Asymptotic analysis of the equations and boundary conditions of thermoelasticity of micropolar thin plates

(1)

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А А А А А А

60, №3, 2007

539.3

А

А А

А

. ., . .

ч : , ,

.

Key words: micropolar, thermoelasticity, thin plate, asymptotic method . Ա. , . .

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Ա

, և

, , : Ք

( ),

: ,

: Ա

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:

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, `

ՙ ՚ :

և և :

S. A. Vardanyan, S. H. Sargsyan

Asymptotic analysis of the equations and boundary conditions of thermoelasticity of micropolar thin plates In the framework of the asymmetrical momental micropolar theory in the present work the boundary value problem of thermal stresses in a three-dimensional thin plate with independent fields of displacements and rotations is studied on the basis of asymptotic method.

Depending on the values of physical dimensionless constants of the material three applied two-dimensional theories of thermoelasticity of micropolar thin plate are constructed (theories with independent rotations, with constrained rotations and with small shift rigidity).

, ,

. -

( )

. ,

.

,

– , –

“ ” .

.

[1,2]. [3]

[4,5]

. [6]

(2)

[7-11]. [12, 13]

.

,

“ ”.

.

, [12,13],

.

1. ч .

2 h

.

α α

₁

,

₂

,

3

α

.

( )

[1,2]:

–

,

0

ji j

σ

=

,

μ

_{ji j}_,

+

э

_ijk

σ =

_jk

0

(1.1) –

(

)

(

)

(

)

(

)

(

)

ji ji ij kk ij

σ = μ + α γ + μ − α γ + λγ − ϑΘ ⋅δ

⎧⎪

⎨

μ = γ + ε χ + γ − ε χ + βχ δ

⎪⎩

(1.2)

–

= ij

γ

u

_{j i}_,

−

э

_kij

⋅ ω

_k

,

χ =

_ij

ω

_{j i}_, (1.3) –

0

=

ΔΘ (1.4)

,

ij ij

σ μ

– ;

,

ij ij

γ χ

–

-;

u

G

– ;

ω

G

–

-; Θ– ;

, , , , ,

λ μ α β γ ε

– ;

α

_t–

,

ϑ = μ + λ α

(2

3 )

_t;

Δ

– ;

H

_i

i

=

1, 2

.

3

h

α = ±

:

3i

p

i

,

3i

m

i

(

i

1, 2, 3),

T

3

h

± ± ±

σ = ±

μ = ±

=

Θ =

α = ±

(1.5)

,

i i

p

±

m

±

(

i=1,2,3

)

– ,

±

T – .

(

∑=Σ1∪Σ2

)

,

:

1 2

,

;

,

ji

n

j

p

i ji

n

j

m

i

u

∗ ∗

(3)

,

i i

p m

∗ ∗– ;

u

G

_∗

,

ω

G

_∗–

;

n

_i–

.

Σ

,

, (

,

∑

).

(1.1)-(1.6) ,

,

3

α

- (

α

₃

11

,

22

,

33

,

12

,

21

,

13

,

23

,

31

,

32

, ,

u u

1 2

,

3

,

σ

σ σ μ μ μ μ

ω Θ

, –

31

,

13

,

32

,

σ

23

,

μ

11

,

μ

22

,

μ

33

,

μ

12

,

μ

21

,

ω ω

2

,

3

, ;

u

3

– ).

,

(

2 h

<<

a

,

δ =

h a

/

<<

1

– ,

).

:

- ( )

, ,

. ,

, [12–14].

2. , ч ч

.

(1.1)-(1.6)

:

3

1 2

,

h

α

ξ =

η =

ζ =

(2.1)

i i

u

a

=

,

σ =

_ij

σ

ij

μ

,

ij ij

a

μ

μ =

μ

,

T

₀

Θ

Θ =

,

α = α ⋅

_t _t

T

₀ (2.2)

0

T

– .

,

. :

2 2 2

,

α

β

γ

ε

μ

(2.3)

.

( )

0

S s

q s

s

Q

−

Q

=

= δ

∑

δ

(2.4)

Q

– ( ),

(4)

.

(2.3) .

3.

