**7. Conclusion**

From the analysis performed in Section 6, we draw the following conclusions.

Although we considered specific cases of stress state, namely, the out-of-plane strain and the plane strain of two-crystals whose separate crystals consist of one and the same material with cubic symmetry and with different orientations of the principal directions of elasticity, we can state that, in the general case of loading of a polycrystalline body, there are stress concentrations at the corner points of the interface between the joined crystals.

It is well known that the structure of the crystal lattice of a given matter plays a definite role in the process of formation of its mechanical properties and characteristics, in particular, the strength of monocrystals. But in polycrystalline materials, along with this factor, the

12 12 *a a* and are the strain coefficients of homogeneous isotropic parts and *E*<sup>1</sup> and

formed by the

/ 2 . The values

*E*<sup>2</sup> are the Young moduli of the same homogeneous isotropic parts. Figures 2-5 correspond to the first, second, fifth, and seventh versions given in Tables 2 and 3, respectively; curves 1

contact surfaces of homogeneous isotropic parts of the compound body. The values of the

of singularities, are taken from Table 1 presented in (Chobanyan, 1987; Chobanyan &

The graphs show that the order of singularity of the stresses at the corner point of the contact surface of aluminum crystals is larger than the order of singularity of stresses at the corner point of the contact surface of gold crystals. The graphs also show that, for the piecewise homogeneous isotropic bodies under study, the order of singularity of the stresses

Although we considered specific cases of stress state, namely, the out-of-plane strain and the plane strain of two-crystals whose separate crystals consist of one and the same material with cubic symmetry and with different orientations of the principal directions of elasticity, we can state that, in the general case of loading of a polycrystalline body, there are stress

It is well known that the structure of the crystal lattice of a given matter plays a definite role in the process of formation of its mechanical properties and characteristics, in particular, the strength of monocrystals. But in polycrystalline materials, along with this factor, the

 /2, /4, 3 /4 and

<sup>2</sup> , and the corresponding values of the orders

and 2 in the same figures correspond to the four values of the linear angle

 <sup>1</sup> and 

From the analysis performed in Section 6, we draw the following conclusions.

concentrations at the corner points of the interface between the joined crystals.

in Figs. 2–5 are respectively equal to:

is much lower than that for two-crystals of aluminum and gold.

and the Poisson ratios

Fig. 5.

angle 

of the ratio

Gevorkyan, 1971).

**7. Conclusion** 

where 1 2

strength of the joint of crystals and the fact that there are stress singularities at the corner points of the interface between the crystals totally play the decisive role in the process of formation of these characteristics. This can be observed in the process of mechanical fragmentation of polycrystalline materials. They split and form small crystals of certain shape. Of course, the separate crystals are also deformed in this process. The modulus of elasticity and the ultimate strength of a monocrystal with cubic symmetry for simple matters is larger than the corresponding characteristics of the polycrystalline material of the same matter.

In the problem of plane strain, the existence of stress concentration (singularity) at the corner point of the interface between the two joined crystals with cubic symmetry made of the same material, just as the degree of stress concentration (the order of singularity), depends on the parameters 1 1 *a b* , , , and <sup>2</sup> , which are determined in Sections 1–4.

In the case of out-of-plane strain of the two-crystal under study, there is no stress concentration at the corner point of the interface between the two joined crystals.

#### **8. References**


**0**

**3**

**Grain-Scale Modeling Approaches for**

<sup>1</sup>*European Commission, DG-JRC, Institute for Energy and Transport,*

<sup>2</sup>*Jožef Stefan Institute, Reactor Engineering Division, Jamova cesta 39, SI-1000 Ljubljana*

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

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

polycrystalline aggregates are being developed and are increasingly being used.

**1. Introduction**

**Polycrystalline Aggregates**

Igor Simonovski1 and Leon Cizelj2

*P.O. Box 2, NL-1755 ZG Petten*

<sup>1</sup>*The Netherlands*

<sup>2</sup>*Slovenia*

