PSI - Issue 52

J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

629

5

( , ) x

0 u t u t x x (,), (,)

0 y u t x

x

(3)

( , ),

.

u t

=

=



x

x

y

u

(2) Traction boundary conditions

( , ) t x

(,), (,) t t t x x

( , ) , t x

x

0

0 y

(4)

,

t

x x n n t   = + = y xy

x xy n n t   = + = y y



x

x

y

t

0 ( , ) x t t x and

0 (,), (,) x y u t u t x x

0 ( , ) y t t x denote displacement and traction boundary values. The initial conditions

where 0

are given in the domain

( ,0) x

u



(5)

( ) U u = x

( ,0), x

() x

( ,0) x

'

U

u

=

=

x

x

x

x

x

t



and

( ,0) x

u



y

( ) U u = x

( ,0), x

( ) x

( ,0) x

x

'

(6)

U

u

=

=



y

y

y

y

t



where ( ,0) x u x , ( ,0) y u x , ( ,0) x u x and ( ,0) y u x are initial displacements and velocities respectively. In order to eliminate time t in Eq. (13), the Laplace transformation



 L f



= 

( , ) t x

( , ) x

(7)

( , )

, st

t e dt −

f x s

f

=

0

which is applied to the system of governing equations and the Eq.(13) has transformed to

( ) x

( ) x

E

E





              

2

2

u

u

u

u

k

k

1









+

+

+

+

2

2

x

x

x

x

( ) x

( ) x

2

2

k

E x x k E y y    

x

y





1

1

( ) x

( ) x

( ) ( ) , x x

2

u

u

u







E

E

X s





( ) k E x ykE y xkxykE  + + =       x x ( ) 3 2 y y y k

,

1

1

1

( ) , x y

( ) x

( ) x

,

(8)



2

2

u

u

u

u









E

E





k

k

1

k

y

y

y

y

+

+

+

+

1

1

( ) x

( ) x

2

2

k

( ) k k E y x E x y    +     x x ( ) 1 1 x E x x k E y y     E E u u   + ( ) x ( ) x 2 x

x

y





2

( ) ( ) , x x

Y s

2 k u k x y k E  =   3 x

,

2

2

2

( ) , X s s u sU U b     = + + −   x x x , ( ) , Y s 2 ' ( ) ( ) x x x x

2 s u sU U b     = + + −   x x x . Considering the ' ( ) ( ) y y y y

in which

boundary conditions, the nodal displacements can be obtained for certain Laplace parameters s in the Laplace transformed domain. Among several Laplace inverse algorithms, one of the simplest methods was proposed by Durbin (1975) in 1975. Thus, the time-dependent function ( ) f t can be obtained, with 1 K + samples ( ) k f s in Laplace space at k s   / 2 / 0 0 2 1 ( ) ( ) Re ( ) , 2 t T K ikt T k k e f t f s f s e T   =    − +      (9)