D. Raabe. Max-Planck-Institut für Eisenforschung Max-Planck-Str Düsseldorf Germany

Transcription

1 Raabe, Texture and FEM Simulations OPEN DOCUMENT - PROJECT REPORT THE MAPPING OF TEXTURE FUNCTIONS ON FINITE ELEMENT GRIDS FOR ANISOTROPY SIMULATIONS D. Raabe Max-Plank-Institut für Eisenforshung Max-Plank-Str Düsseldorf Germany January 2004, Max-Plank-Soiety Keywords orientation distribution funtion, simulation, rystallographi texture, rystal plastiity, anisotropy, finite elements Raabe, edo Server, Max-Plank-Soiety MPI Düsseldorf

2 Projet Referenes D. Raabe, M. Sahtleber, Z. Zhao, F. Roters, S. Zaefferer: Ata Materialia 49 (2001) Miromehanial and maromehanial effets in grain sale polyrystal plastiity experimentation and simulation D. Raabe, Z. Zhao, S. J. Park, F. Roters: Ata Materialia 50 (2002) Theory of orientation gradients in plastially strained rystals Z. Zhao, F. Roters, W. Mao, D. Raabe: Adv. Eng. Mater. 3 (2001) p.984/990 Introdution of A Texture Component Crystal Plastiity Finite Element Method for Industry- Sale Anisotropy Simulations M. Sahtleber, Z. Zhao, D. Raabe: Materials Siene and Engineering A 336 (2002) 81 87, Experimental investigation of plasti grain interation D. Raabe, P. Klose, B. Engl, K.-P. Imlau, F. Friedel, F. Roters: Advaned Engineering Materials 4 (2002) Conepts for integrating plasti anisotropy into metal forming simulations D. Raabe, Z. Zhao, W. Mao: Ata Materialia 50 (2002) On the dependene of in-grain subdivision and deformation texture of aluminium on grain interation D. Raabe: Annual Review of Materials Researh 32 (2002) p Cellular automata in materials siene with partiular referene to rerystallization simulation D. Raabe: Advaned Materials 14 No. 9 (2002) p Challenges in Computational Materials Siene D. Raabe and F. Roters: International Journal of Plastiity 20 (2004) p Using texture omponents in rystal plastiity finite element simulations D. Raabe, M. Sahtleber, H. Weiland, G. Sheele, and Z. Zhao: Ata Materialia 51 (2003) Grain-sale miromehanis of polyrystal surfaes during plasti straining S. Zaefferer, J.-C. Kuo, Z. Zhao, M. Winning, D. Raabe: Ata Materialia 51 (2003) On the influene of the grain boundary misorientation on the plasti deformation of aluminum birystals 2

3 Projet Abstrat The study disusses how rystal orientation distributions an be mapped on finite element grids for onduting large sale anisotropy simulations whih use anisotropi rystalline onstitutive laws. Methods based on diret pole figure inversion, series expansions of spherial harmonis, or a large sets of disrete orientation values are not appropriate to reprodue rystallographi textures in a suffiiently loalized spherial form onto finite element grids. The texture omponent method whih employs Lorentz or Gauss shaped spherial funtions is better suited for this task. It offers a good ompromise between disreteness (spherial loalization), exatness (approximation of ompliated orientation distribution funtions an be ahieved by a few texture omponents), ompatness (simple funtions), salability (the number of used texture omponents an be systematially varied aording to the desired preision of the texture fit), and physial signifiane (texture omponents are related to mirostrutural mehanisms). 3

