Ductile fracture of aluminum 2024-T351 under proportional and …mohr.ethz.ch/papers/47.pdf · 2016-09-20 · Bao and Wierzbicki (2004) performed experiments on specimens of different

International Journal of Solids and Structures 69–70 (2015) 459–474

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International Journal of Solids and Structures

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Ductile fracture of aluminum 2024-T351 under proportionaland non-proportional multi-axial loading: Bao–Wierzbicki resultsrevisited

http://dx.doi.org/10.1016/j.ijsolstr.2015.05.0060020-7683/� 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: [email protected] (D. Mohr).

Jessica Papasidero a, Véronique Doquet a, Dirk Mohr a,b,⇑a Solid Mechanics Laboratory (CNRS-UMR 7649), Department of Mechanics, École Polytechnique, Palaiseau, Franceb Impact 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 a b s t r a c t

Article history:Received 30 December 2013Received in revised form 24 February 2015Available online 16 May 2015

Keywords:Ductile fractureStress triaxialityLode angleNon-proportional loadingTension–torsion

The effect of stress state and loading path on the ductile fracture of aluminum 2024-T351 is characterizedthrough tension–torsion experiments on tubular specimens. The experimental program includes propor-tional and non-proportional loading paths leading to the onset of fracture at nearly plane stress condi-tions at stress triaxialities between 0 and 0.6. Stereo digital image correlation is used to measure thedisplacements and rotations applied to the specimen shoulders. An isotropic non-quadratic Hosford plas-ticity model with combined Voce–Swift hardening is used to obtain estimates of the local stress andstrain fields within the specimen gage section. The hybrid experimental–numerical results indicate ahigher strain to fracture for pure shear than for uniaxial tension. The calibration of a Hosford–Coulombfracture initiation model suggests that the ductility of aluminum 2024-T351 decreases monotonicallyas a function of the stress triaxiality, whereas it is a non-symmetric convex function of the Lode angleparameter. It is shown that a simple non-linear damage accumulation rule can describe the effect ofnon-proportional loading on the strain to fracture.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The work of Bao and Wierzbicki (2004) has contributed signifi-cantly to the development of ductile fracture initiation models thatdepend on the Lode angle in addition to the stress triaxiality. Baoand Wierzbicki (2004) performed experiments on specimens ofdifferent geometries to quantify the effect of stress triaxiality onthe equivalent strain at the onset of fracture for aluminum2024-T351. Their results suggested that the strain to fracture is anon-monotonic, non-smooth function of the stress triaxiality witha local maximum at a stress triaxiality of about 0.4 and a local min-imum at a stress triaxiality of zero (pure shear). A non-monotonic,non-smooth relationship between the strain to fracture and thestress triaxiality had also been reported by Barsoum and Faleskog(2007) for a Weldox steel after completing a series of tension–tor-sion experiments.

The apparent decrease in ductility for shear-dominated loadinghas been particularly intriguing since conventional Gurson-type of

models predict a monotonically decreasing strain to fracture as afunction of the stress triaxiality. In the shear-modified Gursonmodel proposed by Nahshon and Hutchinson (2008), the evolutionof the void volume fraction is not only a function of the equivalentplastic strain, but also sensitive to the Lode number. As a result, thismodel can predict fracture through shear and normal localization atvery low stress triaxialities. Danas and Ponte Castaneda (2012)made use of a porous plasticity model obtained throughnon-linear homogenization that accounts for the Lodeangle-dependent evolution of the porous mesostructure in anapproximate manner. Their estimates suggest a reduction in ductil-ity as the stress triaxiality decreases to zero due to collapsing voids.

The experimental data of Bao and Wierzbicki (2004) has beenexploited to develop stress-based fracture initiation models.Stoughton and Yoon (2011) made use of their data to show thata Mohr–Coulomb criterion can describe the onset of fracture instress space. Another criterion in stress space was proposed byKhan and Liu (2012) after supplementing Bao and Wierzbicki’s(2004) data with additional experimental data points. Bai andWierzbicki (2010) postulated a Mohr–Coulomb fracture model instress space to come up with the corresponding damage indicatormodel in the combined strain–stress space of equivalent plasticstrain, stress triaxiality and Lode angle parameter. A

460 J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474

micro-mechanism inspired ductile fracture model was proposedby Lou and Huh (2013). Their model had also been partially vali-dated using the data reported for aluminum 2024-T351 by Baoand Wierzbicki (2004).

