PSI- Issue 9

Riccardo Fincato et al. / Procedia Structural Integrity 9 (2018) 136–150 Autho name / Structural Integrity Procedia 00 (2018) 000 – 00

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include an additional internal variable for the description of the damaging behavior. In the next section 2.1 the main features of the DSS model are briefly reported. A detailed description of the constitutive equations goes beyond the purposes of this paper, the reader is referred to Fincato and Tsutsumi (2017a) for an exhaustive discussion.

2.1. The Damage Subloading Surface model

The DSS model is a coupled elastoplastic and damage model that derives it formulation from a previous unconventional plasticity theory, named Extended subloading surface model. The main feature consists in the abolition of the neat distinction between the elastic and plastic domains, stating that irreversible deformations can be generated whenever the loading criterion is satisfied. In order to achieve this goal a second surface, named subloading surface , is generated by means of a similarity transformation from the conventional yield surface, here renamed normal yield-surface . The center of similarity is not fixed in the stress space but it can move following the plastic strain rate. This allows to the creation of close hysteresis loops during the unloading and subsequent reloading, as shown by the authors in Fincato and Tsutsumi (2017b), for a more realistic description of the material ratcheting in cyclic mobility and fatigue problems. The numerical model is developed within the framework of finite elastoplasticity, assuming a hypoelastic based plasticity. The total strain rate D can be additively decomposed into and elastic part e D and a plastic part p D . The coupling of the elasticity with damage is obtained by means of the concept of the effective stress Kachanov (1958) as follows:     1 : 1 : ( ) e p D D      σ E D E D D (1) The coupling of the plastic internal variables with the scalar isotropic damage variable is obtained following the approach suggested by Lemaitre (1985) and modifying the expression of the consistency conditions for the normal yield and subloading surface as follows: ˆ ˆ ( ) (1 ) ( ); ( ) (1 ) ( ); f D F H f D RF H       σ σ σ σ α (2) The scalar function f of the stress tensor in Eq. (2) is assumed as the von Mises criterion. The plastic strain rate can be obtained from Eq. (2) as shown in Fincato and Tsutsumi (2017a), giving the following expression:     / (1 ) / ; ; / (1 ); p D D f H H D               D σ σ (3) In order to complete the set of equations describing the material behavior, the following isotropic and kinematic hardening laws are reported in Eqs. (4)-(5). The isotropic hardening law is a modified version of the law presented in Hashiguchi and Yoshimaru (1995), with an additional linear contribution regulated by the constant K . Eq. (5) was initially proposed by Chaboche (1986), and considered the linear combination of N non-linear contributions. The present paper considers N = 2 contributions. The calibration of the material constants K , h 1 , h 2 , C i , B i will be discussed in section 3.2.     2 0 0 1 ( ) 1 d h H H d F H F K H H F h e           (4)   1 ; , 1,... n p i i i i i i C B n N        α D α α α (5)

2.2. The ductile damage criterion

Following a well-consolidated approach adopted by the continuum damage mechanics an arbitrary stress state can be described by using two dimensionless parameters: the stress triaxiality and the Lode angle parameter (see Figure 1b), defined as in Eq. (6):