materal propertes play an mportant role n the formaton of the shear texture. These shear components have been calculated theoretcally [6-7] usng a smplfed stran state and Taylor model. In case of Taylor model, the number of actve slp system should be 5 n full constrant analyss or less than 5 n relaxed analyss [8-9]9 Rate senstve relaton between resolved shear stress and shear stran rate was frst proposed by Hutchnson [10]. Rate senstve analyss actvated all the slp systems and at the zero rate senstvty gave the same result wth Taylor-Bshop-Hll analyss [11]. The rate senstve analyss was used wdely n the analyss of the texture evoluton durng rollng [12,13], baxal stretchng [14], deep drawng [15,16] and torson [11,17,18]. Recently fnte element analyss was used to analyze the texture evoluton durng rollng. Mathur et al. used the rate senstve consttutve model to develop an Euleran fnte element analyss whch can smulate the evoluton of the crystallographc texture n the steady state bulk formng process [19[. Kaldn predcted the evoluton of the texture n FCC metals wth ntal texture usng rate senstve model [20]. In most analyses of rollng processng whch s combned wth shear deformaton, the sample symmetry has been orthorhombc. In ths study, texture evoluton n FCC metals durng plane stran compresson and combned deformaton (plane stran compresson and smple shear) was analyzed by rate nsenstve and rate senstve model consderng non-orthorhombc sample symmetry. Rate nsenstve analyss was carred out by the full constrant Taylor-Bshop-Hll model [21] and Renouard-Wntenberger method [22]. Rate senstve analyss was carred out by the power law relaton [10] and full constrant boundary condton. The three-dmensonal lattce rotaton felds around major components were determned numercally usng the rate nsenstve and rate senstve models. 2. ANALYSIS 2.1. Rate nsenstve and senstve analyss Rate nsenstve analyss In the Taylor-Bshop-Hll analyss, the deformaton rate of each crystallte s assumed to be dentcal wth the prescrbed deformaton. The prescrbed deformaton rate at crystal coordnate, k~' can be obtaned by the transformaton of deformaton rate, kl~x, prescrbed at the sample coordnate. 9 Cx ~?j =lk Ijl 8kl (1) Usng Taylor-Bshop-Hll model, tle work rate W can be gven by W(%) = ~j ej (2) whch can be expressed as

2 - ~ Rate Senstve Analyss of Texture Evoluton n FCC MetaL, 253 W(~ = - Bkl' + Ak22 + 2F~23 + 2Gk31 + 2He12 (3) A = ( ), B = (0-33- al 1), F= 023, G= 0"31, H = o-~2 where r s the crystallographc stress state. The work rate s calculated for the 56 stress state obtaned by Bshop and Hll [23] and the stress states of maxmum work become the yeldng stress states. Thus the crystallographc stress state for the prescrbed stran rate can be determned. From the selected stress states, the actve slp systems are determned and the shear rate ), of each slp system for the prescrbed stran rate can be determned accordng to the followng equaton. 1 k = y~ -4- (m~ + m-,':)~ s By solvng above lnear equaton, shear rate y, can be obtaned. But there s an ambguty on the selecton of the actve slp system for the calculaton of the shear rate, because the number of actve slp system at the stress state s generally larger than 5 (e.g. 6 or 8 n FCC metal). Ths ambguty can be removed usng "second order plastc work mnmzaton" proposed by Renouard and Wntenberger [22]. Accordng to the method, a small stran s appled to each possble set of slp system and crystal rotaton of each set of slp system s obtaned. The new orentatons due to crystal rotaton gve the correspondng plastc work.e. the second order plastc work. The set of slp systems of whch the second order plastc work s mnmum under the appled small stran, s the actve slp system Rate senstve analyss The deformaton of rate senstve analyss s usually modelled by a power law relatonshp between the shear rate ), and the resolved shear stress r at a slp system s. m m-i (4) : r,, v- (5) m s postve and called as rate senstvty. "co s a reference shear stress and ~, s a reference shear rate. The sgn term n Eq. (5) means that the shear rate has same sgn as the resolved shear stress. The resolved shear stress s related to the devatorc stress tensor % of the crystal through the followng relaton9 r s m~ ~ (6) where m~ s = b~ ~; nj, q s defned by the component of the unt vector n. ~ normal to the slp plane and the unt vector b ~ 1 parallel to the slp drecton of the slp system s. Ignorng the elastc deformaton, the vectors nj,~ and ba S can be regarded as orthogonal. The component of the stran tensor k assocated wth the gven stress tensor ~ can be obtaned as follows k- S 1 j (m~ + mg~ 1 gol~, -~ (mj + mj ) mklo'~ (7) It should be noted that the deformaton rate can be deduced from a stress potental [10]. The stress potental can be gven as follows f(~j) m ~, ]1 ~,+' (m+l) v,, t/m ~s j T(m~+mjg~J _ m (1 0f(~j) (8) k'j- O~j) where W (@j) s the rate of plastc work accordng to the prescrbed deformaton rate k n the Taylor-Bshop-Hll analyss. The stress state whch satsfes the above e- quaton at the gven deformaton rate can be obtaned by the Newton-Raphson method [11,24]. The soluton of Eq. (7) was converged regardless of ntal condton. The stress and stran tensor can be reduced nto the followng vectors durng calculaton. 0-1 = N72 (O': )' ~ = ~ O~:~3, 0"3 : "~f2~o"23 s = ~ (s, C2 = -- E33, E'~ = x/2cv~ ~2 "4~ " -- e 4 = N[2,g31, ~c 5 = "f2e Calculaton of the lattce rotaton ~ s the rotaton rate of crystal due to only slp de- formaton wthout rgd body rotaton and elastc deformaton. Thus &lj s called as glde rotaton rate [11]. ~, = ~ 1 (m~_ mjg~ (10) The lattce rotaton rate s defned as the rotaton rate ~2j of the crystal lattce wth respect to the laboratory. ~j s the rotaton rate of prncpal nterest n texture evoluton and can be gven as follows (9)

3 25A Sh-Hoon CHOI and Kyu Hwan OH Fg. 1. Sample coordnate system, KA and crystal coordnate system, KR. S The lattce rotaton rate can be obtaned from the prescrbed velocty gradent L~j and shear rate ~. ~, s cal- culated from rate nsenstve or rate senstve analyss. The Euler angles (%, ~, %) of the deformed materals should be changed accordng to the lattce rotaton rate as follows [25]. ~ l=(f223sn <p2 + {2 31cos %)/snq~ q~=~23 c~ q~2-~231 sn <P2 (12) 2=.h:- ~,cos'/' 2.3. Numercal calculaton of the texture evoluton The numercal procedure descrbed n above sectons was programmed. 800 crystallte orentatons were used as the ntal random orentaton. The prescrbed deformaton rates of plane stran compresson, combnaton of plane stran and shear stran were taken n ths study. The followng velocty gradent was used durng calculaton. Lj = /7 0 a = L11 0 r I s the stran rate ncrement durng plane stran compresson and set to be 0.01 n ths study, ct values of 0 and 5 were consdered to nvestgate the effect of shear deformaton on the texture evoluton. The total Fg. 2. (111) pole fgure of ntal random texture wth 800 crystal. plane stran of 1.0 was used. The m values of 0.05, 0.1, (/.25 and fl.5 were used n ths study to see the effect of rate senstvty on the texture evoluton durng deformaton. Fg. 1 shows the crystal (KB) and sample (KA) coordnate systems. In ths study, the prncpal axes of deformaton rate tensor are coaxal wth those of the sample coordnate system. 3. RESULT AND DISCUSSION 3.1. Plane stran compresson (o~=0) The ntal orentaton dstrbuton of FCC metals n ths study was descrbed by 800 randomly orented grans. Fg. 2 shows the (111) pole fgure of the ntal random texture. Durng rollng processng, the detbrrnaton n a center regon can be characterzed by plane stran compresson. The major orentaton of rollng textures smulated numercally by Htch et al. [26] was cube {100} <001>, G (Goss) {110} <100>, B (brass) {110} <112% S {123} <523> or {123} <634>, C (copper) {112} <111> and D (Dlamore) {44 11} < >. The deformaton textures of FCC metals durng rollng form ~'o fbres: the ~ (G-B) and the Fg. 3. The calculated (111) pole fgures of plane stran compresson usng rate nsenstve and rate senstve models. rate nsenstve model (b)-(e) rate senstve model.

