.., ..

-

. . . . . . . . . . . . . . . . . . . . . 6

.., .., .. - -

. . . . . . . . . . . . 9

.. -

. . . . . . . . . . . . . . . . . . . . . 13

.., .., .. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

.., .. -

. . . . . . . . . . . . . 22

.., .. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

.., .. -

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

.. -

. . . . . . . . . . . . . . . . . . . . . . 37

.. , -

, . . . . . . . . 42

.., .., .., ..

-

. . . . . . . . . . . . . 47

.., ., .

. . . . . . . . . . . . . . . . . . 52

.. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

.., .. -

. . . . . . . . . . . . . . . . . . . . . 66

.., .. -

. . . . . 71

.., .. -

. . . . . . . . . . . . . . . 76

.., .., ..

-

. . . . . . . . . . . . . . . . . . . . . . . . . . 81

..

, . 86

.., ..

. . . . . . . . . . . . . . . . 90

4 .., .., .. -

, -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

.., ..

. . . . . . . . . 99

.., ..

, . . . . . . . . . . . . . . . 104

.., .., .., ..

. . . . . . . 109

.. -

. . . . . . . . 114

.., .., .. -

. . . . . . . . . . . . . . . . . 119

.., .., ..

. . . . 122

.., ..

. . . . . . . . . . . . . . . . 127

.. -

, . . . . . . . . . . . . . . . . . . . . . 132

.. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

.., ..

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

.., .. -

- . . . . . . . . . . . . . . . . . . . . . . . . 145

.., .. RKDG-

. . . . . . . . . . . . . . . . . . 150

.., .., .. -

: . . . . . . . . . . . . . . . . . 155

.., .. -

2

- ( Res 0). . . . . . . . . . . . . . . . . . . . . . 160

.. -

. . . . . . . . . . . . . . . . . . . . . . . . 165

.., .., .. -

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

..

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

.., .. -

. . . . . . . . . . . . . . . . . . . 179

5 .. -

- . . . . . . . 184

.. . . . . . . . . . . 189

.., .. -

, . . . . . . . . . . . 194

.. -

. . . . . . . . . . . 199

.. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

.., .., ..

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

.., .. -

. . . . . . . . . . . . . . . . . . . . . . . . . . 213

.., .. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

.., .. -

. . . . . . . . . . . 223

.., ..

-

. ..

-

[1, -

, , -

, , -

. -

-

-

, ,

() -

. -

-

[2-6. , -

, -

[7-9. -

- -

.

[10, 11.

. -

[11.

[11, 12.

[13. -

-

.

() , , -

, .

-

, . -

, -

,

. -

: - -

... 7

, (-

) .

-

[11. -

-

.

-

[14.

.

.

.

,

, -

. -

. ,

. , -

,

(-, -,

.). -

, .

,

. -

-

-

.

. -

MPI (Message Passing

Interfae). -

-

. -

- -

.

( 08-01-00500-)

( - 6391.2006.8).

[1 .., ..

. : , 1990. 136 .

[2 .., .., .. -

-

8 .., ..

// . .: . 1989. .30. . 1631.

[3 .., .. -

// . 1989.

1. . 150157.

[4 .., .., ..

// . . 1990. 4. . 733736.

[5 .., ..

// . . -

. 1998. 3. . 1220.

[6 .. -

// . .

. 2004. 2. . 94103.

[7 .., .., .., .., ..

// . -

. 1983. 1. . 90-94.

[8 .., .., .., .. -

, . // .

. . 2001. . 42. 1 . 190195.

[9 .., .., .. -

-

//

. 2005. . 47. 13. . 126133.

[10 .., .. .

// . . . 1990. 2. . 158167.

[11 .., .. -

. .: , 2002. 400 .

[12 .. -

// -. . . . 1963. 6. . 6470.

[13 .. . .: , 1948. 211 .

[14 ..

// . 2005. . 17. 9. . 43

52.

Abrosimov N.A., Kulikova N.A. Identiation of visoelasti strain model for

omposite ylindrial shells under impat loading . A method of material onstants and

funtions analysis dening isotropi and omposite materials relations based on minimization

of omputer and experimental results is onsidered. The program realization of the method

is fullled within the bounds of parallel omputation tehnology. It is shown for the problems

of isotropial and omposite materials with rigid and rheologial harateristis using the

results of numerial-experimental analysis for nonstationary deformation of ylindrial shells

impated by rigid body.

-

.., .., ..

. ..

, -

.

- . .

[1, 2, 3, 4. -

[5, 6.

() () -

[6, 7. [7, 8

- [7, 8.

.

(

s) ui p [7:

Gui,jj +

(K +

1

3G

)uj,ij ( )p,i s2( f)ui = Fi,

sfp,ii

2s

Rp ( )sui,i = a,

=kf

2s2

2s + s2k(a + f),

G,K - , - , k - , - - , , a, f - , -

, Fi, a - . [7:[

ujp

]=

[Usij P sjUfi P f

] [tiq

]d

[T sij QsjT fi Qf

] [uip

]d,

Usij , Psj , U

fi , P

f, T sij, T

fi , Q

sj , T

fi [7.

. --

.

10 .., .., ..

, . -

-

, - .

. -

.

, . 1, :

K = 8 109H/2, G = 6 109H/2, = 2458/3, f = 1000/3, = 0, 19,R = 4, 7 108H/2, = 0, 86, k = 1, 9 1010 4/H, t = 1H/2. : uy(y = 0) = 0, qy(y = 0) = 0,y(y = l) = 1/s, p(y = l) = 0, l = 3.

. 1.

- , . 1, 504 .

. 2, 3 -

.

. 2.

- 11

. 3.

: K = 2, 1 108H/2,G = 9, 8 107H/2, = 1884/3, f = 1000/3, = 0, 48, R = 1, 2 109H/2, = 0, 981, k = 3, 55 109 4/H, ty = 1H/2, l = 10. . 4, 5 .

. 4.

- -

.

.

12 .., .., ..

. 5.

[1 , ..

//. . . . . - 1944. - . 8., 4. - . 65-78.

[2 Biot, M. Theory of propagation of elasti waves in a uid-saturated porous solid. I.

Low-frequeny range // J. Aoust. So. Am. - 1956. V. 28, 2. - P. 168-178.

[3 Biot, M. Theory of propagation of elasti waves in a uid-saturated porous solid. II.

Higher-frequeny range // J. Aoust. So. Am. - 1956. V. 28, 2. - P. 179-191.

[4 , ..

.: , 1996. - 447 .

[5 , .. - -

// - 2006. - 1. - . 121-130.

[6 , .., , .., , ..

. // - 2006. - . 70.

- . 2. - . 282-294.

[7 Shanz, M.Wave Propogation in Visoelasti and Poroelasti Continua - Berlin Springer,

2001. - 170 p.

[8 Shanz, M., Strukmeier V. Wave propagation in a simplied modelled poroelasti

ontinuum: Fundamental solutions and a time domain boundary element formulation

// Int. J. Numer. Meth. Engng. - 2005. - V. 64. - P. 1816-1839.

Amenitsky A.V., Igumnov L.A., Karelin I.S. Boundary element simulation of wave

propagation in Bio-medium. A model of a porous medium with a two-phase internal struture

introdued by Bio is disussed. The related integral equations and a boundary element method

for analyzing them are desribed. A numerial example is given.

..

, --

-

, . -

.

, Flex PDE. -

, -

.

, . -

, . -

.

1.

[1

divD = 0, (1)

. -

() . -

x3, x1,x2 -, . i1, i2, i3. [2 -

:

R(x1, x2, x3) = u1(x1, x2)e1 + u2(x1, x2)e2 + (x3 + w(x1, x2))i3,

e1 = i1 cosx3 + i2 sinx3 , (2)

e2 = i1 sinx3 + i2 cosx3,e3 = i3, (, = const).

R - , ( 1) ,

2. (2) -

. - u1(x1, x2), u2(x1, x2), w(x1, x2) (1) n1D1k + n2D2k = 0,(k = 1, 2, 3). , . , :

14 ..

u1 = x1 + 2u(x1, x2) + ...

u2 = x2 + 2v(x1, x2) + ... (3)

w = w0(x1, x2) + ...

. , [1:

D = 2

[(S1

1 2 1)U1 C + C

], S1 = trU 3, U = (C CT )1/2, (4)

C , - , - . , -

D11 = D22 = 0 = 0, (2) (4), : = 1 ( 1).

3. (3), U = (C CT )1/2 U1 = V . (4) D, = 1:

D11 = 22(

1 2 (u

x1+

v

x2+

1

4x21 +

1

4x22 +

1

4(w0x1

)2 +1

4(w0x2

)2

12x1w0x2

+1

2x2w0x1

) + (u

x1+

1

8(w0x1

)2 +1

4x2w0x1

+1

8x22)),

D22 = 22(

1 2 (u

x1+

v

x2+

1

4x21 +

1

4x22 +

1

4(w0x1

)2 +1

4(w0x2

)2

12x1w0x2

+1

2x2w0x1

) + (v

x2+

1

8(w0x2

)2 14x1w0x2

+1

8x21)),

D33 = 22(

1 2 (u

x1+

v

x2+

1

4x21 +

1

4x22 +

1

4(w0x1

)2 +1

4(w0x2

)2

12x1w0x2

+1

2x2w0x1

) + (1

8(w0x2

)2 14x1w0x2

+1

8x21 +

1

8(w0x1

)2 +1

4x2w0x1

+1

8x22)),

D12 = 22(

v

2x1+

u

2x2+

1

8

w

x1

w0x2

18x1x2 +

3

8x2w0x2

+1

8x1w0x1

),

D21 = 22(

v

2x1+

u

2x2+

1

8

w

x1

w0x2

18x1x2 3

8x2w0x2

18x1w0x1

),

D31 = D13 = (w0x1

x2), D32 = D23 = (w0x2

+ x1)

(5)

4. u1(x1, x2), u2(x1, x2) w(x1, x2) -, Flex PDE, , -

[3

[4. (5)

. -

-

, , = 0.1; 0.2; 0.3 = 0.91...1.1.

... 15

,

D32 d = F3 (6)

. () -

b/a ( a b -), -

. , , = 0, = 1 F3 = 0, b/a = 0.818. , . b/a, - . ,

, ,

-

, .

5. .

(2) :R = R(r) = + zZ = z

(7)

r, , z .R, , Z .

R =0

R = R(r)ereR +R

ree + Reze + ezez (8)

er = i1 cos+ i2 sin,

e = i1 sin+ i2 cos, ez = i3,eR = i1 cos + i2 sin ,

e = i1 sin + i2 cos, eZ = ez = i3, , :

D = 2C + 2

[

1 2 (trU 3) 1] A, (9)

A A = U1 C

0

D = 0 :dDrRdr

+DzR D

r Dz = 0 (10)

(10) R(r), :

16 ..

(R +R

r) +

1 + +

(1 )r

[+ R

r

1]+ 2R

[1 + + R

r

(1 ) 1]= 0 (11)

0 r 1, =(+ R

r)2 + 2R2

(10) R(0) = 0, DrR(1) = 0

(1 )R + = 1 + , (12)

r = 1 = 0, R = R0, R = r, = 1 +

R = r + 2U0(r)... (13)

(13) (11) ,

:

U0(r) = 0.117857r + 0.017857r3

6. ,

.5, .4,

(5). -

, = 1 [5 :

X1 = U1(x1, x2) cosx3 U2(x1, x2) sinx3,X2 = U2(x1, x2) cosx3 + U1(x1, x2) sinx3,

X3 = 0

U1 = x1 + 2U(x1, x2), U2 = x2 +

2V (x1, x2)

R =X21 +X

22 =

U21 + U

22 =

(x1 + 2U)2 + (x2 + 2V )2 =

= r + 2(Ux1 + V x2)

x21 + x22

,

= 1 + =

0 r 1, 1 x1 1, 1 x2 1, = 1, = 0.3 , -

Flex PDE. .4 5

U0(r) F (x1, x2)

F (x1, x2) =Ux1 + V x3

x21 + x22

U0(r) F (x1, x2)

... 17

r/x1, x2 F (x1, x2) U0(r)0 0 0

0.1 -0.0011 -0.0011

0.5 -0.057 -0.0578

0.9 -0.0943 -0.0943

1 -0.1 -0.1

[1 .. . .: , 1980.

[2 .. -

// . 1983. . 270. 4. . 827831.

[3 . ., . . -

.// X -

. 2006. . 21-25

[4 .. . .: , 1970.

[5 . ., . . ,

// . 2003. . 191-201

Asotova E.A. The stressed state of prismati bars under nite torsion deformations.

The nonlinear problem of torsion for prismati elasti bodies under small, but nite, angles of

twist is examined. The three-dimensional problem is redused to the nonlenear boundary value

problem at the ross-setion of a bar. The last problem is solved by Signorini disturbane

method with help of Flex PDE pakage. The stressed state for the bar of ompressible half-

lenear material with retangular or ellipti ross-setion is found in limited of the seond-oder

theory. The analytial solution of the problem in ase of irular ylinder is obtained for the

omparison with the numerial results. The appearene of the stress, those are absent at the

lenear theory of torsion, is revealed. The Pointing eet in torsion and the inuene of the

ross-setion form on it are analysed.

.., .., ..

. ..

[1

.

.

1. -

- -

-

() [2, 3. , -

, , -

[4, 5. - ()

[6, 7.

2. -

P (t) = 1000/2 - 11 . 15 . -

: E = 2, 5 108H/2, = 0, 298, = 1884/3. 864 913 . . 1

, [1. 1 -

, ,

2 - [1.

. 1.

P = (H(t) H(t 0, 0085c))H/2 - 221,

. 19

. 96 , -

- - 432

.

: - E = 3 108H/2, = 0, 2, = 2000/3, - E = 1, 38 108H/2, = 0, 35, = 1966/3.

(2,33; 2,33; 0).

. . 2, 3 -

,

. . 2

: 1 ,

2, 3 4 - = 100c1, = 1c1 = 0, 01c1 . . 3 : 1 ,

2, 3, 4, 5 6 - k = 1, = 0, 5, k = 1, = 0, 75, k = 1, = 0, 95,k = 5, = 0, 95, k = 10, = 0, 95 .

. 2.

. 3.

-

(), . 4.

20 .., .., ..

[2. :

P (t) =

p0t/2, 0 < t < 0, 012c

p0/2, 0, 012c < t < 0, 037c

p0(t 0, 025)/2, 0, 037c < t < 0, 049cp0, 0, 049c < t < 0, 061c

p0(t+ 0, 086)/2, 0, 061c < t < 0, 086c0, t > 0, 086c

p0 = 4 106H/2. 226

257 , 940 982 .

B.

. 4.

. 5.

: - E = 3 108H/2, = 0, 2, = 2000/3, - - E = 3 108H/2, = 0, 2, = 2000/3.

. 21

. 5 1 , , -

2 - , -

[2.

[1 Shanz, M. Wave Propogation in Visoelasti and Poroelasti Continua //

. . 2000. 3. . 184188.

[2 , .., , .. -

.: , 2008. - 352.

[3 , .., , .., , .. -

// . . - 2007. - . 69. - . 125-136.

[4 Durbin, F. Numerial inversion of Laplae transforms: an eient improvement to

Dubner and Abate's method // The Computer Journal. 1974. V. 17. 4. P. 371-376.

[5 Zhao, X. An eient approah for the numerial inversion of Laplae transform and its

appliation in dynami frature analysis of a piezoeletri laminate // Int.J. of Solids

and Strutures. 2004. V. 41. P. 3653-3674.

[6 Lubih, C. Convolution Quadrature and Disretized Operational alulus. I //

Numerishe Mathematik. - 1988. - 52. - P. 129-145.

[7 Lubih, C. Convolution quadrature and disretized operational alulus. II //

Numerishe Mathematik. - 1988. - V. 52. - P. 413-425.

Belov A.A., Vasiljev A.A., Pazin V.P. alulation of omposite viso-elasti bodies

dynamis based on diret statement of boundary element method . The alulation results of a

dynami state of viso-elasti bodies are presented based on a boundary element method in

ombination with the onvoation quadrature method [1 and Durbin's method. The solution

of viso-elasti boundary problems expliitly in a 3-dimensional statement is performed

without using step proedures.

.., . .

, . --

, -

, , ,

.

.

, . -

-

. .

-

(, , )

, , .

-

.

[1 [3, [7, -

.

h , , l0. Ox1, Ox3 - .

. pi , - .

:

ij,j + 2ui = 0,

ij = Cijkluk,l,ui|x3=0 = 0, i3|x3=h = pi(x1),

ijnj |l0 = 0, i, j = 1, 3,(1)

, Cijkl -, nj - l0, - , .

(1) ,

[4.

u1, u3, ,

23

Ox1x3. , [5

[a,b

um() = um()

l0

Kim(x, ) (ui(x) ui()) dlx

2 S0

U(m)i (x, )dSx ui(),

um() = ba

pi(x1)U(m)i (0, h, ), i, j,m = 1, 3

(2)

Kim(x, ) = (m)ij (x, )nj(x), x = (x1, x3), = (1, 3),

S0 , l0.

(2) ,

um() , - . (2)

, ,

.

-

(2)

l0

um(y) = um(y)

l0

Kim(x, y)(ui(x) ui(y))dlx

2 S0

U(m)i (x, y)dSx ui(y), i, j,m = 1, 3, y l0.

(3)

(3) ,

[2, [3. (3)

[5.

(1) -

, -

r. :

1 1, < 2 < 1,1 = r/h, 2 = r

/C33,

(3). , -

2 > , , .

x = x0 + r, y = x0 + r, = {cos , sin }, = {cos, sin},

x0 = (x10, x30), , [0, 2].

24 .., ..

(3)

,

(2 = 0) :

Kim(x, y) = K0im(x, y) +K

1im(x, y), i,m = 1, 3 (4)

(4)

[2,[3.

K1im(x, y) = O(22), K

0im(x, y) =

1

1Fim(x, y) +O(1), i,m = 1, 3, x, y l0.

Fim(x, y) .

um(x0 + r) = u0m(x0) + 1 (um,1(x0) cos + um,3(x0) sin ) +O(

22), m = 1, 3. (5)

, (3) (5)

1 -

l0.

u0m(x0) = um(x0),

u1,1(x0) =I2(a

(2)2 , b

(1)2 )u

3,3(x0) + (2 I1(a(1)3 , b(1)3 )u3,1(x0)

1,

u1,3(x0) =I1(a

(1)1 , b

(1)1 )u

3,1(x0) + (2 I2(a(2)3 , b(2)3 )u1,3(x0)

2,

u3,1(x0) =(2 I1(a(1)1 , b(1)1 )u3,1(x0) + I2(a(2)3 , b(2)3 )u1,3(x0)

2,

u3,3(x0) =(2 I2(a(2)1 , b(2)1 )u3,3(x0) + I1(a(1)2 , b(1)2 )u1,1(x0)

1,

1 = (2 I2(a(2)1 , b(2)1 ))((2 I1(a(1)3 , b(1)3 )) I2(a(2)2 , b(2)2 )I1(a(1)2 , b(1)2 ),1 = (2 I1(a(1)1 , b(1)1 ))((2 I2(a(2)3 , b(2)3 )) I1(a(1)1 , b(1)1 )I2(a(2)3 , b(2)3 ),

(6)

I1(a, b), I2(a, b) - . ,

l0 (3)

[2, [3, [5, .

-

,

: 1 = 0.0010.15 2 < 1.

1 l0, -

.

25

. 1. l0

, h = 1 ( ), - (0, h) - . :

r = 0.003h, x10 = h, x30 = h/2. N = 16. .

,

, -

, 4 %,

. (6)

.

26 .., ..

[1 .. .

.: . 2007, 223 .

[2 .., ..

// . -. . . .

. 2006. . 1. . 73 79.

[3 .., .. //

. 2006. 10. . 33-39.

[4 .., ..

. .: , 1976. 319 .

[5 ., ., . . .: . 1987.

524 .

[6 . ., .., ..

. // . . . . . 1989.

2. C. 81 85.

[7 .., . .

// ().

2005, 1. C. 10 16.

Belyak O.A., Vatulyan A.O. The asymptotial approah at the analysis of wave elds

in a layer with a avity of the small size . Problems of distribution of waves in layered

environments with defets suh as raks, avities, inlusions of the various form have the

important appendies in aoustis, seismologies, tehnial diagnostis. Formation of elds in

environments with defets is omplexdiult proess as a result of repeated reverberation

from borders of defets and environment. In the presenttrue work the established antiplane

and plane utuations of a layer with the ylindrial avity whih is not leaving on borders of

a layer are onsideredexamined. Besides a traditional method of data of an initial problem to

systems boundary integrated the equations on the basis of the theory of potential it is oered

asymptotial the approah at the deision of a task in view. Results of numerial experiments

are resulted.

.., ..

, --

-

-

, -

.

-

() ,

. , -

-

, .

[3 .

-

[1

. -

[2,

. -

-

. -

. -

60 , . -

.

x0 y x S (y S) , , , p(x) [5:

p(x) =

S

2pinc(y)

nydS (1)

S p/n|s = 0. pinc(y) - S, - - ( ), ny- S y, k - .

28 .., ..

pinc(y) = |x0 y|1eik|x0y|, = (4)1|x y|1eik|xy|, (2)

k ny

= ikcos()(4)1|x y|1eik|xy|[1 + O 1k|x y| ], (3)

- ny x0 y , |x0 y| |xy| - x0 y S , x y .

-

[3.

(1) [3, -

.

x0 y1 y2 x3 , x0 x3 . y

1 y

2 -

, . p(x3) :

p(x3) =

S

2

2p(y2)(y2; x3)

n2dS2. (4)

p(y2) - y2 S2 y2, S1 y

1.

p(y2) :

p(y2) =

S

1

2pinc(y1)

n1dS1. (5)

, pinc(y1) = |x0 y1|1eik|x0y1| - , ,

:

p(x3) = (

k

2

)2cos(1)cos(2)

L0L1L2

S

2

S

1

eikdS1dS2, (6)

= |x0 y1|+ |y1 y2|+ |y2 x3|, (7)L0 = |x0 y1|, L1 = |y1 y2|, L2 = |y2 x3|.

, -

.

(8) (6) -

[4.

p(x3) = cos(1)cos(2)exp

{i[k(L0 + L1 + L2) +

4(4 + 4)

]}L0L1L2

|det(D4)| , (8)

29

D4 = (dij), i, j = 1, 2, 3, 4 - dij , i j [2. 4 = sign(D)4 - .

(8) p(x3)

. p(x3) - , , -

, -

, , -

,

y1 y2. -

. (8)

kd >> 1, kR(m)1 >> 1 , kR

(m)2 >> 1 , d -

, R(m)1 , R

(m)2 (m = 1, 2) -

y1 y2.

