**Grain-Scale Modeling Approaches for Polycrystalline Aggregates**

Igor Simonovski1 and Leon Cizelj2

<sup>1</sup>*European Commission, DG-JRC, Institute for Energy and Transport, P.O. Box 2, NL-1755 ZG Petten* <sup>2</sup>*Jožef Stefan Institute, Reactor Engineering Division, Jamova cesta 39, SI-1000 Ljubljana* <sup>1</sup>*The Netherlands* <sup>2</sup>*Slovenia*

## **1. Introduction**

48 Polycrystalline Materials – Theoretical and Practical Aspects

Alexandrov V. M., "Longitudinal Crack in an Orthotropic Elastic Strip with Free Faces," Izv.

Aleksandrov V.M., "Two Problems with Mixed Boundary Conditions for an Elastic

Belubekyan V. M., "Is there a Singularity at a Corner Point of Crystal Junction?" in

Chobanyan K. S. and Gevorkyan S. Kh., "Stress Field Behavior near a Corner Point of the

Chobanyan K. S., *Stresses in Compound Elastic Bodies* (Izd-vo Akad. Nauk Armyan. SSR,

Galptshyan P.V., "On the Existence of Stress Concentrations in Loaded Bodies Made of

Knuniants I. L. et al. (Editors), Short chemical encyclopedia, Vol. 1 (Sovietskaya

Lekhnitskii S. G., *Theory of Elasticity of an Anisotropic Body* (Nauka, Moscow, 1977; Mir

Love A. E. H., *A Treatise on the Mathematical Theory of Elasticity*, 4th ed. (Cambridge Univ.

Osrtrik V. I. and Ulitko A. F., "Contact between Two ElasticWedges with Friction," Izv.

Pozharskii D. A. and Chebakov M. I., "On Singularities of Contact Stresses in the Problem of

Pozharskii D. A., "Contact with Adhesion between Flexible Plates and an ElasticWedge,"

Timoshenko S. P. and Goodyear J. N., *Theory of Elasticity* (McGraw-Hill, New York, 1951;

Ulitko A. F. and Kochalovskaya N. E., "Contact Interaction between a Rigid and Elastic

Vainstein B. K. et al. (Editors), *Modern Crystallography*, Vol. 4 (Nauka, Moscow, 1981) [in

Ukrainy. Ser. Mat. Estestvozn., Tekhn.N., No. 1, 51–54 (1995).

No. 5, 72–77 (1998) [Mech. Solids (Engl. Transl.) 33 (5), 57–61 (1998)]. Pozharskii D. A., "The Three-Dimensional Contact Problem for an Elastic Wedge Taking

Akad. Nauk.Mekh.Tverd.Tela, No. 3, 93–100 (2000) [Mech. Solids (Engl. Transl.)

a Wedge-Shaped Punch on an ElasticCone," Izv. Akad. Nauk.Mekh. Tverd. Tela,

Friction Forces into Account," Prikl. Mat. Mekh. 64 (1), 151–159 (2000) [J. Appl.

Izv. Akad. Nauk. Mekh. Tverd. Tela,No. 4, 58–68 (2004) [Mech. Solids (Engl.

Wedges at Initial Point Contact at Their Common Vertex," Dokl. Nats. Akad. Nauk

41 (1), 88–94 (2006)].

Erevan, 1987).

(Engl. Transl.) 70 (1), 128–138 (2006)].

(Aiastan, Erevan, 2000), pp. 139– 143.

Nauk Armyan. SSR. Ser. Mekh. 24 (5), 16–24 (1971).

Feodos'ev V. I., *Strength of Materials* (Nauka, Moscow, 1979) [in Russian].

(2008) [Mech. Solids (Engl. Transl.) 43(6), 967-981 (2008)].

encyclopedia, Moscow, 1961) [ in Russian].

Press, Cambridge, 1927; ONTI,Moscow, 1935).

Math. Mech. (Engl. Transl.) 64 (1), 147–154 (2000)].

Publishers, Moscow, 1981).

Transl.) 39 (4), 46–54 (2004)].

Nauka, Moscow, 1975).

Russian].

35 (3), 79–85 (2000)].

Akad. Nauk. Mekh. Tverd. Tela,No. 1, 115–124 (2006) [Mech. Solids (Engl. Transl.)

Orthotropic Strip," Prikl. Mat. Mekh. 70 (1), 139–149 (2006) [J. Appl.Math. Mech.

*Investigations of Contemporary Scientific Problems in Higher Educational Institutions* 

Interface in the Problem of Plane Strain of a Compound Elastic Body," Izv. Akad.

