**3. As-measured spatial structures**

Models of polycrystalline aggregates can also be created from the experimental data, where the grain shapes and orientations are measured, i.e. the microstructure is being characterized. A widely used method for characterizing the microstructure in order to get the data for creating a finite element model is serial sectioning (Lewis & Geltmacher, 2006; Qidwai et al., 2009; Spowart et al., 2003). With this approach a surface of the specimen is characterized and then a thin layer of material is removed to be able to characterize the next layer. The procedure is consecutively repeated, thus obtaining the data at different depths. The 3D 2 Polycrystalline Materials

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

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

In material science Voronoi tessellations are extensively used to model grain geometry with the purpose of calculating the properties of polycrystalline aggregates (Cailletaud et al., 2003; Kovaˇc & Cizelj, 2005), modeling short crack initiation and propagation or modeling intergranular stress corrosion cracks (Kamaya & Itakura, 2009; Musienko & Cailletaud, 2009; Simonovski & Cizelj, 2011b). A Voronoi tessellation is a cell structure constructed from randomly positioned points, also referred to as Poisson points. For polycrystalline aggregates we can think of these points as points where the solidification starts and then uniformly extends in all directions. A solidification front then expands until it collides with another one, thus creating a grain boundary. In geometrical terms the grain boundary is obtained by introducing lines perpendicular to lines connecting neighboring Poisson points. The result is a set of convex polygons/polyhedra, see Fig. 1. Examples of 2D Voronoi tessellations can be seen in Fig. 2. Additional details on mathematical background and applications are available

Three dimensional Voronoi tessellations can be created using the same approach. A number of mathematical and programming packages like Matlab have the option of constructing such tessellations. Fig. 3 shows examples of 3D Voronoi tessellations generated using Qhull algorithm (*Qhull code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace*

Models of polycrystalline aggregates can also be created from the experimental data, where the grain shapes and orientations are measured, i.e. the microstructure is being characterized. A widely used method for characterizing the microstructure in order to get the data for creating a finite element model is serial sectioning (Lewis & Geltmacher, 2006; Qidwai et al., 2009; Spowart et al., 2003). With this approach a surface of the specimen is characterized and then a thin layer of material is removed to be able to characterize the next layer. The procedure is consecutively repeated, thus obtaining the data at different depths. The 3D

these models may still run for several days on today's high performance clusters.

improving it are considered.

**2. Analytical models of structures**

in (Aurenhammer, 1991; Okabe et al., 2000).

**3. As-measured spatial structures**

*Intersection about a Point*, n.d.) and implemented in (Petriˇc, 2010).

Fig. 1. Construction of a Voronoi tessellation in 2D: a) Poisson points, b) perpendicular lines are introduced to lines connecting neighboring Poisson points and c) final Voronoi tessellation.

Fig. 2. Examples of 2D Voronoi tessellations.

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

*Diffraction spot (bright)*

*Extinction spot (dark)*

*Transmitted beam and absorption image of sample*

*Image coordinate system*

x

y

*Detector plane 2048*×*2048 pixels*

Grain-Scale Modeling Approaches for Polycrystalline Aggregates 53

Y X Z

Fig. 5. An example of a measured grain shape. Full resolution is used.

models can be built in a reasonable time.

One can see that the available level of detail is very high and that the obtained geometry is extremely complex. This results in highly challenging preparation of finite element (FE) models and computationally extremely demanding simulations. These two constraints have so far limited the development and use of the image-based models. However, with suitable simplifications and parallel pre-processing (Simonovski & Cizelj, 2011a), appropriate FE

?

*Set-up coordinate system*

> *Grain aligned for diffraction*

Fig. 4. A scheme of X-ray diffraction contrast tomography (DCT), after (Johnson et al., 2008).

orientations. A rotating polycrystalline sample is exposed to a monochromatic X-ray wave while the projection images are recorded, Fig. 4. Since the sample is rotating, each grain will pass through Bragg diffraction alignments several times. A detector system, significantly bigger than the sample, captures low index reflections. In absence of orientation and strain gradients within the grains, the diffracted beams form 2D spots that can be treated as parallel projections of the grains' volume (King et al., 2010). The shape of each grain can then be reconstructed in 3D using algebraic reconstruction techniques (Gordon et al., 1970). The resolution of the technique is in the order of 1 *μ*m. An example of a measured grain shape of a 400 *μ*m diameter stainless steel wire is given in Fig. 5. The complete wire is depicted in

2?

2

ωθ

*Synchrotron beam*

Fig. 6.

Fig. 3. Examples of 3D Voronoi tessellations.

procedures. Through DCT for example, grain shapes and orientations can be measured and even crack initiation and growth can be monitored (Herbig et al., 2011). Since for some of the presented cases the data has been acquired using DCT, the next section gives an overview of this technique.