The non-uniformity of the stress and strain fields in ductile frac-ture experiments requires a hybrid experimental–numerical anal-ysis of the measurements (e.g. Mohr and Henn, 2007; Barsoumand Faleskog, 2007; Dunand and Mohr, 2010; Faleskog andBarsoum, 2013). The data of Bao and Wierzbicki (2004) for alu-minum 2024-T351 was obtained using a J2-plasticity model withpower-law hardening. As demonstrated for a variety of aluminumalloys (e.g. Barlat et al., 1991; Bron and Besson, 2004; Giagmouriset al., 2010), non-quadratic yield functions and their anisotropiccounterparts appear to be more suitable for describing the plasticresponse of polycrystalline FCC materials. Another source of uncer-tainty in the hybrid experimental–numerical analysis of fractureinitiation experiments is the speculation on the location of incipi-ent fracture. For instance, as alluded to by Nahshon andHutchinson (2008), in case significant barreling occurs in an upset-ting test, fracture may not start under uniaxial compression at thespecimen center, but under biaxial loading at the specimen surface.

Ductile fracture under proportional loading has been the sub-ject of numerous recent investigations (e.g. Gao et al., 2010;Nielsen and Tvergaard, 2010; Lecarme et al., 2011; Gruben et al.,2012; Graham et al., 2012; Lou et al., 2013; Faleskog andBarsoum, 2013; Xue et al., 2013). However, only little is knownon the effect of non-monotonic or non-proportional loading pathson the onset of ductile fracture. Bao and Treitler (2004) performedreverse loading experiments on notched axisymmetric barAA2024-T351 specimens with compression followed by tensionall the way to fracture. They observed a substantial increase in duc-tility due to pre-compression. Bai (2008) discusses the phe-nomenological modification of the accumulation rule of damageindicator models to account for the effect of non-proportional load-ing histories on the onset of ductile fracture. He also shows impor-tant effects of non-linear loading paths on ductile fracture duringthe crushing of prismatic columns. Benzerga et al. (2012) per-formed axisymmetric unit cell computations to demonstrate thatthe strain to fracture is strongly path-dependent and cannot berepresented as a function of the stress state only.

The effect of the loading path on ductile failure has been inves-tigated in detail in the context of sheet metal forming (e.g.Kleemola and Pelkkikangas, 1977; Wagoner and Laukonis, 1983;Hosford and Caddell, 2014). It is now well established that the fail-ure strains, i.e. the strains at the onset of localized necking, arestrongly loading path dependent. More recent studies onbi-axially loaded aluminum tubes (Korkolis et al. (2010), Korkolisand Kyriakides, 2009; Korkolis and Kyriakides, 2011a; Korkolisand Kyriakides, 2011b) insinuate a loading path dependency ofthe forming liming diagram (FLD) in both strain and stress space.Furthermore, they confirmed the non-quadratic form of the yieldsurface of aluminum. The results of Korkolis and Kyriakides(2009) provide also some insight into the loading path dependencyof ductile fracture. In a first approximation, the stress state in theirradial loading experiments may be considered as constant up tothe point of fracture initiation (rupture). In their experiments withcorner loading paths, the specimens were preloaded under uniaxialtension, before continuously increasing the stress triaxialitythroughout plastic loading up to that of selected radial loadingexperiments. Due to the isotropic hardening of the tested Al6220-T4 alloy, it may already be inferred from their plot of the rup-ture points in stress space that a higher equivalent plastic straincould be achieved in the non-proportional loading experiments(e.g. compare the limit stresses shown in Fig. 3 of their paper forradial loading at rx=rh ¼ 0:75, with those for x! h corner loadingwith path change at the axial failure stress for rx=rh ¼ 0:9; both

experiments feature a stress triaxiality of about 0.65 at the instantof rupture).

The work of Haltom et al. (2013) is directly concerned with themeasurement of the strain to fracture in tension–torsion experi-ments on aluminum 6061-T6 tubes with uni-lateral thicknessreduction. Their surface strain measurements suggest a monotonicdecrease of the equivalent plastic strain as a function of the stresstriaxiality for plane stress states between pure shear and uniaxialtension, thereby challenging Bao and Wierzbicki’s (2004) qualita-tive observation of a pronounced increase in ductility of aluminum2024-T351 over the same range of stress states. Even thoughFaleskog and Barsoum (2013) deal with Weldox steel only, it isworth noting that their most recent experimental results from ten-sion–torsion experiments also suggest a decrease in ductility overthe stress triaxiality interval [0,0.33]. Similarly, our tension–tor-sion experiments on a 36NiCrMo16 high strength steel(Papasidero et al., 2014b) revealed a higher ductility for pure shearthan uniaxial tension.

