PSI - Issue 47

Marzak Zerouki et al. / Procedia Structural Integrity 47 (2023) 915–918 Marzak Zerouki et al./ Structural Integrity Procedia 00 (2019) 000–000

917

3

4. Numerical modeling The GTN model taking into account the anisotropy is expressed in equation 1.

2

3 2

q

p

  

  









2    3 q f 1

, ,  

1 q f 2 cosh

0

f

kq



 

(1)

2

2





Where q and k incorporate the effects of anisotropy, Benzerga and Besson (2001) and Benzerga et al. (2019)

3 2





2

2 h h h h h       2 2 2 2 2 1 11 2 22 3 33 4 23 5 31 5 31 6 12 2 2 2 2 h h 2

(2)

q

3

5

k



(3)

2 2

1 2 h h h   2

3

1 4

33 3 2 h H  ;

1 2









2 2 2 H H H H   

; 2

(4)

h

h

44 H H 





1

11

22

66

33

3

55

The extension of the GTN model to take into account the effect of strain rate is described in equation 5.

   

 

1

m



   

p q     

q 

0               0  

1 

N



 

E

  

p

0

  



(5)

  





1 0 1     N

1

N



m t 

N

f





The porosity increment is given by the following equation 6.

p       q 

q 





p

1 A        f f

(6)

  





 

p

1

f





The GTN model, extended to take into account viscoplasticity and anisotropy, was implemented in the Abaqus explicit computational code using the VUMAT subroutine using an elastic predictor-plastic correction algorithm, Aravas (1987), Ould Ouali (2018). 5. Numerical result Fig. 2 shows the boundary conditions imposed on the specimens as with the chosen mesh (C3D8R). Fig. 3 presents a comparison of numerical and experimental results for temperature 25°C and strain rate 10 -4 s -1 according to the three orthotropic directions 0°, 45° and 90°. These comparisons shows that the proposed GTN model extension replicates successfully the effect of strain rate and anisotropy on the mechanical behavior of the study material.

Fig. 2. Boundary conditions applied to the specimens.