Issue 52
O. Kryvyi et alii, Frattura ed Integrità Strutturale, 52 (2020) 33-50; DOI: 10.3221/IGF-ESIS.52.04
x t
y
y
y
y
1
32 3 1 q
33 4 1 q
5
q
( , ) x y
q
q
{
[
]
31 1
2 32 2
21 1
2
2
2
r
r
r
r
r
0
0
0
0
0
x t
y
y
y
1
33 5
5 33 1 [ q
q
34 6 2 q
1 2 d d q }
q
q
]
,
22 2
35 7
36 8
2
2
2
r
r
r
r
r
0
0
0
0
0
x t
y
1
1
1
1
44 6
6
3
5 2 ]
x y q
q
q
q
( , )
{
[
]
[
41 1
42 2
43 4 1
2
2
r
r
r
2
r
r
0
0
0
0
0
1
ln( )} , r dtd
q
q
45 7
46 8
0
r
0
2 q
2 q
1
1
55 7
66 8
56
65
7
8
8 , ( , )
,
x y q
dtd
x y q
dtd
( , )
(A.10)
7 3
r
r
0
0
where
6
p
p
2 ) ( ( x t y
2 ) ,
*
,
,
,3 , s s q q q q q q , kn np kp kp kp kp k
r
q
,
kp
k
0
,3
n
1
22 22 11 23 , . q q q q q q q q q q q q 23 12 32 32 21 33 33 22 , ,
3 ( ), have readily verifiable properties
Differentiation operators in space
x t
x t
x t
y
y
y
, x t y
,
,
2
,
2
1
2
1
1
2
2
2
2
2
r
r
r
r
r
r
0
0
0
0
0
0
x t
y
1
1 1
1
1
2
2 2
, DD , D r
i
,
1
1
2
0
1
2
3
r
r
r
r
r
r
r
0
0
0
0
0
0
0
1 1 D D 0,
2
0 ( ) ( , x t i y r
.
(A.11)
3
4
2
5
i
u
i
,
,
ratio
Using properties (A.11), with respect to complex combinations of jumps and sums:
(A.10) can be rewritten as
q
q
11 1 q
8 [ ],
13
12 (D [ ] D [ ])
1
6 DD [ ]
15 7 16 q q
(D D ) u u
q
14
2
2
q
q
q
q
22 D [ ]
2
23
23
22
6
1
q
D D [ ] q
DD [ ] u
D [ ] u
24
21
0
2
2
2
2
[ ],
D [ ] q q
7
25
26 0 8
q
q
q
q
22 D [ ]
2
23
23
22
6
1
q
D D [ ] q
DD [ ] u
D [ ] u
24
21
0
2
2
2
2
[ ],
D [ ] q q
7
25
26 0 8
49
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