**2. Statement of the problem**

We assume that there are two crystals with rectilinear anisotropy and cubic symmetry, which are rigidly connected along their contact surface (Fig. 1). The crystal contact surface forms a dihedral angle with linear angle whose trace is shown in the plane of the drawing. The contact surface edge passes through point *O*. The *z* -axis of the cylindrical coordinate system *r z* , , coincides with the edge of the dihedral angle. The coordinate surfaces and 0 and 2 coincide with the faces of the dihedral angle. Thus, the first crystal (1) occupies the domain 0; and the second crystal (2) occupies the domain 2 ;0 . In this case 0 2 and 0 *r* .

For simplicity, we assume that the crystals have a single common principal direction of elasticity coinciding with the *z* - axis . The other two principal directions 1 *x* and 1 *y* of the first crystal make some nonzero angles with the principal directions 2 *x* and 2 *y* of the

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

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

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,

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

We assume that there are two crystals with rectilinear anisotropy and cubic symmetry, which are rigidly connected along their contact surface (Fig. 1). The crystal contact surface

drawing. The contact surface edge passes through point *O*. The *z* -axis of the cylindrical

 0; 

 

For simplicity, we assume that the crystals have a single common principal direction of elasticity coinciding with the *z* - axis . The other two principal directions 1 *x* and 1 *y* of the first crystal make some nonzero angles with the principal directions 2 *x* and 2 *y* of the

whose trace is shown in the plane of the

and the second crystal (2) occupies

coincides with the edge of the dihedral angle. The coordinate

and 0 *r* .

2 coincide with the faces of the dihedral angle.

to their interface, then the compound wedge behaves as a homogeneous wedge.

cross-section lies at the edge of the contact surface of the two bodies.

2002, 2006; Alexandrov & Kalyakin, 2005).

forms a dihedral angle with linear angle

Thus, the first crystal (1) occupies the domain

 

2 ;0 . In this case 0 2

**2. Statement of the problem** 

coordinate system *r z* , ,

0 and

same material.

surfaces and

the domain

second crystal. By 1 we denote the angle between 1 *x* and the polar axis 0 , and by <sup>2</sup> , the angle between 2 *x* and the axis 0 . In this case, 1 2 , 2, . If 1 2 0 , then we have a homogeneous medium, i.e., a monocrystal with cubic symmetry, one of whose principal directions *xx x* 1 2 coincides with the polar axis 0 . In this case, the equations of generalized Hooke's law written in the principal axes of elasticity , , *x y z* have the form

$$\begin{aligned} \varepsilon\_{\chi} &= a\_{11}\sigma\_{\chi} + a\_{12} \left(\sigma\_{\chi} + \sigma\_{z}\right), & \qquad \gamma\_{yz} &= a\_{44}\tau\_{yz}, \\ \varepsilon\_{\chi} &= a\_{11}\sigma\_{\chi} + a\_{12} \left(\sigma\_{z} + \sigma\_{\chi}\right), & \qquad \gamma\_{z\chi} &= a\_{44}\tau\_{z\chi} \\ \varepsilon\_{z} &= a\_{11}\sigma\_{z} + a\_{12} \left(\sigma\_{x} + \sigma\_{y}\right), & \qquad \gamma\_{xy} &= a\_{44}\tau\_{xy}. \end{aligned} \tag{1}$$

where , , ..., *x y xy* are the strain components, , , ..., *x y xy* are the stress components, and 11 12 44 *aaa* , , are the strain coefficients.

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 described in (Lekhnitskii, 1981).

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 properties, the equations of generalized

Strength of a Polycrystalline Material 31

where the above form of anisotropy is used. From now on, the first crystal is denoted by the

In the case of cubic symmetry of the material, we have the following dependencies between

11 12 11 12 11 12 11 12 44

*aa a a aa a a a*

In the isotropic medium, we have *a aa* 44 11 12 2 and 44 11 12 2*A* . For cubic

In the case of longitudinal shear along the direction of the axis *z* , we have the following

 and i *<sup>r</sup>* <sup>z</sup> 

*z z ur u r rz z*

0,

1 1 <sup>1</sup> , .

*i i*

*u u*

 

*r r a a*

 

44 44

Substituting (3) into the differential equations of equilibrium, we obtain <sup>0</sup> *<sup>i</sup> uz* , where

Since the crystals are rigidly joined, on the interface between the two crystals the

12 1 2 ,0 ,0 , , , 2 , *ur ur ur ur zz z z*

, 0 ,0 , u ,2 , . *ur ur ur z zz a a <sup>r</sup>*

*a a*

 1 21 2 1 1 44 44 z 2 2 44 44

*i i z z <sup>z</sup> rz i i*

 

 

 *z* 

<sup>1</sup> , , 2 2

2 44 11 12 is called a parameter of elastic anisotropy in

*u u u ur <sup>r</sup>* 

 

> 

 

*z z*

*<sup>i</sup>* , Eqs. (2) correspond to generalized Hooke's law written for

, we call *a* the coefficient of elastic anisotropy.

.

, not identically zero, are related to

According to Hooke's law

(3)

the moduli of elasticity 11 12 44 , , and the strain coefficients 11 12 44 *aaa* ,, :

*a a a*

components of the displacement vector: 0, 0, , *i i ii*

*<sup>u</sup>*z by the Cauchy equations: , . *ii ii* 

*r zr*

 11 12 12 11 12 44

index *i* 1 , and the second, by *i* 2 .

 

For small strains, the strain components *i*

For *a* 0 , we have an anisotropic medium in Eqs. (2).

monocrystals and referred to the principal axes of elasticity.

(Vainstein et al., 1981). In contrast to

We also note that for 0

**3. Out-of-plane strain** 

(2), this implies that

is the Laplace operator.

displacements are continuous,

and the contact stresses are continuous,

i

crystals, the ratio

Hooke's law for these coordinate systems have the same form. In this case, the values of the strain coefficients are the same in both systems: / // / <sup>11</sup> <sup>11</sup> 12 13 12 13 <sup>66</sup> <sup>66</sup> *a a a aa a a a* , , , ... , .

Using the formulas of transformation of strain coefficients under the rotation of the coordinate system about the axis / *z z* (Lekhnitskii, 1981), we obtain their new values expressed in terms of the old values (before the rotation of the coordinate system (, , ) *xyz* ).

Comparing the strain coefficients in the same coordinate system / // (, , ) *xyz* , we obtain, 11 12 44 55 *a aa a* , , 13 23 16 45 26 36 *a aa a a a* , 0 .

Successively rotating the coordinate system (, , ) *xyz* about the axes *x* and *y* by the angle 90 and repeating the same procedure, we finally obtain (1).

The transformation formulas for the strain coefficients under the rotation of the coordinate system about the *x* -and *y* -axes can also be obtained from the transformation formulas for the strain coefficients under the rotation of the coordinate system about the *z* -axis in the case of anisotropy of general form.

For example, to obtain the transformation formulas under the rotation of the coordinate system about the *x* -axis, it is necessary to rename the principal directions of elasticity as follows: the *x* -axis becomes the *z* -axis, the *y* -axis becomes the *x* -axis, and the *z* -axis becomes the *y* -axis. In this case, in the equations of generalized Hooke's law referred to the coordinate system (, , ) *xyz* , <sup>22</sup> *a* plays the role of 11 *a* , <sup>23</sup> *a* plays the role of 12 *a* , and 24 *a* plays the role of 16 *a* . In a similar way, in the equations of generalized Hooke's law referred to the coordinate system / // (, , ) *xyz* , / <sup>22</sup> *<sup>a</sup>* plays the role of / <sup>11</sup> *<sup>a</sup>* , / <sup>23</sup> *<sup>a</sup>* plays the role of / <sup>12</sup> *a* , and / <sup>24</sup> *<sup>a</sup>* plays the role of / <sup>16</sup> *a* . This implies that, in the case of an orthotropic body, 24 *a* 0 under rotation of the coordinate system about the *x* -axis, but, in contrast to the case of rotation of the coordinate system about the *z* -axis, / <sup>24</sup> *a* is generally nonzero.

