PSI - Issue 52
J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646
636 12
Author name / Structural Integrity Procedia 00 (2019) 000 – 000
for left
1 =− and right edge
1 =+ , and
1
1
+
+
M N
( ) 1 u mn n T
( ) ( ) T T d u = m m '
(
) ( ) m T d '
0
, 1
,
(47)
x
0 0
m n
= =
1
1
−
−
1
1
+
+
M N
( ) 1 v mn n T
( ) ( ) T T d u = m m '
(
) ( ) m T d m '
0 y
, 1
. ' 0,1,2, , , M =
(48)
0 0
m n
= =
1
1
−
−
0 y u are the displacement specified on the boundary. For
1 =+ edges. In which, 0 x u ,
1 =− and top
for bottom
weak form traction boundary conditions, we have
1
1
+
+
0 0 M N m n = =
( ) 1
( ) ( ) T T F d T 11
( ) 1
( ) ( ) T T F d
12
u mn m T
+
,
'
,
'
n
n
u
m
n
n
u
1
1
−
−
1
1
+
+
0 0 M N m n = =
( ) 1
( ) ( ) T T F d T 21
( ) 1
( ) ( ) T T F d
22
v mn m T
+
+
,
'
,
'
n
n
u
m
n
n
u
1
1
−
−
1
+
(
) ( ) n T d n '
0 = t
(49)
1,
, ' 0,1,2, , , N =
x
1
−
1
1
+
+
0 0 M N m n = =
( ) 1
( ) ( ) T T F d T 11
( ) 1
( ) ( ) T T F d
12
u mn m T
+
,
'
,
'
n
n
v
m
n
n
v
1
1
−
−
1
1
+
+
0 0 M N m n = =
( ) 1
( ) ( ) T T F d T 21
( ) 1
( ) ( ) T T F d
22
v mn m T
+
+
,
'
,
'
n
n
v
m
n
n
v
1
1
−
−
1
+
( 0 y t =
) ( ) n T d n '
(50)
1,
, ' 0,1,2, , , N =
1
−
for the left-hand side and right-hand side edges, or
1
1
+
+
M N
( ) 1 u mn n T
( ) ( ) m T T F d T 11 , ' m u
( ) 1
( ) ( ) T T F d 12 m m u '
+
,
n
0 0
m n
= =
1
1
−
−
1
1
+
+
M N
( ) 1 v mn n T
( ) ( ) m T T F d T 21 , ' m u
( ) 1
( ) ( ) T T F d 22 m m u '
+
+
,
n
0 0
m n
= =
1
1
−
−
1
+
(
) ( ) m T d m ' ,
0
, 1
' 0,1,2, , , M =
(51)
t
=
x
1
−
1
1
+
+
M N
( ) 1 u mn n T
( ) ( ) m T T F d T 11 , ' m v
( ) 1
( ) ( ) T T F d 12 m m v '
+
,
n
0 0
m n
= =
1
1
−
−
1
1
+
+
M N
( ) 1 v mn n T
( ) ( ) m T T F d T 21 , ' m v
( ) 1
( ) ( ) T T F d 22 m m v '
+
+
,
n
0 0
m n
= =
1
1
−
−
1
+
(
) ( ) m T d m ' ,
0 y
, 1
' 0,1,2, , , M =
(52)
t
=
1
−
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