A modified Gurson model to account for the influence of the Lode parameter at high triaxialities

doi:10.1016/j.euromechsol.2015.09.010

European Journal of Mechanics - A/Solids

Volume 56, March–April 2016, Pages 31-44

https://doi.org/10.1016/j.euromechsol.2015.09.010 Get rights and content

Highlights

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Modification of the Gurson model taking into account the Lode angle at high triaxialities.
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Formulation, integration and implementation of the enhanced constitutive model.
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Computational cell to prescribe both macroscopic triaxiality and Lode parameter.
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Agreement between J2 voided cell and modified Gurson model cell.

Abstract

The influence of the Lode parameter on ductile failure has been pointed out by different authors even at high triaxiality stress states. However, one of the most widely used model for ductile damage, like the Gurson–Tvergaard (GT) model, systematically disregard the role played by the third stress invariant. In this paper, an improvement of the classical Gurson–Tvergaard model is proposed. The new relation takes into account the effect of triaxiality and Lode parameter through the q₁ and q₂ GT parameters. The convexity of the proposed yield surface has been examined and ensured. The integration of the new constitutive equations as well as the consistent tangent modulus have been formulated and implemented in a Finite Element code. A computational 3D cell has been used to prescribe both macroscopic triaxiality and Lode parameter during loading. Numerical simulations are presented for Weldox 960 steel with different initial porosities and for different prescribed macroscopic triaxialities and Lode parameters using a computational 3D cell methodology. The results are compared with those obtained with a J₂ voided cell. These comparisons show that the improved model captures adequately the Lode effect on the stress-strain curves and on the void growth.

Introduction

The ductile fracture phenomenon in metals and alloys usually follows a failure mechanism involving nucleation, growth and coalescence of voids. Pioneering micromechanical studies of this phenomenon were carried out by McClintock, 1968, Rice and Tracey, 1969, considering the growth of isolated cylindrical or spherical voids driven by plastic deformation of the surrounding rigid perfectly plastic matrix material. To analyze the ductile failure of porous materials, the Gurson–Tvergaard (GT) damage model (Gurson, 1977, Tvergaard, 1981, Tvergaard, 1982) is the most widely used approach. Tvergaard, 1981, Tvergaard, 1982 modified the Gurson model by introducing the q₁ and q₂ parameters to more accurately describe the void growth kinetics observed in unit cell computations. Faleskog et al. (1998) and Gao et al. (1998) have shown that these values are not constant but depend on both strength and strain-hardening properties. More recently, Kim et al. (2004) and Vadillo and Fernández-Sáez (2009) have pointed out that the q_i parameters also depend on the triaxiality of the stress field, as well as on the initial porosity, and highlighted the importance of a proper choice of q₁ and q₂ for the correct modelling of the void growth process.

Various extensions of the Gurson model have been developed and provided elsewhere in order to better represent the response of ductile metals (Gologanu et al., 1997, Gărăjeu et al., 2000, Pardoen and Hutchinson, 2000, Zhang et al., 2000, Benzerga, 2002, Flandi and Leblond, 2005, Monchiet et al., 2008). These modifications make all the assumption of axisymmetric cavities remaining spheroidal during plastic deformation. For a review on constitutive models developed to simulate ductile failure see Besson, 2010, Pineau and Pardoen, 2007.

In the last years, several researchers (Zhang et al., 2001, Kim et al., 2003, Kim et al., 2004, Bao and Wierzbicki, 2004, Wen et al., 2005, Gao and Kim, 2006, Kim et al., 2007, Xue, 2007, Barsoum and Faleskog, 2007, Xue, 2008, Bai and Wierzbicki, 2008, Brünig et al., 2008, Gao et al., 2009, Gao et al., 2011, Barsoum and Faleskog, 2011, Barsoum et al., 2011, Jackiewicz, 2011, Danas and Ponte-Castañeda, 2012, Benallal et al., 2014) outlined that the stress triaxiality measure by itself is insufficient to characterize plastic yielding, and highlighted the role of the third invariant of the deviatoric stress tensor, on void growth rates and other aspects of void behaviour which play an important role in strain softening and localization.

At high triaxialities, where the controlling damage mechanism is the void growth, the influence of Lode parameter can be also important (Barsoum et al., 2011). This effect cannot be properly accounted for with the classical GT model. At low triaxialities, the source of the instability cannot be identified with a void growth mechanism (Yamamoto, 1978). The GT model was recently modified to introduce a Lode dependent softening term for low triaxialities (Nahshon and Hutchinson, 2008). By construction, this modification is inconsistent with mass conservation (Danas and Ponte-Castañeda, 2012).

