Magnetoelasticity of Thin Plates and Shells on the basis of the Asymmetrical Theory of Elasticity

(1)

Ս ՈՒԹ ՈՒ Ղ

63, №3, 2010

539.3

*)

. ., . .

ч : , , , ,

.

Key words: micropolar theory, electroconductivity, nonferromagnetic, thin shells, plates

Ս Ս. ., Ս .Ս.

, , “ ”

:

Sargsyan S.H., Sargsyan L.S.

Magnetoelasticity of Thin Plates and Shells on the basis of the Asymmetrical Theory of Elasticity

The present paper is aimed at constructing general mathematical models with independent and constraint rotation and models with “small shift rigidity” of thin elastic electro conductive micropolar shells and plates.

ё « ё ».

.

[1-5]. С ,

, ,

- ,

[6-10]. ,

, ,

.

[5]

-. [11-16]

, .

.

( ё

*)

«Euromech Colloquium 510. UPMC, Paris, France, May 13-16, 2009, Mechanics of Generalized Continua: A hundred years after the Cosserats».

(2)

) ё ю , ю

,

, ,

.

, ,

. ,

« ».

1. ч .

h

2 ё ( )

ё [5,17].

{

01 02 03

}

0

,

,H H

H

H = .

[6-10].

( )

– :

2 2

n n

mn n mn nmk

m m mk

V

f

,

e

J

t

∂

∂ ω

∇

+

= ρ

∇

+

=

∂

(1.1)

(

)

(

)

(

)

(

)

mn mn nm kk nm

σ = μ + α γ + μ − α γ + λγ δ

μ = γ + ε κ + γ − ε κ + βκ δ

(1.2)

ш

,

k

mn m

V

n

e

kmn mn m n

γ = ∇

−

ω

κ = ∇ ω

(1.3)

( )

4

1 rot

h

j

, rot

E

h

, div

h

0, div

E

4

_e

c

c t

π

∂

=

= −

=

= πρ

∂

(1.4)

( )

( ):

( ) ( )

1

( ) ( ) ( )

rot

0, rot

, div

0, div

0

e

e e

h

e e

h

E

h

c

t

∂

=

= −

=

∂

(1.5)

; 3 , 2 , 1 ,n=

m

σ

nm

,

μ

nm–

;

γ

_mn

,

κ

_mn–

- ; V_n–

( )

V ,

n

(3)

{

₁ ₂ ₃

}

₁ 0

,

, j H

c f f f

F = = × (1.6)

F– ;

0

1 V

j

E

H

c

t

⎛

∂

⎞

= σ

_⎜

+

×

_⎟

∂

⎝

⎠

(1.7)

j

– ;

E E

,

( )e –

, ,

;

h h

,

( )e – ,

, ;

ρ

_e–

;

λ μ α β γ ε

, , , , ,

– ,

ρ

–

,

j

– ;

σ −

;

c

–

, ;

δ

nm– ,

nmk

e

– -Ч ы.

:

(

)

3

1, 2,3

k h

q

k

± α =±

σ

=

_∓

=

(1.8)

Σ

,

, ,

.

Э ,

, :

[ ]

4 ˆ

,

[ ]

0,

[ ]

0,

[ ]

0

k nmk k nmk

k e m k k m k

n

E

₋

= πρ

e

n

h

₋

=

n h

₋

=

e

n

E

₋

=

(1.9)

, [⋅]_– (

);

ρ

ˆ

_e– ;

n

k

(

k

=

1, 2, 3

)

–

.

, к ы

ы

:

( )

1

( )

1

0 ,

0

e e

E

h

r

⎛ ⎞

=

_{⎜ ⎟}

=

_{⎜ ⎟}

⎝ ⎠

r

→ ∞

, 0

≤ < ∞

t

(1.10)

r

– .

,

t

=

0

, ,

, .

2. х х

.

