Issue 52
O. Kryvyi et alii, Frattura ed Integrità Strutturale, 52 (2020) 33-50; DOI: 10.3221/IGF-ESIS.52.04
To solve the Riemann problem (A.7) using the method proposed in [13], we obtain
1 k ir
k
0
j
1 2 ( , )
1
jk
(A.8)
j
j
j
3
j
r
k
0
2
2
j j i
where 0
,
r
1 2 ( , ) jk unknown functions;
0,1, 2, 3 j .
,
1
2
0 jk , following [15], are expressed through transformants of jumps of thermoelastic characteristics: , which allowed, using formulas (A.3), to obtain such expressions for the transformant
Unknown functions
1 2 2 ( , ) F [ ( , )] j j x y
1 2 ( , , ) V z j
( ) [ ] z F
:
j
2
3
1 k z e
1 V z
) )
3, (( q i
) )
1, q r k
2, q r 1, k
1
1
2 2 )
(
1 3
1 4 )
(
2 5
i
i
i
( )
( )
{
((
k
k
1 1,
1,
1
4, q r 1, k
5,
6, q r 1, k
6
7
8
q
},
k
1,
3
3
[ (
2 V z
2
2, 1 2, k k
2 1, e q 1 3 1,3 )
2
2, e q e q 1, 1 2, k k 3 1,3
2
( ){ z r
(
1 ] ( )( i i
2
2
2
3
i
e q
i
r
( )
)
)
[
]
k
k
1
1
3
3
[ (
k
2 2, 1 3 1,3 ) e q
1
3, e q e q 2, 1 2, k k 3 1,3
1
1 e q k
3, 2,
2
(
1 ( )( i i
2
4
5
2
i
r
r
i
]
)
[
]
)
k
k
1
1
3
1, e q i 1 2, [ ( k k
1
4, 2, q i ( k
5, 2, q i ( k
1
6, 2, q i ( k
2
2
1
2 6 )
2
7
2
8
r
r
r
)
)
)
]},
k
1
3
3
4 V z
3
2, e q e q 1, 1 3, k k 3 2,3
3
2, e q e 1 3, k k 3
2
2 1,
2 ( ){( )( z i i
( i
1
2
3
1
r
r
i
q
( )
)
[
]
[(
)
)
]
2 2,3
k
k
1
1
3
3
3, e q e q 2, 1 3, k k 3 2,3
2
3, 1 3, k k
2 2, e q 2 3 2,3 )
2
2
(
2 ] ( )( i i
4
1
1
5
r
i
e q
i
r
[
]
[(
)
)
k
k
1
1
3
1, e q i 1 3, [ ( k k
2
4, 3, q i ( k
1
5, 3, q i ( k
2
6, 3, q i ( k
3
1
1
1
6
1
7
1
8
r
r
r
r
)
)
)
)
]},
k
1
3
z e
6 V z
1
2
2 2,
1, q r k
1
((
2
2
(
1
i
r
i
r
q
( )
( )
{
)
)
)
k
k
1 4,
3 4,
k
1
1
1
3, q q )
4, 4,
5, 4, q r k
1
6, q r k
2
1
4
(
2
6
8
i
r
i
r
((
)
)
},
k
k
5 4,
7 4,
3
3
z e
z e
7 V z
1
5 1 7 6, r q {
5 q r
6
6
}, q V z
q
( )
( )
{
( )
( )
}
(A.9)
k
k
k
k
k
k
1 7 5,
8 5,
8
8 6,
k
k
1
1
where
2, 30
2, 31
* * 11 12 s
* s s s
jk s
* s
, k i rz
, S S S S
*
1
k e e
,
,
S S
jk j k
1,
1,
*
, 1,2
30
31
21 22
47
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