PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 75–85 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

80 6

where  , 2  and 3  , respectively. Segments, 1  , 2  and 3  , coincide with the cross-sections of the upper and lower crack arms, and the cross-section of the un cracked beam portion ahead of the left-hand crack tip, respectively. First, the J -integral is solved in segment, 1  . The following expression is used (Broek (1986)): 1  J , 2  J and 3  J are the J -integral values in segments, 1

  

     

  

x p u x  

x p v y  



J

u

ds

a cos 0  −

=

+

,

(20)



1

1



1

where  is the angle between the outwards normal vector to the contour of integration and the crack direction, x p and y p are the components of the stress vector, u and v are the components of the displacement vector and ds is a differential element along the contour of integration. The components of (19) are obtained as  = x p , (21) 0 = y p , (22) 1 dz ds = − , (23)

x u

 

 =

,

(24)

cos 1 =  ,

(25)

/ 2 h −  

1 1 z h

/ 2

. By using (21) – (25), the J -integral (20) takes the form

where

1

h

1

2  h

(

) 1 dz

J

0 −  = − +  1 1 1 u a

.

(26)

2

In segment, 2  , of the integration contour, the components of the J -integral are written as

p

LW  = ,

(27)

x

0 = y p ,

(28)

2 dz ds = − ,

(29)

x u

 

 =

,

(30)

LW

cos 1 =  ,

(31)

where LW  and LW  are the stress and strain in the lower crack arm. By substituting of (27) – (32) in (26), one obtains