(4)

(

)

(

)

3 3 3

3 3

3 3 3

3 3

3 3 3 3

33 33 33

1 2 1 2

1 ,

1

3

1 ,

1

3

1

1 ,

1

ii ii ij ij i i

j j j

ii ii ij ij i i

j j j

i

R

v

i

R

v

R

⎛

_α

⎞

⎛

_α

⎞

⎛

_α

⎞

τ = +

_⎜

⎜

_⎟

⎟

σ τ = +

_⎜

⎜

_⎟

⎟

σ τ = +

⎜

_⎜

⎟

_⎟

σ

↔

⎝

⎠

⎝

⎠

⎝

⎠

⎛

_α

⎞

⎛

_α

⎞

⎛

_α

⎞

= +

_⎜

⎜

⎟

_⎟

μ

= +

_⎜

⎜

⎟

_⎟

μ

= +

⎜

_⎜

⎟

_⎟

μ

↔

⎝

⎠

⎝

⎠

⎝

⎠

⎛

α

⎞⎛

α

⎞

⎛

α

⎞⎛

α

τ = +

_⎜

_⎟⎜

+

_⎟

σ

= +

_⎜

_⎟

+

⎝

⎠⎝

⎠

⎝

⎠⎝

33

⎞

μ

⎜

⎟

⎠

(2.1)

i j

,

1,2,

i

≠

j

.

(2.1) :

2

13 31 3 31 3 3 1

1 2 03 2 02 3

1 1 3 1 2

2

23 32 3 32 3 3 2

2 2 03 1 01 3

2 2 3 1 2

33 3

3 1

1

1 α

α

V

L

H j

R

α

R

t

c

α

V

L

H j

R

α

R

t

c

α

L F

α

R

⎛

⎞

⎛

⎞⎛

⎞ ⎧

⎫

+

∂

+

+ +

_⎜

_⎟

= +

_⎜

_⎟⎜

+

_⎟

_⎨

ρ

−

+

_⎬

∂

_⎩

∂

_⎭

⎝

⎠

⎝

⎠⎝

⎠

⎛

⎞

⎛

⎞⎛

⎞ ⎧

⎫

+

∂

+

+ +

_⎜

_⎟

= +

_⎜

_⎟⎜

+

_⎟

_⎨

ρ

+

−

_⎬

∂

_⎩

∂

_⎭

⎝

⎠

⎝

⎠⎝

⎠

⎛

⎞

∂

− + +

= +

_⎜

_⎟

∂

_⎝

_⎠

( )

(

)

2 3 3

02 1 01 2

2 2

2

3 3 3 3 3 3 3

3 3 2

3 1 2

33 3 3 3 3

12 21

3 1 1 2

1

2

j

i i i i

i j j

i i j

α

V

H j

R

t

c

α

K

J

R

α

R

t

α

-K

Φ

α

R

⎛

₊

⎞ ⎧

_ρ

∂

₋

₊

⎫

⎨

⎬

⎜

⎟

_∂

⎩

⎭

⎝

⎠

⎛

⎞

⎛

⎞

⎛

⎞⎛

⎞

+

∂

∂ ω

+

+ +

⎜

⎟

_∂

− −

⎜

_⎜

+

⎟

_⎟

−

= +

⎜

⎟⎜

+

⎟

_∂

⎝

⎠⎝

⎠

⎝

⎠

⎝

⎠

⎛

⎞

⎛

⎞⎛

⎞

∂

⎛

⎞

+ +

_⎜

_⎟

− +

_⎜

_⎟

= +

_⎜

_⎟⎜

+

_⎟

∂

_⎝

_⎠

_⎝

_⎠

_⎝

_⎠⎝

_⎠

2 3 2

J

t

∂ ω

∂

(2.2)

-( )

3 3 3

33

3 3 3 3 3

33 11 22

1 2 3 1 2

3 3 3 3

3

1

4

i ii jj

j i j

j j

j i j i

α

v

α

v

e

R

E

R

E

R

E

α

V

v

α

v

α

R

α

E

R

E

R

α

t

ω

R

α

R

⎛

⎞

⎛

⎞

⎛

⎞

+

=

+

−

+

−

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

_⎝

_⎠

⎜

⎟

⎝

⎠

⎝

⎠

⎛

₊

⎞⎛

₊

⎞

∂

₌

₋

⎛

₊

⎞

₋

⎛

₊

⎞

⎜

⎟⎜

⎟

_∂

⎜

⎟

⎜

⎟

⎝

⎠⎝

⎠

⎝

⎠

⎝

⎠

⎛

⎞

⎛

⎞

⎛

⎞

μ + α

⎛

⎞

+

− −

+

=

+

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

_⎝

_⎠

⎜

⎟

_μ

_⎝

_⎠

⎝

⎠

⎝

⎠

3

1

4

ij ji j

α

R

⎛

⎞

μ − α

−

⎜

_⎜

+

⎟

_⎟

μ ⎝

⎠

( )

3 3

3

1

4

j i

j i i

j j

α

V

α ω

R

α

R

α

⎛

⎞

_∂

⎛

⎞

_{μ + α}

_{μ − α}

+

− −

+

=

−

⎜

⎟

⎜

⎟

⎜

⎟

_∂

⎜

⎟

_μ

⎝

⎠

⎝

⎠

( )

