A texture evolution model in cubic-orthotropic polycrystalline system

Transcription

1 Internationa Journa of Pasticity 21 (25) A texture evoution mode in cubic-orthotropic poycrystaine system D.S. Li a, H. Garmestani a, *, B.L. Adams b a Schoo of Materias Science and Engineering, Georgia Institute of Technoogy, 771 Ferst Drive, N.W. Rm 175, Atanta, GA 3332, USA b Department of Mechanica Engineering, Brigham Young University, Provo, UT 862, USA Received in fina revised form 28 September 23 Avaiabe onine 7 January 25 Abstract A new methodoogy based on a conservation principe in the orientation space is deveoped to simuate the texture evoution in a cubic-orthotropic poycrystaine system. A east squares error method was used to improve the accuracy of the simuation resuts obtained from the texture evoution function. The mode is appied to uniaxia tension, compression and roing for a arge deformation of more than 5% using a singe evoution parameter. The vaidity and appication range of this new mode are discussed by simuating and predicting texture evoution during different oading conditions. The new methodoogy provides a famiy of texture evoution paths and streamines which empowers the materias designer to optimize the desired microstructure. Ó 2 Esevier Ltd. A rights reserved. Keywords: A. Microstructure; A. Thermomechanica processes; B. Poycrystaine materia; B. Processing path; B. Streamine 1. Introduction A major goa for materias design is the seection and optimization of microstructures for a specified set of properties and mechanica constraints. To achieve * Corresponding author. Te.: ; fax: E-mai address: hamid.garmestani@mse.gatech.edu (H. Garmestani) /$ - see front matter Ó 2 Esevier Ltd. A rights reserved. doi:1.116/j.ijpas

2 1592 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) this goa, it is necessary to choose the right and most appropriate processing path. The soution may not be unique and the optimum path may depend on a number of parameters incuding cost and reiabiity. Traditiona design methodoogies attempt to reate properties to off-the-shef materias whie the range of microstructures (texture, grain size, deformation history, etc.) for each materia of choice is usuay ignored. Such variation in the microstructure of a specific materia provides a arge range of properties that may meet specific design constraints not readiy avaiabe in the conventiona off-the-shef materias design. To make a direct inkage to microstructures, Adams et a. (21, 2) proposed a nove methodoogy, known as microstructure sensitive design (MSD), which shifted the current paradigm for designer materias. The new methodoogy, as was introduced originay, is concentrated on texture (orientation distribution function) in poycrystaine materias as a basic variabe for design. The other microstructure attributes such as grain size, grain boundary distribution, morphoogy, second phase partices and precipitates may be incuded in the framework using two and higher order statistica distribution functions (Garmestani and Lin, 2, 21; Lin et a., 1998, 2; Jefferson et a., 2). The inkage to property encosure may require the correct form of the constitutive reations based on better understanding of the physics and the underying deformation mechanisms. The representation of microstructure in the form of texture may assume the most basic form and by itsef can cover anywhere from a singe crysta (as a imiting form of the distribution function) to bi-crystas and random poycrystas. Appreciabe anisotropic properties are usuay possessed by singe crystas which are expensive to obtain. Poycrystaine materias are usuay ess expensive to produce and more readiy avaiabe. The incentive to cut the cost by utiizing the state of anisotropy to maximize the potentia appication of poycrystaine materia prompts materia scientists to seek the means to achieve the optimized texture by thermomechanica processing. In homogeneous poycrystaine materias texture determines the anisotropy in mechanica, therma, magnetic and eectrica properties. MSD in its present form is an approach to baance the requirements of severa different properties by optimizing texture (Adams et a., 21, 2; Kaidindi et a., 2). To achieve this goa, a quantitative description of materia microstructure as a set of texture coefficients, which is associated with properties, is introduced as a variabe in design. Texture is commony represented as a Fourier series of generaized spherica harmonics weighted by appropriate texture coefficients (Bunge, 1965). Using the set of texture coefficients as a descriptor, microstructures can be represented as points in a mutidimensiona space with coordinates as texture coefficients (Adams et a., 21, 2). The dimension depends on the number of the texture coefficients used. Each point in this Fourier space stands for a unique texture, associated with corresponding properties. Texture coefficients are important in determining the properties of poycrystaine materias. The properties of poycrystaine materias can be represented as a summation of the product of property coefficients and spherica harmonics:

3 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) e ¼ X1 ¼ X MðÞ X NðÞ m¼ n¼ e v _ T mn ðgþ: ð1þ Whie the property coefficients of a singe crysta, e k coefficients of poycrystaine materia, e v coefficients, F kv e v ¼ 1 2 þ 1 X MðÞ k¼1 e k, are constants, the property, are dependent directy on the texture F kv ; 6 6 r; ð2þ where r is the imit of order which depends on the crysta symmetry, sampe symmetry and the property e. It is very important to have a fu mathematica representation of texture evoution during thermomechanica processing. In this paper, texture evoution is defined as a continuous ine in the texture hu to represent the texture evoution. Texture is conventionay measured using X-ray diffraction as a set of poe figures. The compete representation of texture requires a number of poe figures depending on the crysta group and sampe symmetry. For cubic-orthotropic case, three poe figures are usuay needed to fuy characterize texture. Using ODF pots has some other imitations. They distort the distribution geometricay in Euer space. Random distribution ooks ike textured in the ODF pots. Aso there is a mutipicity in ODF pots, especiay for high symmetry materias. Athough texture can be represented in a number of ways, the spectra representation can introduce a new geometrica representation in the texture hu such that every point represents one specific microstructure (texture). Modeing texture evoution is an important component of MSD. To modify the properties through thermomechanica deformation requires correct representation of the evoution of microstructures. A conservation principe in the orientation space proposed by Cement and Couomb (1979, 1982) was used in this work to mode the texture evoution. This principe refers to an infinitesima voume eement in the orientation (Euer) space during processing. Based on CementÕs formaism and the continuity equation, Bunge and Esing (198) studied the fow fied of singe orientations of face centered cubic (fcc) metas using a crysta pasticity formuation based on sip activity on the {1 1 1}Æ1 1æ sip system. In a ater work (Kein and Bunge, 1991), a numerica integration methodoogy was used to obtain a reationship between the texture coefficients and the deformation step. In a recent study by Li and Garmestani (23a,b) an aternate approach using poycrystaine texture representation rather than singe crysta orientation description was used to describe texture evoution. This approach estabished a inear reationship between the rate of change of the texture coefficients and the texture coefficients. Further progress is made in the present study for a direct reationship between texture coefficients and deformation parameter. A processing path function is proposed to describe the evoution of texture coefficients in the form of processing parameters and initia texture coefficients. To examine the accuracy and range of appicabiity for this approach, a modified Tayor mode proposed by Kaidindi et a. (1992a, 1992b) was used for comparison. In Tayor mode, it is assumed that

4 159 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) a the individua grains in poycrystaine materias undergo the same deformation gradient as the macroscopic one. This simpification satisfies the oca compatibiity, but often vioates equiibrium. Athough the Tayor mode ignores many compexities embedding in mechanica 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 (Kaidindi et a., 1992a, 1992b; Garmestani et a., 22). 2. Evoution of texture coefficient during pastic deformation Using the conservation principe in the orientation space, a set of reationships for the evoution of texture coefficients can be derived. A formuation is presented here for the streamines as an anaytica form among the texture coefficients for any specific thermomechanica process. In the formuation presented here, g is used as an appropriate metric of the process. For exampe, in the case of uniaxia tension, g represents the drawing strain and in the case of compression, g represents the compression ratio. Texture descriptor used in this study is a set of F mn ðgþ coefficients that changes as a function of g. Iff(g,g) is used to represent texture as a function of orientation g and processing parameter g, texture at any g can be expressed as a series of generaized spherica harmonic functions in which F mn s are the weights (coefficients) of these harmonics, as shown in the foowing equation: f ðg; gþ ¼ X1 ¼ X MðÞ X NðÞ m¼ n¼ F mn ðgþ T _ mn ðgþ: ð3þ Here T _ mn ðgþ is the symmetric generaized spherica harmonics for the corresponding sampe and crysta symmetry. For a singe crysta orientation distribution in which a the crystas are oriented aong g i, texture coefficients F mn are cacuated directy from the spherica harmonics: F mn ¼ð2þ1Þ T _ mn ðg i Þ: ðþ Texture measured from X-ray diffraction or eectron backscatter diffraction (EBSD) gives the voume fraction of materia with orientation g i (Garmestani, 1998; Garmestani et a., 1999). F mn can then be cacuated from f(g) by inear combination of the texture coefficients of singe crysta orientations: F mn ¼ð2 þ 1Þ X i f ðg i Þ _ T mn ðg i Þ: ð5þ In CementÕs work (1982), the texture evoution is regarded as a fuid fow in orientation space. Three Euerian anges compose the orthogona coordinates of this space. For any point represented by g, the density is 1 f ðg; gþ sin / and the fow rate 8p 2 is R(g). According to the conservation principe in the orientation space, the sum of the increase of the quantity of matter in an eement of voume dv and the materia

5 moving across the surface S shoud be zero. The continuity equation can then be represented as: I 1 f ðg; gþ sin /R ^ndr þ o Z Z Z 1 f ðg; gþ sin /dv ¼ : ð6þ 8p 2 ot 8p 2 Thus, S of ðg; gþ þ 1 div½f ðg; gþ sin /RðgÞŠ ¼ : ot sin / ð7þ For an infinitesima voume eement in the orientation space, dv =du 1 dudu 2, the first term in Eq. (7) is the increase of the quantity of matter per unit time and the second term describes the quantity of matter moving out of the infinitesima voume eement. By simpifying Eq. (7), the continuity equation for the conservation of quantity of matter in a voume eement in the Euer space is: of ðg; gþ þ div½f ðg; gþrðgþš þ ctg/f ðg; gþrðgþ ¼: ð8þ ot Using the expression for texture in Eq. (1), Eq. (8) is expanded in a series of spherica harmonics: X mn df mn ðgþ dg _ T mn ðgþþ X krq F rq k V ðgþ div T _ rq k þ ðgþrðgþ ctg/ T _ rq k ðgþrðgþ ¼ : ð9þ The second summation can be further expanded into a series of generaized spherica harmonics: div T _ rq k þ ðgþrðgþ ctg/ T _ rq k ðgþrðgþ ¼ X A mnrq k _ T mn ðgþ: ð1þ mn Here A rqmn k is introduced as the coefficients of the spherica harmonics. Substituting this back into Eq. (9), a inear reationship between the texture coefficients and their rate of change is derived: df mn ðgþ ¼ X dg krq D.S. Li et a. / Internationa Journa of Pasticity 21 (25) A mnrq k F rq k ðgþ: This inear reationship was used by Bunge and Esing (198) and Kein and Bunge (1991) to predict the texture evoution in the orientation space. In this present work, a texture evoution function obtained by the integration of Eq. (11) was used to describe the evoution of the texture coefficients with the deformation parameter: F ðgþ ¼e Ag F ðg ¼ Þ: ð12þ The coefficients represented by the sixth rank tensor A, can be rearranged as the eements of a matrix and wi be caed texture evoution matrix in this work. In this study, Eq. (12) was used to simuate the texture evoution of fcc materias with random texture to generate a process path. If the number of usefu texture coefficients ð11þ