In the case 1 2 , the equations of generalized Hooke's law in the cylindrical coordinate system (, , ) *r z* have the form

 2 11 12 2 11 12 11 12 44 11 12 44 11 12 ( ) ( ) sin 2 sin 4 , ( ) ( ) sin 2 sin 4 , ( ), 2 4 , 2 4, *i i ii ii i r r z r ir i i i ii ii i zr r ir i i i ii zz r i i i i z z z z i i i i zr zr rz zr i r aa a aa a a a a aa a a aa a* 2 11 12 11 12 44 2 () sin 4 4 cos 2 , 4 2 , , *i ii i r r ir i i i aa a a aa a* (2)

Hooke's law for these coordinate systems have the same form. In this case, the values of the

Using the formulas of transformation of strain coefficients under the rotation of the coordinate system about the axis / *z z* (Lekhnitskii, 1981), we obtain their new values expressed in terms of the old values (before the rotation of the coordinate system (, , ) *xyz* ).

Comparing the strain coefficients in the same coordinate system / // (, , ) *xyz* , we obtain,

Successively rotating the coordinate system (, , ) *xyz* about the axes *x* and *y* by the angle

The transformation formulas for the strain coefficients under the rotation of the coordinate system about the *x* -and *y* -axes can also be obtained from the transformation formulas for the strain coefficients under the rotation of the coordinate system about the *z* -axis in the

For example, to obtain the transformation formulas under the rotation of the coordinate system about the *x* -axis, it is necessary to rename the principal directions of elasticity as follows: the *x* -axis becomes the *z* -axis, the *y* -axis becomes the *x* -axis, and the *z* -axis becomes the *y* -axis. In this case, in the equations of generalized Hooke's law referred to the coordinate system (, , ) *xyz* , <sup>22</sup> *a* plays the role of 11 *a* , <sup>23</sup> *a* plays the role of 12 *a* , and 24 *a* plays the role of 16 *a* . In a similar way, in the equations of generalized Hooke's law referred

under rotation of the coordinate system about the *x* -axis, but, in contrast to the case of

 

 

 

> 

*i ii i r r ir i*

2 () sin 4 4 cos 2 ,

*i i ii ii i r r z r ir i i i ii ii i*

 

<sup>22</sup> *<sup>a</sup>* plays the role of /

, the equations of generalized Hooke's law in the cylindrical coordinate

( ) ( ) sin 2 sin 4 , ( ) ( ) sin 2 sin 4 ,

 

 

*zr r ir i*

  <sup>11</sup> *<sup>a</sup>* , /

<sup>24</sup> *a* is generally nonzero.

<sup>16</sup> *a* . This implies that, in the case of an orthotropic body, 24 *a* 0

2

 

   

 

 

2

   

 

2

<sup>23</sup> *<sup>a</sup>* plays the role of /

<sup>12</sup> *a* ,

(2)

<sup>11</sup> <sup>11</sup> 12 13 12 13 <sup>66</sup> <sup>66</sup> *a a a aa a a a* , , , ... , .

strain coefficients are the same in both systems: / // /

11 12 44 55 *a aa a* , , 13 23 16 45 26 36 *a aa a a a* , 0 .

case of anisotropy of general form.

to the coordinate system / // (, , ) *xyz* , /

rotation of the coordinate system about the *z* -axis, /

*i i ii zz r*

44 11 12 44 11 12

 

*a aa a a aa a*

*i i i i z z z z i i i i zr zr rz zr*

*aa a aa a*

 

 

 

 

4 2 , ,

*aa a*

( ),

 

2 4 , 2 4,

 

 

 

 

*i i*

 

have the form

11 12

11 12 11 12

*a a*

 

*a aa a*

11 12 11 12 44

   

 

 

 

<sup>24</sup> *<sup>a</sup>* plays the role of /

1 2 

*i r*

and /

In the case

system (, , ) *r z* 

90 and repeating the same procedure, we finally obtain (1).

where the above form of anisotropy is used. From now on, the first crystal is denoted by the index *i* 1 , and the second, by *i* 2 .

In the case of cubic symmetry of the material, we have the following dependencies between the moduli of elasticity 11 12 44 , , and the strain coefficients 11 12 44 *aaa* ,, :

$$\mathbf{A}\_{11} = \frac{a\_{11} + a\_{12}}{\left(a\_{11} - a\_{12}\right)\left(a\_{11} + 2\right.a\_{12}\right)}, \qquad \mathbf{A}\_{12} = -\frac{a\_{12}}{\left(a\_{11} - a\_{12}\right)\left(a\_{11} + 2\right.a\_{12}\right)}, \quad \mathbf{A}\_{44} = \frac{1}{a\_{44}}$$

In the isotropic medium, we have *a aa* 44 11 12 2 and 44 11 12 2*A* . For cubic crystals, the ratio 2 44 11 12 is called a parameter of elastic anisotropy in (Vainstein et al., 1981). In contrast to , we call *a* the coefficient of elastic anisotropy. For *a* 0 , we have an anisotropic medium in Eqs. (2).

 We also note that for 0 *<sup>i</sup>* , Eqs. (2) correspond to generalized Hooke's law written for monocrystals and referred to the principal axes of elasticity.

#### **3. Out-of-plane strain**

In the case of longitudinal shear along the direction of the axis *z* , we have the following components of the displacement vector: 0, 0, , *i i ii u u u ur <sup>r</sup> z z* .

For small strains, the strain components *i z* and i *<sup>r</sup>* <sup>z</sup> , not identically zero, are related to i *<sup>u</sup>*z by the Cauchy equations: , . *ii ii z z ur u r rz z* According to Hooke's law (2), this implies that

$$\begin{aligned} \sigma\_r &= \sigma\_\phi = \sigma\_z = \tau\_{r\phi} \equiv 0, \\ \tau\_{\phi z}^{(i)} &= \frac{1}{a\_{44}^{(i)}} \frac{1}{r} \frac{\partial u\_z^{(i)}}{\partial \sigma}, \quad \tau\_{r\mathbb{Z}}^{(i)} = \frac{1}{a\_{44}^{(i)}} \frac{\partial u\_z^{(i)}}{\partial \ r}. \end{aligned} \tag{3}$$

Substituting (3) into the differential equations of equilibrium, we obtain <sup>0</sup> *<sup>i</sup> uz* , where is the Laplace operator.

Since the crystals are rigidly joined, on the interface between the two crystals the displacements are continuous,

$$
\mu\_z^{(1)}(r,0) = \mu\_z^{(2)}(r,0), \quad \mu\_z^{(1)}(r,\ \alpha) = \mu\_z^{(2)}(r,\ \alpha - 2\,\,\pi),
$$

and the contact stresses are continuous,

$$\frac{\partial \operatorname{u}\_{\mathbf{z}}^{(1)}(r,0)}{\partial \operatorname{\boldsymbol{\varrho}}} = \frac{\operatorname{a}\_{44}^{(1)}}{\operatorname{a}\_{44}^{(2)}} \frac{\operatorname{\boldsymbol{\varepsilon}} \operatorname{u}\_{\mathbf{z}}^{(2)}(r,0)}{\operatorname{\boldsymbol{\varepsilon}} \operatorname{\boldsymbol{\varrho}}},\\\frac{\operatorname{\boldsymbol{\varepsilon}} \operatorname{u}\_{\mathbf{z}}^{(1)}(r,\boldsymbol{\alpha})}{\operatorname{\boldsymbol{\varepsilon}} \operatorname{\boldsymbol{\varrho}}} = \frac{\operatorname{a}\_{44}^{(1)}}{\operatorname{a}\_{44}^{(2)}} \frac{\operatorname{\boldsymbol{\varepsilon}} \operatorname{u}\_{\mathbf{z}}^{(2)}(r,\boldsymbol{\alpha}-2\pi)}{\operatorname{\boldsymbol{\varepsilon}} \operatorname{\boldsymbol{\varrho}}}.$$

Strength of a Polycrystalline Material 33

4 4 4 3

 

4 44 2 22 3

*i ii i i*

 

2 2

1 2

 

*r r*

 

1 11 sin 2 2

*i i i i*

 

 

*i*

2 2

 

> 

 

*i i*

 

 

 

(7)

(8)

(9)

  (10)

2 11 3sin 2 2 15sin 2 8

*rr r r r r*

3 14 12 sin 4 0,

The rigid connection of the crystals along their contact surface implies the continuity

If we set *a* 0 in problem (7)–(9), then we obtain a plane problem for the homogeneous

2 2 2 i 2

*r rr r r r r*

1 1 1 1 sin 2 sin 4 ,

*i i i i*

 

2 2 2 2

, ,2 , , 2, .