щ щ . ,

(2.3) :

2 2 2

~ 1,

~ 1

α

β

γ

ε

μ

(3.1)

(2.4)

q

:

ζ

(

):

(

)

(

)

(

)

(

)

(

)

3 3 3

33

2 :11, 22, 12, 21 , , , 1, 2 , , , 1 , 1, 2 , :11, 22, 33, 12, 21 , , 1, 2 , 0 ;

mn i i i

i i mn i

q mn u i

q i mn u i

q

= σ μ μ = ω Θ

= σ σ = μ ω =

= σ

(3.2)

-

ζ

( ):

(

)

(

)

11 22

3 3 3 1 2

2 , ,

1 _i , _i 1, 2 , , , , _mn :11, 22, 33, 12, 21

q

q i u mn

= σ σ Θ

= σ σ = ω ω μ

(

)

11 22 ₃₃ ₁₂ ₂₁ ₃ ₃ ₃

0 ,

,

_i

,

_i

,

_i

1 2 ,

q

=

σ

μ

u i

=

,

ω

(3.3)

:

– :

u

₁

,

u

₂

3

ω

,

u

₃

1

,

2

ω ω

–

ζ

;

(

)

:11,12, 21, 22

mn

σ

μ

kl

:13, 23, 31, 32

(

kl

)

,

σ

_kl

33

,

mn

μ

–

ζ

;

σ

₃₃

ζ

;

– :

u

₃

ζ

,

ω ω

₁

,

₂ ,

3

ω

–

ζ

;

(

:13, 23,32, 31

)

kl

σ

ζ

,

(

)

33

,

:11,12, 21, 22

mn

μ

,

μ

kl–

ζ

,

1

,

2

u

σ σ

mn

,

33 ζ

.

, (3.2), (3.3)

. Э (2.3) ( (2.1),

(5)

, .

:

(

)

3 12 12 3 3 1 3

1

, 11 22 33 , , ,

2

h h h h

ii ii kl kl

h h h h

T d ii : , , S d L d T d

h

− − − −

= σ α

∫

= σ α

∫

= μ α

∫

=

∫

α

3 3

,

3 3

,

33 33 3 3

h h h

kl kl mn mn

h h h

M

d

− − −

= σ α α

∫

Λ = μ α α

∫

Λ = μ α α

∫

(3.4)

:

3

,

3 3

,

33 33 3 3

h h h

kl kl mn mn

h h h

N

d

M

d

M

d

− − −

= σ α

∫

= σ α α

∫

= σ α α

∫

(3.5)

3 33 33 3 3 3 2 2 3 3

3 ,

,

2

h h h h

mn mn kl kl

h h h h

L

d

L

d

T

d

h

− − − −

= μ

∫

α

= μ

∫

α

Λ = μ α α

∫

=

∫

α α

,

0

=

s

:

щ :

(

)

(

)

(

)

(

)

(

)

(

)

(

)

11 21 1 2 2 1

1 2 11 22 12 21 1 1

1 2 2 1 1 2

12 22 1 2 2 1

1 2 12 21 22 11 2 2

1 2 2 1 1 2

13 23 1 2 2 1

1 2 13 23 12 21 3 3

1 2 2 1 1 2

T S H H H H

H H T T S S p p

H H

S T H H H H

H H S S T T p p

H H

L L H H H H

H H L L S S m m

H H

+ −

∂ ₊ ∂ ₋ ∂ ₋ ₋ ∂ ₊ _{= −} ₊

∂α ∂α ∂α ∂α

∂ ∂ ∂ ∂

+ − + − − = − +

∂α ∂α ∂α ∂α

∂ ₊ ∂ ₋ ∂ ₋ ∂ ₊ ₋ _{= −} ₊

∂α ∂α ∂α ∂α

(3.6)

[

]

( )

(

)

(

)

( )

11 2 11 22 1 12 12 21

2 2

- 1 2 , 2 1 2

1

1 t

Eh Eh

T Т S h

-← ←

→ ⎡ ⎤ →

= + υΓ = _⎣ μ + α Γ + μ − α Γ _⎦

− υ

(

)

13 13 31

4

2

1

2 L

=

h

γε

k

+

γ − ε

L

→

γ + ε

(3.7)

( )

(

)

