Issue34

Y. Sumi, Frattura ed Integrità Strutturale, 34 (2015) 42-58; DOI: 10.3221/IGF-ESIS.34.04

 

 

( ) ( ), h   

( ) 1

(3)

1

Following the same method as used by Banichuk [1], Goldstein and Salganik [2, 3], and Cotterell and Rice [4], the perturbation solution of the stress field can be expressed by Muskhelishvili’s analytic functions [18] given by

Figure 2 : Coordinate systems for a kinked and curved crack.

2[ '( ) 22 11 12 z i         11 22 2  





'( )], z

(4)

2[(  

) ''( ) 

( ) z   



z z z

'( )], z

where z = x 1 following form:

+ ix 2 . The analytic functions   ( z ) and  ( z ) are expanded in terms of  (  1

) up to the second order in the

3 ( ), 















z O 

'( ) z

0 ' ( ) z

1 ' ( ) z

2 ' ( )

(5)

3 ( ), 



0  ( ) z     ( ) z ( ) z O 1  2 

( ) z

( z ) are of the zero-th order,  

( z ) are of the first order, and  

( z ) are of the

 z  and  

 z  and  

 z  and  

in which  

 . The boundary conditions on the crack surfaces, at z =  1

+ i    

 become

second order in    

2 '( ) Z z e     (6) where  is the angle of the slope on the crack line. It is assumed that the crack surfaces are subjected to normal and shear tractions T n and T s on the upper surface and - T n and - T s on the lower surface, respectively. We also assume that tractions on the crack surfaces, T n and T s are bounded and integrable in the defined range. Approximate Description of a Slightly Kinked and Curved Extension of a Straight Crack The second order approximation of the stress distribution in the vicinity of the original crack tip can be expressed by 1 2 3 ( , 0) ( , 0) ( , 0) ( ), 1 ,2 1 ,22 1 2 x x x O ij ij ij ij            (7) in which I 1 ( , 0) ( ), 11 1 I 1 2 2 1 k x x T b O x x        '( ) [( ) ''( )  ( ) z   '( )] z (    ), i n s z z z  T iT    

k

x

I

1









( , 0) x

b

O x

(8)

( ),

22 1

I

1



2

2 1 x 

k

x

II

1









( , 0) x

b

O x

( ),

12 1

II

1



2

2 1 x 

where k I are also determined from the solution of the boundary-value problem prior to the crack extension. The surface tractions on the extended crack surfaces, T n and T s and k II are the stress intensity factors, and the coefficients T , b I , and b II

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