**1. Introduction**

24 Will-be-set-by-IN-TECH

26 Polycrystalline Materials – Theoretical and Practical Aspects

Tjahjanto, D. D. (2008). *Micromechanical modeling and simulations of transformation-induced plasticity in multiphase carbon steels*, Phd thesis, Delft University of Technology. Wechsler, M., Lieberman, T. & Read, T. (1953). On the theory of the formation of martensite,

Wei, Y. & Anand, L. (2004). Grain-boundary sliding and separation in polycrystalline metals:

application to nanocrystalline fcc metals, *Journal of the Mechanics and Physics of Solids*

*Trans. AIME J. Metals* 197: 1094.

52: 2587.

There are numerous polycrystalline materials, including polycrystals whose crystals have a cubic symmetry. Polycrystals with cubic symmetry comprise minerals and metals such as cubic pyrites (FeS2), fluorite (CaF2), rock salt (NaCl), sylvite (KCl), iron (Fe), aluminum (Al), copper (Cu), and tungsten (W) (Love, 1927; Vainstein et al., 1981).

It is assumed that many materials can be treated as a homogeneous and isotropic medium independently of the specific characteristics of their microstructure. It is clear that, in fact, this is impossible already because of the molecular structure of materials. For example, materials with polycrystalline structure, which consist of numerous chaotically located small crystals of different size and different orientation, cannot actually be homogeneous and isotropic. Each separate crystal of the metal is anisotropic. But if the volume contains very many chaotically located crystals, then the material as a whole can be treated as an isotropic material. Just in a similar way, if the geometric dimensions of a body are large compared with the dimensions of a single crystal, then, with a high degree of accuracy, one can assume that the material is homogeneous (Feodos'ev, 1979; Timoshenko & Goodyear, 1951).

On the other hand, if the problem is considered in more detail, then the anisotropy both of the material and of separate crystals must be taken into account. For a body under the action of external forces, it is impossible to determine the stress-strain state theoretically with its polycrystalline structure taken into account.

Assume that a body consists of crystals of the same material. Moreover, in general, the principal directions of elasticity of neighboring crystals do not coincide and are oriented arbitrarily. The following question arises: Can stress concentration exist near a corner point of the interface between neighboring crystals and near and edge of the interface?

To answer this question, it is convenient to replace the problem under study by several simplified problems each of which can reflect separate situations in which several neighboring crystals may occur.

A similar problem for two orthotropic crystals having the shape of wedges rigidly connected along their jointing plane was considered in (Belubekyan, 2000). They have a common vertex, and their external faces are free. Both of the wedges consist of the same material. The wedges have common principal direction of elasticity of the same name, and the other elastic-equivalent principal directions form a nonzero angle. We consider longitudinal shear (out-of-plane strain) along the common principal direction.

Strength of a Polycrystalline Material 29

. In this case,

then we have a homogeneous medium, i.e., a monocrystal with cubic symmetry, one of

equations of generalized Hooke's law written in the principal axes of elasticity , , *x y z* have

 

 

 

 

 

are the strain components, , , ...,

1 *y*

O

*x x y z yz yz y y z x zx zx zz x y xy xy*

*a a a a a a a a a*

11 12 44 11 12 44 11 12 44

Equations (1) can be obtained from the equations of generalized Hooke's law for an orthotropic body written in the principal axes of elasticity , , , *x y z* using the method

Rotating the coordinate system (, , ) *xyz* about the common axis / *z z* by the angle

 90 , we obtain a symmetric coordinate system *x*, , *y z* . Since the directions of the axes , , *x y z* and / // *xyz* , , of the same name are equivalent with respect to their elastic

whose principal directions *xx x* 1 2 coincides with the polar axis

 

 

1 we denote the angle between 1 *x* and the polar axis 0

, , , , , ,

 

 

> 

 

 

 

> 

 *x y xy* 

> 

2 *x*

2

 *,* 2

*r*

1 *x*

(1)

1

0

1 2 , 2, 

 . If 1 2

, and by

 0 ,

0 . In this case, the

(1)

are the stress components,

<sup>2</sup> ,

second crystal. By

where , , ..., *x y xy* 

the form

Fig. 1.

the angle between 2 *x* and the axis 0

 

described in (Lekhnitskii, 1981).

and 11 12 44 *aaa* , , are the strain coefficients.

properties, the equations of generalized

2 *y*

(2)

In (Belubekyan, 2000), it is shown that if the joined wedges consist of the same orthotropic material but have different orientations of the principal directions of elasticity with respect to their interface, then the compound wedge behaves as a homogeneous wedge.

The behavior of the stress field near the corner point of the contour of the transverse crosssection of the compound body formed by two prismatic bodies with different characteristics which are welded along their lateral surfaces was studied in the case of plane strain in (Chobanyan, 1987). It was assumed there that the compound parts of the body are homogeneous and isotropic and the corner point of the contour of the prism transverse cross-section lies at the edge of the contact surface of the two bodies.

In (Chobanyan, 1987; Chobanyan & Gevorkyan 1971), the character of the stress distribution near the corner point of the contact surface is also studied for two prismatic bodies welded along part of their lateral surfaces. The plane strain of the compound prism is considered.

There are numerous papers dealing with the mechanics of contact interaction between strained rigid bodies. The contact problems of elasticity are considered in the monographs (Alexandrov & Romalis, 1986; Alexandrov & Pozharskii 1998). In (Alexandrov & Romalis, 1986), exact or approximate analytic solutions are obtained in the form convenient to be used directly to verify the contact strength and rigidity of machinery elements. The monograph (Alexandrov & Pozharskii 1998) presents numericalanalytical methods and the results of solving many nonclassical spatial problems of mechanics of contact interaction between elastic bodies. Isotropic bodies of semibounded dimensions (including the wedge and the cone) and the bodies of bounded dimensions were considered. The monograph presents a vast material developed in numerous publications. There are also many studies in this field, which were published in recent years (Ulitko & Kochalovskaya, 1995; Pozharskii & Chebakov, 1998; Alexandrov & Pozharskii, 1998, 2004; Alexandrov et al., 2000; Osrtrik & Ulitko, 2000; Alexandrov & Klindukhov, 2000, 2005; Pozharskii, 2000, 2004; Aleksandrov, 2002, 2006; Alexandrov & Kalyakin, 2005).

In the present paper, we study the problem of existence of stress concentrations near the corner point of the interface between two joined crystals with cubic symmetry made of the same material.