Polycrystalline Materials," Izv. Akad. Nauk. Mekh. Tverd. Tela, No 6, 149-166

In polycrystalline aggregates microstructure plays an important role in the evolution of stresses and strains and consequently development of damage processes such as for example evolution of microstructurally small cracks and fatigue. Random grain shapes and sizes, combined with different crystallographic orientations, inclusions, voids and other microstructural features result in locally anisotropic behavior of the microstructure with direct influence on the damage initialization and evolution (Hussain, 1997; Hussain et al., 1993; King et al., 2008a; Miller, 1987). To account for these effects grain-scale or meso-scale models of polycrystalline aggregates are being developed and are increasingly being used.

In this chapter we present some of the most often used approaches to modeling polycrystalline aggregates, starting from more simplistic approaches and up to the most state-of-the art approaches that draw on the as-measured properties of the microstructure. The models are usually based on the finite element approach and differ by a) the level to which they account for the complex geometry of polycrystalline aggregates and b) the sophistication of the used constitutive model. In some approaches two dimensional models are used with grains approximated using simple geometrical shapes like rectangles (Bennett & McDowell, 2003; Potirniche & Daniewicz, 2003) and hexagons (Sauzay, 2007; Shabir et al., 2011). More advanced approaches employ analytical geometrical models like Voronoi tessellation in 2D (Simonovski & Cizelj, 2007; Watanabe et al., 1998) and 3D (Cailletaud et al., 2003; Diard et al., 2005; Kamaya & Itakura, 2009; Simonovski & Cizelj, 2011a). In the most advanced approaches, however, grain geometry is based on experimentally obtained geometry (Lewis & Geltmacher, 2006; Qidwai et al., 2009; Simonovski & Cizelj, 2011b) using methods such as serial sectioning or X-ray diffraction contrast tomography (DCT) (Johnson et al., 2008; Ludwig et al., 2008). These approaches are often referred to as "image-based computational modeling" and can also embed in the model measured properties such as crystallographic orientations. The acquired information is of immense value for advancing our understanding of materials and for developing advanced multiscale computational models. The rather high level of available details may render extremely complex geometries, resulting in highly challenging preparation of finite element (FE) models (Simonovski & Cizelj, 2011a) and computationally extremely demanding simulations. These two constraints have so far limited the development and use

...............................................................................................................................

×

×

...............................................................................................................................

×

.

Fig. 2. Examples of 2D Voronoi tessellations.

.................................................................................................................................... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

×

.................................. .. . . ... .

. . . .. . . .. . . . . .. . . .. . . . . . . . . .. . . . . . . . .

.

. ... ... ..

. ... ... .. .. . ... . .. . ... . . .. ... .. ............................................................................................ ..... ..... ..... ..... ..... .

... .....

........................................................................................................................................................................................................................................................ ......................................................................................................................................................................................................................................................

.. .. .... .. .. .. .... .. .. .. .... .. .. .. .. .. .. .. .. .... .. .. .. .... .. .. .. .... .. .. .. ... ................ ... .

Randomly positioned points

×

×

× ..

×

.......... .................................................. .................... ..............

......................................................................................................

×

................................................................................ ...............................

are introduced to lines connecting neighboring Poisson points and c) final Voronoi

. .. . . . . .. . . . . .. . . . . . . . . . . . .. . . . . .. . . . . .. . . . . . ........................................................................................

×

..

.... ........ .................................................................................................................................. ........ .... ........ .... .... ........

×

×

a) b)

..............................

c) Fig. 1. Construction of a Voronoi tessellation in 2D: a) Poisson points, b) perpendicular lines

a) 212 grains b) 525 grains

shapes of the grains are then reconstructed from the data from the 2D layers. However, the problem with this approach is that the specimen is destroyed during the measurement procedure. In last years new experimental techniques have enabled non-destructive spatial characterization of polycrystalline aggregates. Differential aperture X-ray microscopy (Larson et al., 2002), 3D X-ray diffraction microscopy (3DXRD) (Poulsen, 2004) and X-ray diffraction contrast tomography (DCT) (Johnson et al., 2008; Ludwig et al., 2008) are examples of these

........... ............. .

........... ............. ........... . ............ ............ ........ ........ ........ ........ ........ ........ .... .... .... .... .... .... .... .... .... ... .. .. .. .... .. .. .. .. .. .... ..

.

..........

...............................................................

................................................... .. . . . . . . . . . . . . .. . . .. . . ........................................................... . ... .... .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ................................................... .. .. .. .... .. .... .. .. .... .. .. .... .. .. .. .. .. .. .. .. .. .. .... .. .. .... .. ........................................................................ .. .. .. .. .. .. .. .... .. .. .. .. .. .... .. .. .. .. .... .. .. .. .. .. .... .. .. .. .. ..

...............................................................................................................................

×

...... .... .. .. .. .. .. .... .. .. .. .. .... .. .. .. .. .. .. .. .. .. .. .. ..

..

