o método dos elementos de contorno aplicado à ... - Sistemas SET

[ r i r k nk + r ni]
∂ ∂
( ) ( )
∂x
∂n π
qp
w
D * ⎛ ⎞
⎜ , ⎟ = , , ln( )
⎝ ⎠
− 1
4
i
[ ij r i r j r k nk r i nj r j ni]
35
(3.39)
2
∂ ∂
1
( ) ( δ 2 ) ( )
∂x ∂x
∂n 4π
qp
w
Dr
* ⎛ ⎞
⎜ , ⎟ = − , , . , + , + ,
(3.40)
⎝ ⎠
i j
2
∂ ∂
( )
∂x ∂x
∂n 2π
qp
w r k nk
Dr
* ⎛ ⎞ ,
⎜ , ⎟ = (3.41)
⎝ ⎠
k k
∂ ∂ ∂
( ) [ ( ) ]
∂x
∂x ∂x
∂n π
qp
2
w 1
2r
i r k nk n
2
i
2 Dr
*
⎛ ⎛ ⎞ ⎞
⎜ ⎜ , ⎟ ⎟ = , , − (3.42)
⎝ ⎝ ⎠ ⎠
∂M
∂x
i k k
*
ns
i
( )
[
1 − ν
( qp , ) = n r, s s r, n
4πr
( 1 ν)
i( l l ) + i( k k)
− 2r, i( r, k n k)(
r, l sl
)] (3.43)
{ [ 2( δ ij + 4r i r j )( r l sl ) − 2sir j −2r
i sj] ( r k nk)
2
∂ M
2
∂x ∂x
4π
qp
*
ns
( , ) = , , , , , ,
r
− −
∂
∂x
i j
i
2 * ⎛ ∂ M
⎜
⎝ ∂x ∂x
( , j i , i j)( , ) . i j i j}
− 2r n + r n r s + ns + sn
2 *
∂ M
∂x ∂x
ns
k k
ns
k k
l l (3.44)
( 1 − ν)
( sr )( r n )
{ l l k k }
( qp , ) =
, ,
2
πr
( 1 − ν)
⎞
( qp , ) ⎟ = [ 4r,
i ( r, k sk)( r, n ) ni( r k s
3
l l − , k)
+
⎠ πr
(3.45)
− si ( r, l nl
)] (3.46)
{ ( ν) r ( )( i ν r k nk) [ r i( r k nk) ni]
}
*
∂M
n 1
( qp , ) = 1+ , −2 1 − , , , − (3.47)
∂x4πr i
{
2
∂ M
1
2
∂x ∂x 4π
qp
*
n
( , ) =− + − , , + 21 ( − ν)[ nn i j +
r
i j
( 1 ν)( δij
2r
i r j )
( )
2
− 2r, n r, n −( − 2r,
r, )( r, n ) +
j i k k
δ ij j k k k
[ ]
− ( ) −
⎤ ⎫
2r, i r, k n k n j r, j ( r, k n k)
⎬
(3.48)
⎦⎥
⎭
2
[ k k ]}
2 *
∂ M n
1
( qp , ) =−
⎧
2 ⎨21−
1−2 ,
∂x ∂x 4πr
⎩
k k
( ν)
( r n )
(3.49)