Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 282 1 1 , 0 : i ij j i i i             (5) 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: i i E x      (6) Moreover, if 0 0 0 , , j j t u w and 0  are prescribed values per unit area, the mechanical boundary conditions can be written as: 0 1 j ij i n t on     (7) 0 2 j j u u on    (8) and the electric ones are in the following forms: 0 3 i i n D w on     (9) 4 0 on     (10) The piezoelectric plate is supposed to be clamped by 2 ;   and n represents the unit outer normal to .  If the body force and extrinsic bulk charge are assumed to be negligible, and D  are divergence-free, i.e. 0 ij j in x      (11) 0 i i D in x     (12) On the other hand, we have:   0 i ij ij S n on        (13)   0 i i S u u on      (14)   0 S on        (15)   0 i i i S n D D on      (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)