, ,2 , ,2 , .

1 2

*r r*

*b r r r rr r*

3 2 2 2 3 22 2 3

*i i i*

, 0 ,0 , 0 , 0 , ,

1 2 2 2 1 2

,0 ,0 ,0 ,0 , ,

*ur ur ur ur r r r r*

1 2 2 2 1 2

 

 

*ur ur ur ur r r r r*

2 4 4

<sup>4</sup> 1 1 11sin 2 6 2

*r r r r r*

3 2 2 43 3 4

 

*i*

*rr r r r*

4 2 3 33

*i i ii*

2

<sup>2</sup> 2 2

 

1 1 .

2 22

*r r r r*

conditions for the displacements on this surface.

*r r*

*r r*

and the continuity conditions for the contact stresses,

*r r*

*r r*

2 2

 

*r ii i*

1 1

*<sup>u</sup> b b r r r r r*

1 2 2 2 2

1 2

1 2

According to (4), (5), and (6), we have

isotropic body.

*i*

*a*

 

2 2

*i i*

1

*a*

3

2

*r*

2

6

Since 1 2 44 44 <sup>44</sup> *aaa* , this implies that, in the case of out-of-plane strain, the two-crystal composed of monocrystals of the same material behaves as a monocrystal corresponding to the case 1 2 .

Thus, in the case of longitudinal shear in the direction of the *z* -axis , there is no stress concentration at the corner point of the interface between the two joined crystals regardless of the orientation of the principal directions *x*<sup>1</sup> and *x*<sup>2</sup> .

#### **4. Plane strain**

In this case, we have

$$
\mu\_r^{(i)} = \mu\_r^{(i)}(r, \,\varphi), \qquad \mu\_\varphi^{(i)} = \mu\_\varphi^{(i)}\left(r, \,\varphi\right), \quad \mu\_z^{(i)} \equiv 0.
$$

Hence the following strain components are nonzero:

$$\varepsilon\_r^{(i)} = \frac{\partial \, u\_r^{(i)}}{\partial r}, \quad \varepsilon\_\phi^{(i)} = \frac{1}{r} \frac{\partial \, u\_\phi^{(i)}}{\partial \, \phi} + \frac{u\_r^{(i)}}{r}, \qquad r\,\gamma\_{r\phi}^{(i)} = \frac{\partial \, u\_r^{(i)}}{\partial \, \phi} + r\frac{\partial \, u\_\phi^{(i)}}{\partial \, r} - u\_\phi^{(i)}.\tag{4}$$

Hooke's law (2) has the form

$$\begin{aligned} \varepsilon\_r^{(i)} &= b\_1 \, \sigma\_r^{(i)} + b\_2 \, \sigma\_\phi^{(i)} - a \left[ \left( \sigma\_r^{(i)} - \sigma\_\phi^{(i)} \right) \sin^2 2a\_i + \tau\_{r\phi}^{(i)} \sin 4a\_i \right] \\ \varepsilon\_\phi^{(i)} &= b\_2 \, \sigma\_r^{(i)} + b\_1 \sigma\_\phi^{(i)} + a \left[ \left( \sigma\_r^{(i)} - \sigma\_\phi^{(i)} \right) \sin^2 2a\_i + \tau\_{r\phi}^{(i)} \sin 4a\_i \right] \\ \gamma\_{r\phi}^{(i)} &= 2 \left( a\_{11} - a\_{12} \right) \tau\_{r\phi}^{(i)} - a \left[ \left( \sigma\_r^{(i)} - \sigma\_\phi^{(i)} \right) \sin 4a\_i + 4 \tau\_{r\phi}^{(i)} \cos^2 2a\_i \right] \\ b\_1 &= a\_{11} - \frac{a\_{12}^2}{a\_{11}}, \quad b\_2 = a\_{12} - \frac{a\_{12}^2}{a\_{11}}. \end{aligned} \tag{5}$$

In the absence of mass forces, we satisfy the differential equations of equilibrium by expressing , *i i r* and *<sup>i</sup> r* via the Airy stress function *<sup>i</sup>* :

$$
\sigma\_r^{(i)} = \frac{1}{r^2} \frac{\partial^2 \Phi\_i}{\partial \rho^2} + \frac{1}{r} \frac{\partial \Phi\_i}{\partial r}, \quad \sigma\_\phi^{(i)} = \frac{\partial^2 \Phi\_i}{\partial r^2}, \quad \tau\_{r\phi}^{(i)} = -\frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial \Phi\_i}{\partial \rho} \right). \tag{6}
$$

By substituting (5) into the strain consistency condition

$$\frac{\partial^2 \chi\_{r\phi}^{(i)}}{\partial r \, \partial \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\begin{Bmatrix} -\hat{r} \, \hat{\mathcal{E}}\_{\phi}^{(i)} \\ \hat{r} \, \hat{\mathcal{E}}^2 \end{Bmatrix} - \frac{1}{r} \frac{\partial^2 \,\mathcal{E}\_r^{(i)}}{\partial \!\!\!\!\!\!\/} + \frac{1}{r} \frac{\partial \chi\_{r\phi}^{(i)}}{\partial \!\!\!\!\!\/\!\!\!\/} - 2 \frac{\partial \,\mathcal{E}\_{\phi}^{(i)}}{\partial \!\!\!\!\/\!\!\/} + \frac{\partial \,\mathcal{E}\_{r}^{(i)}}{\partial \!\!\!\/\,} = 0$$

after several simplifying transformations, according to (6), we obtain the basic equation of the problem:

composed of monocrystals of the same material behaves as a monocrystal corresponding to

Thus, in the case of longitudinal shear in the direction of the *z* -axis , there is no stress concentration at the corner point of the interface between the two joined crystals regardless

> , , , , 0. *ii ii i u ur u u r u r r*

 <sup>1</sup> ,, . *i i i i i i i r r r i i*

*r rr r*

*u u u u u*

 

 

 

(4)

2

( ) sin 2 sin 4 ,

 

( ) sin 2 sin 4 ,

 

 

 

1 1 2 0

   

  2

 

*<sup>z</sup>*

*r ru*

 

 

> 

> 

> >

(5)

(6)

i 2

 

 

*r r*

2 2 12 12

*i i ii*

 

11 11

<sup>2</sup> <sup>2</sup>

2 2 2

2 2

 

*a a*

, .

 

*i i i ii i r r r ir i*

 

*i i i ii i*

 

11 12 r

*r rr i i*

In the absence of mass forces, we satisfy the differential equations of equilibrium by

via the Airy stress function *<sup>i</sup>* :

1 1 <sup>1</sup> ,, . *<sup>i</sup> i i i i <sup>i</sup> <sup>i</sup> r r r r r r r r*

2 2 2

*i i i i i i r r r r <sup>r</sup> r r r rr r*

after several simplifying transformations, according to (6), we obtain the basic equation of

 

( ) ( )

 

2 ( ) sin 4 4 cos 2 ,

*r r ir i*

 

 

of the orientation of the principal directions *x*<sup>1</sup> and *x*<sup>2</sup> .

Hence the following strain components are nonzero:

1 2

 

 

2 1

1 11 2 12

 and *<sup>i</sup> r* 

By substituting (5) into the strain consistency condition

 

 

*a a b a ba*

Hooke's law (2) has the form

expressing , *i i r*

the problem:

*bba*

*b ba*

*aa a*

 

44 44 <sup>44</sup> *aaa* , this implies that, in the case of out-of-plane strain, the two-crystal

Since 1 2

1 2 .