4 Rate Senstve Analyss of Texture Evoluton n FCC Metals (B-S-C/D) fbres. Fg. 3 shows the calculated (111) pole fgures of plane stran compresson usng rate nsenstve analyss (Taylor-Bshop-Hll model and RW method) whch s equvalent to the result of m=0 and rate senstve models for m=0.05, 0.1, 0.25 and 0.5 (b-e). The calculated pole fgures for m=0.05 and 0.1 are smlar to those of rate nsenstve model. In order to nvestgate texture evoluton, ODF (orentaton dstrbuton functon) was calculated usng deformed crystal orentatons and flterng procedure n the Euler space [27-29]. Fg. 4 shows the calculated ODF secton ((P2= 45") for plane stran compresson at varous rate senstvty. Trclnc sample symmetry was consdered durng ODF analyss to compare wth the results of the combned deformaton. The sample symmetry n Fg. 4 can be characterzed as orthorhombc sample symmetry. Fg. 5 shows the calculated orentaton densty, f(g) along fbre A ((p~= = 0'~-'', (2: = 45'D. As rate (0 0 1) [L l o ] (.4 4 I I) ( I I) (l 0) [ ~ ~ 8~.... [] I 2] [0 [1..--o--- : rate nsenstve --,<>-- : rn=o,05 ~P1= 9 0 ~ : m=o. 1 : rn= = ~2=-450 : m=0.5 f(g) 20-6"0.... Co) (I ~ o) l 40 o] (1 I O) (I I O) (I [lll I~2] [0 - - o - - : rate nsenstvty Contour, Lnes : , c-- :::> % m=o.05 : m=o. 1 r 3 ~ ~ q,z--45 ~ : m= ~ 2 --o--- : m=0.5 f(g) z0 o (c) (b) 0 (44u) (2 I 3) (] 0 1) ---o.-- : rate nsenstvty ---o-- : m=o (c) --.a,-- ; m=o.1 9 : rn= : m=0.5 fg) (d) o q~2 (e) Fg. 4. The calculated ODF secton (q3:=45~ for plane stran compresson. Fg. 5. The calculated orentaton densty, f(g) along fbre A. (c~=d), (b) The calculated orentaton densty, f(g) along fbre C. (o~=0), (c) The calculated orentaton densty, f(g) along fbre ~. (o~=0)

5 256 Sh-Hoon CHOI and Kyu Hwan OH 9 (4 4 11)~11 II I-] (b) (c) (d) Fg. 6. The calculated (111) pole fgures usng ODF for plane stran compresson. senstvty of rate senstve models ncreases, the maxmum orentaton densty of D component {44 11} < > decreases. Fg. 5(b) shows the calculated often- oo ~so ~ -:+==~,~ I : _.... [I, -- ~ ]~`~)~U``;`~`~;``~;~:``~`~;;~5~``~-~.~:~````:```~`~;~:'~:`~.::7~:~;~`t.~ -"~"~"'";;;;;;"~::"'""~ F~'~'~'~::~-~U'';;;;;;;'';';'''''~c-'''' l,... I., I , ]..... :: : ',::..::::2!!!!{!!!7:::::::'::::::',::.:::::!!!!![!!::T::::::!/ (b)!!!~!!!!!~!f!lllll[l!!!!!!g{!!!~!!!!!!!!!!!!!t++f!lll{l!!!!!!!!!!!!!! (c)!!!!!}[!!!!!!!!!!!!!!!!!! (d)!!!!j!!+ + :((((l!!! ll ll l ++![l (e) 9 (+ + ~;)[F 9 F B] : D (1 I o)[1 'T 2] : B Fg. 7. Lattce rotaton rates n the q02=45" secton for plane stran compresson. taton densty, f(g) along fbre C(q~t = 0"-", ~ = ", q~2= 45+'). The maxmum orentaton densty for rate senstvty of B component {110} <112> ncreases wth ncreasng rate senstvty. Fg. 5(c) shows the change of orentaton densty along ]3 fbre at varous rate senstvty. Wth ncreasng rate senstvty, the orentaton densty of D and S component decreases and that of B component ncreases. Fg. 6 shows (111) pole fgures recalculated from ODF. The predcted texture n rate nsenstve and rate senstve models for m=0.05, 0.1 and 0.25 (b-d) can be characterzed by plane stran texture of {44 11} < >. The calculated texture n rate senstve model for m=0.5 (e) cart be characterzed by {110} <112> and {44 11} < >. To nvestgate the stablty of a partcular orentaton, lattce rotaton felds are calculated for plane stran deformaton. Fg. 7 shows the lattce rotaton felds for the ODF secton (q~2=45"). The drecton of the arrows ndcates the orentaton change and the length ndcates the total rotaton rate. Most orentatons around brass and D move toward the exact