Simulation of textures and Lankford values for face centered cubic polycrystaline metals by using a modified Taylor model

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1 Ma, Roters, Raabe, exture and Ansotropy n CC Metals Smulaton of textures and ankford values for face centered cubc polycrystalne metals by usng a modfed aylor model A. Ma,. Roters, D. Raabe Max-Planck-Insttut für Esenforschung Max-Planck-Str Düsseldorf Germany or further detals please contact: raabe@mpe.de or roters@mpe.de Raabe, edoc Server, Max-Planck-Socety - - MPI Düsseldorf

2 Ma, Roters, Raabe, exture and Ansotropy n CC Metals ABSRAC hs report presents a modfed aylor model s presented whch statstcally consders gran nteracton n a polycrystallne aggregate n terms of a standard devaton for the symmetrc part of the velocty gradent. he model can be solved usng a Newton teraton method. We smulate crystallographc rollng textures and the ansotropy arsng from unaxal tenson tests (ankford values for dfferent drectons n the rollng sheet plane). he results reveal n part a good agreement wth expermental data. Keywords texture, ansotropy, r-value, mcromechancal modellng, rollng, plastc deformaton, CC metals, smulaton Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

3 Ma, Roters, Raabe, exture and Ansotropy n CC Metals Introducton. Motvaton for ths texture study Crystallne engneerng materals mostly occur n polycrystallne forms where each gran has a dfferent crystallographc orentaton, shape, and volume fracton. he dstrbuton of the gran orentatons s referred to as crystallographc texture. he dscrete nature of crystallographc slp along certan lattce drectons on preferred crystallographc planes together wth the occurrence of pronounced textures wth certan preferred orentatons ental an overall hghly ansotropc elastcplastc response of such polycrystallne samples under mechancal loads. An mportant am of polycrystal research conssts n developng mechancal models for understandng and predctng the evoluton of texture and crystallne ansotropy partcularly at large plastc strans. In ths context mechancal homogenzaton models (such as aylor-type models) play an mportant role due to ther consderable success n provdng n part excellent texture approxmatons on the bass of relatvely smple and, therefore, computatonally very effcent consttutve concepts [for overvews see -5]. he current study s concerned wth the ntroducton of a modfed aylor model whch consders gran nteracton n a polycrystallne aggregate n a statstcal fashon by ntroducng a standard devaton for the symmetrc part of the mposed velocty gradent. We use t for the smulaton of rollng textures and ankford values of face centered cubc (CC) metals..2 undamentals of plastcty homogenzaton models for textured polycrystals wo early versons of texture homogenzaton models were proposed by Sachs [6] and aylor [7]. he Sachs model s a no-stran-constrants (NC) model n whch the external stresses are consdered homogeneous for each ndvdual gran of a polycrystallne aggregate. Snce t neglects stran compatblty the Sachs model represents a lower-bound result for the stresses of a mechancally loaded polycrystallne sample [3]. he aylor model s a full-stran-constrants (C) approach n whch the external strans are equally vald for each sngle gran. Snce t neglects Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

