Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 289 0 1 1 1 0 1 * ( ) ( ) 0 k k ij ikl Y Y j j i l l i YC u u w w dy e dy x y y x y y w V                                                                       (72) All these results can then be summarized through the following proposition: Proposition3. Under the expansions (47) and (48) for the solution ( ( , ), ( , )) u x y x y    of Problem (46), the first term 0 0 ( ( ), ( )) u x x  satisfies Eqs. (62)-(63) and appropriate boundary conditions. Furthermore, for given 0 0 ( ( ), ( )), u x x  the field 1 1 ( ( , ), ( , )) u x y x y  is the solution of the nonlinear problem ( LHP ; Eqs. 70-71), and ( 0 ij   , 0 i D  ) is therefore, defined as functions of 0 0 ( ( )) ( ( )). x x grad u x and grad x  So, Eqs. (70)-(71) represent a nonlinear piezoelectric law. A NALYSIS OF THE ( STRAIN , ELECTRIC POTENTIAL )-( STRESS , ELECTRIC DISPLACEMENT ) LAW Remark1. Problem ( LHP ) can be written as in the following simplified form: Find 1 1 * * ( , ) u u YC YC u inV V     such that we obtain, for given 0 0 ( ) ( ) u x and x  :     0 1 1 0 1 1 * ( ) ( ) ( ) ( ) ( ) ( ) 0 (.) (.) (.) (.) ; (.) ; (.) u ijkl klx kly jiy Y u ijk kx ky ijy Y k k klx kly kx l l l u u YC a h u h u h w u dy e h h h w u dy h h h x y x w V                                (73) and     0 1 1 0 1 1 * ( ) ( ) ( ) ( ) ( ) ( ) 0 (.) (.) ij jx jy iy ikl klx kly iy Y Y iy k YC h h h w dy e h u h u h w dy h y w V                               (74) Remark2. Denoting 0 1 1 1 0 0 ( ); ( ); ; ; ; kl klx kl kly ij ij i i H h u h h u u u D D           (75) Problem ( LHP ) can then be formulated as follows: Find * * ( , ) u u YC YC u inV V     such that we obtain, for given 6 3 kl k H and H   R R :     * ( ) ( ) ( ) ( ) 0 u u ijkl kl kl ji ijk k k ij Y Y u u YC a H h u h w u dy e H h h w u dy w V                  (76)