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Journal of Materials Processing Technology 216 (2015) 385–404

Contents lists available at ScienceDirect

Journal of Materials Processing Technology

jo ur nal ho me page: www.elsev ier .com/ locate / jmatprotec

comparative study of three ductile damage approaches for fracturerediction in cold forming processes

.-S. Caoa,∗, C. Bobadillab, P. Montmitonneta, P.-O. Boucharda

Mines ParisTech, CEMEF – Center for Material Forming, CNRS UMR 7635, BP 207, 1 rue Claude Daunesse, 06904 Sophia Antipolis Cedex, FranceArcelorMittal Long Carbon R&D, Gandrange, France

r t i c l e i n f o

rticle history:eceived 9 April 2014eceived in revised form 7 October 2014ccepted 10 October 2014vailable online 22 October 2014

eywords:ire drawingire flat rolling

racture predictionuctile damageode parameterixed FEM

a b s t r a c t

Damage growth and ductile fracture prediction is still an open question for complex stress state applica-tions. A lot of models, both phenomenological and micromechanical, have been extensively developed.There is a real need to compare them to choose the best suitable for complex loading applications. This isdone here taking examples in cold metal forming, namely wire drawing and wire flat rolling. In the presentstudy, the prediction of damage for the ultimate wire drawing and the wire flat rolling processes of a highcarbon steel is investigated, using three different approaches of ductile damage: uncoupled phenomen-ological models (or fracture criteria), coupled phenomenological models (accounting for the softeningeffect of damage), and micromechanical models (accounting for damage associated microstructure evo-lution). These models were first implemented in a finite element code dedicated to forming processsimulations, then calibrated via different mechanical tests exhibiting different stress states. Numericalresults of the applications of these models to the two above-mentioned forming processes simulationswere compared with experimental ones. These applications help comparing different approaches forfracture prediction in multi-stage forming processes and also in the process that involves important

shear effect. The present study supplies important data for the characterization of ductile failure in form-ing processes, as well as an effective assessment of different phenomenological and micromechanicalmodels, characterizing their performance for different stress states. It also suggests the use of “modular”models for complex loading cases, by combining different driving factors of damage accumulation atdifferent stress states.

. Introduction

Fracture prediction in real size structures subjected to complexoading conditions has been of utmost interest in the scientific andngineering community in the past century. Numerical simulationsith nonlinear finite element (FE) codes allow investigating vari-

us complicated problems for damage and fracture prediction ineal scale models, which is an important topic in many industries,ncluding metal forming industry. For all industrial cold formingrocesses, the ability of numerical modeling to predict ductile frac-ure is crucial. However, this ability is still limited because of theomplex loading paths (multi-axial and non-proportional loadings)

nd important shear effects in several forming processes wherehe stress triaxiality is nearly zero. Moreover, since forming pro-esses involve large strain, the use of a suitable FE code with robust

∗ Corresponding author at: Mines ParisTech, Centre des matériaux, UMR CNRS633, BP 87, F-91003 Evry Cedex, France. Tel.: +33 6 68 26 46 11.

E-mail address: trong-son.cao@mines-paristech.fr (T.-S. Cao).

ttp://dx.doi.org/10.1016/j.jmatprotec.2014.10.009924-0136/© 2014 Elsevier B.V. All rights reserved.

© 2014 Elsevier B.V. All rights reserved.

damage and fracture prediction models is essential to obtain realis-tic results for both geometry precision and mechanical properties.Regarding ductile damage models, three main approaches havebeen extensively used and developed for fifty years: uncoupledphenomenological damage models (or fracture criteria), coupledphenomenological models and micromechanical models. The roleof microvoids in ductile failure was firstly modeled by the studyof McClintock et al. (1966), which analyzed the evolution of anisolated cylindrical void in a ductile elastoplastic matrix. Rice andTracey (1969) studied the evolution of spherical voids in an elastic-perfectly plastic matrix. In these studies, the interaction betweenmicrovoids, the coalescence process and the hardening effects wereneglected and failure was assumed to occur when the cavity radiuswould reach a critical value specific for each material. These resultsshowed that the voids growth is governed by the stress triaxi-ality, which is the ratio between the mean stress and the von

Mises equivalent stress. Gurson (1977), in an upper bound anal-ysis of a finite sphere containing an isolated spherical void ina rigid perfectly plastic matrix, employed the void volume frac-tion f (or porosity) as an internal variable to represent damage

386 T.-S. Cao et al. / Journal of Materials Process

D1, D2, D3, D4, D5, D6 material constants in the Bai &Wierzbicki model

E, � Young’s modulus and Poisson’s ratioEM, �M Young’s modulus and flow stress of undamaged

materialY, w(D) energy density release rate and weakening function

(Lemaitre model)˛1, ˛2, �1, �2, �D0, A additional material constants in the LEL

model�p equivalent plastic strain rate�f0, pL, q, k, m, ˇ, � , �DX, Dc material constants in the Xue

model� stress triaxiality�f equivalent plastic strain at fracture�p equivalent plastic strain� von Mises equivalent stress�0 flow stress�1, �2, �3 three principal stresses, �1 ≥ �2 ≥ �3�m mean or hydrostatic stress, �m = (�1 + �2 + �3)/3�, � Lode angle and Lode parameterb, S, Dc, h, �D material constants in the Lemaitre modelf, FD plastic potential (yield function in associative flow),

damage dissipative potentialp hydrostatic pressureq1, q2, SN, fN, �N, fc, ff material constants in the GTN modelq∗ , q additional material constants in the modified GTN

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smith cross”, which led them to a conclusion that these zones weredanger zones in such a process. They then used the Lemaitre dam-age model to study the localization of damage but it failed to predict

3 4model

nd its softening effect on material strength. This model was thenxtensively improved to account for different aspects: predictionccuracy (Tvergaard, 1981), void nucleation and void coalescenceTvergaard and Needleman, 1984), void shape effect (e.g. Gologanut al., 1993), void size effect (e.g. Wen et al., 2005), void/particlenteraction (e.g. Siruguet and Leblond, 2004), isotropic strain hard-ning (e.g. Leblond et al., 1995), kinematic hardening (e.g. Muhlichnd Brocks, 2003), plastic anisotropy (e.g. Benzerga and Besson,001), rate dependency (e.g. Tvergaard, 1989), shear effect (e.g. Xue,008).

On the other hand, the Continuum Damage Mechanics (CDM)odels have been developed within a consistent thermodynamic

ramework, in which the evolution of the phenomenological dam-ge parameter is obtained through a thermodynamic dissipationotential. This class of models has been continuously developednd widely used, especially the Lemaitre model Lemaitre (1986) –ee Besson (2010) for a complete review of continuum models ofuctile fracture).

In addition to the phenomenological CDM models andicromechanics-based damage models, uncoupled phenomeno-

ogical models have been increasingly developed, especially forndustrial applications. Uncoupled models employ an indicatorariable to predict material failure when its critical value is reached.his variable is often taken as a weighted cumulative plastic strain,n which the weighting function accounts for the effect of stresstate on the fracture initiation.

The early ductile damage models used only the stress triaxialityn order to account for the influence of stress state. Several recenttudies (e.g. Barsoum and Faleskog, 2007) also demonstrated themportant effect of the third stress invariant in damage evolution,specially at low stress triaxiality; the Lode angle parameter is gen-

rally used to include it. This parameter combines the second andhird invariants of deviatoric stress tensor. Xue (2007) developed

damage-plasticity model, which accounts for the influence ofydrostatic pressure and the Lode angle. Bai and Wierzbicki (2008)

ing Technology 216 (2015) 385–404

constructed an asymmetric fracture locus using a weighting func-tion of the stress triaxiality and the Lode parameter. More recently,the same authors transformed the stress-based Mohr–Coulombfailure criterion into the space of the stress triaxiality, the equiv-alent plastic strain and the Lode parameter (Bai and Wierzbicki,2010). The common idea of these works is to account for the wholestress state in damage model formulation, which is defined by thestress triaxiality, the von Mises equivalent stress, and the Lodeparameter. Gurson-based models have also been enhanced to bet-ter describe ductile damage for low stress triaxiality (e.g. Nahshonand Hutchinson, 2008).

Despite their increasing developments, among numerous duc-tile damage models proposed in the literature, very few wereactually applied and validated on complex industrial applications,such as multi-stage forming processes. The comparison of the threeabove-mentioned damage approaches on real complex multi-stagemanufacturing processes is important to clarify the advantages anddrawbacks of each one. In the present study, multi-stage ultimatewire drawing and wire flat rolling processes were chosen to servefor this purpose.

Regarding damage in wire drawing process, defects in drawnwire come from both the initial defects from the preform and thedeformation process itself. The common defect observed in draw-ing is chevron cracking or central burst (also called “cupping” – cupand cone fracture). However, under certain conditions of materialand microstructure states (e.g. large defects on the surface of theinitial wire), fracture can initiate at the surface due to the pres-ence of important shear effect at this position. Interested readersare invited to the recent work of Cao et al. (2014d) for a study ofductile fracture in multi-stage drawing, and references therein fora literature review on ductile damage in this process.

Concerning the wire flat rolling process, numerous studies werecarried out by Kazeminezhad and Karimi Taheri. Kazeminezhad andTaheri (2005a) performed experimental studies on the rolling forceand the deformation behavior of rolled wire. They found that therolling force depends on rolling speed and rolling reduction, butlubricant has a negligible effect on both rolling force and geometry(width of contact area and lateral spread). An analytical relation-ship was proposed by these authors (Kazeminezhad and Taheri,2005b) to evaluate the lateral spread for both low and high car-bon steels, which is a function of the ratio between initial and finalheights of flattened wire. In terms of stresses and strain analy-ses, Kazeminezhad and Taheri (2006a) used analytical analyses andfound that there exists a maximum in the pressure distribution sim-ilar to that observed in strip rolling. They also revealed shear bandson the cross section of the flattened wire in form of a cross (theso-called “blacksmith” cross1) by using combined finite and slabelement methods and metallographic observation (Kazeminezhadand Taheri, 2006b). In addition, the deformation pattern was foundinhomogeneous, both on longitudinal and transverse cross sec-tions. Vallellano et al. (2008) showed that this inhomogeneity ofdeformation has strong impacts on contact and residual stresses.

The above studies principally concentrated on geometry (e.g.spread) or loading (e.g. rolling force) predictions, there are fewstudies in the literature dealing with damage and fracture pre-diction in cold rolling of long products. Recently, (Massé et al.,2012) performed Scanning Electron Microscope (SEM) observa-tions of damage state in a wire flat rolling process. These authorsreported a higher void density in the wire core and in the “black-

1 The concentration of the strain rate on the diagonals of the section, in the shapeof a cross, when the section height and width are of the same order.

