PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 428–442 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

434

7

The results is

  

  

   

   





  u a z 

1 l a h h z

  

.

(21)

1 D n

2

F

U

n

2 2

1

2

After substituting of  and U  in (18) – (21), the four equations are solved with respect to D  , U  , 1 1 n z and 2 2 n z by using the MatLab computer program. It should be mentioned that  and U  which are involved in equations (18) – (21) are continuous functions of the time (refer to formulae (13) and (17)). Therefore, equations (18) – (21) can be used to determine the curvatures and the coordinates of the neutral axes at various values of the time.

/ 0.25 1  h h , curve 2 –

Fig. 4. The strain energy release rate in non-dimensional form plotted against the non-dimensional time (curve 1 – at

at / 0.50 1  h h and curve 3 – at / 0.75 1  h h ). By using the equation for equilibrium of the elementary forces in the cross-section of the lower crack arm, the force, F , that is involved in (5) is written as

D A    ( )

F

dA

.

(22)

From (5), the strain energy release rate is obtained as

1

u

a U





b G F 



F

.

(23)

a b





After substituting of (6), (7), (14), (21) and (22) in (23), one derives