International Journal of Solids and Structures 67–68 (2015) 40–55

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsolst r

Micromechanically-motivated phenomenological Hosford–Coulombmodel for predicting ductile fracture initiation at low stress triaxialities

http://dx.doi.org/10.1016/j.ijsolstr.2015.02.0240020-7683/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Solid Mechanics Laboratory (CNRS-UMR 7649),Department of Mechanics, École Polytechnique, Palaiseau, France. Tel.: +33 16933 5801.

E-mail address: mohr@mit.edu (D. Mohr).

Dirk Mohr ⇑, Stephane J. MarcadetSolid Mechanics Laboratory (CNRS-UMR 7649), Department of Mechanics, École Polytechnique, Palaiseau, FranceImpact and Crashworthiness Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

a r t i c l e i n f o

Article history:Received 27 January 2015Available online 23 February 2015

Keywords:Ductile fractureLow stress triaxialityLode angleMohr–CoulombHosfordShear localization

a b s t r a c t

A phenomenological ductile fracture initiation model for metals is developed for predicting ductilefracture in industrial practice. Its formulation is based on the assumption that the onset of fracture isimminent with the formation of a primary or secondary band of localization. The results from a unit cellanalysis on a Levy–von Mises material with spherical defects revealed that a Mohr–Coulomb type ofmodel is suitable for predicting the onset of shear and normal localization. To improve the agreementof the model predictions with experimental results, an extended Mohr–Coulomb criterion is proposedwhich makes use of the Hosford equivalent stress in combination with the normal stress acting on theplane of maximum shear. A fracture initiation model is obtained by transforming the localization crite-rion from stress space to the space of equivalent plastic strain, stress triaxiality and Lode angle parameterusing the material’s isotropic hardening law. Experimental results are presented for three differentadvanced high strength steels. For each material, the onset of fracture is characterized for five distinctstress states including butterfly shear, notched tension, tension with a central hole and punch experi-ments. The comparison of model predictions with the experimental results demonstrates that the pro-posed Hosford–Coulomb model can predict the instant of ductile fracture initiation in advanced highstrength steels with good accuracy.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Ductile fracture is a well-known physical process that leads tothe formation of cracks in metals due to the nucleation, growthand coalescence of voids. In cases where macroscopic localizationprecedes void coalescence, the ductile fracture process may bedescribed as follows: as the material deforms plastically, pre-existing (primary) voids evolve and new ones nucleate (stage s

in Fig. 1). Due to the increase in porosity and the decrease inmacroscopic strain hardening, the conditions for a discontinuityin the macroscopic strain field along a planar interface may bemet. The result is the formation of a primary band of localizationat the mesoscale (stage t in Fig. 1). As stated by Pardoen andHutchinson (2000) with reference to Tvergaard (1981), the widthof a primary localization band is expected to be of the order ofthe inter-void spacing. Subsequently, the material inside the bandexperiences accelerated void growth and nucleation (stage u in

Fig. 1). As a result, the porosity and/or the number of voids withinthe band increase sharply and the mechanical fields around indi-vidual primary voids begin to interact. The nucleation of secondaryvoids (which are often several orders of magnitude smaller thanprimary voids) is also possible at this stage. The final coalescencephase sets in when the deformation begins to localize withinsecondary bands of localization at the microscale (stage v inFig. 1). In other words, inside the primary localization band, thereis a transition from diffuse to localized plastic flow, which ulti-mately leads to primary void coalescence and the formation ofcracks through internal necking or void sheet fracture of the liga-ments between primary voids (stages w and x in Fig. 1). As dis-cussed by Tekoglu et al. (2015), polycrystalline materials mayalso fail due to (i) localized plastic flow only (e.g. necking up tozero cross-sectional area), (ii) localization of plastic flow after dam-age-free deformation, followed by void growth and nucleationinside the primary band of localization, and (iii) direct coalescence(secondary localization) without any prior occurrence of primarylocalization.

Research on ductile fracture has addressed several aspects ofthe ductile fracture process. A wealth of literature deals with por-ous plasticity, i.e. the effective large deformation behavior of

Fig. 1. Eulerian illustration of the ductile fracture process with coalescence through (a) internal necking, and (b) void sheet fracture. The mesoscopic primary band oflocalization is highlighted in gray color, the microscopic secondary band of localization is highlighted in red. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55 41

mildly porous metals (e.g. Gurson (1977), Tvergaard (1981), Mearand Hutchinson (1985), Gologanu et al. (1993), Leblond et al.(1995), Benzerga and Besson (2001), Molinari and Mercier(2001), Monchiet et al. (2008), Nahshon and Hutchinson (2008),Danas and Ponte Castañeda (2012)). The formation of primarybands of localization is expected to come out naturally when solv-ing boundary value problems with accurate porous plasticity mod-els. In other words, there is no need to introduce any localizationcriterion. However, the computation of the stress and strain fieldsafter the initiation of localization bands is challenging due to theassociated loss of ellipticity of the governing field equations. Theuse of non-local formulations (e.g. gradient plasticity) appears tobe necessary to regularize the mathematical problem in the postlocalization regime (e.g. Anand et al. (2012)).

As an alternative to using highly accurate porous plasticitymodels, the formation of primary bands of localization can alsobe predicted using conventional non-porous plasticity models inconjunction with localization criteria. This approach is motivatedby the fact that non-porous plasticity models provide an excellentapproximation of the multi-axial stress–strain response of metalsup to the point of localization (e.g. Dunand and Mohr (2011a,b)).Furthermore, from an engineering perspective, the formation of aprimary band of localization is considered as the onset of fracture.Accordingly the engineering literature does not differentiatebetween fracture initiation models and criteria predicting the for-mation of primary bands of localization.

Localization analysis with porous plasticity models providesvaluable insight into the effect of stress state on the formation ofprimary bands of localization (e.g. Rudnicki and Rice (1975), Rice(1977)). Recent examples are the localization analysis with aband-like defect in a shear-sensitive Gurson solid (Nahshon andHutchinson, 2008), and the investigation of the loss of ellipticityand peak load by Danas and Ponte Castañeda (2012) using ahomogenization-based porous plasticity model. The main

limitation today is the accuracy of the advanced porous plasticitymodels. The above models are able to capture first order effects,but to the best of the authors’ knowledge, the above localizationestimates have not yet been utilized to predict the onset of local-ization in real structures.