. 1.

-

, -

(. 1) 2a , . -

30 .., ..

. 1 a, ,

. -

60 .

:

L0 = L1 = L2 = a3.

, -

(8) ,

: p(x3) = 1/(L0 + L1 + L2).

p(x3) -

p(x3) S2

eik|y2x3|[S1

eik(|x0y1|+|y1y2|)dS1

]dS2,

dS = dxdy. - .

i1 j1 , - - i2 j2. S1 ( ) S2( ). S1 S2 : y

1 -

, y2 - . ,

ka, , (ka)max = 550. , a/ = ka/2 =88, - . 8 , N = 8a/ 700 i1, j1, i2, j2. N4 = 7004 .

:

p(x3) = p1 + ip2 =N

i2=1

Nj2=1

[cos(k|x3 yi2,j2|) + i sin(k|x3 yi2,j2|)

]

N

i2=1

Nj2=1

{[cos(k|x0 (y1)i1,j1|+ |(y1)i1,j1 (y2)i2,j2|)

]+

+i sin(k|x0 (y1)i1,j1|+ |(y1)i1,j1 (y2)i2,j2|)}, A(p(x3)) = |p1 + ip2|.

(9)

p(x3) . 100 ka 550). A ka 1 .2, .

, -

.

31

. 2. ( ) -

[1 . ., . . ,

// . 2003. . 392. 5. . 614 -

617.

[2 . . -

// . . . 2005.

5. . 65 - 80.

[3 . ., . . . .: , 1978.

248 .

[4 . . . .: , 1977. 368 .

[5 . . . - .: . - 1972. - 352

.

Boev N.V., Sumbatyan M.A. On the a

uray of Kirhho's physial diration theory

in the ase of re-reeted waves. We give a omparative analysis for diret numerial treatment

of diration integral for a doubly reeted high-frequeny aousti wave along the trajetory,

whih is a broken straight line.

..

, ..

. ..

, --

, --

() .

PIC16.

() -

, ,

, , .

, -

: -

,

() -

. ,

-

. -

, ,

, () [1-6.

st ( ) t ( I1t) () , -

i1t :

I1t = st/(t) = I1t = 1; i1t = |(t st)|/(t) = 0. (1)

u1u - i1u:

I1u = su/(u) = I1u; i1u = |(I1u I1u)|/I1u = 0. (2)

su u - -, I1u - , 0,8 [1,3-6. -

(1) (2)

.

... 33

.

, -

. 10 -

. -

[6.

,

. . 1. ,

1

, 0,7. ,

, 2

( 0,4% 0,7%), -

, , ""

. -

.

. 1. . -

. - .

,

,

, -

, , ( ) .

-

-

PIC16, PIC18 dsPIC [7. ,

. -

34 .., ..

-

-

: t, (t)2, u, u2, ut.

t = t/n; st =[(t)2 (t)2/n]/(n 1). (3)

u = u/n; su =[u2 (u)2/n]/(n 1). (4)

LCD HD44780.

USB. -

AD8307 "Analog Devies".

. 2 , -

.

AD8307,

. -

0,4 2,5

80 . ,

. -

16F877 PIC 16 8 10

. RA3 -

. 20 .

200 , ,

400 .

. 2. .

-

.

... 35

.

. -

. -

, -

. RA1/AN1

,

,

(3), . -

.

, -

.

(, 1) -

.

4- .

8 .

. TX RX

COM-USB ( ).

, (, ).

-

-

10-20 ,

( -

10000 !) .

,

. -

-

, .

-

,

, .

-

( 06-08-01039).

[1 .. - -

- . --:

, 2008. 192 .

[2 Builo S.I.Use of Invariant Combinations of Parameters Charaterizing Aousti Emission

in Diagnostis of Prefrature States of Solid. // Rus. J. of Nondestrutive Testing. 2002.

Vol. 38. 2. pp 116120.

36 .., ..

[3 ..

//

. 7- . ., --. 2002. . 2. . 7983.

[4 Builo S.I. Diagnostis of the Predestrution State Based on Amplitude and Time

Invariants of the Flow of Aousti-Emission Ats. // Rus. J. of Nondestrutive Testing.

2004. Vol. 40. 8. pp 561564.

[5 .. - -

// 11- .

" "OMA-11, (), 10-15 2008 .,

--, , 2008, . 102105.

[6 .. -

//

: 16- -

. . ., 1-5 2008,: 2008. . 8486.

[7 . PIC.

: . . : "- 2008. 544 .

Builo S.I., Orlov S.V.Aousti Emission Invariants Method in Diagnostis of Predestrutive

State and Its Hardware Realization .The method of diagnostis of the predestrution state

based on deviations of the amplitude and time invariant relationships between the parameters

of the ow of a

ompanying aousti emission (AE) pulses and their stable values is

desribed. Hardware realization of the aousti emission invariants method based on Pi16

miroontrollers is oered.

. .

,

, --

-

, , -

, .

,

, -

-

(,

, ..). -

-

.

, , -

-

, - . -

.

, -

. () -

. -

-

[2-4. [5,6,7 ,

-

.

-

.

V - S .

ij,j + 2ui = 0 (1)

38 ..

mj = Cmjkluk,l (2)

ui |Su= 0, ijnj |S= pi (3) Cijkl - , -

; -

V -, - , nj- S.

(1)- (3) -

[1,2 -

. -

,

, , , . -

.

,

. ,

,

ui |S= fi, [1, 2] (4) - -

. (1)-(4)

, -

, , -

(1-4).

, ; -

( )

, .

-

[3,4. -

. , , -

,

, -

[5. , [6

( ),

-

.

.

V , - ,

(1)-(2) (1-3).

1.

39

V

2L(ui, vi, Cijkl, )dV +

S

pividS = 0, (5)

2L(ui, vi, Cijkl, ) = 2uivi Cijkluk,lvi,j. ,

2L(ui, vi, Cijkl, ) . 1 (1) -

vi V . , - , (5).

, (5)

. vi = ui , = 0 (5) [1.

(4) , ,

S (5), -, ,

Cijkl, ( , ui Cijkl, , [2) :

V

(2uiui Cijkluk,lui,j)dV +S

piuidS = 0, [1, 2]. (6)

, (6,7), -

-

. (6)

, ui, Cijkl, , [8.

2. V

2L(ui, ui, Cijkl, )dV S

piuidS = 0, (7)

Cijkl, . ui, Cijkl, (1)(4). (6); (6)

, V

2(uiui + 2uiui) Cijkluk,lui,j 2Cijkluk,lui,jdV +S

piuidS = 0, (8)

, , -

V

(2uiui Cijkluk,lui,j)dV S

piuidS = 0,

[2. (6)

40 ..

, (7)

(6),

. , , -

, u(n)i , C

(n)ijkl,

(n) . - C

(0)ijkl

(0)( -

-

, [7). -

. u(n1)i

(1)(3) C(n1)ijkl

(n1),

V

2L(u(n1)i , u

(n1)i , C

(n)ijkl,

(n))dV S

pi(fi u(n1)i )dS = 0, [1, 2], (9)

(7), (9)

1- , -

.. [2,9.

. , Cijkl(x) (x) (x), , , Cijkl(x) , (x). - (9) (Cijkl = 0)

2V

(n)u(n1)i u

(n1)i dV

S

pi(fi u(n1)i )dS = 0, [1, 2], (10)

, V = [0, l] F , p1 = p,p2 = p3 = 0, u1 = u(x, ), u2 = u3 = 0 (10)

2l

0

(n)(x)(u(n1)(x, ))2dx+ p(f() u(n1)(l, )) = 0, [1, 2], (11)

[5,7 .

, , 3 (

(x), (x) (x)), . , -

, , 3

( E(x) , G(x) (x) ), - ,

, -

, . -

l0

E(n)(x)(u(n1)(x, ))2dx2l

0

(n)(x)(u(n1)(x, ))2dxp1(f1()u(n1)(l, )) = 0,

(12)

41

l0

E(n)(x)(w(n1)(x, ))2dx c2l

0

(n)(x)(w(n1)(x, ))2dx

p2(f2() w(n1)(l, )) = 0,

[1, 2] (13)

, .

[1 . . .: , 1975. 872 .

[2 -. . .: , 1984. 472 .

[3 Isakov V. Inverse problems for PDE. Springer-Verlag. 2005. 284 p.

[4 . . . .: , 1994. 206 .

[5 . . . .:

, 2007.224 .

[6 . . //

, . 2007. 4 (54). . 93103.

[7 . ., . .

// , . . . . . 2008, 3. . 3337.

[8 . .

// . 2008. . 422. 2. . 182184.

[9 .., ..

. .: , 2004. 480 .

Vatulyan A. O. Variational methods in problems of identiation

nonhomogeneities in elasti bodies. Within the bounds of model of the linear

nonhomogeneous theory of elastiity the basi identity linking possible states in whih

not only omponents of elds of displaement and stresses but also omponents of

modules of elastiity and density vary is formulated.

On the basis of the variation equation examples of build-up of the operator equations

and iterative proesses in problems of restoration of modules of elastiity and a density

at the steady-state vibrations of nite bodies, and also at identiation of alloation

of porosity in models of the adaptive theory of elastiity are presented.

,

,

.

..

. .. . . --.

-

. -

. -

-

. ,

. -

.

.

.

. . -

:

R = R(), = () (1)

, ..

.

, (.. -

) .

. -

.

-

. ,

. , , -

.

.

: = u A, u - , A - .

u =

V

W (1, 2, 3) dV, A = P (V v). (2)

. 43

W (1, 2, 3) - , 1, 2, 3 - , -

.

,

W (1, 2, 3) = hW(1, 2,

11

12

)(3)

h - , W - .

:

1 =R() cos(())

cos(), 2 =

R()2 +R()2()2 (4)

:

= 2

/2/2

cos()W [1, 2] d 23

P

h

/2/2

R3 cos()d 2

(5)

x3. - (4) (5) ,

. -

.

,

-

,

:

W1

cos() + W2

R()2 cos()(R)2+R2()2

dd

(R cos()

(R)2+R2()2W2

) P

hR2 cos() = 0

RW1

sin() + dd

(R2 cos()(R)2+R2()2

W2

) P

hR2R cos() = 0

(6)

:

(2

)=

2, R

(2

)= 0. (7)

.

. -

(6),

: -, -

, , , . :{R = R0 + () = + ()

(8)

44 ..

R0 - , , - - , , - , - .

, = 0, - , -

.

:

= APn(sin()), = BPn(sin()) cos() (9)

(9) -

, -

. ,

:

(D n (n+ 1)F ) (J Gn (n + 1)) C2n (n + 1) = 0 (10)

D,F, J,G, C R0 P . .

(10) P , - R0. P (R0)

. (10), -

:

f (R0, n) = 0 (11)

.

, (11)

P (R0), . . -

(9) n = 1. , n = 1 (11) . ,

, -

(11).

(11) n = 2, 3 . . . , .

:

1. .

2.

.

3. : -,

-

.

. 45

4. : -, , -

, .

5. .

1 -

(

, ).

. 1.

.

-

. :

R = Rk + cB1

2(3 sin2() 1), = + 3B cos() sin() (12)

Rk , c - A B.

B (12) (5), -

. ,

. -

. -

. ,

.

2,

.

46 ..

. 2. 1 - , 2, 3 -

[1 . . . . ,

// . - .

. 2006. 1. . 30-34.

