PSI - Issue 52

641 17

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

M N  m n

,

2 , ) ( T  n T          + + + T  ( ) ( n T ) + + ( ) u mn m 11 , m 12

( )

 

u P 

=





,

x

0 0 M N  0 0 = = = =

, u mn ( ) ( )          + ( ) ( ) ,  m n T T T T   n 1 11 , m 12 ,

−





m n M N  m n

,

2 , ) ( T  n T         + + +  T  ( ) ( n T ) + + ( ) u mn m 21 , m 22

( )

 

u P 

=





,

y

0 0 M N = =  0 0 m n = =

, u mn            + 1 21 , m 22 , ( ) ( ) ( ) ( ) .  n m n T T T T

(74)

−





Therefore, we have the following relationship ( ) ( 1 , , 1 ( ) , ( ) x x x y y y k E u u       = + = x x

)

(

)

2 , x y    + ( ) k E u u x

(75)

,

,

, x x    + k E u u







=

, y y

, y x

xy

Static stress intensity factor is determined by

1/2

   

   

  

  

E

(

)



'

,

(76)



x x y y + − − x x   t u t u u t u t d         y y  

K

0

=



I

2 2(1 ) −

a

 

'



where

,

. Therefore the transformed stress intensity factor, for a plane strain



y xy     + n

y 

x xy  

y y  

t

n

t

n

n



=



=

+

x

x x

problem, yields

  

 

E

(

)

(

)





0 t u t u u t u t d       + − − 0 0 0

( )

'  +

' . 

K s 

x 

y 

x 

y 

u X u Y d   +

0

=



(77)

I

x

y

x x

y y

2 2(1 ) −

K a 

 

I

'

'





Consider a circular contour shown in Fig. 3(a), we have

1/2

   

   

   

   

(2)

n 

E

(

)

 

(78)

,



x x y y n tu tu ut utId           y y + − − x x  

K

0 2



=

I

(1 ) −

a

 

1

n

=

(1)

n 

2

2

x                +

y x

x

x

y

y

y

,    











sin

cos , 

sin

cos , 

(79)

I

R

R

R

R

=

=−

+

=−

+





n

   





  







where (1)

(2) n  indicate the starting and ending angles of block n. The real case transformed stress intensity factor

n  and

can be evaluated

   

   

(2)

(2)





R

E

n

n

(

)

(

)







 

0

0

( )

,

(80)

x y y n tu tu ut utId       + − − x  y y  x x

x 

y 

K s

u X u Y J rdrd   +

0

=

+





I

n

2 2(1 ) −

*

aK

 

1

1

n

n

=

=

I

(1)

(1)

0





n

n

where

n J is Jacobs of block n due to domain transformation.