Asymptotic solutions of non-stationary problems of thermal conductivity and thermoelasticity with nonclassical boundary conditions for the two-layer plates

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68, №4, 2015

536.21; 539.3

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Keywords: The non-classical boundary value problem, asymptotic method, the internal task.

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Gevorgyan R.S.

Asymptotic solutions of non-stationary problems of thermal conductivity and thermoelasticity with nonclassical boundary conditions for the two-layer plates

The practical importance of the nonclassical solutions is well known. The solutions of the nonclassical boundary value problems of stationary heat conduction and nonconnected thermoelasticity theory are constructed for the orthotropic two-layer plates. Examples are given with their analysis.

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(5)

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k

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(6)

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

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1 2 1 2 1

(2) 33

1

s

s s s s s s

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z

A

q

x

y

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1

s s s s s s

z

B

A

B

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1

s s s

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W

A

x y

x

W

x

y

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0

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0

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u

z

u

z

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

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* *

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y

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

, ё (1.10), (1.11) ё

(2.1),(2.2) ,

(

S

)

O



в а ых а ах щ ях

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

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_

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0

, ,

S i s i

s

Q

x y z

Q

x y z





. (2.3)

,

ё



_k

 

x y

,



h

_k



const

,

k



0,1, 2

,

const,

k

 

const,

k

W



:

1

const,



 

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

*

1

,

q u

1

,

j

,

j

x y z

, ,

  





(2.4)

ё (2.1)–(2.4)

ё (1.10)–(1.11), ,

,

, :





2

h

z

(7)









(1) (1) (1) (1) (1)

(1) (1) (1) (1)

1 1 1 13 1 11 1

,

0 ,

( , ;1, 2)

x x y y xz yz xy

zz xx

u

g z

h

gA

z

h

x y

 



     

  



  



  

(2.5)





2 ₍₁₎

_

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



2

1 1

(1) (1) 33 (1)

1 33 (1) 33 1 1 (1) 1

33 33

2

6

z z

z

h

z

h

u

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u

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q





_

 

z



h





 





W















_

_

–





 







_

_

2

0 1 0 1 0

0 1 0

2 1 (2) (1) (2) (1) 1

33 33 33 33

2 0

2 2 1 1 0 1 0 2 (2)

33

2 ,

(

)

2 h

z

h

z

h

q

W

h

z

W

q

h

h W

h

z W

 





_









   

_



_













_



_





_

_















(2.6)





















(2) (2) (2) (2) (2)

1 0 1 2 0

(2) (2) (2) (2)

1 13 1 0 1 2 0 11 2

0,

,

, ( , ;1, 2),

xz yz xy zz x x

xx y y

g h

h

g z

h

u

gA

g h

h

g z

h

x y

u



     

  



 





  





 



  







2 ₍₂₎

_

_





2

0 0

(2) 33 (2)

2 33 (2) 33 0 (2) 2

33 33

(1)

0 1 0 1 0

2

6 (

),

(

),

(

)

z z

z

h

z

h

u

gA

q

z

h

W

u

z

h

z

h

q

q z

h



















_





_

 





_

 



_







_

_



  



, (2.5), (2.6) (2.4)

( ), .

,

z

 

₂

( , )

x y



h

₂



const

(2.6)

2

(

)

cal

Q





Q z



h

,

Q

 



( 2)

,

q

(2),

  

(2)_xz

,

(2)_yz

,

(2)_zz ,

u

_x(2)

,

u

(2)_y ,uz( 2)



(2.7) [2,6,9,10], (2.5),

(2.6) (2.4)

:

1 1 1

(2) (2)

2 2 2 2

:

const,

( , )

const,

, ,

:

,

0 ( , ),

j

cal cal cal

xz xz xz zz

z

h

u

x y

j

x y z

z

h

q

x y

 

     

  

 



  





 





 

(2.8)

,

(

S

)

O



ё

.

(8)

А А

1. А А. ., . ., А. .

А GPS/

. А А . . 2003. .56. №3. .3-13.

2. Aghalovyan L.A., Gevorgyan R.S., Sahakyan A.V. Mathematical simulation of collision of arabian and euroasian plates on the base of GPS data // . А А . . 2005. .58. №4. .3-9.

3. А. .

. // . А А . .

2006. .59. №4. .24-31.

4. . ., . ., Ш . .

. .: , 1980. 286 .

5. . .

« А »

( ) // II . :

А . : 2010. I.

.182-186.

6. Aghalovyan L.A. On one class of three-dimensional problems of elasticity thory for plates //Proceedings of A. Razmadze Mathematikal Institute of Georgia. 2011. Vol.155, pp. 3-10.

7. А . А. А . :

. , 1997. 414 .

8. А .А., . .

, . : , 2005. 468 .

9. А .А., . ., . . ё

GPS

// А А . 2012. .112. №3. .264-270

10. А. . . .: , 1952. 392 .

11. . ., Э.А. А

. .: - , 1978. 224 .

:

− , . - . ,

А А. .: (37410) 270828; e-mail: gevorgyanrs@mail.ru