4 Raabe, Texture and FEM Simulations Raabe, edo Server, Max-Plank-Soiety MPI Düsseldorf

5 1 Introdution Modern experimental and theoretial methods for investigating rystallographi orientation distributions (textures) failitate and in part neessarily entail the aumulation of data sets onsisting of huge sets of disrete rystallographi orientation values (mirotexture, nanotexture) or pole figure projetions of statistial orientation distributions (marotexture). In either ase quantitative standard measures must be identified to utilize these data in subsequent polyrystal anisotropy simulations whih inorporate orientation dependent onstitutive laws. Examples are the mapping of rystallographi textures into mathematial methods for preditions of elasti and elasti-plasti anisotropy, ferromagneti response, thermal expansion, and the resistive eletrial ondutivity of textured polyrystalline matter ontaining large numbers of rystals and its integral response to mehanial and/or eletromagnetial loadings. The seond even more demanding task along this line is the predition of the hange of rystalline anisotropy during loading, as for instane ourring during elasti-plasti deformation where the rystals individually rotate and subdevide during deformation owing to the antisymmetry of the displaement gradients reated by rystal slip. This study deals with the integration of rystallographi texture data into suh large sale anisotropy simulations. Partiular pronuniation is plaed on reduing redundant texture information to a level where suffiient details an be reovered without loosing physial signifiane. This rather omplex problem an be split into two quite separate tasks (Figure 1 [1]). The first one is the formulation of a basi solution method whih uses texture as a state variable. This is typially ahieved by formulating an orientation dependent onstitutive law whih maps the requested physial anisotropy at the single rystal sale and by embedding this formulation into a finite element ode. The numerial implementation then takles the interation of the differently oriented volume portions and thereby predits the integral response of the sample under loads. Any suh formulation requires a disrete representation of the orientation distribution funtion or a portion of it at eah integration point. Therefore, the seond task onsists in mapping single orientations diretly on the Gauss points of the finite element mesh or in deomposing orientation distributions in suh a way that they an be subsequently mapped on a mesh in a manner that they orrespond to the initial overall distribution. Integral anisotropy preditions of polyrystalline matter an in priniple be onduted by diretly feeding sets of disrete orientations onto the integration points of finite element models, i.e. by simply assigning eah orientation to a separate integration point. However, suh a mapping method is not pratiable when aiming at the simulation of speimens whih onsist for instane of disrete rystals. A seond limitation omes from the neessity to formulate a salable method. Salability in this ontext means the requirement to map large as well as small texture data sets with equal mathematial onsisteny. Both aspets suggest the introdution of a ompat mathematial form to map and update textures properly in anisotropy preditions. The present study disusses the advantages and disadvantages of some funtions available for the representation of textures under the speial boundary onditions mentioned above. This means in partiular the requirement for a disrete representation of the orientation distribution funtion or a portion of it at eah integration point. 5

6 2 Basi approahes for the reprodution of the orientation distribution funtion 2.1 Introdution The orientation distribution of the rystals assembling a polyrystalline aggregate, f(g), an be reprodued from two-dimensional entro-symmetri projetions whih are referred to as pole figures or sets of single orientations [2-5]. In either ase different mathematial methods an be used. In the first ase the orientation distribution is ommonly desribed in terms of the diret inversion of the pole figures or in terms of Fourier-based series expansion methods whih use spherial harmonis as library funtions. In the seond ase one ommonly reprodues the orientation distribution by use of large sets of disrete single grain orientations with idential satter and amplitude or by use of the texture omponent method where eah preferred orientation has individual satter and amplitude. 2.2 Series expansion methods In the Fourier-type series expansion methods the orientation distribution funtion is µν approximated by omputing the oeffiients C l of its orthogonal expansion from the expansion oeffiients F ν l ( h i ) of experimentally deteted pole figures [2-5]. Sine the entrosymmetri pole figures are expanded with a series of even order spherial harmonis, only the µν even order oeffiients C l of f (g) are generated, rendering the so determined funtion an approximate orientation distribution funtion ontaining only the even part of f (g) whih is ~ referred to as f ( g ). The omplete funtion, f (g), is named true orientation distribution ~ ~ funtion. The real orientation distribution is hene given by the equation f ( g) = f ( g) + f ( g), ~ ~ where f ( g ) represents any funtion whih an be added to f ( g) without hanging ~ orresponding projetions, i.e. pole figures [6]. The even funtion f ( g ) usually auses texture dependent errors whih are referred to ghost omponents. They an lead to positive or negative deviations from the true orientation distribution funtion. For reduing ghost intensities in the orientation distribution funtion advaned series expansion methods use the non-negativity ondition whih led to the development of an iterative series-expansion method [5, 7-9]. 2.3 Diret pole figure inversion The methods whih use diret inversions of pole figure projetions were formulated by Williams [10], Ruer and Baro [11], Imhof [12], Matthies and Vinel [13], and Pawlik [14]. They approximate textures in terms of the diret integration of the fundamental equation of texture analysis using a set of experimental pole figures. Sine the diret inversion methods diretly use the disrete form of the fundamental equation for pole figure inversion they work in real spae and not in Fourier spae. This means that these approahes make use of the fat, 6

