D. Raabe and F. Roters

HOW DO 10 BILLION CRYSTALS CO-DEFORM? D. Raabe and F. Roters Max-Plank-Institut für Eisenforshung Max-Plank-Str. 1 40237 Düsseldorf Germany January 2004, Max-Plank-Soiety http://www.mpg.de http://www.mpie.de http://edo.mpg.de/ KEYWORDS: anisotropy, rystal, plastiity, simulation, texture, finite element method, mehanis, rystal, anisotropy, metals, polymers, geology Raabe, edo Server, Max-Plank-Soiety - 1 - MPI Düsseldorf

Projet referenes D. Raabe, M. Sahtleber, Z. Zhao, F. Roters, S. Zaefferer: Ata Materialia 49 (2001) 3433 3441 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) 421 440 Theory of orientation gradients in plastially strained rystals Z. Zhao, F. Roters, W. Mao, D. Raabe: Advaned Engineering Materials 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, Z. Zhao, F. Roters: Steel Researh 72 (2001) 421-426 A Finite Element Method on the Basis of Texture Components for Fast Preditions of Anisotropi Forming Operations D. Raabe, P. Klose, B. Engl, K.-P. Imlau, F. Friedel, F. Roters: Advaned Engineering Materials 4 (2002) 169-180 Conepts for integrating plasti anisotropy into metal forming simulations D. Raabe, Z. Zhao, W. Mao: Ata Materialia 50 (2002) 4379 4394 On the dependene of in-grain subdivision and deformation texture of aluminium on grain interation D. Raabe and F. Roters: International Journal of Plastiity 20 (2004) 339-361 Using texture omponents in rystal plastiity finite element simulations D. Raabe, M. Sahtleber, H. Weiland, G. Sheele, and Z. Zhao: Ata Materialia 51 (2003) 1539-1560. Grain-sale miromehanis of polyrystal surfaes during plasti straining S. Zaefferer, J.-C. Kuo, Z. Zhao, M. Winning, D. Raabe: Ata Materialia 51 (2003) 4719 4735. On the influene of the grain boundary misorientation on the plasti deformation of aluminum birystals J.-C. Kuo, S. Zaefferer, Z. Zhao, M. Winning, D. Raabe: Advaned Engineering Materials, 2003, 5, (No.8) 563-566 Deformation Behaviour of Aluminium-Birystals Raabe, edo Server, Max-Plank-Soiety - 2 - MPI Düsseldorf

Report abstrat Crystalline materials reveal highly anisotropi mehanial behavior 1-6. Examples are metals, geologial substanes, semiondutors, superondutors, or semi-rystalline polymers. Large sale rystalline anisotropy has four major soures. First, rystals reveal intrinsi elasti reversible anisotropy due to the orientation dependene of the atomi bonds. Seond, they reveal individual irreversible anisotropy due to rystallographi material translations along preferred diretions and planes 1-6. Third, rystalline matter usually ours in polyrystalline form with typially more than 10 billion interating rystals 4-6 eah of whih has a different orientation, shape, and size 7. Fourth, rystals undergo re-orientations during loading 1-6. Our report presents for the first time a method whih predits mehanial anisotropy of polyrystals onsisting of huge numbers of grains taking full aount of these mehanisms. The novelty of the approah onsists in the integration of a small set of spherial rystallographi orientation distribution funtions 8,9 into a non-linear finite element model with an elasto-plasti anisotropi onstitutive law 10-12. The method is suited for simulating anisotropial mehanial response of rystalline samples as enountered in materials siene, geology, prodution tehnology, solid state physis, and ivil engineering. Raabe, edo Server, Max-Plank-Soiety - 3 - MPI Düsseldorf

Mehanial anisotropy is a general property of materials whih onsist of one or more rystals. It proeeds from the intrinsi elasti-plasti anisotropy of eah single rystals on the one hand and from the mehanial interation and re-orientation of the rystals on the other hand (Fig. 1). Anisotropy may be desired suh as in many funtional materials (promotion of quasi single rystalline superondutivity in high-t C superondutors or transformer steels) or not desired suh as in many partially rystalline thermoplasti polymers. Figure 1a Researh on anisotropy of materials has been a topi for 4000 years. The reasons for that are obvious: Engineers want to save material, obtain similar mehanial properties everywhere in the material, avoid failure, and minimize elasti bak stresses in formed parts. Sientists want to understand the nano- and miromehanis of interating rystals. Geologists and polymer hemists want to understand the origin of anisotropy in rystalline or partially rystalline material ontaining large numbers of rystals. Raabe, edo Server, Max-Plank-Soiety - 4 - MPI Düsseldorf

