Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30

0 ij      Y          x y

   

  

   

   

   





0 k u u x y         1 k

1

1

1

   







(   w

(   w

)

)







dy

e

dy

0

ikl

 





y

y

Y

(72)

 



j

j

i

l

l

i







*

  

w V

YC

All these results can then be summarized through the following proposition:

Proposition3. Under the expansions (47) and (48) for the solution ( ( , ), ( , )) u x y x y   

of Problem (46), the first term

0 0 ( ( ), ( )) u x x 

satisfies Eqs. (62)-(63) and appropriate boundary conditions. Furthermore, for given 0

0 ( ( ), ( )), u x x 

the

D  ) is therefore, defined as

field 1

1 ( ( , ), ( , )) u x y x y 

is the solution of the nonlinear problem ( LHP ; Eqs. 70-71), and ( 0 ij   , 0 i

0

0 x  ( ( )).

( ( )) x grad u x and grad x

functions of

So, Eqs. (70)-(71) represent a nonlinear piezoelectric law.

A NALYSIS OF THE ( STRAIN , ELECTRIC POTENTIAL )-( STRESS , ELECTRIC DISPLACEMENT ) LAW

Remark1. Problem ( LHP ) can be written as in the following simplified form:

* inV V     u YC

u

1 1 ( , ) u

*

such that we obtain, for given 0

0

( ) ( ) u x and x  :

Find

YC





      

0 ( ) a h u h u h w u dy   1 ( ) 1 ( ) u



ijkl

klx

kly

jiy

Y





u

0 ( ) 

1 ( ) 

1







)  

e h

h

( h w u dy

0

ijk kx

ky

ijy

Y

(73)







(.)

(.)

(.)

k

k

(.)   klx h





h

h

;

(.)

;

(.)

kly

kx







x

y

x

l

l

l



u

u

*

  

w V

YC

and









  





0 ( ) 

1

1

0 ( ) e h u h u h w  1 ( ) (

1







( ) ( 



)   dy 







h

h

iy h w

dy

)

0

ij

jx

jy

ikl

klx

kly

iy

Y

Y

(.)    

h

(74)

(.)

iy

y w V  

    

k



*

YC

Remark2. Denoting

0 ( ); H h u h h u u u   1 ( );

1

1

0

0

;  



D D 

;   



;

kl

klx

kl

kly

ij

ij

i

i

(75)

Problem ( LHP ) can then be formulated as follows: Find * * ( , ) u u YC YC u inV V    

6

3

kl k H and H   R R :

such that we obtain, for given









u

u

    



( ) ( 

( ) a H h u h w u dy    ( )

ijk k k e H h 

 

) h w u dy

0

ijkl

kl

kl

ji

ij

Y

Y

(76)



u

u

*

  

w V

YC

289