**4. Plane strain**  In this case, we have

the case

 4 4 4 3 2 2 4 44 2 22 3 1 3 2 2 2 3 22 2 3 2 4 4 2 4 2 3 33 3 2 1 11 sin 2 2 2 11 3sin 2 2 15sin 2 8 <sup>4</sup> 1 1 11sin 2 6 2 6 *i i i i i i i ii i i i i i i a b r r r rr r rr r r r r r r r r r r* 3 2 2 43 3 4 <sup>2</sup> 2 2 2 2 22 3 14 12 sin 4 0, 1 1 . *i i ii i rr r r r r r r r* (7)

The rigid connection of the crystals along their contact surface implies the continuity conditions for the displacements on this surface.

$$\begin{aligned} \frac{\partial u\_r^{(1)}(r,0)}{\partial r} &= \frac{\partial u\_r^{(2)}(r,0)}{\partial r}, \qquad \frac{\partial^2 u\_\phi^{(1)}(r,0)}{\partial \boldsymbol{r}^2} = \frac{\partial^2 u\_\phi^{(2)}(r,0)}{\partial \boldsymbol{r}^2},\\ \frac{\partial u\_r^{(1)}(r,a)}{\partial r} &= \frac{\partial u\_r^{(2)}(r,a-2\pi)}{\partial r}, \qquad \frac{\partial^2 u\_\phi^{(1)}(r,a)}{\partial \boldsymbol{r}^2} = \frac{\partial^2 u\_\phi^{(2)}(r,a-2\pi)}{\partial \boldsymbol{r}^2}.\end{aligned} \tag{8}$$

and the continuity conditions for the contact stresses,

$$\begin{aligned} \Phi\_1(r,0) &= \Phi\_2(r,0), & \frac{\partial \Phi\_1(r,0)}{\partial \varphi} &= \frac{\partial \Phi\_2(r,0)}{\partial \varphi}, \\ \Phi\_1(r,a) &= \Phi\_2(r,a-2\pi), & \frac{\partial \Phi\_1(r,a)}{\partial \varphi} &= \frac{\partial \Phi\_2(r,a-2\pi)}{\partial \varphi}. \end{aligned} \tag{9}$$

If we set *a* 0 in problem (7)–(9), then we obtain a plane problem for the homogeneous isotropic body.

According to (4), (5), and (6), we have

$$\begin{split} \frac{\partial u\_{r}^{(i)}}{\partial r} &= b\_{1} \Big( \frac{1}{r^{2}} \frac{\partial^{2} \Phi\_{i}}{\partial \rho^{2}} + \frac{1}{r} \frac{\partial \Phi\_{i}}{\partial \boldsymbol{r}} \Big) + b\_{2} \frac{\partial^{2} \Phi\_{i}}{\partial \boldsymbol{r}^{2}} \\ &- a \Big[ \Big( \frac{1}{r^{2}} \frac{\partial^{2} \Phi\_{i}}{\partial \rho^{2}} + \frac{1}{r} \frac{\partial \Phi\_{i}}{\partial \boldsymbol{r}} - \frac{\partial^{2} \Phi\_{i}}{\partial \boldsymbol{r}^{2}} \Big) \sin^{2} \boldsymbol{2} a\_{i} + \Big( \frac{1}{r^{2}} \frac{\partial \Phi\_{i}}{\partial \rho} - \frac{1}{r} \frac{\partial^{2} \Phi\_{i}}{\partial \rho \partial \boldsymbol{r}} \Big) \sin 4a\_{i} \Big], \end{split} \tag{10}$$

Strength of a Polycrystalline Material 35

/// //

2 2 2 5 sin 4 4 1 2 cos 4 } 0

1 12

*<sup>r</sup> <sup>i</sup> <sup>i</sup>*

*<sup>u</sup> r bF b b F*

*ra F F*

1 // /

*i i*

2 2 /2 2

 

2 0 sin 4 1 0 1 cos 4 ,

*i ii i*

// //

2 02 1 0 0 1 cos 4

*F aa b aab aa*

 

5 8 1 cos 4 2 0 1 2 sin 4 2 0

2 2///

0 1 cos 4 4 0 1 sin 4 0 3 1

 

20 1 1 2 ,

*i i i i ii*

*F F*

// //

 

11 12 2 11 12 1 12 11

*F b F*

*i i i i*

 

*i i i i i*

/// // / 2

 

5 6 12 , ,, 2 , *XF XF i ii i i <sup>i</sup>*

2 sin 4 1 1 cos 4 , *i i i i i i i i*

According to (13), (16), and (17), the continuity conditions (8) and (9) acquire the form

*F aab aab aa*

 

2 [ 1 1 2.

*i ii ii i*

<sup>1</sup> 1 1 cos 4 .

*i i i i*

*F F*

2 2 2 // 2 2 // 2 <sup>2</sup>

 

*i i i i*

 

2

 

*F F r bF*

11 12 1 11 12 2 12 11

// /// 2 ///

1 2 1 2 ( 1,2,...,8), 0 , 0 , *XX j XF XF j j ii i <sup>i</sup>* (18)

 

<sup>1</sup> 1 cos 4 sin 4

*i i i i*

<sup>1</sup> 1 2 sin 4 3 15 8 1

 

*F F*

 

1

 

> 

> >

/

 

 

> 

 

1

 

   

 

   

(16)

(17)

1

  (15)

2 1 1 {[ 2 1 1 ] sin 2

*i i i i i i i*

1

*<sup>a</sup> F F FF F F b*

*IV IV*

 

 

whose general integral has the form (14) for *a* 0 .

*i*

*r*

After the substitution of (13) into (10) and (12), we can write

2

2

3 1 1 2

/ 2

*F F*

 

 

*X bF b b F a F*

*ii i i ii*

*X aF F F*

/

 

/ 2

2 2 1 1cos 4

 

*X bF b b F a F*

*i*

2

*u*

*r*

/

4

/

7 1 1 2

 

 *i*

 

*i*

2

2 2 sin 4 4 1 2 cos 4

1 //

2

*r aF F*

<sup>1</sup> cos 4 2 1 sin 4 1 cos 4

 

*F*

*i i i*

 

 

2 / 2

$$\begin{split} \frac{\partial u\_r^{(i)}}{\partial \rho} + r \frac{\partial u\_\rho^{(i)}}{\partial r} - u\_\phi^{(i)} &= 2(a\_{11} - a\_{12}) \left( \frac{1}{r} \frac{\partial \Phi\_i}{\partial \rho} - \frac{\partial^2 \Phi\_i}{\partial \rho \partial r} \right) \\ &+ a \left[ \left( r \frac{\partial^2 \Phi\_i}{\partial r^2} - \frac{1}{r} \frac{\partial^2 \Phi\_i}{\partial \rho^2} - \frac{\partial \Phi\_i}{\partial r} \right) \sin 4a\_i - 4 \cos^2 2a\_i \left( \frac{1}{r} \frac{\partial \Phi\_i}{\partial \rho} - \frac{\partial^2 \Phi\_i}{\partial \rho \partial r} \right) \right]. \end{split} \tag{11}$$

Differentiating (10) with respect to and (11) with respect to *r* and eliminating the derivative <sup>2</sup> *<sup>i</sup> u r <sup>r</sup>* , we obtain

$$\begin{split} \frac{\partial^2 u\_0^{(1)}}{\partial r^2} &= a \left[ \frac{\partial^3 \Phi\_i}{\partial r^3} \sin 4a\_i + \frac{1}{r^3} \frac{\partial^3 \Phi\_i}{\partial \rho^3} \sin^2 2a\_i + \frac{1}{r} \frac{\partial^3 \Phi\_i}{\partial \rho \partial r^2} \left( 4 - 5 \sin^2 2a\_i \right) \right] \\ &- \frac{2}{r^2} \frac{\partial^3 \Phi\_i}{\partial \rho^2} \sin 4a\_i - \frac{2}{r} \frac{\partial^2 \Phi\_i}{\partial r^2} \sin 4a\_i \\ &+ \frac{4}{r^3} \frac{\partial^2 \Phi\_i}{\partial \rho^2} \sin 4a\_i + \frac{1}{r^2} \frac{\partial^2 \Phi\_i}{\partial \rho \partial r} \left( 13 \sin^2 2a\_i - 8 \right) \\ &+ \frac{2}{r^2} \frac{\partial \Phi\_i}{\partial r} \sin 4a\_i + \frac{1}{r^3} \frac{\partial \Phi\_i}{\partial \rho} \left( 8 - 12 \sin^2 2a\_i \right) \Bigg] \\ &- b\_1 \frac{1}{r^3} \frac{\partial^3 \Phi\_i}{\partial \rho^3} - \frac{1}{r} \frac{\partial^3 \Phi\_i}{\partial \rho \partial r} \left( a\_{11} - a\_{12} + b\_1 \right) \\ &+ \frac{1}{r^2} \frac{\partial^2 \Phi\_i}{\partial \rho \partial r} \left( a\_{11} - a\_{12} - b\_2 \right) + \frac{2}{r^3} \frac{\partial \Phi\_i}{\partial \rho} \left( a\_{12} - a\_{11} \right). \end{split} \tag{12}$$

We use the expressions (10) and (12) to represent the continuity conditions (8) via the stress function .

