Simulation of Crystallographic Texture and Anisotropie of Polycrystals during Metal Forming with Respect to Scaling Aspects

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1 Raabe, Roters, Wang Simulation of Crystallographi Texture and Anisotropie of Polyrystals during Metal Forming with Respet to Saling Aspets D. Raabe, F. Roters, Y. Wang Max-Plank-Institut für Eisenforshung, Max-Plank-Straße Düsseldorf, Germany (raabe@mpie.de) Abstrat We present a method to map and trak textures in rystal plastiity finite element simulations using texture omponents. The use of suh funtions allows us to ondut forming simulations with full anisotropy update on all size sales ranging from the mirosopi to the large-sale regime. The artile presents the onept and some appliations to the investigation of saling aspets assoiated with texture and anisotropy during metal forming. 1 Introdution Our main aim in polyrystal researh lies in developing methods for mapping rystallographi anisotropy into mathematial methods for prediting large strain plasti deformation. A seond even more demanding goal is the predition of the hange in rystalline anisotropy during deformation. This is neessary sine the rystals rotate during deformation owing to their elasti-plasti spins. The mirostrutural proesses involved during these reorientation proesses of the grains in polyrystalline matter annot be aptured in terms of simple empirial onstitutive laws but require the use of physially-based onepts. In this ontext the rystal plastiity finite element onstitutive methods have gained momentum [1-6]. This artile addresses the question how textures an be merged with rystal plastiity finite element onstitutive desriptions in a rigorous, salable, and effiient way. A partiular hallenge in this ontext lies in the redution of redundant texture information to a level where suffiient details an be reovered without loosing physial signifiane. This problem of representing large texture sets in plastiity simulations an be split into two separate tasks. The first one is the formulation of a basi solution method whih uses rystallographi orientation as a state variable. This is 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. 1

2 Max-Plank-Report on Simulation of Texture and Anisotropie, MPI Düsseldorf For this we use the approah of Raabe et al. [2-8] and Kalidindi [1]. 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 feeding one single rotation matrix (rystal orientation) diretly on eah Gauss point of the finite element mesh. This amounts to mapping or respetively deomposing orientation distributions in suh a way that they an be subsequently mapped on a mesh in a disrete manner thereby mathing the initial overall distribution. For this task we use the texture omponent method [9,10] whih approximates the orientation distribution funtion by a superposition of sets of simple standard funtions with individual spherial oordinates, orientation density, and satter in orientation spae. 2 Theoretial bakground We use the texture omponent method for reproduing orientation distributions. This mean that the texture is approximated by a superposition of model funtions with individual height and individual full width, i.e. 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 (ODF) and F is the random texture omponent. The intensity I desribes the volume fration of all rystallites belonging to the omponent. The ODF is defined by 2 dvg f ( g) dg = 8 π whih implies f ( g) 0 (2) V where V is the volume and d Vg the volume of all rystals with an orientation g within dg=sin(φ) dφ dϕ 1 dϕ 2. Normalization requires C f g) dg = 1 whih implies = 0 ( I As a rule texture omponents require positivity, i.e. = 1 (3) f ( g) 0 for all g G where G is the orientation spae. For the omponents we use spherial Gauss-funtions whih are desribed by and ( S osω ~ ) I > 0 (4) f ( g) = N exp (5) 2

3 Raabe, Roters, Wang where N is a normalization fator and ω ~ the orientation distane. Further details on texture omponents are given in [9,10]. The onstitutive rystal plastiity model was also desribed in previous works [1-8]. 3 Mapping texture omponents in the rystal plastiity finite element onstitutive model One important hallenge of polyrystal plastiity simulations lies in identifying an effiient way of mapping statistial orientation distributions on the integration points of a grid of a rystal plastiity finite element model. This applies in partiular when aiming at the simulation of larger parts typially ontaining more than rystals. The new onept we suggest for this task is based on mapping small sets of spherial Gaussian texture omponents on the integration points of a rystal plastiity finite element model. Fig. 1: Priniple of the texture omponent rystal plastiity finite element method. Fig. 1 shows the priniple of the new approah. After reovering texture omponents from experimental or theoretial data they are mapped onto the integration points of a finite element mesh. This is onduted in two steps. First, the disrete preferred orientation g (enter orientation) is extrated from eah of the texture omponents and assigned in terms of its respetive Euler triple (ϕ 1, φ, ϕ 2 ), i.e. in the form of a 3