#### **3.1 X-ray diffraction contrast tomography (DCT)**

DCT (King et al., 2008a; Ludwig et al., 2008) is a measurement procedure jointly developed by the European Synchrotron Radiation Facility (ESRF) and University of Manchester, Materials Performance Centre, School of Materials. It combines the X-ray diffraction imaging and image reconstruction from projections to obtain the data on the grain shapes and crystallographic 4 Polycrystalline Materials

100 grains 500 grains

1000 grains 5000 grains

procedures. Through DCT for example, grain shapes and orientations can be measured and even crack initiation and growth can be monitored (Herbig et al., 2011). Since for some of the presented cases the data has been acquired using DCT, the next section gives an overview of

DCT (King et al., 2008a; Ludwig et al., 2008) is a measurement procedure jointly developed by the European Synchrotron Radiation Facility (ESRF) and University of Manchester, Materials Performance Centre, School of Materials. It combines the X-ray diffraction imaging and image reconstruction from projections to obtain the data on the grain shapes and crystallographic

Fig. 3. Examples of 3D Voronoi tessellations.

**3.1 X-ray diffraction contrast tomography (DCT)**

this technique.

Fig. 4. A scheme of X-ray diffraction contrast tomography (DCT), after (Johnson et al., 2008).

orientations. A rotating polycrystalline sample is exposed to a monochromatic X-ray wave while the projection images are recorded, Fig. 4. Since the sample is rotating, each grain will pass through Bragg diffraction alignments several times. A detector system, significantly bigger than the sample, captures low index reflections. In absence of orientation and strain gradients within the grains, the diffracted beams form 2D spots that can be treated as parallel projections of the grains' volume (King et al., 2010). The shape of each grain can then be reconstructed in 3D using algebraic reconstruction techniques (Gordon et al., 1970). The resolution of the technique is in the order of 1 *μ*m. An example of a measured grain shape of a 400 *μ*m diameter stainless steel wire is given in Fig. 5. The complete wire is depicted in Fig. 6.

Fig. 5. An example of a measured grain shape. Full resolution is used.

One can see that the available level of detail is very high and that the obtained geometry is extremely complex. This results in highly challenging preparation of finite element (FE) models and computationally extremely demanding simulations. These two constraints have so far limited the development and use of the image-based models. However, with suitable simplifications and parallel pre-processing (Simonovski & Cizelj, 2011a), appropriate FE models can be built in a reasonable time.

The right-hand-side of Fig. 6 displays wire data-set after the treatment above.

on the implemented approach can be found in (Simonovski & Cizelj, 2011a).

Geometries of individual grains need to be reconstructed from the voxel-based data. This is usually achieved through reconstructing the surfaces of individual grains. Surface reconstruction from voxels is available in a number of commercial visualization tools. The origins of these tools can mainly be traced to the field of medical visualization. The tools were later further developed for the application to material science. In this work surfaces are reconstructed as sets of triangles with Amira package (Visage Imaging GmbH, 2010). A label is assigned to each measured point, defining to which grain this point belongs. The labels are equal to the index of the crystallographic orientation. Label 1 refers to grain 1 with

Grain-Scale Modeling Approaches for Polycrystalline Aggregates 55

Amira's built-in SurfaceGen tool with unconstrained smoothing option is used. This tool partitions the bounding volume into 362 grains depending on the number of different labels in the 8 vertices of a given voxel. Near the triple points between the grains and near the grain boundaries vortices of a given voxel will be distributed among several grains. In these cases the voxel is subdivided into up to 6<sup>3</sup> sub-vortexes to give a topologically correct representation of the implicitly defined separating surfaces (Westerhoff, 2003). If two adjacent sub-vortexes are of different grains, their common face is added to the list of boundaries between the two grains. A comprehensive explanation of the procedure is given in (Stalling et al., 1998; Westerhoff, 2003) and was later implemented in Amira. Described approach automatically increases the resolution near the triple points between the grains and near the grain boundaries where vortices of a given voxel are distributed among several labels/grains. This is especially important since stress increases at these points can be expected due to different crystallographic orientations of the adjacent grains. Further details

The density of the triangles forming the reconstructed surfaces is limited by the resolution of the experimental data. At full resolution the number of triangles is 4 758 871, resulting in FE model with 51 211 552 finite elements. The number of triangles therefore needs to be decreased. This is done using Amira's built-in surface simplification tool (Zachow et al., 2007). The simplification decreases the details as well as resolution at the triple lines between the grains, see Fig. 7 where the number of triangles has been decreased to 30 000 (case 30K),

The complexity of the reconstructed surfaces, together with a rather large number of grains, essentially prevents the direct use of the finite element meshing capabilities of either Amira or even professional mesh generators such as for example ABAQUS/CAE (Simulia, 2010). A framework for the automatic meshing has been therefore developed (Simonovski & Cizelj, 2011a) using the Python scripting language and ABAQUS/CAE meshing tools, which are

The framework can be applied to both analytical models of structures (e.g. 3D Voronoi tessellations) and to data obtained from experimental techniques. In both cases the surface structures are defined by the triangle-based surfaces, bounding the volume of individual grains. In the case of voxel-based data these surfaces are reconstructed with Amira. For

**3.3.2 Surface reconstruction**

crystallographic orientation index 1 and so forth.

150 000 (case 150K) and 300 000 (case 300K).

**4. From surfaces to FE models**

fully accessible through Python.

Fig. 6. Reconstructed geometry of a 400 *μ*m diameter stainless steel wire. Left: holes in the original experimental data. Right: wire after the hole treatment, surface grooves present.