4 Ma, Roters, Raabe, exture and Ansotropy n CC Metals stress equlbrum the aylor model represents an upper-bound result for the strans of a polycrystal [3]. Early modfcatons of ths full-constrants aylor model were ntroduced by ncorporatng stran relaxatons between neghborng grans, leadng to the so called relaxed-constrants aylor model varants (RC models) [8-]. In these approaches some of the external shear components are not transferred nto the gran,.e. the RC models allows for shear strans at the gran-scale. A relaxaton of the stran component ε 3 corresponds to a shear n longtudnal drecton (so called lath RC model), whle ε 23 denotes the transverse shear. he relaxaton of both ε 3 and ε 23 s referred to as pancake RC model. Allowng relaxatons for these shears locally leads to dstnct changes n the reorentaton rates and, therefore, n the texture development when compared to the predctons of the C aylor model. Although the texture predctons obtaned by RC models prncpally yeld better results than those of the NC or C approaches, devatons were observed partcularly n the large stran regme. herefore, dfferent approaches were suggested to render the classcal aylor or Sachs homogenzaton models physcally more plausble and n better accord wth expermental data. Conceptual modfcatons of the consttutve descrptons consst essentally n the ntroducton of gran-nteractons or respectvely nteracton penalty measures as well as n stran relaxaton schemes whch depend on the correspondng gan n the deformaton energy resultng from certan types and amounts of stran relaxatons. hese terms quantfy the elastc-plastc msmatch between neghborng grans or wthn larger gran clusters where the nteractng grans are typcally selected statstcally from a large set of sngle orentatons whch map the nherted texture. Detals about these model varants can be found n [2-8]. Smlar efforts were made for nstance by Mao [9] to modfy the Sachs model. He calculated the deformaton by actvatng prmary slp and assumng shear stran mpedment n terms of back stresses generated by the surroundng grans. In ths modfed Sachs model the nduced reacton stresses may ental the actvaton of addtonal slp systems whch compensate for the prmary deformaton. Related concepts are the self-consstent polycrystal models [e.g. 2]. In these approaches each crystal s treated as an elastc-plastc ncluson whch s embedded n an otherwse homogeneous effectve medum whch has the average propertes of the polycrystal. Self-consstent polycrystal models are solved n an teratve fashon for the deformaton and the stress. Although Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

5 Ma, Roters, Raabe, exture and Ansotropy n CC Metals self-consstent models satsfy both, stran compatblty and stress equlbrum at the same tme they represent one-ste approxmatons. Modfyng them nto N-ste approxmatons by consderaton of some local neghborhood would render them less effcent. Based on the aylor model, a neghborhood complance model was suggested by Sarma and Dawson [2]. In ths model a neghborhood s determned for every sngle crystal, whch contans a certan number of crystals around ths man sngle crystal. A new complance tensor for each sngle crystal embedded n such a gran cluster can be determned through a aylor calculaton and a local stress homogenzaton procedure. hs early approach of Sarma and Dawson to explctely ncorporate gran neghborhood effects nto texture models nspred our current study n whch we ntroduce a statstcal neghborhood for each gran of a polycrystal through a pseudorandom functon mposed for the velocty gradent tensor..3 Basc ntroducton to the crystallographc textures of rolled CC metals Owng to the cubc symmetry of the CC crystal system and the orthorhombc sample system whch s set up by the rollng drecton (RD), normal drecton (ND), and transverse drecton (D) of the sample, textures of rolled CC polycrystals are typcally presented n the reduced Euler space where an orentaton s gven by the three Euler angles ϕ, φ, and ϕ 2, ( ϕ, φ, ϕ 2 9 ). We use the Bunge notaton for the Euler angles throughout ths report [22]. Crystal orentatons can also be convenently descrbed by the use of Mller ndces {hkl}<uvw>. In ths concept the trple {hkl} descrbes the crystallographc plane parallel to the sheet surface whereas <uvw> ndcates the crystal drecton parallel to RD. Important texture components are on the fcc fber whch comprses all orentatons wth a common crystallographc fber axs <> parallel to the normal drecton ncludng major components {}<> (Goss-component, ϕ =, φ=45, ϕ 2 = ), {}<2> (Brass component, ϕ =35, φ=45, ϕ 2 = ), {}<>, and {}<> (9 about the normal rotated Goss-component, ϕ =9, φ=45, ϕ 2 = ) and the less symmetrc β skeleton lne ncludng major Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

