PSI - Issue 33

6

Author name / Structural Integrity Procedia 00 (2019) 000–000

Derouiche Sami et al. / Procedia Structural Integrity 33 (2021) 996–1006

1001





                 

                 

31

  21

1 1 E

0

0

0

E E

2

3



1

32



0

0

0

E

E

2

3

1

0

0

0

E

  S

3



(10)

1

0

0

G

23

1

sym

0

G

13

1

G

12

Where E 1 , E 2 and E 3 are young modulus;  21 ,  31 and  32 are Poisson’s ratio and G 23 ,G 13 and G 12 are shear modulus. For the three dimensional problem it is a 6x6 matrix, and for the two dimensional problems it is reduced to 3x3 matrix as:

       

        



1

21 0 1 0 2



E E

1

  S

 

(11)

E

2

1

sym

G

12

The virtual crack extension method associated with the RMQ-7 element is used to calculate the energy release rate G in the case of kinking. A single discretization is enough to evaluate the energy release rate for the case of co-linear extension of the crack as demonstrated by Bouzerd(1992). In the present work, the same procedure is adopted with some modifications added in order to take into account the non co-linear extension of the crack as showed in Fig. 2 Bouzerd and al., (2011). By considering a crack extension  a following a specified direction with an angle  with the initial direction Fig. 2 (left), the new situation can be represented by an angled section starting at node 1 of the upper crack tip element, and following the path of the new position of the crack tip after extension, as indicated in Fig. 2 (Right). This estimation is quite acceptable, as long as  a is small. That is why the choice of  a has an important role in our study. Theoretically, the value of  a must be taken as small as possible, in order to numerically represent the Eqs. (10)-(11).