Issue 42

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

(5)

1     , i 

ij j i   



1 i i   

0 :

where 1  and 0  are constants; a superimposed bar denoting the complex conjugate. But, under the quasi-electrostatic approximation [9], there exists an electric scalar potential  such that:





 

E

(6)

i



x

i

t u w and 0  are prescribed values per unit area, the mechanical boundary conditions can be written as:

Moreover, if 0 0 0 , , j j

0

 

 

n

t

on

(7)

j ij

i

1

0

u on   

u

(8)

j

j

2

and the electric ones are in the following forms:

0

3 i i n D w on    

(9)





 

0 on

(10)

4

;   and n represents the unit outer normal to

The piezoelectric plate is supposed to be clamped by 2 and extrinsic bulk charge are assumed to be negligible,

.  If the body force

and D 

are divergence-free, i.e.





ij j

0   in

(11)



x



0 i D in x i

 

(12)



On the other hand, we have:   0 i ij ij n on       

(13)

S

 

 0 on      i i u u  0       on 

(14)

S

(15)

S



 0

i S n D D on      i i

(16)

The variational problem (VP) corresponding to Eqs. (7) to (12) is obtained by introducing the following spaces:   1 0 2 2 ; ( ), i i i V u u H u u       (17)

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