6 Ma, Roters, Raabe, exture and Ansotropy n CC Metals components {2}<> (Copper component, ϕ =9, φ=35, ϕ 2 =45 ), {23}<634> (S component, ϕ =6, φ=32, ϕ 2 =65 ), and the Brass component {}<2> (ϕ =35, φ=45, ϕ 2 = ) [,2,4,8,9,23-25]..4 Macroscopc parameters for plastc sheet ansotropy Sheet rollng s related to a plane-stran compresson process [-4, 26]. he plastc ansotropy of a rolled sheet s typcally characterzed n terms of the ankford coeffcent [27] or R-value, whch s P P defned as the rato of the wdth plastc stran, ε W, to the through-thckness plastc stran, ε,.e. R = ε / ε (.) P W P Although ths smple defnton offers at frst vew an adequate and straghtforward means for assessng the n-plane and through-thckness flow and shape ansotropy of sheets, experments reveal that the R-value shows a sgnfcant stran dependence even at very small strans [2-4,28-3]. As addtonal parameters Kocks et al. [4] have suggested to use the rato of the wdth component of the stran rate, D, to ts through-thckness component,.e. r = D W / D (.2) and the rato of the wdth component of the stran rate to the length component q = D W / D. (.3) Mathematcally, the latter parameter q s a very helpful one because t wll not vary from to nfnty as the conventonal r parameter. Another parameter for the quantfcaton of the sheet resstance aganst thnnng and ts ansotropy was suggested by Spoldor [3] as Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

7 Ma, Roters, Raabe, exture and Ansotropy n CC Metals dy x ρ = (.4) dx y where x and y are the actual thckness and the wdth of the specmen, respectvely. hs parameter s desgned to take nto account the change of the loadng path durng a mechancal test. akng an emprcal theoretcal perspectve Rees [26] has developed an analytcal functon for the evoluton of the r value durng stranng whch, however, s based on the assumpton that the rollng ansotropy can be expressed by Hll's yeld functon. In our study we choose the q - parameter to study the planar ansotropy of rolled CC sheets. 2 Consttutve equatons for conventonal elastcplastc aylor modelng 2. Elastc rule Based on the somorphy assumpton [3], the actual elastc law of a sngle crystal can be represented by the referental elastc law through a plastc transformaton P. he elastc law for the current confguraton s )] 2 2PK = PK[ ( P CP I P (2.) C = (2.2) where s the deformaton gradent, 2PK s the second Pola-Krchhoff stress tensor, C s the rght Cauchy-Green stran tensor, I s the second order dentty tensor, and the fourth order tensor K s the elastcty tensor wth respect to the undstorted confguraton. rom equaton (2.), the Cauchy stress can be calculated as J 2 = K[ ( I)] (2.3) Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

8 Ma, Roters, Raabe, exture and Ansotropy n CC Metals wth and = P (2.4) J = det( ). (2.5) 2.2 low rule he velocty gradent can be expressed as = & s used to express the deformaton process. Usng equaton (2.4) t + P = & (2.6) P = P P&. (2.7) Usng d for the slp drecton and n undstorted confguraton, one obtans the dyadc product the slp plane normal drecton wth respect to the M = d n as the Schmd tensor for the slp system. or CC crystals we assume slp to occur on the 2 octahedral systems {}<>. he plastc porton of the velocty gradent, p, can then be formulated as wth P 2 = & γ M (2.8) = D m τ & γ = & γ sgn( τ ). (2.9) τ he vscoplastc rule gven by equaton (2.9) s used to calculate the ndvdual shear rates. In ths D expresson γ& s the reference shear rate, m s the stran rate senstvty, τ the drag stress, and τ s the resolved shear stress of slp system, whch can be approxmated by τ K [ 2 ( I)] M (2.) Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

