PSI - Issue 80
Miroslav Hrstka et al. / Procedia Structural Integrity 80 (2026) 471–492 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000 – 000
491 21
( ) 1 i
( ) 2 i
( ) 3 i
( ) ( )
( ) ( )
( )
( )
i
i
i
i
i
i
i
diag =
e
,
e
,
e
,
Z
R
R
R
1
2
3
(A2)
( ) 1 i
( ) 2 i
( ) 3 i
( )
( )
i −
i −
i −
i
i
i
i
diag =
e
,
e
,
e
,
Z
R
R
R
1
2
3
where
( ) (
) ( 2
) ( 2
)
2
'
'' k
cos = + + sin sin ,
1,2,3 ,
(A3)
R
k
=
k
k
'' k
' sin + sin k
arctan
for
−
( ) = k
(
)
(A4)
,
1,2,3
k
=
cos
for
−
=−
The symbols '
'' k denote real and imaginary part of the material eigenvalue k , which is the root of the following
k ,
characteristic equation
( ) ( ) ( ) l m −
( ) 2 3
(A5)
0,
l
=
2
4
2
' 55 D = = ' 11 ˆ ˆ S S D
ˆ ˆ S
ˆ
( ) ( )
2
' D D '
2 2
,
l l
45 S S +
−
2
44
(
)
' D D ' 12 66 ˆ
ˆ
ˆ
4
'
3
2
' D D '
D
2 + +
2
,
S S
26 S S +
−
−
4
16
22
(A6)
(
) ( 2
)
( ) ( )
'
3
'
'
'
'
'
ˆ m g
ˆ g − +
ˆ g
ˆ g + +
ˆ g
ˆ g
,
11 =
−
3
21 16
12 26
22
ˆ
ˆ
' ˆ , 22
'
2
'
2 =− + −
2
11
12
and matrices A and L in Eq.(17) are
11 12 13 a a a a a a a a a 21 22 23 31 32 33
1 − 2 3 3 3 1 1 1 − − − − −
,
,
A
L
(A7)
=
=
1
2
where
(
) ( ,
2 ' ˆ
' D D 11 12 ˆ
ˆ S
)
'
'
'
D
ˆ g g − ˆ
1,2
a
S S
k
=
k
+ − +
16 k k k
=
1
11 21
k
(
)
2 ' ˆ
' D D 12 22 ˆ
' 26 D k k + ˆ S k
(
) 1,2 ,
'
'
ˆ g g − ˆ
/
,
a
S S
k
=
k
+ −
k
=
2
12 22
k
(
)
ˆ
ˆ
(
)
2 '
'
' ' + − + 12 22 / , k ˆ g k k k ' 2 6
ˆ g g
ˆ + −
1,2
a
k
=
k
=
3
21 22
k
(A8)
(
)
2 ' ˆ
' D D 11 12 ˆ
' 3 16 D ˆ S
' ' 11 21 ˆ
ˆ g g
,
a
S S
=
+ −
+ −
3
3 3
13
( (
)
2 ' ˆ
' D D 12 22 ˆ
3 3 3 26 D ˆ S '
'
'
ˆ g g ˆ
/
,
a
S S
=
3
+ −
+ −
3
23
12 22
)
ˆ
ˆ
2 '
'
3 3 3 − + 3 ' 26 12 22 / ˆ g ' '
ˆ g g
ˆ + −
,
a
=
3
33
21 22
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