1 2 1 2 2 1

11 1 2 12 1 1 3

1 1 2 1 1 2

3

1 1 2

21 2 2 3 13 1

2 2 1 1

1 2 ,

,

1

2 V

H

V

H

V

H

V

O

H

O

V

H

V

O

k

H

→ ←

→

∂

Γ =

−

Γ =

+

−

∂α

∂

Γ =

+

=

∂α

(3.8)

(

m m

)

(

)

, T

(

T T

)

. h

(6)

(

)

2

13 23 1 2 2 1

1 2 13 23 3 3

1 2 2 1 1 2

1

2 T

N N H H H H

H H N N p p M

H H

+ −

∂ ∂ ∂ ∂

+ − − = − + + ∇

∂α ∂α ∂α ∂α (3.10)

(

)

(

)

(

)

(

)

(

)

(

)

11 21 1 2 2 1

1 2 11 22 12 21 23 32 1 1

1 2 2 1 1 2

12 22 1 2 2 1

1 2 12 21 22 11 31 13 2 2

1 2 2 1 1 2

L L H H H H

H H L L L L N N m m

H H

L L H H H H

H H L L L L N N m m

H H

+ −

∂ ∂ ∂ ∂

+ − − − + + − = − +

∂α ∂α ∂α ∂α

∂ ∂ ∂ ∂

+ − + − − + − = − +

∂α ∂α ∂α ∂α

(

)

(

)

(

)

( )

(

)

_{( )}

13 13 31

12 12 21

11 11 22 33

4

2

1

2

2 1 2

4 ₂

2 1 2

2

2 N

h

N

L

h

k

L

h

k

L

→ ←

μα

μ − α

=

Γ +

→

μ + α

⎡

⎤

=

_⎣

γ + ε

+ γ − ε

_⎦

⎡

γ γ + β

γβ

⎤

β

=

_⎢

+

_⎥

+

γ + β

⎣

⎦

(3.11)

13 1 2 23 2 1

1 2

,

W W

H ∂ H ∂

Γ = + Ω Γ = − Ω

∂α ∂α (3.12)

( )

1 2 1 2 2 1

11 1 2 12 1 1

1 1 2 1 1 2

1 2 ,

1 2

H

k

H

k

H

→ →

← ←

∂Ω

∂

∂Ω

∂

=

−

Ω

=

+

Ω

∂α

(

)

(

)

3

3 1, 2

2

T

i i i i

i

M

N

=

h p

+

−

p

−

+

H

∂

i

=

∂α

(

)

(

)

2

33 3 3

,

3 1

T t

Eh

M

T

L

h m

m

(

х

)

+ − + −

= −

−

=

−

(3.13)

,

.

щ

щ :

1)

11 12 13

1 3 , 2 3 , 3 3

h h h

T S L

p d∗ p d∗ m d∗

− − −

Γ Γ Γ

=

∫

α =

∫

α =

∫

α (3.14)

2) -

1 2 3

0 , 0 , 0

V V

Γ Γ Γ

Ω

= = = (3.15)

щ

1)

13 11 12

3 3 1 3 2 3

1

1 ,

,

2

h h h

T T

h h h

N

M

L

М

p dx

m dx

x

∗ ∗ ∗

− − −

Γ Γ Γ Γ Γ

∂

=

−

=

−

∂

(7)

2)

,

1 2

2

4

T

W

M

h

Γ Γ

α − μ

=

μα

11

1 3

,

h

L

m dx

∗

− Γ

=

_∫

12

2 3

h

T

h

L

М

m dx

∗

−

Γ Γ

=

_∫

−

(3.17)

3) -

,

1 1

2 4

Т

W

M

h

Γ Γ

= −

μ

1

0,

Γ

Ω

=

2

0

Γ

Ω

=

(3.18)

(3.6)-(3.8) (3.14), (3.15)

.

(3.10)-(3.12) (3.16)-(3.18)

.

,

. 4.

щ .

,

(2.3) :

2 2 2

~ 1,

~

,

~

,

~

α

β

_{δ β}

γ

_{δ γ}

ε

_{δ ε}

μ

(4.1)

β

,

γ

,

ε

– .

