Issue 52
O. Kryvyi et alii, Frattura ed Integrità Strutturale, 52 (2020) 33-50; DOI: 10.3221/IGF-ESIS.52.04
P
2
2
d
q s
f
3 a
a
2
4
3 a
4
z
i
i
00
0 11
*
, ( )
m
s
q
s m
xy M e
de
Re(
)
Re(
),
0
11 0
11 1
3
q
q
2
15
4
65
65
* b g b g 21 1 * 21 1
*
s
g b
(
)
4
4
1
21
1 22
m m
m
m
g
,
,
,
.
2
3
2
4
1
* b g
q
(
45
q
2(
)
)
65
21 1
23
In the case of a penny-shaped inclusion of thickness h
m m
( )
s
h
z
0
1
11
( , ),
s v
11 5
2
2
2
2
a
a
hs
2
) ,
u
i
i
2
2
2
2
2 2
2 2 ) 3 ) 2 ] m a m
21
(1 ) a a
a
Re( q de
m a a
e
)[ (5 (
Re(
xy
0
2
3
4
) .
i
i
2
2
2 m a m
2
2 2
2 ))
u a
(3 5 (
Im( q de
a a
m
e
)(
Im(
xy
0
3
2
4
q
8 ( cos
T
2
2
2
2
0
00 d a
d
d
a
{2
sin )
}
10
01
q
3
65
For z , x , y using conditions (6), the following expressions are obtained
P b
M
M
8 z
y
21
2
2
2
x
z
* 5 f m a q d m 0 00 6
y
7 0 10 8 m a q d m
x
7 0 01 8 m a q d m
(19)
,
,
,
3
3
a
a
a
where
*
*
(
2 11 21 11 21 s b b s
)
*
*
2 m s b b s 11 21 11 21 ( 5
m
),
,
6
q
2
6
65 m
* b g
* b s
* * b g 21 1
( b g s )
3(
)
4
7
21 1
11 21
21 1 11
m
m
,
.
7
8
* * b b b g b b * * * 12 21 12 1 11 22
q
15
4(
)
(
)
65
Note that the exact solutions obtained are rigorously justified using the theory of generalized functions and the Riemann boundary value problem. The reliability of the results obtained also confirms their coincidence with the results of [20], in which a particular case of this problem for a homogeneous transversely isotropic medium is considered.
N UMERICAL RESULTS AND THEIR ANALYSIS .
sing the solution obtained, we investigate the behavior the jump of normal stresses in the neighborhood of inclusion. We assume that the load applied to the inclusion is linearly distributed, and in polar coordinates has the form: , in this case, the resulting force and moments allow representations: , , . Calculations were performed for the combination of materials. Cadmium (material ), Magnesium (material ), Al 2 O 3 (material ), Zn (material ) Tab. 1 shows the values of thermoelastic constants of these materials. Calculations were performed with , for different values Figs. 2-13, for the combination of materials m1 (upper half space), m2 (lower half space), Figs. 14-19, for the combination m3 m4. Figs. 2, 4, 6, 8, 10, 12 shows the dependences on the polar angle or different values of the polar radius , Figs. 3, 5, 7, 9, 11, 13 show dependencies on the polar radius or different values of the polar angle. In addition, z 0 0 01 10 ( , ) (1 ( sin( ) cos( )) p P s s 2 0 z P a P 4 0 10 0.25 y M a P s 4 0 01 0.25 x M a P s 1 m 2 m 3 m 4 m 01 10 1 3, 1 7, d d 01 10 1 5, 1 7, s s 1 a 0 , q 0 , P z z U
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