On problem of electroelastic wave propagationin trigonometrically graded piezoelectric layer

72

Ա ԱՍ Ա Ի Ի ՈՒԹ ՈՒ Ի Ա Ա Ի Ա Ա ԻԱ Ի Ղ Ա Ի

Ц

67, №1, 2014

539.3

А АЧ А А

Ч

Щ А.А.

ч ы : , ,

Key Words: electroelastic waves, inhomogeneous piezoelectric medium, tension distribution over the thickness of the layer

Ք Ա.Ա.

[4]- : Ո

-:

-, և :

Kamalyan A.A.

On problem of electroelastic wave propagationin trigonometrically graded piezoelectric layer

This paper is a continuation of paper [4]. Here the propagation of electroelastic waves in trigonometrically graded layer (over thickness) made of 6mm piezoceramics is investigated. Electroelastic and mechanical characteristics of the layer (metalized and tension free surface) are variations of the trigonometric function, which is derived here. The dispersion of shear wave is analized and the numerical data is added.

[4],

.

6mm - .

, [4].

1.

Ox

( 6mm),

( ; ; )

x y z

   

x

; 0

 

y

2 ;

h

    

z

.

Oz

,

xOy

,

0;

2

y



y



h

.

0;

0

yz

73

[5].

ё .

, :

2 0 2

yz

xz

W

x

y

t





_

_{ }









(1.2)

0

grad .

y

x

D

D

x

y

E





_

_





 





(1.3)

( ),

a y

a

(0) 1, (0)



a



0

[6]. - ( :

C

44– ,

15

e

– ,



₁₁ –



–

) :

0 0

15 15 11 11

0

44 44 0

( )

( ) ,

( )

( )

( )

( ) ,

( )

( )

e

y

e a y

y

a y

C

y

C a y

y

a y





 





 

(1.4)

0 0 0 44

,

15

,

11

,

0

C

e

 

.

y



0

0 0 0 44

,

15

,

11

,

0

C

e

 

ё .

6mm (1.4)

,

xz yz

 

D D

_x

,

_y

:

0 0 0 0

44

( )

15

( )

,

44

( )

15

( )

xz yz

W

W

C a y

e a y

C a y

e a y

x

x

y

y









 



 











(1.5)

0 0 0 0

15

( )

11

( )

,

15

( )

11

( )

x y

W

W

D

e a y

a y

D

e a y

a y

x

x

y

y











 



 









(1.6)

(1.5), (1.6) (1.2),(1.3),

,

.

2

2 0

0 2

44

(1

0

)

a

W

W

W

a

y

C

t

















 



(1.7)

0

2 15 2

0 11

e

a

a

W

W

a

y

a

y













  



_





_





_



_

(1.8)

0 2 15 0 0 0

44 11

e

C

 



–

74

ё . .1

-

6mm.

1 CdS

 



₁

[3]

ZnO

 



2 [1]

PZT-7

 



3 [9]

PZT-5H

 



4 [10]

PZT-4

 



5 [1]

10 2

44

10

C



Н

1.49 4.25 2.5 2.3 2.56

3 3

10

г



4.82 5.68 7.8 7.5 7.5

2 15

e



К

-0.21 -0.59 13.5 -17.0 12.7

11 11

10

Ф



 

7.99 7.38 1710 22.71 6.45

0



0.03 0.11 0.42 0.55 0.97

t

C



с

1784 2881 2133 2180 2593

(1.7)-(1.8) :

1 2

0

( )

( ) exp (

)

W



W y a



y

i kx

 

t

(1.9) 

1 2

0

( )

y a

( ) exp (

y

i kx

t

)



  

 

(1.10) 

(1.9) (1.7), :

2

2 0

0

2

1

2

0

W

k

W

y

k





_



_

_



_













(1.11)

2 2

t

V C



 

, (1.12)

0

2 44 0

0

(1

)

t

C

C



 



– ,

2 2

V

k



 

  

 

–

 

2 2

1

1

2

4

a

a

a

a





 





.

( )

a y

,

 

2

2

1

1

0

2

4

a

a

a

a





 







, (1.13)

ё 2

2

k

 



,



– ,

ё ,

 

0

, [4]. (1.11) :

2

2 2 0

0

2

0

W

k

W

y



 



75









0 1

sinh

2

cosh

W



A

k y





A

k y



,

1

,

2

A A

–





1 2 2

1

     

.

0

  

1,

, ё :

2

0

    

1

2

0

  

1

. (1.15)

(1.9), (1.8):









 

 

0 15

0 0 1 2 3 4

11

sinh

cosh

sinh

cosh

e

A

k y

A

k y

A

k y

A

k y





 

_

 



_



 





2

1

 

 

.

2.

a y

 

,

(1.13),

:





2

2 1

( )

cos

2

a y



C



_



y



C



_

,

1

,

2

C C

– .

(0) 1,

(0)

0

a



a





ё

C

₁



0,

C

₂



1,

a y

( )



cos (

2



y

)

.