(1.1)-(1.7) (3.2),

(3.3) . . (3.4)

q

:

(

)

(

)

(

)

(

)

(

)

(

)

33 3 3 3

3 3

3

0 ,

,

1, 2 ,

3 ,

1, 2

1 ,

1, 2 ,

11 22 33 12 21

2 11 22 12 21 ,

1, 2 ,

,

i i i

i i mn

mn i

q

i

q

u

i

q

i

mn : , , , ,

q

mn : , , ,

u

i

=

σ

μ

=

ω

=

σ

=

μ

=

σ

=

ω

Θ

(4.2)

(4.1), (4.2) (

-): )

.

) ;

, .

.

33

μ

,

μ

31,

μ

32

ζ

,

ξ

,

η

.

(8)

(

)

(

)

(

)

₍

₎

(

)

(

) (

)

(

)

(

)

13 23 1 2 2 1

1 2 13 23 3 3

1 2 2 1 1 2

11 12 21 22 1 2

1 2 11 12 22 21

1 2 2 1

2 1

12 11 21 22 23 1 1 2 2

1 2

12 11 22 21 1 2

1 2 12

1 2 2 1

N

H

N

p

H

L

M

L

M

H

L

M

L

M

H

L

M

L

M

N

m

h p

p

H

L

M

L

M

H

L

H

+ −

+ − + −

∂

+

−

= −

+

∂α

∂

−

∂

−

∂

+

−

∂α

∂

−

+

−

+

= −

+

−

∂α

∂

+

∂

+

∂

+

−

∂α

(

)

(

)

(

) (

)

11 21 22

2 1

22 21 11 12 13 2 2 1 1

1 2

M

L

M

H

L

M

L

M

N

m

h p

p

H

+ − + −

+

−

∂

−

+

−

+

−

= −

+

−

∂α

(4.3)

-(

)

( )

(

)(

)

( ) 3 2

1 2 1 2 1 2

11 2 1 2 2 1 t 2

1 1 2 2 2 1

3

1 1 3 3

2 2 1 1 1 2

12 1 1 2 2

1 1 2 2 2 1 1

3

0 2

21 1

2 2

, 1 2 3 1 3(1 ) - th 2 -3(1 ) 3(1 )

H H H H

Eh Eh

M H H T

H H

h k k m m

H H H H

Eh

M H H

H H k

Eh

M H

→ ←

+ −

⎡⎛ ∂β ∂ ⎞ ⎛ ∂β ∂ ⎞⎤

= _⎢_⎜ − β _⎟ + υ_⎜ − β _⎟_⎥− α

∂α ∂α ∂α ∂α − υ

− υ _⎣_⎝ _⎠ _⎝ _⎠_⎦

−

⎛ ∂β ∂ ∂β ∂ ⎞

= _⎜ + β + + β _⎟

+ υ _⎝ ∂α ∂α ∂α ∂α _⎠

∂β =

+ υ ∂α

(

)(

)

(

)

(

)

( )

1 1 3 3

2 1 1 1 2

1 2 2

1 1 2 2 2 1 1

2 2 1 1 1 2

12 1 1 2 2

1 1 2 2 2 1

- th

2 1 2

h k k m m

H H H H

H

H H k

H H H H

L h H H

H H

+ −

→ ←

−

⎛ ₊ ∂ _{β +} ∂β ₊ ∂ _{β +}⎞

⎜ _∂α _∂α _∂α ⎟

⎝ ⎠

⎡ ⎛ ∂Ω ∂ ⎞ ⎛ ∂Ω ∂ ⎞⎤

= ⎢ γ + ε ⎜ _∂α + _∂α Ω + γ − ε⎟ ⎜ _∂α + _∂α Ω ⎟⎥

⎝ ⎠ ⎝ ⎠

⎣ ⎦

(

) ( )

1 2 1 1

11 1 2 3 3

1 1 2 1

th

4 1 2

2 H

H

h

k

L

h

H

m

H

k

+ − →

←

⎛

∂Ω

∂

⎞

β

=

γ

_⎜

−

Ω +

_⎟

−

∂α

γ + β

⎝

⎠

(4.4)

1 2 1 2 1 2

1 2

,

W

h

hH

∂

h

hH

∂

β = Ω = −

β = − Ω = −

∂α

(4.5)

