PSI - Issue 2_A

Benjamin Werner et al. / Procedia Structural Integrity 2 (2016) 2054–2067 Author name / Structural Integrity Procedia 00 (2016) 000–000

2064

11

f f f     

(6)

g

n

comprises an increment of the growing process of existing voids as well as an increment of nucleation. The incremental character of the values is represented by the time derivative. The incremental growth of the existing voids   1 g kk f f      (7)

is calculated from the first invariant of the plastic strain kk   which corresponds with the volume dilatation. The increment of nucleation



   

  

f

  



(8)



exp 1 2   

f



n

n

 

 

n

s

2

s





n

n

is calculated from the plastic strain rate   and the existing void volume fraction through nucleation f n . Furthermore, the equation includes the normal distribution specified by Gauss with the standard deviation s n and the average value ε n . Nahshon and Hutchinson (2008) extended the Gurson-Tvergaard-Needleman damage model for softening under shear stress. Softening under shear load occurs solely through the deformation and rotation of voids and is therefore different from the process of softening under tensile load with growth and nucleation of voids. Nahshon and Hutchinson (2008) defined the incremental growth of voids



s









   

1 k f         kk f

f

ij

(9)

g



vM

as an effective damage value and extended g f  by a second mathematical term describing the damage due to shear load. In equation (9) controls the material parameter k ω the magnitude of damage. The equation also includes the void volume fraction f , the function ω ( σ ), the deviatoric stress s ij , the plastic strain rate   , as well as the equivalent von Mises stress σ vM . The function

Fig. 12. Experimentally and numerically determined force-displacement curves using the Gurson damage model of experiment (a) K1 and (b) K2