7 that the disrete hemisphere of the pole figure orresponds to families of projetion tubes in the f (g) spae, i.e. the non-negativity ondition for the orientation distribution funtion is automatially taken into aount. The solution is ahieved by iteration. The diret methods are like all texture reprodution methods whih use starting data of higher symmetry (i.e. 2D projetions) than the orientation distribution funtion affeted by the inversion symmetry of the pole figure data entailing positive or negative texture portions without physial signifiane (texture ghosts) [5,14-16]. In the disrete form, ells in the pole figures and in the f (g) spae are diretly related by linear equations. The density in eah pole figure ell is the sum of the orresponding ell densities in the f (g) spae. Sine different f (g) sums lead to the same pole density, the set of linear equations is underdetermined whih leads to so alled ghost errors in the orientation distribution funtion. The seond disadvantage of the diret inversion methods is that they do not provide Fourier oeffiients. Although this shortoming is inevitably onneted with all diret approahes it an be avoided by subsequently fitting an orientation distribution funtion derived by diret inversion using a series expansion or a texture omponent based method (for the latter approah see next setion). Fourier oeffiients of texture funtions are generally desirable beause they failitate the alulation of anisotropi behavior suh as diretional elasti, magneti, or eletrial properties of polyrystalline aggregates from texture data. Although diretional physial properties are of ourse not neessarily onneted to Fourier oeffiients, their employment permits high speed alulations of integral properties of textured samples partiularly in ases where homogenization theory is used to ouple texture and properties. 2.4 Texture omponent methods Mapping disrete portions of a statistial orientation distribution requires the redution of the information ontent to a level at whih omplex integral anisotropy problems arising from the interation of large numbers of intrinsially anisotropi rystals an be simulated at reasonable omputation osts. Suh an approah is provided by the texture omponent method [17-22]. It goes bak to the early texture studies where experimental and predited pole figures were mostly interpreted in terms of the evolution and physial signifiane of single disrete texture omponents [23]. Classial terms introdued in these early studies on rystallographi orientation distributions were for instane the Copper texture omponent, the Brass texture omponent, and the Taylor texture omponent. The use of preferred orientations prevailed in texture researh until the late sixties of the last entury, i.e. statements about texture evolution were made pratially exlusively on the basis of pole figures and estimated preferred omponents (ideal positions, texture omponents). Modern approahes whih desribe texture omponents in orientation spae [17-22] approximate the orientation distribution funtion by a superposition of sets of Gauss- or Lorentz-shaped model funtions with individual height and individual full width at half maximum as a measure for the strength and satter of a rystallographi texture omponent in orientation spae. Suh a disrete representation of a preferred orientation is referred to as a texture omponent. In ontrast to the use of global symmetri Wigner funtions for instane in the Fourier-type series expansion methods, the texture omponent method is based on using loalized spherial normalized standard funtions. The desribed properties learly qualify the texture omponent approah as a key method for diretly extrating texture information in a ompat fashion from experiment or theory and subsequently feeding it into finite-element based anisotropy simulations whih involve large numbers of rystals. The following setions will give a onise mathematial review of the 7