(4.3), (4.4),

:

(

)

₍

₎

(

)

₍

₎

(

)

₍

₎

(

)

₍

₎

1 1 2 2

2 2 1 2 2 1

1 1 1 2 2 2

1 2 1 2 1 2

2 2 ₁ ₂ 1 1 ₂ ₁

1 2 2 2 1 1

1 2 1 2 1 2

p p _H _H p p _H _H

D W h H p p H p p

H H

m m _H _H m m _H _H

H m m H m m

H H + − + − ∗ + − + − + − + − + − + − ⎛ ∂ − _∂ ∂ − _∂ ⎞ ⎜ ⎟ ∇ ∇ = − − + − − +

⎜ ∂α ∂α ∂α ∂α ⎟

⎝ ⎠

∂ + _∂ ∂ + _∂

+ − + − + + +

∂α ∂α ∂α ∂α

(

)

₍

2

₎

2

3 3 2

2 3 1

t

Eh

p

+

p

−

T

+

α ∇

− υ

(

4.6)

∗

D – :

3

2

2 2 (

),

3 1

Eh

D

h

D

( -

х

)

∗

₌

₊

_{γ + ε}

₌

(4.7)

(9)

, , (4.6) [15].

-.

.

33

μ

,

μ

₃_n

1, 2

(

n

=

)

:

(

)

2

2 33

1 33

2

33 3

0

1 k

m

±

⎧∂ μ

− μ =

⎪ ∂ζ

⎨

⎪μ ζ = ± = ±

⎩

(

)

(

)

(

)

2 ₁ ₂

2 3

2 3 1

2

3

,

sh

,

1

n

n n

n

n n

k

F

k

F

m

±

⎧ ∂ μ

−

μ = ζ

ξ η +

ζ ⋅

ξ η

⎪ ∂ζ

⎨

⎪μ ζ = ± = ±

⎩

(4.8)

,

: 1)

(

)

₍

₎

(

)

₍

₎

13 2 12 11

3 2 3 2 1 3

2

12 11

3 1 2 3

h

N H M L

p H p m d

L M

p m d

∗ ∗ ∗

− Γ

∗ ∗

− Γ

⎛ ₊ ∂ ₋ ⎞

⎡ ∂ ⎤

⎜ _∂α _{⎟ =} ₊ _α ₋ _α

⎢ ⎥

⎝ ⎠ _∂α

⎣ ⎦

+

= α + α

∫

(4.9)

2)

(

12 11

)

3 1 2 3

0,

(

)

h

w

L

M

p

∗

m d

∗ −

Γ Γ

+

=

_∫

α

+

α

(4.10)

3) -

1 1 ,

0

0 W

H

W

Γ

∂

∂α

=

(4.11)

(4.3), (4.4) (4.9)–(4.11)

. Э (

)

.

, (4.9)

[16].

5. “

”.

,

(10)

.

(2.3)

, , . (2.3)

:

2

2 2 2

~

,

~ 1,

~ 1

α

_{δ α}

β

γ

ε

μ

(5.1)

. (2.4) q :

(

)

(

)

(

)

(

)

(

)

(

)

33 3 3 33

3 3 3

3

0 , 1 , 1, 2 ,

2 11 22 12 21 , 1, 2 , , 1, 2 , , 3 , 1, 2 , : 11, 22, 12, 21

i i

mn i i i

i mn

q q i

q mn : , , , u i i

q u i mn

⎧ = σ = σ σ = μ

⎪

= σ = μ μ = ω Θ

⎨

⎪ = ω = μ

⎩

(5.2)

:

u

₃

ω ω

₁

,

₂

,

u

₁

,

u

₂

ω

₃–

ζ

;

σ

kl

(

kl

:13, 23, 32,31

)

-(

)

33

,

:11,12, 21, 22

mn

μ

,

33

,

mn

σ

μ

_kl–

ζ

.

, (5.1), (5.2)

“ ”

, . “ ”

, “ ”

.

