**6. Study of the roots of the characteristic equation**

 

Table 1 shows the values of the dimensionless ratio 1 *a b* for some cubic crystals at room temperature. Moreover for all the cosidered materials 1*b* 0 and with the exception of cubic pyrits *FeS*<sup>2</sup> , for which 11 1 2 2 *b b* 0,00365798 10 Pa , 0 .

11 20

 

cos 1 sin .

The least value of the ratio is attained for the niobium crystal (Nb) and the largest, for the sodium crystal (Na). In absolute value, / 1. <sup>1</sup> *a b*

To study the roots of Eq. (19) in the interval l 0 Re 1, in Table 1 we choose six real materials and two imaginary materials for which <sup>5</sup> / 10 <sup>1</sup> *a b* . To investigate whether

4 1 2 2cos 4 1

20 11 3 3 1 13 2

 

 

 

4 1 sin sin sin 1 4 1 1

 

2

22 3 1 2 1 9

8 33 1

2

 

2 2 *b b* 0,00365798 10 Pa , 0 .

 

cos 1 sin .

 

20 11 11 12 0

 

 

 

5 3 1 2 2 cos 4

 

sin 1 sin cos 1 sin 8 1 1 4 sin

17 12 15

 

 

sin 1 sin 4 sin

4 1 cos cos 1 cos cos

Table 1 shows the values of the dimensionless ratio 1 *a b* for some cubic crystals at room temperature. Moreover for all the cosidered materials 1*b* 0 and with the exception of cubic

The least value of the ratio is attained for the niobium crystal (Nb) and the largest, for the

materials and two imaginary materials for which <sup>5</sup> / 10 <sup>1</sup> *a b* . To investigate whether

1 sin cos 1 cos 1

 

11 11 1

 

11 20

2 22 1 3 11

 

 

4 1 1 4 1 cos4 sin

 

22 2 33 1 12 2 33

 

2 1 cos4 1 sin 1 1

12 13 1 3 2 21 1 2

2 1 1 sin cos 1 ,

 

2 2

 

 

> 

 

 

2 2

 1

 

2

 

   

> 

> > 2

in Table 1 we choose six real

   

   

> 

 

 

2

 

 

> 

 

 

 

   

4 1 cos 4 sin

21 1

 

 

 

2

 

11 12 0

4 1

8 1 sin 4 2 1

 

   

 

 

> >

 

 

 

21

 

23

   

1

 

 

 

2

 

 

3 12 5

 

 

1 1 19

in cos

11 13 2 3 3

 

2 1 4 sin 1 s

 

2

 

 

12 13 1 3 2

4 1 cos 4 sin 2 1

 

 

 

3 12 5

 

 

4 1 cos cos 1 cos cos

 

 

2

 

**6. Study of the roots of the characteristic equation** 

To study the roots of Eq. (19) in the interval l 0 Re 1,

 

 

 

pyrits *FeS*<sup>2</sup> , for which 11 1

sodium crystal (Na). In absolute value, / 1. <sup>1</sup> *a b*

 

2 1 2 1 cos 4 ,


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 the parameters 1 , and <sup>2</sup> , which are given in Tables 2 and 3. For example, the first

Table 2.

Strength of a Polycrystalline Material 43

the modulus of elasticity of titanium carbide, which is equal to 460 GPa. We note that the

 3 4 

 3 4 

 2

1 6

 <sup>2</sup> 3

*i*0.0739984 0.656204

*i*0.064893 0.702898

0.9960624 *i*0.0057440

0.993404

0.0154580 *i*0.094053 0.56112

1 6

 <sup>2</sup> 3

0.9906207 *i*0.1403553 0.995563

 0.991197 *i*0.1402922 0.999734 *i*0.004690

0.7444930

0.6977614

 0.451447 0.655004

1 4

<sup>2</sup> 0

Mo 0.710072 0.0775947 0.0411225 0.957018 0.0094110

TiC 0.774834 0.0542824 0.0365724 0.95741 0.0071750

W - - - - -

 <sup>4</sup> 8.82199 10 0.9905998 *i*0.1403189 0.995236

<sup>5</sup> 1.0567 10 0.9276334 0.9912171 0.0011696 0.0013220

Au 0.0644867 0.8730987 0.0779069 0.1780295 0.4007251

C 0.1243433 0.8400085 0.0557915 0.332059 0.5246499

0.7677329

0.7224011

0.0512975 0.612502 0.65982

*i*0.236169 0.2491514

*i*0.140329 0.99971 *i*0.005039

titanium carbide is a compound matter.

 4

0.993177 <sup>4</sup> 7.87311 10

0.927667 *i*0.1746807 0.999978 *i*0.0029137

*i*0.174718 0.997109

*i*0.272280 0.4864447

*i*. 0.113575

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

 2

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

Table 3.

