PSI - Issue 2_A

Wiktor Wcislik et al. / Procedia Structural Integrity 2 (2016) 1676–1683 Author name / Structural Int grity Procedia 00 (2016) 00 –000

1678

3

(1)

V f V V 

Original Gurson (1977) model is described by the following equation:

2

  

1       f 2

  

   

R σ

σ

(2)

e

e m

2 2 cosh 3 f

0

 

R

e

Assuming f = 0, the Gurson criterion reduces to the classic Huber – Mises model. Original Gurson model was frequently modified. Tvergaard (1981) introduced q i coefficients providing the better convergence of numerical and experimental results. The Gurson Tvergaard constitutive model is defined as follows:

2

  

   

  

1       q f 2 3

2 R σ q

R σ

(3)

e

e 2 m

1 q f

0

2

cosh 3

 

e

In Tvergaard (1981) it is suggested to adopt: q 1 = 1.5, q 2 = 1.0, q 3 = 2.25. The further research of Faleskog et al. (1998) revealed that q i coefficients depend on material parameters (yield stress, Young modulus, hardening exponent). Tvergaard and Needleman (1984) replaced the porosity function f with its modified version f * , which takes into account the influence of voids coalescence on material behavior (the authors assumed that coalescence starts when the distance between voids is equal to their diameter). Gurson Tvergaard Needleman (GTN) constitutive model is expressed by the following formula:

2

  

   

  

1     

R σ

2 R σ q

*

*2

e e

e 2 m

(4)

1 q f

3 q f

2

cosh 3

0

 



The porosity evolution is given as follows:

      

f

or

f

f

f



c

f

f



*

 F c F c f

f

f

 f f

or

f

  f

f

(5)

(

) f





c

c

F

c

f

f

or

f

f

f



F

F

where:

2 1 1 q q q q  

3

(6)

f



F

3

The GTN model, by introducing porosity as the softening parameter, makes it possible to simulate reduction in load carrying capacity caused by voids and microdefects development. The model is often implemented in commercial FEM software, which promotes its use in engineering. The GTN model is recommended in the literature, e. g. Sedlacek et al.(1998), for the analysis of structural members prior to failure. As shown in many studies, f c , f F and  N significantly affect the results obtained with the use of Gurson Tvergaard Needleman model. The issue was discussed in Richelsen and Tvergaard (1994), Rakin et al. (2000), Jackiewicz (2012), Wcislik (2014 b, c), Kossakowski and Wcislik (2014), Kossakowski (2015).