9 Ma, Roters, Raabe, exture and Ansotropy n CC Metals 3 Consttutve equatons for the modfed aylor model he shortcomng of the classcal aylor model conssts n the dsregard of elastc-plastc gran nteracton. Sarma and Dawson [2] assumed n ther analyss that for one crystal whch s embedded n an aggregate of grans only about 2 neghborng grans wll nfluence the stran rate and stress of that crystal. hs means that a relatvely small set of grans suffces to compose a neghborhood for a gran n a polycrystal. Sarma and Dawson showed that for the case of an sotropc texture the varous components of the deformaton rates of all ndvdual grans n the aggregate establsh normal dstrbutons about the mean value [2]. or the case of a plane stran compresson texture Sarma and Dawson observed a skewness of the normal dstrbuton. hese theoretcal results on the heterogenety n the gran-togran dstrbuton of the accumulated plastc strans n mechancally loaded polycrystals were recently confrmed by experment [32-33]. In our present approach we adopt these observatons of Sarma and Dawson for the formulaton of a modfed aylor approach. Snce we am at preservng the concept of the classcal aylor models, namely, to treat each gran n an solated fashon wthout explct ncorporaton of a local gran neghborhood, we ntroduce (postve and negatve) standard devatons from the mean value for each component of the local velocty gradent tensor. In our model we assume that the devaton dstrbuton can be expressed n terms of a pseudorandom functon for all stran rate components. We also assume that the local velocty gradent s of the same prncpal type as the external one, the only dfference beng the pseudorandom dstrbuton for each tensor component. Usng the pseudorandom number functon f (seed) the local velocty gradent tensor of each sngle gran can be modfed based on the formulaton for the global velocty gradent,, accordng to = + SD f (seed) (3.) j j = tr ( ) I. (3.2) 3 In expresson (3.) SD s the standard devaton and seed s the seed for the random number Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

10 Ma, Roters, Raabe, exture and Ansotropy n CC Metals generator. A smlar approach was recently suggested by Engler [34] who ntroduced wth consderable success statstcal varatons of the stran rate state nto a aylor model. 4 Numercal modellng of sheet rollng 4. Sheet rollng knematcs and materal constants Sheet rollng can be dealzed as a plan stran compresson state. et the base vector e be n the compresson drecton (ND), e 2 n the transverse drecton (D), and e 3 n the rollng drecton (RD). Plane stran compresson can then be approxmated by the followng sochorc moton x = (exp( t) e e + exp( 33t) e3 e3 )[ X] (4.) In the present case we used =. 77 and 33 =. he materal was assumed as OHC D copper wth the elastc constants K = 68GPa, K22 = 2GPa, K22 = 75GPa, τ =6 MPa. urther constants were γ& =., m =. 2. s 4.2 Smulaton of the unloadng process he am of the unloadng smulaton conssts n a relaxaton step whch reduces the macroscopc Cauchy stress to zero. he velocty gradent tensor for unloadng s obtaned as follows. rst, the macroscopc stffness tensor, K, s used for calculatng the complance tensor S U for the unloadng, S U = ( K) (4.2) Raabe, edoc Server, Max-Planck-Socety - - MPI Düsseldorf

11 Ma, Roters, Raabe, exture and Ansotropy n CC Metals Second, the complance s used for calculatng the unloadng deformaton, E,accordng to U EU = SU []. (4.3) nally, the unloadng velocty gradent tensor s assumed to be parallel to the unloadng deformaton U E U (4.4) he begnnng of the frst unloadng stage s not purely elastc,.e. the macroscopc stress cannot drectly relax to zero durng unloadng. Hence, a second unloadng step s requred to reach complete unloadng. In ths study more than two unloadng steps are used to unload and relax the macroscopc Cauchy stress. 5 aylor-based smulaton of unaxal tenson testng for polycrystals Classcal varants of the aylor model use the velocty gradent to descrbe the loadng process,.e. n terms of ther boundary condtons they are models whch are strctly deformaton controlled. he unaxal tenson test, however, s a stress controlled process, whch contradcts the basc approach taken by aylor models. In the followng we show how one can crcumvent ths problem and conduct a aylor-based tenson test smulaton. Unaxal tenson can be acheved by modfyng the velocty gradent of smple tenson. Usng X and x for the ntal and the current locaton of a materal pont, respectvely, the smple tenson movement can be wrtten Raabe, edoc Server, Max-Planck-Socety - - MPI Düsseldorf