6 1596 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) F mn is imited to N, then texture data at N + 1 different strains wi be needed to obtain a soution for texture evoution coefficients A rqmn k. It wi be shown that redefining the sixth rank tensor coefficients to an N N matrix wi substantiay simpify the numerica scheme. The fina anaytica form for the texture evoution as a function of A wi represent the texture evoution. The introduction of a microstructure parameter A can be used as a methodoogy to predict microstructure evoution in a arge set of poycrystaine materias. The sixth order tensor A as derived in Eq. (11) (continuity reation in the Euer space) is the underying parameter in this basic and simpe aw. The sixth order tensor maybe a function of many different other features of the microstructure. A mnrq k ¼ A mnrq k ðs 1 ; s 2 ; s 3 ;...Þ; ð13þ where s i may represent a number of microstructure features such as grain size and distribution, precipitates, disocation density, second phase partices which can affect the evoution. A Tayor type crysta pasticity mode is used in this paper to cacuate the sixth order A. The appication of Tayor may introduce a imitation in the range of appications but within the imits of Tayor, many of these microstructura parameters can be incorporated. As a first exercise, this paper concentrates on Tayor and produces enough resuts which wi show that such continuity reations and the consequent streamines (derived in this paper) are vaid for a arge range of deformation processes. The incorporation of the other aternative (crysta pasticity homogenization) methodoogies can be easiy incorporated at a ater time. Texture is defined as preferred orientation distribution and is a macroscopic and average representation of the microstructure. A formuation based on texture does not take into account grain structure and grain boundary character and aso the grain to grain interaction. Texture as an average representation is ony a onepoint distribution function. Higher order statistics can incorporate the additiona detais of the microstructure (Garmestani et a., 2, 21; Lin et a., 1998, 2; Jefferson et a., 2; Adams et a., 1989; Torquato and Ste, 1985) and can be used for the evoution of the microstructure. It is cear that the evoution of texture is a function of the detais and physics of the microstructure and the underying deformation mechanism. At a first gance it seems that such detais are negected in the formuation presented in the paper. A these detais are however embedded in the sixth order tensor A. It sounds very optimistic to expect that such a parameter can incorporate a these effects but the main goa of this paper is to investigate whether such a caim is vaid and to what degree. It wi be shown that the evoution can be taken care of using conservation principe for more than 5% deformation. 3. Simuation resut of texture coefficient evoution The Tayor Mode is used in the present study to provide the input data of evoution of F mn at corresponding strains. Severa other methods were deveoped to simuate the texture evoution based on different assumptions. Such methods incude the sef-consistent mode (Lopes et a., 23), the finite eement anaysis method

7 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) (Nakamachi et a., 2; Raabe and Roters, 23; Kaidindi, 21; Demire et a., 23; Houtte et a., 22) and the Constrained Hybrid mode (Parks and Ahzi, 199; Ahzi et a., 199). This work is neither to verify the correctness of the Tayor mode, nor to deveop a new mode in terms of deformation mechanisms. Factors that wi be considered in physica modes, such as the constitutive reations in the sip systems, strain hardening and strain rate, are not considered at this stage. Prediction from Tayor mode simuation and not experimenta data are used since the Tayor mode gives accurate estimates at different strains. The goa of our study is to propose an anaytica form for the texture evoution and check its vaidity, imitations and appicabiity. Using Eq. (11), the soution for A rqmn k was obtained from known texture coefficients at different deformed states. To check the vaidity of this approach, texture coefficients F mn at other strains cacuated from the resutant A rqmn k according to Eq. (12) are compared with those predictions from the Tayor mode. For convenience, the initia texture in this study is arbitrariy assumed to be an aggregate of crystas eveny distributed in the orientation space. The corresponding (1 ), (1 1 ) and (1 1 1) poe figures of this data set are iustrated in Fig. 1. The maximum intensity is very sma, ess than 1.5 times random. F 11 ; F 11 ; F 12 for this texture condition are 1.,.,.5, and.2, respectivey. A the other texture coefficients are zeros. The vaues are very cose to the idea random texture where ony F 11 is 1 and a the other texture coefficients are zero. For the present study, face centered cubic crysta system and orthotropic sampe symmetry are assumed. In this system, the higher order texture coefficients F mn with > (Eq. (2)) wi not infuence eastic properties (Bunge, 1982). As a resut, a minimum vaue of is chosen for the order of rank in Eq. (3). This greaty reduces the compexity of our probem. Furthermore, F 11 is a constant (aways 1.). This reduces the number of texture coefficients to ony three terms: F 11 ; F 12. That is to say, N =3. From Eq. (11), we have: df 11 ðgþ XLmax X NðkÞ ¼ dg k¼ r¼ X MðkÞ q¼ A mnrq k F rq k ðgþ ¼ A 1111 F 11 þ A1111 F 11 þ A1112 F 12 þ A1113 F 13 þ A F 11 6 þ A F 12 6 þ; ð1:1þ Fig. 1. (1 ), (1 1 ) and (1 1 1) poe figures of initia simuated random texture by crystas.

