Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 290 and     * ( ) ( ) ( ) ( ) 0 ij j j i ikl kh kl i Y Y YC H h h w dy e H h u h w dy w V                        (77) Therefore, 0 ij ij    and 0 i i D D  can be written as follows:         ( ) ( ) ( ) ( ) ij ijkl kl kl ijk k k i ij j j ikl kl kl a H h u e H h D H h e H h u                (78) So, the homogenized (strain, electric potential)-(stress, electric displacement) law is characterized by the function defined by: ( , ) ( , ) kl kl ij i H h D     (79) Nevertheless, for the study of (79), let us introduce the following functions, defined from 6 3  R R towards R , by:         1 ( , ) ( ) ( ) 2 1 ( ) ( ) 2 sm s ijkl ij ij lm lm Y ijk i i jk jk Y W H H a H h u H h u dy Y e H h H h u dy Y          (80)        * 1 ( , ) ( ) ( ) 2 1 ( ) ( ) 2 sm s ij i i j j Y ikl ik ik l l Y W H H H h H h dy Y e H h u H h dy Y             (81) Moreover, the proposition that follows presents the main result of this analysis: Proposition 4. The functions defined above, through (80) and (81), are of class C 1 , positive; and D    satisfying the following relations: * 1 2 1 2 ij ij i i W H W D H               (82) Proof. As u and  are continuous functions of     1,2,3 , 1,2,3 ij i i i j H H and H H     , defined from 6 3 ( . ) resp R R towards ( . ), u YC YC V resp V  * W and W are then of class 0 . C Let us now introduce: * * ( ) ( ) ij ij ij i i i h h u H and h h u H          (83)