0.9970875 *i*0.00590

0.8890054 *i*0.4002118

Al 0.215732 0.0206982

version, where /2, /4, 0, 1 2 concerns the case in which the interface between two crystals is formed by the plane of elastic symmetry of the second crystal but not of the first crystal. In the fourth version 4, 0, 4 1 2 , the part 1 1 0 of the interface is the plane of elastic symmetry of the first crystal, and the other part <sup>2</sup> 4 is the plane of elastic symmetry of the second crystal.

For all materials given in Tables 2 and 3 and for all versions, we found, in general, all realand complex roots of Eq. (20) with 0 Re 1 , including all (without any exception) rootswith minimum positive real part.

It follows from Tables 2 and 3 that, for all two-crystals except tungsten and for all the versions, there are stress concentrations near the corner point of the interface between the crystals. If we compare the two crystals of molybdenum *Mo* and titanium carbide *TiC* for which <sup>1</sup> *a b* 0 , then it follows from the results obtained for seven versions that, in general, the stress concentration degree (the order of singularity) of molybdenum is less than that of titanium carbide. It is of interest to note that the ultimate strength of polycrystalline molybdenum is larger than the ultimate strength of polycrystalline titanium carbide, which is an integral characteristic of strength. In Table 2, we present the ultimate strengths under tension at temperature 200C for molybdenum and titanium carbide.

For the two-crystal of tungsten *W* , we have 1 *a b* 0 and hence, according to (20), there is no singularity of stress concentration near the corner point of the interface between two crystals. This may be one of the causes of the fact that the polycrystalline tungsten materials have very high ultimate strength.

In Table 2, we present the ultimate strengths under tension of the polycrystalline tungsten annealed wire (1100 МPа) and unannealed wire (from 1800 МPа to 4150 МPа, depending on the diameter). We draw the reader's attention to the fact that the ultimate strength of the diamond monocrystal at temperature 20 *C* is equal to 1800 МPа.

Note that for the polycrystalline metals listed in Table 2 there is a correspondence between the ultimate strength and the modulus of elasticity *E* (here the quantity *E* is treated as an integral characteristic of elasticity of a metal). The moduli of elasticity of the polycrystalline metals *Mo W Au* , , and *Al* listed in Table 2 are, respectively, equal to (285- 300) GPа, (350-380) GPа, 79 GPa, and 70 GPa. The ultimate strength is larger for a metal with larger modulus of elasticity.

All numerical values of strength limit brought in the table (2) as well as elastic modulus for the discussed materials considered to be a published data taken from various sources. For example, these data for tungsten (W) are taken from the book (Knuniants and etc. 1961).

Strength limit of unannealed tungsten wire is depended from the diameter and could be explained by the existence defects of crystal lattice.

Here we also note that there is no such correspondence if molybdenum and titanium carbide are compared. Although the ultimate strength of molybdenum is larger than the ultimate strength of titanium carbide, the modulus of elasticity of molybdenum is less than

two crystals is formed by the plane of elastic symmetry of the second crystal but not of the

 

For all materials given in Tables 2 and 3 and for all versions, we found, in general, all

It follows from Tables 2 and 3 that, for all two-crystals except tungsten and for all the versions, there are stress concentrations near the corner point of the interface between the crystals. If we compare the two crystals of molybdenum *Mo* and titanium carbide *TiC* for which <sup>1</sup> *a b* 0 , then it follows from the results obtained for seven versions that, in general, the stress concentration degree (the order of singularity) of molybdenum is less than that of titanium carbide. It is of interest to note that the ultimate strength of polycrystalline molybdenum

larger than the ultimate strength of polycrystalline titanium carbide, which is an integral characteristic of strength. In Table 2, we present the ultimate strengths under tension at

For the two-crystal of tungsten *W* , we have 1 *a b* 0 and hence, according to (20), there is no singularity of stress concentration near the corner point of the interface between two crystals. This may be one of the causes of the fact that the polycrystalline tungsten materials

In Table 2, we present the ultimate strengths under tension of the polycrystalline tungsten annealed wire (1100 МPа) and unannealed wire (from 1800 МPа to 4150 МPа, depending on the diameter). We draw the reader's attention to the fact that the ultimate strength of the

Note that for the polycrystalline metals listed in Table 2 there is a correspondence between

an integral characteristic of elasticity of a metal). The moduli of elasticity of the polycrystalline metals *Mo W Au* , , and *Al* listed in Table 2 are, respectively, equal to (285- 300) GPа, (350-380) GPа, 79 GPa, and 70 GPa. The ultimate strength is larger for a metal with

All numerical values of strength limit brought in the table (2) as well as elastic modulus for the discussed materials considered to be a published data taken from various sources. For example, these data for tungsten (W) are taken from the book (Knuniants and etc. 1961).

