PSI - Issue 23

Petr Kubík et al. / Procedia Structural Integrity 23 (2019) 15–20 Petr Kubík et al. / Structural Integrity Procedia 00 (2019) 000 – 000

16 2

Nomenclature a

plasticity related material constant

f

yield function

1 I 2 J 3 J

first invariant of stress tensor

second invariant of deviatoric stress tensor third invariant of deviatoric stress tensor

k

yield correction function

  1  2  3  y   0 

plasticity related material constant

equivalent stress

maximum principal stress middle principal stress minimum principal stress

yield stress

normalized Lode parameter plasticity related material constant

I

     

(1)

1

1

2

3

where 2 3      are the three principal stresses ordered according to their magnitude. Next, another widely used measures are the second and third invariants of deviatoric stress tensor, 2 J and 3 J , respectively, which are written as       2 2 2 2 1 2 2 3 3 1 J               (2) 1

1 6





 

  

  

  

I

I

I

J

1    

2 



3 



1

1

1

(3)

3

3

3

3

The yield criterion according to von Mises is one of the most utilized one for polycrystalline metal materials (Mises, 1913). This criterion is dependent only on the second invariant of deviatoric stress tensor. The yield locus forms a circle in the deviatoric plane and the yield surface is prismatic in the Haigh – Westergaard space. Many ductile metals exhibit distinct flow behavior for different loadings. The yielding of von Mises criterion is identical for all types of loadings, therefore it is not available to capture these differences. One of the first yield criteria, which is not dependent only on the second invariant of deviatoric stress tensor, is the one of Coulomb (1776), which was later modified by Mohr (1900), and Drucker and Prager (1952). The yield criterion of Mohr – Coulomb is dependent on all the above mentioned stress invariants. The Drucker – Prager yield criterion is dependent on the first invariant of stress tensor, apart from the dependence on the second invariant of deviatoric stress tensor. Both mentioned models of plasticity are mainly used for description of polymers, rocks and soil. The yield criterion according to Tresca (1864) is dependent on the second and third invariants of deviatoric stress tensor. Hershey (1954), later followed by Hosford (1972) to whom the work is sometimes attributed, developed a generalized yield criterion having one parameter, which can degenerate into the one proposed by von Mises or Tresca. Similar model behavior is seen at the yield criterion proposed by Kroon and Faleskog (2013), which is dependent on the second and third invariants of deviatoric stress tensor. The most complex model mentioned within this short literature review can be the one proposed by Bai and Wierzbicki (2008) with dependence on all the above mentioned stress invariants. An alternative to these plasticity models may be a directional distortional plasticity (Španiel et al., 2017). Nevertheless, the list is not exhaustive.