PSI- Issue 9

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 – 000

2

2 2

Riccardo Fincato et al. / Procedia Structural Integrity 9 (2018) 126–135 of the mechanical performance during the material loading. The present paper offers a comparison between two ductile da age criteria. The first one considers a modified version of the Lemaitr ’s law, presented by the authors in Fincato and Tsutsumi (2017a). The second one consists in a modified version of the Mohr-Coulomb criterion, extensively used in the rock/soil mechanics but already successfully adapted to the investigation on ductile materials (Algarni et al., 2015; Bai and Wierzbicki, 2010). The two damage law are comp red reproducing the failure behavior of a series of notched bars and flat grooved plates under monotonic loading. Nomenclature σ Cauchy stress α back stress s similarity centre σ conjugate Cauchy stress α conjugate back stress E tensor of the elastic constants F isotropic-hardening function F 0 initial size of the normal-yield surface H isotropic hardening variable D H cumulative plastic variable (i.e. equivalent plastic strain) R similarity ratio λ plastic multiplier D ductile damage scalar variable σ m mean stress Mises  von Mises stress η stress triaxiality  Lode angle  Lode angle parameter (-1 <  < 1) f  equivalent strain t fracture Y energy release rate (Lemaitre, 1985a) a b  Heaviside step function: 0 if 0; 1 if 0 a b a b a b a b         The elasto-plastic and damage model presented in this paper, the Damage Subloading Surface model (i.e. Damage S-S), was develope by coupling a scalar ductile damage variable D with the constitutive equations of the Extend d Subloading Surface (i.e. S-S model) (Fincato and Tsutsumi, 2017b), following the approach suggested by Lemaitre. The choice of the unconventional plasticity model was due to the ability of the subloading surface to catch a realistic accumulatio f plastic deformation under cyclic loading, as shown in previous works (Fincato and Tsutsumi, 2017 , 2017d; Hashiguchi et al., 2012; Hashiguchi and Tsutsumi, 1993; Tsutsumi et al., 2016, 2010, 2006). The Damage S-S maint ins the same structure of the origi al S-S model, abolishing the distinction between the elastic and plastic domains and allowing the development of plastic deformation for every stress state that satisfies the loading riterion (Hashiguchi, 1994) during the material loading. For that purpose a new surface is created, the subloading surface, always passing through the current stress state and contracting and expanding following the loading or unloading of the material, see Figure 1. The subloading surf ce is created by means of a similarity transf rmation from the c nventional plastic potential, here renamed norm l-yield surface. Moreover, the introductio of a mobile si ilarity center, that follows the plastic strain rate, allows the description of close hysteresis loops during the unloading reloading of t e material (Fincato and Tsutsumi, 2017c). The detail of the elastoplastic S-S model can be f und in Hashiguchi (2009). Hereafter the main features of the Damage S-S model will be introduced, leavi g the details to Fincato and Tsutsumi (2017b). 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 ductile damage variable D is coupled with the elasticity through the concept of the effective stress, allowing to write the following expression that relates the corotational rate of the Cauchy stress with the elastic strain rate:     1 : 1 : ( ) e p D D      σ E D E D D (1) of the mechanical p rformance during the mat rial loading. The present paper offers a comparison between two d ctile damage criteria. The first one considers a m dified versio of the Lemaitre’s law, presented by the authors in Fincato and Tsutsumi (2017a). The second one consists in a modified version of the Mohr-Coulomb criterion, extensively used in the rock/soil mechanics but already successfully adapted to the investigation on ductile materials (Algarni et al., 2015; Bai and Wierzbicki, 2010). The two damage law are compared r producing the failure behavior of a series of notched bars and flat grooved plates under monotonic loading. Nomenclature σ Cauchy stress α back stress s similarity centre σ conjugate Cauchy stress α conjugate back stress E tensor of the elastic constants isotropic-hardeni g function F 0 initial size of the normal-yield surface isotropic hardening variable D H cu ulative plastic variable (i.e. equivalent plastic strain) R similarity ratio λ plastic multiplier D ductile damage scalar variable σ m mean stress Mises  von Mises stress η stress triaxiality e l  Lode angle parameter (-1 <  < 