“ ” :

(

)

(

)

(

)

(

)

(

)

(

)

11 21 1 2 2 1

1 2 11 22 12 21 1 1

1 2 2 1 1 2

12 22 1 2 2 1

1 2 12 21 22 11 2 2

1 2 2 1 1 2

L L H H H H

H H L L L L m m

H H

L L H H H H

H H L L L L m m

H H

+ −

∂ ∂ ∂ ∂

⎧ ₊ ₋ ₋ ₋ ₊ _{= −} ₊

⎪ _∂α _∂α _∂α _∂α

⎪

⎨ _∂ _∂ _∂ _∂

⎪ ₊ ₋ ₊ ₋ ₋ _{= −} ₊

⎪ ∂α ∂α ∂α ∂α

⎩

(5.3)

[

]

( )

(

)

( )

12 12 21 11 11 22

4 2

2 ( ) ( ) 1 2 ; 2 1 2

2 2

L = h γ + ε k + γ − ε k →_← L = h⎡_⎢ γ γ + β k + γβ k _⎥⎤ →_←

γ + β γ + β

⎣ ⎦

(5.4)

( )

1 2 1 2 2 1

11 1 2 12 1 1

1 1 2 1 1 2

1 2

H

k

H

, k

H

→ ←

∂Ω

∂

∂Ω

∂

=

−

Ω

=

+

Ω

∂α

(5.5)

22 21 12 11,k ,k ,k

k –

-, Ω1,Ω2–

α

1

α

2

(11)

“ ” :

(

)

(

)

(

)

(

)

(

)

13 23 1 2 2 1

1 2 13 23 3 3

1 2 2 1 1 2

11 1 2 2 1

31 1 2 11 22 1 1

1 2 2 1 1 2

22 1 2 2 1

32 1 2 22 11 2 2

1 2 2 1 1 2

2

N N H H H H

H H N N p p

H H

M H H H H H

N H H M M H h p p

H H

M H H H H

H

N H H H M M h p p

H H

+ −

∂ ∂ ∂ ∂

+ − − = − +

∂α ∂α ∂α ∂α

⎡ ∂ ∂ ∂ ∂ ⎤

−_⎢ + − − − _⎥= −

∂α ∂α ∂α ∂α

⎣ ⎦

⎡ ∂ ∂ ∂ ∂ ⎤

−_⎢ + − − − _⎥= −

∂α ∂α ∂α ∂α

⎣ ⎦

(5.6)

(

)

(

) ( )

(

)

(

) ( )

13 13 31 23 23 32

2 2

11 ₂ 11 22 2

2 2

22 ₂ 22 11 2

2

12 21 12

2

4

2

4

2

2 3 1

3 1

2

2 3 1

3 1

4

3

t

N

h

N , N

h

N

Eh

M

K

T

Eh

M

K

T

h

M

H

K

=

⋅ Γ +

=

⋅ Γ +

= −

− υ

−

α

− υ

= −

− υ

−

α

− υ

μ

=

(5.7)

(

)

2

−

+₋

= T T

T

2

2 1

13 1 2 23 2 1 11 1 1

1 2 1 1 1 2 2

, , H H

W W W W

H H K h H H

H

⎡ ⎛ ⎞ ∂ ⎤

∂ ∂ ∂ ∂ ∂

Γ = + Ω Γ = − Ω = _⎢ _⎜ _⎟− _⎥

∂α ∂α _⎣ ∂α _⎝ ∂α _⎠ ∂α ∂α _⎦

2

1 2

22 2 2

2 2 2 1 1

H

W

K

h H

H

⎡

∂

⎛

∂

⎞

∂

⎤

=

_⎢

_⎜

_⎟

−

_⎥

∂α

_⎝

∂α

_⎠

∂α ∂α

⎣

⎦

(5.8)

1 2

12 1 2 2 2 1 1 2

1 2 2 1 2 1 1 2

H H

W W W W

K = −h H⎡_⎢ ∂ ⎛_⎜H ∂ _⎟⎞+H ∂ ∂ +H ∂ _⎜⎛H ∂ ⎞_⎟+H ∂ ⎛_⎜H ∂ ⎞_⎟_⎥⎤ ∂α _⎝ ∂α _⎠ ∂α ∂α ∂α _⎝ ∂α _⎠ ∂α _⎝ ∂α _⎠

⎣ ⎦

,

W

–

α

₃( ),

Γ

₁₃

,

Γ

₂₃–

,

K

₁₁

, ,

K

₂₂

K

₁₂– -

.