Strength limit of unannealed tungsten wire is depended from the diameter and could be

Here we also note that there is no such correspondence if molybdenum and titanium carbide are compared. Although the ultimate strength of molybdenum is larger than the ultimate strength of titanium carbide, the modulus of elasticity of molybdenum is less than

is equal to 1800 МPа.

and the modulus of elasticity *E* (here the quantity *E* is treated as

/2, /4, 0, 1 2 concerns the case in which the interface between

 

, including all (without any exception)

0 of the

 <sup>2</sup> 4

> is

 4, 0, 4 1 2 , the part 1 1

version, where

 

first crystal. In the fourth version

rootswith minimum positive real part.

have very high ultimate strength.

the ultimate strength

larger modulus of elasticity.

diamond monocrystal at temperature 20 *C*

explained by the existence defects of crystal lattice.

 

is the plane of elastic symmetry of the second crystal.

realand complex roots of Eq. (20) with 0 Re 1

temperature 200C for molybdenum and titanium carbide.

 

interface is the plane of elastic symmetry of the first crystal, and the other part

the modulus of elasticity of titanium carbide, which is equal to 460 GPa. We note that the titanium carbide is a compound matter.


Table 3.

Strength of a Polycrystalline Material 45

*r* is the ratio of the coordinate *r* to the characteristic dimension of the two-crystal). Curves 3 and 4 correspond to the two-crystal of gold *Au* and the two-crystal of aluminum *Al* , respectively. Curves 1 and 2 correspond to a two-component piecewise homogeneous

and to the two-component piecewise homogeneous isotropic body with

*GG a a* / / 0.05 and the Poisson ratios 1 2

1 2 44 44

*GG a a* / / 20 and Poisson ratios

 0.2, 0.3 ,

isotropic body with shear moduli ratio 2 1

1 2 44 44

11 22 12 12

 *Ea Ea* , ,

shear moduli ratio 2 1

respectively. Moreover, 1 2

Fig. 3.

1 2

 0.2, 0.4 

( \*

Fig. 4.

Discussing the results obtained for two-crystals of gold *Au* and aluminum *Al* (Tables 2 and 3), for which <sup>1</sup> *a b* 0 , we conclude that, according to the root of Eq. (20) obtained for seven versions, the stress concentration degree (the order of singularity) near the corner point of the interface between two crystals is larger for the two-crystal of aluminum. Here we also note that the ultimate strength and the modulus of elasticity of polycrystalline gold are larger than those of polycrystalline aluminum. In Table 2, we present the ultimate strengths under tension for polycrystalline aluminum annealed wire (50 МPа) and coldrolled wire (115 МPа).

For a two-crystal of diamond *C* , the stress concentrations near the corner point of the interface between two crystals are rather large (see Tables 2 and 3).

Depending on the choice of the coordinate axes, the modulus of elasticity of the diamond monocrystal varies from 1049.67 GPа to 1206.63 GPа, and, as was already noted, the ultimate strength is approximately equal to 1800 MPa. But for diamond polycrystalline formations (edge, aggregate), we did not found the corresponding integral characteristics of elasticity and strength in the literature. We assume that these characteristics, numerically, must be less than the modulus of elasticity and the ultimate strength of the diamond monocrystal, because there is no stress concentration in the interior of a polycrystalline body.

As follows from Tables 2 and 3, for the imaginary materials with the ratios <sup>5</sup> <sup>1</sup> *a b* <sup>10</sup> , there are very strong stress concentrations for some of the versions.

In Figs. 2–5, we present graphs of variation of the function Re 1 \**r* as \**r* approaches the pole *r* 0

Fig. 2.

Fig. 3.

<sup>1</sup> *a b* <sup>10</sup> ,

as \**r* approaches the pole

44 Polycrystalline Materials – Theoretical and Practical Aspects

Discussing the results obtained for two-crystals of gold *Au* and aluminum *Al* (Tables 2 and 3), for which <sup>1</sup> *a b* 0 , we conclude that, according to the root of Eq. (20) obtained for seven versions, the stress concentration degree (the order of singularity) near the corner point of the interface between two crystals is larger for the two-crystal of aluminum. Here we also note that the ultimate strength and the modulus of elasticity of polycrystalline gold are larger than those of polycrystalline aluminum. In Table 2, we present the ultimate strengths under tension for polycrystalline aluminum annealed wire (50 МPа) and cold-

For a two-crystal of diamond *C* , the stress concentrations near the corner point of the

Depending on the choice of the coordinate axes, the modulus of elasticity of the diamond monocrystal varies from 1049.67 GPа to 1206.63 GPа, and, as was already noted, the ultimate strength is approximately equal to 1800 MPa. But for diamond polycrystalline formations (edge, aggregate), we did not found the corresponding integral characteristics of elasticity and strength in the literature. We assume that these characteristics, numerically, must be less than the modulus of elasticity and the ultimate strength of the diamond monocrystal, because there is no stress concentration in the interior of a polycrystalline

As follows from Tables 2 and 3, for the imaginary materials with the ratios <sup>5</sup>

\**r* 

interface between two crystals are rather large (see Tables 2 and 3).

there are very strong stress concentrations for some of the versions.