Figure 1b Researh on anisotropy in terms of multisale modelling. This progress report is about a new method whih is apable of prediting the mehanial anisotropy of polyrystalline material onsisting of an arbitrary number of rystals, its origin, and its development under loads. The method takes full aount of rystalsale reversible and irreversible anisotropy, the mehanial interations among rystals, and their individual re-orientations. The approah an help to better understand and predit the anisotropy of samples onsisting of huge numbers (e.g. 10 10 ) of rystals. Classial rystal elastiity and plastiity finite element models represent exellent tools for simulating the mehanis and re-orientations of rystals under realisti boundary onditions 10-12. However, these models are limited by the relatively small number of rystals they an handle (usually less than 10 4 ). This limit is due to the fat that rystal-sale finite element approahes up to now required a disrete representation of the rystalline orientation information at eah integration point. When dealing with small numbers of rystals disrete mappings of their orientations are ahieved by a simple one-to-one approah, where eah Raabe, edo Server, Max-Plank-Soiety - 5 - MPI Düsseldorf

Gauss point in the finite element mesh is oupied by one disrete rystallographi orientation. This approah, however, is inappropriate when simulating samples whih ontain muh larger numbers of rystals. The key hallenge of our new approah, therefore, lies in identifying a way of plaing orientation distributions rather than disrete sets of orientations onto a finite element mesh. This new tehnique is the main topi of this progress report, i.e. we introdue a new effiient method of mapping a representative rystallographi orientation distribution, omprising a huge number of grains, on the Gauss points of a finite element mesh using a ompat mathematial form whih permits rystalline re-orientation during loading. For this purpose we use the orientation omponent method 8,9. This is a tehnique of approximating the orientation distribution funtion of large numbers of rystals in the form of disrete sets of symmetrial spherial entral funtions whih are defined in orientation spae. Suh funtions have individual height and full width at half maximum as a measure for the strength and satter of the rystallographi orientation omponent they represent. The mathematial reprodution of the orientation distribution funtion by omponent funtions an be expressed by the superposition C f ( g) = F + I f ( g) = I = 1 C = 0 f ( g) where I 0 = F, f 0 ( g) = 1 (1) 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). The intensity I desribes the volume fration of all rystallites belonging to the orientation omponent. The orientation density of the omponent is desribed by a entral funtion, i.e. its value dereases isotropially with inreasing orientation distane ~ ω = ~ ω ( g, g ) from the maximum. This means that f ( g) only depends on ~ ω = ~ ω ( g, g ), but it is independent on the rotation axis n ~. In our approah we deompose an initial orientation distribution funtion by a set of spherial entral Gauss funtions whih are desribed by Raabe, edo Server, Max-Plank-Soiety - 6 - MPI Düsseldorf

( S osω ~ ) f ( g) = N exp (2) where S ln 2 = 1 os( b / 2) and N = I 0 ( S 1 ) I 1 ( S ) (3) with I l (x) being generalized Bessel funtions. The value b is the full width at half maximum whih orresponds to the mean diameter of a spherial omponent in orientation spae. The orientation omponent method is well suited for an inorporation of orientation distributions into rystallographi finite element methods. This advantage is due to the fat that the method is based on using sets of loalized spherial normalized standard funtions whih are haraterized by simple parameters of physial signifiane (Euler angle triple for the main orientations, volume frations, full widths at half maximum). Typially only a few orientation omponents are required to desribe the orientation distribution funtion whih in turn an represent the texture of any rystal assembly whatever size it may have. The seond step of the new method is that the orientation omponent method must now be onneted to a suited rystal elastiity and plastiity onstitutive model. In our approah we use the large-strain onstitutive rystal model suggested by Kalidindi 12. In this formulation one assumes the stress response at eah marosopi ontinuum material point to be potentially given by one rystal or by a volume-averaged response of a set of rystals omprising the respetive material point. Details of the onstitutive law are given in 12. The third element of our new approah onsists in the integration of the orientation omponent funtions into the rystal finite element method. More preisely, the main task of the new onept is to represent sets of spherial Gaussian orientation omponents on the integration points of a finite element mesh designed for rystalline onstitutive laws. This proedure works in two steps: In the first step the disrete preferred orientation g (enter orientation, mean orientation) is extrated from eah of the orientation omponents and assigned in terms of its respetive Euler triple (ϕ 1, φ, ϕ 2 ), i.e. in the form of a single rotation matrix, onto eah integration point (Fig. 2). In the seond step, these disrete orientations are re-oriented in suh a fashion that their resulting overall distribution reprodues the Raabe, edo Server, Max-Plank-Soiety - 7 - MPI Düsseldorf

texture funtion whih was originally presribed in the form of a Gaussian orientation omponent (Fig. 2). In other words the orientation satter desribed initially by a texture omponent funtion is in the finite element mesh represented by a systematially re-oriented set of orientations, eah assigned to one integration point, whih reprodues the original spherial satter presribed by that 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. Figure 2 The first step of the deomposition of an orientation omponent onsists in extrating the preferred orientation (enter or mean orientation) from the orientation funtion and assigning it in terms of its respetive Euler angle triple, i.e. in the form of a single idential rotation matrix, onto eah integration point. In this state the sample is a single rystal. In the seond step all orientations are re-oriented to give the initial orientation distribution. The desribed alloation and re-orientation proedure is formulated as a weighted sampling Monte Carlo integration sheme in orientation spae. Loal homogenization allows one to map more than one preferred rystallographi orientation on eah integration point and to assign to eah of them an individual volume fration. This means that the proedure of mapping and rotating single orientations in aord with an initial orientation omponent satter width is individually onduted for all presribed omponents as well as for the random bakground extrated from initial experimental or theoretial data. After deomposing and representing the initial orientation omponents as a lateral and spherial Raabe, edo Server, Max-Plank-Soiety - 8 - MPI Düsseldorf

