PSI - Issue 18

Riccardo Fincato et al. / Procedia Structural Integrity 18 (2019) 75–85 Author name / Structural Integrity Procedia 00 (2019) 000–000

79 5

H 

( ) F H F KH F h e      1

 

D

2 h H

;   vp vp

vp

;

2 3









0

0 1

 





(4).

n

 

 









C α D 

, α α α 

vp

vp

,

1, ... ;

B

n N 





i

i

i i

i





1

i



Where F 0 is the initial size of the normal-yield surface (i.e. initial macroscopic yield stress), vp  is the viscoplastic multiplier and K , h 1 , h 2 are material constants. The kinematic hardening evolution law in Eq. (4) 2 was introduced by Chaboche (1986), where the back stress co-rotational rate is evaluated by a linear combination of N non-linear contributions, dependent on the material parameters C i and B i . Next, the similarity centre evolution law can be formulated as:

  

 

σ 

1

1

  

  

dF

 

 

N 



α  



ˆ s 

ˆ H if R s

c s D 

vp

1      

1





R 

i

d

dH F









1

n



d

(5).



ˆ 1 σ

  

 

1

  

  

dF

N 



α  

H 

ˆ    s

ˆ s 

c s D 

vp

1

if R





R 

i

d

dH F









1

n



d



Where c is a parameter regulating the speed of the similarity and  is a constant introduced to avoid theoretical and computational inconsistencies, a detail explanation of these two variables is offered in Hashiguchi (2017). The details for the inclusion of the damage effect in Eq. (5) can be found in Fincato and Tsutsumi (2017c). The last terms to be defined are the elastic and viscoplastic strain rates introduced in Eq. (2) 1 . In this work, an associative flow rule for the viscoplastic strain rate is adopted, assuming that the inelastic deformation is generated along the tensor N normal to the plastic potential in the current stress state. In particular, D vp is defined analogously as:









n R R 

n R R 

1

1

e

e





d

d

1

1





(6).

1 D E σ :  

; N σ E D

: E N

:  



















1

1

m d D R R 

m d D R R 









Where  is the viscoplastic coefficient and n is a non-dimensional material constant regulating the amount of viscoplastic strain. 2.1. Similarity ratios definition A key aspect of the DOSS model is the definition of the similarity ratios R and R d during loading and unloading conditions. During the viscoplastic loading R d can always be defined by means of the formula reported in Eq. (2) 2 . A different definition is given to the subloading surface ratio. Whenever the stress satisfy the loading criterion (Hashiguchi, 1994) it is possible to compute R with the analytical formula proposed by Hashiguchi (2009):   1 0 0 2 1 cos cos exp 2 1 2 1 vp vp e e e e e R R R R u R R R                                   (7). Where R e is a material parameter defined by Tsutsumi et al. (2006) to introduce a small elastic domain for a better modelling of the material behavior. In detail, R e ≥ 0.2 in metals, and this constant can be set close to zero for granular materials. Under the initial conditions R = R 0 and 0 vp vp    , defining vp vp dt    D . During the elastic unloading the size of the subloading surface must be kept constant until the dynamic loading surface coincides with the subloading surface, otherwise the viscoplastic strain rate in Eq. (6) 1 is overestimated and the model would predict an incorrect material behaviour. Whenever the dynamic loading ratio satisfies the condition R d ≤1 during elastic unloading the subloading similarity ratio can be computed by the analytical formula proposed by Fincato and Tsutsumi (2018b). The following Eq. (8) reports the expression of R and R d in case of elastic unloading.