T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404 387

twim

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Fig. 1. Initial microstructure of steel wire at patenting state.

he above-mentioned danger zones. These critical areas coincideith “sheared” zones. The use of more advanced damage models

n this context is crucial to obtain more precise results in terms ofaximum damage location prediction.The present study aims at providing a critical assessment

f different phenomenological and micromechanical models andharacterizing their performance for complex loading configura-ions: multi-stage ultimate cold wire drawing (Section 2.3.1) andire-flat rolling (Section 2.3.2) processes. The objective is to sup-ly a critical comparison between six damage models of the threebove-mentioned approaches on these two forming processes: Baind Wierzbicki (Bai and Wierzbicki, 2010 – uncoupled phenomen-logical approach); Lemaitre (Lemaitre, 1986), Xue (Xue, 2007)nd Lode dependent enhanced Lemaitre Cao et al. (2014b) (cou-led phenomenological approach); GTN and modified GTN by Xue2008) (micromechanical approach). From this comparison, theoal is to discuss the applicability of a selection of models to diverserocesses with completely different states of stress. The paper isrganized as follows. Section 2 describes experimental techniquessed for damage models identification and experiments on formingrocesses. This section also presents experimental results of dam-ge occurring during the ultimate wire drawing and wire flat rollingrocesses. Section 3 presents the numerical techniques and damageodels developed and used. Section 4 provides numerical results

f damage on the two applications: multi-stage wire drawing andolling, obtained with the six damage models. From these results,ection 5 discusses the performance of the three approaches inhe above-mentioned applications and also suggests a researchrend that could be followed for other applications. Appendix Aeviews all the damage models implemented and used in thisork. The identification of these models is discussed in Appendix. Appendices C and D present additional results on simulations ofhe processes.

. Experimental techniques

.1. Material

The material used in the present study is a high carbon steelrade, which presents a fine pearlite structure after a patenting pro-ess. All the specimens used in the mechanical tests are extracted

rom the longitudinal direction of steel rods of maximum diameterf 17 mm. The mechanical properties of this steel grade can be con-idered isotropic in the patented state. The structure is pearlitic:

Fig. 2. Representations of mechanical tests carried out in the space of initial stresstriaxiality and Lode parameter.

ferrite with lamellae of cementite (see Fig. 1, which shows cemen-tite lamellae in white and ferrite in dark).

The material microstructure at this state is nearly homogeneous,with equiaxial grains and random crystallographic texture. It isinteresting to note that such a fine pearlite is an extremely ductilematrix. As a consequence, damage is observed to be of the usualtype, formation and growth of voids around inclusions. Therefore,the usual classes of ductile fracture models will be investigated.

2.2. Mechanical tests

The objectives of the experimental campaign are to identify theplasticity and damage parameters with a series of mechanical teststhat cover a large range of stress triaxiality and Lode angle param-eter. The stress triaxiality (�) is the ratio of the mean stress (�m)to the von Mises equivalent stress (�); while the Lode angle (Lode,1925) is defined as

�L = tan−1(

1√3

2�2 − �1 − �3

�1 − �3

)(1)

where �1 ≥ �2 ≥ �3 are the three principal stresses and−�/6 ≤ �L ≤ �/6. This parameter can be normalized to obtainthe so-called Lode parameter � (−1 ≤ � ≤ 1), which is defined by

� = 1 − 6�

�= − 6

��L with � = �L + �/6 (2)

Fig. 2 represents all the tests used in the present study in the spaceof initial stress triaxiality and initial Lode parameter. Tests veloci-ties were selected to have similar strain-rate for these tests (around0.7–1 s−1). Nine type of tests were performed: (1) compression testson cylinders (11 mm of height and 8 mm of diameter); (2) torsiontest (specimens diameter is 6 mm, the gauge length is 30 mm); (3)tensile tests on round bars (RB) (specimens diameter is 6 mm);(4–6) tensile tests on notched round bars (NRB) with differentnotch radii: 4 mm (NRB-R4), 6 mm (NRB-R6) and 9 mm (NRB-R9)(specimens minimum diameter is 6 mm); (7–9) tensile tests on flat-grooved (FG) specimens, with different groove radii: 2 mm (FG-R2),5 mm (FG-R5) and 7 mm (FG-R7) (specimens minimum thicknessis 1.5 mm).

In addition to the mechanical tests, in situ micro-tomography

tensile tests were also performed, which were used to identifymicromechanical damage models used in the present study (seeCao et al., 2014c for more details).

3 rocessing Technology 216 (2015) 385–404

2

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Table 1Recapitulation of damage models implemented and used in the present study. Thebullet symbol (•) indicates the full dependency. For the modified GTN model, thecoupling is accounted though the void volume fraction f, not through the damageparameter D.

Models Pressure Lodeangle

Couplingpossibility

B&W (Bai and Wierzbicki, 2008) • •Lemaitre (Lemaitre, 1986) • •Xue (Xue, 2007) • • •Enhanced Lemaitre (Cao et al., 2014b) • • •GTN (Tvergaard and Needleman, 1984) • •

kept constant for all the simulations.

88 T.-S. Cao et al. / Journal of Materials P

.3. Experimental tests on forming processes

In addition to the mechanical tests for material characterizationnd damage models parameters identification, forming processesests were also performed to validate the prediction ability of dam-ge models. An “ultimate” wire drawing test was conducted, forhich an initial wire of 21 mm diameter was drawn until failure

final diameter of 4.6 mm). This process consisted in 14 passes ofrawing, whose section reduction ratio per pass was about 16–22%.he drawing speed increased from the first to the last passes toeflect the real industrial condition (maintain the material flux)lthough the present ultimate drawing test was performed with

single pass drawing bench. Regarding the wire-flat rolling pro-ess, the experimental results of Massé et al. (2012) are considered,n which the authors performed detailed observations of damagetate at the end of the rolling process, using Scanning Electronicroscope (SEM).

.3.1. Ultimate wire drawingAs mentioned above, in the present study, fracture occurred

fter 14 passes of drawing, in the wire center. Fig. 3a shows theamage state observed after four passes of drawing, near the wireenter (total average deformation � = ln(S4/S0) ≈ 0.81; where S0nd S4 are the initial cross section and the cross section after fourasses). As can be observed, deformable inclusions are oriented fol-

owing the drawing direction. At this stage, inclusion cracking islso observed but negligible.

In the longitudinal view of microstructure after four passesFig. 3b), pearlite colonies are aligned in the drawing direction. Thenter-lamellae distance has decreased strongly, compared to thosefter patenting (Fig. 1). The damage state at the end of drawing (14asses) was also observed but is not shown here. Interested readersan refer to (Massé, 2010) for detailed analyses on microstruc-ure evolution during ultimate drawing of high carbon pearliticteels.

.3.2. Wire flat rollingFig. 4a represents the schematic of the studied wire flat rolling

rocess.As the wire passes through the rolls, all material points across

he width experience some tendency to expand laterally (trans-erse direction); this is called “spread”. The edges are thus strainedn tension, which may lead to edge cracking. The edge centerends to expand laterally more than the upper and lower sur-aces, which produces barreled edges similar to those observed inompression of a cylinder. Therefore, theoretically, in this process,he barreled areas are critical zones where cracks tend to initi-te first. Massé et al. (2012) carried out an experimental studyf damage in rolling process. Damage was identified at a micro-copic scale through SEM observations, which showed decohesionround non deformable inclusions (matrix-inclusion debonding)nd fragmentation of deformable inclusions. The FEM analysis ofhe strain map was also presented, which showed a localizationn form of a cross (the “blacksmith cross”). This strain localization

ay explain the evolution of microstructure during rolling (seeassé et al., 2012 for more details). These authors then carried

ut SEM observations of the transverse as well as the longitudi-al cross sections to obtain the 3D view of damage localizationt the end of rolling. The experimental sketch of damage, super-osed on a strain map obtained with FEM analysis, is presented inig. 4b.

As stated above, the rolling process involves a global vertical

ompression with a significant transverse flow, the expansion ofecohesion thus follows the transverse direction. Moreover, the

nclusions located in the branches of the blacksmith cross seemo have a preferential orientation due to rotation associated with

Modified GTN (Xue, 2008) • • •

shear strain in this area. The authors mentioned a higher voiddensity as well as a higher flattening of cavities in the wire corethan in the edge. Therefore, from this experimental observation,the rolling process principally affects the voids in the wire coreand on the blacksmith cross. Voids located in the barreled zonewere also expanded laterally due to rolling, but the expansion issmaller.

3. Numerical models and techniques

3.1. Finite element model

Implicit finite element (FE) simulations of all experiments areperformed with Forge2009®, which is based on an implicit mixedFE formulation of velocity and pressure. The updated Lagrangianformulation is adopted, which allows using the small strainapproach. The local integration of constitutive equations is solvedby backward Euler method. Since the mesh is distorted at largedeformation, an automatic adaptive remesher allows Forge2009®

dealing with large strain simulations (e.g. forming processes simu-lations). The present simulations are carried out with the 3D solver,in which the so-called MINI element (P1+/P1) is used. This linearisoparametric tetrahedron element has a velocity node added atits center ensuring the stability condition – the Brezzi/Babuskacondition of existence and uniqueness of solution. Exploiting thesymmetry of the geometries and loading conditions, only onefourth of the wire is modeled in the simulations of the formingprocesses.

In this work, six damage models are implemented andemployed: the uncoupled Bai & Wierzbicki (B&W) damage model(or fracture criterion), the Lemaitre and the Xue coupled damagemodels, the Lode dependent enhanced Lemaitre model (LEL); theGTN and modified GTN “micromechanical” models2 (see Table 1for the recapitulation). The detailed formulations of these mod-els are summarized in Appendix A. Their identification using theabove-mentioned mechanical tests is discussed in Appendix B. Itshould be noted that, for coupled damage models, the identifica-tion was based on the softening effect of damage. This softening,however, depends on the mesh size as stated in several publicationsin the literature (e.g. Jirásek, 1998): a finer mesh leads to a fasterdamage accumulation. Several methods have been proposed in theliterature in order to overcome this limitation (e.g. a non-local for-mulation as used in Peerlings et al., 1996). In the present study, noparticular technique has been employed; the mesh size was thus

2 The GTN model is a micromechanics-based model but its extension proposedby Xue (2008) makes this model become phenomenological.

T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404 389

Fig. 3. SEM images of microstructure in the drawing direction (horizontal direction) after four passes wire drawing: (a) damage state and (b) microstructure.