4 Max-Plank-Report on Simulation of Texture and Anisotropie, MPI Düsseldorf single rotation matrix, onto eah integration point (Fig. 2a). In the seond step, the mapped single enter orientations of the texture omponents are rotated in suh a fashion that the resulting overall distribution of all rotated orientations reprodues exatly the texture funtion whih was originally presribed in the form of a ompat texture omponent (Fig. 2b). a) b) Fig. 2: Main steps of the spherial deomposition of a texture omponent. In other words the orientation satter individually desribed by eah texture omponent funtion is mapped onto the finite element by systematially modifying the orientations at eah point in a way whih exatly imitates the satter presribed by the texture omponent. This means that the satter whih was originally only given in orientation spae is now represented by a distribution both, in real spae and in orientation spae, i.e. the initial spherial distribution is transformed into a spherial and lateral distribution. 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 and to assign to them different volume fration (Fig. 3). 4

5 Raabe, Roters, Wang Fig. 3: Mapping of a set of texture omponents on a mesh. This means that the proedure of mapping and rotating single orientations in aord with the initial texture omponent satter width is individually onduted for all presribed omponents as well as for the random bakground extrated from initial experimental or theoretial data by use of the omponent method. After having mapped the texture omponents by deomposing them into a group of single orientations whih are arranged in the form of a lateral and spherial distribution on the mesh, the texture omponent onept is no longer required in the further proedure. During the subsequent rystal plastiity finite element simulation eah individual orientation originally pertaining to one of the texture omponents an undergo individual orientation hange as in the onventional rystal plastiity methods. This means that the texture omponent method loses its signifiane during the simulation. In order to avoid onfusion one should, therefore, underline that the texture omponent method is used to feed textures into finite element simulations on a strit physial and quantitative basis. The omponents as suh, however, are in their original form as ompat funtions not traked during the simulation. On the other hand the mapped orientation points whih were extrated from the omponents must not be onfused with individual grains, but they mark points of an exat distribution funtion. 5

6 Max-Plank-Report on Simulation of Texture and Anisotropie, MPI Düsseldorf 4 Simulation Results and Experimental Results In the following we present some up drawing appliations of the new texture omponent rystal plastiity finite element simulation method, Fig. 4. Simulations of up drawing tests depend on details of the ontat between tool and speimen. The present 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 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. Fig. 4: Shape hange during drawing of an aluminum sample ontaining about rystals. The gray sale sheme represents the von Mises equivalent stress. Different frition properties (µ=0 to 0.2) were heked and the results showed that frition properties had under these ontat onditions only little influene on the relative ear height. This is an important aspet ompared to onventional J2-based ontinuum plastiity simulations whih generally reveal stronger dependene on frition. Consequently the µ =0 ase was seleted to save omputing time. 6

7 Raabe, Roters, Wang Fig. 5 shows simulation results for a speimen the texture of whih was approximated using a volume fration of % 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 texture realulated 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 relative earing height reveals a very good orrespondene with the simulation result. Fig. 5: Simulation and experiments for earing in an aluminum sample the texture of whih was approximated by a volume fration of % of a omponent lose to ube (Euler angles at Gauss maximum: ϕ 1 =197.87, φ=6.47, ϕ 2 = ) and the rest as random texture bakground omponent. The realulated texture shows good agreement with the original experimental pole figure. The predited distribution of the relative earing height reveals a very good orrespondene with the simulation result. 5 Conlusions The study presented a new finite element method whih inludes and updates rystallographi texture during forming simulations. The method is based on feeding disrete loalized spherial texture omponents onto the Gauss points of the mesh of a finite element simulation whih uses a rystal plastiity onstitutive law. The method was tested and the results were ompared to experimental data. 7

8 Max-Plank-Report on Simulation of Texture and Anisotropie, MPI Düsseldorf Aknowledgements The work was funded by the Deutshe Forshungsgemeinshaft DFG within the Shwerpunktprogramm 1138 (Modellierung von Größeneinflüssen bei Fertigungsprozessen) 6 Referenes [1] S. R. Kalidindi, C. A. Bronkhorst, L. Anand: Journal Meh. Phys. Solids Vol. 40 (1992) p [2] D. Raabe, M. Sahtleber, Z. Zhao, F. Roters, S. Zaefferer: Ata Materialia 49 (2001) p [3] D. Raabe, Z. Zhao, S. J. Park, F. Roters: Ata Materialia 50 (2002) p [4] D. Raabe, P. Klose, B. Engl, K.-P. Imlau, F. Friedel, F. Roters: Adv. Engineering Materials 4 (2002) p [5] D. Raabe, Z. Zhao, W. Mao: Ata Materialia 50 (2002) p [6] D. Raabe, F. Roters: Intern. Journal of Plastiity, in press [7] Z. Zhao, F. Roters, W. Mao and D. Raabe: Adv. Eng. Mater. 3 (2001), p [8] D. Raabe, Z. Zhao, F. Roters: Steel Researh 72 (2001), p [9] K. Lüke, J. Pospieh, K. H. Virnih, J. Jura: Ata Metall. 29 (1980) p [10] K. Helming., R.A. Shwarzer, B. Raushenbah, S. Geier, B. Leiss, H. Wenk, K. Ullemeier, J. Heinitz, Z. Metallkd. 85 (1994) p