Issue 32

I. Telichev, Frattura ed Integrità Strutturale, 32 (2015) 24-34; DOI: 10.3221/IGF-ESIS.32.03

Using the expansion of the function u 2

(ξ) in terms of Lagrange interpolation polynomials over the Chebyshev’s nodes we

obtain the expression for the function * * 2 2 ( ) g x :   1 * * * 1 1 ( ) (1) N g x g u       2 2 2 2 k

  

  

  

1 N  

      r k r T T  





1 2 



d

(6)

N

2   1 k  

1



r

1

* 2

x

After integration

1 



d

  * 2



arccos x

2





1

* 2

x

and

  

1 

T

1 

  * 2

  

 

1 r



d

sin r arccos x

r

2





* 2

x

we get





  

  

  

  

1

1

1 N  

N 

  * 2

  * 2

  k 

  k T sin r arccos x   r

 

* * 2 2 g x g ( )

* 2

(1)      



u

arccos x

(7)

2



2

N

r





k

r

1

1

Analogously we obtain the expressions for * * 0 0

( ) g x and * * 1 1

* 0 0 / ,   x x l x L   and 0 0 0

* 1 / ,     x x l x L   : 1 1 1 1

( ) g x at





  

  

  

  

1

1

1 N  

N 

  * 0

  * 0

  g x g l  0 0 0 0  

  k 

  k T sin r arccos x   r

 

* * 0 0 g x g ( )

* 0 (1)  

 



 

 

l

u

arccos x

/

2

(8)



0

0

N

r





k

r

1

1





  

  

  

  

1

  k T sin r arccos x   1 2 r

1 N  

N 

  * 1

  * 1

  k 

  g x g l  1 1 1 1  

 

* * 1 1 g x g ( )

*   (1)







 

 

u

arccos x

(9)

l

/



1

1

1

N

r





k

r

1

1

From Eq. (2) in the symmetric case we have        ' ' ' v v v x x x          

   ' g x

1 æ 

(10)









G

4

v v v( ) 2 x   



(1 ) ( ) æ g x 



C  , where n is a segment number. The constants of

Integrating we obtain the relation:

n

n

G

4

integration C n

are determined by displacement at the end of the segment:

2 C 

0

  2

2 G     æ g l æ g l

(1 ) 4 (1 )



C

1

  1

(1 ) æ 

  1    2 g l

  2

1



C  

 



C

1 g l

0  Thus the crack opening displacement (COD) for the segment L n 1 4 4 G G

is defined as following

* * n n n æ l g x

(1 ) 

( )

* ( ) 2v( ) x  *





COD x

C

2

(11)

n

n

n

G

2

Since for the plane stress (1 ) 2 4 æ G E  

* ( ) n COD x takes the form

, the expression for





  

  

u

 n Y

l

4

S

 

1

  1 N

 N

  * n

  * n

   k T sin r arccos x r

 



n k

* ( ) 2 COD x C

 



 ] ,    0,1,2 n

arccos x

(12)

[

2

 

n

n



EN

S

r





k

r

1

1

Y

30