8 1598 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) df 12 ðgþ ¼ A 1111 F 11 dg þ A1211 F 11 þ A1212 F 12 þ A1213 F 13 þ; ð1:2þ df 13 ðgþ ¼ A 1311 F 11 dg þ A1311 F 11 þ A1312 F 12 þ A1313 F 13 þ; ð1:3þ or in matrix form, 2 6 df 11 df 11 df 12 =dg 3 2 =dg 7 5 ¼ 6 =dg df 13 =dg A 1111 A 1111 A 1112 A 1113 A 1111 A 1111 A 1112 A 1113 A 1211 A 1211 A 1212 A A 1311 A 1311 A 1312 A F 11 F 11 F 12 F : ð1:þ 7 5 Here A mnrq k are components of a sixth order tensor reduced to a second rank tensor. Note that since F 11 is aways equa to 1, the rate of change of this coefficient is aways. It is cear that the summation over the right hand side does not need to be imited to L max = and actuay depends on the character of the texture. Most of the components of this tensor are negigibe because most of F mn s are either or not of our interest. When L max =, ony 16 of the nonzero A mnrq k coefficients are shown in the matrix above. These coefficients wi be used to describe the rate of change of texture coefficients as a first estimate. The contribution from the higher order terms up to L max = 8 wi aso be examined in the atter part of this work. The truncation error introduced by terminating at a finite vaue k = L max wi be studied in a subsequent work. For simpification, ony L max = is considered here for mathematica expansion, Eq. (1) is then rewritten in the foowing contracted form: df 1 =dg A 1 A 11 A 12 A 13 F 1 df 2 =dg 6 7 df 3 =dg 5 ¼ A 2 A 21 A 22 A 23 F A 3 A 31 A 32 A 33 5 F 3 5 : ð15þ df =dg A A 1 A 2 A 3 F As described above, the sixth order tensor of texture evoution coefficients is reduced to a second order matrix by fitering texture coefficients to ony the ones which are of interest. After the cacuation of the texture evoution matrix A, texture coefficients aong the deformation history can be cacuated by the texture evoution function: F ðgþ ¼e Aðg g Þ F ðg Þ: In the present anaysis, the texture coefficients at severa different strains under uniaxia tension were obtained by the Tayor mode. From these input data, texture evoution coefficients A rqmn k are cacuated by soving Eq. (15). Next these coefficients wi be used to simuate the evoution of texture coefficients at other strains in the deformation history according to Eq. (16). These simuated texture coefficients were compared with the Tayor prediction. Two methodoogies are used to cacuate the A coefficients from the Tayor mode. In the first attempt the data from the simuations ð16þ

9 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) are cacuated for five different strains and the four simutaneous sets of equations are soved for the A coefficients. In the second method, texture data at a arge number of strains are used and a east squares error method is used to cacuate the A coefficients Determined systemõs approach A number of points in the texture evoution paths are chosen and the data for the texture coefficients are cacuated using the Tayor mode. In the first set, a strain step dg of 5% was used. The F mn, input data, at strains of 15%, 2%, 25%, 3% and 35% were cacuated from Tayor mode. Here these strains combined are caed the initia strain set. From the rate of change of texture coefficient df mn =dg and the texture coefficients F mn at different strains, A rqmn k were cacuated. Using the texture evoution matrix A, the simuated vaues of texture coefficients F mn at strains from % to 5%, were obtained. The evoution of these three texture coefficients with the strain is shown in Fig. 2. In the strains between 5% and 5%, the recacuated F 11 from A rqmn k are very cose to those obtained from Tayor mode according to Fig. 2(a). When the strain is higher than 5%, the recacuated F 11 begins to deviate from the Tayor prediction. The same trend is observed in the evoution curves of F 12 from Fig. 2(b) and (c). In the range of initia strain set, from 5% to %, the resuts from the inear mode have a first order agreement with the Tayor prediction. In a typica case when the strain is 3%, the difference between recacuated vaues and Tayor prediction is negigibe. It deviates from the prediction of Tayor mode when the strain is far from the initia strain set. In the second set, a strain step of 2% was used. The initia strain set incudes 22%, 2%, 26%, 28% and 3%. Fig. 2 shows that the texture evoution function using A from this strain set works we in the strains from 2% to %. If the strain is smaer than 2%, F 11 and F 12 are under-predicted is over-predicted. A simiar procedure was appied with a strain step of 1%. The initia strain set incudes 23%, 2%, 25%, 26% and 27%. Texture evoution matrix A obtained from this procedure works we in the strain range cose to the initia strain set. When the strain is arger than 3%, the simuated texture coefficients evoution curve deviates from the raw data curve. At arger strains, the predictions for F 11 ; F 12 are competey out of the acceptabe range. It is cear from the resuts above that the procedure introduced in this paper provides the best resuts when used for interpoation. This means that if the texture coefficients are predicted at a strain which is within the range of initia strain set, the error is negigibe. If this prediction is extended to strains out of the range of the initia strain set, the farther the strain is from the range of the initia strain set, the worse the prediction of the texture evoution. The choice of the initia strain set is critica in improving the accuracy of the modeing for the texture evoution. In practice, the main appication for this methodoogy is to cacuate the A coefficients from the experimenta data. One major probem for this appication is that the experimenta data for texture at different strains are usuay not widey avaiabe. Furthermore, it may be necessary to predict the texture data outside of the range of

10 16 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) F11.1 Evoution of F11 during uniaxia tension strain % 1% 2% 3% % 5% (a) raw data east squares error method strain step = 5% strain step = 2% strain step = 1% F12. Evoution of F12 during uniaxia tension raw data east squares error method strain step = 5% strain step = 2% strain step = 1% -.1 strain % 1% 2% 3% % 5% -.2 (b) -.3 F Evoution of F13 during uniaxia tension raw data east squares error method strain step = 5% strain step = 2% strain step = 1%.1.5 % 1% 2% 3% % 5% strain (c) -.5 Fig. 2. Simuated processing path function during uniaxia tension of (a) F 11 12, (b) F and (c) F 13 using texture evoution matrix A obtained by strain step of 1%, 2% and 5%, respectivey. Evoution curves of F 11 ; F 12 of Tayor prediction and simuation curve by east squares error method are aso iustrated.