3 3 3 3 3

3 3

1

4

j

i j i i

j i j i i

α

g

ω

R

α

R

α

R

⎛

₊

⎞

_{+ −}

⎛

₊

⎞

⎛

₊

⎞

₌

μ+α

⎛

₊

⎞

₋

μ−α

⎛

₊

⎞

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

_⎝

_⎠

⎜

⎟

_μ

_⎝

_⎠

_μ

_⎝

_⎠

(5)

(

)

(

)

(

)

(

)

(

)

(

)

3 3 3

33

3 3 3 3 3

33 11 22

1 2 3 1 2

1 1 1

3 2 2 3 2 2 3 2

1 1 1 1

3 2 2 3 2 2 3 2

i ii jj

j i j

α α α

R R R

α α ω α α

R R α R R

⎛ ⎞ + ⎛ ⎞ ⎛ ⎞

+ κ = + − + −

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ _γ ₊ _⎝ _⎠ _γ ₊ ⎜ ⎟ _γ ₊

⎝ ⎠ ⎝ ⎠

⎛ ₊ ⎞⎛ ₊ ⎞∂ ₌ + ₋ ⎛ ₊ ⎞ ₋ ⎛ ₊ ⎞

⎜ ⎟⎜ ⎟_∂ _γ ₊ _γ ₊ ⎜ ⎟ _γ ₊ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

3 3 3

3 3

3

3 3 3

3 3

1

4

1

1 .

4

i

j ij ji i i

j i j

i i i

j i i

α

ε

α

ε

α

ω

ε

n

,

R

ε

R

ε

R

α

ε

α

_θ

ε

α

ε

α

R

ε

R

ε

R

⎛

₊

⎞

₌

+

⎛

₊

⎞

₋

−

⎛

₊

⎞

∂

₌

+

₋

−

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

_γ

_⎝

_⎠

_γ

⎜

⎟

_∂

_γ

⎝

⎠

⎝

⎠

⎛

⎞

+

⎛

⎞

−

⎛

⎞

+

=

+

−

+

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

_γ

_⎝

_⎠

_γ

_⎝

_⎠

⎝

⎠

3 3 2 2 3 2 3

1 03 02

2 2 2 3 2 2

1

4

1

1 h

h

V

E

H

A

R

c

t

⎛

⎞

⎛

⎞

∂

_{− +}

α

∂

₋

_{= +}

α

πσ

⎛

₊

⎛

∂

₋

∂

⎞

⎜

⎟

⎜

⎟

⎜

⎟

∂α

⎝

⎠

∂α

⎝

⎠

⎝

∂

⎠

3 1 1 3 3 3 1

2 01 03

1 3 1 1 1 1

1

4

1

1 h

h

1 E

H

V

H

V

R

A

R

c

t

⎛

₊

α

⎞

∂

₊

₋

∂

_{= +}

⎛

α

⎞

πσ

⎛

₊

⎛

∂

₋

∂

⎞

⎜

⎟

_∂α

⎜

⎟

⎜

_⎝

_∂

⎟

_⎠

⎟

⎝

⎠

⎝

⎠

⎝

⎠

3 2 2 3 3 1 1 3

2 1

1 2 1 1 2 1 1 2 1 2 1 2 2 2

3 3 1 2

3 02 01

1 2

1 1 1 1

4 1

1 1 1

h A h A

h h

A R A A R A R A A R

V V

E H H

R R c c t t

A

⎛ α ⎞∂ ∂ ⎛ α ⎞ ⎛ α ⎞ ∂ ∂ ⎛ α ⎞

+ + + − + − + =

⎜ ⎟_∂α _∂α ⎜ ⎟ ⎜ ⎟_∂α _∂α ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ α ⎞⎛ α ⎞ πσ⎛ ⎛ ∂ ∂ ⎞⎞

= +_⎜ _⎟⎜ + _⎟ _⎜ + _⎜ − _⎟_⎟

∂ ∂

⎝ ⎠

⎝ ⎠⎝ ⎠

3 1 2 3 3 2 1 3

1 2

1 2 1 1 2 1 1 2 1 2 1 2 2 2

3 3 3 3 3 3 3

1 2 3 2 1 1 2

1 1 1

1 1 1 1

h A h A

h h

R A A R A R A A R

h h h

R R R R R R

⎛ α ⎞∂ ∂ ⎛ α ⎞ ⎛ α ⎞ ∂ ∂ ⎛ α ⎞

+ + + + + + + +

⎜ ⎟_∂α _∂α ⎜ ⎟ ⎜ ⎟_∂α _∂α ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ α ⎞⎛ α ⎞ ∂ ⎛ α ⎞ ⎛ α ⎞ + +_⎜ _⎟⎜ + _⎟ + +_⎜ _⎟ + +_⎜ _⎟ =

