PSI - Issue 52
J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646 Author name / Structural Integrity Procedia 00 (2019) 000 – 000
631
7
, y
,
(13)
=
+
=
+
11
12
21
22
x
where
1 J
1 J
1 J
1 J
y
y
x
x
x y x y −
(14)
,
,
,
,
.
J
=
=−
=−
=
=
11
12
21
22
J represents a Jacobian of transformation for each block. From the Eq. (4) and Eq. (5), the second order partial derivatives can be derived as
2
=
+
+
+
11
12
11
12
2
x
x x
x
x
2
2
11 11
=
+
+
+
11
11
11
12
12
2
2
2
2 ,
(15)
+
+
+
12 11
+
12
12
11
12
12
2
=
+
+
+
11
12
11
12
x y y
y y
y
2
2
=
+
+
11 21
+
11
11
21
22
22
2
2
2
2 ,
(16)
+
+
+
+
12 21
12
12
21
22
22
2
=
+
+
+
21
22
21
22
2
y
y y
y
y
2
2
=
+
+
+
21
21
21
22
21 21
22
2
2
2
2 .
+
+
+
+
22
22
21
22
22 21
22
(17)
/ and
/ are given
in which all of the partial differentials
ij
ij
ij
ij
1
1
J
J
,
,
J
J
(18)
=
−
ij
=
−
ij
2
2
J
J
y
y
x
x
(19)
,
,
,
,
=
=−
=−
=
11
12
21
22
and
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