PSI - Issue 33

8

Derouiche Sami et al. / Procedia Structural Integrity 33 (2021) 996–1006 Author name / Structural Integrity Procedia 00 (2019) 000–000

1003

By substituting Eq. (14) in Eq. (13), the energy release rate G relation becomes:

 

  

 ne

 ne

1



  T

  ) (



  { ( )} [ ( )] ( ) T i i v a K a v a







 v a a K a a v a a    (

   )

G

(15)

i

i

i

   i a

2

i



i

1

1

Taking into account Eq. (12), the relation (15) can be written as follows:

  1 1 ( ne i a  



    k a v a a   (







T i

 



 v a a K a a  )

  

 

     

G

)

(14)



i

2

i

i

Because, only the elements of crack tip and the elements immediately linked to them are disturbed (Fig. 3), then G can be recalculated by means of the following relation:                                  1 1 ( ) ( ) 2 nf t f f f f f G v a a K a a K a v a a a (17) Where nf is the number of elements concerned by the disturbance δa , following the inclined extension of the crack. In Fig. 3, we notice that nf � � . The Eq. (17) demonstrate that the energy release rate G is calculated using only the elements implicated be the perturbation caused by the crack. Following this, it is necessary to evaluate their elementary matrices in the configuration « a » and the energy release rate is calculated using Eq. (17); by means of a unique discretization, after a difference calculation of the elementary matrices of the concerned elements only; representing the states « a � δa » and « a » . The relation (17) can be written as:                 1 1 2 nf T f f f f K G v v a (18) In practice, the discretisation of the cracked structure is done in the context « a � δa » . While, the “a” setup is acquired in the same way of computing and storing the elementary matrices of the elements involved by the crack and consider as null. Then, at the resolution phase, the nodal values of the involved elements are extracted and the energy release rate G can be calculated using the Eq. (18). 3. Numerical example In order to prouve the efficiency of the method on orthotropic media, one problem taken from the work of Maïti (1986) (Maiti 1986) is been investigated. Consider a center-edge cracked plate under uniaxial tension Fig. 4 with the geometric and the mechanical properties shown in the Table 1.