[2 . . . --

.: - . 1982.

[3 . . .

..: . 2000.

[4 . . . .: . 1980.

[5 . . . . . .: .

1987.

[6 Muller I. Struhtrup H. Inating a Rubber Balloon. // Mathematis and Mehanis of

Solids. 2002. 7. 569-577.

Galaburdin M.V. The behaviour of an losed spherial shell loaded internal pressure,

after loss of stability . The problem of instability of a thin losed spherial shell under a

strethed strain was investigated. The spetrum of ritial pressures was found by method of

linerization for row ordinary hyperelasti unompressible models of materials. Postbukling

behavior was studied by Ritza's method on basis variant Lagrandge's priniple.

.., .., .., ..

,

-

, . -

-

- ,

.

,

, . -

. ,

, -

,

, .

1.

(, , -

), ,

, , , .. -

-

,

,

-

,

().

2. -

ueit(u = (ux, uz)) , = {|x| 1. (2)

, i , -: ui 6 0 i < 0 ui > 0 i > 0. , -

.

(2) - -

x = x(t) (., , [4)

D [ui] =

[iui1 + s

], i = 1, . . . , n, D =

dx(t)

dt, [f ] = f+ f, (3)

D , f+ = f(x(t)+0, t), f = f(x(t)0, t).

62 ..

, x = x(t) - , k- (., ,[5)

k Dk +k , k = 1, . . . , n,k1 Dk +k+1, k = 1, . . . , n.

(4)

k = k(u(x(t) 0, t)), +k = k(u(x(t) + 0, t)), k Aij = j(iui/(1 + s)), j = /j .

ui|t=0 =

0, x < x0

u0i , x0 < x < x1,

0, x > x1

i = 1, . . . , n. (5)

(2)(5) -

.

2. . ,

[2, (2)(5) Ri, ,

R1t

+R21R212

R1x

= 0,R2t

+R1R

22

12

R2x

= 0; (6)

D

[2

R1R2(1 R1)(1 R2)

]= [(1 R1)(1 R2)],

D

[1

R1R2(2 R1)(2 R2)

]= [(2 R1)(2 R2)];

(R1 )2R2

12 D1 (R

+1 )

2R+212

, D1 R+1 (R

+2 )

2

12,

R1 (R2 )

2

12 D2 R

+1 (R

+2 )

2

12,

(R1 )2R2

12 D2;

R1|t=0 =

1, x < x0

R1 , x0 < x < x11, x > x1

, R2|t=0 =

2, x < x0

R2 , x0 < x < x12, x > x1

u = u(R)

u1 =2(R1 1)(R2 1)

R1R2(1 2) , u2 =1(R1 2)(R2 2)

R1R2(2 1) .

R = R(u)

(1 + u1 + u2)R2 (1 + 2 + 1u2 + 2u1)R + 12 = 0.

63

(2)

F (u1, u2) = (1 + 2 + 1u2 + 2u1)2 4(1 + u1 + u2)12.

F (u1, u2) > 0, 1 + s > 0 - (2) , F (u1, u2) < 0 (2) (. . 1).

.

1

1

u1

u2

A

B

1+s=

0

C

F (u1, u2) = 0A

(0,2 1

1

)B

(1 2

2, 0

)C

(2

2 1,

1

2 1

)

. 1. .

3. .

u01 6 0, u02 > 0.

. 2a.

x1 x2 , . 2b. (xs) D1, D2

x1s = x2 +D1t, D1 =R1 R

2

2; x2s = x1 +D2t, D2 =

R1 R2

1.

- , (6)

( )

Ri(z) =iz, z =

(x xi)t

, i = 1, 2.

(xl) (xr) -

x1l = x1 + 1t, x1r = x1 +

(R1 )2

1t, x2l = x2 +

(R2 )2

2t, x2r = x2 + 2t.

(T1, X1),

T1 =12(x2 x1)R1 R

2 (2 1)

, X1 = x1 +R1 R

2

1T1,

64 ..

Ri2

xx2x1

1

R2

R1

Ri 2

xx2lx1r

1

R2

R1

x1l x2s x

1s x

2r

b)

a)

Ri 2

xx2lx1r

1

R2

R1

x1l x2sx

1s x

2r

c)

Ri 2

xx2l

1

R2

x1l x2sx

1s x

2r

d)

Ri 2

x

1

x1l x2sx

1s x

2r

e)

D2 D1

D2D1

D2D1

D1 D2

x2x1

u2

u02

x

u1

u01

x

u2

u02

x

u1

u01

x x2x1

u2

x

u1

x1r1x1l1x

1s

x2s x2l x

2r

2R2R

2

1R1R

1

2R2R

2

1R1R

1

x2s x1s x

2l x

2r

x1l x1r x

2s x

1s

u2

xx2s x2l x

2r

u1

x1l1x1s

x

R2

R1

R2

R1

R1

R2

R2

R1

x

u2

xx2s x2r

u1

x1l1x1s

x

. 2. .

x = x2s x = x1s.

. 2.

xis = Xi +Di(t Ti), Di = Ri i = 1, 2.

, t = T1 (. . 2). , 2 R2 R1 1, (T2, X2) x = x

1r x = x

1s

(. . 2d), (T3, X3) x = x2s

65

x = x2l (. . 2e).

T2 =12(x2 x1)(R2 1)R1 R

2 (2 1)(R1 1)

, X2 = x1 +(R1 )

2

1T2,

T3 =12(x2 x1)(2 R1 )R1 R

2 (2 1)(2 R2 )

, X3 = x2 +(R2 )

2

2T3.

. . 2d

k = 1. . 2e k = 2.

Dk = k +1t

(k(Xk+1 xk) k

Tk+1

),

xks = xk +(

kt+Xk+1 xk

kTk+1

)2.

-

,

-

.

07-01-00389, 07-01-

92213-.

[1 .., .. . //

. 1982..267, 2. .334-338.

[2 .. . // , 1984. .24,

4. .549-565.

[3 .. . --:

, 2005. 215 .

[4 .., .. . .: -

, 1978. 668 .

[5 Lax P.D. Hyperboli systems of onservation law II. // Comm. Pure Appl. Math. 1957.

10. P. 537-566.

Elaeva M.S. Evolution of mixture omponent under ation of an eletri eld .

Separation of two-omponent mixture under ation of an eletri eld is investigated. We

assume that ondutivity is depended on onentration omponents of mixture. Diusionless

approximation of the model is transformed to Riemann invariants is analyzed by the method

of harateristis. We onsider interation between two shok waves and between shok wave

and rarefation wave. Solutions for eah stage of separation proess is obtained.

..

, ..

-

. , -

. ,

- -

.

.

-

, .

. [1: -

, -

, ; -

.

. 10 1.

Tn > Tg, Tg . -

.

, Tg - , Tg , . . . , = 0 , = 1 .

,

.

,

.

,

. [2

: , -

.

(T0), . T0 , , , . -

, .

[3 -

, .

, . . -

67

. T1(r, t) , :

T1(r, t)

t= a1

(2T1 (r, t)

r2+

1

r

T1 (r, t)

r

)r0 < r < rc, t > 0 (1)

T1(r, t) = T0, r0 < r < rc, t = 0. (2)

,

.

, = 2c21c1

1+2d/2T21+2d/T2

h2 =22T1

11+2d/T2

.

T1(r, t)

r= h1 [T1(r, t) 1] , r = r0, t > 0,

T1(r, t)

r= d(1 + 1/2k)

a1

T1(r, t)

t h3 [T1(r, t) 2] , r = rc, t > 0. (3)

h1 =1T1

-

; h3 = h2(1 + k) -

; d ; T1 , T2 ; 1,2 ; c1, c2 ; a1 = T1/1c1 ;k = d

rc; = r

rc .

(3) T (r, s) =0

T1(r, t)estdt

(2),

T 1(r, s) T0T0 =

(h1a+ h3b)I0

(sa1r

)+ (h1c+ h3d)K0

(sa1r

)s {[qI1(qr0) h1I0(qr0)] [qK1(qrc) (Cs+ h3)K0(qrc)]

c [qK1(qr0) + h1K0(qr0)]} (4)

c = qI1(qr0) + (Cs+ h3)I0(qrc);C = (1 + 0.5k) d

a1; q =

sa1.

-

(

) [4. -

, :

(1)zz (r, t) =T1E11 1

2r2c r20

rcr0

T1(, t)d T1(r, t)+ + 2

22 , (5)

68 .., ..

= T12

R2r2c

Rrc

[T2(, t) Tg]d T12

r2c r20

rcr0

[T1(, t) Tg]d

= 1E1 2

E2; = r

2cR2r2cr20

; = (1 +

r20

r2c

)1E1

+(1 R2

r2c

)(2E2 1

E1

)(1 + R

2

r2c

)1E2; R =

rc+d; T1 , T2 ; E1, E2 ;1, 2 .

(5)

(4),

(1)zz (r, s). :

zz =

{(1)zz (r, t) + ( + 2) (T2 T1) Tg22

}(1 1)

T1E1 (T0 )= Bi2

n=1

ey2nFoAnBn

(6)

An =cn

[cn

Bi1Bi2

+ an (1 + k)]

(Bi21 +m2y2n) c

2n

{[Bi2 (1 + k) k

(1 + k

2

)y2n]2+ y2n

[1 + 2k

(1 + k

2

)]}a2n

,

Bn = bn

[2

(1m2) y2nen J0 (yn)

] an

[2

(1m2) y2nfn Y0 (yn)

] 1 1

E1

+ 2 22

[bn

{2

(1m2) y2nen J0 (yn)

} an

{2

(1m2) y2nfn Y0 (yn)

}],

=T1T2

; en = ynJ1 (yn)mynJ1 (myn); fn = ynY1 (yn)mynY1 (myn);an = mynJ1 (myn) +Bi1J0 (myn); bn = mynY1 (myn) +Bi1Y0 (myn);cn = ynJ1 (yn)

[Bi2 (1 + k) k

(1 + 1

2k)y2n]J0 (yn);

dn = ynY1 (yn)[Bi2 (1 + k) k

(1 + 1

2k)y2n]Y0 (yn);

Bi1 = h1r0;C =

a1d

(1 +

1

2k

);Bi2 = h2rc;

J0(y), J1(y), Y0(y), Y1(y) - , .

yn (n = 1, 2, 3,...) , ,

andn bncn = 0. (7) .

-

. (1)zz (r, s) =0

(1)zz (r, t)estdt -

st, t s .

69

, Iv(z) Kv(z) .

, :

zz = B52m

1m22

Fo+B5

2m2

Bi1(1m2)[1 e(Bi1m

Fo)

2

(Bi1m

Fo

)]

2Bi2 (1 + k) z1rc/z2rc(z1rc)

2 + (z2rc)2

{B5

2

1m2 z2rc2

Fo

[B5

2

1m22z1rcz2rc

(z1rc)2 + (z2rc)

2

B4T2 (1 1)T1E1

z2rc

](1 u

(z2rc

Fo, z1rc

Fo))

z2rc

1

e(

12Fo

)2

(e

(1

2Fo

)2(1 2Fo

) u

(z2rc

Fo,

1 2Fo

+ z1rcFo

))+

+

[B5

2

1m2(z1rc)

2 (z2rc)2(z1rc)

2 + (z2rc)2

B4T2 (1 1)T1E1

z1rc

]v(z2rc

Fo, z1rc

Fo)+

70 .., ..