#### **5. Solution method**

For *a* 0 , from (7) we derive the biharmonic equation and, solving it by separation of variables, obtain the following solution (Chobanyan, 1987; Chobanyan & Gevorkyan, 1971):

$$\Phi\_i(r,\,\varphi) = r^{\hat{\lambda}+1} F\_i(\,\mathcal{X};\,\varphi)\_{\prime} \tag{13}$$

$$\begin{aligned} F\_i(\mathcal{L}; \boldsymbol{\varrho}) &= \mathbf{A}\_i \sin(\mathcal{L} + 1) \, \boldsymbol{\varrho} + \mathbf{B}\_i \cos(\mathcal{L} + 1) \, \boldsymbol{\varrho} \\ &+ \mathbf{C}\_i \sin(\mathcal{L} - 1) \, \boldsymbol{\varrho} + D\_i \cos(\mathcal{L} - 1) \, \boldsymbol{\varrho}. \end{aligned} \tag{14}$$

where *λ* is a parameter and , , *iii B C* and *Di* –are integration constants.

For *a* sufficiently small in absolute value, we replace the solution of Eq. (7) by the solution of the biharmonic equation (13). By substituting (13) into (7), we obtain a fourth-order ordinary differential equation for *Fi* ; :

  2

2 2 2

 

1 1 sin 4 sin 2 4 5sin 2

*i ii i i*

 

> 

*i*

2

2

*r rr r r*

*ii i*

 

2

 

and (11) with respect to *r* and eliminating the

 

  (11)

(12)

1 1 sin 4 4cos 2 .

*i i*

2 2

 

*a r*

Differentiating (10) with respect to

*a*

*r u aa*

*i i*

*u u*

derivative <sup>2</sup> *<sup>i</sup> u r <sup>r</sup>*

*i*

*u*

function .

1971):

**5. Solution method** 

differential equation for *Fi*

11 12

<sup>1</sup> <sup>2</sup>

*r rr*

2 2

2 3 3 3

3 2 2 2 2 2 2

2 2 sin 4 sin 4

i 32 2

*rr r r*

2 3 3 3

*r r r*

1 3 3

*b*

1 1

*r r*

2

*F*

 

; :

2 3 33 2

*rr r r r*

*i i*

*i i*

*i i*

*i i*

*r r r*

*r r r*

4 1 sin 4 13sin 2 8

11 12 2 12 11 2 3 1 2 . *i i*

 

*i i i*

*i i*

*i i*

We use the expressions (10) and (12) to represent the continuity conditions (8) via the stress

For *a* 0 , from (7) we derive the biharmonic equation and, solving it by separation of variables, obtain the following solution (Chobanyan, 1987; Chobanyan & Gevorkyan,

> <sup>1</sup> , ;, *i i r rF*

 ; sin 1 cos 1 sin 1 cos 1 .

For *a* sufficiently small in absolute value, we replace the solution of Eq. (7) by the solution of the biharmonic equation (13). By substituting (13) into (7), we obtain a fourth-order ordinary

 

 

(13)

(14)

 

*ii i*

 

*i i*

*C D*

where *λ* is a parameter and , , *iii B C* and *Di* –are integration constants.

 

*aab aa*

11 12 1 2

*aab*

2 1 sin 4 8 12sin 2

*r*

 

, we obtain

*r i i i*

 

$$\begin{aligned} &F\_i^{IV} + 2\left(\lambda^2 + 1\right)F\_i^{IV} + \left(\lambda^2 - 1\right)^2 F\_i - \frac{a}{b\_1}\left\{\left[F\_i^{IV} + 2\left(\lambda^2 + 1\right)F\_i^{IV}\right] + \left(\lambda^2 - 1\right)^2 F\_i\right\} \sin^2 2a\_i \\ &- 2\left(\lambda - 2\right) \sin 4a\_i \, F\_i^{//} + 4\left(\lambda - 1\right)\left(\lambda - 2\right) \cos 4a\_i \, F\_i^{//} \\ &+ 2\left(\lambda - 2\right) \left[\left(\lambda - 2\right)^2 - 5\right] \sin 4a\_i \, F\_i^{~} + 4\left(\lambda^2 - 1\right)\left(\lambda - 2\right) \cos 4a\_i \, F\_i \right] = 0 \end{aligned} \tag{15}$$

whose general integral has the form (14) for *a* 0 .

After the substitution of (13) into (10) and (12), we can write

$$\frac{\partial u\_r^{(i)}}{\partial r} = r^{\lambda - 1} \left[ \begin{array}{c} b\_1 F\_i^{//} \left( \wp \right) + \left( b\_1 + \lambda \, b\_2 \right) \left( \lambda + 1 \right) F\_i \left( \wp \right) \right] \right. \\\\ -r^{\lambda - 1} a \left[ \frac{1}{2} F\_i^{//} \left( \wp \right) \left( 1 - \cos 4a\_i \right) \, -F\_i^{//} \left( \wp \right) \lambda \sin 4a\_i \right. \\\\ -\frac{1}{2} \left( \lambda^2 - 1 \right) F\_i \left( \wp \right) \left( 1 - \cos 4a\_i \right) \right]. \end{array} \tag{16}$$

$$\frac{\partial^2 u\_{\rho}^{(i)}}{\partial r^2} = r^{\lambda - 2} a \left\{ F\_i(\rho) \left( \lambda^2 - 1 \right) \left( \lambda - 2 \right) \sin 4a\_i + \frac{1}{2} F\_i^{'} \left( \rho \right) \left[ 3 \lambda^2 + 1 + \left( 5 \lambda^2 - 8 \lambda - 1 \right) \right. \\\\ \left. + 3 \lambda^2 + 1 + \left( 5 \lambda^2 - 8 \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 1 \right) \left( \lambda - 1 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 1 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda - 2 \right) \left( \lambda$$

According to (13), (16), and (17), the continuity conditions (8) and (9) acquire the form

$$\begin{array}{ll} \mathcal{X}\_{1j} = \mathcal{X}\_{2j} & \left(j = 1, 2, \dots, 8\right), \quad \mathcal{X}\_{i1} = F\_i\left(0\right), \quad \mathcal{X}\_{i2} = F\_i^{\prime}\left(0\right), \\\\ \mathcal{X}\_{i3} = 2 \, b\_1 F\_i^{\prime \prime \prime}\left(0\right) + 2 \left(b\_1 + \lambda \, b\_2\right) \left(\lambda + 1\right) F\_i\left(0\right) - a \left[F\_i^{\prime \prime \prime}\left(0\right) \left(1 - \cos 4\theta\_i\right)\right. \\\\ \left. + 2 \, F\_i^{\prime \prime}\left(0\right) \lambda \sin 4\theta\_i - \left(\lambda^2 - 1\right) F\_i\left(0\right) \left(1 - \cos 4\theta\_i\right)\right], \end{array}$$

$$\begin{split} X\_{i4} &= a \left\{ F\_{i}^{\prime\prime\prime} \left( 0 \right) \left( 1 - \cos 4\theta\_{i} \right) + 4 F\_{i}^{\prime\prime\prime} \left( 0 \right) \left( \lambda - 1 \right) \sin 4\theta\_{i} + F\_{i}^{\prime} \left( 0 \right) \right\} 3\lambda^{2} + 1 \\ &+ \left( 5\lambda^{2} - 8\lambda - 1 \right) \cos 4\theta\_{i} \right\} - 2F\_{i} \left( 0 \right) \left( \lambda^{2} - 1 \right) \left( \lambda - 2 \right) \sin 4\theta\_{i} \right\} - 2b\_{1} \; F\_{i}^{\prime\prime\prime\prime} \left( 0 \right) \\ &+ 2F\_{i}^{\prime} \left( 0 \right) \left[ \left( \lambda + 1 \right) \left( a\_{11} - a\_{12} - b\_{2} \right) - \lambda \left( \lambda + 1 \right) \left( a\_{11} - a\_{12} + b\_{1} \right) + 2 \left( a\_{12} - a\_{11} \right) \right] .\end{split}$$

