Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 285     * ; ; 0 V V         (39) And where a, b, c, and d are bilinear forms on * * * * * , , , u u u V V V V and V V      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 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 C   is then defined as follows:   1 2 3 ( , , ); C C x x x x x y Y Y C            (41) 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)     * ( ); ; 0 u u i i i i V u u u V u N       (43)   1 ; ( ); 0 C V H            (44)     * ; ; 0 V V          (45) The corresponding variational formulation ( FVP  ) of such a piezoelectric problem in , C   is then defined as follows: W