Multi-axial experiments provide the only viable alternative tocomputational localization analysis (e.g. Hancock and Mackenzie(1976), Mohr and Henn (2007), Haltom et al. (2013)). Here, themain challenge is the analysis of heterogeneous mechanical fieldsdue to the localization at the structural level (localized necking)which often precedes the formation of primary bands of localiza-tion. In particular, the highest strains within a specimen are oftenreached below the specimen surface. Except for rare cases wheretomography-based 3D digital image correlation is possible (e.g.Morgeneyer et al. (in press)), the strains can only be estimatedthrough statistical analysis of grain deformation in micrographs(e.g. Ghahremaninezhad and Ravi-Chandar (2012)) or hybridexperimental–numerical analysis (e.g. Dunand and Mohr (2010)).

Bao and Wierzbicki (2004) provide a comprehensive overviewon functional relationships between the equivalent plastic strainand the stress state (derived from the works of McClintock(1968), Rice and Tracey (1969), LeRoy et al. (1981), Cockcroft andLatham (1968), Oh et al. (1979), Brozzo et al. (1972), and Cliftet al. (1990)) that may be interpreted as localization or fracture ini-tiation criteria for proportional loading. To account for the effect ofnon-proportional loading, the above functions are typically inte-grated into a damage indicator framework. Early examples of dam-age indicator models for predicting the onset of fracture are thestress triaxiality dependent model of Johnson and Cook (1985)and the stress triaxiality and Lode parameter dependent model ofWilkins et al. (1980). Recent examples are the modified Mohr–Coulomb model proposed by Bai and Wierzbicki (2010) and amicro-mechanism inspired damage indicator model proposed byLou et al. (2012) and Lou and Huh (2013). It is worth noting that

42 D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55

there is a significant difference between damage indicator modelsand Continuum Damage Mechanics (CDM). In the latter frame-work, loss in load carrying capacity is modeled through an internaldamage variable while the constitutive equations are derived fromthe first and second principles of thermodynamics for continuousmedia (e.g. Lemaitre (1985), Chaboche (1988)). In particular, inCDM models, the elasto-plastic material response is affected bydamage, whereas the plastic response remains unaffected by dam-age evolution in damage indicator models.

Unit cell analyses provide another means for studying the for-mation of localization bands in metals. Since the pioneering worksof Needleman and Tvergaard (e.g. Needleman (1972), Tvergaard(1981)), numerous unit cell analyses have been performed consid-ering a wide range of unit cell configurations and loading condi-tions (Koplik and Needleman, 1988; Brocks et al., 1995;Needleman and Tvergaard, 1992; Pardoen and Hutchinson, 2000;Barsoum and Faleskog, 2007; Tvergaard, 2008, 2009; Scheyvaertset al., 2011; Nielsen et al., 2012; Rahman et al., 2012; Tekogluet al., 2012). The most recent studies now consider 3D unit cellmodels with general 3D boundary conditions (e.g. Barsoum andFaleskog (2011), Dunand and Mohr (2014)). All of the above unitcell analyses are performed considering a single void only alongwith periodic boundary conditions which limits their capabilityto predicting secondary localization only.

Coalescence models have been developed to describe the tran-sition from diffuse plastic flow to localized plastic flow within theinter-void ligament of neighboring primary voids within themesoscopic band of localization. Simple mechanical models of aperiodic array of square cuboidal voids have been used byThomason (1968, 1985) to analyze the process of internal neckingin an approximate manner, while more advanced models consid-ering spheroidal voids and shear deformation have been devel-oped later (e.g. Benzerga (2002), Pardoen and Hutchinson(2000), Tekoglu et al. (2012)). Coalescence models predict the for-mation of secondary bands of localization. The mechanical systemat this stage of the ductile fracture process is characterized by astrong interaction of neighboring voids. Consequently, coales-cence criteria incorporate information on void size and spacing.In contrast, the formation of primary bands of localization isthe outcome of an instability of the material response at themacroscopic level.

In the present work, a phenomenological fracture initiationmodel is proposed for predicting the onset of ductile fracture inengineering practice. The backbone of the proposed model is alocalization criterion for radial loading in terms of the Hosfordequivalent stress and the normal stress acting on the plane of max-imum shear. Using the isotropic hardening law associated with thematerial’s plastic behavior, the criterion is transformed from prin-cipal stress space to the space of equivalent plastic strain, stress tri-axiality and Lode angle parameter. As a final result, a fractureinitiation model is obtained which preserves the underlying phys-ical meaning of the stress-based localization criterion. The resultsfrom fracture experiments for five different stress states are pre-sented for three different advanced high strength steels (DP590,DP780 and TRIP780). It is shown that the Hosford–Coulomb (HC)fracture initiation model can accurately describe the experimentaldata for all materials and experiments including pure shear,notched tension and equi-biaxial tension.

2. Preliminaries

2.1. Description of the stress state

The stress state is described by the stress triaxiality and theLode angle parameter. The stress triaxiality is defined as the ratioof the mean stress rm and the von Mises stress �r,

g ¼ rm

�r: ð1Þ

It may be interpreted as a measure of the ratio of the first and sec-ond stress tensor invariants. The Lode angle parameter on the otherhand measures the ratio of the third and second stress tensorinvariants,

�h ¼ 1� 2p

arccos3ffiffiffi3p

2J3

ðJ2Þ3=2

" #: ð2Þ

According to the above definition, the Lode angle parameter variesbetween �1 (axisymmetric compression) and 1 (axisymmetric ten-sion). Fig. 2a provides a graphical interpretation of the modifiedHaigh–Westergaard coordinates fg; �h; �rg. For plane stress condi-tions, the stress space is reduced from 3D to 2D which results in afunctional relationship between the Lode angle parameter and thestress triaxiality (Fig. 2b). Based on the modified Haigh–Westergaard coordinates, the ordered principal stresses(rI � rII � rIII) may be reconstructed as

rI ¼ �r gþ f 1ð Þ; ð3Þ

rII ¼ �r gþ f 2ð Þ; ð4Þ

rIII ¼ �r gþ f 3ð Þ; ð5Þ

with the Lode angle parameter dependent trigonometric functions

f 1½�h� ¼23

cosp6ð1� �hÞ

h i; ð6Þ

f 2½�h� ¼23

cosp6ð3þ �hÞ

h i; ð7Þ

f 3½�h� ¼ �23

cosp6ð1þ �hÞ

h i: ð8Þ

2.2. Plasticity model

Before developing the fracture initiation model, we briefly out-line the constitutive equations of a non-associated quadratic plas-ticity model with isotropic hardening. The combination of a vonMises yield surface with a Hill’48 flow rule is chosen as it providesa good approximation of the large deformation response ofadvanced high strength steels (Mohr et al., 2010). The reader isreferred to Stoughton (2002) for the proofs of the uniqueness ofthe stress distribution, the stability of plastic flow and the unique-ness of the stress and strain state.

