PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 115–124 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

119 5

z

1 i       1 1 i 1 1 i z i n i

M b 

z dz

.

(21)

1 1

After substituting of stress in (20) and (21), the two equations are solved with respect to the curvature and the coordinate of the neutral axes by using the MatLab computer program. Since the beam in Fig. 2 is statically undetermined structure, the bending moments and the axial forces in the upper crack arm and in the un-cracked beam portion, a x l   4 , can not be obtained directly (these bending moments and axial forces are needed in order to determine the curvatures and the coordinates of the neutral axes). Therefore, first, the static indeterminacy should be resolved. Two equations are composed for resolving the static indeterminacy by using the fact that the axial displacement and the angle of rotation of the left-hand end of the upper crack arm are zero. For this purpose, by applying the integrals of Maxwell-Mohr, one derives     0 2 2 2 3 3 3 2 2            l a h h z l a z a n n    , (22)   where 2  and 3  are the curvatures of the upper crack arm and the un-cracked beam portion, n z 2 and n z 3 are the coordinates of the neutral axes of the upper crack arm and the un-cracked beam portion, respectively. Four additional equations are written by considering the equilibrium of the elementary forces in the cross-sections of the upper crack arm and the un-cracked beam portion ) 0 ( 3 2    l a a   , (23)

z

1 i        i        2 1 i 2 1 z z i n n i 1 2 1 i 2 1 z i n n i

N b H 

dz

,

(24)

2

upi

1

M b 

z dz

,

(25)

2 2

H

upi

1

z

1 i       3 1 i 3 z i n i

N F b  

dz

,

(26)

3

H

uni

1

z

1 i       3 1 i 3 z i n i

   

 

   

1 M N h h M F h h b         2 1 1 2 2 2 2 H H

z dz

.

(27)

3 3

uni

After substituting of stresses in (24) – (27), equations (22) – (27) are solved with respect to 1 H M ,

1 H N , 2  , 3  ,

n z 2 and n z 3 by the MatLab computer program. The integrals of Maxwell-Mohr are applied to obtain

F u and  which are involved in (16). The result is

  









u

   1 1 z a n 

z l a

2 2 1 l a h h

       3 



,

(28)

3 3

F

n

( 3 a       1

)

l a

.

(29)

The strain energy release rate is found by substituting of (17), (28) and (29) in (16). The result is                  2 2 1 3 3 3 1 1 h h z z b G F n n         3 1   b M

z

z

1 i      3 z z i n i

      1 1 i n i i

      1 1 i n n i i

1 1

2 1

3 1 i

u dz i

u dz i

02 u dz i





.

(30)

01 1

2

03 3

z

z

1

2

i

i

The MatLab is used to solve the integrals in (30). The delamination in the beam in Fig. 2 is analyzed also by the method of J -integral (Broek (1986)). For this