Issue 42
J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30
u
*
*
inV V
( , ) u
Problem ( FVP ): Find
such that:
u
*
( , ( , ) ( , ) ( , d u ) 0 ) 0 a u v u b v u v V c
(46)
*
V
In order to study the asymptotic behavior of the solution when tends to zero, we use the classical following expansions, both for the unknown and the test functions:
0
1
2 2
( ) u x u x u x y ( ) ( , )
( , ) ...
u x y
(47)
0
1
2 2
( ) x
( , ) ... x y
( ) x
( , ) x y
(48)
0
1
2 2
( ) v x v x v x y ( ) ( , )
( , ) ...
v x y
(49)
0
1
2 2
( ) x
( , ) ... x y
( ) x
( , ) x y
(50)
Introducing the following spaces: ( ); u YC i i
1
( ); C V v v v H Y Y periodic
(51)
( ); YC V v v v V v ; u u YC i
0
(52)
u
u
*
( ); i V v v v V v N on C ; 0 YC i YC i
(53)
* YC V v v v V v * ( ); ; u u YC i
0
(53)
1
;
( ); C H Y Y periodic
V
(54)
YC
V
; ; V
0
(55)
YC
YC
*
;
V
V
;
0
(56)
YC
YC
V
*
*
;
V
;
0
(57)
YC
YC
Comparing (47)-(50) with (46), we get the following relations:
0 0
1 1
0 0
(
(
( u j
v u
v u
v u
0 k
0 k
1 k
)
)
)
u
u x
j
j
j
j
j
dx a
a
dx a
dx
ijkl
ijkl
ijkl
x x
y
y
x
l
l
i
l
i
i
1 1
1 k ( u j
v u
0 0
1 1
)
0
0
(
(
v u
v u
)
)
j
i
i
i
i
dx e
a
dy
e
dx
ijkl
ijk
ijk
x x
y
y
y
x
(58)
l
i
k
j
k
j
0 0
1 1
1
1
(
v u
v u
(
)
)
0 1 1 0; , v v ( ( )) H
3
i
i
i
i
dx e
e
dy
k y
ijk
ijk
0
y y
x
j
k
j
286
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