(5.6)-(5.8) “ ”

(

1

,

2

)

W

α α

:

2 2 1 2

1 1 2

1 2 1 1

2 1

2 2 1

2 1 2 2

8 H

H

W

D

W

h

H

W

H

⎡⎛

∂

⎞⎛

∂

⎞

∇ ∇

+ α

_⎢⎜

−

_⎟⎜

+ Ω +

_⎟

∂α

⎝

⎠⎝

⎠

⎣

⎤

⎛

∂

⎞⎛

∂

⎞

+

_⎜

−

_⎟⎜

− Ω

_⎟

_⎥

=

∂α

⎝

⎠⎝

⎠⎦

(

)

1 2

(

)

3 3 1 1 1

1 2 1

H

p

h H

p

H

+ −

⎛

∂

⎞

+ −

= −

+

_⎜

−

_⎟

−

+

∂α

(12)

(

)

₍

2

₎

2

2 1

2 2 2 2

2 1 2

2 3 1

t

H

Eh

H

p

T

H

+ −

⎛

∂

⎞

+

_⎜

−

_⎟

−

+

α ∇

∂α

− υ

⎝

⎠

(5.9)

8 h

α

- , ,

α

–

μ

,

(5.1)

“ ”.

, (5.1)

.

(5.3)-(5.5) (5.6)-(5.8):

1) :

11 12

1 3 2 3

12

13 2 11

3 3 2 3 3 1 3 3

2

,

h h

h h h

L

m d

M

N

H

M

p d

p

d

p

d

∗ ∗

− −

Γ Γ

∗ ∗ ∗

− − Γ −

Γ

=

α

=

α

⎛

∂

⎞

+

⎜

_∂α

_{⎟ =}

_{α +}

_{α α}

₌

_{α α}

⎝

⎠

∫

(5.10)

2)

11

1 3 3

,

0 ;

h

W

M

p

∗

d

−

Γ Γ

=

∫

α α

11

1 3

,

h

L

m dx

∗

− Γ

=

∫

12

2 3

.

h

L

m dx

∗

− Γ

=

∫

(5.11)

3) - :

1 1

,

0

0 W

W

H

Γ Γ

∂

=

∂α

,

1

0,

Γ

Ω

=

2

0

Γ

Ω

=

(5.12)

(5.3)-(5.8) (5.10)-(5.12)

“ ”.

, . .

.

ч .

. ,

,

. ,

(13)

1. . . // . . 1967. . III. . 1. . 3-17.

2. Nowacki W. Theory of Asymmetric Elasticity. Pergamon Press. Oxford. New York. Toronto. Sydney. Paris. Frankfurt. 1986. P. 383.

3. . . . :

-, 1999. 214 .

4. . . . .: , 1967. 266 .

5. . . . .: , 1974. 446 .

6. . .

// . . . 2002. . 55. №3. . 20-33.

7. Green A. E. Boundary-layer equations in the linear theory of elastic shells// Proc. Roy. Soc., Ser. A. 1962. Vol. 269. №1339. P. 39-49.

8. . .

// .: I . Ш

. : . .

-, 1975. . 51-149.

9. . . . .: , 1976. 510 .

10. . . :

, 1997. 414 .

11. . . .

: . , 1992. 260 .

12. Sargsyan S.H. On some Interior and Boundary Effects in Thin Plates Based on the Asymmetric Theory of Elasticity // Lecture Notes in Applied and Computational Mechanics. Vol. 16./Theories of Plates and Shells. Critical Review and New Applications. Springer. 2004. P. 201-210.

13. . .

// . 2008. . 72. . 1.

14. Sargsyan S. H., Vardanyan S. A. Asymptotic Theory of Thermoelastic Micropole Thin Plates// Proceedings of the 5-th International Congress on Thermal Stresses and Related Topics. 8-11 June 2003, Blacksburg. V. A. Vol. 1. MA5-3-1 – MA5-3-4.

15. . . // .

. 1966. .2. . 10. . 36-43. 16. Х .

// . . . . - .- . .

1964. . 31, №4. . 149-150.

.