PSI- Issue 9

Riccardo Fincato et al. / Procedia Structural Integrity 9 (2018) 136–150 Author name / Structural Integrity Procedia 00 (2018) 000 – 0 0   1.0 6 m Mises          (6) In particular, the Lode angle parameter field of existence covers all the possible loading conditions, from the uniaxial extension to the uniaxial compression passing by the pure shear (or plane strain) condition where it assumes null value. Therefore, every pair of variables   ,   , representing a unique stress state, can be used in the formulation of fracture envelope in the MC criterion. This criterion has been widely and successfully used in the prediction of the failure behavior of granular materials such as rock, soil and concrete. However, recent works (Algarni et al., 2015; Bai and Wierzbicki, 2010; Rousselier and Luo, 2014), applied the MC criterion to metallic materials due to its characteristic of being able to catch an exponential decay of the ductility with the stress triaxiality and its Lode angle dependency. The full expression of the MC requires the calibration of eight material parameters, however, if a von Mises potential is accounted for, the failure envelope can be defined by the calibration of only four material constants   1 2 , , , A N c c (Bai and Wierzbicki, 2010). The use of a von Mises potential should be limited to cases where the effect of the Lode angle on plasticity and the pressure effect on the yield-surface can be neglected. Nonetheless, it can give a good first approximation of the damaging behavior of the material. Therefore, in the present paper we adopted the following form of the failure envelope: 139 4

1



    

    

   

   

N

  

   

6       

      

c

2

1



A

1

1 c    

cos

sin

f 



(7)

1

c

3

3 6

2

Based on the previous Eq. (7) the ductile damage increment can be written as:     3 1 3 2 3 ; 2 3 , p t i p p f T D H d H T           D D D D

D T

H H

 

(8)

Where d 1 is an additional material parameter and H represent the Heaviside step function, introduced by the authors to allow the damage to evolve whenever the cumulative plastic strain exceeds the parameter d 1 . The term at the numerator in Eq. (8) has been modified to take into account an additional inelastic term generated by the stress rate component tangential to the plastic potential as in Figure 1a. H T represents the cumulative inelastic strain that actively contributes to the damage. Non-proportional loading, in fact, triggers a non-negligible component of the stress rate that is directed tangentially to the yield surface. The idea is to consider the contribution of the strain rate associated with the tangential component of the stress rate in the damage accumulation. Previous work of the authors (Fincato and Tsutsumi, 2017c; Hashiguchi and Tsutsumi, 2001; Momii et al., 2015; Tsutsumi and Kaneko, 2008) considered a similar approach to overcome the excessive stiffness predicted by the extended subloading surface model with associate flow rule during non-proportional loading. In this case, the effect of the tangential deviatoric strain rate t D affects only the damage evolution and its contribution can be regulated by the material constant 3 T 3 (0 1) T   . The computation of the deviatoric tangential stress rate is done similarly as in Fincato and Tsutsumi (2017c) and in Momii et al. (2015). Briefly, it is assumed that the deviatoric tangential strain rate is linearly related to the component of the stress rate tangential to the yield surface t  σ . The advantage of this hypothesis lies in its very simple form, suitable for the application to boundary-value problems and general loading conditions.

2 G T R   1 T R 1 2 1 T

 

A

ˆ

t

; σ σ I σ  

t 

t 

D

A

(9)

;



 

2     1 T

G

2