In the present paper, an improvement of the Gurson–Tvergaard model that accounts for the influence of the Lode parameter at high triaxiality stress states is presented. The modification consists on incorporating the Lode parameter effect into the GT yield surface through q₁ and q₂, which depend not only of the stress triaxiality T, but also on the third invariant of the deviatoric stress tensor J₃. This new term is calibrated to ensure the convexity of the yield surface. The integration of the new constitutive equations has been implemented using a full implicit Euler-backward scheme combined with the return mapping algorithm. Additionally, the consistent tangent modulus has been formulated. For validation purposes, a 3D extension of the computational cell model employed by Xia and Shih, 1995a, Xia and Shih, 1995b, Xia and Shih, 1996 has been developed extending the prescription to both macroscopic triaxiality and Lode parameter. Numerical simulations using the Finite Element code ABAQUS/Standard (Simulia, 2014) are presented for Weldox 960 steel considering different initial porosities and various prescribed macroscopic triaxialities and Lode values. The obtained results using the new continuum damage model are compared with those found with a J₂ voided cell for both the void growth and the stress-strain response of the material.

Section snippets

The unit-cell model

Under the assumption of a periodic microstructure, a porous material can be approximated by representative volume elements (RVE), each containing a void. The axisymmetric cell model is a very convenient way to simplify the problem, because it requires only two-dimensional calculations, so is the most frequently way to analyse the material behaviour. Those authors who only deal with axisymmetric conditions ignore the influence of other possible Lode parameter values in the response of the

Numerical cell results for the voided J₂ and for the classical Gurson–Tvergaard model

Many authors (Zhang et al., 2001, Kim et al., 2007, Gao et al., 2005, Xue, 2008) show that a voided cell subjected to the same stress triaxiality ratio, would tends to react differently when Lode parameter is different. In this section, we will discuss the macroscopic stress-strain evolution and the growth of the porosity (until coalescence) of a voided J₂ cell subjected to prescribed triaxiality and Lode parameter values during the deformation history. For this purpose, the range of high

Constitutive equations

One of the major limitations of the GT model is that, although it is extensively used, it can only handle the growth of spherical voids remaining spherical, which is only approximately true for L = −1 and at triaxialities around 1.5. However, at high triaxialities, it is possible to calibrate q₁ and q₂ parameters in GT model to reproduce the behavior of the material in these stress situations. These calibrated q_i values are not constants, but dependent on the material, the stress triaxiality

Integration procedure

In the context of the Finite-Element method, the integration process is local in space and occurs at each quadrature points of the finite elements. The incremental integration of the constitutive equations is a strain-driven process in which the total strain tensor increment at each quadrature point, $\dot{E}$ , is given at a time (n) and both the stress tensor and the state variables should be updated at time (n + 1).

To integrate the set of non-linear constitutive Eqs. (26), (27), (28), (29), (30),

Comparison between the voided J₂ cell and the modified continuum GT cell

In order to analyse the accuracy of the proposed model, a selection of different loading and initial void volume fractions for Weldox 960 material will be studied in this section. The stress-strain and void volume fraction evolution within the deformation range 0 ≤ E_e ≤ E_c have been compared for the two RVE cell model approaches using three different triaxiality values (T = 1, 1.5, 2) and two Lode parameters (L = 0, 1) for the initial porosities f₀ = 0.005 and f₀ = 0.01. The case corresponding

Concluding remarks

The salient feature of the present paper is the proposition of an improved GT model that accounts for the triaxiality and Lode effects through q₁ and q₂. We also present the finite element implementation of the modified GT model using return mapping method (Euler-backward integration technique) and the formulation of the consistent tangent modulus. An extension of the computational cell model employed by Xia and Shih, 1995b, Xia and Shih, 1995a, Xia and Shih, 1996 has been developed to

Acknowledgements

The authors are indebted to the Ministerio de Economía y Competitividad (Projects DPI2011-23191 and DPI/2011-24068) for the financial support received which allowed condicting part of this work.

The authors would like to thank Professor J. Faleskog, Royal Institute of Technology (KTH) of Sweden, and Dr. J.A. Rodríguez-Martínez and Dr. J. Zahr, University Carlos III of Madrid (UC3M), for valuable discussions.

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Citation Excerpt :
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A modified Gurson model to account for the influence of the Lode parameter at high triaxialities

Highlights

Abstract

Introduction

Section snippets

The unit-cell model

Numerical cell results for the voided J2 and for the classical Gurson–Tvergaard model

Constitutive equations

Integration procedure

Comparison between the voided J2 cell and the modified continuum GT cell

Concluding remarks

Acknowledgements

Int. J. Plast.

Int. J. Mech. Sci.

Int. J. Solids Struct.

Int. J. Solids Struct.

Procedia Eng.

Eur. J. Mech. A/Solids

J. Mech. Phys. Solids

Int. J. Plast.

Int. J. Solids Struct.

Eur. J. Mech.

Eng. Fract. Mech.

Int. J. Solids Struct.

Int. J. Solids Struct.

Int. J. Plast.

Int. J. Plast.

Comput. Methods Appl. Mech. Eng.

J. Mech. Phys. Solids

Eng. Fract. Mech.

Int. J. Solids Struct.

Eng. Fract. Mech.

Int. J. Solids Struct.

Int. J. Solids Struct.

Int. J. Plast.

Eur. J. Mech. A/Solids

Numerical cell results for the voided J₂ and for the classical Gurson–Tvergaard model

Comparison between the voided J₂ cell and the modified continuum GT cell