The von Mises yield surface is expressed as

f ½r; k� ¼ �r� k ¼ 0; ð9Þ

with k denoting a deformation resistance that controls the size ofthe elastic domain. The direction of plastic flow is assumed to bealigned with the stress derivative of a flow potential function g[r],

dep ¼ dk@g@r

; ð10Þ

where deP denotes the plastic strain tensor increment in materialcoordinates;dk � 0 is a scalar plastic multiplier. The potential func-tion is defined as an anisotropic quadratic function in stress space

g2½r� ¼ r211 þ G22r2

22 þ ð1þ 2G12 þ G22Þr233 þ 2G12r11r22

� 2ð1þ G12Þr11r33 � 2ðG22 þ G12Þr22r33 þ G33r212

þ 3r213 þ 3r2

23; ð11Þ

with the anisotropy coefficients G22,G12 and G33. The above functioncorresponds to a special case of an orthotropic Hill’48 flow potentialfunction which accounts for the planar anisotropy associated with

σΙ = σΙΙ

= σΙΙΙ

Fig. 2. (a) Illustration of the stress triaxiality g and the Lode angle parameter �h in principal stress space {rI, rII, rIII}. Selected Lode angle parameter values are only shown forthe 60� segment of p-plane where the principal stresses satisfy the order {rI P rII P rIII}. For the other five 60� segments, the same labeling applies because of thesymmetries of the unordered principal stress space; (b) non-linear relationship between �h and g for plane stress. The blue, black and red curves shows the relationship forbiaxial compression (two negative principal stresses), biaxial tension–compression (one positive and one negative principal stress), and biaxial tension (two positive principalstresses), respectively. Open dots highlight the special cases of cases of uniaxial compression, pure shear, uniaxial tension, plane strain tension, and equi-biaxial tension’’. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55 43

direction-dependent Lankford ratios. For G22 ¼ 1, G12 ¼ �0:5 andG33 ¼ 3, the above potential reduces to the von Mises potential.Isotropic hardening is introduced into the model through thefunction

k ¼ k½�ep�; ð12Þ

with the equivalent plastic strain defined as work-conjugate to thevon Mises equivalent stress,

�ep ¼Z

g½r��r

dk: ð13Þ

3. Fracture initiation model for proportional loading

3.1. Motivation

A fracture initiation model is developed to predict the onset ofductile fracture in a macroscopically defect-free solid. In particular,

the goal is to predict the strain to fracture, i.e. the macroscopicequivalent plastic strain that can be achieved before the formationof a primary or secondary band of localization. At the macroscopiclevel (average material response over several inter-void spacing),there is actually no noticeable difference between the strains atthe onset of localization and those at the instant of void coales-cence as all deformation localizes within a narrow band betweenthese two events. It is therefore assumed that the onset of fracturecoincides with the formation of a primary or secondary (whicheveroccurs first) band of localization. It is emphasized that the model-ing of ductile fracture, i.e. the modeling of the propagation ofcracks, requires the careful modeling of the conversion of bulk tosurface energy and is beyond the scope of the present work.

The presence of voids may be neglected when describing themacroscopic elasto-plastic response of sheet metal at low stresstriaxialities (e.g. Dunand and Mohr (2011a)). However, voids playan important role in triggering the onset of localization. Unit cellanalyses (e.g. Barsoum and Faleskog (2007)) and stability analyses

Lode angle parameter

Fig. 3. Localization analysis results: (a) relationship between the shear and normalstress acting on the plane of localization; (b) macroscopic equivalent plastic strainat the onset of localization as a function of Lode angle parameter and stresstriaxiality; each dot represents the result from a unit cell analysis for a particularstress state, the solid curves correspond to the predictions of the Mohr–Coulombmodel. Note that both plots have been prepared using the same MC modelparameters (friction c1 = 0.13, cohesion c2 = 666 MPa).

1 The Hosford criterion becomes non-convex for a < 1. This requires special carehen using the Hosford function as yield surface, but there is no restriction withspect to convexity when it is used as localization criterion.

44 D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55

with advanced porous plasticity models (e.g. Nahshon andHutchinson (2008)) have demonstrated that shear and normallocalization at low stress triaxialities can be predicted when takingthe effect of voids (and void shape changes) into account. A mech-anism-based model for predicting the onset of ductile fracturewould thus require (i) a void nucleation model, (ii) void volumefraction and shape evolution equations, and (iii) a void volumefraction and shape dependent shear/normal localization criterion.In theory, a comprehensive porous plasticity model would satisfyall these requirements since the onset of localization could be pre-dicted by analyzing the loss of ellipticity of the incremental moduliassociated with the current state of the material. Both Danas andPonte Castañeda (2012) and Nahshon and Hutchinson (2008) indi-rectly pursued this approach. However, given the sensitivity oflocalization analysis to small changes in the constitutive modelformulation and the number of approximations necessary duringnon-linear homogenization (see e.g. Ponte Castañeda (2002)), itis still questionable whether accurate predictions of the equivalentplastic strain at the onset of fracture will be obtained in the nearfuture using a porous plasticity approach.

A different approach is pursued here. Instead of predicting theonset of localization by means of an advanced porous plasticitymodel, we make use of a conventional non-porous plasticity modelalong with a localization criterion. The localization criterion forproportional loading is transformed from stress space to a mixedstress–strain space, before inserting the resulting functional intoa damage indicator model framework to account for the effect ofnon-proportional loading.

