PSI - Issue 52

640 16

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

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

η

crack tip

3

4

7



a

b



y

R

P



δa

3

crack tip

'  

7

δη

' 

' P

δξ

ξ

8

α

4

6

( ) ' P P

6

8

O

1

5

2

x

1

2

5

Fig. 3. The integral contour in a block. (a) variation of integral contour; (b) integral contour in physical domain.

Combining Eqs. (68) and (69), we can obtain

  

  

  

  

8

8

8

8

11 21   i i N x      =− + i

22   i i

/ , N y J

/ , N y J 

N x     =− +

(70)

12

i

i

i

1

1

1

1

i

i

i

i

=

=

=

=

where J is given in Eq. (14). The simplest option is that the variation of nodes of block, as shown in Fig. 3(b),

3 cos( ) x a    = ,

3 sin( ) y a    = , at the crack tip and

6 cos( ) /2 x a    = ,

6 sin( ) /2 y a    = ,

7 cos( ) /2 x a    = and

7 sin( ) /2 y a    = . Then

(

)

( , ) N x N x N x    + + ( , )

(               ) ( 7 7 21 3 7 7 22 3 ( , ) ( , )

11 3      =− 

3

6

6

)

( , ) N y N y N y J       + +  ( , ) ( , ) / ,

+

3

6

6

7

7

(

( , ) N x N x N x    + + ( , )

12 3      =− 

3

6

6

)

( , ) N y N y N y J       + +  ( , ) ( , ) / .

(71)

+

3

6

6

7

7

In this case, the variations of the field point (

, ) P     + + can be simplified as

cos ,       = + = + = + = + ' , ' . R       

sin , 

R

(72)

The increments of displacement and their derivatives in intrinsic space can be determined at the field point ( , ) P  

0 0   0 0 M N m n M N = = m n = =

 

, 2 1 mn m n T T        + + − ( ) ( n T ) ( ) ( ) ,  u u mn m T ,

( )

x  u P



=



 

, 2 1 mn m n T T        + + − ( ) ( n T ) ( ) ( ) ,  v v mn m T ,

( )

y  u P

(73)



=



and the increment of their derivatives in intrinsic space can be determined as