PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 134–144 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

141

8

replacing of  with un  in (31) where the stress, un  , in the un-cracked beam portion is obtained by (20). For this purpose,  is replaced with un  . The distributions of the strains along the thickness of the upper crack arm and the un-cracked beam portion is expressed as   n z z 2 1 2     (32) and   n un z z 3 2 3     , (33) respectively. The curvatures, 1  and 2  , and the coordinates of the neutral axes, n z 2 and n z 3 , are determined in the following manner. First, two equations are written by using the fact that the axial force in the upper crack arm and the un-cracked beam portion are zero

h

2

2  h

0

b

dz





,

(34)

2

2



2

h

2 

0

b

dz





.

(35)

3

un

2 h



One equation is written by considering the equilibrium of the bending moments in the upper crack arm and the un-cracked beam portion

h

2 h 

2

2  h

2 2 b z dz b  

z dz



.

(36)

3 3

un

2 h

2





2

Finally, one equation is written by calculating the angle of rotation of the free end of the upper crack arm   2 1    l a a    . (37) After substituting of the stresses in (34) - (36), equations (34) – (37) are solved with respect to 1  , 2  , n z 2 and n z 3 by the MatLab computer program. The bending moment, M , involved in (29) is found as

h

2

2

2 2 2 M b z dz h   

.

(38)



2

Combining of (29), (30), (37) and (38), one derives

h

h

2 h 

2

2

2

2  h



 h      2 2

G

z dz

u dz

u dz

 





.

(39)

1

2 2

01 2

02 3

2 h

2





2

2

The MatLab is used to solve the integrals in (39). The solution of the strain energy release rate (39) is verified by applying the method of the J -integral (Broek (1986)). The integration is carried-out along the contour, B , shown by dashed line in Fig. 1. The solution of the J integral is derived as