8 texture omponent method and explain how texture omponents an be mapped on a finite element grid in ases where the underlying onstitutive model has been formulated in an orientation dependent fashion [24-27]. 3 Basi formulation of the texture omponent method Following the work of Helming [20-22] the mathematial reprodution of the orientation distribution funtion by texture omponent funtions whih are loally restrited in orientation spae an be expressed by the superposition C C 0 0 f ( g) = F + I f ( g) = I f ( g) where I = F, f ( g) = 1 (1) = 1 = 0 where g is the orientation, f (g) is the orientation distribution funtion and F is the volume portion of all randomly oriented rystals (random texture omponent). F may be understood as the intensity of the only global omponent used in the model, whih is given by f ( g) = 1 for eah orientation point in Euler spae, g G.The intensity I desribes the volume fration of all rystallites belonging to the omponent. Figure 2 shows a shematial sketh of a spherial texture omponent whih is desribed in terms of a maximum orientation density at a preferred orientation g and satter width b. The orientation density of the omponent dereases with inreasing orientation distane ~ ω = ~ ω ( g, g ) from the maximum, i.e. f ( g) only depends on ~ ~ ( g, g ω = ω ) and is independent on the rotation axis n ~. The orientation distribution funtion is defined by dv 2 g f ( g) dg = 8π whih implies f ( g) 0 (2) V where V is the sample volume and d Vg the volume of all rystals with an orientation g within the orientation portion dg=sin(φ) dφ dϕ 1 dϕ 2. Normalization requires C g) dg = 1 whih implies = 0 f ( I = 1 (3) As a rule texture omponents require positivity, i.e. f ( g) 0 for all g G and I > 0 (4) where G is the orientation spae. Equation 4 an also be derived by using Equation 2 and the assumption that the texture omponents do not overlap in orientation spae and that an orientation distribution funtion an be desribed by one single texture omponent. Distribution funtions whih have a maximum at a preferred orientation g and derease with inreasing orientation distane ~ ω = ~ ω ( g, g ) are referred to as entral funtions. Suh funtions, inluding orresponding pole figures, an be generally represented in the form of series expansions of χ funtions or respetively Legendre polynomials. More pratial approximations of texture omponents have been introdued on the basis of spherial Gaussand Lorentz-funtions. The examples presented later in this work made use of Gauss-shaped 8

9 model funtions for the deomposition of the orientation distribution funtion whih are desribed by f ( g) = N exp( S osω ~ ) (5) where and and S ln 2 1 = and N = (6) 1 os( b / 2) I ( S ) I ( S ) ( g, b, y) N exp( S sin( υ / 2) ) I ( S os( υ / 2) ) 9 0 P M h = (7) 0 ( g y) 1 osυ = h, (8) I l (x) are generalized Bessel funtions. The value b is the halfwidth and an be interpreted as the mean diameter of a spherial omponent in orientation spae [20-22]. Equation 7 shows that the orresponding pole figure projetions P M ( g, b y) h, are given by losed analytial expressions. The omponents desribing f(g) an be determined by the best M P ~ y N with the realulated pole figures fit of the experimental pole figure input data h ( ) i r hi ( g, b y ) ~ M Ph I, r. The index r marks the measured sample diretions y r. The omponent parameters I, g and b and the normalization solving the least squares problem N h of the pole figures are obtained by i ~ ~ M wir Ph ( yr ) N h I Ph ( g, b, yr ) Min. i i (9) i i, r where w ir are weight fators. Usually the parameters g and b must be alulated by a nonlinear algorithm. First estimates are required, whih may be obtained manually from the graphial representation of the differene pole figures whih are alulated aording to ~ ~ M y = P y I P g, b y (10) h ( r ) h ( r ) h ( r ) i i i, Depending on experiene in interpreting rystallographi textures the user an speify the position, height, and satter of the texture omponents within ertain bounds before the minimization. This makes partiularly sense, when the number of texture omponents initially presribed to math an experimental texture is small or when a ertain satter width of the omponents should not be exeeded. Further details on the method are given in the works of Helming et al. [20-22]. 4 Main texture omponents for body entered and fae entered ubi rystals The texture omponent method provides a small set of ompat funtions whih are haraterized by simple parameters of physial signifiane (Euler angles, satter, volume fration). Usually, only a few texture omponents together with a random bakground omponent are required for representing textures in a preise mathematial form and for desribing the integral anisotropy of a speimen. 2

