Surface electroelastic shear waves in a system of piezoactive layer and half-space

(1)

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А А А А А А

60, №3, 2007

539.3

А

( 6mm)- А ( 43m)

ч . .

чевые в : , , , , .

Key words: electroelastic, shear, wave, layer, half-space.

. . Խ

և (6mm

)-(43m )

`(43m )

(6mm ): և

, :

և և

:

և

:

V.M. Khachatryan

Surface electroelastic shear waves in a system of piezoactive layer and half-space

A piezoactive half-space with a piezoactive layer on its surface is considered. Material of the layer is piezoelectric of class 6mm and material of the space is piezoelectric of class 43m. The layer and the half-space are in a condition of the sliding contact, where shear mechanical stresses are taken to be zero. Interaction between shear waves in the layer and the half-space is due to continuity of electrical field on their boundary. Features of existence of surface electroelastic shear waves are analyzed.

( 43m)

( 6mm). ,

.

. .

1.

- ,

, ( ., [1,2]). ,

, [3-5]. [6]

,

, . [7]

.

6mm

. .

.

, ,

,

43m , –

(2)

6mm

43m .

.

( , , )

x y z

x

−∞ < < ∞

,

0 ≤ < ∞

y

,

−∞ < < ∞

z

, –

x

−∞ < < ∞

,

− ≤ <

h

y

0

,

−∞ < < ∞

z

.

−∞ < < ∞

x

,

y h

−∞ < < − ,

−∞ < < ∞

z

.

( )

[8]:

(1) 2 2

2 14 1 1

1 1 2

1

2 e

W

a

W

x y

t

∂ ϕ

∂

∆

+

=

ρ ∂ ∂

∂

,

2 (1) 1 11

1 (1) 14

2 W

x y

e

∂

ε

=

∆ϕ

∂ ∂

(1.1)

( )

[9,10]:

2

2 2

2 2 2

W

a

W

t

∂

∆

=

∂

,

( )

(2) 11 2 ₂ 2

15

W

e

ε

∆

=

∆ϕ

(1.2)

(1.1) (1.2) (1) 2 44 1

1

c

a

=

ρ

,

(2) 2 44 2 2

2

(1

)

c

a

=

+ χ

ρ

,

(2) 2 15 2 (2) (2)

11 44

[

e

]

c

χ =

ε

(1.3)

1

W

₂ – ,

ϕ

₁

ϕ

₂ – ,

, ,

c

44(1)

(2) 44

c

–

,

χ

₂ –

,

e

₁₄(1)

e

₁₅(2) – , ,

.

:

(1) 23

0 σ =

,

σ =

(2)₂₃

0 ,

ϕ = ϕ

₁ ₂

,

D

₂(1)

=

D

₂(2)

y

=

0

(1.4) (1)

23

σ

(2)23 – ,

(1) 2

D

2(2) –

.

y

= −

h

(2) 23

0 σ =

,

ϕ =

₂

0

(1.5)

2

0 ϕ =

.

(1.1) (1.2),

(1.4), (1.5) :

1

lim

0

y→∞

W

=

,

lim

y→∞

ϕ =

1

0

(1.6)

(1.1) ,

(1.6), :

1

[

1

exp(

1

)

1

exp(

2

)]exp (

)

(3)

(1) 44

1 (1) 1 1 1 1 2 2

1 2

14

1 1

[ ( ) exp( ) ( ) exp( )]exp ( ) 2

c

A p kp y B p kp y i t kx

p p

ie

− η − η

ϕ = − − + − − ω −

0 < η <

1

. (1.8)

(1.7)

2 2

1 1 1 1

1

2 4 8 (2 ) 16 2

p = − η + χ + η + χ − η + χ

2 2

2 1 1 1

1

2

4 8 (2

) 16

2 p

=

− η + χ − η + χ

− η + χ

2

2 2 1

k a

ω

η =

,

(1) 2 14 1 (1) (1)

11 44

[

e

]

c

χ =

ε

–

.