 / 5 6 12 , ,, 2 , *XF XF i ii i i <sup>i</sup>* 

$$\begin{split} \mathbf{X}\_{i7} &= 2\mathbf{b}\_{1}\mathbf{F}\_{i}^{//}\left(\boldsymbol{\beta}\_{i}\right) + 2\left(\boldsymbol{b}\_{1} + \lambda\boldsymbol{b}\_{2}\right)\left(\lambda + 1\right)\mathbf{F}\_{i}\left(\boldsymbol{\beta}\_{i}\right) - a\left\{\left[1 - \cos 4\left(\alpha - \theta\_{i}\right)\right]\mathbf{F}\_{i}^{//}\left(\boldsymbol{\beta}\_{i}\right)\right\}, \\ &- 2\lambda\sin 4\left(\alpha - \theta\_{i}\right)\mathbf{F}\_{i}^{//}\left(\boldsymbol{\beta}\_{i}\right) - \left(\lambda^{2} - 1\right)\left[1 - \cos 4\left(\alpha - \theta\_{i}\right)\right]\mathbf{F}\_{i}\left(\boldsymbol{\beta}\_{i}\right) \end{split}$$

Strength of a Polycrystalline Material 37

polynomial of the first or the second degree, i.e., various approximations to Eq. (19). We also note that for *a* 0 , from the above system of algebraic equations, just as from Eq. (19), we

Preserving only terms up to the first or second degree in 1 *a b* in Eq. (19), we finally obtain

2 1 cos cos 1 cos cos sin 1

 

2

 

 

sin 1 sin sin 1 sin 8 1

 

sin 1 sin cos 1 sin 2 1

4 1 cos 4 sin cos 1 2 1

 

 

 , 3 21 

> 

2 cos sin sin cos ,

 

 

 

1 sin 1 1 sin 1 ,

 

sin cos 1 cos 1 8 sin

 

3 1 9 2 22 1 11

 

 

cos cos 1 cos cos sin 1 sin 5 3 1 4 1 1 cos4

> 

1 11 3

 

 

4 1 cos cos 1 cos cos

 

cos sin 1 sin cos 1 si

 

cos 1 sin 2 1 cos cos 1 cos

1

*a b*

sin sin sin 2sin cos 1 cos

determining the eigenvalues *<sup>k</sup>*

 

 

> 

 

 

 

 

 

1 9 13 2 4 3 5

cos 1 8 1 sin

 ,

> <sup>2</sup>

 sin 4 sin 4 , 

> <sup>11</sup>

 <sup>3</sup> 

 

 

44 1 11 1

12 1 sin 4 sin 4 

 

2 2 cos 4 1 8 sin 4 35 2

   

22

2

3 2 39

1 sin cos 1 12 1

17 3

 

 

> 

 <sup>2</sup> 

 

 421 

> 

2 cos cos ,

 

> 

2sin sin ,

2sin cos 1 ,

2 sin cos sin cos ,

 

 

3 5

 

 

 

 

 

 

 

 

 

 

> 

 

> 

 

> 

> > 4

  (20)

 

> 

 

 

> 

 

n

 

 

 

 

> 

> > 4

 

> 

 cos 4 cos 4 , 

> 

 

11 18

 

 

*k k* **N** which

obtain the equation sin 1 0

 

 

> 

 

 

> 

 

 

 

22 1 cos 4 cos 4 

> 

 

> 

2sin sin ,

2 sin cos sin cos ,

 

 

4

 

sin cos 1

 <sup>1</sup> 

1 

 

 

 

8 2

 

 

 

4 1 cos4

18

 

 

4

 

11

 <sup>3</sup> 

 <sup>22</sup> 

 

 

2 44

 

21 23

 

 

> 

> >

3 4

 

 

 

cos 1 sin 2 1 3 5

 

 

 

> 

 

 

 

cos 1 sin 0, 

 

 

 

 

1 31 11

 

correspond to the plane strain of a homogeneous isotropic body.

$$\begin{split} X\_{i8} &= a \left\{ F\_i^{//} \left( \beta\_i \right) \left[ 1 - \cos 4 \left( a - \theta\_i \right) \right] - 4 \, F\_i^{//} \left( \beta\_i \right) \left( \lambda - 1 \right) \sin 4 \left( a - \theta\_i \right) \right\} \\ &+ F\_i^{//} \left( \beta\_i \right) \left[ 3 \, \lambda^2 + 1 + \left( 5 \, \lambda^2 - 8 \, \lambda - 1 \right) \cos 4 \left( a - \theta\_i \right) \right] \\ &+ 2 \, F\_i \left( \beta\_i \right) \left( \lambda^2 - 1 \right) \left( \lambda - 2 \right) \sin 4 \left( a - \theta\_i \right) \right) - 2 \, b\_1 \, F\_i^{//} \left( \beta\_i \right) \\ &- 2 \, F\_i^{//} \left( \beta\_i \right) \left[ \lambda \left( \lambda + 1 \right) \left( a\_{11} - a\_{12} + b\_1 \right) \right] \\ &- \left( \lambda + 1 \right) \left( a\_{11} - a\_{12} - b\_2 \right) - 2 \left( a\_{12} - a\_{11} \right) \right]. \end{split}$$

By substituting (14) into (18), we obtain a homogeneous system of linear algebraic equations for the constants , , *iii B C* and *Di* .

After some cumbersome calculations, from the existence condition for the nonzero solution of this system, we obtain the following characteristic equation for , which determines the stress concentration degree (6) see in (Galptshyan, 2008):

$$f(\mathcal{A}; a\_{11}, a\_{12}, a, \theta\_1, \theta\_2, \alpha) = 0\tag{19}$$

Equation (19) contains six independent parameters 11 12 1 2 *aaa* , ,, , and .


#### Table 1.

For certain specific values of these parameters, it follows from (6) and (13) that the stress components at the pole *r* 0 have an integrable singularity if 0 Re 1 . In this case, the order of the singularity is equal to Re 1 .

Thus, studying the singularity of the stress state near the corner point of the interface between two crystals in the case of plane strain is reduced to finding the root of the transcendental equation (19) with the least positive real part.

A structural analysis of Eq. (19) shows that its left-hand side is a polynomial of degree 18 in <sup>1</sup> *a b* . The absolute value of 1 *a b* is sufficiently small. Therefore, preserving only terms up to the first or the second degree in (19), instead of a polynomial of degree 18, we obtain a

*ii ii i i i*

[3 1 5 8 1 cos 4

*i i i*

*F b F*

11 12 2 12 11

1 2.

*aa b a a*

 

*F a a b*

of this system, we obtain the following characteristic equation for

Equation (19) contains six independent parameters 11 12 1 2 *aaa* , ,, ,

 

2 1 2 sin 4 2

 

/// //

/2 2

*X aF F*

2 1

/

 

*F*

*i i*

 

stress concentration degree (6) see in (Galptshyan, 2008):

Al 0. 1403437

order of the singularity is equal to Re 1

Table 1.

 

8

for the constants , , *iii B C* and *Di* .

1 cos 4 4 1 sin 4

 

> 

*i i i i i*

 

11 12 1

By substituting (14) into (18), we obtain a homogeneous system of linear algebraic equations

After some cumbersome calculations, from the existence condition for the nonzero solution

11 12 1 2 *f aaa* (; , ,, , , ) 0

Nb − 0. 6423463 MgO 0. 2276457 CaF2 − 0.4838456 Si 0. 2498694 FeS2 − 0. 4066341 Ge 0. 275492 KCl − 0. 2682469 Ta 0. 2874998 NaCl − 0. 2154233 LiF 0. 3094264 V − 0. 2139906 Fe 0. 4637442 Mo − 0. 1877868 Ni 0. 4804368 TiC − 0. 0664576 Ag 0. 5856406 W 0 Cu 0. 593247 Au 0. 0556095 Pb 0. 7026827 C 0. 0965294 Na 0. 8089901

For certain specific values of these parameters, it follows from (6) and (13) that the stress

Thus, studying the singularity of the stress state near the corner point of the interface between two crystals in the case of plane strain is reduced to finding the root of the

A structural analysis of Eq. (19) shows that its left-hand side is a polynomial of degree 18 in <sup>1</sup> *a b* . The absolute value of 1 *a b* is sufficiently small. Therefore, preserving only terms up to the first or the second degree in (19), instead of a polynomial of degree 18, we obtain a

components at the pole *r* 0 have an integrable singularity if 0 Re 1

transcendental equation (19) with the least positive real part.

