UNCORRECTED PROOF ARTICLE IN PRESS. , Scott Schoenfeld b. SMM 4402 No. of Pages 5, DTD = July 2003 Disk used

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1 SMM 2 No. of Pages 5, DTD =.3. 2 Juy 23 Disk used 2 Evoution of crysta orientation distribution coefficients 3 during pastic deformation D.S. Li a, H. Garmestani a, *, Scott Schoenfed b 5 a Schoo of Materias Science and Engineering, Georgia Institute of Technoogy, 77 Ferst Drive, N.W. Rm 75, Lov., 6 Atanta, GA 33, USA 7 b Aberdeen Proving Ground, Maryand, MD , USA 8 Received 5 Apri 23; received in revised form 6 Juy 23; accepted 7 Juy Introduction Scripta Materiaia xxx (23) xxx xxx Properties of poycrystaine materias are de- 2 pendent on the orientation distribution of crysta grains, aso caed texture. It is usuay represented as a Fourier series of generaized spherica har- 5 monics weighted by appropriate texture coefficients 6 []. Microstructures can be represented as points in 7 a mutidimensiona space (microstructure hu) with 8 coordinates as texture coefficients [2]. Microstruc- 9 ture sensitive design as proposed earier by Adams 2 et a. [2] introduces a new methodoogy to estabish 2 processing paths in the microstructure hu. Opti- 22 mization of texture in microstructures requires 23 correct representation of these processing paths. 2 Cement and Couomb [3,] proposed a principe of 25 conversation of quantity of matter referred to a 26 given crysta orientation during processing of mi- 27 crostructures in the orientation space. This can be 28 used to produce processing paths in the micro- 29 structure hu for a certain process from the initia 3 texture. Based on CementÕs formaism and the 3 continuity equation, Bunge and Esing [5] studied the fow fied of singe crystas of face centered cubic (fcc) metas according to ( )hi sip. The present study proposes an aternate approach using a poycrystaine materias description rather than singe crysta orientation description. This approach estabishes a inear reationship between the rate of change of the texture coefficients and the initia texture. To examine the accuracy and range of appicabiity of this inear approach, a modified TayorÕs mode proposed by Kaidindi etc. [6,7] was used for comparison. In TayorÕs mode, it is assumed that a the individua grains in the poycrystaine materia undergo the same deformation gradient as the macroscopic one. This simpification satisfies the oca compatibiity, but often vioates equiibrium. Athough TayorÕs mode ignores many of the compexities of materiaõs deformation, it provides a fairy accurate approximate soution for the texture evoution of singe phase, highy symmetric attice structures, such as fcc poycrystas, during arge pastic deformation [7]. * Corresponding author. Te.: ; fax: / E-mai address: hamid.garmestani@mse.gatech.edu (H. Garmestani) Rate of change of texture coefficient during pastic deformation At any point during deformation, texture can be represented by a set of Fourier coefficients F, /$ - see front matter Ó 23 Pubished by Esevier Ltd. on behaf of Acta Materiaia Inc. doi:.6/s59-662(3)3-3

2 SMM 2 No. of Pages 5, DTD =.3. 2 Juy 23 Disk used 2 D.S. Li et a. / Scripta Materiaia xxx (23) xxx xxx 58 where g is an appropriate metric of the processing 59 step [5]. For exampe, in a drawing step, g repre- 6 sents the drawing. The orientation distri- 6 bution function at a specified processing step g, 62 f ðg; gþ of the poycrystaine materias can be 63 represented as a series of generaized spherica 6 harmonic functions: f ðg; gþ ¼ X ¼ X MðÞ X NðÞ F m¼ n¼ T T _ ðþ 66 _ T T are symmetric generaized spherica har- 67 monics for the corresponding sampe and crysta 68 symmetry. The continuity reations obtained from 69 the principe of conservation in the orientation 7 space can be expanded in a series of spherica 7 harmonics [5]: X F rq k divð T T _ rq k RÞ krq þ X df _ T T ¼ ð2þ 73 Here R defines the rotation fied. The first term 7 describes the increase of the quantity of matter in 75 the orientation space moving out of a infinitesima 76 voume in the orientation space. The second 77 term describes the decrease of quantity of matter 78 per unit time. The divergence function in the first 79 term can be expanded further into a series of 8 generaized spherica harmonics: divð T T _ rq k RÞ ¼ X A rq k _ T T ð3þ 82 Substituting this expansion into Eq. (2), we find 83 the inear reationship between the texture coeffi- 8 cients and their rate of change: df ¼ X krq A rq k F rq k ðþ This inear reationship was used by Bunge to predict the texture evoution for singe crysta orientations. In this study, we wi use this reationship to simuate the texture evoution of fcc materias with random texture and examine its accuracy. If the number of texture coefficients F is imited to N, then texture data for N þ different s wi be needed to obtain a soution for the texture evoution coefficient A rq k. Further the sixth rank tensor coefficients are represented by an N N matrix as outined ater in the paper. 3. Resuts TayorÕs mode is used in the present study to provide the corresponding F and their change for different s, g. Using these coefficients in Eq. (), we obtain the soution for A rq k. To check the vaidity of this approach, we use the resutant A rq k to predict texture at other s and compare the resuts with that predicted by TayorÕs mode. The initia texture in this study is assumed to be an aggregate of crystas eveny distributed in the orientation space. The corresponding ( ), ( ) and ( ) poe figures of this data set are iustrated in Fig.. The maximum intensity is very sma, ess than.5 times random. For the present study, face centered cubic crysta system is assumed aong with orthotropic sampe symmetry. The texture coefficients F with > do not affect eastic properties for cubic poycrystaine materias [8] and as a resut a Fig.. ( ), ( ) and ( ) poe figures of initia simuated random texture by crystas.

