PSI- Issue 9

Riccardo Fincato et al. / Procedia Structural Integrity 9 (2018) 126–135 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 Author name / Structural Integrity Procedia 00 (2018) 000 – 00

3 3 3

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The definition of the plastic strain rate passes through the inclusion into the consistency condition of Eq. (2) following the approach in Fincato and Tsutsumi (2017b) and in Lemaitre (1985b). This allows consideration of the damaging process which develops from around the impurities in the material and leads to the formation of micro-voids, their coalescence, and finally in the formation of cracks. The definition of the plastic strain rate passes through the inclusion into the consistency condition of Eq. (2) following the approach in Fincato and Tsutsumi (2017b) and in Lemaitre (1985b). This allows consideration of the damaging process which develops from around the impurities in the material and leads to the formation of micro-voids, their coalescence, and finally in the formation of cracks. The definition of the plastic strain rate passes through the inclusion into the consistency condition of Eq. (2) following the approach in Fincato and Tsutsumi (2017b) and in Lemaitre (1985b). This allows consideration of the damaging process which develops from around the impurities in the material and leads to the formation of micro-voids, their coalescence, and finally in the formation of cracks.

. . .

Figure 1 Sketch of the conventional yield surface, here rename normal-yield surface, and of the subloading surface. Figure 1 Sketch of the conventional yield surface, here rename normal-yield surface, and of the subloading surface. Figure 1 Sketch of the conventional yield surface, here rename normal-yield surface, and of the subloading surface. ˆ σ ˆ σ (2) Eqs. (2) report the expression of the normal-yield and subloading surface in the Damage S-S model, where f is a generic yield-stress function (herein, the von Mises criterion is assumed), and ˆ σ is the Cauchy stress on the normal-yield surface observed from the back stress (see Figure 1). Finally, the expression of the plastic strain rate is obtained by some mathematical manipulation from the second of Eqs. (2):     / (1 ) / ; ; / (1 ); p D D f H H D               D σ σ (3) ˆ ˆ ( ) (1 ) ( ); ( ) (1 ) ( ); f D F H f D RF H       σ σ σ σ α (2) Eqs. (2) report the expression of the normal-yield and subloading surface in the Damage S-S model, where f is a generic yield-stress function (herein, the von Mises criterion is assumed), and ˆ σ is the Cauchy stress on the normal-yield surface observed from the back stress (see Figure 1). Finally, the expression of the plastic strain rate is obtained by some mathematical manipulation from the second of Eqs. (2):     / (1 ) / ; ; / (1 ); p D D f H H D               D σ σ (3) ˆ σ σ α (2) Eqs. (2) report the expression of the normal-yield and subloading surface in the Damage S-S model, where f is a generic yield-stress function (herein, the von Mises criterion is assumed), and ˆ σ is the Cauchy stress on the normal-yield surface observed from the back stress (see Figure 1). Finally, the expression of the plastic strain rate is obtained by some mathematical manipulation from the second of Eqs. (2):     / (1 ) / ; ; / (1 ); p D D f H H D               D σ σ (3) The material hardening is described by an isotropic Swift law. Where K and n are two material constants assumed as in Li et al. (2016). The material hardening is described by an isotropic Swift law. Where K and n are two material constants assumed as in Li et al. (2016). ( ) (1 ) ( );   ( ) (1 ) ( );   ( ) (1 ) ( ); D RF H   σ ( ) (1 ) ( ); D RF H   σ f f D F H f D F H f     σ σ α The material hardening is described by an isotropic Swift law. Where K and n are two material constants assumed as in Li et al. (2016). ˆ

( ) n F H F KH   ( ) n F H F KH   ( ) n F H F KH  

(4) (4) (4)

0 0 0

The calibration of the elastoplastic and damage parameter will be discussed in the following section 3.1. The calibration of the elastoplastic and damage parameter will be discussed in the following section 3.1. The calibration of the elastoplastic and damage parameter will be discussed in the following section 3.1.

2.1. The Lemaitre’s ductile damage evolution law 2.1. The Lemaitre’s ductile damage evolution law 2.1. The Lemaitre’s ductile damage evolution law

Lemaitre’s theor y is based on the concept of effective stress and the hypothesis of strain equivalence , prerequisites that were also adopted in the formulation of the elastoplastic and damage constitutive equations of the Damage S-S model. Under this assumptions the ductile damage evolution law was formulated as follows: Lemaitre’s theor y is based on the concept of effective stress and the hypothesis of strain equivalence , prerequisites that were also adopted in the formulation of the elastoplastic and damage constitutive equations of the Damage S-S model. Under this assumptions the ductile damage evolution law was formulated as follows: Lemaitre’s theor y is based on the concept of effective stress and the hypothesis of strain equivalence , prerequisites that were also adopted in the formulation of the elastoplastic and damage constitutive equations of the Damage S-S model. Under this assumptions the ductile damage evolution law was formulated as follows:

1 1 1 D s          s Y D s          D s          s Y s Y

2 2 2

D D D

  

(5) (5) (5)

(1 ) (1 ) (1 )

Where s 1 and s 2 are two material constants and Y is the so-called damage energy release rate corresponding to ‘the variation of internal energy density due to damage growth at constant stress’ (de Souza Neto et al., 2008). Eq. (5) considers the exponential decay of the ductility with the stress triaxiality, accounted for in Y , however, it cannot consider the effect of the Lode angle. The authors in Fincato and Tsutsumi (2017a) suggested a modification of Eq. (5) in order to avoid this inconvenience, reformulating the previous expression in: Where s 1 and s 2 are two material constants and Y is the so-called damage energy release rate corresponding to ‘the variation of internal energy density due to damage growth at constant stress’ (de Souza Neto et al., 2008). Eq. (5) considers the exponential decay of the ductility with the stress triaxiality, accounted for in Y , however, it cannot consider the effect of the Lode angle. The authors in Fincato and Tsuts mi (2017a) suggested a modification of Eq. (5) in order to avoid this inconvenience, reformulating the previous expression in: Where s 1 and s 2 are two material constants and Y is the so-called damage energy release rate corresponding to ‘the variation of internal energy density due to damage growth at constant stress’ (de Souza Neto et al., 2008). Eq. (5) considers the exponential decay of the ductility with the stress triaxiality, accounted for in Y , however, it cannot consider the effect of the Lode angle. The authors in Fincato and Tsutsumi (2017a) suggested a modification of Eq. (5) in order to avoid this inconvenience, reformulating the previous expression in: