PSI - Issue 18

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

80 6

2 3

  

  

2

2









2

2 2

ˆ  

ˆ  

ˆ s σ  



σ s 

σ s 

(1 )

tr

tr

D F



 





R



d

2 3

  

2   

2 2

ˆ s

(1 ) 

D F







2 3

  

  

2

2









2

2 2

ˆ  

ˆ  

ˆ s σ  



ss σ s 

ss σ s 

(1 )

tr

tr

D F



 



ss



(8).

  σ σ s 

;

if

1

R

R





ss

ss

d

2 3

  

2   

2 2

ˆ s

(1 ) 

D F







1

if

1

R

d R 



2.2. Ductile damage evolution law The ductile damage variable is coupled in Eq. (2) 2 with the analytical definition of the dynamic loading surface with the purpose to induce a progressive material softening during the generation of inelastic deformation. This paper adopts the Lemaitre (1985) approach for the formulation of the ductile damage evolution law, based on the concept of effective stress and strain equivalence. An initial work of the authors (Fincato and Tsutsumi, 2016) also adopted the original Lemaitre law, however, a subsequent formulation was proposed to include the effect of the Lode angle (Fincato and Tsutsumi, 2017a). More recent works of the authors adopted a different definition of the damage evolution based on the Mohr-Coulomb criterion, due to its simplicity and the accurate prediction of the material failure (Fincato and Tsutsumi, 2018c; Fincato et al. , 2018). This study aims to show the feature of the DOSS model, therefore, a discussion on the most suitable damage evolution law fall outside the purpose of the paper. The damage rate is here defined as:

s

1         vp Y s 

2

D 

(9).

Where Y is the damage energy release rate term, s 1 and s 2 are two material parameters (see Lemaitre (1985) for details). For sake of simplicity, the damage accumulates in the same manner during tension as well as in compression. In general, a different mechanism should be considered.

3. Numerical integration scheme Due to the nature of the overstress theory, the integration algorithm for the viscoplasticity is formulated by means of the fully implicit closest-point projection scheme (i.e. CPP) and it cannot be formulated by means of the cutting plane algorithm (i.e. CP) since there is not a base state, such as the normal-yield or the subloading surface, to which the stress has to be returned for the fulfilment of the local equilibrium. In case of the rate-independent elastoplasticity, the subloading surface can be integrated with both schemes (Fincato and Tsutsumi, 2018). In a previous work, the authors (i.e. Fincato and Tsutsumi, 2017c) integrated the rate-independent elastoplastic constitutive equations with the damage variable through the CP algorithm, i.e. the DSS model. The DSS algorithm is used in section 4 to evaluate the response of the DOSS algorithm under a quasi-static cyclic loading condition. The CPP method is based on a multi-equation Newton-Raphson scheme, where the unknowns represents the roots of the equations to minimize. A detailed description of the method for the elastoplastic problem is offered in Fincato and Tsutsumi (2017b). Here, the description of the algorithm is omitted.