∂α

⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0

3 3 2 2 3 1

2 2 2 3 2 2

3 1 1 3 3 2

1 3 1 1 1 1

1

1 E

E

h

A

R

c

R

t

E

h

R

A

c

R

t

⎛

⎞

⎛

⎞

∂

_{− +}

α

∂

₋

_{= −}

₊

α

∂

⎜

⎟

⎜

⎟

∂α

_⎝

_⎠

∂α

_⎝

_⎠

∂

⎛

₊

α

⎞

∂

₊

₋

∂

_{= −}

⎛

₊

α

⎞

∂

⎜

⎟

_∂α

⎜

⎟

_∂

⎝

⎠

⎝

⎠

(2.4)

3 2 2 3 3 1

2

1 2 1 1 2 1 1 2 1 2

3 3 3 3

1

1 2 2 2 1 2

3 1 2 3

1

1 2 1 1 2 1 1

1

1 E

A

E

A

R

A A

R

A

R

h

A

E

A A

R

c

R

t

E

A

E

A

R

A A

R

A

⎛

₊

α

⎞

∂

₊

∂

⎛

₊

α

⎞

₋

⎛

₊

α

⎞

∂

₋

⎜

⎟

_∂α

⎜

⎟

⎜

⎟

_∂α

⎝

⎠

⎝

⎠

⎝

⎠

⎛

α

⎞

⎛

α

⎞⎛

α ∂

⎞

∂

−

_⎜

+

_⎟

= −

_⎜

+

_⎟⎜

+

_⎟

∂α

_⎝

_⎠

_⎝

_⎠⎝

_⎠

∂

⎛

₊

α

⎞

∂

₊

∂

⎛

₊

α

⎞

₊

⎜

⎟

_∂α

⎜

⎟

⎝

⎠

⎝

⎠

3 2 1 3

2

2 1 2 1 2 2 2

3 3 3 3 3 3 3 3 3

1 2 3 2 1 1 2 1 2

1

4

e

E

A

E

R

A A

R

E

R

⎛

₊

α

⎞

∂

₊

∂

⎛

₊

α

⎞

₊

⎜

⎟

_∂α

⎜

⎟

⎝

⎠

⎝

⎠

⎛

α

⎞⎛

α ∂

⎞

⎛

α

⎞

⎛

α

⎞

⎛

α

⎞⎛

α

⎞

+ +

_⎜

_⎟⎜

+

_⎟

+ +

_⎜

_⎟

+ +

_⎜

_⎟

= +

_⎜

_⎟⎜

+

_⎟

πρ

∂α

(6)

3 1 2 3 3 2 1 3

1 2

1 2 1 1 2 1 1 2 1 2 1 2 2 2

3 3 3 3 3 3 3

1 2 3 2 1 1 2

1 1 1 1

1 1 1 1 0

j A j A

j j

A R A A R A R A A R

j j j

R R R R R R

⎛ α ⎞ ∂ ∂ ⎛ α ⎞ ⎛ α ⎞ ∂ ∂ ⎛ α ⎞

+ + + + + + + +

⎜ ⎟_∂α _∂α ⎜ ⎟ ⎜ ⎟_∂α _∂α ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ α ⎞⎛ α ⎞ ∂ ⎛ α ⎞ ⎛ α ⎞ + +_⎜ _⎟⎜ + _⎟ + +_⎜ _⎟ + +_⎜ _⎟ =

∂α

⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

3 3

2 1

1 1 03 02 2 2 01 03

1

1 ,

V

j

E

H

j

E

H

c

t

c

t

⎛

∂

⎞

⎛

∂

⎞

=σ

_⎜

+

_⎜

−

_⎟

= σ

_⎜

+

_⎜

−

_⎟

∂

⎝

⎠

⎝

⎠

⎝

⎠

⎝

⎠

(2.5)

1 2

3 3 02 01

1 V

V

j

E

H

c

t

⎛

∂

⎞

= σ

_⎜

+

_⎜

−

_⎟

∂

⎝

⎠

⎝

⎠

3

h

α = ±

:

(

)(

)

(

)(

)

3 3

3 33

3

3 3 1 3 2

3 33

3

3 3 1 3 2

1

i

j _α _h _α _h

i

j _α _h _α _h

q ,

q

α

/ R

α

/ R

α

/ R

m ,

m

α

/ R

α

/ R

α

/ R

± ±

=± =±

± ±

=± =±

⎡

⎤

⎡

⎤

=

⎢

₊

⎥

⎢

₊

⎥

⎢

⎥

⎣

⎦

⎣

⎦

⎡

⎤

⎡

⎤

=

⎢

₊

⎥

⎢

₊

⎥

⎢

⎥

⎣

⎦

⎣

⎦

∓

(2.6)

:

(

)

(

)

1

ii

1

j

1

ji

1

i

i ii jj ji ij

i i i j i j j i j j

A

L

A

α

A A

α

A

α

A A

α

∂

=

+

−

+

∂

(

)

(

)

13 23

11 22 2 1

13 23

1 2 1 1 1 2 1 2 2 1 2 2

13 23

11 22 2 1

13

1 2 1 1 1 2 1 2 2 1 2 2

1

j ji

ii i

i ii jj ji ij

i i i j i j j i j j

A

L

, F

R

A

α

A A

α

A

α

A A

α

A

K

A

α

A A

α

A

α

A A

α

A

K

,

R

A

α

A A

α

A

α

A A

α

∂

=

+

=

+

∂

=

+

−

+

∂

=

+

Φ =

+

∂

23

(2.7)

3 3

1

j

i i i i

i j i j i

i i i j j i j j i j i i i i

j

i i i i

i j i j i

i i i j j i j j i j i i i i

A

V

A

V

e

V

, t

V , g

A

α

A A

α

R

A

α

A A

α

A

α

R

A

ω

A

ω

, n

ω

,

θ

A

α

A A

α

R

A

α

A A

α

A

α

R

∂

=

+

=

−

=

−

∂

κ =

+

=

−

=

−

∂

3. ч .

(2.2)-(2.5), (1.5), (2.6), (1.9), (1.10)

,

. ,

, , ,

(7)

(

α

_n t):

( )

1

3 0

0

2 1

2

*

0 0

,

p l

i i

l k ij ij k ok

ij ij k ok

k k e

ok ok k k e

h

E

R

t

c

V

H

J

J,

, V

, H

h

E

RE

h

E

h

c

E

c

h

H

h

E

c

E

− − ω

+

κ

α = λ ξ

α = λ ζ

= λ

τ

=

ρ

= λ

=

ρ

= λ

=

ρ =

(3.1)

( 1)

0

0 0

,

l

k i i к

k m i k к

j

c

R

q

m

c

h

R

j

R

, R

, q

, m

, t

c

E

c c

R

E

RE

c

± ±

− ω−

± ±

σ

=

= λ

σ

m

R – ,

ω

-( ) ; p l ;

,

p l

– , l> p≥0;

λ

–

,

h

= λ

R

−l; R− .

. :

2 2 2

, , ,

E R E R E R E

α β γ ε

(3.2) : Rm ~1.

,

, (

1

,

2

,

ξ ξ ζ

τ

)

(

ξ

₁

ξ

₂

τ

).

4. х х ч

щ щ . ,

(3.2) :

2 2 2

~ 1, ~ 1, ~ 1, ~ 1

E R E R E R E

α β γ ε

(4.1)

Ч

ω

, k κ₁ (3.1) ,

.

, :

1

1 , , 2 2

p p

k l p

l l

ω = + = − κ = − − (4.2)

(4.1), (4.2),

( )

p l

Oλ − :

(

0 2 1

)

2

(

0 1

)

0

(

0 1

)

3 3 3

, ,

l p l p c l p c l c l c

i i i ii ii ii

(8)

(

)

(

)

(

)

(

)

(

)

0 2 1 0 0 1 0 0 1

3 3 3

0 2 1 0 2 1

3 3 3 3 3 3

, ,

,

p c l p c l c l c

i i i ij ij ij

p c l p c p c l p c

i i i i i i

ω E

E E

− + − + − − + − +

− + − + − − + − + −

= λ ω + λ ζω ω = λ ω + λ ζω τ = λ τ + λ ζτ τ = λ τ + λ ζτ τ = λ τ + λ ζτ

(

)

(

)

(

)

(

)