+ z1rc

1

e(

12Fo

)2v

(z2rc

Fo,

1 2Fo

+ z1rcFo

)}+

m

e(m2Fo

)2

v(z2rc

Fo, z1rc

Fo)+ z1rc

1

e(

12Fo

)2v

(z2rc

Fo,

1 2Fo

+ z1rcFo

)}+

+

m

e(m2Fo

)2 [e

(1m2Fo

)2(1m2Fo

) e

(1m2Fo

+Bi1Fo)2(1m2Fo

+Bi1Fo

)]+

+B4 (T2 T1) (1 1)

T1E1(8)

B4 =+222; u(x, u), v(x, y) -

; (x) .

. 1

zz Fo, 1 = 2 = 25000 /

2 K, m = 0.96, (6) ( ), a 1 , b 2 , 100 -

(8) ( ) d. ,

, (6) (8) . . 2,

, ,

. , -

( = 1) ( = 0). = 1, = 0.98, = 0.96 , .

[1 .., .., .., .., ..,

.. -

. // . 1966. 2. . 42-53.

[2 .., ..

// . 1968. 3. . 15-21.

[3 .., ..

// . 1972. 3. .100-

108.

[4 .. , -

. : . 2002. 259 .

Zhornik A. I., Prokopenko Yu. A. Nonstationary thermoelasti state of two-layered

ylinder. Nonstationary problem of thermoelastiity for a two-layered relatively long ylinder

is onsidered. It is shown that stresses of two types arise. The rst ones are aused by

temperature gradients in eah of the ylinders, the seond by physial and mathematial

dierene of the material onstants of the ylinders, thermal extension oeients espeially.

Solutions for stresses are obtained in two forms suitable for large and short time periods.

..

, ..

. ,

, ,

. , ,

. -

, .

rc, rb, .

T0 - d, , ,

ra < r < rb < rc, z = /2 Fo > 0 -

q ,

q (r, z, t) = q [ (r ra) (r rb)] (z

2

) (Fo Fo) , (1)

(x) ; (r) .

.

, -

,

[1:

T (r, z, t)

r= S

1 + SdS

1

T (r, z, t), r = rc, z > 0, t > 0, (2)

S -; S, .

:

T (r, z, t)

T0=

n=1

k=1

Anke(y2n+x

2k

r2c2

)FoJ0

(yn

r

rc

)(xkBi0

cosxkz

+ sin xk

z

), (3)

72 .., ..

Ank = 4Bi2

{(xkBi0

sin xk cosxk + 1) q(FoFo)

T0V cV

xkJ1(yn)

e

(y2n+x

2k

r2c2

)Fo

(xkBi0

cosxk2

+ sinxk2

)[rbrcJ1

(ynrbrc

) rarcJ1

(ynrarc

)]}

{(

y2n +Bi2)ynJ1 (yn)

[(1 +

x2kBi20

)(1 x

2k

Bi20

)sin 2xk2xk

+2

Bi0sin2 xk

]xk

}1,

xk

ctg x =x2 Bi202xBi0

; (4)

yn

yJ1(y) = BiJ0(y); (5)

J0,1(r) , ; V ; V -; Fo ; Bi =

rc ; =

s1+sd

s

; Bi0 =0

; 0 . ,

, .. -

( ). -

.

ij = Tij +

Pij , (6)

ui = uTi + u

Pi . (7)

(6), (7), -

, . -

, (z = /2) Trz(r, /2, F o) u

Tz (r, /2, F o) ,

. -

, -

Tzz. :

zz

(r,

2, t

)=

Tzz (r, /2, t) (1 )TET0

=

n=1

k=1

Ankyne(y2n+x

2k

r2c2

)Fo(

y2n + x2kr2c2

) {ynJ0(yn rrc

)

J1 (yn)xkrcI0

(xk

r

)I1(xk

rc

) }( xkBi0

cosxk2

+ sinxk2

), (8)

E ; ; T - .

... 73

( =0,8 /, V = 2550 /3, cV = 833 /, E = 69, 58 103 , = 0,23,T = 90107 1/) rc = 2103 = 5102 (S = 50 /) d = 2104 S 4000 /

2. -

.

,

, -

.

, .. , .

[2. . 1

(8) zz (z = /2) - Fo = r

rc,

Bi = 10; Bi0 , rc = 0, 04, q = 0.

(6) (7)

, .

[2

() KI(, Fo), , - KI (, Fo) :

KI (, Fo) =KI (, Fo) (1 )

2rcETT0. (9)

. 2 KI (, Fo) Fo = r/rc , -, . 1.

74 .., ..

KI (, Fo), -, KIC = 0, 17. , = rb/rc = 0, 96 - Fo 0,018 ( a . 2), (. a) = ra/rc = 0, 75( ).

. -

rb, , ra,

(0,75 r/rc 0,96), - Fo = 0, 018. (3) Q = q

V cV

1T0

= 5103 ( ) Q = 0 ( ) . 3 (z/ = 0,5) = r/rc Fo.

-

.

. 4 ( )

.

[3

q = 7,5 /2. , - , Q = q

V cV

1T0

-

, .

Q = 7, 06 1051/T0.

... 75

T0 . [4 - KIC 7105 /3/2. (9) , KIC = 0, 17, T0 = 251 . , , Q 2, 8 107. -

Fo Q = 5103 ( ) Q = 2, 8107 ( ) . 4. , . 4 Q = 2, 8 107 , , . 1 Q = 0, , - ,

, . ,

( ,

..) .

, .

.. -

(BRHE)

-

(CRDF) ( ... 2.22.3.10012).

[1 .., ..

. : , 2003. 143 .

[2 .., ., .., ..

// -

. X ,

--. 2006. .I. . 115-119.

[3 Shand E.J. / Amer. Cer. So. 1961. Vol. 44. P21.

[4 / .. , ..

, .. // . 1991. .17. 2. . 261-267.

Zhornik V.A., Savohka P.A. Frature energy inuene estimate on ring-shaped

propagation in a ylinder under ooling .

Frature energy inuene on ring-shaped propagation in a ylinder under ooling is

estimated. It is shown that frature energy inuene on temperature stresses aused by

ylinder ooling is negligibly small. However, heat soures aused by other reasons shouldn't

be negleted. In this ase results obtained in the present work may be used.

.., ..

, ,

, --

, -

-

, . -

[1, -

. -

. -

,

, -

, .

. [1,

, , -

-

. ,

, -

, -

-

.

[2, , ,

.

. , -

-

, ,

,

d

dt(2u, 4w)=

(20p+20u+zzuq0,2zp+40w+2zzwqz),div0 u+ zw = 0,

(20+ zz

)= 2q, q =

k

zkck, (1)

2dckdt

+ 2 div0 ik + zIk = 0, (ik, Ik) = Dk(0ck + zkck0, zck + zkckz). v = (u, w) , u , p -, q , , ck ,ik, Ik , ,Dk, zk , - , ,

77

, - , d/dt = t+u0+wz, 0= (x, y), 0=xx+yy.

, -

z = h x - 0, z = const E

out

(. . 2). z = 1 - , ,

wz=1= 0, (zu+

20w)z=1= 0, Ik

z=1= 0, z

z=1= 0. (2)

(1) -

(2).

, , (. (5)(7)).

, Dk -

=Ta2

, Dk =DkTa2

, =EaFC ,

2 =h2

a2 1, T = 1

a

FCE , =FEaRT

. (3)

: [x, y] = a, [z] = h,[t] = T , [u, v] = a/T , [w] = h/T , [ck] = C, [E] = E , [] = Ea, [q] = FC, [p] = FCEa2, a, h ; - ; T , C, FC , ; F ; R ; T ;aE x.

. [2,

f(x, y, t) = 12

11f(x, y, z, t)dz, f = ff (1), (2)

{u, w, p, q, ck, } =({um, wm, pm, qm, cmk , m}+ {um, wm, pm, qm, cmk , m})2m.

(1) (2)

2 : q = q0+O(2), =0+O(2), ck = c

0k+O(

2), 0 = 0, c0k = 0, q0 = 0,

u0, w0 c1k. ,

, O(2)

2d0u

dt+ 0(U U) = 20p+ 20u U , div0 u = 0, U = q0,

0 = q, q =k

zkck,d0dt

= t + u 0, = 2

45, k =

4

945Dk, (4)

d0ckdt

k2 div0(U(U 0ck)) + div0 ik = 0, ik = Dk(0ck + zkck0).

,

O(2), , , , , Dk,. . . , .

(4)

(. . 2). , y = 0 y = Y -

78 .., ..

out

. ,

z = const (. . 2)

(n )y=0, Y

= E0, E0 = out(n Eout). (5)

x = 0, X y = 0, Y , y = 0, Y , x = 0, X 0

u nx=0,X

= 0, u ny=0, Y

= 0, ik ny=0, Y

= 0, x=0

= 0, x=X

= 0. (6)

, ( )

u y=0, Y

= R30 y=0, Y

, u x=0,X

= R30 x=0,X

, (7)

n, , R3 , (. (9)).

, (7) -

. R3 -, ,

y = 0, Y , - (4)(6) u = (u(y), 0), ck = ck(y) = cBke

zk(y), = (y) + Ex, E

,

cBk . , -

(, H+ OH ,cB1 = cB2 = cB, z1 = 1, z2 = 1) , R3 E30 - R3 (. (9)). , 0(U U).

. -

(4), (q = 0) -, .

U ( ).

u = (u, v) - D = {0 6 x 6 X, 0 6 y 6 Y }

tu+ u 0u = 0p+ 0u, div0 u = 0, 0 = 0, (8)(u, v)

x=0,X

=0, (u+R3x, v)y=0, Y

= 0, x=0

= 0, x=X

= 0, yy=0, Y

= E0.

R3 (9) R3 = R3(E0) E30 . , -

, xy=0, Y

, , , uy=0, Y

, -

0, E0, X, Y . y = 0, Y , u . , , -

, ,

(. . 2). [1,

79

1

0.5

0 0.5 1

0.3

0.8

0.55

1

0.5

0 0.5 1

0.015

0.009

0.0

03

1

0.5

0 0.5 1

0.015

0.0

03

0.009

0.019

. 1. (x, y, t) (x, y, t) t = 10 (0.78 c), 30 (2.34 c)

, -

Eout

0. . (8) -

. -

FreeFem++ [3 -

. .

[1,

: 0 = 20, a = 102, E

out

= 30000/, = 106 2/, 0 = 8.85 1012/( ), = 78.3 0, out = 1.0 0, = 103 /3,C = cB = 104 /3, F = 9.65 104/, R = 8.3/( ), T = 293. h = 0.29102 E = 2000/, T = 7.8 102 , - a/T = 0.128/ , R3(a/T ) = 3 102/.

(8), (. (3))

a

T R3 FcB

E313523

(2E2RTcB

)1/2E30h

4, E0 =out

Eout

E , 0 =0aE , (9)

E0 = 0.19, 0 = 1.0, = 0.29, = 7.8 104, R3 = 0.235, X = 1.0, Y = 1.0. 2 0.09 , , , - (4). ,

aR3/T h4, . . .

. 1 -

, .

-

.

(2.5 c)., -

0, E0, X, Y , . 2 0 = 0.1; E0 = 0.19; X = 1; Y = 1. t 200. - . 1, 2 ,

, .

80 .., ..

1

0.5

0 0.5 1

-0.11

-0.10

0.01

0

-0.08

-0.03

1

0.5

0 0.5 1

0

0

0.0

002

0.0

002

0.0024

0.0024

0.0

014

0.0

014

0.0

018

0.0

018

Eout

u > 0

u < 0yx

z=+1

z=1

= 0

= 0

X

Y

. 2. (x, y) () (x, y, t) t = 200 (15.6 c)

(. . 1) ,

[1, . -

, -

2 .

( 3 /), , . ,

, (9), -

, [1 ,

, .

, , ,

, , [1 -

.

, 07-01-00389, 07-01-

92213-, INTAS 04-80-7297 CRDF RUM1-2842-RO-06.

[1 Amjadi A., Shirsavar R., Radja N.H., Ejtehadi M.R. A Liquid Film Motor //

arXiv:ond-mat/0805.0490v2. 2008. 9 p.

[2 Oddy M.H., Santiago J.G.Multiple-speies model for eletrokineti instability // Phys.

Fluids. 2005. 17. P. 064108.1064108.17.

[3 .., .. FreeFem

, . /: . ,

2008. 256 .

Zhukov M.Yu., Shiryaeva E.V. EHD Flow in a Thin Liquid Film. A Liquid Film

Motor . The mathematial model desribing a rotation EHD ow in a thin suspended liquid

lm under ation of an external eletri eld is onstruted and investigated by numerial

methods. For the rst time similar ow was experimentally observed in [1 where deteted

eet is named A Liquid Film Motor. The model presented by us desribes EHD ow within

the lassial framework. The depth-average proedure allows to obtain 2D model and to show,

that the main ontribution in the tangent veloity make the Reynolds average stress.

..

, ..

, ..

,

-

, ,

, -

.

. , -

.

[1, 2 -

z 0. V1, - x0y, , - q(x, y). V2, V2 z = 0, . .

.

x, y, ...yz, - u, v, w. :

z(x, y, 0) = q(x, y), (x, y) V1; z(x, y, 0) = kw(x, y, 0), (x, y) V2;xz(x, y, 0) = yz(x, y, 0) = 0, (x, y) V1 + V2. (1)

(1) [1

z(x, y, z) =1

2

V1

(, ){3z(x )2

51+1 221

[z(y )2

21+(x )2 (y )2

1 + z]+

+

2

0

[(1 + 2 tz)J0(t) (x )2 (y )22

(1 2 zt)J2(t)]etz tdtt+

}dd,.........................................................................................................................

z(x, y, z) =1

2

V1

(, ){3z3

51+

0

(1 + tz)J0(t)etz tdt

t+ }dd,

.........................................................................................................................

yz(x, y, z) =1

2

V1

(, )(y ){3z51

+z

0

J1(t)etz t

2dt

t+ }dd (2)

82 .., .., ..

, ,

u(x, y, z) = 1 + 2E

V1

(, )(x )

{ 1 21(1 + z)

z31

0

(1 2 tz)J1(t)etz dtt+

}dd,

v(x, y, z) = 1 + 2E

V1

(, )(y )

{ 1 21(1 + z)

z31

0

(1 2 tz)J1(t)etz dtt+

}dd,

w(x, y, z) =1 +

2E

V1

(, )

{2(1 )1

+z2

31

0

(2 2 + tz)J0(t)etz dtt+

}dd. (3)

(2)-(3) , 1

=(x )2 + (y )2, 1 =

(x )2 + (y )2 + z2.

[3

(x, y) = q(x, y) +

V1

(, )G(x , y )dd, (x, y) V1, (4)

G(x, y) =

2

0

J0(rt)tdt

t+ , r2 = x2 + y2. (5)

-

,

. (1)

R,, z. , z - , ,

V1, R,, q = q(R,). :

z(R,, 0) = q(R,), (R,) V1; z(R,, 0) = kw(R,), (R,) V2;Rz(R,, 0) = z(R,, 0) = 0, (R,) V1 + V2. (6)

x = R cos, y = R sin, z = z. (7)

83

R = x cos2 + y sin

2 + 2xy cos sin,

= x sin2 + y cos

2 + 2xy cos sin, z = z,

R = (y x) sin cos+ xy(cos2 sin2 ),Rz = xz cos+ yz sin, z = xz sin+ yz cos. (8)

x, y, ...yz (2) x, y R, (7), ,

= R1 cos1, = R1 sin1, (R1, 1) V1 (9)

. 1.

(3), -

w(R,, z) . ,

.

(. 1),

= |u2 + v2| cos, (10)

+, u v , , u v .

tg =u

v. (11)

, (10)

= |u2 + v2| cos( ) (12)

84 .., .., ..

(10) -

cos sin. (3) u v (10)-(12) x, y, , (7),(9).

, (8).

, (8),

- , -

, : -

.

z(R, 0) = q0(R), R a; z(R, 0) = kw(R), R a;Rz(R, 0) = 0, 0 R . (13)

v , uR w .

uR = 1 + E

a0

(R1)T1(R R1, z)(R R1)R1dR1 (14)

T1, (14),

T1(R, z) =1 2

(RR1)2 + z2((R R1)2 + z2 + z)

z((R R1)2 + z2) 32

|R R1| 0

(1 2 tz)etzJ1(|R R1|t) dtt+

. (15)

, 1

= |R R1|, 1 =(RR1)2 + z2. (16)

(3) (16)

w = 1 + E

0

(R1){ 2(1 )(R R1)2 + z2

+z2

((RR1)2 + z2) 32+

+

0

(2 2 tz)etzJ1(|R R1|t) dtt+

}R1dR1. (17)

, , -

, , -

.

(8).

:

R(R, z) =

a0

(R1){3z(R R1)2

51+

1 21(1 + z)

+

85

+

2

0

{(1 + 2 tz)J0(|RR1|t) (1 2 zt)J2(|R R1|t)}etz tdtt+

}R1dR1,

(R, z) =

a0

(R1){1 21

{ z21 11 + z

}+ 2

0

{(1 + 2 tz)J0(|R R1|t)+

+(1 2 zt)J2(|RR1|t)}etz tdtt+

}R1dR1,

z(R, z) =

a0

(R1){3z3

51+

0

{(1 + tz)J0(|RR1|t)etz tdtt+

}R1dR1,

Rz(R, z) =

a0

(R1)|RR1|z{3z51

+

|R R1| 0

t2etz

t+ }J1(|RR1|t)dt}R1dR1.

(18)

(R)

(R) = q0(R) +

a0

(R1)G(|R R1|)R1dR1, R < a, (19)

G(R) =

0

J0(Rt)tdt

t+ .

[1 .. -

// . .14. 2007. . 7482.

[2 .., .., .. -

// . -

. - --, : " 2006. - . 120124.

[3 .., ...

// . - 1980. - .251. 6.- . 1338

1341.

Zaletov V.V., Storoshev V.I., Hapilova N.S. Solution of the mixed problem of

elastiity theory for isotropi half-spae in ylindrial oordinate system. The analytial

solution of the mixed problem about the deformation of isotropi half-spae is reeived in the

ase, when on the boundary tangent stresses are absent, in nite domain of the border plane

the distributed load ats, outside of its normal stresses and displaements are proportionately.

The problem is solved in ylindrial oordinate system. The private ase is investigated when

in irular domain distributed load does not depend on angular oordinate.

,

..

,

, -

, , . -

,

.

-

y > 0, |x|

87

k - .

-

:

x = 11x + 12y,

y = 12x + 22y, (3)

xy = 66xy.

ij

11 =1 1331

E1, 22 =

1 2332E2

=1

E2 1331

E1, (4)

12 = 12 + 1331E1

= 21 + 2332E2

, 66 =1

G3.

C (1),(3) [2]:

224

x4+ (212 + 66)

4

x2y2+ 11

4

y4= 0, (5)

-:

x =2

y2, y =

2

x2, xy =

2

xy. (6)

[1,4, -

(1),(2),(3) :

x(x, y) =

V

Gx(x , y)()d,

y(x, y) =

V

Gy(x , y)()d,

xy(x, y) =

V

Gxy(x , y)()d, (7)

u(x, y) =

V

Gu(x , y)()d,

v(x, y) =

V

Gv(x , y)()d.

(x)

(x) = p(x) +

V

()G(x )d, (8)

88 ..

Gx(x, y) =1

r1r2r1 r2

[r21y

x2 + r21y2 r

22y

x2 + r22y2+

+(r1Ei(r1y ix) + r2Ei(r2y ix)

)],

Gy(x, y) =1

r1r2r1 r2

[y

x2 + r21y2 yx2 + r22y

2+

+(1

r1Ei(r1y ix) 1

r2Ei(r2y ix)

)], (9)

Gxy(x, y) =1

r1r2r1 r2

[ xx2 + r21y

2 xx2 + r22y

2+

+(Ei(r1y ix) + Ei(r2y ix)

)],

Gu(x, y) =1

r1r2r1 r2

[(11r1 12

r1

)Ei(r1y ix)

+

(11r2 12

r2

)Ei(r2y ix)

],

Gv(x, y) =1

r1r2r1 r2

[(12r1 22

r21

)Ei(r1y ix)

+

(12r2 22

r22

)Ei(r2y ix)

],

G(x) =

(2sin(x) Si(x) sin(x) Ci(x) cos(x)

).

Si(x) - , Ci(x) - ,

= k22r1 + r2r1r2

, Ei(x) = Ei(x)ex, Ei(x) =

x0

ex

xdx,

r1, r2 =

(212 + 66)

(212 + 66)2 41122211

.

.2 y(x, y) -. , -

. E1 = 1104, E2 = 0, 5104,12 = 13 = 31 = 0, 2, 21 = 0, 1, G3 = 0, 1 104, k = 0, 05 104/l, p(x) = P0.

(7)-(9) .

, ,

. -

, - -

.

89

. 2. y(x, y)/P0 = const

[1 .., .. -

// . 2000. 5. . 165172.

[2 .. . .: , 1977. 415 .

[3 ../ . : . ,

1992. 232 .

[4 .. - -

, ,

// . 2004. 9. . 7680.

[5 .. - ,

,

//

. 2008. 16. . 8892.

[6 .., .. -

. // .-. . 1977.

5. . 4853.

Zenhenkov A.V. The mixed problem of the theory of elastiity for an isotropi half-plane

lying on the elasti foundation, by ation on boundary of distributed load .

With the help of an integral Fourier transformation the mixed problem of the theory of

elastiity for the orthotropi half-plane lying on the punhed elasti foundation is solved in a

ase when on a nal segment of boundary the normal load is applied. The analytial formulas

for omponents of stresses, whih at in an elasti half-plane, are reeived.

.., ..

, --

.

, .

, .

-

, -

. -

.

[1,

. , [1,

,

.

,

, -

.

, , - , x3= b/f(t), f 2- , , b . s = (cos, 0, sin).

dv

dt= q +v, div v = 0,

dT

dt= Pr1T,

dC

dt= Sc1(C T ), (1)

w = + s, divw = 0. x3=(x1, x2, t)

(v, ) =

t, =

( x1

, x2

, 1

),

iknk (q +Ga Re2

(w2

2+

z(w, )

))ni = (2)

= 2K

(Cp Ma1

PrT Ma2

ScC

)ni +

Ma1Pr

T

xi+Ma2Sc

C

xi, i = 1, 2, 3,

T

n Bi1T = 11, C

n Bi2C = 12, = 0.

91

x3 = 1

T

x3+B01T = 21,

C

x3+B02T = 22, wn = 0. (3)

v, q, T , C , , , - . : Pr, Sc, Ga, Ma1,Ma2, Bi1, Bi2, Cp, , , , , ,

Re2 =b2h2

2f 2.

