PSI - Issue 52
J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646 Author name / Structural Integrity Procedia 00 (2019) 000 – 000
633 9
( ) x
( ) x
E
E
( ) I E k = x
,
k
k
k
k
k
+
+
+
+
+
21
21
11
11
2
1 21
1 22
2 11
2 12
2
11 1
21
x
y
( ) x
( ) x
E
E
( ) J E k = x
,
(23)
k
k
k
k
k
+
+
+
+
+
22
22
12
12
2
1 21
1 22
2 11
2 12
2
12 1
22
x
y
in which,
( ) x
( ) x
E
E
( )
( )
, ,
, .
E
E
(24)
=
+
=
+
11
12
21
22
x
y
For a rectangular plate of 2 2 a b in homogenous materials, we have 2 2 1 1 0 1 2 0 1 3 0 1 1 1 1 1 1 1 2 2 2 3 0 2 2 0 2 1 0 2 2 2 2 2 2 2 / , / , / , C 0, / , / , / , 0, A kEa B kEb H kEab D E F G I J CkEabFkEaGkEb ABDEH I J = = = = = = = = = = = = = = = = = = = =
(25)
where 0 E is constant Young’s modulus. 2.3. Chebyshev polynomial interpolation The Chebyshev polynomial interpolation is employed in this work, the function ( , ) f x y in mapping domain can be approximated as
M N =
( ) , f x y f
( )
( ) ( ) mn m n T T
(26)
, :
,
0 0
m n
= =
where mn donates the unknown coefficients. ( ) k T z is the k -th order Chebyshev polynomial of the first kind, which is defined as ( ) ( ) ( ) cos arccos , , . k T z k z z = = (27)
Some relationships can be obtained in (Rivlin, 1974)
( )
m T z
( )
( )
(28)
,
, m z T z
mU z
=
=
1
m
−
z
(
) ( ) m m T z U z + − ( ) 1 m
( )
2
m T z
( )
,
(29)
, m zz T z
m
=
=
2
2
1
z
z
−
( ) k U z is the k -th order Chebyshev polynomial of the second kind
in which
(
)
( ) z
( ) ) sin 1 arccos sin arccos k z + (
( )
.
U z
(30)
=
k
In Laplace transform domain, the displacements can be written as follows
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