10 The most important of these omponents in fae entered ubi metals are the Cubeomponent ({001}<100>, ϕ 1 =0, φ=0, ϕ 2 =0 ), the Goss-omponent ({011}<100>, ϕ 1 =0, φ=45, ϕ 2 =0 ), the Brass-omponent ({011}<211>, ϕ 1 =35, φ=45, ϕ 2 =0 ), the Copperomponent ({211}<111>, ϕ 1 =90, φ=35, ϕ 2 =45 ), and the S-omponent (~{123}<634>, ϕ 1 =60, φ=32, ϕ 2 =65 ). In body entered ubi metals the most important texture omponents are the Rotated Cube omponent {001}<110> (ϕ 1 =0, φ=0, ϕ 2 =45 ), the Inverse Brass-omponent {112}<110> (ϕ 1 =0, φ=35, ϕ 2 =45 ), the {111}<110> omponent (ϕ 1 =0, φ=54.7, ϕ 2 =45 ), the {111}<112> omponent (ϕ 1 =30, φ=54.7, ϕ 2 =45 ), and the Goss-omponent ({011}<100>, ϕ 1 =0, φ=45, ϕ 2 =0 ). 5 Mapping texture omponents onto disrete spatial grids The main hallenge of diretly prediting integral polyrystal anisotropy on the basis of loal single rystal behavior lies in identifying an effiient way of mapping an originally statistial and representative rystallographi orientation distribution in a disrete fashion on the integration points of a spatially disrete mesh (as for instane required for finite element implementation). This boundary ondition favors the use of sets of loalized spherial texture omponents. Figure 3 explains how texture omponents an be embedded in disrete rystal anisotropy simulations. The example given in Figure 3a refers to a simulation of the integral elasti stiffness of a textured polyrystal. In this ase the orientations are mapped as independent state variable on eah Gauss point and the single rystal stiffness modulus ats as a onstitutive law (see also Figure 1). The example given in Figure 3b refers to a simulation of the integral elasti-plasti response and the orresponding rystallographi reorientation rates of a textured polyrystal. In this ase the orientations are also mapped as independent state variable on eah Gauss point and the single rystal stiffness and slip systems inluding some plasti hardening rule at as onstitutive laws. Determination and mapping of the texture omponents proeeds as follows: In the first step the texture omponents must be reovered from experimental or theoretial data. This an be done by applying the texture omponent method formulated by Helming [20-22] as outlined in setion 3. Mapping the omponents in a disrete fashion onto the integration points of a finite element mesh is onduted in two separate steps. First, the disrete preferred orientation (enter orientation, mean orientation) of eah texture omponent is equally assigned in terms of its respetive Euler triple (ϕ 1, φ, ϕ 2 ) onto eah of the integration points. It is important in this ontext, that the use of the Taylor assumption loally allows one to map more than one preferred rystallographi orientation on eah integration point. In the seond step, the mapped enter orientations of the texture omponents are statistially rotated in suh a fashion that the resulting distribution of all the rotated orientations reprodues exatly the desired texture omponent funtion. In other words the orientation satter individually desribed by eah texture omponent funtion aording to the texture omponent method is mapped onto the finite element mesh by systematially modifying the orientations at eah point in a way 10