11 the avaiabe experimenta data. Fuy utiizing the strain range of the experimenta data wi increase the accuracy of this mode to predict the texture evoution behavior outside of the experimenta data Least square errors method To improve the modeõs texture evoution predictions, the fu range of the experimenta data in the initia strain set is utiized. This was achieved by using the east squares error method to obtain the texture evoution coefficients. If M + 1 is the dimension of the initia strain set, there are M simutaneous sets of equations. N is the number of the rate of change of F mn s of interest. P is then the number of F rq k coefficients which are required to cacuate each of the change rate of F mn. From M + 1 texture data (M > N) at different strains, the texture evoution coefficient matrix A was cacuated from the over-determined system of equations as shown beow: df mn ðgþ ¼½AŠ½F rq k dg ðgþš: ð17þ df mn ðgþ=dg is an N M matrix. A is an N P matrix and F rq k ðgþ is an P M matrix. Here the number of points M is chosen as 1; the initia strain set varies from 3% to 1%, and the strain step dg is 1%. The range of this initia strain set is smaer than that used with strain step of 5%, but arger than those used with the strain step of 2% and 1%. From the texture data at these strains, A rqmn k coefficients were cacuated by the east squares error method. The resutant A rqmn k s were used to recacuate the evoution of texture coefficients during the deformation from 2% to 5%. To iustrate the advantage of the east squares error method, the resuts are aso shown in Fig. 2. It can be seen that the east squares error method describes the behavior better than using a strain step of 5%. The first resut has a sma initia strain set range (from 3% to %) whie the second resut has a reativey arge initia strain set range (from 25% to 5%). The agreement with the Tayor prediction of the simuated resuts using east squares method is amost the same as using the strain step of 5% in the strain range of 25% to 5%. This range is incuded in the initia strain set. When extrapoated outside of the initia strain range, the recacuated texture coefficients evoution curves using the east squares error method are coser to the curves from the Tayor prediction than any other simuated curves. The texture evoution coefficients obtained using east squares method give a more accurate description of the texture evoution behavior during mechanica deformation. An error parameter, error mn, defined as the absoute vaue of the difference between the simuated texture coefficients and the Tayor prediction is introduced to demonstrate the accuracy of the procedure. error mn D.S. Li et a. / Internationa Journa of Pasticity 21 (25) ðgþ ¼absððF mn Þ simuated ðf mn Þ rawdata Þ: ð18þ To compare the errors in a strain range, a mean error error mn parameter is defined as the average of the errors as cacuated in Eq. (18) aong the strain range:

12 162 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) error mn ¼ 1 X error mn : ð19þ N Fig. 3 iustrates the mean error parameter for the three nonzero texture coefficients obtained from different approaches. In the strain range from 2% to 5%, the simuation from over-determined system using east squares error method fits best to the Tayor prediction. The mean errors of F 11 ; F 12 from the over-determined system (east square errors method) are a smaer than the mean errors of the corresponding texture coefficients using the determined system method. The simuation from strain step of 5% is the next most accurate foowed by the 2% strain step. For a strain step of 1%, the mean error becomes very arge. This arge error occurs because the error is averaged in a strain range which is 1 times the range of its initia strain set. The anaysis of the mean error corresponds to the anaysis of simuation behavior of the previous texture evoution function Infuence of truncation imit for texture coefficients In a above simuations the maximum number of coefficients used in Eq. (3) was L max =. Since texture representation requires a arger number of coefficients, there maybe a truncation error which may be reduced by increasing the number of coefficients. To check the infuence of truncation error, the number the resuts for L max = was compared to L max = 8 by increasing the number of coefficients to 8. The initia strain set incuded 1%, 2%, 3%,...,5%. The simuated resuts for the evoution of texture coefficient F 11 is presented in Fig.. The simuated evoution curve for F 11 from L max = (using the same initia strain set) is aso shown here for error.25 Mean error in simuating texture coefficients.2.15 east squares error strain step=5% strain step=2%.1.5 F11 F12 F13 Fmn Fig. 3. error mn, mean error of texture evoution of F 11 ; F 12 during uniaxia tension when the simuated simuation resuts are obtained using different methods.