. 2 ///

 

1

 

 and .

<sup>1</sup> *a b* <sup>1</sup> *a b*

(19)

. In this case, the

, which determines the

polynomial of the first or the second degree, i.e., various approximations to Eq. (19). We also note that for *a* 0 , from the above system of algebraic equations, just as from Eq. (19), we obtain the equation sin 1 0 determining the eigenvalues *<sup>k</sup> k k* **N** which correspond to the plane strain of a homogeneous isotropic body.

Preserving only terms up to the first or second degree in 1 *a b* in Eq. (19), we finally obtain

$$\begin{split} &2^{2}\left(\lambda^{2}-1\right)\left[\left(\cos\lambda\pi\cos a-\left(\lambda+1\right)\cos\lambda\left(a-\pi\right)\cos\left(\pi\cdot\lambda+a\right)+\sin\left[\left(a-\pi\right)\left(\lambda-1\right)\right]\right) \\ &\times\sin\lambda\pi\right]\sin\left(\pi\cdot\lambda+a\right)\sin\lambda\left(a-\pi\right)+2\sin^{2}\lambda\pi\cos\left[\left(\lambda-1\right)\left(a-\pi\right)\right]\cos\left[\left(a+\lambda\left(\pi-a\right)\right)\right] \\ &\times\cos^{2}\left[\left(\lambda+1\right)\left(a-\pi\right)\right]\sin^{4}\lambda\pi+\frac{a}{b\_{1}}\left(\left.2^{4}\left(\lambda-1\right)\right]\cos\lambda\pi-\left(\lambda+1\right)\cos\lambda\left(a-\pi\right)\right) \\ &\times\cos\left(\pi\cdot\lambda+a\right)\sin\left[\left(a-\pi\right)\left(\lambda-1\right)\right]\sin\lambda\pi\right]\rho\_{2}\left(\lambda\right)\cos^{2}\left[\left(\lambda+1\right)\left(a-\pi\right)\right]\sin^{4}\lambda\pi \\ &-\rho\_{21}\left(\lambda\right)\sin\left[\left(\lambda-1\right)\left(a-\pi\right)\right]\sin\lambda\pi-\rho\_{23}\left(\lambda\right)\sin\left[\left(a-\pi\right)\left(\lambda-1\right)\right]\sin\lambda\pi+8\left(\lambda-1\right) \\ &\times\left[\cos\lambda\cos\alpha-\left(\lambda+1\right)\cos\lambda\left(a-\pi\right)\cos\left(\pi\cdot\lambda+a\right)+\sin\left[\left(a-\pi\right)\left(\lambda-1\right)\right]\sin\lambda\pi\right] \\ &\times\left[\eta\_{3}\left(5\cdot3\lambda\right)\chi\_{1}\left(\lambda\right)+\left(\lambda+1\right)\rho\_{3}\left(\lambda\right)\chi\_{2}\left(\lambda\right)-4\chi\_{2$$

 18 4 17 3 1 9 13 2 4 3 5 44 1 11 1 4 4 1 cos cos 1 cos cos sin 1 sin cos 1 sin 2 1 2 2 cos 4 1 8 sin 4 35 2 4 1 cos 4 sin cos 1 2 1 sin cos 1 11 18 11 cos 1 8 1 sin cos 1 sin 0, 

$$
\eta\_1 = \sin 4 \left( \alpha - \theta\_2 \right) - \sin 4 \left( \alpha - \theta\_1 \right) \dots
$$

22 1 cos 4 cos 4 , 3 21 sin 4 sin 4 , 421 cos 4 cos 4 , 

 <sup>1</sup> 2 cos sin sin cos , <sup>2</sup> 2 cos cos ,

$$\begin{aligned} \mathcal{L}\_3(\lambda) &= 2 \left( \lambda \sin \lambda a \cos a + \sin a \cos \lambda a \right), \quad \chi\_{11}(\lambda) = 2 \sin \lambda \pi \cos \left[ \left( \lambda + 1 \right) \left( a - \pi \right) \right], \\\\ \chi\_1(\lambda) &= \left( \lambda + 1 \right) \sin \left( \lambda - 1 \right) a - \left( \lambda - 1 \right) \sin \left( \lambda + 1 \right) a, \quad \chi\_2(\lambda) = 2 \sin a \sin \lambda a \end{aligned}$$

 <sup>22</sup> 2sin sin , <sup>3</sup> 2 sin cos sin cos ,

Strength of a Polycrystalline Material 39

 

 

3 sin 4 3sin 4 1 sin 1 2

 

 

 

 

 

12 3 1 4 2

 

 

 

 

4 1 1 cos 4 ,

11 4 11 3 12 2 3 11 2 4 2

12 11 11 1

 

 

15 17 12

 

 

cos 1 1 1 cos 4 sin 1

4 1 cos4 2 1 sin cos 1

 

 

4 2 2cos 4 1

 

14 13 1 13 1 3 2

3 sin 4 5sin 4 cos 1 2 4sin 4

1 3 4cos 4 5cos 4 1 sin 1 2

11 2 1 1

 

2 1 sin sin 1 ,

4 1 cos 4 1 1

 

31 4 3 3 1 cos 4

3 4cos 4 3cos 4 1 1 cos 1 2

 

> 

> >

 

2

2 1 cos 4 sin ,

 

> 

 

 

 

> 

> >

 

> 

 

 

> 

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2 2

 

> 

> >

 

> 

 

 

2

 

 

> 

1 3 1

 

 

 1 1 <sup>2</sup> ,

> 

 

1 1

 <sup>2</sup> 0 3 3 4 2 1 12

1 11 1

1 12 0

 

 

<sup>2</sup>

1 12 1 1 11 5 12

 <sup>2</sup> 15 7 11 11

 

<sup>17</sup> 2 1 4 11 3 12 cos 1 6 1

 

2 3 11 3 2 4 3 2

31 1

1 cos 4 1 cos 1 4 1 sin 4 sin 1 ,

1 3 1 2 4 1 1 cos 4 ,

 

> 

 

3 11 2

 

 

 

4 sin ,

4 sin ,

3 11 4 12 sin 1 2 11 16 12 6 ,

2 4 1 cos4 sin 1 1

 

 

> 

 

2 1 sin cos 1

 

> 

2 12 11 2

 

> 

 

 

> 

 

 

 

 

 

 

 

18 11 1 1 13 3 2

 

 

 

 

 

> 

21 1 12

 

4 2 12

 

17 11

5 42 

 

 

 

 

> 

 

 

  

 

 

 

> 

 

 

 

> 

 

 

 

$$\begin{aligned} \mathcal{R}\_{3}\left(\{\lambda\}-(\ell-1)\sin\left[\left(\lambda-1\right)\left((\alpha-2\pi)\right)\right]-\left(\lambda+1\right)\sin\left(\lambda+1\right),\\ \mathcal{R}\_{3}\left(\lambda\right)-(\lambda-1)\sin\left[\left(\lambda-1\right)\left(2-2\pi\right)\right]+\left(\lambda+1\right)\sin\left(\lambda+1\right),\\ \mathcal{R}\_{3}\left(\lambda\right)-2\cos\left[\left(\alpha-2\pi\right)\cos\left(\lambda+\alpha\right),\ \lambda^{2}\right]-2\sin^{2}\left[\left(\lambda+1\right)\left(2-\pi\right)\right],\\ \mathcal{R}\_{3}\left(\lambda\right)-2\cos\left[\left(\lambda-1\right)\sin\left(\lambda+1\right)+\left(\lambda+1\right)\sin\left[\left(\lambda-1\right)\left(2-\pi\right)\right],\\ \mathcal{R}\_{3}\left(\lambda\right)-\left(\lambda+1\right)\sin\left[\left(\lambda-1\right)\left(2-2\pi\right)\right]-\left(\lambda-1\right)\sin\left(\lambda+1\right),\\ \mathcal{R}\_{3}\left(\lambda\right)-\left(\lambda+1\right)\sin\left[\left(\lambda-1\right)\left(2-2\pi\right)\right]-\left(\lambda-1\right)\sin\left(\lambda+1\right),\\ \mathcal{R}\_{3}\left(\lambda\right)-\left(\lambda-1\right)\left(4+\lambda\right)\left(4-\pi\lambda\right),\\ \mathcal{R}\_{3}\left(\lambda\right)-\left(\lambda-1\right)\left(4+\lambda\right)\left(4-\pi\lambda\right)\\ \mathcal{R}\_{3}\left(\lambda\right)-\left[\lambda\pm1\right]\cos\left(\lambda-\lambda\right)-\left(4+\lambda\right)\left[\left(4-\pi\lambda\right)\left(2-\pi\lambda\right)\right]\\ \mathcal{R}\_{3}\left(\lambda\right)-\left[\lambda\pm1\right]\cos\left(\lambda-\lambda\right)\left[\lambda\pm1\right]$$