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carbon from cementite into the ferrite matrix) at large deformation(above 2–3, see also Zelin, 2002). The elastic constants are E = 210(GPa) (Young’s modulus), � = 0.3 (Poisson’s ratio).

ig. 4. (a) Schematic of wire flat rolling and (b) sketch of the damage state at the eictures show flattened deformable inclusions in the wire core and in the blacksmeferences to color in this figure legend, the reader is referred to the web version of

.2. Recapitulation of developed damage models

.2.1. Models implementationAll the models implemented and used in the present study are

ecapitulated in Table 1. In this table, the bullet symbol indicateshe full dependency.

These models have been implemented in Forge® (see Cao,013), which required adaptation of algorithms to its mixedelocity–pressure formulation and to its finite element (P1+/P1).he uncoupled B&W model was implemented by a user subrou-ine. For the coupled phenomenological models (Lemaitre, LEL andue), they were implemented through a user subroutine, with aweak coupling” between damage and material behavior: the dam-ge variable at time step n − 1 is used to solve the mechanicalquations at time step n. Regarding the micromechanical models,he implementation procedure was detailed in Cao et al. (2013c).

.2.2. The choice of the cutoff value of stress-triaxialityFor the Xue model, the damage-deactivation threshold is

efined by the limiting pressure (pL), which can somehow be linkedo a cutoff value of stress triaxiality (e.g. for J2 plasticity and nooftening effect of damage, −pL/�0 defines the corresponding cut-ff triaxiality, below which there is no damage). There is no needo introduce a cutoff value of stress triaxiality in this model. As dis-ussed in Appendix A.2.1, the cutoff value of −1/3 has been adopted

or several damage models (e.g. by Bouchard et al., 2011a; Massét al., 2012 for the Lemaitre model; by Brünig et al., 2008 for theiructile damage criterion). Regarding the LEL model, this value ofutoff has also been used as a first approximation. The B&W model

rolling superimposed to strain map obtained by FEM analysis (color). The zoomedoss (SEM pictures – adapted from (Massé et al., 2012)). (For interpretation of therticle.)

(Appendix A.1) does not have a natural cutoff value, the value of−1/3 was also adopted for this model to couple the cutoff stress tri-axiality and the strain to fracture formula. It should be noted that forproportional loadings, this choice does not lead to an infinite valueof the “fracture strain” at the cutoff stress triaxiality.3 Therefore,for proportional loadings, further improvements of the uncoupledmodel used would be carried out to obtain more consistent for-mula of strain to fracture (as done by Lou et al., 2014). Regardingmicromechanically based models used in the present study, no cut-off value has been introduced since the void growth is deactivatedunder compression (fgrowth = 0 if � < 0). The impact of the choice ofcutoff value on damage localization is discussed later in Section 5.1.

3.3. Simulations data

For numerical simulations, the stress–strain curve was iden-tified, which was validated up to large strain. This curve can beapproximated by a Ludwik law: �0 = 1043(1 + 0.282�1.335) (MPa).The convex shape (exponent >1) of the strain-hardening functionis due to the high ductility and peculiar hardening mechanismsof pearlitic steels. These include cementite lamellae orientation atmoderate strain (up to 1) and “cementite dissolution” (release of

3 The fracture strain is expected to increase gradually as the stress triaxialitydecreases and tend to infinite at the cutoff stress triaxiality.

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Therefore, the contribution to total damage (for which the criticalvalue is Dc = 1) is small with respect to the shear term. It raises aquestion about the validity of the phenomenological nature of the

4 From Fig. 7e, for the modified GTN model, the contribution of the original GTNdamage can be obtained from q1f, which is around 1.5 × 4.92 × 10−6 = 7.38 × 10−6

on the surface and around 9.9 × 10−6 in the wire core. To obtain the lower bound oftotal damage of the modified GTN model, supposing that there is no void growth onthe wire surface during the second pass (i.e. f = 4.92 × 10−6); q∗

3 = 0.85 and q4 = 1/3(from Cao et al., 2014c); the equivalent plastic strain is around 0.5 at the wire sur-face at the end of the second pass; g� is around 0.8–0.9 during the deformationprocess; the shear contribution at the wire surface can be analytically calculated:Dshear ≈ 1.6 × 10−3, which is significantly higher than the original GTN contribution

90 T.-S. Cao et al. / Journal of Materials P

It should be noted that temperature and strain rate influencesere not accounted for in this study, neither on work-hardening

aw, nor on damage models. The temperature remains low for thesewo processes because of the following reasons: (1) wire drawingas performed on a single-pass bench, there was insignificant heat

ccumulation and (2) for rolling, the flattening wire was sprayedith cold lubricant, the temperature was lower than 200 ◦C (Massé

t al., 2012). Regarding the strain rate effect, mechanical tests forodels parameters identification were carried out at a strain rate

round 1s−1.Friction laws for processes simulations were also identified,

hich were the same as used in Massé et al. (2012) and Bobadillat al. (2007) for these two processes.

.4. Mesh size sensitivity

To avoid partly the mesh-dependency due to the use of a localoupled damage model, mesh size in the gauge section of theacroscopic mechanical tests was fixed (0.25 mm). This mesh sizeas selected based on numerical tensile tests on notched round bar

as done in Cao et al., 2013b). For this numerical test, different meshizes were tested and the mesh size of 0.25 mm was chosen for aompromise between the accuracy and the CPU time. The meshizes used for the processes simulations were chosen to have theame “mesh gradient”.

. Numerical results and analyses

.1. Wire drawing

.1.1. Mechanical analysesIn order to illustrate the mechanical state of material during the

rocess, the evolutions of strain and stress states of drawn wireuring the fourth pass are studied. For this pass, the section reduc-ion ratio is 20.69% and the semi-angle is 0.105 rad. The drawingpeed is around 600 mm/s.

Fig. 5a and b represents the equivalent plastic strain rate andtrain fields at steady state for the fourth pass. The strain-rate isigher at inlet and outlet contact points and also in the wire core,nd is equal to zero outside the working zone (represented by aectangular box in Fig. 5a). Note that damage accumulation processccurs only in this working zone (with a non-zero strain-rate).

As can be seen in Fig. 5a, the strain-rate map is strongly het-rogeneous both in radial and axial directions. Shear bands canlso be observed. The equivalent strain is larger on the wire sur-ace than in the wire core due to a stronger shear effect on theire surface (see Fig. 5b). The difference of these values is quasi-

ndependent of the section reduction and can be approximated as:� = tan(˛)/2, where is the inlet semi-angle. The stress triaxial-

ty and Lode parameter maps are shown in Fig. 6a and b. Fig. 6aepresents only values greater than −1/3 (the cutoff value of stressriaxiality proposed by Bao and Wierzbicki, 2005), the rest appearsn gray.

The representation of the stress triaxiality helps capturingotential locations for damage growth. In the working zone, thetress triaxiality is greater in the wire core and has positive valuest these positions. Since the strain-rate is positive in the wire core,amage models based only on triaxiality (e.g. Lemaitre, GTN) tendo predict higher damage at this position (see Section 4.1.2).

The Lode parameter (see Eq. (2)) is equal to 1 in the wire core inhe working zone, the wire core is under a stress state equivalent

o uniaxial tension (plus a certain superimposed pressure). Beforend after entering the working zone, the Lode parameter is nearlyqual to −1 in the wire core, which suggests that the wire core isnder a stress state equivalent to uniaxial compression. Therefore,

ing Technology 216 (2015) 385–404

a tendency of “compression-tension-compression” of material inthe wire core during this process can be observed.

4.1.2. Results of damageFirst, qualitative results of damage (i.e. damage distribution) are

presented. For this process, six damage models were used: B&W,Lemaitre, Xue, LEL, GTN and GTN modified by Xue (see Table 1).

As expected, most models predict higher damage in the wirecore than in the wire skin. Fig. 7a–d shows damage distributionson the longitudinal and transverse cross sections, at the secondpass, with four phenomenological models: Bai & Wierzbicki, Xue,Lemaitre and LEL, which confirm high localization of damage in thewire core. Moreover, between the Lemaitre and Xue coupled mod-els, a stronger localization in the wire center can be observed withthe Lemaitre model, while for the Xue model, damage changes grad-ually from the wire core to the wire surface. In addition, regardingthe LEL model, which accounts for the influence of the Lodeparameter, damage accumulation is also predicted on wire surface,although less intense than in the wire core. It means that, for themodels based on both the Lode parameter and the stress triaxiality(i.e. B&W, Xue and LEL), the influence of shear, which is maximumon the wire surface, is accounted for with a stronger weight thanfor the stress triaxality-based model (e.g. Lemaitre). Moreover, inthe working zone, since the stress triaxiality in the wire surface isnegative (see Fig. 6a), damage accumulation predicted by the stresstriaxiality-based Lemaitre model is not noticeable at this location.

Regarding micromechanically based models, the GTN model(Fig. 7e) predicts damage higher in the wire core than for othermodels (although at this stage, the void volume fraction is stillvery small), while the GTN modified by Xue (Fig. 7f) predicts dam-age higher on the wire surface. The reason is, for the modifiedGTN model, the damage variable is the sum of two terms: onecorresponds to the GTN original contribution (q1 f ), the other corre-sponds to the shear contribution (Dshear = q∗

3f q4 �p�p[1 − (6|�L|/�)]– see Eq. (A.29)). In the working zone, the Lode parameter is nearlyequal to 1 at the wire center (i.e. �L = − �/6 – see Fig. 6b), the shearcontribution is nearly zero at this position, damage accumulationat this position is principally due to the original contribution ofthe GTN model. At the wire surface, since the Lode parameter isaround 0–0.2 (i.e. 6|�L|/� ≈ 0 −0.2), damage accumulation at thisposition has a significant contribution from Dshear term. From ananalytical calculation, it is straightforward to show that at the wiresurface, the shear contribution prevails over the GTN original con-tribution (q1fsurface); and the value of total damage induced at thewire surface is higher than that in the wire core (q1fcore).4 Oneof the reasons for this result is the void volume fraction identi-fied for the GTN model using tomography observations (Cao et al.,2014c) is significantly smaller than the one used by Xue (2008)when deriving analytical formulation for the modified GTN model.

of 7.38 × 10−6. In addition, as explained above, in the wire core, the shear contri-bution is nearly zero, the total damage in the wire core obtained with the modifiedGTN model is approximately the original GTN contribution (9.9 ×10−6). From thesecalculations, the total damage in the wire core is significantly smaller than that onthe wire surface, which is consistent with the damage map in Fig. 7f.