12 Ma, Roters, Raabe, exture and Ansotropy n CC Metals wth exp(.533t) x = exp(.5 33 ) t e e j exp( 33t), j =,2,3. [ X] (5.) Snce crystallne response to the loadng s ansotropc the aylor assumpton for such deformaton does not, as a rule, ental an unaxal stress state. herefore, we use a Newton procedure to terate the proper velocty gradent tensor,, whch produces a macroscopc smple tenson stress state for the tme step.hs means that the followng equaton should be solved G ( ) = ( ) σ e3 e3 =. (5.2) In ths expresson the unaxal tenson drecton s along the e 3 drecton. rom ths equaton the consstent tangent J G = can be derved as outlned n detal n the Appendx of ths report. he algorthm of the teraton procedure conssts of the followng steps: rst, one starts wth a gven ntal value = and a tme ncrement t. Second, G and velocty gradent = ( J ) [ G ]. hrd, the new value for to check whether G J are calculated to fnd the new G s calculated. nally, one has Err. If ths apples the tme step s fnshed. If t does not apply for a fxed teraton number the tme step s reduced by a factor of 2 and the algorthm s repeated. In ths study the rato η j of the dsspaton rate, P dss, and the total deformaton energy densty rate, P total, s used to express the deformaton amount for the sngle crystal j P tr( ) dss p η j = =. (5.4) Ptotal tr( ) j j or determnng whether the polycrystal yelds we use the volume average of ths rato accordng to Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

13 Ma, Roters, Raabe, exture and Ansotropy n CC Metals N V η j = j j j j = η = N (5.5) V where V j are the sngle volume elements. We analyze the q value along dfferent drectons to nvestgate the planar ansotropy of rolled copper sheets. he loadng drectons of the smple tenson tests are screened by usng the fxed longtudnal drecton e 3 and stepwse rotatatons about the normal drecton of the rollng plane e usng a rotaton tensor R. he new lattce poston for the crystal s gven by = R (5.6) R where s the elastc part of the deformaton gradent, whch s not orthogonal after the rollng and the unloadng deformatons, and R s the new value of ths tensor after the rotaton. Conductng sheet normal rotatons n 5 steps yelds 9 separate smple tenson test smulatons whch are suffcent to study the ansotropy of rolled sheets. Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

14 Ma, Roters, Raabe, exture and Ansotropy n CC Metals 6 Evaluaton of the texture predctons gs. and 2 show that the ntroducton of a standard devaton for the velocty gradent by use of a pseudorandom dstrbuton functon drastcally decreases the texture sharpness known from the classcal aylor theory. hs can be attrbuted to the fact that the new approach consders the nteracton of the grans at least n a statstcal fashon as outlned above. Some postons of the man texture components are also n better accordance wth expermental data than the predctons of the classcal aylor theory, though the agreement s not yet suffcently convncng. or nstance the poston of the Brass component, as predcted by the modfed aylor model, s n better accord wth the experment when compared to the C aylor model. he scatter on the <> texture fber between the Goss and the Brass components, however, s larger than observed n experment. he postons of the Copper and of the S components are n the new approach smlar as n the aylor C smulaton, but ther sharpness s much smaller. he observaton that the agreement between the new aylor model and the expermental data s not yet suffcently good can n part be attrbuted to the fact that we ntroduced the nteracton effect for the modfcaton of the symmetrc part of the velocty gradent tensor, but the correspondng pseudorandom modfcaton of the skew symmetrc part of ths tensor was not ncluded although t mght be of relevance for the calculaton of the reorentaton rates. Another aspect whch offers room for mprovement of the model s that we ntroduced the standard devaton equally for all components of the velocty gradent. Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

15 Ma, Roters, Raabe, exture and Ansotropy n CC Metals (a) (b) (c) g. {} pole fgures for (a) the modfed aylor model (standard devaton=.2, 9% thckness reducton); (b) aylor C model (9% thckness reducton); (c) Expermental data for rolled Cu [data taken from Kocks, 4], rolled to 96%. (a) (b) (c) g. 2 Orentaton dstrbuton functon for (a) the modfed aylor model (standard devaton=.2, 9% thckness reducton); (b) aylor C model (9% thckness reducton); (c) expermental data for Cu-2%Zn rolled to 96% thckness reducton [data taken from Savoe, 35]. he functons are gven n the reduced Euler space n the form of ϕ 2 =5 sectons runnng from ϕ 2 = to ϕ 2 =9. Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