,

 



/

k

, ,

,

2

( )

cos (

)

a y



k y



(2.1)

, ,

W x y t

( , , )





x y t

, ,



– :









1 2

(

sinh

cosh

)

exp (

)

cos(

)

A

k y

A

k y

W

i kx

t

k y

 





 



(2.2)













 

 





0 1 2 15 0 11 3 4

sinh

cosh

exp (

)

cos(

)

sinh

cosh

exp (

).

cos(

)

A

k y

A

k y

e

i kx

t

k y

A

k y

A

k y

i kx

t

k y

 



 

  





 





 



(2.3)

(2.1), (2.2) (2.3) (1.5), :

0 0

44 15

0 0 0 0

1 44 0 1 2 44 0 2 3 15 3 4 15 4

( )

( )

,

(1

)

(1

)

,

yz

yz

W

C a y

e a y

y

y

A C

A C

A e

A e





 







 

   

   

 



(2.4)

















1 2 3 4

cosh[

]cos(

)

sin(

) sinh[

]

sinh[

]cos(

)

sin(

) cosh[

]

cosh[

]cos(

)

sin(

) sinh[

]

sinh[

]cos(

)

sin(

) cosh[

]

k

k y

k y

k y

k y

k

k y

k y

k y

k y

k

k y

k y

k y

k y

k

k y

k y

k y

k y

 





  





 





  





 





  





 





  





76

(1.1)

k





kh

,

det

A

_i



0,

i



1,..., 4

:

2 2

0 0 0

2 2 2 2 2

0 0

2 0

2 2

0

( ,

, , )

2

1

(1

)(cosh[2

]cosh[2 1

] 1)

((1

)

(1

) ) sinh[2

]sinh[2 1

]

tan[2

] (

1) sinh[2 1

]cosh[2

]

tan[2

]

1

cosh[2 1

]sinh[2

] 0

T

k

k

k

k

k

k

k

k

k

k

k

 

     

 



 

 



    

 



 



 

   

 

 

 

 

 

 

 























(2.6)

0

 

ё

[4].



,

.

(

k





0

),

, (2.6) :

2 2

0 0

(1

)(1

/ (1

)) 1

   

 

 

  

. (2.7)

(

k



 

), ,

2 2 2 2

0 0

(1

)(1

/ (1

) ) 1

   

 

 

  

. (2.8)

(2.6) ,

.

ё



0 ,



k



(2.6).

(2.2) (2.3), ,

0

2

k y



    

. П ав яя а а ь ач

y



2

h

я а а в



k





kh

, ч в :

0

4

k



    



(



– а ч а ь ч ). (2.9)

.2 (2.9)

.



2

0.15

 

 

0.3

 

0.5

 

0.7

5.23

k





k





2.67

k





1.57

k





1.21

,

 

0.3

k



(0,2.67).

.3 ё



0



( .1)



(

k



( . 2).

“-“

77

3

0.15

 

1





₂



₃



₄



₅

0.1

k





0.9709 0.9011 0.7049 0.6459 0.5084

0.5

k





0.9730 0.9077 0.7203 0.6629 0.5271

1

k





- 0.9227 0.7572 0.7041 0.6353

2

k





- 0.9502 0.8291 0.7858 0.6699

5

k





- 0.9765 0.8925 0.8546 0.7124

0.3

 

0.1

k





- 0.9011 0.7049 0.6459 0.5084

0.5

k





- 0.9077 0.7204 0.6630 0.5273

1

k





- - 0.7588 0.7058 0.5757

2

k





- - 0.8338 0.7910 0.6753

0.5

 

0.1

k





- - 0.7049 0.6459 0.5084

0.5

k





- - 0.7207 0.6634 0.5277

1

k





- - - 0.7092 0.5797

1.5

k





- - - - 0.6314

ч и .

.

.

,

C

_t2 .1

, ё .3.

[4],

, ,

, .

-

( (2.2) (2.3)).

1 2 0

W a

 1 2 0

a





,

. .

 

k y



. ,

(1.15) (2.8)

, [4]

78

1. . .

.// . . . 1988.

.41. №5. .34-40.

2. . ., . .

.// . . . 1994. .47. №3-4. .31-36.

3. . .

.// . . . . 1987. .40. №1. .24-29.

4. . .

.// . .

. 2013. .66. №4. .38-48.

5. Belubekyan M.V., Belubekyan V.M. Surface waves in piezoactive elastic system of a layer on a semi-space. Proceedings of the Yerevan State University, Physical and Mathematical Sciences 2013, Mechanics, № 3, p. 45–48.

6. Collet Bernard, Destrade Michel, Maugin Gerard A. Bleustein-Gulyaev waves in some functionally graded materials.// European Journal of Mechanics A/Solids 25 (2006) 695-706.

7. Hasanyan D.J., Piliposian G.T., Kamalyan A.H., Karakhanyan M.I. Some dynamic problems for elastic materials with functional inhomogeneties: anti-plane deformations. Continuum Mech. Thermodyn. (2003) 15: 519-527.

8. . .

. // . . . 1999. .52. №1. .12-16.

9. Li Peng, Feng Jin. Bleustein-Gulyaev waves in a transversely isotropic piezoelectric layered structure with an imperfectly bonded interface//Smart Mater. Struct 21 (2012) 045009 (9pp).

10. CAO Xiaoshan, Jin Feng, WANG ZiKun & Lu TianJian Bleustein-Gulyaev waves in a functionally graded piezoelectric material layered structure//Sci China Ser G-Phys Mech Astron, Apr. 2009, vol.52, no. 4,613-625.

11. . ., . ., . ., . .

. .: , , 1991. 416 .

12. . ., . . . :

, 1982.

13. . ., . ., . .

. Editorial , 2003. 336 .

и :

А и А ич – ,

.: (094)90-96-92, (060)44-71-70 E-mail: [email protected]