 

 

 

<sup>13</sup> 1 sin 1 1 sin 1 2 ,

 

9 4 1 4 1 cos 4 ,

1 41 32 3 1 4 <sup>11</sup> 1 cos 4 <sup>1</sup> ,

 

3 sin 4 3 sin 4 1 cos 1 2

1 cos 4

 

 

 

1 1

1 1

1 1

 

1 1

 

 

3 sin 4 5sin 4 sin 1 2

 

 

1 1

1 cos 4 cos 1 4 1 sin 4 sin 1 ,

 

 

 

7 4 1 3 2 6 11

   

 

   

3 4cos 4 3 cos 4 1 1 sin 1 2

1 1 1 sin 1 4 1 sin 4 cos 1 ,

3 5 4 1 cos 4 <sup>4</sup> <sup>1</sup> ,

cos 4 1 cos 4 1 1 sin 1 2 1 cos 4 1 sin 1 1 cos 1 2 ,

 

 

 

 

 

 

1 cos 4 1 cos 4 sin 1 2 1 cos 4 sin 1 1 cos 1 2 ,

1 cos 4 1 cos 4 cos 1 2 1 cos 4 cos 1 1 sin 1 2 ,

 

 

 

1 3 4cos 4 5cos 4 1 cos 1 2

2

 

 

3 1 24 1 cos 4 <sup>1</sup> ,

cos 4 1 cos 4 1 cos 1 2 1 cos 4 cos 1 1 sin 1 2 ,

 

 

 

1 sin 1 2 1 sin 1 ,

1 sin 1 2 1 sin 1 ,

1 sin 1 2 1 sin 1 ,

 

> 

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1

> 

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2

     

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<sup>12</sup> 2sin sin 1 ,

 

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2 cos cos ,

22 1 2 11 2

> <sup>13</sup>

14 1 2

 

 

> 

 

 

 

> 

> >

16 1 2

6 12

812

19 2 1 2

 

11 2

 

 

 

 

 

 

 

 <sup>31</sup> 

 <sup>44</sup> 

> 

 <sup>21</sup> 

 

 

 <sup>33</sup> 

 

> 

> >

> > >

 

 

 

2

 

 2 12 11 2 1 1 3 4cos 4 3cos 4 1 1 cos 1 2 3 sin 4 3sin 4 1 sin 1 2 1 cos 4 1 cos 1 4 1 sin 4 sin 1 , <sup>2</sup> 0 3 3 4 2 1 12 1 3 1 2 4 1 1 cos 4 , 5 42 31 4 3 3 1 cos 4 1 1 <sup>2</sup> , 2 3 11 3 2 4 3 2 1 11 1 2 12 3 1 4 2 1 12 0 3 11 2 4 1 cos 4 1 1 31 1 4 1 1 cos 4 , 11 4 11 3 12 2 3 11 2 4 2 1 12 1 1 11 5 12 1 3 1 2 1 cos 4 sin , <sup>2</sup> 12 11 11 1 4 sin , <sup>2</sup> 15 7 11 11 4 sin , <sup>17</sup> 2 1 4 11 3 12 cos 1 6 1 3 11 4 12 sin 1 2 11 16 12 6 , 2 18 11 1 1 13 3 2 2 2 14 13 1 13 1 3 2 21 1 12 4 2 2cos 4 1 2 4 1 cos4 sin 1 1 4 1 cos4 2 1 sin cos 1 

$$\begin{aligned} &+2\left(\begin{array}{c}\lambda+1\\\end{array}\right)\,\,\rho\_{15}\left(\begin{array}{c}\lambda\\\end{array}\right)+\rho\_{17}\left(\begin{array}{c}\lambda\\\end{array}\right)\chi\_{12}\left(\begin{array}{c}\lambda\\\end{array}\right)\left[\sin\lambda\pi\cos\big|\begin{array}{c}\left(\begin{array}{c}\lambda-\pi\\\end{array}\right)\left(\begin{array}{c}\lambda-1\\\end{array}\right)\right]\,\,\big|\\ &+2\,\rho\_{17}\left(\begin{array}{c}\lambda\\\end{array}\right)\chi\_{11}\left(\begin{array}{c}\lambda-1\\\end{array}\right)\left(\begin{array}{c}\lambda-\pi\\\end{array}\right)\sin\lambda\pi\sin\big|\begin{array}{c}\left(\begin{array}{c}\lambda-\pi\\\end{array}\right)\left(\begin{array}{c}\lambda-1\\\end{array}\right)\end{aligned}\,\big|\end{aligned}$$

$$\begin{split} \rho\_{4}\left(\lambda\right) &= \left(\lambda - 1\right) \left[ \left(3\eta\_{2}\lambda + 4\cos 4\left(a - \theta\_{1}\right) - 5\cos 4\left(a - \theta\_{2}\right) + 1\right) \sin\left[\left(\lambda - 1\right)\left(a - 2\pi\right)\right] \right] \\ &- \left(3\eta\_{1}\lambda + \sin 4\left(a - \theta\_{1}\right) - 5\sin 4\left(a - \theta\_{2}\right)\right) \cos\left[\left(\lambda - 1\right)\left(a - 2\pi\right)\right] + 4\sin 4\left(a - \theta\_{1}\right) \\ &\times \cos\left(\lambda + 1\right)a \left[ + \left(\lambda + 1\right)\left[ 1 - \cos 4\left(a - \theta\_{1}\right) \right] \sin\left(\lambda + 1\right)a \end{split}$$

Strength of a Polycrystalline Material 41

there is a singularity in the stress concentration at the corner point of the interface between the two joined crystals, for each of the materials, we choose seven versions of variations in

> Mo 0.1877868 800 1200 0.647029 0.0174393 *i*0.058343 0.688156

> TiC 0.0664576 560 0.0153193 0.01012946 0.6899690 *i*0.047990 0.72254

W 0 1100 - -

 *i* 0.0107712 0.9276061 *i*0.1747073

Au 0.0556095 140 0.0497266 0.0809312 0.422350 0.2043483 0.4714287

C 0.0965294 0.032592 0.5889015

Al 0.1403437 50 0.0284796 0.0522492 115 0.560425 *i*0.073474 0.617351

0.5279915

<sup>5</sup> 10 <sup>6</sup> 7. 3815 10 0.996981

<sup>5</sup> 10 0.994154 0.000786231

, *МПа*

0. 987524

1800 4150

<sup>2</sup> , which are given in Tables 2 and 3. For example, the first

 4

1 4

> <sup>2</sup> 0

0.9546085 *i*0.216914

 2

1 4

> <sup>2</sup> 0

the parameters 1

Table 2.

, and 

> 1 *a b*

 2 2 20 11 3 3 1 13 2 1 1 19 22 2 33 1 12 2 33 2 2 12 13 1 3 2 21 1 2 11 12 0 4 1 2 2cos 4 1 8 1 sin 4 2 1 2 1 cos4 1 sin 1 1 4 1 1 4 1 cos4 sin 2 1 4 sin 1 s in cos 1 2 11 13 2 3 3 2 22 1 3 11 2 1 2 12 13 1 3 2 2 21 1 3 12 5 4 1 4 1 cos 4 sin 2 1 4 1 sin sin sin 1 4 1 1 4 1 cos 4 sin 2 1 1 sin cos 1 , 22 3 1 2 1 9 8 33 1 5 3 1 2 2 cos 4 2 1 2 1 cos 4 , 2 23 17 12 15 2 20 11 11 12 0 2 3 12 5 4 1 cos cos 1 cos cos sin 1 sin cos 1 sin 8 1 1 4 sin 1 sin cos 1 cos 1 2 21 2 11 11 1 4 1 cos cos 1 cos cos sin 1 sin 4 sin 