single orientation distribution in the mesh, the texture omponent onept is no longer required in the further proedure. This is due to the fat that during the subsequent rystal finite element simulation eah individual orientation originally pertaining to one of the orientation omponents an undergo an individual orientation hange. This means that the orientation omponent method loses its signifiane during the simulation. In order to avoid onfusion one should, therefore, underline that the orientation omponent method is used to feed rystal orientations into finite element simulations on a stritly physial, saleable, and quantitative basis. The omponents as suh, however, are in their original form as ompat funtions not traked during the simulation. It must also be noted that the orientation points whih were originally obtained from the omponents do not represent individual grains but portions of an orientation distribution funtion. Figure 3 Example of a simulated large strain forming operation (99.99% aluminum) inluding orientation information and rotation of the rystals during straining. The importane of the rystallographi orientation an be seen from the resulting shape revealing the so alled earing phenomenon. The olor sheme indiates the wall thikness. Raabe, edo Server, Max-Plank-Soiety - 9 - MPI Düsseldorf

Figure 4 Shape hange during drawing of an aluminum sample ontaining about 10 10 rystals. The olor sheme represents the von Mises equivalent stress. Fig. 3 shows an example of a up drawn aluminum sample and a orresponding experiment. The simulation predits very well the final ear shape and the thikness distribution of the drawn sample. The original sample ontained about 10 8 rystals whih were all inluded in the simulation via our new orientation omponent finite element method. Fig. 4 shows an example of a large strain drawing simulation of an aluminum sheet ontaining about 10 10 rystals. The last part of the figure shows details of the elasti spring-bak effet, i.e. the elasti plasti relaxation of the material after removal of the tool. The urrent progress report gave an introdution into a novel finite element method whih inludes and updates orientation distributions in physially based anisotropy simulations. The method is based on feeding spherial orientation funtions onto the Gauss points of a finite element mesh whih uses a elasti-plasti onstitutive law taking full aount of single rystalline anisotropy. The major progress of the approah onsists in its ability to feed large numbers of rystals in terms of mathematially ompat orientation omponent funtions into finite element simulations on a stritly physial, saleable, and quantitative basis. Raabe, edo Server, Max-Plank-Soiety - 10 - MPI Düsseldorf

Referenes 1. Taylor, G. I. Plasti strain in metals. J. Inst. Metals 62, 307 328 (1938). 2. Bishop, J. F. W. & Hill, R. A theory of the plasti distortion of a polyrystalline aggregate under ombined stresses. Phil. Mag. XLVI, 414 427 (1951). 3. Hill, R. Generalized onstitutive relations for inremental deformation of metal rystals by multislip. J. Meh. Phys. Solids 14, 95 102 (1966). 4. Hill, R. & Havner, K. S. Perspetives in the mehanis of elasto-plasti rystals. J. Meh. Phys. Solids 30, 5 22 (1982). 5. Hosford, W. F. The Mehanis of Crystals and Textured Polyrystals, Oxford University Press (1993). 6. Koks, U. F., Tomé, C. & Wenk, H.-R. Texture and Anisotropy. Preferred Orientations in Polyrystals and Their Effet on Material Properties. Cambridge University Press, Cambridge, England (1997). 7. Bunge, H. J., Texture analysis in materials siene. Butterworths, London, England (1982). 8. Lüke, K., Pospieh, J., Virnih, K. H. & Jura, J., On the problem of the reprodution of the true orientation distribution from pole figures. Ata Metall. 29, 167 185 (1981). 9. Helming, K., Shwarzer, R. A., Raushenbah, B., Geier, S., Leiss, B., Wenk, H.-R., Ullemeier, K. & Heinitz, J., Texture estimates by means of omponents. Zeitshr Metallkunde 85, 545 553 (1994). 10. Piere, D., Asaro, R. J. & Needleman, A. Material rate dependene and loalized deformation in rystalline solids. Ata Metall. 31 1951-1976 (1983). 11. Asaro, R. J. & Needleman, A. Texture Development and Strain Hardening in Rate Dependent Polyrystals. Ata metall. 33, 923 941 (1985). 12. Kalidindi, S. R., Bronkhorst, C.A. & Anand, L. Crystallographi texture evolution during bulk deformation proessing of f metals. J. Meh. Phys. Solids 40, 537 569 (1992). Raabe, edo Server, Max-Plank-Soiety - 11 - MPI Düsseldorf