Issue 52
O. Kryvyi et alii, Frattura ed Integrità Strutturale, 52 (2020) 33-50; DOI: 10.3221/IGF-ESIS.52.04
The latter makes it possible, after reducing the matrix to a diagonal form, to obtain a solution to system (13):
2
n
* { 0.5 b с 1 j
*
( )
2 12 1 n (
3 0 2 )} n
U
b
jn
n n
j
2
2
a
5, V ( ),
* * n n j ,
W [
( )
( )]
2 b F
1 s F j
s
j
1, 2;
n
n
j
n
1
n
(16)
c
n n
U ( )
,
n
3
23 q a
2
2
2
q d
4 3
0
n
U ( )
0
1 )) n
1
[ ( d
id
1 )]}
{2
(
n
n
n
n
4
00
10 1
01
q
2
2
2
2
a
a
65
where
(2 1)(2 )!! , n n s
*
*
*
*
n
1 24 b q b q s 2 44 , j j
1 25 b q b q j j
.
j
j
1
1
2 45
(2 1)!! n
q d a
4 3
5, V ( )
2
2
2
2
0
1, )) n
a
[ ( d
id
{2
(
)]}
n
n
n
n
n
00
0,
10 1,
01 1,
1,
q
65
, W ( , ) n n n n ( )
1
3 ( ) R , and representation (17) allow to write expressions for the
Equality
which is true in 0, 1, 2, n , 0
coefficients in the expansion (12) so (
a )
1,
3,
3,
V ( ) U , V ( ) W [U U ], V ( )
1, n n W [U U ]. n n 2 3
(17)
1, n n
n
n n
n
n
n
1
2
3
The number of compounds in the decomposition (12) depends both on the heat flux on it and on the form of inclusion, and is generally limited. In particular, for the axial symmetric inclusion and heat flux (10), three components will remain in decomposition (12), . The jumps in normal stresses and tangential displacements in this case can be represented as follows : 2 2 0 0 ( , ) ( ), x y x y 0, ( ) [ ] ( ) n n n F F 1, 0,1 n
m m
11 s L F
( )
z
'
0
1
( [ ( )])
( , ),
s v
0
11 5
2
2
a
2 { [ ( )] s
u
2
2 }
21
L F
f a
n
) ,
i
i
2 a
2
2 2
2 2 ) 3 ) 2 ] m a m
Re( q de
m a a
e
)[ (5 (
Re(
(18)
xy
0
2
3
4
) .
i
i
2
2
2 m a m
2
2 2
2 ))
u a
(3 5 (
xy e
Im( q de
a a
m
Im(
)(
0
3
2
4
q
8 3
T
2
2
2
2
0
00 d a
10 ( cos d
d
a
{2
sin )
}
01
q
65
Here we use the notation
, d d id M M iM
y
x
i
,
,
xy
y
x xy
10
01
a
a
t
( ) tF t dt
( )
F
t
d
1
0
0
0
0
, [ ( )] L F
,
f
dt
d
*
0
a
dt
2 2
2
2
2
2
a t
t
t
38
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