.

(1)(3)

v0 = 0, T0 = z, C0 = z, q0 =Re2

2cos2 ,

w0 = (cos, 0, 0), 0 = x3 sin, 0 = 0.(4)

(4) -

Ma1(,Ma2) : Cr = (PrC)1 = 0.01, Pr = 0.01, Sc = 10,Bi1 = Bi2 = 0, B01 = B02 = 0, Ga = 0, (Re sin)2/C = 1. , -

Ma2 < 0 , Ma2 > 0 (..1, .2). -

.

.1

-

. ,

92 .., ..

.2

[2. [3.

, ,

[2, [4. , -

.

, -

.

[5.

.

Lv = L2v, Pr = L v, ScC = L(C SrT ) SrLe1v,z = 0 : v = Pr, D2v + 2v =

Ma

2

( + C + (1 + Sr)),

Cr((32 + )Dv D3v) = 2 (2 +BO + th) ,D Bi( + ) = 0, DC SrDT = 0,

(5)

z = 1 : v = Dv = 0, D +B0 = 0, DC SrDT = 0. Sr = 2

1 ,

Le =Pr

Sc . (5)

. , Sr > 0 - , . . , ,

Sr , , = 0, Sr = 0. , - Cr = 0.033, Pr = 0.01, Sc = 10,Bi1 = 0, 1, Bi2 = 0, B01 = B02 = 0, Ga = 14.848, .1 .3.

93

Ma

Sr = 0 Sr = 0.01 Sr = 0.01 Sr = 0.015 Sr = 0.015 = 0 = 0 = 2.247 = 0 = 24.023

5,00 206,71 18,54 18,55 12,74 12,75

4,00 137,72 12,50 12,51 8,60 8,61

3,00 86,66 8,28 8,31 5,70 5,73

2,00 49,59 5,69 5,78 3,94 4,02

1,00 18,92 4,04 4,44 2,90 3,22

0,80 13,58 3,65 4,23 2,68 3,14

0,60 8,94 3,16 3,97 2,39 3,08

0,40 5,33 2,53 3,62 2,00 3,04

0,20 3,17 1,90 3,14 1,58 3,01

0,15 3,01 1,84 3,04 1,54 3,01

0,12 3,09 1,87 3,01 1,56 3,01

0,10 3,31 1,95 3,02 1,61 3,01

0,05 6,12 2,67 3,31 2,09 3,02

0,01 100,90 4,58 4,59 3,10 3,13

0,001 9880,73 4,80 4,80 3,20 3,20

.1

, Sr = 0.01 = 2.247, Sr = 0.015, = 24.023. , 0 Ma =

0.048

Sr+O(2), .

.3

94 .., ..

. , -

, -

.

( 07-0100099- 07-01-92213-

- " ").

[1 .., .. -

// . 2002. .

66. . 573-583.

[2 .., . ., .. .

: . 2000. 280 .

[3 .., .. -

. //

VII " ". . 2000. . 248

261.

[4 . ., .., ..

. .: , 1989. 318 .

[5 Gershuni G. Z., Kolesnikov A.K., Legros J. C., Myznikova B. I. On the vibrational

onvetive instability of a horizontal binarymixture layer with Soret eet. // J. Fluid

Meh. 1997 Vol. 330, P. 251269.

Zenkovskaya S.M., Shleykel A.L. On the inuene of an admixture and

thermodiusion on vibration Marangoni onvetion. Double-diusive Marangoni onvetion

in a binary mixture layer under the ation of high-frequeny vibration is investigated on

the basis of the analysis averaged problems. The ase when thermodiusion is take into

a

ount is onsidered also. Numerially and asymptotially spetral problems of stability are

investigated, neutral urves are plotted.

,

.., .., ..

. ..

. .

, .

() , , -

: -

(). -

,

, , -

[1, 2.

- [2, 3, 4, 5.

-

-,

[1. -

[1.

[2, 3, 4:

y = q p, y(nt) =n

k=0

nk(t)g(kt), n = 0, 1, ..., N,

n(t) =Rn

L

L1l=0

q((Reil2L1)/t)einl2L

1,

kxn+k+k1xn+k1+...+0xn = t[k(sxn+k+g((n+k)t))+...+0(sxn+g(nt))],

(z) =0p

k + ...+ k0pk + ...+ k

, x(t, p) =

t0

ep(t)g()d

q - q. , k (k = 1, 2, ..., L),

0 = 0, L = 2, :

0 =Rn

2

Lk=1

k+1k

Re[f()]d, n =Rn

2

Lk=1

k+1k

[f()]eind t > 0.

96 .., .., ..

a = k, b = k+1 , n(b a) >> 1, f() - . m m >> n(b a)/ [6:

ba

Lmeind = Snm(f) =

b a2

einb+a2

mj=1

Dj

(nb a2

)f(j),

Dj(w) =

11

(k 6=j

dkdj dk

)eiwd, j =

b+ a

2+b a2

dj, j = 1, ..., m, w = nb a2

.

Rm(f) =

ba

(f() Lm())einkd,

Rm(f) b

a

|f() Lm()|d D(d1, ..., dm)(max[a,b]

|f (m)()|)(

b a2

)m+1.

-

,

-.

p = 1/2 - , . -

: = 7850/3, = 0, E = 2, 11 1011H/2. (p = 1, = 0, 5, = 0, E = 1). - : - 126 .

. 1 , N = 500, L = 500 0 2 t = 0, 01. 1 , - , 2 - , -

f(), 3 - , f(). , ,

.

. 2 ( ) , -

: 0 /2 125 , /2 3/4 - 20 , 3/4 2 -125 . - L=270, N=500, t=0,01. , . ,

, ,

, e 97

. 1.

. 2.

. 2.

(L=270) 1 . 1 (L=500).

,

. , , N , - 2L, - , ( -

N > L). .3 - 0 /2 - 70 , /2 3/2 - 20 , 3/2 2 - 70 .

, -

. -

.

98 .., .., ..

. 3.

[1 , .., , .. -

- .: , 2008. - 352.

[2 Shanz, M. Wave Propogation in Visoelasti and Poroelasti Continua.// Berlin

Springer, 2001. - 170 p.

[3 Lubih, C. Convolution quadrature and disretized operational alulus. I // Numerishe

Mathematik. - 1988. - 52. - P. 129-145.

[4 Lubih, C. Convolution quadrature and disretized operational alulus. II //

Numerishe Mathematik. - 1988. - V. 52. - P. 413-425.

[5 Shanz, M., Steinbah O. Boundary element analysis - Berlin Springer, 2007. - 354 p.

[6 , .., , .., .. - .: -

, 2001. - 632 .

Igumnov L.A., Litvinhuk S.U., Markin I.P. Convoation quadrature method, Durbin

method and boundary element method in dynami problems of elasti bodies. A sheme of a

boundary element method is given in ombination with a onvoation quadrature method. The

modiations of the onvoation quadrature method are onsidered. The results of numerial

experiments showing the advantages of the obtained modiations are presented.

.., ..

-

-

.

-

. .. , .. -

. , ,

. -

. ,

.

[1 -

-

, :

:

u(2)i ,

(2)i :

(+ )u(2)1 + 2(2(2)3 3(2)2 ) = 0,

(+ 2)22u(2)2 + (+ )33u(2)2 + (+ )23u(2)3 + 23(2)1 = (1)3 ,

(+ 2)33u(2)3 + (+ )22u(2)3 + (+ )23u(2)2 22(2)1 = (1)2 ,

( + )(2)1 4(2)1 + 2(2u(2)3 3u(2)2 ) = 0, (1)

( + 2)22(2)2 + ( + )33(2)2 + ( + )23(2)3 4(2)2 + 23u(2)1 =

= 2((1)3 + x2

(1)1 ),

( + 2)33(2)3 + ( + )22(2)3 + ( + )23(2)2 4(2)3 22u(2)1 =

100 .., ..

= 2((1)2 x3(1)1 ), :

(+ )

nu(2)1 + 2e1n

(2) = ( ){n(1) + (1)1 e1xn},

n2{(+ 2)2u(2)2 + 3u(2)3 }+ n3{(+ )3u(2)2 + ( )2u(2)3 + 2(2)1 } =

= n2((1)1 + e1(1) x),

n3{(+ 2)3u(2)3 + 2u(2)2 }+ n2{(+ )2u(2)3 + ( )3u(2)2 2(2)1 } =

= n3((1)1 + e1(1) x),

( + )

n(2)1 = ( )n(1) , (2)

n2{( + 2)2(2)2 + 3(2)3 }+ n3{( + )3(2)2 + ( )2(2)3 } = n2(1)1 ,

n3{( + 2)3(2)3 + 2(2)2 }+ n2{( + )2(2)3 + ( )3(2)2 } = n3(1)1 .

:

u(2)2 = u

(2)2

(1)1

2(+ )x2 (2)1 x3 +

(1)3

+

(

+ 2x22 +

2

x23

)+

+( ) +

(1)2 x2x3 +

(1) v

(2) ,

u(2)3 = u

(2)3

(1)1

2(+ )x3 +

(2)1 x2

(1)2

+

(

+ 2x23 +

2

x22

)

( ) +

(1)3 x2x3 +

(1) v

(3) ,

u(2)1 = u

(2)1 x(1) + k(1)1 v(1)1 , (2)1 = (2)1

+

x(1) +

(1)

(3) ,

(2)2 = (1)3

(1)1

2( + )x2 +

(1)1

(2)1 ,

(2)3 =

(1)2

(1)1

2( + )x3 +

(1)1

(3)1 . (3)

d

ds~F + ~ ~F = 0, d

ds~M + ~ ~M + ~e ~F = 0, (4)

... 101

[2

Fi =

1id, Mi =

{eikxk1k + 1i}d ; (5)

d

ds s

.

-

(1) (5), (5) -

.

(3) v(j)i ,

(j)i

.

-

[3,4. v(j)i ,

(j)i

.

v(j)i ,

(j)i , (3).

(1)i ,

2

+ + (+ 2)22(2)3 + (+ )

+

+ (+ )23(2)3 (+ ) +

+

+ (+ )23(3)3 2 +

+ 23(3)3 = 0,

(+ 2)22(2)2 + (+ )23(2)2 + (+ )23(3)2 + 23(3)2 = 0,

(+ ) (1)1 + 2

(2(3)1 +3(2)1

)= 0,

2 +

+ (+ 2)23(3)2 (+ ) +

+ (+ )22(3)2 (+ ) +

+

+ (+ )23(2)2 + 2 +

22(3)2 + = 0,

(+ 2)23(3)3 + (+ )22(3)3 + (+ )23(2)3 + 23(3)3 = 0,

( + ) (3)2 + 2

(2(3)2 3(2)2 2(3)2

)= 0,

( + ) (3)3 + 2

(2(3)3 3(2)3 2(3)3

)= 0, (6)

102 .., ..

( + 2)22(2)1 + ( + )23(2)1 + ( + )23(3)1 +

+

(2

+ 1)x2 4(2)1 + 23(1)1 = 0,

( + 2)23(3)1 + ( + )22(3)1 + ( + )23(2)1 +

+

(2

+ 1)x3 4(3)1 22(1)1 = 0.

v(j)i ,

(j)i :

n2(2)2 + n3

(3)2 = 0, n22(3)2 n33(3)2 = 0, n22(3)3 + n33(3)3 = 0,

n2 ( + 2)2(2)1 + n3 ( + )3(2)1 = 0, ( )n2(3)3 ( + )n3(3)2 = 0, (7)

n2 ( )2(3)1 + n33(3)1 = 0. (6) v

(j)i ,

(j)i , -

-

v(j)i ,

(j)i

.