3 SMM 2 No. of Pages 5, DTD =.3. 2 Juy 23 Disk used D.S. Li et a. / Scripta Materiaia xxx (23) xxx xxx 3 6 minimum vaue of is chosen for the order of rank 7 of the Fourier series F in Eq. (). For this case 8 there are four nonzero F coefficients in which F 9 is a constant (aways.). This reduces the number 2 of nonzero texture coefficients to ony three terms: 2 F, F 2 and F. That is to say, N ¼ From Eq. (), we have: df ¼ A F þ A 2 F 2 þ A F þ ¼ A F þ A 2 F 2 þ A F ð5:þ df 2 ¼ A 2 F þ A 22 F 2 þ A 2 F ð5:2þ df ¼ A F þ A 2 F 2 þ A F ð5:3þ 26 or, 2 df = 3 2 A df 2= A 2 A 5 ¼ A 2 df = A 22 A 2 5 F 3 F 2 5 A A 2 A ð6þ F 28 Here A rq k are components of a sixth order tensor. 29 These cumbersome indices emphasize the charac- ter of the tensors. To simpify the indices, Eq. (6) is rewritten in the foowing contracted form: df = A A 2 A df 2 = 5 ¼ A 2 A 22 A 23 5 F 3 F 2 5 ð7þ df 3 = A 3 A A 33 F 3 3 In this way we simpify the soution of a sixth order tensor of texture evoution coefficient to a 5 second order matrix operation by reducing the 6 number of texture coefficients to ony the ones 7 which are of interest. 8 For the anaysis, first the texture evoution co- 9 efficients A rq k are cacuated from N þ texture data obtained by TayorÕs mode at different s. Then these coefficients are used to simuate 2 the rate of change of the texture coefficients at other points in the deformation path. These rates are compared with the predictions from TayorÕs 5 mode. In the first set, a step of % is used. 6 The F s at s of 3%, 3%, % and 33% are 7 cacuated from TayorÕs mode. Here these s 8 are caed the initia set. From the texture 9 coefficient rates df = and the texture coeffi- cients F at different s, A rq k s are cacuated. Using the sixth rank tensor evoution parameters A rq k, the rates of change of texture coefficients, df = at s from 2% to 5%, were computed. This process was repeated for a step of 2% by changing the initia set to 28%, 3%, % and 3%. In the third set a step of 5% was used and the initia set incuded 25%, 3%, 35% and %. To demonstrate the vaidity of the procedure, an error parameter, error is introduced in the form error df ¼ ¼: ca df ¼: ð8þ Tayor Fig. 2 shows a pot of error as a function of 62 for three different initia sets. It is 63 cear that the errors of the rate of change of F 6 from the simuation are ow when the is 65 taken as 2%. Since a of 3% is incuded in 66 the initia set for the fitting, the corresponding 67 error is zero. The error is maintained at a rea- 68 sonaby ow vaue for a of % whie it is 69 dramaticay increased for a of 5%. The 7 same trend is refected in Fig. 2(b) which iustrates 7 the errors in the rate of change of F step is within % or 2%, the simuation 73 produces a arge error (>.2) when compared to 7 TayorÕs mode at a of 2%. When the 75 is within 3% or %, the error is within an ac- 76 ceptabe range of. (except at a step of 77 %). But with the increase of, the error in- 78 creases to an unacceptabe range, arger than. 79 at a of 5%. Fig. 2(c) shows the variation of 8 the error parameter for df =. 8 smaer than.6 in the range of s from 2% 82 to % for a the three steps. When the 83 is increased to 5%, the error is increased 8 sighty to.8 (for a step of 5%), ).39 (for 85 a step of 2%) and.38 (for a step of 86 %), respectivey. Overa, we can see that the 87 simuation is very cose to TayorÕs mode in the 88 range of 2 %. It deviates from the pre- 89 diction of TayorÕs mode when the is far 9 from the initia set. It is cear from the re- 9 suts above that the procedure introduced in this 92