1) f  equivalent strain at fracture Y energy release rate (Lemaitre, 1985a) a b  Heaviside step function: 0 if 0; 1 if 0 a b a b a b a b         2. The numerical model The elasto-plastic and damage model presented in this paper, the Damage Subloading Surface model (i.e. Damage S-S), was developed by coupling a scalar ductile damage variable D with the c nstitutive equations of the Extended Subloading Surface (i.e. S-S model) (Fincato and Tsutsumi, 2017b), following the approach s ggested by Lemaitre. The choice of the unconventional plasticity model was due to the ability of the subloading surface to catch a realistic accumulation of plastic deformation under cyclic loading, as shown in previous works (Fincato and Tsutsumi, 2017c, 2017d; Hashiguchi et al., 2012; Hashiguchi and Tsutsumi, 1993; Tsutsumi et al., 2016, 2010, 2006). The Damage S-S maintains the same structure of the original S-S model, abolishing the distinction between the elastic and plastic domains and allowing the development of plastic deformation for every stress state that satisfies the loading criterion (Hashiguchi, 1994) during the material loading. For that purpose a new surface is created, the subloading surface, always passing through the current stress state and contracting and expanding following the loading or unloading of the material, see Figure 1. The subloading surface is created by means of a similarity transformation from the conventional plastic p tential, here renamed normal-yield surface. Moreover, the introducti n of a mobile similarity center, that follows the plastic strain rate, allows the description of close hysteresis loops during the u loading reloading of the material (Fincato and Tsutsumi, 2017c). The detail of the elastoplastic S-S model can be found in Hashiguchi (2009). Hereafter the main features of the Damage S-S model will be introduced, leaving the details to Fincato and Tsutsumi (2017b). The numerical model is developed within the framework of finite elastoplasticity, assumi g hypoelastic based plasticity. The total strain rate D can be additively de omposed into and elastic part e D and a plastic part p D . The ductile damage variable D is coupled with the elasticity through the concept of the effectiv stress, allowing to write the following expression that relates the corotational rate of the Cauchy stress with the elastic strain rate:     1 : 1 : ( ) e p D D      σ E D E D D (1) 2 Aut o nam / Structural Integrity P ocedia 00 (2018) 000 – 000 of the mechanical performance during the material loading. The present paper offers a comparison between two ductile damage criteria. The first one considers a modified version of the Lemaitre’s law, presented by the authors in Fincato and Tsutsumi (2017a). The second one consists in a modified version of the Mohr-Coulomb criterion, extensively used in the rock/soil mechanics but already successfully adapted to the investigation on ductile materials (Algarni et al., 2015; Bai and Wierzbicki, 2010). The two damage law are compared reprodu ing the failure behavior of a series of notched bars and flat grooved plates under monotonic loading. Nomenclature σ Cauchy stress α back stress s similarity centre σ conjugate Cauchy stress α conjugate back stress E tensor of the elastic constants F isotropic-hardening function F 0 initial size of the normal-yield surface H isotropic hardening variable D H cumulative plastic variable (i.e. equivalent plastic strain) R similarity ratio λ plastic multiplier D ductile damage scalar variable σ m mean stress Mises  von Mises stress η stress triaxiality  Lode angle  Lode angle parameter (-1 <  < 1) f  equivalent strain at fracture Y energy release rate (Lemaitre, 1985a) a b  Heaviside step function: 0 if 0; 1 if 0 a b a b a b a b         2. The numerical model The elasto-plastic and damage model presented in this paper, the Damage Subloading Surface model (i.e. Damage S-S), was developed by coupling a scalar ductile damage variable D with the constitutive equations of the Extended Subloading Surface (i.e. S-S model) (Fincato and Tsutsumi, 2017b), following the approach suggested by Lemaitre. The choice of the unconventional plasticity model was due to the ability of the subloading surface to catch a realistic accumulation of plastic deformation under cyclic loading, as shown in previous works (Fincato and Tsutsumi, 2017c, 2017d; Hashiguchi et al., 2012; Hashiguchi and Tsutsumi, 1993; Tsutsumi et al., 2016, 2010, 2006). The Damage S-S maintains the same structure of the original S-S model, abolishing the distinction between the elastic and plastic domains and allowing the development of plastic