(5), Mi-:

M1 = B11+A11, M2 = B222+B233+A23, M3 = B312+B333+A32, (8)

Ai (9) -, ,

-. B23 B31, B23 = B31 = 0 - - . ,

-

i, . , - (Ai = 0) (8) , .

Ai , A1 . , -

( ) , , -

M1. A2 A3:

... 103

A2 =

( )2(3)3 d, A3 =

( + )3(3)2 d,

:

A2 A3 =

(( )2(3)3 ( + )3(3)2

)d = 0. (9)

.

A2 = A3 = A. (10)

(10) , - -

.

(8) Mi - - .

:

M1 = B11, M2 = B22 + A3, M3 = B33 + A2, (11)

B1 = B1 + A1. (11) -

, -

, -

, .

[1 .., .. //

. - . . 2001. . . 9294.

[2 .. .

: , 1979. 216 .

[3 .. . .

1980. 6. C. 111117.

[4 .. // .

. TT. 1994. 3. C. 181190.

Ilyukhin A.A., Timoshenko D.V. In addition to results of work of A.A. Ilyukhin,

N.N. Shhepin To moment theories of elasti rods are reeived losing parities for system of

Kirhho equations. Kinemati parameters with whih it is neessary to involve that together

with system of dierential Kirhho equations to reeive the losed system are speied. Other

geometrial sizes are found from parities dening them. Conditions with whih should satisfy

fators in losing parities are reeived. For the one-dimensional theory the deision at presene

is speied to symmetry. The reeived results were interpreted within the limits

of the mehanial approah to denition of ongurations of moleules of DNA.

,

.., ..

, . --

, -

, . ,

.

P (t) .

P - , . t = 0 (- ) v. .

, -

[1. w(r, t) -,

r = 0, -

w +k

Dw +

h

D

2w

t= q(r, t)

= /r2 + (1/r)/r , D = Eh3/12(1 2) - , E , , a , h - , k , h - , Q = ha2 , P ,q(r, t) . , -

, , . q(r, t) = 0.

w(a, t) = 0;w

r(a, t) = 0. (1)

t = 0

w(r, 0) = 0;w

t(r, 0) = 0; r 6= 0. (2)

r = 0 t = 0 Z = (t). t

F = P Pg

2w

t2+ Z (3)

105

= r/a (1)(4),

(

2+

1

)(2w

2+

1

w

) + (L4 +

p2

v2)w = 0, (4)

a4

DF =

a4

D(P P

gp2w + Zp), (5)

L4 = a4k/D, v2 = gD/ha4. ,

, , ( = 0). (6) (a4/D)G1(r, 0, p). F (6) :

G(, p) =a4

D[P + Zp P

gp2G(0, p)]G1(, 0, p) (6)

= 0, G(0, p), (7)

G(, p) =a4

D

(P + Zp)G1(, 0, p)

1 + (a4P/gD)p2G1(0, 0, p)(7)

G(, p) pk =ivk. G1(0, 0,iv) = G(0, 0, ), k

1 Pa2

Q2G(0, 0, ) = 0 (8)

(8) -

w(, t) = w

+a4

D

k=1

(P + Zp)G1(, 0, pk)epkt

[pk + (P/g)(a4/D)p3kG1(0, 0, pk)]pk

, (9)

w

= (Pa4/D)G(, 0, L). pk ikv -

w(, t) = w

a4

D

i=1

(P cos vit Zvi sin vit)G(, 0, i)0.5[i (Pa2/Q)3iG(0, 0, i)]i

. (10)

(5), 2 > L4. :

w(, ) = A1J0(u) + A2I0(u) + A3N0(u) + A4K0(u), (11)

u = 42 L4. J0(z) , I0(z)

, N0(z) , K0(z) -

, H(1)n (z) K n(z) =

12ie

1

2niH

(1)n (iz).

106 .., ..

(12), G(, 0, ) = (a2/4u2D)K(, 0, u), , = 0,

K(0, 0, u) =(4/u) + I1(u)N0(u) + I0(u)N1(u) + (2/)[K0(u)J1(u)K1(u)J0(u)]

2[J0(u)I1(u) + J1(u)I0(u)]

G(0, 0, ) K(0, 0, u), (9) - i

1 P4Q

2

u2K(0, 0, u) = 0 (12)

(11) , .. = 0,

w(0, t) =Pa2

4L2DK(0, 0, L) a

2

4D

4Q

P

k=1

u2k2kD(uk)

[P cos vkt Zvk sin vkt], (13)

D(uk) = u2k

2k

2(u2k)+

P4k

16Qu3k

K uk(0, 0, uk).

K u(0, 0, u), - [2,

K u(0, 0, u) =2

u

[I0(u) J0(u)]2[J0(u)I1(u) + I0(u)J1(u)]2

u :

Jp(u) = [Pp(u) cos(u 2p+14 )Qp(u) sin(u 2p+14 )]

2u,

Np(u) = [Pp(u) sin(u 2p+14 ) +Qp(u) cos(u 2p+14 )]

2u,

Ip(u) =e42u

Sp(2u),ip+1H

(1)p (iu) =

2ueuSp(2u),

Pp(u) = 1 (4p21)(4p29)2!(8u)2

+ ...

Qp(u) =4p218u

(4p21)(4p29)(4p225)3!(8u)3

+ ...

Sp(u) = 1 +4p211!4u

+ (4p21)(4p29)2!(4u)2

+ ...

(13) - :

1 P2

8Qu2cos u+ U1 sin(u 4 ) + U2 cos(u 4 )sin u+ U2 sin(u 4 ) U1 cos(u 4 )

= 0, (14)

U1 =12[ 964u2

+ 39256u3

+ 30032768u4

+ ...], U2 =12[ 14u

+ 964u2

30032768u4

+ ...].

, D(u)

K u(0, 0, u) =1

2

1 + 14u

+ 532u2

+ 21128u3

+ ...

[sin u+ U2 sin(u 4 ) U1 cos(u 4 )]2.

107

, (15) ,

w(0, t) . , ,

s w(14):

s = + w. (15)

s

mgd2s

dt2= P (t), (16)

m , P (t) . (16), (s|t=0 = 0,dsdt|t=0 = v0),

s = v0t 1mg

t0

t10

P (t2)dt2dt1. (17)

(16) (17),

..:

v0t 1mg

t0

t10

P (t2)dt2dt1 = (t) +

t0

P (t1)wdt1. (18)

, :

[3.

= kP2

3, k = ( 9

161

r12

)2

3, ,

r . :

- = bP2

3 ;P < supP ()|t < P1- = (1 + )cP

1

2 + (1 )Pd; supP ()|t >P1; supP ()|t = P

- = b

P2

3 + p[sup()|t]; supP ()|t > P1; supP ()|t > P P1 =

3[3r/4E]2, = k, k = 2, E = E1E2[(1 21)E2 + (1 22)E1]1,

b = (1/r)1

3 (3/4E)2

3, c = 3

1

2E1

8, d = 0.5(r)1, b

= (1/R

)1

3 (3/4E)2

3,

p = (1 )Pmax(2Rp)1, R = (e Pmax )1, e = 0.75P1

21

2E1,Rp = (r

1 e Pmax )1,r , E , E1 E2 -

, 1 2 , k - , = 5.7 , -, = 0.33 , Pmax , - , P1 - . . 1 -

-

.

108 .., ..

-

1 = 7800 /3, 2 = 8900 /

3 , 1 = 0.28,

2 = 0.32, E1 = 2.18e11 , E2 = 1.2e11 , k = 1.2e8 , = 0.33, a = 1 - ,m = 3 , r = 0.045 .

. 1

. 1 , -

, ,

, , -

, . ,

. P (t) P (t) . -

[4.

[1 .. . .: , 1970.

736 .

[2 ., . ,

. .: , 1979. 832 .

[3 .., .., .. -

. // . 1984. 1. .16-26.

[4 .., .. .

.: , 1977. 240 .

Kadomev I.G., Lapin A.G. Elastioplasti impat of the massive body above the round

plate, situated on the foundation. The problem of the impat of the massive body above

the round plate, rigidly xed over the ontour, situated on the foundation, is investigated.

The Green's funtion is derived, the problem of elasti and inelasti impat is solved. Time

dependene graphs of ontat interation fore P (t) is onstrutid.

..

, ..

, ..

,

..

. .. , --

. .. ,

(),

.

, ,

, , ,

- , .

(, -

, , ) ()

(), -

, - () -

- ,

, ,

, -

.

-

[1. ,

,

,

, - -

, [2.

-

, , , ,

,

-

.

-

, -

( ) -

(. 1), -

30 , - -

.

110 .., .., .., ..

3 -

- . ,

7/4 -

- -

. , ,

- ()

[3. , ,

- .

(sinXX

)4, X = N

(fFiFi

), N .

. 2 . , -

30 .

. 1.

. 2. : 1

; 2

- -

. . 3 , -

. .

, . ,

, -

.

,

, -

111

. 3. ,

. , 30 ,

.. 60 , ,

.

- -

, -

(. 4).

-

, .

, -

- ,

- . -

- -

,

, -

(

- ),

- .

. 4.

-

, , ,

Nk2 >> 1, - , (k2

112 .., .., .., ..

). , -

, .

. ,

.

.

-

,

.

. 5.

. 6. (1),

(2) (3)

, -

(. 5). -

, (14-20

), n ( - ). , 5-6

n = 10. ,

113

,

-

. . 6 ,

,

- ( 1). 2

, - -

, 3 - .

, -

.

, , ( )

,

, -

.

07-08-00583.

[1 .., ..

// . 2006. 5. C. 5354.

[2 .., ., .., .. -

,

// XI

" . 2007 .

[3 .., .., .., ..

. .. 1699327 15.08. 1991 .

Karapetyan G. Ya, Dneprovski V.G., Bagdasarian A.S., Bagdasarian S.A. The

physial value sensor on the base of narrow band lters blok and delay lines on surfae aousti

waves for remote monitoring .

The problem of the designing the sensor on surfae aousti wave (SAW) is onsidered, in

whih data about measured physial value will not depend on distanes and mutual loation

of the antennas of the sensor and reader. This o

urs due to the reeiving from sensor not

one reeted pulse, but two and more, value whih, exept one, depends on values of the

load, due to the installation additional reetive interdigital transduers, adjusted on dierent

frequenies.

..

,

, -

. -

.

.

[13 -

-

, [4 -

. [5 [13 -

.

a b h . .

, , -

.

-

[2, 5

k02w

t2+D2w +

h

(a2 b2)

20

ab

w rdrd + n=1

m=0

nm (r, )

snm20

ab

wnm (r, ) rdrd = q(r, , t). (1)

w|r = a, b

= 0,w

r

r = a, b

= 0.

w(r, , t) ; k0 = 1h1; 1, h1 D -, ; q(r, , t) , ; nm(r, ) knm -

; nm(r, ) = Zm(knmr)(nm cosm + nm sinm), knma = nm,

115

Zm(knmr) = Jm(knmr) + nmYm(knmr), nm = J m(nm)/Y m(nm); nm n- - J

m()Y

m() J m()Y m() = 0 ( = b/a), = 0 (nm = 0)

Jm () = 0; nm -

-

r ; snm = 2 tanh(nm){a2[(1