PSI - Issue 52

J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646

634

10

Author name / Structural Integrity Procedia 00 (2019) 000 – 000

0 0  M N m n = =

0 0  M N m n = =

(31)

u

v

( , , ) s  

( ) ( ) ( ), m n s T T u

( , , ) s

( ) ( ) ( ). m n s T T  

u





   





x

mn

y

mn

Then

u

 

0 0  M N m n = =

0 0  M N m n = =

(32)

u

, mn mn D s   u u

( ) s mU T

( ) ( )  

( ),





=

1

mn

m

n

−



u



0 0  M N m n = =

0 0  M N m n = =

(33)

u

, mn mn D s   u u

( ) ( ) s nT U m n

( )  

( ),





=

1

mn

−





( 1) ( ) m T U  + −

( ) 

2

u



0 0  M N m n = =

0 0  M N m n = =

u 

, D s   u u mn mn

( )

( ) 

( ),

(34)

s m

T



=

m

m

mn

n

2

2

1







−

2

u



0 0  M N m n = =

0 0  M N m n = =

u

, D s   u u mn mn

( ) s mnU U

( ) ( )  

( ),

(35)





=

1

1

mn

m

n

−

−

 

 

( 1) ( ) n T U + −

( )  

2

u



0 0  M N m n = =

0 0  M N m n = =

u 

, D s   u u mn mn

( )

( ) 

( ).

(36)

s n

T



=

n

n

mn

m

2 

2 

1



−

For the convenience of numerical implementation, the first and second orders of the partial derivatives of the displacements can be arranged in vector form as

, u D α u D α u D α , u u = = =

, v D α v D α v D α , v v = = =

(37)

,

,

u

v

0

,

,

0

,

,



















 0









T

T

T

, x y z u u = .

1 2 , x x xM u u u u= ,...,

1 2 , y y yM u u u = v ,...,

1 2 z M    = α , ,..., z z z

,

where

,

,

( 1) ( 1) M M N = +  + ,

0

0

0

/ , ,       =  = D D . Then, we have the following linear algebraic equations, Eqs. (21) and (22) can be rewritten, in matrix form, as ( ) 0      + + + + + = AD BD CD DD ED FD α Q (38) 0 ( 0,1, 2,..., ) k M = . , , , , A B CDE and F are matrices of coefficient from governing equations in Eqs. (21), (22) and boundary conditions in Eqs. (3) and (4). Q is vector consisting transformed body force and initial conditions in Eqs. (5)(6) and boundary values. Moreover, the global number of , 0,1,..., k n M m m M =  + = , and 0,1,..., n N = , and the number of unknowns in total is 0 ( 1) ( 1) M M N = +  + . In this paper, nodal distributions in the normalized square domain 1 ( , ) 1   −   are chosen as , where   , u v k k   = α are the unknown coefficients,

m M

n

(39)

cos

,  

cos

0,1, 2,..., ,

0,1, 2,..., . N

m

M n



=−

=−



=

=

，

m

n

N

Finally, the system equation can be arranged in matrix form ， and the unknown coefficients  can be obtained.

K

α

Q

(40)

.

=

2 2 M M M  2

2

M

0

0

0

0