3.2. Localization criterion in stress space

Dunand and Mohr (2014) subjected a unit cell of a Levy–vonMises material with a central void of 1.2% porosity to combinationsof shear and normal loading to determine the macroscopic equiv-alent plastic strain at the onset of localization as a function ofthe stress state. They performed this type of analysis for more than160 different stress states for stress triaxialities ranging from 0 to 1and Lode angle parameters ranging from �1 to 1. Their resultsdemonstrate that a Mohr–Coulomb criterion,

maxn½sþ c1rn� ¼ c2; ð14Þ

provides a reasonable prediction of the instant of onset of secondarylocalization (Fig. 3a), with s and rn denoting the shear and normalstress on a plane of normal vector n. In terms of the ordered princi-pal stresses, the Mohr–Coulomb criterion may be rewritten as

rI � rIIIð Þ þ c rI þ rIIIð Þ ¼ b; ð15Þ

with

c ¼ c1ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c2

1

q and b ¼ 2c2ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c2

1

q ; ð16Þ

which is fully equivalent to (14).However, the predictions of the Mohr–Coulomb (MC) model do

not always agree well with results from experiments (where pri-mary localization may also precede coalescence). These deficien-cies are partly attributed to the shortcomings of the periodic unitcell model (which can only capture secondary localization).Furthermore, strong simplifying assumptions with regards to theshape and the volume fraction of the defects triggering the local-ization are expected to play a role. Note that the results ofDunand and Mohr (2014) were obtained assuming the same vol-ume fraction of spherical voids irrespective of the stress state.

To improve the agreement of model predictions with experi-mental data (which will be discussed in Section 6), we constructa simple phenomenological model based on the above

micromechanical results. In particular, an extension of the MC cri-terion is proposed by substituting the Tresca equivalent stress in(15) by the Hosford (1972) equivalent stress,

�rHF þ cðrI þ rIIIÞ ¼ b; ð17Þ

with

�rHF ¼12ðrI � rIIÞa þ ðrII � rIIIÞa þ ðrI � rIIIÞa� �� �1

a

; ð18Þ

And 0 < a 6 2 denoting the Hosford exponent1. The abovemodel is referred to as Hosford–Coulomb (HC) model: as postulatedby Coulomb (1776), the material’s deviatoric strength is decomposedinto a cohesion b and a frictional term that is proportional to the nor-mal stress ðrI þ rIIIÞ=2 acting on the plane of maximum shear. TheHC model actually reduces to the MC model for a ¼ 1. However, an

wre

D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55 45

important difference between the MC and HC models becomesapparent for biaxial tension (i.e. plane stress states of loadingbetween uniaxial tension and equi-biaxial tension): since the MCmodel does not dependent on the second principal stress, it reducesto a maximum principal stress criterion (because of rIII ¼ 0), whilethe HC model remains sensitive to the biaxial stress ratio rII=rI .

3.3. Fracture initiation model in mixed strain–stress space

Recall that the onset of ductile fracture is considered to beimminent with the formation of a primary or secondary band oflocalization. The above localization criterion (17) is thereforeemployed to predict the onset of fracture. The results from ductilefracture experiments are typically presented in terms of the equiv-alent plastic strain, the stress triaxiality and the Lode angle param-eter. We transform the localization criterion from the principalstress spacefrI;rII;rIIIg to the mixed strain–stress space fg; �h; �epg.

Fig. 4. Coordinate transformation for plane stress conditions: (a) initial von Mises yieldlocalization locus (blue line) for proportional loading; (b) representation of the same envspace. (For interpretation of the references to color in this figure legend, the reader is re

Using Eqs. 3–5, the criterion (17) is first rewritten in the modifiedHaigh–Westergaard stress space fg; �h; �rg, where it takes the form

�r¼ �rf ½g;�h� ¼b

12 ðf 1� f 2Þ

aþðf 2� f 3Þaþðf 1� f 3Þ

a� �� �1aþcð2gþ f 1þ f 3Þ

: ð19Þ

Fig. 4a and b illustrate this transformation for plane stress condi-tions, with the localization criterion shown as a blue envelope.Von Mises stress iso-contours are shown as dotted lines in both fig-ures. Furthermore, we included the trajectories of constant stressstate as straight dashed lines. Since the stress point is located onthe yield surface when plastic localization initiates, the inverse of

the isotropic hardening law (12), �ep ¼ k�1½�r�, may be used to trans-form the localization criterion from the modified Haigh–Westergaard space to the mixed stress–strain spacefg; �h; �epg,

�eprf ¼ k�1 �rf ½g; �h�

: ð20Þ

envelope (solid black line) with subsequent von Mises stress iso-contours and HCelopes in the modified Haigh–Westergaard space, and (c) in the mixed strain–stressferred to the web version of this article.)

46 D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55

In Fig. 4, this final transformation corresponds to a non-affine map-ping of the ordinate from Fig. 4b and c.

3.4. Illustration of the HC model

Fig. 5 shows the strain to fracture for proportional loading (20)as a function of the stress triaxiality and the Lode angle parameterfor different sets of model parameters fa; b; cg. In all graphs, theparameter b has been adjusted such that the strain to fracture foruniaxial tension equals 0.8.

� For a ¼ 1 (Fig. 5a), we obtain a representation of the Mohr–Coulomb criterion in the mixed strain–stress space. The strainto fracture is a monotonically decreasing function of the stresstriaxiality for c > 0 and a convex function of the Lode angleparameter which exhibits a minimum for generalized shearð�h ¼ 0Þ. The characteristic signature of a normal stress depen-dent criterion is the asymmetry with respect to the Lode angleparameter. Note that the use of the pressure instead of the nor-mal stress in (17) (e.g. a Drucker–Prager type of criterion)would result in a symmetric Lode angle dependency.� For c ¼ 0 (Fig. 5b), the normal stress term is no longer active.

Consequently, the criterion becomes independent of the stresstriaxiality and symmetric with respect to the Lode angle param-eter (Hosford model).� For a ¼ 2 (Fig. 5c), the Lode angle dependence is only due to the

normal stress term. In the limiting case of a ¼ 2 and c ¼ 0 (vonMises model), the criterion becomes independent of both thestress triaxiality and the Lode angle parameter which corre-sponds to a fracture initiation model that depends on the equiv-alent plastic strain only.