In Figs. 2–5, we present graphs of variation of the function Re 1

rolled wire (115 МPа).

body.

*r* 0

Fig. 2.

( \* *r* is the ratio of the coordinate *r* to the characteristic dimension of the two-crystal). Curves 3 and 4 correspond to the two-crystal of gold *Au* and the two-crystal of aluminum *Al* , respectively. Curves 1 and 2 correspond to a two-component piecewise homogeneous isotropic body with shear moduli ratio 2 1 1 2 44 44 *GG a a* / / 20 and Poisson ratios 1 2 0.2, 0.4 and to the two-component piecewise homogeneous isotropic body with shear moduli ratio 2 1 1 2 44 44 *GG a a* / / 0.05 and the Poisson ratios 1 2 0.2, 0.3 , respectively. Moreover, 1 2 11 22 12 12 *Ea Ea* , ,

Fig. 4.

Strength of a Polycrystalline Material 47

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

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),

concentration at the corner point of the interface between the two joined crystals.

In the case of out-of-plane strain of the two-crystal under study, there is no stress

Alexandrov V. M. and Romalis B. L., *Contact Problems in Mechanical Engineering* 

Alexandrov V. M. and Pozharskii D. A., *Nonclassical Spatial Problems in Mechanics of Contact Interactions between Elastic Bodies* (Faktorial,Moscow, 1998) [in Russian]. Alexandrov V. M. and Pozharskii D. A., "On the 3D Contact Problem for an Elastic Cone

Alexandrov V.M., Kalker D. D., and Pozharskii D. A., "Calculation of Stresses in the

Alexandrov V. M. and Klindukhov V. V., "Contact Problems for a Two-Layer Elastic

Aleksandrov V. M., "Doubly Periodic Contact Problems for an Elastic Layer," Prikl. Mat.

Aleksandrov V. M. and Pozharskii D. A., "Three-Dimensional Contact Problems Taking

Alexandrov V.M. and Klindukhov V. V., "An Axisymmetric Contact Problem for Half-Space

526 (2004) [J. Appl. Math. Mech. (Engl. Transl.) 68 (3), 463–472 (2004)]. Alexandrov V.M. and Kalyakin A. A., "Plane and Axisymmetric Contact Problems for a

(2005) [Mech. Solids (Engl. Transl.) 40 (5), 20–26 (2005)].

(2005) [Mech. Solids (Engl. Transl.) 40 (2), 46–50 (2005)].

with Unknown Contact Area," Izv. Akad.Nauk.Mekh. Tverd. Tela, No. 2, 36–41

Axisymmetric Contact Problem for a Two-Layered Elastic Base," Izv.Akad. Nauk.Mekh. Tverd. Tela,No. 5, 118–130 (2000) [Mech. Solids (Engl. Transl.) 35 (5),

Foundation with a Nonideal Mechanical Constraint between the Layers," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 84–92 (2000) [Mech. Solids (Engl. Transl.) 35

Mekh. 66 (2), 307–315 (2002) [J. Appl.Math. Mech. (Engl. Transl.) 66 (2), 297–305

Friction and Non-Linear Roughness into Account," Prikl. Mat. Mekh. 68 (3), 516–

Three-Layer Elastic Half-Space," Izv. Akad. Nauk.Mekh. Tverd. Tela, No. 5, 30–38

Inhomogeneous in Depth," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 55–60

<sup>2</sup> , which are determined in Sections 1–4.

 , and

(Mashinostroenie, Moscow, 1986) [in Russian].

(1998) [Mech. Solids (Engl. Transl.) 33 (2), 29–34 (1998)].

matter.

**8. References** 

depends on the parameters 1 1 *a b* , ,

97–106 (2000)].

(3), 71–78 (2000)].

(2002)].

Fig. 5.

where 1 2 12 12 *a a* and are the strain coefficients of homogeneous isotropic parts and *E*<sup>1</sup> and *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 and 2 in the same figures correspond to the four values of the linear angle formed by the contact surfaces of homogeneous isotropic parts of the compound body. The values of the angle in Figs. 2–5 are respectively equal to: /2, /4, 3 /4 and / 2 . The values of the ratio and the Poisson ratios <sup>1</sup> and <sup>2</sup> , and the corresponding values of the orders of singularities, are taken from Table 1 presented in (Chobanyan, 1987; Chobanyan & Gevorkyan, 1971).

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