Issue 57

S. Derouiche et alii, Frattura ed Integrità Strutturale, 57 (2021) 359-372; DOI: 10.3221/IGF-ESIS.57.26

where σ y is the normal stress applied in the closed crack, Δ l is the small crack extension. Δ v is the displacement of two points on the crack lips along the y-axis (Fig. 2). For Mode II:



l

l  



   , u r 





, 0    l r

G

dr

0 lim 1 2 l  

(13)

II

y

0

where Δ u is the displacement of two points of the crack lips along the x-axis, and τ is the shear stress along the closed crack. The application of the Eqns. (12) and (13) can be executed by computing the stresses for closing the crack and calculate the displacements corresponding, which could be expressed by:



l

1   l

(1)

( 2 ) v dx

(14)







G

I

y

2

0



l

1   l

(1)

( 2 ) u dx

(15)







G

II

xy

2

0

Considering the implementation of the two Eqns. (14) and (15) in the finite element RMQ7 proposed, it is mandatory to define the nodes that will help in getting the value of the ERR. First, the forces that are submitted on the nodes (5, 6, 5’ and 7’) which represent the first step of the crack closer integral technique and are determined by the Eqn. (11), then the second step is to determine the displacement on the nodes (4, 3’, 7 and 6’) (Fig. 3) as:

4            

43' u u u u u u v v v v v v 76' 43' 4 3' 76' 7 6' 7 6' 3'

(16)

Figure 3: The two configurations of the crack closure integral technique occurred in RMQ7: a) Forces that act to extend the crack lips. b) Displacements resulted from the crack extension.

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