Fig. 5. Representation of special cases of the Hosford–Coulomb (HC) model in the modistress conditions. (For interpretation of the references to color in this figure legend, the

� A typical HC surface is shown in Fig. 5d (a ¼ 1:5 and c ¼ 0:1).It exhibits the same stress triaxiality dependence as the MCmodel, but as the comparison of Fig. 5a and d shows, themodels sensitivity to the Lode angle parameter can beadjusted.

All 3D plots include a blue curve which highlights the modelresponse for plane stress conditions. As illustrated in Fig. 2b,the Lode angle parameter is a non-linear function of the stress tri-axiality for plane stress. To shed more light on the effect of themodel parameters a and c, we plotted the strain to fracture as afunction of the stress triaxiality for plane stress conditions inFig. 6:

� The effect of the friction coefficient c on the MC model (a = 1) isshown in Fig. 6a. Note that the curves are in hierarchical orderfor g < 1/3, i.e. the higher the friction coefficient, the higher thestrain to fracture. For g P 1/3 (biaxial tension), the friction coef-ficient has no effect on the strain to fracture predicted by theMC model. This is due to the fact that the MC model reducesto a maximum principal stress criterion for biaxial tension withonly one independent parameter.� In case of the HC model (a – 1), the model response for biaxial

tension can still be adjusted by the Hosford exponent (Fig. 6b)and the friction parameter (Fig. 6c). Note that the ordering ofthe curves with respect to the friction parameter changes atg = 1/3. All curves exhibit an absolute minimum at g ¼ 1=

ffiffiffi3p

which corresponds to transverse plane strain tension for aLevy–von Mises material (rII ¼ 0:5rI). Fig. 6d shows the crite-rion for a ¼ 2 which includes the special case c ¼ 0 (strain tofracture independent of the stress state).

fied Haigh–Westergaard space. The blue lines show the strain to fracture for planereader is referred to the web version of this article.)

Fig. 6. Effect of the parameters of the Hosford–Coulomb (HC) model on the fracture envelope for plane stress loading.

D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55 47

3.5. Comment on model extension for non-proportional loading

The focus of the present work is on monotonic proportionalloading, i.e. loading histories throughout which the stress triaxial-ity and Lode parameter remain constant up to the point of fractureinitiation. A model extension for non-proportional loading isnonetheless included in this paper because of inevitable stressstate variations in many fracture experiments (in particular dueto necking in the case of sheet materials).

When using porous plasticity (e.g. Gurson (1977), Gologanuet al. (1993), Benzerga and Besson (2001), Monchiet et al. (2008),Danas and Ponte Castañeda (2012)) the evolution law for the voidvolume fraction (and other possible microstructural state vari-ables) is loading path sensitive and failure predictions withmicrostructurally-informed coalescence criteria (e.g. Thomason(1985), Pardoen and Hutchinson (2000), Benzeraga (2000),Tekoglu et al. (2012)) are then ‘‘naturally’’ loading path dependent.This is a key advantage of the latter over phenomenological frac-ture criteria which are used in conjunction with non-porous plas-ticity models that do not feature any loading path dependentdamage measure as internal variable such as the void volume frac-tion in porous plasticity.

A first approach to evaluating a phenomenological fracture ini-tiation model for proportional loading based on experimental datawith stress state history variations in the plastic range would be to

postulate that the model holds true for the average stress state his-tory (e.g. Bai and Wierzbicki (2008)). However, as clearly demon-strated by Benzerga et al. (2012), this approach is in strongcontradiction with the results from unit cell coalescence analysisfor non-radial loading paths. A second approach would be to applythe stress-based localization criterion (Eq. (17)) directly even if theloading path is non-proportional (e.g. Stoughton and Yoon (2011)and Khan and Liu (2012)). This corresponds to assuming that thestrain to fracture is independent of the stress state history anddepends on the current stress state only. As will be shown inSection 6, our experimental data includes different loading pathswhich show significantly different fracture strains even thoughthe stress state at the instant of fracture initiation is very similar(see loading paths NT6, NT20 and PU in Fig. 11d).

As an alternative, we make use of Fischer’s integral extension toevaluate our model for proportional loading based on experimentaldata with inevitable necking-induced loading path variations. Theresulting final integral form of the HC fracture initiation modelreadsZ �ef

0

d�ep

�eprf ½g; �h�

¼ 1; ð21Þ

with �ef denoting the equivalent plastic strain at the onset of fractureafter loading along a path characterized by the histories g½�ep� and�h½�ep�. The assessment of the general validity of this integralapproach is deferred to future research as a comprehensive series

Table 1Lankford ratios.

r0 [–] r45 [–] r90 [–]

DP590 0.98 0.84 1.13DP780 0.78 0.96 0.77TRIP780 0.89 0.82 1.01

48 D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55

of non-proportional loading path experiments would needed forthis. Here, we can only justify the use of the integral formulationthrough empirical arguments, i.e. it has been widely used (withoutany justification) in engineering practice for 30 years (see Johnsonand Cook (1985)) and is mathematically similar to the well-estab-lished Palmgren–Miner rule in high cycle fatigue (Palmgren,1924). At the same time, it is noted that the integral extension fornon-proportional loading is conceptually problematic as discussedby Benzerga et al. (2012). We therefore emphasize that even thoughEq. (21) depends on the stress-state history during plastic loading,i.e. g½�ep� and �h½�ep�, it is just introduced as a means to identify the cri-terion for proportional loading from basic fracture experiments,while it should not be understood as a general recommendationfor predicting fracture initiation after non-proportional loading.

3.6. Comment on model consistency

The basis of the proposed fracture initiation model is a localiza-tion criterion in stress space for proportional loading (Eq. (17)).This criterion is transformed from stress space to the mixedstrain–stress space using the material’s plasticity model. Thisapproach is considered as consistent in the sense that the link withthe underlying localization criterion in stress space is preserved. Inother words, the final HC fracture initiation model features onlyparameters that are associated with the localization criterion(17) in stress space.