2 0 2 1 2 0 2 1

0 1 0 1

3 3 3 3 3 3

,

p c l p c p c l p c

ii ii ii ij ij ij

p l c p l c

i i i i i i

v

RE

v

RE

v

RE

v

RE

v

− + − + − − + − + −

− − + − − +

=

λ

+ λ

ζ

=

λ

+ λ

ζ

=

λ

+ λ

ζ

=

λ

+ λ

ζ

(4.3)

(

)

(

)

(

)

2 2

0 1 2 2 2 0 1 2 2

33 33 33 33 33 33 33 33

0 1 0

3 3 2

,

, 2

l c l p c l l c

i i i

E

v

RE

v

h

E

h

E

h

l

p

− + − + − − − +

κ κ

τ = λ

τ + ζτ + λ

ζ τ

=

λ

+ ζ

+ λ

ζ

=

λ

+ ζ

=

λ

κ = − +

2 2 , 2 2 4 , 2 4 .

c= p l− l≤ p c= −l p p≤ ≤l p c= p l≥ p

, .

, :

(

1 3

)

3,

(

1 3

)

3,

(

1 3

)

3 3

h h h

ii j ii ij j ij ii j ii

h h h

T R d S R d G R d

− − −

=

_∫

+α σ α =

_∫

+α σ α =−

_∫

+α σ α α (4.4)

(

1

3

)

3

,

(

1

3

)

3

,

(

1

3

)

3 3

h h h

ii j ii ij j ij ij j ij

h h h

L

R

d

L

R

d

H

R

d

− − −

=

∫

+ α

μ α

=

∫

+ α

μ α

=

∫

+ α

σ α α

,

:

3 3 3

0, 0, 0, 0

i i i i

u =V _ζ= w= −V _ζ= Ω = ω _ζ= Ω = −ω _ζ= . (4.5)

, ,

( )

p l

O

λ

− ,

-

:

(

)

(

)

( )

0 03 22

(

)

1

2

j ji

ii i

ii jj ji ij

i i i j i j j i j j

j _i

j i i

A

S

T

A

T

S

A

α

A A

α

A

α

A A

α

u

j H

h

q

c

t

+ −

∂

₊

₋

₊

∂

₊

∂

+ −

= ρ

+

∂

(

)

(

)

( )

(

3 3

)

2 2

(

)

1

2

j ji

ii i

ii jj ji ij

i i i j i j j i j j

j _i

j j i i

A

L

A

L

A

α

A A

α

A

α

A A

α

N

Jh

m

t

+ −

∂

+

−

+

∂

∂ Ω

+ −

−

=

+

∂

(4.6)

(

) (

)

2

₍

₎

2 13 1 23

11 22

01 02 02 01 2 3 3

1 2 1 2 1 2

1

2 A N

A N

T

w

j H

h

q

R

A A

α

c

t

+ −

∂

⎡

⎤

∂

+

_⎢

+

_⎥

−

+

= ρ

−

+

∂

⎣

⎦

(

)

(

_{) (}

₎

2

₍

₎

2 13 1 23 3

11 22

12 21 2 3 3

1 2 1 2 1 2

1

2 A L

A L

L

S

Jh

m

R

A A

α

t

+ −

∂

⎡

⎤

∂ Ω

+

_⎢

+

_⎥

−

=

−

+

∂

(9)

(

)

(

)

(

)

(

)

₍

₎

₍

₎

(

)

(

)

3 3

2

3 3 3 3

2 ,

2

1

4

2

2 ,

2

4

2 ,

2

ii ii jj ij ij ji

i i i i i i i i

Eh

v

T

v

h q

q

S

h

v

L

h

m L

h

N

h

h q

q

L

h

h m

m

+ −

+ − + −

⎡

⎤

⎡

⎤

=

_⎣

Γ + Γ −

_⎦

−

=

_⎣

μ+α Γ + μ−α Γ

_⎦

−

γ β+γ

⎡

γβ

⎤

β

_⎡

_⎤

=

_⎢

χ +

χ −

_⎥

=

_⎣

γ+ε χ + γ−ε χ

_⎦

β+ γ

⎣

⎦

αμ

μ−α

γε

γ−ε

=−

Γ +

−

=−

χ +

−

α+μ

μ+α

γ+ε

γ+ ε

(4.7)

( )

3

3 3

3

1

j

i i

i j i

i i i i i i

j j

i i i

ii j ij i

i i i j j i i i i j j

u

Ω

w

Г

Ω

,

χ

A

α

R

A

α

R

u

A

w

A

Г

u

,

Г

u

Ω

A

α

A A

α

R

A

α

A A

α

∂

= −

−

+ −

= −

−

∂

=

+

−

=

−

+ −

∂

(4.8)