deformation for every stress state that satisfies the loading criterion (Hashiguchi, 1994) during the material loading. For that purpose a new surface is created, the subloading surface, always passing through the current stress state and contracting and expanding following the loading or unloading of the material, see Figure 1. The subloading surface is created by means of a similarity transformation from the conventional plastic potential, here renamed normal-yield surface. Moreover, the introduction of a mobile similarity center, that follows the plastic strain rate, allows the description of close hysteresis loops during the unloading reloading of the material (Fincato and Tsutsumi, 2017c). The detail of the elastoplastic S-S model can be found in Hashiguchi (2009). Hereafter the main features of the Damage S-S model will be introduced, leaving the details to Fincato and Tsutsumi (2017b). 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 ductile damage variable D is coupled with the elasticity through the concept of the effective stress, allowing to write the following expression that relates the corotational rate of the Cauchy stress with the elastic strain rate:     1 : 1 : ( ) e p D D      σ E D E D D (1) 127 of the mechanical performance during the material loading. The present paper offers a comparison between two ductile damage criteria. The first one considers a modified version of the Lemaitre’s law, presented by the authors in Fincato and Tsutsumi (2017a). The second one consists in a modified version of the Mohr-Coulomb criterion, extensively used in the rock/soil mechanics but already successfully adapted to the investigation on ductile materials (Algarni et al., 2015; Bai and Wierzbicki, 2010). The two damage law are compared reproducing the failure behavior of a series of notched bars and flat grooved plates under monotonic loading. Nomenclature σ Cauchy stress α back stress s similarity centre σ conjugate Cauchy stress α conjugate back stress E tensor of the elastic constants F isotropic-hardening function F 0 initial size of the normal-yield surface H isotropic hardening variable D H cumulative plastic variable (i.e. equivalent plastic strain) R similarity ratio λ plastic multiplier D ductile damage scalar variable σ m mean stress Mises  von Mises stress η stress triaxiality  Lode angle  Lode angle parameter (-1 <  < 1) f  equivalent strain at fracture Y energy release rate (Lemaitre, 1985a) a b  Heaviside step function: 0 if 0; 1 if 0 a b a b a b a b         The elasto-plastic and damage model presented in this paper, the Damage Subloading Surface model (i.e. Damage S-S), was developed by coupling a scalar ductile damage variable D with the constitutive equations of the Extended Subloading Surface (i.e. S-S model) (Fincato and Tsutsumi, 2017b), following the approach suggested by Lemaitre. The choice of the unconventional plasticity model was due to the ability of the subloading surface to catch a realistic accumulation of plastic deformation under cyclic loading, as shown in previous works (Fincato and Tsutsumi, 2017c, 2017d; Hashiguchi et al., 2012; Hashiguchi and Tsutsumi, 1993; Tsutsumi et al., 2016, 2010, 2006). The Damage S-S maintains the same structure of the original S-S model, abolishing the distinction between the elastic and plastic domains and allowing the development of plastic deformation for every stress state that satisfies the loading criterion (Hashiguchi, 1994) during the material loading. For that purpose a new surface is created, the subloading surface, always passing through the current stress state and contracting and expanding following the loading or unloading of the material, see Figure 1. The subloading surface is created by means of a similarity transformation from the conventional plastic potential, here renamed normal-yield surface. Moreover, the introduction of a mobile similarity center, that follows the plastic strain rate, allows the description of close hysteresis loops during the unloading reloading of the material (Fincato and Tsutsumi, 2017c). The detail of the elastoplastic S-S model can be found in Hashiguchi (2009). Hereafter the main features of the Damage S-S model will be introduced, leaving the details to Fincato and Tsutsumi (2017b). 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 ductile damage variable D is coupled with the elasticity through the concept of the effective stress, allowing to write the following expression that relates the corotational rate of the Cauchy stress with the elastic strain rate:     1 : 1 : ( ) e p D D      σ E D E D D (1) 2. The numerical model 2. The numerical model