(1.2) :

2

[

2

exp(

1 )

2

exp(

1 )]exp (

)

W

=

A

k

− θη +

y

B

−

k

− θη

y

i

ω −

t

kx

(1.9)

(2) 15

2 (2) 2 2

11

[ exp(C ky) Dexp( ky) e [A exp(k 1 y) B exp( k 1 y)]]exp (i t kx)

ϕ = + − + − θη + − − θη ω −

ε 2

1 2 2

a

ϑ =

.

(1.7)-(1.9)

A

₁,

A

₂,

B

₁,

B

₂, C,

D

– .

2.

, , (1.4)

(1.5) :

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

44 14 44 15

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

1 2 14 11 15 11

0,

0 ,

W

c

e

c

e

y

x

y

W

e

x

y

∂

∂ϕ

∂

∂ϕ

+

=

+

=

∂

∂ϕ

∂

∂ϕ

ϕ = ϕ

− ε

=

− ε

∂

,

y

=

0

(2.1)

c

₄₄(2)

W

2

e

₁₅(2) 2

0 y

y

∂

∂ϕ

+

=

∂

,

ϕ =

2

0 y

= −

h

(2.2)

(1.7), (1.9) (2.1), (2.2)

( ,

B

1

A

,

A

1, ,

A

2,

B

2,

C

,

D

):

(2) (2)

44 44

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

15 15

(1

)

c

(1

)

c

0 C

D

A

B

e

− + − χ γ

− + χ γ

=

(2.3)

(2) (2)

15 15

2 2 2 2

(2) (2)

11 11

exp(

−ζ

)

C

+

exp( )

ζ

D

+

e

exp(

−γ ζ

)

A

+

e

exp(

γ ζ

)

B

=

0 ε

ε

(2.4)

(2) (2)

44 44

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

15 15

exp( )C exp( )D (1 ) c exp( )A (1 ) c exp( )B 0

e e

(4)

(2) (2)

(2) (2) 15 15

11 11 (2) 2 (2) 2

11 11

(

L

+ ε

)

C

+

(

L

− ε

)

D

+

L

e

A

+

L

e

B

=

0 ε

ε

(2.6)

(1) 2 3 3

11 2 2 1

1 1 1

2

2 1 1 2 1

(

1 )

[1 (2

)(1

)

]

2 (

)

p

L

p

ε

+ − η

−

=

+ χ − η − γ +

γ

−

γ

+ γ

,

1

1 γ =

− η

,

γ =

₂

1 − ϑη

,

ζ =

kh

(2.7)

(2.4) (2.5)

C

D

A

₂

B

₂

: (2)

15 2 2

2 2 2 2 2 2

(2)

2 2

11 (2)

15 2 2

2 2 2 2 2 2

(2)

2 2

11

1

1 [(1

) exp(

)

(1

) exp(

)

]

2

1

1 [(1

) exp(

)

(1

) exp(

)

]

2 e

C

A

B

e

D

A

B

e

−ζ

ζ

+ χ

= −

+ γ

−γ ζ

+ − γ

γ ζ

χ

ε

+ χ

= −

− γ

−γ ζ

+ + γ

γ ζ

χ

ε

(2.8)

(2.8) (2.3) (2.6) 2

A

B

₂:

2 2 2 2 2 2

2 2 2 2

2 2 2 2 2 2

2 2 2 2

(1 ) (1 )

[[ ]exp( ) (1 )]

2 2

(1 ) (1 )

[[ ]exp( ) (1 )] 0

2 2 A e e B e e ζ −ζ ζ −ζ

χ − γ + χ χ + γ + χ

− −γ ζ + γ + χ +

χ + γ + χ χ − γ + χ

+ − γ ζ − γ + χ =

(2) (2)

11 2 2 2 11 2 2 2

2 2

(2) (2)

11 2 2 2 11 2 2 2

2 2

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

[[ ]exp( ) ]