The derivation of the so-called modified Mohr–Coulomb (MMC)model (Bai and Wierzbicki, 2010) is mathematically similar to theHC model, but it is usually used as inconsistent model. As opposedto the isotropic hardening law provided by Eq. (12), Bai andWierzbicki (2010) made use of a rather unconventional stress tri-axiality and Lode angle dependent hardening rule

k ¼ g½g; �h�h½�ep�; ð22Þ

to transform the Mohr–Coulomb criterion from stress space

�r ¼ �rf ½g; �h�; ð23Þ

to strain space,

�eprf ¼ h�1 �rf ½g; �h�

g½g; �h�

� �: ð24Þ

For most engineering materials, the hardening rule does not depen-dent on the Lode angle or the stress triaxiality. Consequently,g½g; �h� ¼ 1 must be used in (22) to describe the isotropic strain hard-ening. However, when applying the MMC model, the functiong½g; �h�is not set to unity in Eq. (24) even if g½g; �h� = 1 is assumed to modelstrain hardening (e.g. Bai and Wierzbicki (2010), Luo and Wierzbicki(2010), Beese et al. (2010), Dunand and Mohr, 2011a,b). This prac-tice is considered as inconsistent modeling. It is mostly done toobtain a better fit of the model to experimental data. In additionto the two Mohr–Coulomb parameters, the parameters describingthe function g½g; �h� can be adjusted to improve the predictions ofthe strain to fracture. The main difference between inconsistentand consistent modeling is that the latter approach preserves thelink with the underlying stress-based fracture initiation model.For proportional loading, the application of the HC model (21) pro-vides the same result as the direct use of the underlying stress-based criterion (17). In the case of the inconsistent MMC model,the application of Eq. (24) does not provide the same result as theMohr–Coulomb criterion in stress space (Eq. (23)). In other words,the link with the original Mohr–Coulomb criterion is lost whenusing the inconsistent MMC model.

Another particular feature of the present work is the use of anon-associated flow rule model to account for the anisotropy inthe plastic flow. It is emphasized that the hardening law relates

the von Mises stress to its work-conjugate equivalent plastic strain.As a result, the HC fracture model is an isotropic criterion in boththe stress and plastic strain space. In other words, the anisotropyof the flow rule does not alter the isotropic response of the fractureinitiation model.

4. Fracture experiments

We make use of the experimental data for three differentadvanced high strength steels to calibrate and validate the HC frac-ture initiation model.

4.1. Materials

Two Dual-Phase (DP) steels and one TRIP-assisted steel areconsidered:

� 1.4 mm thick DP590 steel sheets from ArcelorMittal� 1.0 mm thick DP780 steel sheets from US Steel� 1.4 mm thick TRIP780 steel sheets from POSCO

The experimental data for the TRIP780 steel is taken from theliterature (Dunand and Mohr, 2011a), while new experimentalresults are reported here for dual phase steels.

4.2. Uniaxial tension experiments

Tension experiments are performed on different specimengeometries. Dogbone specimens featuring a 10 mm wide gage sec-tion are used to characterize the material response for uniaxial ten-sion up to the onset of necking. Specimens are extracted alongthree different sheet directions (rolling, transverse and 45�-direc-tion). All experiments are performed at an axial strain rate of about10�3=s. Throughout the experiments, the in-plane displacementfields are monitored using planar Digital Image Correlation (DIC).In particular, the evolution of the width strain is determined as afunction of the axial strain using virtual extensometers of about9 and 20 mm length for the respective directions. After computingthe logarithmic plastic strains in the width and thickness direc-tions (assuming plastic incompressibility), the Lankford ratios aredetermined from the average slopes,

r ¼ depw

depth

: ð25Þ

Table 1 summarizes the measured Lankford ratios for all threematerials and material orientations. The axial true stress versus log-arithmic plastic strain curves for the DP steels are shown in Fig. 7.As reported by Mohr et al. (2010) for the TRIP780 steel and anotherDP590 steel, the axial stress–strain curves for different material ori-entations lie approximately on top of each other which motivatesthe use of an isotropic yield condition.

4.3. Other tensile experiments

In addition to uniaxial tension experiments, tension experi-ments are performed on specimens with circular cut-outs (notchedtension) and a central hole (‘‘CH’’-specimen). We use the labels‘‘NT6’’ and ‘‘NT20’’ to refer to specimens with notch radii of

Fig. 7. Measured true stress versus logarithmic plastic strain curve for uniaxialtension along different material directions up to the point of necking for (a) DP590,(b) DP780. Note that each graph shows the curves for three different specimenorientations (0�, 45� and 90�), but they lie exactly on top of each other and thus onlyone curve is visible.

D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55 49

R ¼ 6:67 mm and R ¼ 20 mm, respectively (Fig. 8). All specimenshad been extracted from the sheets using abrasive water-jet cut-ting, with the specimen tensile axis aligned with the sheet rollingdirection. The hole in the CH-specimens is introduced using CNCmachining to minimize the effect of the cutting technique on theonset of fracture at the hole boundaries (see Dunand and Mohr(2010)). The specimens are tested in a hydraulic universal testingmachine with custom-made high pressure clamps. All experimentsare performed under displacement control at a constant cross-headspeed of about 1 mm=min. The relative vertical displacement ofpoints positioned on the lower and upper specimen boundary ismeasured using a DIC-based virtual extensometer. The recordedforce–displacement curves are shown as solid dots in Figs. 9 and10. Note that only one curve is shown per experiment because ofthe remarkable repeatability of the experimental measurements.

4.4. Shear experiments

Butterfly specimens (Fig. 8d) are subject to tangential loading tocharacterize the fracture response for pure shearðg ffi 0, �h ffi 0). Thelatest specimen design as optimized by Dunand and Mohr (2011b)is used. It features clothoidally shaped specimen shoulders, convexlateral boundaries and a gage section of 0.5 mm thickness whilepreserving the original sheet thickness in the shoulder regions.The specimen is tested in a dual actuator system using the samehigh pressure clamps as for the above tension experiments. Theshear experiments are performed under combined displace-ment/force control, i.e. the horizontal actuator applies a constanttangential velocity of 0:2 mm=min while keeping the vertical forceequal to zero (see Mohr and Oswald (2008) for details on theexperimental procedure). The relative tangential and normal dis-placement of two points located on the upper and lower specimenshoulder is measured using DIC. The dotted curves in Figs. 9 and 10show the measured force–displacement curves for shear loading.All curves increase monotonically up to the point of fracture. Thelocation of onset of fracture is assumed to coincide with the loca-tion of the highest equivalent plastic strain within the gage section.