3

1

i

1

i

1

j

1

i

ii j ij i

Ω

A

Ω

A

χ

Ω

,

χ

Ω

A

α

A A

α

R

A

α

A A

α

∂

=

+

−

=

−

∂

[5]:

( )

( ) ( ) ( )

( )

1 1 2 1

01 02

01 2

2

03 02

,

2 ,

,

2 ,

PQ PQ

j

Q t e Q e P

j

Q t e

Q e P

h

j

P t

d

c

t

R

u

P t

w P t

h

H

P

c

t

Ω

⎡

⎤

σ ∂

= −

⎢

+

⎥

Ω +

∂ ⎢

_⎣

⎥

_⎦

∂

⎛

⎞

σ

+

_⎜

−

_⎟

∈Ω

∂

⎝

⎠

∫

( )

( ) ( ) ( ) ( ) ( ) ( )

( )

2 1 2 2

01 02

02 2

1

01 03

,

2 ,

,

2 ,

PQ PQ

j

Q t e

P e Q j

Q t e Q e

P

h

j

P t

d

c

t

R

w P t

u P t

h

H

P

c

t

Ω

⎡

⎤

σ ∂

= −

⎢

⎥

Ω +

∂ ⎢

_⎣

⎥

_⎦

∂

⎛

⎞

σ

+

_⎜

−

_⎟

∈Ω

∂

⎝

⎠

∫

(4.9)

−

3 , , _ij _i

ii S N

T ; L_ii ,L_ij ,L_i₃−

; Ãii,Ãij,Ãi₃− ,

3

, ,

ii ij i

χ χ χ −

-. e1,e2–

α

1

α

2

;

R

_PQ– (1.5)

( )

3

R [5];

R

_PQ –

Q

∈Ω

P∈R3 (

(4.9),

P

∈Ω

).

Γ

(10)

* * *

11 1 3 12 2 3 13 3 3

* * *

11 1 3 12 2 3 13 3 3

, S

,

h h h

T

p d

N

p d

L

m d

L

m d

L

m d

Γ Γ Γ

− − −

Γ Γ Γ

− − −

=

α

=

α

= −

α

=

α

=

α

= −

α

∫

(4.10)

Э Γ [5]:

0 , 0 02 01= j =

j (4.11)

(4.6)-(4.9) (4.10),(4.11) .

, (4.6)-(4.9), (4.10),(4.11)

.

5. х х ч

ё щ . ,

(3.2) :

2 2 2

* * *

2 2 2

~ 1 l , ,l l

α _, ε

E R E R E R E

− − −

= λ β = λ γ = λ ε , β γ ε_*, ,_* _*~ 1 (5.1)

ω

, k κ₁ (5.1)

1

1 , , 2 2

2

p p

k l p c

l l

ω = + = − κ = − − + , (5.2)

2 2 , 0 2 .

c= p l− l≤ p c= l≥ p

( )

p l

Oλ − :

(

)

(

)

(

)

(

)

(

)

0 0 2 1 0 0 2 1 0 2 1

0 1 2 2 2 0 2 0 1

3 3 3 3 3 3 33 33 33

0 0

3 3

, ,

, , , ,

l p c l p c l p l p c

ii ii ii ij ij ij i i i

p c l c l p p c l p c

i i i i i i

l c p c c

i i ii ii

E E V h V V

E E E

V h V v RE v

− + − − + − − − + −

− − + − + − − + −

− − −

τ = λ τ + λ ζτ τ = λ τ + λ ζτ = λ + λ ζ τ = λ τ + λ ζτ + λ τ τ = λ τ τ = λ τ + ζτ

= λ ω = λ ω = λ

(

)

(

)

(

)

(

)

0 0 0 0 2 1

3 3 3

0 2 1 0 1 2 2 2 2

3 3 3 33 33 33 33

, ,

3 ,

c l p c

ij ij

p l p c c l c l p

i i i

v RE v

v RE v v i v RE v v v

− − + −

− − + − − − + − +

= λ ω = λ ω + λ ζω

= λ + λ ζ ↔ = λ + λ ζ + λ ζ

(

)

2 0 1 2 0 0 1 0

3 3 2

,

, 2

,

i i i k k

c

h

E

h

E

h

l

p

j

E

j

c

κ κ κ

=

λ

+ ζ

=

λ

κ = − +

=

σ

λ

(5.3)

,

.