..

. ... .

..

. ... . .. .

..

.

.

.

.

. .... . .... . ..

. ... . .. . .. . ... .

. ... . ... . ... .

..

..

.

..

..

.

..

.

.

..

.

.

..................... .. .... . .. . . .. . .. ... .

...... .......... ...... ...

×

...............................................................................................................................

×

........................................................... ............................................................

. ... .. ... . .. ........................................ .................. .................. .................. .................. ........................................................................................... .......... .......... .. .... .. ...... .... ............ .... .. .... ...... ............

.. .. .. .. .. .... .. .. .. .. .. .. .... .. .. .. .... .. .. .... .. .. .. .. ......... ..... ..... ..... ..... .. .... .. .. .. .. .. .. .. .... .. .. .. .. .. .. .. .. .... .. ..

........................................................................................................................................................................................................................................................ ......................................................................................................................................................................................................................................................

............. ............. ............. ............. .. .. .. .. .. ... .. .. .. .. .. ...

×

×

×

...................................

..................................................................................................................................................................................................... .................................................................

............. ............. ....

.............................................................................................................

×

..................

..... ..... ..... ..... ...... ...... . ...... ...... . ...... ...... . ...... .

..............................................................................................

× <sup>×</sup> <sup>×</sup>

..

. ... . .. .

....................................................................... ..................................................

. ... . .. . ... . ... .

. .... . ... .

. .... . .. .

. .... .

.

. .... .

. .... . .. .

. .... . .. .

. .... .

. ... . ... .

. ... .

.. .. .. .. .. .. .. . .. .. .. .. .... .. .. .. .. .. .. .. . .... .. .. .. .. .. .. .. .. .. .. .. .. ........................... .. .... .. .. .. .. .. .. ...... .. .. .. .. .... .. .. .. .... .... .. .. .... .. .. .... .... .. .. .. ...... .. .. .. .. .... .. .. .. .. ..

.. .. .. .. .... .. .. .. .. .. .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .... .

........................................................................................................ ... .. .. .. .. .... .. .. .. .. .. .. .. .... .. .. .. .... .. .. .. .. .. .. .. .... .. .. .. .... .. .. .. .. .. .. .. .. ...... .... .... .... .... .... .... .... .... .... ............................................

............................ .............................

............................................................ . ...........................................................

......................................................

×

.............................................................................................................................................................. .. .... .. .. .. .. .. .. .... .. .. .. .. .. .. .... .. .. .. .. .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .... .. .. .. .. .. .... .. .. .. .. .. .... .. ..

......................................................................................................

×

× <sup>×</sup> <sup>×</sup>

.....................................................................................................

×

×

...............................................................................................................................

..............................................................................................................................

×

× <sup>×</sup> <sup>×</sup>

×

.

. .. .

. .

.

. ... . .. . ... . .. .

.. ... .... .. .. .. .. .. .. .. .... .. .. .. .... .. .. .. ....... ......... ......... ....... .... .. .. .... .. .. ............ ..

..

. .. .. .

..

. .

. ... . .. .

. ...... ..

..

.

. ... . .

..

... .......................................

.

.

.

.

..

. ...... ... .. . ...... ...... . ..... .... .... . .... . .. .

. . . ......... ..

×

...................................................................................................................................... ............................................................................................................................. ................................................................................. .................

×

...................................................................................................................................................... .......... .. ..

.. .. .. .. .. .. .. .. .. .... .. .. .. ...... ..... ..... ..... ..... ..... ..... ... ..

...............................................................

.....................................................................................................

×

...

Grain-Scale Modeling Approaches for Polycrystalline Aggregates 51

×

×

×

...............................................................................................................................

..............................................................................................................................

×

...

×

tessellation.

of the image-based models. Steps aimed at obtaining a 'reasonable' size model in the terms of computational times are presented. The term 'reasonable' should be taken in relative terms as these models may still run for several days on today's high performance clusters.

Specifically, two analytical approaches (two and three dimensional Voronoi tessellations) for modeling grain geometries as well as an approach based on the X-ray diffraction contrast tomography (DCT) (Johnson et al., 2008; Ludwig et al., 2008) are presented. The DCT enables spatial non-destructive characterization of polycrystalline microstructures (King et al., 2010). The process of building a finite element model from either analytical spatial structures or as-measured spatial structures is explained. The meshing procedure, including ensuring conformal mesh between the grains as well as grain boundary modeling are discussed and explained. The most used constitutive models and the issues related to the modeling of the grain boundaries using the cohesive zone approaches are discussed. In the last section the effects of grain structure to inhomogeneous stress/strain distribution is demonstrated. Initiation and development of intergranular stress corrosion cracks is outlined and discussed for different constitutive models. Also, the stability of the simulations and measures aimed at improving it are considered.