brass and D orentaton, respectvely. For m=0.5 (e), most orentatons around brass and D move slower than for m=0.05, 0.1 and 0.25 (b-d). From the lattce rotaton felds, durng plane stran compresson, orentatons move frst nto the o~ fbre and nto the ]3 fbre, then along c~ to 13. At the lower rate senstvty (a-c), the total rotaton rate of most orentatons are large. Thus most of orentaton are dstrbuted between S and D as shown n Fg. 5(c). However, at the hgher rate senstvty over m=0.25 (d-e), the total rotaton rate of orentatons around a and ]3 fbre are small. Thus most of the orentatons move frst nto the ~z and 13 fbre and move slowly along ]3 fbre. At hgh value of rate senstvty (e), the orentaton densty along ~ fbre shows the unform dstrbuton and have the maxmum densty at {110} <112> and {44 11} < >, as shown n Fg. 5(c) Combnaton of plane stran compresson and smple shear (a=5) Durng rollng of FCC metals, the deformaton state n

6 Rate Senstve Analyss of Texture Evoluton n FCC Metals 257 Fg. 8. The calculated (111) pole fgures for combned stran deformaton usng rate nsenstve and rate senstve models. a surface regon dffers from plane stran compresson due to the frctonal nteracton between roll and metals. The deformaton state at the surface regon can be characterzed by the combnaton of plane stran compresson and smple shear. Ths deformaton state gves the nhomogeneous texture through thckness drecton. Fg. 8 shows the calculated (111) pole fgures of the combned deformaton, usng rate nsenstve and rate senstve models for m=0.05, 0.1, 0.25 and 0.5 (b-e). The calculated (111) pole fgures for m=0.05 and 0.1 show the smlar results wth that of rate nsenstve model. Fg. 9 shows the calculated ODF secton (q02=45~ (o o ) [l 1 O] (] II) [~ ~ 2] (4 4 II) Ill ]l 8] (l O) [o ~] -.-.o--- : rate nsenstve --o-- : m:o.05 ' rn=o. 1 <P1=~ 9 : m=0.25 e2 = 4 5 ~ 40 f(g) 20 O) (1 1 l) (l I l) (1 I I) (I [ l o] [ 1 2 I] [0 11 [! 2 I) : rate nsenstvty --o-- : m=o.05 r : m=o = : m= : m:o.5 ~ ~2=45 ~ f(g) 10 q)l Fg. 9. The calculated ODF secton (q)2=45~ for combned stran deformaton. Fg. 10. The calculated orentaton densty, f(g) along fbre A. (ct=5), (b) The calculated orentaton densty, f(g) along fbre B. (~=5)

7 258 Sh-Hoon CHOI and Kyu Hwan OH Fg. 11. The calculated (111) pole fgures usng ODF for combned stran deformaton. for the combned deformaton. Trclnc sample symmetry was consdered to evaluate the ODF. The sample symmetry for combned deformaton can not be characterzed as orthorhombc or monoclnc. The major components are dfferent from those of plane stran compresson shown n Fg. 4. The maxmum orentaton densty around major component for m=0.5 (e) s lower than that of plane stran compresson. The calculated orentaton densty shows more complcate dstrbuton than that of plane stran compresson. Fg. 10 shows the calculated orentaton densty along fbre A (q01= ~ q~=0%", tp2=45'~). The maxmum densty orentaton s between RC (rotate cube) component and D component (001) [110] and D component (44 ll) [ ]. Wth ncreasng rate senstvty, the maxmum densty orentaton moves toward RC component and the value of maxmum densty decreases. Fg. 10(b) shows the calculated orentaton densty along fbre B (q)l=0~ ~ (I) ~ q0z=45"). The fbre B can be characterzed by { 111 }// ND texture. In the values of m > 0.1, the orentaton densty of {111}//ND texture components s neglgble. Fg. 11 shows the calculated (111) pole fgures for combned deformaton at varous rate senstvty. The calculated texture n rate nsenstve and rate senstve models lot m=0.05, 0.1 (b-c) can be characterzed by shear type texture components of (119) [14 143], (557) [ ] and (557) [ ]. For the m=0.25 (d), major component s characterzed by shear type texture component, (119) [14143]. These shear type texture components (119) [14 143] and (557) [ ] have 20 ~ and 15" rotatonal relatonshps wth (44 11] [11 118] and (110) [001] orentatons about TD, respectvely. For m= 0.5 (e), ntensty of pole fgure has the value of about 1 and the major component can not be seen as expected from Fg. 9(c). Fg. 12 shows lattce rotaton felds for combned stran deformaton. Durng combned deformaton, orentatons around shear type texture components move toward the shear type texture components. 4. CONCLUSION Fg. 12. Lattce rotaton rates n the q02:45 ~ secton for combned stran deformaton. The texture evoluton of FCC metals durng plane stran compresson and combned stran deformaton (plane stran compresson and shear stran) was analyzed by rate nsenstve and rate senstve models. Durng the plane stran compresson, the major orentatons were classfed by {4411} < >,. D component and

8 Rate Senstve Analyss of Texture Evoluton n FCC Metals 259 {110} <112>, B component. The D component was stable n rate nsenstve and rate senstve for m=0.05, 0.1 and The B and D components were the major components n rate senstve model for m=0.5. For combned deformaton, shea r type texture components ((1 19) [14 143], (557) [8 II 14] and (557) [ ] were major orentatons. ACKNOWLEGMENT Ths work was fnancally supported by the Korean Mnstry. of Educaton through the Advanced Materals Research Program n REFERENCES 1. I. L. Dllamore and W. T. Roberts,.L h~st. Metals (1963/1964). 2. W. Truszkowsk, J. Krol and B. Major, Metall. Trans. 11, 749 (1980). 3. T. Kamjo and H. Fukutom, n Proc. of the 16th Rso Int.,~vmp. oll Mat. Sc., 377 (1995). 4. C. H. Cho, J. W. Kwon, K. H. Oh and D. N. Lee, Acre metal, n press (1997). 5. S. H. Cho, J. W. Kwon and K. H. Oh, Metals and Materals 2, 133 (1996). 6. C. S. Lee and B. J. Duggan, MetaH Trans. 4, 67 (1991). 7. C. S. Lee and R. E. Smallman and B. J. Duggan, Mat. Sc. Tech. 10, 149 (1994). 8. P. Van Houtte, Mat. Sc. Eng. 55, 69 (1982). 9. P. Van Houtte, Textures and Mcrostructures, 8-9, 313 (1988). 10. J. W. Hutchnson, Proc. R. Soe. A348, 101 (1976). 11. L. S. Toth, P. Glormn and J. J. Jonas, Acta metall. 36, 77 ( Y. Zhou, K. W. Neale and L. S. Toth, Acta metall. 39, 29 (1991). 13. Y. Zhou, L. S. Toth and K. W. Neale, Acta metall. 40, 3179 (1992). 14. Y. Zhou and K. W. Neale, Acta metall. 42, 2175 (1994). 15. J. Savoe, Y. Zhou, J. J. Jonas and S. R. Macewen, Acta metall. 44, 587 (1996). 16. Y. Zhou, J. J. Jonas and K. W. Neale, Acta metall. 44, 7 (1996). 17. K. W. Neal, L. S. Toth and J.J. Jonas, hlt. J. Plast. 6, 45 (19). 18. L. S. Toth, J.J. Jonas, P. Glormn and B. Bacrox, Int. J. Plast., 6, 83 ( K. K. Mathur and P. R. Dawson, Int. J. Plast. 4, 67 (1989). 20. S. R Kaldnd and L. Anand, J. Mech. Phys. Solds. 42, 459 (1994). 21. G. I. Taylor,,L hst. Metals 63, 7 (1938). 22. M. Renouard and M. Wntenberger, C. R. Aead. Sc. Pars. B292, 385 (1981). 23. C. N. Red, Deformaton Geometly for Materals Scentsts, Pergamon Press, (I973). 24. G. R. Canova, C. Fressengeas, A. Molnar and U. F. Kocks, Acta metall. 35, 1961 (1988). 25. A. Clement, Mater. Sc. Eng. 55, 203 (1982). 26. J. Hrsch and K. LOcke, Acta metall. 36, 2883 (1988). 27. S. Matthes, Phys. Stat. Sol. 101, 111 (1980). 28. S. Matthes and F. Wagoner, Phys. Star. Sol. 107, 591 ( S. Matthes, J. Muller and G. W. Vnel, Tertures and Mcrostructures 10, 77 (1988).