11 whih exatly imitates the orientation satter of the texture omponent. This means that the satter whih was originally only given in orientation spae is now mathed by an equivalent satter both, in real spae and in orientation spae. This proedure is individually onduted for all presribed texture omponents extrated from initial experimental or theoretial data. The random sattering bakground omponent an be mapped by adding to eah integration point a randomly hosen orientation using the volume fration suggested by the texture omponent fit so that the total set of all these orientations amounts to a random distribution with the orret volume fration. This method is useful sine during plasti deformation additional texture omponents an gradually build up from this random omponent as is known from orresponding experiments whih show that the random texture portion hanges during forming. 6 Example of a polyrystal plastiity simulation In the following we present results obtained from a up drawing simulation using a rystal plastiity finite element method in onjuntion with the texture omponent method outlined above. Simulations of up drawing, partiularly those whih predit details of the shape hange, depend on details of the ontat situation between sample and tool. The up drawing simulations were onduted under the assumption that the irular blank being drawn had an initial radius of 100 mm and an initial thikness of 0.82 mm. The blank was modeled using 432 elements of type C3D8 and 80 elements of type C3D6. The interation between the blank and the blank holder was assumed as a soft ontat to impose the appropriate lamping pressure in the thikness diretion of the element between blank, die, and blank holder. The simulations used an exponential soft ontat funtion. Figure 4 shows the simulated and experimentally observed development of earing in a up drawn aluminium sample together with the experimental and reprodued pole figures. The diagram shows simulation results for a speimen the texture of whih was approximated using a volume fration of 29 % of an orientation lose to the ube omponent (Euler angles at Gauss maximum: ϕ 1 =197.87, φ=6.47, ϕ 2 = ) and the rest (71%) as random texture bakground omponent. The texture reprodued by the omponent method given in terms of {111} and {200} pole figure projetions shows good agreement with the original experimental data. The pole figures are shown in stereographi projetions using 1.0, 2.0, 3.0, 4.0, 7.0 ontour levels. The predited distribution of the earing height reveals a very good orrespondene with the simulation result. 7 Conlusions The study disussed various reprodution methods of rystallographi orientation distributions with respet to their appliability in disrete simulation methods of integral polyrystal anisotropy properties whih use the rystal orientation as an independent state variable in their underlying onstitutive laws. We disussed diret pole figure inversion, Fourier-type series expansions, and the texture omponent method. The latter method whih employs Lorentz or Gauss shaped spherial funtions is well suited for the desribed task. It 11

12 offers a good ompromise between disreteness (spherial loalization), exatness (texture approximation an be ahieved by a few texture omponents), ompatness (simple funtions), salability (the number of used texture omponents an be varied aording to the desired preision of the texture fit), and physial signifiane (texture omponents are related to mirostrutural mehanisms). The integration of the texture omponents into property simulations was demonstrated based on feeding disrete loalized spherial texture omponents onto the Gauss points of the mesh of a finite element simulation whih used a rystal plastiity onstitutive law. The method was tested and the results were ompared to experimental data. 12

13 8 Referenes [1] K. HELMING, Materials Siene Forum, , 363 (1994). [2] R.J. ROE, Journ. apl. phys. 36, 2024 (1965). [3] H.J. BUNGE, Z. Metallkunde 56, 872 (1965). [4] H.J. BUNGE, Texture Analysis in Materials Siene, Butterworths, London, England, [5] H.-R. WENK, H.J. BUNGE, J.S. KALLEND, K. LÜCKE, S. MATTHIES, J. POSPIECH, and P. Van Houtte, Orientation distributions: Representation and determination. Summary Report from a panel disussion at the Eighth Int. Conf. on Textures of Materials, Proeedings Eighth Int. Conf. on Textures of Materials (ICOTOM 8), The Metallurgial Soiety of AIME., Warrendale, Pennsylvania, USA, 17 (1988). [6] S. MATTHIES, H.-R. WENK, and G.W. VINEL, J. appl. ryst. 21, 285 (1988). [7] M. DAHMS and H.J. BUNGE, J. appl. ryst. 22, 439 (1989). [8] M. DAHMS, Textures and Mirostrutures 19, 169 (1992). [9] D. RAABE, Textures and Mirostrutures 23, 115 (1995). [10] R.D. WILLIAMS, J. appl. phys. 39, 4329 (1968). [11] D. RUER and R. BARO, Adv. X-ray anal. 20, 187 (1977). [12] J. IMHOF, Z. Metallkunde 68, 38 (1977). [13] S. MATTHIES and G.W. VINEL, phys. stat. sol. (b) 112, K (1982). [14] K. PAWLIK, phys. stat. sol. (b) 134, 477 (1986). [15] D. RAABE and K. LÜCKE, phys. stat. sol. (b) 180, 59 (1993). [16] D. RAABE and K. LÜCKE, Mater. S. Forum , 413 (1994). [17] K. LÜCKE, J. POSPIECH, J. JURA, and J. HIRSCH, Z. Metallkunde 77, 312 (1986). [18] K. LÜCKE, J. POSPIECH, K.H. VIRNICH, and J. JURA, Ata Metall. 29, 167 (1981). 13

