Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 288 H OMOGENIZED LOCAL PROBLEM IN THE PERIOD Y et us choose the fields 1 1 v and  as: 1 1 1 * ( , ) ( , ) ( ) ( , ) ( ); 0 1; u u u u u u YC v x y u x y w y u x y D w V                     (65) 1 1 1 * ( , ) ( , ) ( ) ( , ) ( ); 0 1; YC x y x y w y x y D w V                              (66) where ( ) D  represents the set of the infinitely derivable functions with compact support in  . When we take (64) and (65) into account, then the comparison of (58) and (59) with (60) and (61) respectively, gives the following equations, 1 0 1 1 0 1 ( ) ( ) 0 ( ); 0 1 u u j j u u k k i i ijkl ijk l l i k k j u u w u u u w u a dy e dy x y y x y y D                                                                          (67) 0 1 1 1 0 1 ( ) ( ) 0 ( ); 0 1 k k ij ikl j j i l l i u u w w dy e dy x y y x y y D                                                                                   (68) Therefore, we locally obtain, respectively: 1 0 1 1 0 1 * ( ) ( ) 0; u u j j u u k k i i ijkl ijk YC l l i k k j w u u u w u a e w V x y y x y y                                                          (69) 0 1 1 1 0 1 * ( ) ( ) 0; k k ij ikl YC j j i l l i u u w w e w V x y y x y y                                                                 (70) and the local homogenized problem ( LHP ) in Y, then follows: Problem (LHP): Find 1 1 * * ( , ) u u YC YC u inV V     such that we obtain, for given 0 0 ( ) ( ) u x and x  : 1 0 1 1 0 1 * ( ) ( ) 0 u u j j k k i i ijkl ijk Y Y l l i k k j u u YC w u u u w u a dy e dy x y y x y y w V                                                                (71) and L