4.5. Punch experiment

Circular discs are extracted from the sheets for punch testing.A hemispherical punch of a diameter of 45 mm is used to applythe loading, while clamping the specimen on a 127 mm diameterdie. Four about 0.05 mm thick Teflon layers with grease are posi-tioned between the specimen and the punch to reduce the effectof friction. After clamping the specimens with sixteen M10screws, the experiments are performed at a constant punch veloc-ity. The experiments are stopped as soon as the punch forcereaches its maximum. Subsequently, the specimens are cut in halfto be able to measure the final specimen thickness at the apex ofthe punched specimens. The measured thickness reductions forthe DP590, DP780 and TRIP780 steels were 67%, 61% and 59%,respectively.

5. Identification of the loading path to fracture

5.1. Plasticity model parameter identification

The plasticity model requires the identification of the isotropichardening law and of the anisotropic flow potential function. Theparameters of the latter are uniquely determined from the threeLankford ratios r0, r45 and r90, using the analytical relationships

G12 ¼ �r0

1þ r0; G22 ¼

r0

r90

1þ r90

1þ r0and

G33 ¼1þ 2r45

r90

r0 þ r90

1þ r0: ð26Þ

The isotropic hardening law for the dual phase steels is identified ina two-step procedure. Firstly, we approximate the true stress versuslogarithmic plastic strain curve up to the point of necking using anexponential law (Voce, 1948),

kV ¼ k0 þ Q 1� e�b�ep� �

: ð27Þ

Secondly, the same experimental curve is approximated using apower law (Swift, 1952),

kSW ¼ A �ep þ e0� �n

: ð28Þ

As suggested by Sung et al. (2010), the final hardening curve isapproximated by the linear combination of the exponential andpower law,

Fig. 8. Specimen drawings for flat tension specimens with (a) a central hole, (b)–(c) different notches, and (d) the butterfly specimen for shear testing.

50 D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55

k ¼ ð1� aÞkV þ akS: ð29Þ

The weighting factor a plays an important role in the post-neckingrange and needs to be determined through inverse analysis. TheNT20 experiment is chosen to identify a. Unlike in dogbone speci-mens, the location of the through-thickness necking zone is prede-termined by the notched specimen geometry. Furthermore, themechanical system for the NT20 experiment does not lose any sym-metry in the post-necking range. Hence, the modeling of one eighthof the specimen without any artificial imperfections is sufficient forsimulating a notched tension experiment. We follow closely themodeling guidelines given by Dunand and Mohr, 2010: eight ele-ments in thickness direction and at least 105 explicit time steps,while using the user material subroutine developed by Mohr et al.(2010) for the non-associated quadratic plasticity model.

The identification of a is posed as a minimization problem. Forthis, we introduce the residual

w½a� ¼XNp

i¼1

FNUMi ½a� � FEXP

i

FEXPi

; ð30Þ

to quantify the difference between the simulated and measuredforce–displacement curve for notched tension (NT20), with FNUM

i

and FEXPi denoting the respective computed and measured forces

corresponding to the same displacement ui of the discrete experi-mental force–displacement curve with Np data points. After

minimizing w using a derivative-free simplex optimization algo-rithm, the values a ¼ 0:70 and a ¼ 0:79 are obtained for theDP590 and DP780 materials, respectively. A summary of all param-eters describing the isotropic hardening of the DP steels is given inTable 2.

The black solid lines in Figs. 9 and 10 show the simulated force–displacement curves for notched tension (NT20 and NT6), tensionwith a central hole (CH), and the butterfly shear specimens (SH).In the latter case, only one half of the butterfly specimen is mod-eled with symmetry boundary conditions applied to the specimenmid-plane (about 40,000 first-order solid elements). The goodagreement of the simulations (solid lines) and experiments (dots)partially validates the applicability of the calibrated plasticitymodel.

5.2. Loading paths to fracture

The calibration of a fracture initiation model requires knowl-edge of the stress and deformation history at the material pointwithin the specimen where fracture initiates:

For notched tension (NT6 and NT20) and shear loading (SH), theloading paths to fracture are extracted at the integration point ofthe element with the highest equivalent plastic strain. As can beseen from the contour plots in Figs. 9 and 10, this location corre-sponds to the very center of the NT specimens.

Fig. 9. Measured and simulated force displacement curves for selected fracture experiments on the DP590 steel. The star symbols represent the experimental curves, whilethe simulation results are shown as solid lines. A contour plot of the equivalent plastic strain at the onset of fracture is shown below each figure.

D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55 51

For tension with a central hole (CH), the loading path isextracted from the element on the specimen mid-plane that ispositioned at the root of the hole.

The punch experiment (PU) does not exhibit any necking andwe therefore assume that the stress state remains equi-biaxial ten-sionðg ¼ 0:67, �h ¼ �1) throughout the entire deformation history.According to the non-associated flow rule, the equivalent plasticstrain to fracture can be determined from the final thicknessreduction

�e fp ¼ �

ln½t=tini�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ G22 þ 2G12p : ð31Þ

The determined loading paths to fracture are shown as black solidlines in Fig. 11. Each row of figures corresponds to a different

material. The left plots show the evolution of the equivalent plas-tic strain as a function of the stress triaxiality, while the centerplots show the same evolution as a function of the Lode angleparameter. For tension with a central hole (CH), the stress triaxial-ity and Lode angle parameters remained more or less constant.Under notched tension (NT), the stress state is typically constantup to the onset of through-thickness necking; thereafter, an out-of-plane stress builds up which causes an increase in stress triax-iality and a decrease in the Lode angle parameter. The deformedmeshes at the instant of onset of fracture for notched tensionare shown next to the plots of the loading paths in Figs. 9 and10. The color contours are chosen such that the maximum value(red) corresponds to the equivalent plastic strain at the instantof onset of fracture.

Fig. 10. Measured and simulated force displacement curves for selected fracture experiments on the DP780 steel. The star symbols represent the experimental curves, whilethe simulation results are shown as solid lines. A contour plot of the equivalent plastic strain at the onset of fracture is shown below each figure.

