Issue 42

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





u

on

0

(28)

i

  0 i u N onC  i

(29)

 



(30)

0 on

  0 on C  

(31)

, ijkl k l j a u n e E n  kij k j

) ( 

, ijkl k l j a u n e E n  kij k j

 

, NNkl k l a u e 

) E N on C

(

)

(

(32)

kNN k

i

1

1

2

, NNkl k l a u e 

0 E on C 

(33)

kNN k

(34)

 



D n

D n

on C

0

i i

i i

1

2

where N and n represent the unit normal to C, outer to its side 1, see (Fig. 2), and the unit outer normal to the open domain , C C     respectively.

Figure 2 : Representation of the fissured piezoelectric structure C  .

These relations express a compression on C according to (33), as well as the normality of the force which acts; with an opposition between the action and the reaction, according to (32). Consequently, the variational formulation (FVP) for the problem of such a fissured piezoelectric structure can be stated as follows:

* ( , ) u u inV V    *

Problem (FVP): Find

such that:

u

*

( , ( ,                  ) ( , ) ( , d u  ) 0 ) 0 a u v u b v u v V c     

* V 

(35)

where





u

1

( ); C i V u u u H u   ( i i

);   

0

(36)











u

u

*

( ); i V u u u V u N     ; i i i

0

(37)







1

;         ( ); C H

V

0

(38)

284