Cap tulo 2 Análisis de medios anisótropos y

5 0
, .
.
*
9 5
> , .
1
> 3 % 0+> 6
+ %0F+ ( 4) > 9 .
.
B 2C> + 1
.
%0 B4C .
%0 0
.
3 = B &C> 1
- BC B C % ,
,
31
9 > ->
- + - ,
.
.
.
1
> 5 > 5
> .
1
9 =
$
'
σij + .
εkl , "
σij = Cijklεkl
( )
* Cijkl ,
> + - + 5
"
Cijkl = Cjikl = Cijlk = Cklij
()
9
9 9 1
+ 9 9 , % .
9,
.
9 B&C
- > 9- 9 σ13
+ σ23 x3 % - ,
> + BC 8 G ( 4 &) ( )> "
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
σ11
σ22
σ33
σ23
σ13
σ12
⎞
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟=⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
C11 C12 C13
0
0
C16
C12 C22 C23
C13 C23 C33
0
0
0
0
C26
C36
0
0
0 C44 C45
0
0
0 C45 C55
0
C16 C26 C36 0
⎞⎛
⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎜
0 ⎟
⎟⎜
⎜
0 ⎟
⎠⎝
C66
ε11
ε22
ε33
2ε23
2ε13
2ε12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
()
9 x1 − x2 5
(
) > +
+ 9
"
:
#
⎛
⎞
σ11
⎛
C11 C12 C16
⎞⎛
ε11
⎞
⎜
⎟ ⎜
⎟⎜
⎟
⎝ σ22 ⎠ = ⎝ C12 C22 C26 ⎠ ⎝ ε22 ⎠
σ12
C16 C26 C66
2ε12
:
σ23
σ13
=
C44 C45
C45 C55
2ε23
()
2ε13
()
$
6
.
+ , 1
εij = (ui,j + uj,i)
2
(#)
.
- ,
)
*
G σij,j + bi = 0
* bi , .
($)
, + -" σij = σji - . 1
+ -> 9 .
.
(. - ) + ,
> .
- (. - ,) - -> , +
- 8 .
> 9 ,
1
- 9 5
.
BC
'
,
"
* " u(x1 , x2 )
" σ(x1 , x2 )
$
:
-" ϕ(x1 , x2 )
/
-" E(x1 , x2 )> 9 -
* - -
" D(x1 , x2 )
:
-" φ(x1 , x2 )
/
-" H(x1 , x2 )> 9 -
-
" B(x1 , x2 )
,
3 "
σij,j = −bi
(2)
Di,i = fe
(4)
Ei (x1 , x2 ) = −ϕ,i (x1 , x2 )
Bi,i = fm
Hi (x1 , x2 ) = −φ,i (x1 , x2 )
( &)
(
)
( )
( )
bi > fe + fm .
, > -
+ -
> 1
,
3 , + 3
+
,
*
9 (4) + (
) 3 9>
( &) + ( ) + (--) + (--) 1
> , + 3 3 9
,
+ -
> 3 B C
, 3 "
⎧
⎪
⎨ ui , H >>I
uI =
ϕ, HI
⎪
⎩
φ, H
2
( )