Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 287 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ij ij ij j i j i j i k k ij ikl ikl j i l i i i dx dx dx x x x y y x u u dy e dx e dx y y x x xl y e                                                                                                                 1 1 0 0 1 1 0 1 1 0 ( ) ( ) 0; , ( ) k k kl ikl l i l i u u dx e dy H y x y y                                                        (59) with  represents the operator average defined on any Y-periodic function f(y) by: 1 ( ) f f y dy YY    (60) In the particular case where 1 1 1 1 , v u and     we get: 0 0 0 1 0 0 0 1 0 1 1 3 0 ( ) ( ) 0; , ( ( )) j j k k i i ijkl ijk l l i k k j v u u u v u a dx e dx x y x x y x v v H                                                    (61) 0 1 0 0 0 0 0 1 0 1 1 0 ( ) ( ) 0; , ( ) k k ij ikl j j i l l i u u dx e dx x y x x y x H                                                           (62) Consequently, the corresponding homogenized equations in which there are no more fissures then follow: 0 0 1 0 1 0 0 ij j k k ij ijkl ijk l l k k x u u a e x y x y                                               (63) and 0 0 1 0 1 0 0 i i k k i ij ikl j l l D x u u D e xj y x y                                             (64)