4 SMM 2 No. of Pages 5, DTD =.3. 2 Juy 23 Disk used D.S. Li et a. / Scripta Materiaia xxx (23) xxx xxx error in simuating change rate of F error in simuating change rate of F 2 error.5 2% 3% % 5% 2% 3% % 5% error % error % - - error 5% error 5% error east squares (a) -.5 error east squares (b) paper provides the best resuts when used for in- 9 terpoation. This means that if the texture coeffi- 95 cients are predicted at a which is within the 96 range of initia set, the error is minimum. If 97 this prediction is extended to s out of the 98 range of the initia set, the farther the 99 is from the range of the initia set, the worse 2 the prediction of the texture evoution is. To 2 compare the errors from different ranges, we 22 define mean error error : error ¼ X error N ð9þ 2 Fig. 3 iustrates the mean error parameter for the 25 three nonzero texture coefficients obtained from 26 different steps. In the range from 2% 27 to 5%, the simuation from step of 5% fits 28 best to the TayorÕs mode. The simuation from 29 step of 2% foows next. If the step is 2 as sma as %, the simuation in the arge 2 range wi resut in a arge error. The mean error of 22 the evoution rate parameter F 2 is arger than the 2 other two texture coefficients. Fig. shows the 2 evoution of these three texture coefficients during 25 deformation process obtained from TayorÕs error.5 - (c) -.5 error 2 2% 3% % 5% error in simuating change rate of F error % error 5% error east squares Fig. 2. (a) error, (b) error2, (c) error as a function of when the simuation resuts are obtained using step as %, 2% and 5%, respectivey. The error of simuation obtained from east squares method is aso iustrated. mode. Note that the magnitude of F 2 is sma compared to the other two, which resuts in arger variations in the error. One of the contributions to the error is truncation error from Eq. (3). This inear approach works we in the deformation history cose to the initia set. The A rq k obtained from the initia set does not predict the texture coefficients when the is far from the initia set. To overcome this imitation, different A rq k mean error F mean error in simuating change rate of F step % step 2% step 5% east squares F 2 F Fig. 3. error, mean error of texture evoution rate of F, F 2 and F when the simuated resuts are obtained using different methods

5 SMM 2 No. of Pages 5, DTD =.3. 2 Juy 23 Disk used D.S. Li et a. / Scripta Materiaia xxx (23) xxx xxx % 3% % 5% C Fig.. Evoution of texture coefficients F during deformation process using TayorÕs mode. 226 may be used in different deformation stages. For 227 the deformation up to %, five matrices form of 228 A rq k combined can be used to describe the pro- 229 cessing path each covering a range of 2%. 23 This approach is sti a simpe and efficient way in 23 describing the processing path and streamines for 2 the deformation processes for a reativey arge 233 range. 23 Another approach to obtain the texture evou- 235 tion coefficient is to use east squares estimates 236 method. From M þ texture data (M > N) at 237 different s, the texture evoution coefficient 238 matrix A is cacuated from the over-determined 239 system beow: bdf =c ¼½AŠ½F Š ðþ 2 df = is an N M matrix. M þ is the di- 22 mension of the initia set. N is the number of 23 texture coefficients we are interested in. A is an 2 N N matrix and F is an N M matrix. 25 Here we choose M as, the initia set 26 from 3% to %, and the step as %. From 27 the texture data at these s, A rq k is cacuated 28 by east square fitting method. Using the resutant 29 A rq k, we simuate the evoution rate of texture 25 coefficients during the deformation from 2% to 25 5%. The resut is aso shown in Fig. 2. To dem- 252 onstrate the advantage of this method, the resut 253 of east squares method is compared with those 25 from other three methods discussed before. Fig shows ceary that the A rq k obtained from inear C 2 C soution at step of % or 2% does not describe the behavior as we as the A rq k obtained from the east squares method. The comparison of the east squares method with the inear method using a step of 5% is a itte compicated. The atter one has a arger initia set range, from 2% to %, whie the first one has a reativey smaer initia set range, from 3% to %. The fit for east squares method is not as good as the inear fit in the ow range of around 2%, which is incuded in the initia set for the atter one but not in the initia set for the east squares method. The mean error in the range from 2% to 5% of the east squares method is sti ower than that of the atter one. It is shown in the mean error iustrated in Fig. 3. Using east squares method we may obtain the texture evoution coefficients which can better describe the texture evoution behavior during mechanica deformation. Acknoweements This work has been funded under the AFOSR grant # F and Army Research Lab contract # DAAD7-2-P-398 and DAAD References [] Bunge HJ. Z Metakde 965;56:872. [2] Adams BL, Henrie A, Henrie B, Lyon M, Kaidindi SR, Garmestani H. J Mech Phys 2;9:639. [3] Cement A. Mater Sci Eng 982;55:23. [] Cement A, Couomb P. Scripta Mater 979;:899. [5] Bunge HJ, Esing C. Scripta Mater 98;8:9. [6] Kaidindi SR, Bronkhorst CA, Anand L. J Mech Phys 992;:537. [7] Garmestani H, Kaidindi SR, Wiiams L, Bacatchuk CM, Fountain C, Lee EM, Se-Said OS. Int J Past 22;8:73. [8] Bunge HJ. Texture anaysis in materias science: mathematica methods. London: Butterworth & Co; 982. p