16 Ma, Roters, Raabe, exture and Ansotropy n CC Metals 7 Evaluaton of the ansotropy predctons We studed the n-plane ansotropy for a rolled Copper sheet as a functon of the precedng deformaton by use of the new aylor model. We use the volume average parameter η to descrbe the deformaton. or purely elastc stranng η s almost zero. or larger deformatons t saturates at.. g. 3a shows the nfluence of the deformaton on the q value as a functon of the angle between stranng drecton and the former rollng drecton for a standard devaton SD =. When the deformaton s small (e.g. η =.5), only a small porton of the grans begns to yeld whle the others are loaded only nto the elastc regme. he q value does at ths stage, therefore, not represent the complete plastc ansotropy of the rolled sheet. As the deformaton ncreases the q value gradually converges. he numercal result s based on the smulated texture shown n g. 2b. In g. 3b the smulated ankford coeffcents for dfferent SD at η =. are compared to expermental data taken from the work of Kocks [4]. he curve ndcated by Exp was a rolled and annealed Copper specmen whle the curve ndcated by Exp2 was rolled and not subsequently heat treated. he comparson shows that the ansotropy predctons obtaned by the modfed aylor model reveal a good agreement wth the expermental data for angles above 5. he dscrepancy of experment and smulaton for angles below 5 can essentally be attrbuted to the devaton between the expermentally observed and smulated nput textures. g. 3b also shows the nfluence of the standard devaton on the q value as a functon of the angle between stranng drecton and the former rollng drecton. Although the match between experments and smulatons n g. 3b, such as for SD =.2. 5 and η =., s somewhat better than the C aylor model n g. 3a for SD = and η =. a prncpal shft of the predctons towards larger values s stll apparent. Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

17 Ma, Roters, Raabe, exture and Ansotropy n CC Metals (a) g. 3 (a) he q-value for dfferent deformaton stages predcted by use of the C aylor model; η =.5,.,.2,.4,.6,.8,., SD =. (b) he q-value for dfferent standard devatons; SD =.,.,.2,.3,.4,.5 ( 3 ), η =.. Exp: Experment for Cu rolled to 93% and annealed at 6 C (data taken from Kocks [4]). Exp2: Experment for Cu rolled to 96% (data taken from Kocks [4]). (b) Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

18 Ma, Roters, Raabe, exture and Ansotropy n CC Metals Conclusons We presented a modfed aylor model whch statstcally consders gran nteracton n a polycrystallne aggregate n terms of a standard devaton for the symmetrc part of the velocty gradent. he model was solved by usng a Newton teraton scheme. We smulated crystallographc rollng textures and the elastc-plastc ansotropy arsng from unaxal tenson tests. he results reveal n part a good agreement wth expermental data. In partcular we observed a consderable drop n the predcted texture sharpness. he smulaton results obtaned for unaxal tenson showed that the texture sharpness has consderable nfluence on the plan ansotropy of rolled sheets. Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