14 [19] S. MATTHIES, G.W. VINEL, and K. HELMING, Standard Distributions in Texture Analysis, Vols. I III, Akademie Verlag, Berlin, Germany ( ). [20] K. HELMING, Texturapproximation durh Modellkomponenten (in German), Habilitation Thesis, Tehnial University Clausthal, Germany, Cuvillier Verlag Göttingen, Germany (1996). [21] K. HELMING and T. ESCHNER, Cryst. Res. Tehnol. 25, K203 K208 (1990). [22] K. HELMING, R.A. SCHWARZER, B. RAUSCHENBACH, S. GEIER, B. LEISS, H.- R. WENK, K. ULLEMEIER, and J. HEINITZ, Z. Metallkunde 85, 545 (1994). [23] G. WASSERMANN and J. GREWEN, Texturen metallisher Werkstoffe (in German), Springer-Verlag Berlin, Germany (1969). [24] H.J. BUNGE, Mater. S. Forum , 3 (1998). [25] D. RAABE, Z. ZHAO, and F. ROTERS, Steel Researh 72, 421 (2001). [26] Z. ZHAO, F. ROTERS, W. MAO, D. RAABE, Adv. Eng. Materials 3, 984 (2001). [27] D. RAABE, Z. ZHAO, and F. ROTERS, Materials Siene Forum, in press (2002). 14

15 9 Figures Figure 1 The predition of integral polyrystal anisotropy an be split into two quite separate tasks [1]. The first one is the formulation of a solution method whih uses texture as a state variable. This is typially ahieved by formulating an orientation dependent onstitutive law whih maps the requested physial anisotropy at the single rystal sale and by embedding this formulation into a finite element ode (elastiity in the present ase). The numerial implementation then takles the interation of the differently oriented volume portion and thereby predits the integral response of the sample under loads. Any suh formulation requires a disrete representation of the orientation distribution funtion or a portion of it at eah integration point. Therefore, the seond task onsists in mapping single orientations diretly on the Gauss points of the finite element mesh or in deomposing orientation distributions in suh a way that they an be subsequently mapped on a mesh in a manner that they orrespond to the initial overall distribution. Figure 2 Shematial presentation of a spherial texture omponent with a preferred orientation g and satter width b. f ( g) only depends on ~ ω = ~ ω ( g, g ), i.e. it is independent on the rotation axis n ~ [20-22]. Figure 3a Priniple of the texture omponent method for the alulation of the integral anisotropy of polyrystalline matter. The example in Figure 3a refers to a simulation of the integral elasti stiffness of a textured polyrystal. In this ase the orientations are mapped as independent state variable on eah Gauss point and the single rystal stiffness modulus ats as a onstitutive law. Figure 3b Priniple of the texture omponent method for the alulation of the integral anisotropy of polyrystalline matter. The example given in Figure 3b refers to a simulation of the integral elasti-plasti response and the orresponding rystallographi reorientation rates of a textured polyrystal. In this ase the orientations are also mapped as independent state variable on eah Gauss point and the single rystal stiffness and slip systems inluding some plasti hardening rule at as onstitutive laws. Figure 4 Simulation and experimental results for earing in an aluminum sample the texture of whih was approximated using a volume fration of 29 % of an orientation lose to the ube omponent (Euler angles at Gauss maximum: ϕ 1 =197.87, φ=6.47, ϕ 2 = ) and the rest as random texture bakground omponent. The reprodued texture shows good agreement with the original experimental pole figure. The predited distribution of the earing height reveals a very good orrespondene with the simulation result. 15

16 Figure 1 16

17 Figure 2 17

18 Figure 3a 18

19 Figure 3b 19

20 Figure 4 20