Issue 42
J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30
H OMOGENIZED LOCAL PROBLEM IN THE PERIOD Y
L
et us choose the fields 1
1 v and as:
u u
1
1
1
( , ) v x y u x y ( , )
( ) w y u x y ( , )
(65)
( ); 0 1; u u u D w V
u
*
YC
w y
1
1
1
( , ) x y
( , ) x y
( , ) x y
( )
(66)
( ); 0 1; D w V
*
YC
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,
u
1
0 k u u x y ( ); 0 ijkl l l u u a D ( ); 0 ij j j x y D 1 k 0 1
(
u
w u
1
)
0
1
(
w u
)
j
j
u
u
i
i
dy
e
dy
0
ijk
(67)
y
x y
y
i
k
k
j
1
0 k u u x y 1 k
1
1
( w
( w
)
)
dy
e
dy
0
ikl
(68)
y
y
i
l
l
i
1
Therefore, we locally obtain, respectively:
u
1
(
u
w u
0 k u u x y
1 k
1
)
0
1
(
w u
)
j
j
u
u
*
i
i
0;
a
e
w V
(69)
ijkl
ijk
YC
y
x y
y
l
l
i
k
k
j
0 k u u x y 1 k
0
1
1
1
x y
( w
( w
)
)
*
0;
e
w V
(70)
ij
ikl
YC
y
y
j
j
i
l
l
i
and the local homogenized problem ( LHP ) in Y, then follows:
* inV V u YC
u
1 1 ( , ) u
*
such that we obtain, for given 0
0
( ) ( ) u x and x :
Problem (LHP): Find
YC
u
1
(
u
w u
0 k u u x y
1 k
1
)
0
1
(
x y
w u
)
j
j
i
i
a
dy
e
dy
0
ijkl
ijk
y
y
Y
Y
(71)
l
l
i
k
k
j
u
u
*
w V
YC
and
288
Made with FlippingBook Ebook Creator