13 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) comparison. Using a higher truncation imit (L max = 8), the predictions for the texture coefficients are coser to TayorÕs resuts. The texture coefficients for the initia microstructure has been seected near the origin of the texture hu, which represents a random state. The spherica harmonics method is efficient in representing the random or weak textured microstructures. This efficiency may have contributed to the sma differences using the different truncation imits in Fig.. The resuts may be different if the initia microstructure is seected as a highy textured materia. A fu study of these other factor wi be presented in a ater work. 3.. Appication of the texture evoution mode for other processing paths This mode has shown a remarkabe success in representing texture evoution in uniaxia tension. It can aso be appied to other processing paths incuding uniaxia compression test. From the same random texture, the evoution of the texture coefficients is iustrated in Fig. 5 for compression. The Tayor prediction of F 11 ; F 12 are shown in Fig. 5(a), (b) and (c), respectivey. In a simiar process, a strain step dg of 5% whose initia strain set incudes 5%, 1%, 15%, 2% and 25% is used. Aso, a east squares error method was used with the initia strain set of 1%, 2%, 3%,...,3%. Fig. 5 shows that the resutant curves from these two simuation methods agree we with the curves from the predictions from the Tayor mode. Least squares error method resuts in a sma improvement in the prediction. In Fig. 6, mean error is used to compare the Evoution of F11 during uniaxia tension F strain 35% 36% 37% 38% 39% % raw data Lmax= Lmax= Fig.. Simuated processing path function of F 11 during uniaxia tension using texture evoution matrix A obtained using L max as and 8, respectivey.

14 16 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) F11.25 Evoution of F11 during uniaxia compression F11 raw data strain step= 5% east squares error method.5 (a) strain % 5% 1% 15% 2% 25% 3% F12 Evoution of F12 during uniaxia compression strain -. % 5% 1% 15% 2% 25% 3% F12 raw data strain step= 5% east squares error method (b) -.7 F13 Evoution of F13 during uniaxia compression strain % 5% 1% 15% 2% 25% 3% F13 raw data strain step= 5% east squares error method (c) -.8 Fig. 5. Simuated processing path function during uniaxia compression of (a) F 11 12, (b) F and (c) F 13 using texture evoution matrix A obtained by strain step of 5%, respectivey. Evoution curves of F 11 ; F 12 of Tayor prediction and simuation curve by east squares error method are aso iustrated.

15 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) mean error.18 Mean error in modeing of texture coefficients during uniaxia compression strain step= 5% east squares error..2 F11 F12 F13 texture coefficient Fig. 6. error mn, mean error of texture evoution of F 11 ; F 12 simuated simuation resuts are obtained using different methods. during uniaxia compression when the deviation from the Tayor prediction. In the strain range from % to 3%, the mean error using east squares error method is ony one-fifth to one-tenth of that using strain step dg of 5%. Another state of stress used for the deformation process is roing. From the same initia random texture, the evoution of the texture coefficients during roing for a strain range between % and 3% is shown in Fig. 7. The resutant simuated curves for the evoution of the texture coefficients are aso iustrated in Fig. 7. The resuts show that the texture evoution mode aso fits the Tayor prediction in roing and the simuation resuts from the east squares error method are in better agreement than that using strain step dg of 5%. The mean errors of these two simuation methods for the strain range from % to 3% shown in Fig. 8 gives the same trend. Predictive range of the texture evoution mode is not very impressive when it is appied to the Tayor simuation. It works we in the range cose to the initia strain set. If the texture evoution matrix is appied to highy deformed microstructures or those quite different from the initia texture, the error is increased. This deviation may be attributed to the imitation of the Tayor mode which is based on the assumption of homogeneous deformation rate throughout the materia. This imitation eads to the imited predictive range of the conservation principe. The imitation in using the simuation from Tayor is aso reveaed in the restriction of the predictive range of the texture evoution matrix A for other points in the texture hu. The idea texture evoution matrix A shoud work for any microstructure with different textures as ong as the same processing path is used. It is cear from this investigation that predictive range of A is arge but imited. For exampe, A for uniaxia tension obtained from the initia strain set of 3%, 31%,..., % using the east squares error method is appied to a randomy textured sampe which was

16 166 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) F11.5 Evoution of F11 during roing strain % 5% 1% 15% 2% 25% 3% F11 raw data strain step = 5% east squares error method (a) -.2 Evoution of F12 during roing F strain % -.6 5% 1% 15% 2% 25% 3% F12 raw data strain step = 5% east squares error method (b) Evoution of F13 during roing F13 strain % 5% 1% 15% 2% 25% 3% F13 raw data strain step = 5% east squares error method (c) -.7 Fig. 7. Simuated processing path function during roing of (a) F 11 12, (b) F and (c) F 13 using texture evoution matrix A obtained by strain step of 5%, respectivey. Evoution curves of F 11 ; F 12 of Tayor prediction and simuation curve by east squares error method are aso iustrated.

17 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) mean error.3 Mean errors in modeing texture coefficients strain step = 5% east squares error.1.5 F11 F12 F13 texture coefficients Fig. 8. error mn, mean error of texture evoution of F 11 ; F 12 during roing when the simuated simuation resuts are obtained using different methods. roed to 5%. As shown in Fig. 9, the deviation of the present mode for the texture evoution is very arge when compared to TayorÕs mode. More work is needed to improve the predictive capabiity of the texture evoution matrix. Increasing the truncation imit for the texture coefficients may improve the error as mentioned in the earier section. texture coefficients.5 Texture evoution during uniaxia tension for a sampe roed 5% strain % 5% 1% 15% 2% F11 raw data F12 raw data F13 raw data F11 from A F12 from A F13 from A Fig. 9. Soid curves iustrate Tayor prediction of F 11 ; F 12 during uniaxia tension of sampe roed 5% first from random texture status obtained from Tayor mode. Dashed curves iustrate simuation resut of F 11 ; F 12 from A cacuated from the initia strain set of 3%, 31%,...,% (uniaxia tension strain start from a random status) using the east squares error method.

