Issue 42

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





      ;   ; V 



*

V

0

(39)

* u u V V V V and V V      * * * * , , , u

And where a, b, c, and d are bilinear forms on

respectively.

Proposition2. Problem (FVP) is equivalent to Eqs. (26) to (34). The proof is analogous to that of proposition 1, by taking into account Eqs. (29) and (31).

H OMOGENIZED EQUATIONS -F ORMAL EXPANSION

W

e now consider a linear piezoelectric plate with a   periodic distribution of fissures, so that, the period Y of 3 , R admits a smooth fissure C verifying: C Y     (40)

Figure 3 : Representation of the period Y with a smooth fissure C.



The fissured material denoted by

is then defined as follows:

C 





1 2 3 ( , , ); C x x x x x y Y Y C    

    

(41)



C



And we assume that, there is no fissure intersecting the boundary  of the open .  Introducing the following spaces:   1 ( ); ( ); 0 u i i C i V u u u H u         (42)









u

u

*

( ); i V u u u V u N      ; i i i



0

(43)







1

;      ( H



 

V

);

0

(44)





C





      ;   ; V 



*

V

0

(45)



The corresponding variational formulation ( FVP  ) of such a piezoelectric problem in , C  

is then defined as follows:

285