19 Ma, Roters, Raabe, exture and Ansotropy n CC Metals REERENCES. Van Houtte P. In: Preferred Orentatons n Deformed Metals and Rocks. An Introducton of Modern exture Analyss, Vol. Ed.: H. -R.Wenk. Academc Press, New York, Aernoudt EP, Van Houtte P, effers. Chapter 3 n: Deformaton and extures of Metals at arge Stran, Volume 6, Vol. ed.: H. Mughrab: Plastc Deformaton and racture of Materals of Materals Scence and echnology A Comprehensve reatment, Ser. eds.: R. W. Cahn, P. Haasen, and E. J. Krämer. VCH, Wenhem Hosford W. he Mechancs of Crystals and extured Polycrystals. Oxford Unversty Press Kocks U, óme CN, Wenk HR. exture and Ansotropy, Cambrdge Unversty Press Raabe D. Computatonal Materals Scence, Wley-VCH, Wenhem Sachs GZ. Ver. Dtsch. Ing. 928; 72: aylor GI. J. Inst. Metals 938; 6: Honneff H, Meckng H. Proc. 6th Int. Conf. on ex. of Mat. ICOOM 6, ed. S. Nagashma, Iron and Steel Inst. of Japan 347, Van Houtte P. n Proc. 6th Int. Conf. on 'exture of Materals' (ICOOM 6), (ed. S. Nagashma), 428; 98, Iron and Steel Inst. of Japan.. Kocks U, Chandra H. Acta Metall. 982; 3:695.. Raphanel J, van Houtte P. Acta Metall. 985; 33: Wagner, P., Ph. D. hess, RWH Aachen, Germany, Schmtter, U., Dplomarbet RWH Aachen, Germany, Van Houtte P, Delannay, Samajdar I. extures Mcrostructures 999; 3:9. 5. Raabe D. Acta Metall. 995; 43: Raabe D. Mater. Sc. Eng. A 995; 97:3. 7. Raabe D, Zhao Z., Mao W. Acta Mater. 22; 5: Crumbach M, Pomana G, Wagner P, Gottsten G. Recrystallzaton and Gran Growth Proceedngs of the rst Jont Internatonal Conference Eds. G. Gottsten and D.A. Molodov, Sprnger-Verlag, Vol. II, 2, Mao W. Mater. Sc. Eng. A 998; 257:7. 2. Molnar A, Canova GR, Ahz S. Acta metall. 987; 35: Sarma G, Dawson PR. Int. J. Plast. 996 ; 2 :23. Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

20 Ma, Roters, Raabe, exture and Ansotropy n CC Metals 22. Bunge, HJ. exture analyss n materals scence. Butterworths, ondon, England Wassermann G, Grewen J. exturen metallscher Werkstoffe (n German), Sprnger-Verlag Berln, Germany Hrsch J, ücke K. Acta Metall. 988; 36: Hölscher M, Raabe D., ücke K. Acta metall. Mater. 994; 42: Rees DWA. Appl. Math. Modellng, 997; 2: ankford W, Snyder SC, Bausher JA. rans. ASM 95; 42: Ktamura K, Myazak S, Iwa H, Kohl M. Mater. Sc. Eng. A 999; : ee WB, o S. J. Mater. Proc. echn. 995; 48: Spoldor A. J. Mater. Sc. 996; 3: Bertram A. Int. J. Plast. 999; 5: Raabe D, Sachtleber M, Zhao Z, Roters, Zaefferer S. Acta Materala 2; 49: Sachtleber M, Zhao Z, Raabe D. Mater. Sc. Engn. A 22; 336: Engler O. Adv. Engn. Mater. 22, 4:8 35. Savoe J. Ph.D. Dssertaton hess, RWH Aachen, 99. Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf

21 Ma, Roters, Raabe, exture and Ansotropy n CC Metals Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf Appendx : Dervaton of the consstent tangent modulus he volume average of Cauchy stress can be smplfed for the case that every sngle crystal has the same volume = = N t N t ), ( ), (. (A.) or every tme step t the followng equaton should be satsfed for every sngle crystal H = = t n n n n ), ( ) ( &. (A.2) here are two varables n and n ths functon, so that ther dervatve relaton can be set H H H = + = n n (A.3) H H = n n (A.3) Based on these equatons the consstent tangent module J can be wrtten as ), ( ), ( G = = = N t N t (A.4) G J = = = = = = N N N N (A.5)

22 Ma, Roters, Raabe, exture and Ansotropy n CC Metals Wth ths result the equaton (2.3) can be smplfed for small elastc deformatons 2 K[ ( I)] (A.6) and the dervatve of wth respect of s d δ K ( I)] K[ ( I)] δ ak K kldncn d = ac [ + bc + abcd 2 db 2 ad bl (A.7) wth δ j = = j j. Replacng by and by n equaton (A.7) and combnng wth equaton (A.5), one can get the consstent tangent module J n explct form. Raabe, edoc Server, Max-Planck-Socety MPI Düsseldorf