(4.4) (4.5), ,

(5.1),(5.2),

( )

p l

Oλ −

-ё

:

(

)

(

)

( )

0 03 22

(

)

1 1 1 1 1

1 j 2

j ji

ii i i

ii jj ji ij j i i

A S

T A u

T T S S j H h q q

A α A A α A α A A α c t

+ −

∂ ∂

∂ ₊ ₋ ₊ ₊ ∂ ₊ _{+ −} _{= ρ} ∂ ₊ ₊

(11)

(

)

(

)

2

₍

₎

2 13 1 23

11 22

01 02 02 01 2 3 3

1 2 1 2 1 2

1 1 1

2

A N A N

T T w

j H j H h q q

R R A A α α c c t

+ −

∂ ∂

⎡ ⎤ ∂

+ + _⎢ + _⎥− + = ρ − +

∂ ∂ ∂

⎣ ⎦ (5.4)

( )

(

)

(

( )

)

(

( )

)

(

( )

)

( )

(

)

(

( )

)

3

( )

2 2

(

) (

)

1 1 1

1 1 1 1

1

1 1 1 2

j j j j j

ii ij ii ij jj ji ji jj

i i i j i j j

j j j j

i

ji jj ij ii i j j i i

i j j

A

G L G L G L H L

A α A A α A α

A

H L H L N Jh m m h q q

A A α t

+ − + −

∂

∂ _{− −} ₊ _⎡ _{− −} ₋ _{+ −} _⎤₋ ∂ _{+ −}

⎣ ⎦

∂ ∂ ∂

⎡ ∂ Ω ⎤

∂ ⎡ ⎤

− _∂ _⎣ + − + − − _⎦− = − ⎢ _∂ + + ⎥− −

⎢ ⎥

⎣ ⎦

[

] ( )

j

(

)

12 21 3 3

2

1 ,

-1

1

2

ii ii jj ij

Eh

T

v

S

m

v

+ −

⎡

⎤

=

_⎣

Γ + Γ

_⎦

=

Γ + Γ +

+

−

+

(5.5)

( )

[

]

( )

(

)

(

)

₍

₎

₍

₎

3 3

j

12 21 3 3 33

2

33

2

1 , -1

3 1

2 3 1

4

2

2 ,

2

ii ii jj ij

ii ii jj ij ij ji

Eh

G

K

vK

H

K

m

L

v

L

h

L

h

+ −

⎡

⎤

⎡

⎤

=−

_⎣

+

_⎦

=

+

_⎣

+

_⎦

+

−

γ β+γ

⎡

γβ

⎤

β

_⎡

_⎤

=

_⎢

χ +

_⎥

=

_⎣

γ+ε χ + γ−ε χ

_⎦

β + γ

β+ γ

⎣

⎦

1

j

i i i

ii j ij i

u

A

w

A

Г

u

,

Г

u

A

α

A A

α

R

A

α

A A

α

∂

=

+

−

=

−

∂

(5.6)

( )

(

)

(

)

3

3 21 12

1

1 ,

,

1

1 ,

2

j

i i i i

ii j ij i i

i i i j j i i i j j i i i

j

i i i

ii j ij i

j

i j j

A

w

u

K

A

α

A A

α

A

α

A A

α

A

α

R

Ω

A

Ω

A

χ

Ω

,

χ

Ω

A

α

A A

α

R

A

α

A A

α

∂β

∂

=

+

β

=

−

β

β =

+

∂

=

+

−

=

−

∂

Ω = − −

ψ − β

Ω =

Γ − Γ

( )

1

₍

₎

( )

1

₍

₎

33 11 22 3 3

1 1

th

1 4

hk

L

h

m

hk

+ −

⎛

⎞

=

_⎜

−

_⎟

γ χ + χ

−

⎝

⎠

.

(5.4)-(5.6)

(4.9) j10,j20

.

Γ

[14,15]:

(

)

(

)

* * * *

11 1 3

, ,

12 2 3 12 11 3 1 2 3

h h h

T

_Γ

p d

S

_Γ

p d

L

G

_Γ

p

m d

− − −

=

∫

α

=

∫

α

−

=

∫

α

+

α

(5.7)

(

)

*

(

* *

)

13 12 11 3 3 2 1 3

2 2 2 2

1

h

1

h

-N

H

L

p

m

d

A

Γ ₋

A

⎡

∂

⎤

⎡

∂

⎤

+

−

=

+

α

−

α

⎢

_∂α

⎥

⎢

_∂α

⎥

⎣

⎦

∫

⎣

⎦

Э Γ (4.11).

(5.4)-(5.6),(4.9), (5.7), (4.11)

ё