18 168 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) Texture evoution path in texture hu As described before, for the cubic-orthotropic system, three nonzero texture coefficients, F 11 ; F 12 are sufficient to describe the texture in studying eastic properties. In a 3D space whose Cartesian coordinate axes are composed of these three texture coefficients, any point represents a microstructure (with a unique texture) that is reated to its corresponding properties. The microstructure at any deformation state can then be represented by a point with coordinates represented by the texture coefficients. Texture evoution path is then a curve connecting the points during a singe deformation process. Fig. 1 shows the texture evoution paths of Fig. 1. Processing path from the Tayor prediction and simuated resut presented in the texture hu whose coordinates are three texture coefficients F 11 ; F 12. Soid ine is Tayor prediction. Dash ine denoted by symbo $, very cose to the back ine, is from simuated resut using east squares error method. Dash ine for the simuation from strain step of 5% and 2% are denoted by symbo and respectivey.

19 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) cubic-orthotropic system from TayorÕs simuation and the texture evoution path mode. The resuts describe the texture evoution from a strain of 2% to a strain of 5% during uniaxia tension. Using the mode proposed in this paper, the evoution of texture coefficients as a function of the deformation is obtained. This mode describes a simpe but effective methodoogy to connect the evoution of microstructure and processing. The soid texture evoution path from the Tayor prediction is indistinguishabe from the red dashed texture evoution path ines simuated using east squares error method. The dashed texture evoution path ine denoted by symbo represents the simuation resut by a strain step of 5%. It deviates a itte from the texture evoution path as constructed from the Tayor prediction. The texture evoution path ines simuated by a strain step of 2% shows a arger deviation. The appication of texture evoution path in MSD becomes very convenient and more understandabe when it is constructed in the texture hu. Iustrated in Fig. 11, the texture hu is a compact convex subspace. A textures (representing microstructures) can ony exist inside this convex subspace such that no point can exist outside of the wire-framed texture hu. The texture evoution path in Fig. 11 shows the simuated texture evoution from a textured state ðf 11 ¼ 1; F 12 ¼ ; F 13 ¼ Þ to a strain Fig. 11. Processing path of uniaxia tension from an initia state ff 11 ; F 12 ; F 13 g¼f1; ; g to a strain of 5% from the simuated resut using east squares error method in the wire-framed texture hu. The staring point at a strain of % is cose to origina point. Yeow circe abes the microstructure at a strain of 5%.

20 161 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) of 5% during uniaxia tension. The texture evoution matrix was obtained from the east squares error method for this simuation. The starting point is very cose to origin of the coordinate system because the first three texture coefficients of the origina random state are a zero. Using MSD, properties cosure (Adams et a., 21, 2; Kaidindi et a., 2) can be obtained by the intersection of severa hyperpanes and may resut in a subspace in the texture hu. Simiary, properties aong the texture evoution path can be obtained by simpe arithmetic averages of these microstructure components using TayorÕs mode as upper bound and SachÕs mode as ower bound. TayorÕs mode assumes a uniform deformation gradient for a grains in the poycrystaine materias and identica to the macroscopic deformation gradient. Sachs mode assumes uniform stress throughout a the grains. Constrained-hybrid mode adds the assumption of zero extension aong some specific crysta directions (Parks and Ahzi, 199). These modes a require a cear understanding of the underying deformation mechanism. They are appied in different materias and different deformation stages. The mode proposed here in this study can simuate the texture evoution from the experimenta measurement of textures without fuy understanding the underying deformation mechanisms. The mode can ceary work best for interpoation as ong as the texture evoution matrix (A) can be determined from the experimenta data. Extrapoation seems to give arge deviation from the expected resuts when textures are predicted farther away from the initia strain set. Improving the deformation mechanisms may add some constraints and improve the predictive capabiity. In this work ony uniaxia tension, compression and roing were studied. The same methodoogy can be impemented in obtaining texture evoution functions for other processing methods, such as biaxia tension, processing in magnetic fied and so on. With the broadened knowedge, the processing path from one initia microstructure to a desired microstructure can be achieved by a combination of these texture evoution path functions. 5. Streamines for the evoution of texture coefficients The texture evoution function, as described earier, is a function of the processing parameter h. This may impose a restriction on the use of the present mode to a variety of optimization processes. In the spirit of Microstructure Sensitive Design, it is desirabe to get the famiy of a texture evoution path functions for a specific processing path. This means that independent of the initia texture (microstructure), a materias designer may wish to expore a the different texture evoution paths to examine the famiy of microstructures that may be achieved by a singe process (roing, etc.). In this section, the streamine functions wi be derived from the texture evoution functions. If the evoution of texture coefficients for poycrystaine materias in the texture hu is considered as a fuid fow, streamines can then describe the texture evoution independent of the path parameter, h. In the foowing, the streamine for the evoution of the texture

21 coefficients wi be derived. Suppose the eigenvaues of A are a 1, a 2,...,a n and the corresponding eigenvectors are: ~a 1 ; ~a 2 ;...;~a n. 2 3 a 1 a 2 LetL ¼ ð2þ. 5 and D.S. Li et a. / Internationa Journa of Pasticity 21 (25) a n P ¼ ½~a 1 ~a 2 ~a n Š: ð21þ Now it is cear that the evoution matrix A, (or the sixth rank tensor) can be decomposed as: A ¼ PLP 1 : ð22þ Substituting this equation back into the texture evoution path function (12), we obtain: F ðgþ ¼e PLP 1g F ðg ¼ Þ: Further use of Eqs. (2) and (21), we have: 2 F ðgþ ¼Pe Lg P 1 F ðg ¼ Þ ¼P 6 e a 1g e a 2g... e ang ð23þ P 1 F ðg ¼ Þ: ð2þ The same operation is executed on both sides of Eq. (2) to get: 2 3 e a 1g e a 2g P 1 F ðgþe m ¼ P 1 F ðg ¼ Þe m : ð25þ e m is a unit vector whose eements are except the mth eement. That is: e ang P 1 F ðgþe m P 1 F ðg ¼ Þe m ¼ eamg : ð26þ From here the texture evoution path parameter is obtained:

22 1612 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) n g ¼ P 1 F ðgþe m P 1 F ðg ¼Þe m a m : ð27þ Substituting the evoution path parameter back to the texture evoution path function, we get the expression of the streamine of the texture coefficients: n F ðgþ P 1 F ðgþe m P 1 F ðg Þe m a m 1 AAF ðg ¼ Þ: ð28þ Fig. 12 shows the streamines for texture evoution in ff m g space during uniaxia tension. The ony variabe is the initia texture. The five streamines shown here are from five different microstructures, whose ff 11 ; F 12 ; F 13 g are {.2,, }, {.,, }, {.6,, }, {.8,, } and {1.,, }, respectivey. The same texture evoution matrix A cacuated by the east squares error method in Section 3.2 is used for the cacuation of these five streamines. The texture evoution can now be compared to the steady-state condition. There is no change in the direction of the veocity vector at any point. This means that the streamine is unique if the initia texture and processing method are the same. The continuity reations in the Euer space can reduce a arge set of constitutive reations and a its detais into a sixth order tensor A for reativey arge strains. Such a reduction and its consequent set of streamines can make the optimization for the texture evoution path rea simpe and save the designer a arge amount of time. The foowing exampe is provided to expain the main utiity of the texture evoution path parameter. Texture can be represented by a set of texture coefficients as in Eq. (3). The first two nonzero texture coefficients are used in a two dimensiona pot as in Fig. 13(a). An off-the-shef materia (manufactured by casting and roing) maybe represented by point A in this two-dimensiona pot. The materia represented by point A, however, may not be the desired materia and we may want to process a desired microstructure represented by point B. The question that may be raised is how the materia represented by A can be processed to become a materia represented by B. Based on the present pasticity formuations we may have to depend on a arge data base and the correct answer may ie in a arge number of crysta pasticity set of simuations. The present paper provides a time saving and robust methodoogy to get to the materia B with texture represented by Eq. (3) by seecting one or a mutipe number of paths. If the famiy of a possibe deformations paths are considered for both A and B (as in Fig. 13(a)) the soution may be a combination of texture evoution paths connecting the two points (Fig. 13(b)). The soution is obviousy not unique but using the streamines formuation deveoped in the present paper we can choose a path which can best fit the designers objective.

23 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) Fig. 12. Streamine of uniaxia tension cacuated from the texture evoution matrix obtained from TayorÕs mode. Texture is considered a macroscopic manifestation of the microstructure crysta grain orientation through the continuity reations in the Euer space. The resuts show that the continuity reations work in a arge strain range which is a very important factor proving the accuracy of this principe. Remember that we use the continuity reations in the materias coordinate as part of the fied equations in many of the formuations without regards to the

24 161 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) Fig. 13. (a) Materias design probem presented in a simpified two dimensiona materias space. (b) Streamine grid in the materias spaces present the soution for materias design. underying mechanisms for the deformation processes. It is cear that the continuity reations in the materias coordinate may aso reach its imits and can ony appy when the materia can be assumed a continuous medium throughout the deformation process. The continuity reations in the orientation space can aso reach its imits of usefuness when there is a discontinuity in the orienta-

25 D.S. Li et a. / Internationa Journa of Pasticity 21 (25) tion space. This wi occur during recrystaization and may occur in a number of other situations. 6. Concusion and perspectives A conservation principe for texture evoution has been utiized to get a texture evoution path function. The soution aso provides streamines during any mechanica deformation process (roing forging, drawing, etc.). Using texture coefficients as a descriptor, microstructures can be represented as points in the mutidimensiona space whose axes are these texture coefficients. Texture evoution path, a trace of texture coefficients during deformation history, is used to represent the microstructure evoution in this mutidimensiona space. The texture evoution matrix A is a critica parameter in the processing path function deveoped here. It is cacuated from texture coefficients at different eves of deformation defined as the initia strain set. The simuation works we as a predictive too if the desired microstructure (texture) is cacuated for deformations in the range of the initia strain set. Increasing the range of the initia strain set wi improve the performance of simuated resut in a arger strain range. If more data are avaiabe to cacuate the texture evoution matrix, utiization of the east squares error method wi increase the accuracy of simuated texture evoution path. Presenting the resuts of the texture evoution path in the texture hu shows how the microstructure evoves with the deformation parameter. Streamines obtained from the texture evoution path functions give a representation of the texture evoution independent of the deformation parameter. Streamines can simpify the optimization of the processing path and provide the mechanica designers with an important too in Microstructure Sensitive Design. Acknowedgements This work has been funded under the AFOSR Grant # F and Army Research Lab Contract # DAAD17-2-P-398 and DAAD B.L. Adams acknowedges support of Army Research Office, Proposa No MS. The authors express their sincere gratitude to Professor Surya Kaidindi, for hepfu suggestions and correspondence for the derivation of the processing path functions. References Adams, B.L., Canova, G.R., Moinari, A., A statistica formuation of viscopastic behavior in heterogeneous poycrystas. Textures Microstruct. 11, Adams, B.L., Henrie, A., Henrie, B., Lyon, M., Kaidindi, S.R., Garmestani, H., 21. Microstructuresensitive design of a compiant beam. J. Mech. Phys. Soids 9,