T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404 391

Fig. 5. Equivalent plastic strain-rate and strain distributions at steady state during the fourth drawing pass: (a) strain rate and (b) strain.

rth pa

mrtb

Fv

Fig. 6. Stress state at steady state during the fou

odification proposed by Xue to account for micromechanical voidotation mechanism, and also the accuracy of the present parame-ers identification for the shear contribution. Cao et al. (2014c) usedoth macroscopic tensile tests and in situ X-ray micro-tomography

ig. 7. Damage at steady state of second drawing pass for six damage models: (a) Bai & Walues are represented on the longitudinal and transverse cross sections.

ss: (a) stress triaxiality and (b) Lode parameter.

tensile tests to obtain the parameters of the original GTN model.These parameters are thus reliable since their identification wasbased on microscopic observations. However, for Xue’s extensionterm, they used only the macroscopic torsion test to identify the

ierzbicki, (b) Xue, (c) Lemaitre, (d) LEL, (e) GTN and (f) modified GTN by Xue. The

3 rocess

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92 T.-S. Cao et al. / Journal of Materials P

arameters. The modified GTN model by Xue with the identifiedarameters fails to predict higher damage in the wire core becausef the above reasons.

Since damage was higher in the wire core (except the modi-ed GTN model), numerical sensors were placed at this positiono follow damage evolution as well as to determine the instant ofracture (i.e. when damage variables reach their critical values –uantitative results). Fig. 8 represents the increase of damage vari-bles with the accumulated plastic strain during multi-pass wirerawing. Note that since remeshing took place after each drawingass (to avoid mesh distortion), the values may slightly change dueo the error of field transfer from old mesh to new mesh. For thiseason, in Fig. 8, a small jump after each pass may be observed.

As can be seen in Fig. 8a, for the B&W uncoupled model (Fig. 8a),amage increases proportionally with the cumulative plastic strain.racture is predicted at the fourth pass with this model when theamage variable reaches its critical value (D(B&W)

c = 1). The Xue,he Lemaitre and the LEL models predict fracture at fifth pass (seeig. 8b, c, and d, respectively). With these three models, damage isnly activated from a certain level of plastic strain due to the pres-nce of plastic strain threshold for damage initiation (see Appendix). Between the Lemaitre and LEL models, similar tendency cane observed although damage increases a little faster with the LELodel. Another difference is that the LEL model also predicts signif-

cant damage accumulation on the wire surface although it is lessntense than that of the wire core (Fig. 7d)).

In Fig. 8e, for the GTN model (see Cao et al., 2014c), from therst to the fourth drawing passes, porosity increases with the plas-ic strain but the value is still small (since the initial porosity wasmall, f0 = 4.92 × 10−6). The porosity then increases sharply with thequivalent plastic strain (pass 5 and pass 6) because nucleation isaximum at this strain level (the “nucleation strain” �N is obtained

hanks to Eq. (A.24) and the stress triaxiality level at the wire cen-er is around 0.15 in the working zone – Fig. 6a). After this strongucleation period, the porosity increases slowly with the equiva-

ent plastic strain and reaches its critical value at the 13th pass. Fig. 9epresents the contour plot of the void volume fraction for differentrawing passes, showing the strong localization of damage in theire center for the GTN model throughout the process. However,

imilar to the stress triaxiality-based Lemaitre model, this modelannot capture the damage accumulation on the wire surface. Its worth recalling that all the parameters of damage models weredentified from laboratory mechanical tests and they were appliedas is” to forming processes simulations (they are not re-identifiedrom these processes).

.1.3. Comparison with experimental results – discussionsThe above comparisons between six identified models lead to

he following conclusions concerning qualitative (localization) anduantitative results (instant of fracture):

Five identified damage models (B&W, Xue, Lemaitre, LEL andGTN) show a good agreement with experimental observation interms of damage localization (in the wire core). The modified GTNmodel also reveals damage in this zone but the most danger-ous zone is predicted on the wire surface because the identifiedparameters give a too strong weight for the “shear” contribution,thus favor damage accumulation on the wire surface.The four studied phenomenological models (B&W, Xue, Lemaitreand LEL) cannot predict accurately the instant of fracture initi-ation. These models predict fracture at the fourth or the fifth

passes, far from experimental results (the 14th pass). For the GTNmodel, this model predicted damage at the 13th pass, which iscomparable with the experimental result. However, this resultmust be taken with caution since in the present authors opinion,

ing Technology 216 (2015) 385–404

the use of a Gaussian distribution to describe the nucleation forthis material is not the most suitable (see the above discussionsin Section 4.1.2).

• For the drawing process, since the wire center is often the mostdangerous zone, where the stress state during the deformationprocess (i.e. inside the working zone) is nearly a tensile state(superimposed to a certain hydrostatic pressure), the use ofstress-triaxiality based damage models could be suitable to pre-dict damage localization at the wire center. For the same reason,the results in terms of the prediction of dangerous zone obtainedwith the models, with or without Lode-dependency, are simi-lar. The exception is the modified GTN model, which is mostlikely due to the identification process and the phenomenologicalassumption of the model. However, stress-triaxiality based dam-age models cannot capture the damage accumulation on the wiresurface, where strong shear influence can be observed. Whenthe wire surface has a large amount of defects, these defects candevelop to cause fracture initiation at the wire surface (see e.g.Cao et al., 2014d). The use of a model accounting for shear effect(e.g. B&W, Xue, LEL or modified GTN) is more suitable to capturethe development of damage on the wire surface. In the next sec-tion, another application with stronger shear effects is presentedto illustrate the important role of the Lode-dependency.

4.2. Wire flat rolling

4.2.1. Mechanical analysesIn the real process, after four passes of drawing, the wire is sub-

jected to three rolling passes. In the present study, the first rollingpass is considered for qualitative numerical results of damage. Forthis reason, the mechanical history of wire drawing is not taken intoaccount in order to focus on the effect of a different stress states,rather than try to compare quantitatively with experiments.

The schematic of the studied wire flat rolling process was pre-sented in Fig. 4a. The studied process is one stage cold rolling ofwire, with the height reduction (H) of 3 mm and the roll rota-tion speed is 20 rpm (≈2.09 rad/s). The four drawing passes beforerolling are the same first four drawing passes in the ultimate draw-ing (Section 4.1).

Fig. 10a shows the strain-rate distribution on the longitudinalcross section when wire passes through the rotating rolls. Strainrate is higher at the entrance contact point and in the flat wirecenter. Fig. 10b shows its distribution on the cross section (A–A)perpendicular to the rolling section, where strain rate is higher inthe blacksmith cross.

Fig. 10c and d represents the contour plots of the stress triaxial-ity and the Lode parameter in the cross section (A–A), the positionwhere wire was in contact with rolls. The central zone, except thewire core, has a stress triaxiality smaller than −1/3; while max-imum, positive triaxiality is found in the barreling zone (lateralfree surface). For this reason, as it will be shown in the following(Section 4.2.2), damage models based only on the stress triaxialitytend to predict maximum damage in this zone. Regarding the Lodeparameter, a cross-shaped band (“shear” band) where � ≈ [−0.2, 0]can be observed in Fig. 10d.

4.2.2. Results of damageFig. 11 represents damage distribution for different models on

a cross section perpendicular to the rolling direction at the end ofrolling.

First, the Xue model predicts higher damage in the wire core andon the blacksmith cross (Fig. 11a), where strain localization takes

place (see Fig. 10b). With this model, damage and strain localize atthe same position. This cross was also the position where the Lodeparameter was zero and the stress triaxiality was negative duringthe deformation process (i.e. when wire passed through rolls – see

T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404 393

F for dif

Fc

Lbig

antae

ig. 8. Evolution of damage variables of a material point located at the wire center

ig. 10c and d). A secondary maximum of damage in the lateral zonean also be observed.

Stress triaxiality- (or pressure-) based damage models (i.e.emaitre and GTN), as expected, predict maximum damage in thearreling zone (Fig. 11b and d), where the stress triaxiality is pos-

tive when deformation takes place (Fig. 10c). Note that porosityrowth is extremely small with the GTN model.

For the B&W model, damage is higher in the barreling zone andlso in the center (Fig. 11c). Secondary maximum is also observed

ear the barreling zone. Although the B&W model also accounts forhe third stress invariant, it cannot predict the significant damageccumulation in the blacksmith cross. It could be due to the pres-nce of the cutoff value of the stress triaxiality (see Fig. 10c for the

ferent models: (a) Bai & Wierzbicki, (b) Xue, (c) Lemaitre, (d) LEL and (e) GTN.

map of stress triaxiality, where the stress triaxiality is smaller than−1/3 in the zone corresponds to the blacksmith cross). Moreover,in the construction of such a model, there is no “explicit” param-eter that could control whether the “tensile” or “shear” states aredominant factors (such as � in the Xue model, ˛1 in the LEL model,or q∗

3 in the modified GTN by Xue).As presented in Section 2.3.2, the experimental result of Massé

showed a higher density of voids in the wire core and in theblacksmith cross as well as several voids at lateral surface (see

Fig. 4b). Among the above-studied models, the Xue model givesrelatively good results concerning damage localization duringrolling. With this model, damage is localized on the blacksmithcross, and there is less significant but nevertheless noticeable

394 T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404

g pas

dhsd(dar

4s

paaad

F(

Fig. 9. Contour plot of the void volume fraction at different drawin

amage in the barreling zone (Fig. 11a), where in reality, voidsave expanded slightly. The models based on the pressure or thetress triaxiality (Lemaitre, GTN) cannot provide the localization ofamage in the shear zone (Fig. 11b and d). For the B&W modelFig. 11c), it can be considered as an intermediate case, whereamage is significantly higher in the core and nearly equivalentt other positions except the position under contact surface witholls.

.2.3. Qualitative results of damage with enhanced models forhear-dominated loadings

As the coupled Xue model accounts for the influence of the Lodearameter and it gives relatively correct results in terms of damage

ccumulation, the modifications of GTN and Lemaitre models thatccount for the Lode-dependency (i.e. the modified GTN by Xuend the LEL model) are investigated. Fig. 12a and b represents theamage localization obtained with these two models.

ig. 10. Mechanical fields in rolling process: (a) Strain-rate in the longitudinal cross sectiod) Lode parameter on the (A–A) cross section.

ses, showing damage localization at the wire center (GTN model).

As can be observed, damage is localized in the blacksmithcross (Fig. 12a) and in the wire center (Fig. 12b). For the mod-ified GTN model, since the shear contribution prevails over theoriginal GTN contribution as explained in Section 4.1.2, damageaccumulation in shear-dominated zone (i.e. blacksmith cross) issignificantly higher than that of tensile zone (i.e. barreling zone).For this reason, although starting from a same ingredient to repre-sent Lode-dependency as the Xue model, the modified GTN modeldoes not predict a secondary maximum of damage in the barrelingzone as for the Xue model (Fig. 11a). This localization is qualitativelyin agreement with experimental results concerning the influenceof rolling process on damage.

Regarding the LEL model, Fig. 12b shows that since at the wire

center the stress triaxiality is higher than −1/3 and the LEL modelaccounts for the Lode dependency, damage accumulation is higherat this position. This model also predicts a weak secondary max-imum at the barreling zone (which is the most dangerous zone

n; (b) strain rate in the (A–A) section; (c) stress triaxiality in the (A–A) section; and

T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404 395

F rent mp

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5

5

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ig. 11. Damage on a cross section perpendicular to the rolling direction with differevious drawing passes is not accounted for.

redicted by the Lemaitre model). However, this damage localiza-ion is most likely due to the presence of the cutoff value of stressriaxiality (−1/3) since in the central zone (including the blacksmithross), except the wire center, the stress triaxiality is smaller than1/3.

From these results, it can be observed that the modified GTNodel proposed by Xue can predict the damage localization in the

lacksmith cross. The LEL model (and also the B&W model), with thessumed cutoff value of the stress-triaxiality (−1/3) cannot provide

good damage localization. However, this is only a qualitative con-lusion since the parameters of shear-extension terms proposed byue for the GTN model were identified only from the torsion test. Inddition, the results of Lode-dependent phenomenological (B&W,EL) seems to be strongly impacted by the presence of the cutoffalue of stress triaxiality. This point is discussed in the next section.

. Discussions

.1. Role of the cutoff value of triaxiality

As discussed above, based on series of tests on various types ofteel under pressure by Bridgman (1952) and Bao and Wierzbicki

ig. 12. Damage localization in wire flat rolling: (a) the modified GTN model and (b) the

or the LEL model.

odels: (a) Xue, (b) Lemaitre, (c) B&W and (d) GTN. The mechanical history of four

(2005) suggested a cutoff value of stress triaxiality of −1/3, belowwhich fracture does not occur. In the present study, this value hasbeen used for the Lemaitre model from the work of Bouchard et al.(2011a). However, from several experiments results (e.g. Lou et al.,2014) this value has been shown to be far from universal. Lou et al.(2014) proposed the following relation between the cutoff value ofstress triaxiality and the Lode parameter:

�cutoff = −C − 3 − L

3√

L2 + 3(3)

where L = (2�2 − �1 − �3)/(�1 − �3) is another definition of the Lodeparameter, L ≈ −� for a general stress state and L = −� for severalconventional stress state (e.g. uniaxial tension, uniaxial compres-sion, torsion, etc.); C is a parameter. Lou et al. (2014) tested C = 1/3and C = 0 for their applications. With these chosen values of C, thecutoff value of stress triaxiality for different values of Lode param-eter are tabulated as in Table 2.

Although it was based on phenomenological grounds, this study

revealed that the choice of the cutoff value of stress triaxiality isstill an open question. For forming processes, this choice is of greatimportant since the stress triaxiality in numerous forming processis often negative or just slightly positive. In Xue (2007), the author

LEL model. Note that the cutoff value of stress triaxiality of −1/3 has been adopted

396 T.-S. Cao et al. / Journal of Materials Process

Table 2The cutoff value of the stress triaxiality with different values of Lode parameter.

Lode parameter L = −� = 1 L = −� = 0 L = −� = −1

ita(i2vacftetrocc

fl−rb

ofsi

ttete0onttatsiFm

oie

vo(aos

A

�cutoff , C = 1/3 −2/3 −0.91 −1�cutoff , C = 0 −1/3 −0.577 −2/3

nvestigated the fracture locus (the strain to fracture as a function ofhe stress triaxiality) for plane stress state for the same Aluminumlloy 2024-T351 as in Bao and Wierzbicki (2004) and Lou et al.2014). The author reported a cutoff value of approximately −2/3nstead of −1/3 used by Bao and Wierzbicki (2004) (see Fig. 5 of Xue,007). In addition, for the Xue model (see Section A.3), the cutoffalue is not presented through the stress triaxiality, but through

limiting pressure pL. From this limiting pressure, an equivalentutoff stress triaxiality can be deduced, which is defined as −pL/�0or von Mises plasticity (without damage-induced softening). Withhe value identified pL = 1735 MPa (Cao et al., 2013a), and the hard-ning law used (see Section 3.3), the cutoff value varies from −0.55o −0.33 when the equivalent plastic strain varies from 0 to 2 (ineality, due to the softening effect of damage, the calculated cut-ff value is smaller than these values). From these analyses, theonstant value for the cutoff triaxiality of −1/3 seems to be tooonservative.

In order to study the influence of the cutoff value, the wire-at rolling is taken as an example,5 by using the cutoff value of2/3 instead of −1/3 for the Lemaitre, LEL and B&W models. The

esults in terms of damage localization are shown in Fig. 13a–c (toe compared with Figs. 11b and c and 12b, respectively).

Regarding the Lemaitre model (Fig. 13a), no clear difference isbserved, damage is still higher in the barreling zone (see Fig. 11bor the result with �cutoff = −1/3). However, for the LEL model,trongest damage accumulation takes place in the wire core andn the blacksmith cross. The reason is that, for the modified model,

he terms 1/(˛1 + ˛2�2) plays an important role: for a given stress

riaxiality ratio, damage rate is maximum when the Lode param-ter is equal to zero. Fig. 10d shows the Lode parameter map inhe working zone. As can be observed, the Lode parameter near thends of each branch of the blacksmith cross is in the range [−0.2;], while this value is in the range [−0.4; −0.2] at other positionsf the cross. Therefore, damage accumulation inside cross is higherear the ends of each branch, as can be observed in Fig. 13b. Inhe wire center, damage is slightly higher than the neighbor posi-ion, it is most likely due to a slightly higher strain rate (Fig. 10a)nd stress triaxiality (Fig. 10c) at the wire center. These observa-ions explain such a localization of damage observed in Fig. 13b. Ithould be noted that, by adjusting the parameters ˛1, damage local-zation could be totally consistent with experimental observations.or the results presented in Fig. 13b, the value of ˛1 identified fromechanical tests was kept.For the uncoupled B&W model (Fig. 13c), with the cutoff value

f −2/3, this model predicts higher damage in the wire core andn the blacksmith cross, which is qualitatively in agreement withxperimental observations.

Regarding the wire drawing process, the change of the cutoffalue of stress triaxiality from −1/3 to −2/3 has negligible impactn the prediction of dangerous zone and the instant of fracturesince the most critical zone is the wire core, where the stress tri-xiality is positive during the deformation process). For the results

f damage localization in this case (B&W, Lemaitre and LEL models),ee Appendix D.

5 This process is a very good example of complex stress state applications (seeppendix C for an analysis on the loading paths at different material points).

ing Technology 216 (2015) 385–404

5.2. Role of stress state characterization variables in damagemodels constructions

As shown in Sections 4.1 and 4.2, the choice of a suitable damagemodel for a forming process depends strongly on the loading stateof that process (i.e. the stress state to which a material point is sub-jected during the process). For the wire drawing process, since thestress state in the wire core (the most critical location in forming)is nearly a tensile state with a certain superimposed hydrostaticpressure, the damage models that account for only the stress tri-axiality could give correct results in terms of damage localizationprediction. For this reason, the identified GTN and Lemaitre modelscan predict higher damage in the wire core (as well as the modelsthat account for the Lode parameter: B&W, Xue and LEL). However,these two models cannot capture the damage accumulation on thewire surface, at which shear force is dominant.

Regarding the wire flat rolling process, since the loading pathsare complex for different material points at different positions (seeAppendix C), the use of damage models that account for the wholestress state in their formulation is necessary. For this process, onlythe models that account for both stress triaxiality and Lode param-eter (B&W, Xue, GTN modified by Xue and LEL), with a suitablecutoff value of stress triaxiality, can give relatively correct resultsin terms of damage localization prediction. These results suggestthat, in order to obtain correct results for complex loading applica-tions, damage models must account for different stress componentsin their formulation, represented by the stress triaxiality and theLode parameter.

5.3. Three approaches of ductile damage

In the present study, three approaches of ductile damage havebeen investigated. The uncoupled model (B&W in the presentstudy) has a major advantage of being easy to implement and use. Inaddition, since no coupling is taken into account (i.e. no softeningdue to damage), there is negligible influence of mesh size. How-ever, this model suffers from several limitations, mainly its purelyphenomenological nature (i.e. no physical parameters involved).The coupled damage models (Lemaitre, Xue, and LEL) also sufferthe same limitation, although these models account for the soften-ing due to damage. This softening effect is sometimes important,especially at very large strain encountered in multi-stage formingprocesses (i.e. in the last stages of damage, just before failure). Inaddition, among the three models, the Lemaitre model is derivedwithin the framework of continuum damage mechanics, whichensures the thermodynamic consistency (although this property isnot enough for a material model to be accurate and predictive). TheLEL model meanwhile is rather a heuristic extension of the Lemaitremodel. Its thermodynamic consistency in a strict sense is not alwayssatisfied (see Cao et al., 2014b). However, this shortcoming does notaffect the prediction ability of this model.

For the drawing processes, although all the models used givecorrect results in terms of maximum damage location, only themicromechanical GTN model identified from both mechanical testsand microscopic observations (see Cao et al., 2014c) can give a com-parable result in terms of the instant of fracture. This suggests thatfor complex microstructure materials (as the case of the studiedpearlitic high carbon steel), identification of damage models shouldbe based on both macroscopic and microscopic observations toobtain reliable parameters. In addition, the model itself must becapable of accounting for the real microstructure parameters asso-

ciated with the physical problems (e.g. voids for ductile damagephenomenon). It also underlines the important role of the identi-fication process: a well-constructed damage model can only givecorrect prediction if its parameters are accurately calibrated.

T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404 397

F axialit

irdvibtTvdDmGTetovsmtmd

5

ffidds

ig. 13. Damage localization in wire flat rolling with the cutoff value of the stress tri

Regarding micromechanical approach, although the GTN models a micromechanical model, it is still based on an ideal rep-esentation of voids: they are spherical and keep their shapesuring the deformation process. For the present applications,oids are elongated in the drawing direction in the wire draw-ng process and voids are flattened and rotated following the shearand in the rolling process. The above-mentioned assumption forhe GTN model is therefore not fulfilled in real-life applications.he use of a purely micromechanical model, which accounts foroid shape change and void rotation, is necessary. Several recentevelopments devoted to this problem have been carried out byanas and Aravas (2012) based on the nonlinear homogenizationethod, or by Cao et al. (2014a) based on the combination of theurson-like approach and the nonlinear homogenization method.hese authors considered general ellipsoidal voids embedded in anlastoplastic matrix subjected to 3D deformation, and investigatedhe evolution of the void volume fraction, voids aspect ratios andrientations. In addition, loading is not necessarily aligned withoids axes. Although it is out of scope of this paper, the use ofuch a model for ductile fracture prediction in forming processesight be a subject for future studies. However, in order to apply

o structural computations and especially industrial applications,odels parameters identification must remain tractable. It is briefly

iscussed in the following for a general framework.

.4. Models parameters identification

A well-constructed damage model can only give correct resultsor complex applications if its parameters are accurately identi-

ed. For complex stress state applications (as in forming processes),amage models should be calibrated from mechanical tests atifferent loading configurations, covering a large range of stresstates. It is the case of the present study, where nine different

y equal to −2/3: (a) the Lemaitre model; (b) the LEL model; and (c) the B&W model.

tests were performed (see Fig. 2 for the recapitulation). By usingthese tests, hardening and damage models parameters can beidentified and validated for various stress states. In addition, formicromechanics-based models, since they are based on micro-mechanical considerations, their calibration should be based onboth macroscopic tests and microscopic observations (as done byCao et al., 2014c for the GTN model and the modified GTN modelby Xue). However, using a large number of tests leads to laboriouscalibration tasks, which requires a robust method for optimiza-tion procedure. In several studies in the literature, e.g. Dunand andMohr (2010), parameters of uncoupled models were identified inpost-processing using the strains to fracture, by supposing the pro-portionality of all the tests used. This method is quite simple butit suffers a limitation that only few tests satisfy the proportion-ality assumption. In the present study, all models, both coupledand uncoupled, are identified using multi-objective optimizationby inverse analyses. Optimization was based on an evolution strat-egy algorithm (see Bouchard et al., 2011 for more information).In addition, the inverse analysis computations were performed inparallel, which reduced significantly the CPU time.

5.5. Toward modular models

As shown in Section 4.2, the original Lemaitre and GTN modelscannot predict damage localization in the blacksmith cross for therolling process. However, by accounting for the third stress invari-ant as in the modified GTN model by Xue and the LEL model, thesemodels become capable of predicting such a localization of dam-age. Such construction of damage model leads to the definition of

“modular” models. The idea is, regarding its physical origin, a givendamage model can be enhanced by adding suitable “ingredients”(e.g. a Lode-dependent term, or a stress triaxiality dependent strainthreshold for damage accumulation) to improve its prediction

3 rocess

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6

aalcs

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98 T.-S. Cao et al. / Journal of Materials P

bility. If all the ingredients are well-constructed, different modelsay converge to a same prediction in terms of damage localization

as in the case of the present rolling process). Several examplesan be taken to illustrate this idea: the B&W model is obtainedrom the well-known Rice and Tracey formulation for fracture atigh triaxiality, by introducing a quadratic dependency of strain to

racture to the Lode parameter; the modified GTN model for shearoading is obtained by adding a Lode dependent term to the evo-ution of the void volume fraction; or the LEL model is obtainedrom the Lemaitre model by adding a parabolic Lode dependenterm. It should be noted that, with this kind of approach, all mod-ls become phenomenological. To the best of the present authors’nowledge, no purely micromechanical model has been shown toe capable of capturing both damage mechanisms at high and lowtress triaxiality in real multi-stage shear-dominated manufactur-ng processes. Again, the development of such a model might bef interest for future studies. Anisotropy of material matrix (dueo material processing) and morphological anisotropy (initial non-pherical particles, or induced by void shape) have been showno have important effects on ductile fracture (as pointed out byenzerga et al., 2004). Furthermore, it is also important to incor-orate the rate influence on both plasticity and damage behaviors.hese problems could be subjects of future studies.

. Closure remarks

This paper presents the application of three ductile damagepproaches to predict fracture in cold forming processes. Dam-ge occurring in multi-stage wire drawing and shear-dominatedoading wire flat rolling processes is investigated numerically andompared with experimental results. The main conclusions can beummarized as

Six damage models, namely the B&W model (uncoupled phen-omenological approach), the Lemaitre, enhanced Lemaitre andXue models (coupled phenomenological approach) and the GTNand modified GTN models (“micromechanical” approach) areimplemented in a FE code based on a mixed velocity–pressureformulation. These models were identified from mechanical testsat different stress states.Application to the “ultimate wire drawing” test (14 passes) showsthat, except the modified GTN model, all the models used cangive correct results in terms of maximum damage location (insidethe wire core), which is consistent with experimental results.However, only the GTN model can give relatively correct resultin terms of the instant of fracture (at 13th pass, compared toexperimental result of 14th pass). Since the GTN model is amicromechanical model and its identification for the studied steelwas carried out by using both mechanical tests and microscopicobservations, the result suggests that for complex microstruc-ture materials (as the case of the studied pearlitic high carbonsteel), identification of damage models should be based on bothmacroscopic and microscopic observations to obtain reliableparameters. Such an identification procedure, however, is quiteheavy, both for experimental and numerical tasks. A robust opti-mization process should be used as in the present work.Qualitative application to one pass wire flat rolling processreveals that only the models that account for both stress triax-iality and Lode parameter (B&W, Xue, GTN modified by Xue andLEL) can give relatively correct results in terms of damage local-ization prediction (compared with the experimental results of

Massé et al., 2012). For Lode-dependent phenomenological mod-els that do not have a natural cutoff value of stress triaxiality(B&W, LEL), the choice of the cutoff value is shown to have strongimpact on damage localization. If the value of −1/3 is used (from

ing Technology 216 (2015) 385–404

the study of Bao and Wierzbicki, 2004), higher damage is pre-dicted in the wire core (both B&W, LEL) and at the barreling zone(B&W). If the value of −2/3 is used (as suggested in Xue (2007)),both models predict damage higher in the blacksmith cross and inthe wire center, which is qualitatively in agreement with exper-iments. This suggests that, in order to obtain correct results forcomplex loading applications, damage models must account fordifferent stress state characterization variables in their formula-tion, represented by the stress triaxiality and the Lode parameter.In addition, the cutoff value of triaxiality of −1/3 used in differentstudies in the literature seems too conservative and care shouldbe taken to choose such a threshold.

• From the results obtained, the present study also suggests the useof modular damage models. The idea is, regarding its physical ori-gin, a given damage model can be enhanced by adding suitableingredients to improve its prediction ability (e.g. a Lode depend-ent term was added to the GTN model and the Lemaitre modelby Xue, 2008 and Cao et al., 2014b to improve their predictionfor shear-dominated loading). With such an approach, all modelsbecome phenomenological. In spite of the diversity of the modelsinvestigated here, due to their phenomenological grounds, nonecan describe damage completely satisfactorily for both processes.For this reason, the development of a purely micromechanicalmodel that can capture damage mechanisms at both high and lowstress triaxialities is desirable and might be a subject for futurestudies.

Acknowledgments

The financial support from ArcelorMittal, Cezus-Areva andUgitech via the METAL project is appreciated.

Appendix A. Damage models used

A.1. Uncoupled approach: Bai & Wierzbicki fracture criterion

Bai and Wierzbicki (2008) constructed a 3D fracture locus inthe space (�f , �, �), which defines the strain to fracture as a func-

tion of the stress triaxiality (�) and the Lode parameter (�). Thisfunction is based on three limiting cases: �(−)

f (corresponding to

� = −1), �(0)f (corresponding to � = 0) and �(+)

f (corresponding to

� = 1). By adopting, on phenomenological ground, a parabolic func-tion to represent the effect of the Lode parameter on fracture locus,the fracture envelope �f = f (�, �) is thus defined as

�f (�, �) =[

12

(�(+)

f + �(−)f

)− �(0)

f

]�

2 + 12

(�(+)

f − �(−)f

)� + �(0)

f

(A.1)

From the early studies of McClintock et al. (1966) and Rice andTracey (1969), for each limiting bound, the influence of stress triaxi-ality on material ductility can be introduced through an exponentialfunction, hence: �(+)

f = D1e−D2�, �(0)f = D3e−D4�, �(−)

f = D5e−D6�. Eq.(A.1) can be rewritten as

�f (�, �) =[

12

(D1e−D2� + D5e−D6�) − D3e−D4�]

�2

1 −D � −D � −D �

+2

(D1e 2 − D5e 6 ) � + D3e 4 (A.2)

where D1, D2, D3, D4, D5, D6 are six positive material parameterswhich need to be identified. A linear incremental relationship is

rocess

as

D

waf(ttt(

AL

A

wwdtt(mfp

�

wctt

D

Y

w�d

F

So

D

wt√gwanmaidra

T.-S. Cao et al. / Journal of Materials P

ssumed between the damage variable D and the equivalent plastictrain �p:

(�p) =∫ �p

0

d�p

�f (�, �)(A.3)

here the triaxiality (� = �(�p)) and the Lode parameter (� = �(�p))re functions of the equivalent plastic strain. It should be noted that,or non-proportional loadings, the function �f (�, �) defined in Eq.A.2) should not be considered as the strain to fracture function. Inhis case, it is rather a stress dependent function, which is then usedo calculate the phenomenological damage parameter. Throughouthe present study, this model is referred to as Bai and WierzbickiB&W) model.

.2. Coupled phenomenological approach: Lemaitre, enhancedemaitre and Xue models

.2.1. The Lemaitre damage modelThe Lemaitre model is derived from the thermodynamics frame-

ork of continuum damage mechanics. The scalar D (0 ≤ D ≤ 1),hich is an internal variable, is adopted to describe the isotropicamage (D is often interpreted as the ratio of damaged area SD tohe total surface S: D = SD/S.). The effective stress is used to describehe impact of damage on the macro-behavior of material (see Eq.A.9)). This stress is the one that should be applied to an undamaged

aterial, in order to get the same strain tensor as the one obtainedrom the damaged material under actual stress (strain equivalencerinciple). The effective stress is defined as

′ij = �ij

1 − D(A.4)

here � ′ij

is the component of the effective stress tensor, �ij is theomponent of the actual stress tensor. The effective stress is used inhe constitutive equations instead of the Cauchy stress to describehe damage impact on the macroscopic behavior of materials.

The energy density release rate (Y), the variable associated with, is derived from the state potential as

= �2

2E(1 − D)2

[23

(1 + �) + 3(1 − 2�)�2]

(A.5)

here � is the von Mises equivalent stress, E is Young’s modulus, is the Poisson ratio, p is the hydrostatic pressure. The damageissipative potential (FD) is defined as

D = S

(b + 1)(1 − D)

(Y

S

)b+1(A.6)

(MPa) and b are two material parameters (which might dependn temperature). Finally, the damage evolution is given by

˙ = ∂FD

∂Y=

1 − D

(Y

S

)b

= �p

(Y

S

)b

(A.7)

here is the plastic multiplier, which can be deduced fromhe equivalent plastic strain rate (�p) as: = �p(1 − D), with �p =

(2/3)�p : �p (�p denotes plastic strain rate tensor). Lemaitre sug-

ested that there might exist a limit of equivalent strain �D belowhich the damage accumulation does not occur. From the present

uthors’ point of view, the presence of this threshold links withucleation process and depends strongly on the material. For someaterial with less second phase particles, the nucleation of voids

t low plastic strain is less noticeable or its presence does not

nfluence material strength, the coupled damage variable is thusisabled at low plastic strain. When the equivalent plastic straineaches a certain threshold, the presence of voids and its influencere noticeable; damage is activated.

ing Technology 216 (2015) 385–404 399

However, regarding void nucleation, whether nucleation isdriven by particle fracture or by matrix-particle decohesion, thestress level attained in the particle and the buildup energy dependon the loading conditions. Therefore, at high stress triaxiality, voidnucleation can take place faster and earlier (in terms of equiva-lent plastic strain) than at low stress triaxiality as confirmed byseveral tomography observations as in Maire et al. (2008) and theearlier experimental work of Kao et al. (1990). The assumption of aconstant threshold equivalent plastic strain is not realistic. A mod-ification was proposed in Cao et al. (2014b) and is presented inSection A.2.3.

Moreover, based on the observations of Bridgman experimentalresults (Bridgman, 1952; Bao and Wierzbicki, 2005) showed thatthere exists a limit of the stress triaxiality (�cutoff = −1/3) belowwhich, there is no damage. This result was adopted by Bouchardet al. (2011a) for the Lemaitre model and the damage evolution istherefore modified as

D =

⎧⎨⎩ �p

(Y

S

)b

if �p > �D and � > −13

0 otherwise

(A.8)

However, as shown in recent studies, Cao et al. (2013b) by using theModified Mohr–Coulomb model (developed by Bai and Wierzbicki,2010) and Lou and Huh (2013) by using a shear controlled ductilefracture model, the value of cutoff triaxiality can be well below−1/3. The value of −1/3 found by Bao and Wierzbicki (2005) is thusfar from universal. In a recent study, Lou et al. (2014) proposed acutoff value of the stress triaxiality, which is a function of the Lodeparameter. In the present study, the value of −1/3 for the cutoffvalue is kept for the Lemaitre model in a first approximation (butSection 5.1 discusses the role of �cutoff on damage localization inthe wire flat rolling process).

In order to account for the softening by damage accumulation, aweakening function w(D) has been adopted. Its effect is illustratedhere for the J2 plasticity and isotropic uniaxial elastic cases:{

� = w(D)�M

E = w(D)EM

with w(D) ={

1 − D if � > 0

1 − hD if � < 0(A.9)

where �M and EM are the flow stress and Young’s modulus ofundamaged material, h is a material parameter (0 < h < 1), whichaccounts for the “micro-crack closure effect”, i.e. the distinctionbetween “compressive” and “tensile” damages. By measuring thevariation of Young’s modulus in compression and tension, Lemaitreproposed the parameter h equals to 0.2, which is assumed validfor most materials. It should be emphasized that, by accountingfor the “micro-crack closure effect”, the effective stress under com-pression must be modified as: � ′

ij= �ij/(1 − hD). Consequently, Eqs.

(A.5), (A.6) and (A.8) were modified to account for this effect (seeAppendix A.2.2). The difficulty lies in the definition of compressiveand tensile states when 3D stress states are involved. A methodwas proposed by Ladevèze (1983), which consists in dissociatingthe stress tensor into positive (tensile) and negative (compressive)parts (see also Desmorat and Cantournet, 2008). This method wasemployed in Bouchard et al. (2011a) for the Lemaitre model. In thepresent study, a simple way to distinguish compressive and tensilestress states has been used, which was based on the scalar stresstriaxiality ratio.

A.2.2. Lemaitre model – accounting for micro-crack closure effect

For a general case, the effective stress is expressed as

�′ij = �ij

1 − hD(A.10)

4 rocess

wemAd

�

Fv

Y

Ift

�

Ttgw

As

osttmlad

1

2

3

D

{)

wccfatoc

l

00 T.-S. Cao et al. / Journal of Materials P

here h is a parameter accounting for the micro-crack closureffect. For the original Lemaitre model that accounts for theicro-crack closure effect: h = 1 when � > 0 and h = h when � < 0.ccording to Lemaitre, the elastic-damage state potential wasefined as (see Lemaitre, 1986):

∗ed = 1 + �

2E

�ij�ij

1 − hD− �

2E

�2kk

1 − hD(A.11)

rom this potential, under isothermal condition, the associatedariable for damage can be obtained:

= �∂ ∗

ed

∂D= h�2

2E(1 − hD)2

[23

(1 + �) + 3(1 − 2�)�2]

(A.12)

n addition, the equivalent plastic strain rate in this case is obtainedrom the plastic multiplier (due to the use of the effective stress inhe flow potential):

˙p =

1 − hD(A.13)

herefore, in order to account for the micro-crack closure effect,he equation of damage evolution (Eq. (A.7)) must use the aboveeneral formulations for Y and �p (Eqs. (A.12) and (A.13)), with h = 1hen � > 0 and h = h when � < 0.

.2.3. Lode-dependent enhanced Lemaitre model (LEL) forhear-dominated loading

Under shear-dominated loading, the stress triaxiality is zeror slightly positive, Lemaitre damage variable still increases withtrain, but more slowly since it is based principally on the stressriaxiality evolution (Eq. (A.7)). As shown in Cao et al. (2013a,b),he use of this model in shear-dominated forming processes or

echanical test (e.g. torsion) may lead to inaccurate damageocalization and fracture prediction.6 Cao et al. (2014b) proposed

Lode-dependent enhanced Lemaitre model (LEL) for shear-ominated loading by accounting for the following modifications:

. The evolution of damage variable (see Eq. (A.14)) was modifiedto account for the Lode parameter.

. The damage threshold (�D) was modified to account for the influ-ence of the stress triaxiality (Eq. (A.14)).

. The weakening function was modified to better account for theinfluence of damage at low stress triaxialities (Eq. (A.15)).

The governing equations of the LEL model are summarized as

˙ =

⎧⎨⎩ �p

(Y

S

)b1

˛1 + ˛2�2

if �p > �D = �D0 exp(−A�) and � > �cutoff

0 otherwise

(A.14)

� = w(D)�M

E = w(D)EM

with w(D)=

⎧⎪⎨⎪⎩

1 − D if � ≥ �1

1 − (1 − h)� + h�1 − �2

�1 − �2D if �1 > � ≥ �2

1 − hD if � < �2

(A.15

here ˛1, ˛2, �D0, A, �1, �2 are additional parameters; �cutoff is theutoff value of the stress triaxiality to be identified, which could behosen equal to −1/3 from the study of Bao and Wierzbicki (2004)or a first approximation. The parameter ˛2 may be chosen as 1 − ˛1s explained in Cao et al. (2014b). Because experimental observa-ions results were not sufficient, in the present study, quantification

f �D as a function of � was not possible, �D has been supposedonstant.

6 In the present study, the term damage localization refers to the maximum damageocation, unless otherwise indicated.

ing Technology 216 (2015) 385–404

A.3. Xue model

Xue (2007) proposed a phenomenological damage model, whichis based on the definition of the equivalent fracture strain �f as afunction of hydrostatic pressure (p) and Lode angle (�L):

�f = �f 0�p(p)��(�L) (A.16)

where �f0 is the reference fracture strain, determined from tensiontest at constant zero confinement pressure; �p(p) and ��(�L) arethe pressure-dependent function and the Lode angle-dependentfunction respectively. Eq. (A.16) defines a fracture envelope in threedimensional space of pressure, Lode angle and equivalent strain.Since p and �L are orthogonal to each other, they can be givenseparate forms:

�p = 1 − q ln(

1 − p

pL

), �� = � + (1 − �)

[6|�L|

�

]k

(A.17)

where p is the hydrostatic pressure, pL is the limit pressure (abovewhich damage does not occur), �L is the Lode angle, � is the ratiobetween fracture strain under shear loading and fracture strainunder uniaxial tension with the same imposed pressure; k, q > 0are material parameters. From the expression of �p, there exists acutoff value of pressure where �p reduces to zero (so does �f, whichmeans the material fails immediately):

pcutoff = pL(1 − e1/q) < 0 (A.18)

The scalar damage variable DX was used as an internal variableto represent the material degradation. The weakening functionw(DX ) = 1 − Dˇ

X is adopted to describe the damage effect on themacroscopic strength:

DX = m

(�p

�f (p, �L)

)m−1�p

�f (p, �L)and � = (1 − Dˇ

X )�M (A.19)

where is the weakening exponent, �p is the equivalent plasticstrain, �f(p, �L) is the fracture strain, which depends on the currentstress state (p, �L), �M is the flow stress of undamaged material.

Based on the studies of Lemaitre, the damage accumulation pro-cess is activated above a certain “threshold” defined by a physicalparameter (plastic strain or stored energy). For this reason, weadopted this observation for the Xue model by introducing a strainthreshold �DX, from which damage begins to occur. At the other endof the damage evolution, when DX = Dc, a “mesocrack” is initiated.The damage critical value Dc is another material parameter whichneeds to be identified. The coupling between damage and elastic-ity is introduced as: E = (1 − Dˇ

X )EM for a uniaxial case, where EM

is Young’s modulus of undamaged material. This enhanced modelwas recently successfully employed to model the crack formationand growth under different stress states (Cao, 2014). Although sev-eral modifications have been applied to the original model, it is stillreferred to as the Xue model hereafter.

A.4. Micromechanical approach: GTN and modified GTN models

A.4.1. The GTN modelThe yield function of the GTN model (Tvergaard and Needleman,

1984) writes:

� =(

�

�0

)2

+ 2q1f ∗ cosh(

−3q2

2p

�0

)− 1 − q3f ∗2 = 0 (A.20)

rocess

�cf

f

wbtu

f

wiN

f

wSt2nont

�

wstn

l

(

wpa

wtawcb

A

cc

f∫

T.-S. Cao et al. / Journal of Materials P

0 is the flow stress of matrix material; q1, q2, q3 = (q1)2 are materialonstants; f* is the effective void volume fraction, which accountsor the voids’ linkage:

∗ =

⎧⎨⎩

f if f < fc

fc + f ∗u − fcff − fc

(f − fc) if f ≥ fc(A.21)

here fc represents the critical value of f at which void coalescenceegins, ff its value at ductile failure, and f ∗

u = (q1 ±√

q21 − q3)/q3

he corresponding value of f* at failure. The evolution of void vol-me fraction is described as

˙ = fnucleation + fgrowth (A.22)

here fgrowth is defined as fgrowth = (1 − f )trace(�p) and fnucleations often described by a Gaussian curve introduced by Chu andeedleman (1980):

˙nucleation = fN

SN

√2�

exp

[−1

2

(�p − �N

SN

)2]

�p = A(�p)�p (A.23)

ith �N is the value of mean plastic strain at maximal nucleation,N the standard deviation of the corresponding Gaussian distribu-ion; fN the volume fraction of voids that can be nucleated (Besson,010). This parameter is determined so that the total void volumeucleated is consistent with the volume fraction of particles.7 Inrder to account for the influence of the stress triaxiality on theucleation process, Cao et al. (2014c) proposed a formulation forhe strain at maximum nucleation �N as

N = �N0 exp(−B�) (A.24)

here � is the stress triaxiality, which can be taken as the initialtress triaxiality if the loading is nearly proportional; B and �N0 arewo parameters to be identified. This modification allows the voiducleation process to take place earlier at high stress triaxialities.

The evolution of plastic strain is obtained through the equiva-ence of the plastic work, overall and in matrix material:

1 − f )�0d�p = � : d�p → �p = � : �p

(1 − f )�0(A.25)

here � and �p are the stress tensor and the increment of thelastic strain tensor. The evolutions of the two internal state vari-bles (�p and f) are summarized as

�p = � : �p

(1 − f )�0(A.26)

f = (1 − f )�p + A(�p)�p (A.27)

here �p is the volumetric part of the plastic strain tensor. Althoughhis model was successfully applied to predict fracture at high tri-xiality, it still suffers a major limitation in pure shear loading, forhich there is no void growth. In the present study, the modifi-

ation for shear-dominated loadings proposed by Xue (2008) haseen adopted and is detailed in the following.

.4.2. Modified GTN model for shear-dominated loadingsXue (2008) carried out an analytical development for a shearing

ase, where a square cell having length L containing a cylindri-al void of radius R at the center, is subjected to a simple shear

7 The volume fraction that can be nucleated is equal to∫ ∞

0A(

�p

)d�p , whereas

N =∫ ∞

−∞A(

�p

)d�p . If SN is small enough with respect to �N , fN =

∫ ∞−∞

A(

�p

)d�p ≈

∞0

A(

�p

)d�p .

ing Technology 216 (2015) 385–404 401

straining. The author defined the damage associated with the voidshearing, due to void rotation, as8

Dshear = q∗3f q4 �p�p (A.28)

where q∗3 = 6/

√� = 1.69, q4 = 0.5 for 2D case, and q∗

3 = 3(6/�)1/3 =3.722, q4 = 1/3 for 3D problem. For an arbitrary loading case, Xueintroduced a Lode angle dependence function g� into the aboveequation:

Dshear = q∗3f q4 �p�pg� where g� = 1 −

6∣∣�L

∣∣�

(A.29)

In generalized tension, �L = − (�/6), g� = 0, there is no contri-bution of shear damage, while in generalized shear, �L = 0, g� = 1,Eq. (A.28) is resumed. Note that in Xue (2008), the form of Lodeangle dependence function was chosen phenomenologically. In thepresent authors point of view, any function g� = 1 − (6|�L|/�)k couldbe employed (all these functions satisfy the condition g� = 0 in gen-eralized tension and g� = 1 in generalized shear).

Xue proposed a new damage variable (D) which accounts forvoid shearing damage:

D = ıD

(q1 f + Dshear

)(A.30)

where ıD is the damage rate coefficient. For the GTN model, whichaccounts for coalescence (i.e. using fc to define the start of coales-cence stage), this coefficient is defined as

ıD =

⎧⎨⎩

1 if D ≤ Dc = q1fc

f ∗u − fcff − fc

if Dc < D ≤ 1(A.31)

On the other hand, if the coalescence is not accounted for in theoriginal GTN model, this coefficient can be defined as

ıD =

⎧⎨⎩

1 if D ≤ Dc = q1f ∗c

f ∗u − f ∗

c

ff − f ∗c

if Dc < D ≤ 1(A.32)

where f ∗c is not a physical parameter, but rather a phenomenologi-

cal parameter to describe the acceleration of damage. The fractureis assumed to occur when D = 1. This modified damage variable canbe considered as (1) an indicator with no influence on yield function(damage counter approach – Xue (2008)); or (2) a damage yield-ing variable where the effective void volume fraction is replaced bythis variable in the yielding condition (full coupling approach). Inthe present study, the damage counter approach is employed.

Appendix B. Damage models calibration

The calibration of the coupled Lemaitre and Xue models for thismaterial was detailed in Cao et al. (2013a), using the mechanicaltests recapitulated in Fig. 2. The calibration of the micromechan-ical GTN and modified GTN models was presented in Cao et al.(2014c). In this paper, the authors combined macroscopic ten-sile tests on RB and NRB presented in Fig. 2 and in situ X-raymicro-tomography tensile tests on NRB to identify the parame-ters. With this method, the identified parameters are validated forboth macroscopic and microscopic tests. It should be noted that, byusing micro-tomography observations, the initial porosity (or void

−6

volume fraction) for this material is quite small (f0 = 4.92 × 10 ),as well as its value at fracture (ff ≈ 0.0024). For the LEL model,the identification method was detailed in Cao et al. (2014b). The

8 It should be noted that the term “shear” in Dshear refers to damage caused prin-cipally by void rotation, which leads to void shearing failure mechanism (not thedamage caused only by “shear” force).

402 T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404

Table B3Recapitulation of the critical values of damage variables.

tdmt

etpe(tmvua

aoH(pit

Fig. B14. Comparison between experimental and numerical displacements to frac-ture of different tests, with the uncoupled damage model B&W. Note that for the

Ft(

B&W Lemaitre Xue GTN LEL Modified GTN

1 0.126 0.98 0.0024 0.15 0.98

orsion test was used to identify additional parameters of the Lode-ependent term (in addition to the parameters of the Lemaitreodel). The parameters identified for the Lode dependency func-

ion are: ˛1 = 0.2 and ˛2 = 0.8.For the calibration of B&W model, the same method as in Cao

t al. (2013b) was used, which was based on the displacementso fracture for different mechanical tests presented in Fig. 2. Thearameters were identified to minimize the difference betweenxperimental and numerical displacements to fracture for all testsfor the torsion test, the “displacement to fracture” correspondso the “number of rotations to fracture”). The numerical displace-

ents to fracture (dfnum) are the displacements at which the damage

ariable reaches its critical value (Dc = 1) in mechanical tests sim-lations. Fig. B14 presents the comparison between experimentalnd numerical results.

All the details of the calibration procedure can be found in thebove-mentioned references. In the present work, we only focusn the final applications of these models to the forming processes.ere, only the critical values of damage variables for each model

which defines the instant of fracture) is recalled in Table B3. Therincipal objective of the present study is to compare all the models

n forming processes, the use if critical values would be enough forhis assessment.

ig. C15. (a) Numerical sensors at three positions on a quarter of transverse cross sectiohe equivalent plastic strain map. Evolution of stress triaxiality, Lode parameter and stra1); (c) sensor (2); and (d) sensor 3. (For interpretation of the references to color in this fi

torsion test, the “displacement to fracture” corresponds to the “number of rotationsto fracture”.

Appendix C. Loading history in the wire flat rolling process

This section presents a study on the loading paths of differ-ent material points on a wire transverse cross section during thewire flat rolling. The objective is to illustrate the complexity of the

loading paths in this process. Fig. C15a presents numerical sensorsplaced at three different positions: (1) wire center, (2) free edgezone, and (3) blacksmith cross. The color in this figure corresponds

n: (1) center, (2) free edge zone and (3) blacksmith cross. The color corresponds toin rate with equivalent plastic strain during the deformation process for (b) sensorgure legend, the reader is referred to the web version of the article.)

T.-S. Cao et al. / Journal of Materials Processing Technology 216 (2015) 385–404 403

F al damv with

tr

pdtrstwtsetn2tvv

Ad

BtmtdptFo�w

ioi

R

B

B

ig. D16. Damage at steady state of second drawing pass for three phenomenologicalues are represented on the longitudinal and transverse cross sections (compared

o the equivalent plastic strain map at the end of one rolling pass,epresented on a quarter of transverse cross section.

Fig. C15b–d shows the evolution of the stress triaxiality, the Lodearameter and the strain rate with the equivalent plastic strainuring deformation process for the three sensors 1, 2, and 3, respec-ively. As can be observed, among these three positions, the strainate is highest at the wire center (and so is the equivalent plastictrain). The strain rate and the equivalent plastic strain are higher inhe blacksmith cross than in the free edge zone, which is consistentith the strain map at the end of the process (Fig. C15a). The stress

riaxiality is negative at wire center (sensor 1) and in the black-mith cross (sensor 3) except at bite inlet, but is positive in the freedge zone (sensor 2) (see blue curves in Fig. C15b–d). Regardinghe Lode parameter, the values at positions 1 and 3 are small andegative (≈−0.2), but the Lode parameter is positive at position

during the deformation process. These investigations show thathe stress state of material points at different positions would beery different in this process. The use of damage models that arealidated for different stress state is thus vital.

ppendix D. Wire drawing – results of phenomenologicalamage models with �cutoff = −2/3

This section presents the application results obtained with the&W, Lemaitre and LEL models when the cutoff value of the stressriaxiality is equal to −2/3. For this process, the stress state in the

ost dangerous position (i.e. wire core) is equivalent to a uniaxialensile state, on which a certain pressure caused by the convergentie is superimposed (as a whole, the triaxiality is positive in thelastic zone). Consequently, the change of the cutoff value of stressriaxiality does not have strong impacts on damage localization.ig. D16a–c shows the results of damage at the second drawing passbtained with the B&W, Lemaitre and LEL models, respectively (forcutoff = −2/3. As can be observed, damage is higher in the wire core,ith a same magnitude as the case �cutoff = −1/3 (see Fig. 7).

The change of the cutoff value of stress triaxiality has negligiblempacts on the prediction of damage localization and the instantf fracture since the most dangerous zone (i.e. wire core) is lessnfluenced by this change.

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