Table 2Dual steel hardening law parameters.

k0 [MPa] Q [MPa] b [–] A [MPa] e0 [–] n [–] a [–]

DP590 345.9 335.8 24.9 1031.0 0.0013 0.2 0.70DP780 614.0 270.0 32.2 1170.0 3.1 � 10�5 0.11 0.79

52 D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55

6. Model calibration and verification

6.1. Calibration procedure

The HC model parameter identification is done through inverseanalysis. Denote the loading history for each calibration experi-ment i with gi ¼ gi½�ep� and �hi ¼ �hi½�ep�. For a given set of HC model

parameters v ¼ fa; b; cg, the strain to fracture �e fi ¼ �e f

i ½v� for an

experiment i is then determined based on Eq. (21). The formal min-imization problem for the parameter identification reads

v ¼ arg minv

maxi

�e fi ½v� � �eexp

i

� �with i 2 SEXP; ð32Þ

with SEXP denoting a set of calibration experiments. As for the isotro-pic hardening law identification, a derivative-free simplex algo-rithm is used to solve (32) in an approximate manner.

6.2. Model application

Among the loading paths shown in Fig. 11, we chose theresults from the experiments SEXP ¼ fSH; CH;NT6g to calibratethe three parametersfa; b; cg of the HC model. The instants ofonset of fracture predicted by the calibrated HC model are

Fig. 11. Comparison of the Hosford–Coulomb (HC) model predictions (blue dots) with experimental results (end point of the black lines) after calibration based on the SH, CHand NT6 experiments. The predictions of the Mohr–Coulomb (MC) model are shown as red dots. The units of the cohesion b are MPa. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55 53

depicted by blue dots in Fig. 11. The overall comparison of themodel predictions with the end points of the loadings paths(black lines) shows satisfactory agreement for all experimentsand for all three materials. The good agreement for SH, CH andNT6 demonstrates that the mathematical structure of the HCmodel is flexible enough to be fitted to three distinct loadingpaths. Differences between the model predictions and the exper-iments become apparent for the PU and NT20 experiments. Themodel error for punch loading is small for the DP590 steel whichfeatures a high Hosford exponent. For smaller Hosford exponents(see DP780 and TRIP780), the model becomes more sensitive tosmall variations in stress triaxiality in the vicinity of uniaxial ten-sion (g ¼ 0:33, �h ¼ 1:0). Consequently, small uncertainties in theloading path to fracture for CH have a strong effect on the iden-tified model parameters b and c (and hence on the results for PU).A more robust calibration is obtained when including the resultfrom the punch test in the calibration procedure, i.e.SEXP ¼ fSH;CH;NT6; PUg.

The results for the DP780 steel (second row in Fig. 11) elucidatethe effect of the Lode angle parameter. The calibration reveals onlylittle stress triaxiality dependence (c ffi 0), even though the plot ofthe loading paths in the equivalent plastic strain versus triaxialityplane (Fig. 11d) shows significant variations in the strain to frac-ture. According to our model, these variations are only due to thefact that the loading paths feature different Lode angle parameters(see Fig. 11e). The right-most column in Fig. 11 shows the under-lying localization criteria in mixed strain–stress space (Eq. (20)).Observe that their shape varies substantially as a function of thematerial, with a

� Mohr–Coulomb type of surface (a ffi 1) for the TRIP780 steel(both stress triaxiality and Lode angle dependent),� Hosford type of surface (c ffi 0) for the DP780 steel (Lode angle

dependent only)� Mises type of surface (a ffi 2, c ffi 0) for the DP590 steel (nearly

stress state independent).

54 D. Mohr, S.J. Marcadet / International Journal of Solids and Structures 67–68 (2015) 40–55

This result illustrates not only the flexibility of the HC model,but also the importance of validating fracture initiation modelsfor different materials.

To expose the effect of the Hosford extension of the Mohr–Coulomb model, an attempt is made to describe the experimentaldata using the MC model. The model calibration is thus repeatedfor the special case of a ¼ 1 using the same experimental database,SEXP ¼ fSH; CH;NT6g. The corresponding predictions (MC model)are depicted as red solid dots in Fig. 10. Its approximation of thecalibration experiments is only satisfactory for the TRIP780 steel.For the DP steels, the MC model significantly underestimates thestrain to fracture for notched tension. The calibration for NT6 doesnot improve when using two calibration experiments only (e.g.SEXP ¼ fCH;NT6g). This is due to the fact that the two-parameterMC model reduces to a one-parameter model for biaxial tension.In other words, the MC model provides only a poor description ofexperimental data for biaxial tension.

7. Conclusions

A fracture initiation model is proposed to predict the onset offracture in advanced high strength steels at low stress triaxialities.Assuming that the onset of ductile fracture is imminent with theformation of a band of localization at the mesoscale or microscale,a stress triaxiality and Lode angle parameter dependent fractureinitiation model is formulated based on the results from a fullythree-dimensional localization analysis. The resulting model is anextended Mohr–Coulomb model which makes use of the Hosfordequivalent stress and the normal stress acting on the plane of max-imum shear to predict the onset of fracture. A consistent transfor-mation from the principal stress space to the space of equivalentplastic strain, stress triaxiality and Lode angle parameter is per-formed to obtain a fracture initiation model for non-proportionalloading. The particular feature of this transformation is that itmakes use of the material’s isotropic hardening law and thereforepreserves the physical meaning of the underlying stress-basedlocalization model.

Experimental results are reported from ductile fracture experi-ments on three different advanced high strength steel sheets(DP590, DP780 and TRIP780). The experimental program includesa shear experiment, tension with a central hole, notched tensionand a punch experiment, thereby covering a wide range of stressstates. The comparison of experiments and simulations showsgood agreement of the Hosford–Coulomb model predictions withall experiments for all materials. It is worth noting that withoutany shear localization analysis results at their disposal,Stoughton and Yoon (2011) and Bai and Wierzbicki (2010) hadalready correctly hypothesized on the existence of a Mohr–Coulomb type of criterion for predicting the onset of ductile frac-ture. However, the present analysis shows that the direct use of aMohr–Coulomb criterion systematically underestimates the strainto fracture for biaxial loading, while it can be predicted with greataccuracy when using the Hosford–Coulomb model.

Acknowledgments

The partial financial support of MIT Industrial FractureConsortium is gratefully acknowledged. Thanks are due toMatthieu Dunand for valuable discussions and providing the datafor the TRIP780 steel. Thanks are also due to Fabien Ebnoetherfor his contributions to the development of the Hosford–Coulomb fracture initiation model.

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