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
479
9
for piezoelectric material I, and
( ) ( )
( ) ( )
( )
1
1
1
*1
* 1,
* 2, * 2,
* 3,
−
1
−
−
,
σ σ
λ
λ λ
λ
1 H r H r 1
2 H r H r 2
3 H r H r 3
=
+
+
3
2
x
x
x
(31)
2
2
2
( )
1
1
1
*2
* 1,
* 3,
−
1
−
−
,
λ
λ
=
+
+
3
2
x
x
x
1
1
1
for isotropic non-piezoelectric material II, where H i are generalized stress intensity factors and
11 D 12 1
21 D 22 2
1
2
,
.
σ
σ
=
=
(32)
The derivatives of the shape functions (subscripts , 1 and , 2 denote differentiation with respect to x 1 and x 2 ) in Eqs.(30) and (31) are given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 1 1 , 1 1 , 1 * 1 * * * * , 1 * 1 * * * , , 1,2,3 , 1,2,3, , 1,2,3, , 1,2,3, i i i i i i i i i i x i i i i i x i i i i i x i i i i i x i i i i i i i i * λ L Z v L Z w λ L Z μv L Z μw λ B Z v B Z w λ L Z v L Z w − − − − − − − − = + = = + = = + = = + = (33)
where
3 1 0 0 0 2 i i i − = − 1 2 0
0 0
1
0 0
1
*
B
,
0
0 ,
0
0
μ
μ
2
2
=
=
0 0
3
0 0
3
and
( ) 1 i
( ) 2 i
( ) 3 i
( ) ( )
( ) ( )
( )
( )
i
i
i
i
i
i
i
diag =
e
,
e
,
e
,
Z
R
R
R
1
2
3
(34)
( ) 1 i
( ) 2 i
( ) 3 i
( )
( )
i −
i −
i −
i
i
i
i
diag =
e
,
e
,
e
,
Z
R
R
R
1
2
3
(
)
1 −
i
e
0
0
) ( ) 2 i −
(
)
( )
(
1 −
i
* 1 −
2 1 e i
sin e
0 ,
Z
= − −
(
)
1 −
i
0
0 e
(35)
(
)
1
i − −
e
0
0
1
−
) ( ) 2 i − −
(
)
( )
(
1
i − −
*
2 1 e i −
sin e
0 .
Z
=
(
)
1
i − −
0
0 e
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