PSI - Issue 28

Mamadou Abdoul Mbacké et al. / Procedia Structural Integrity 28 (2020) 1431–1437 Mamadou Abdoul MBACKE/ Structural Integrity Procedia 00 (2019) 000–000

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3

implemented models are consistent referring to the previous works cited above. Its only drawback is its need to additional subroutines as URDFIL or UEXTERNALDB for information exchange with Abaqus. Consequently, this last solution is selected in the present work. 3. Constitutive equations and numerical implementation The objective is to control the natively available quasi-static cohesive zone model using field variables. It follows a traction

separation behavior law as shown by the figure 1. The traction is governed by the following equation:

n   

n   

  K                   s s   t  t  

(5)

, n s   and t  are the normal stress and the two local shear stresses respectively, and the quantities , n s   and t 

The quantities

are the corresponding displacement jumps.

(1 ) 

0 (no damage in compression)

D if 





   

n

n

(6)



n

else



n

The normal stress n  is calculated from the stress n  obtained by elastic prediction according to the relation (5).

Fig. 1. Shape of the cohesive zone model used

The idea is to reformulate the damage variable in order to take into account its evolution with the cyclic loading. The damage is calculated incrementally using the Paris’s law as follows:

0 2 

 

f

(1 )

D D





 

1

D

a

 





(7)



0

f

N A

N



 

CZ

With:

   

m

C G

max if G G G  

a C

    

     

th

C

(8)

N G

0

else

0

max 2

  

   

f

(

)

2 

  

max

f

(9)

G

  



0

f

  

max

3 9 32 ( ) E G 

CZ A b  

in conformity with some works available in the literature, A CZ is the area of the cohesive zone, G max is the

0 2

maximum energy release rate, E 3 is the Young modulus of the bulk material perpendicularly to the crack plane, σ

0 is the interfacial