2 2

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

[[ ]exp( ) ] 0

2 2 L L L A e e L L L B e e ζ −ζ ζ −ζ

− ε χ − γ + χ ₊ + ε χ + γ + χ _{−γ ζ −} ₊

χ χ

− ε χ + γ + χ + ε χ − γ + χ

+ + γ ζ − =

χ χ

(2.9)

:

2 2 2

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

( , , , ) L[2 (1 ) ( (1 ) ) sh( ) sh( ) 2 (1 ) ch( ) ch( )]

Γ η ζ χ χ ≡ χ + χ γ + χ + γ + χ ζγ ζ − χ + χ γ ζγ ζ +

(2)

11

(1

2

) [

2 2

ch(

2

) sh( )

2

(1

2

) sh(

2

) ch( )]

0 +ε

+ χ γ χ

ζγ

ζ − γ

+ χ

ζγ

ζ =

(2.10)

3. (2.10) , η = 1

ϑ ,

.

(ζ <<1)

(2.10) :

2

(1

)

1 ϑη + χ =

2 1 (1 ) η =

ϑ + χ (3.1)

(1.8),

(1

)

1

(5)

.

(2.10) ζ → ∞( –

,

) :

(2)

1

( , ,

1

,

2

)

L

[

2

(1

2

) 1

]

11

(1

2

) 1

0 Γ η ∞ χ χ ≡ χ − + χ

− ϑη + ε

+ χ

− ϑη =

(3.2)

2

( , ,

2

)

2

(1

2

) 1

0 Γ η ∞ χ ≡ χ − + χ

− ϑη =

(3.3)

(3.3) :

2

1 2

2 2

[1

]

(1

)

−

χ

η = ϑ

−

+ χ

2 2

1 (1

)

χ

−

< ϑ

+ χ

.

4. 43m

GaAs

(

c

₄₄

=

5.94 10

×

10 / 2,

3

5.307 10

ρ =

×

/ 3,

e

₁₄

= −

0.16

/ 2,

ε =

₁₁

9.73 10

×

−11 / )

6mm

ZnO

(

c

₄₄

=

4.25 10

×

10 / 2,

ρ =

5.68 10

×

3 / 3,

e

₁₅

= −

0.59

/ 2, 11

11

7.38 10

−

ε =

×

/ ).

1

ζ

η

₁

η

₂

η

₃

1 ζ <<

0.669221 - -

0.001 0.669222 - -

0.05 0.669281 - -

0.1 0.66947 - -

0.3 0.672117 - -

0.5 0.170154 0.680539 -

1.0 0.602692 0.768529 -

5.0 0.664186 0.818872 -

10 0.676513 0.752176 0.875203

. 1 (2.10) (2.7)

( )

ζ. ,

(3.1).

(3.2). (3.2)

2 2 2 1

0 a

a

< η <

η =₁ 0.677445.

(3.3) η =₂ 0.736075.

(2.10). , ζ =100 2

2 2 1

0 a

a

< η <

,

1

(6)

1. . ., . .

.

// . . . 1995. .48.

3. .43-52.

2. . ., . . Э . : .

, 2006. 492 .

3. Maerfeld C., Tournois P. Pure shear elastic surface wave guided by the interface of two semi-infinite media. //Appl. Phys. Lett. 1971. V.19.

4. P.117-121

4. . ., . Э

. // . .

. 1994. .47. 3-4.

.31-36.

5. Li. Sh. The electromagneto-acoustic surface wave in a piezoelectric medium: the Bleustein–Gulyaev mode. //J. Appl. Phys. 1996. V.80. №9. P.5264-5267.

6. Zakharenko A. Love-type waves in layered systems consisting of two cubic piezoelectric crystals. //Journal of sound and vibration. 2005. V.285. P.877-886.

7. . ., . .

- . //

. . 2006.

3. .25-30.

8. Cheng N.C., Sun C.T. Wave propagation in twolayered piezoelectric plates. //J. Acoust. Soc. Amer., 1975. V57. №3. P.632-639.

9. . ., . . . : ,

1982. 239 .

10. . ., . ., . .

. .: . 2003. 336 .