The aim of the present paper is twofold: firstly, in an attempt toexplain the apparent qualitative contradictions in existing experi-mental data, we characterize the effect of stress state on the equiv-alent plastic strain to fracture for proportional loading foraluminum 2024-T351 through tension–torsion experiments.Special attention is paid to the measurement of the strains throughstereo digital image correlation and the numerical analysis of allexperiments using a non-quadratic Hosford plasticity model.Secondly, we make use of the tension–torsion technique to per-form experiments for non-proportional loading paths featuringtwo distinct ranges of constant stress state. Aside from reportingand discussing all experimental results, the Hosford–Coulombdamage indicator model (Mohr and Marcadet, 2015) is employedto predict the initiation of ductile fracture for proportional andnon-proportional loading.

2. Experiments

We closely follow the experimental procedure for tension–tor-sion fracture testing proposed by Papasidero et al. (2014b). Using adual actuator system, selected combinations of tension and torsionloading are applied to a stocky tubular specimen.

2.1. Material and specimen

All specimens are extracted from the same square bar of alu-minum 2024-T351 supplied by the distributor McMaster Carr. A3-ft long bar with a 400 � 400 square cross-section is chosen insteadof a plate or tube product in an attempt to minimize the initial ani-sotropy. EBSD analysis of a 6.3 mm � 4.2 mm large area (which cor-responds to about 1000 grains) reveals an isotropic texture (Fig. 1).Possible variations in material properties within the bar sectionhave not been quantified. The full bar cross-sections was cut intofour 1 � 100 sections prior to extracting the tubular specimens.

The geometry of the stocky tubular specimen is shown inFig. 2(a). The specimen axis is always parallel to the longitudinaldirection of the bar. The specimen features an inner diameter of20 mm. The wall thickness is 2 mm in the shoulder areas, whileit is only eini ¼ 1 mm within the 2 mm high gage section. The tran-sition from the 22 mm to the 24 mm outer diameter is achievedthrough a toroidal surface of 1 mm radius. A surface roughness ofRa0.1 is specified for the inner and outer surfaces of the gage sec-tion, along with a coaxiality tolerance of 0.02 mm. Micrometermeasurements of the inner specimen diameter prior to testingreveal variations of �10 lm from specimen to specimen. The ver-ification of the external diameter of the gage section throughstereo DIC revealed variations of �25 lm.

{100} {110}

high

low

Fig. 1. Pole figures for aluminum 2024-T351 obtained from the EBSD analysis of a1 mm � 1 mm large area (about one million points) with the bar axis as normaldirection. The color contour shows the normalized frequencies for the {100} and{110} planes of the polycrystalline FCC material.

J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474 461

2.2. Experimental procedure

The specimens are attached to the testing machine using annu-lar pressure clamps. The mechanical connection between the test-ing machine and the specimen is established through frictionalstresses between the contact surfaces only. The grips (part s inFig. 2(c)) have an annular cavity with the specimen (part r) beinginserted on a central cylindrical rod that is integral to part s. Thealignment of the upper and lower grips is verified prior to testingusing a short tube of constant wall thickness. After degreasing allsurfaces, the specimen is inserted. Due to the tight fit of the spec-imen into the grips (no longer movable by hand), both actuatorscan already be put into a force-control mode to avoid any strainingof the specimen gage section during set-up. Subsequently, theouter pressure ring (parts t and u) is tightened to apply a normalpressure to all connecting surfaces. Note that shear stresses aretransmitted between the specimen and the grips through boththe inner and outer specimen surfaces.

A multi-axial load cell (TemaConcept) with a maximum axialcapacity of 100 kN and a maximum torque of 600 Nm is used torecord the force F and torque MT at the specimen ends. The loadingis characterized through the biaxial loading angle b,

tan b :¼ FRMT

ð1Þ

Fig. 2. (a) Specimen geometry, (b) axisymmetric FE model (c) specimen clamps with speeight screws around the circumference.

with R denoting the gage section radius. The history of b is con-trolled by the actuator control algorithm.

2.3. Stereo digital image correlation

The displacement fields are measured by stereo-correlation ofdigital images (3DIC). Two cameras (F505B, model PIKE,2452 � 2054 pixel sensor) equipped with 90 mm lenses (Tamron,model SP AF 90 mm F/2.8 Di) are used to monitor the specimensurface from two different radial perspectives at a frequency of1 Hz. The cameras are positioned at an angle of about 16� withrespect to each other, and at a distance of about 600 mm fromthe specimen axis. An F11 aperture is used. In addition to the gagesection, the relative displacement of the specimen shoulders is alsomonitored with 3DIC. Except for the area that is inserted into themachine clamps, a thin layer of white paint is applied on the spec-imen surface along with a fine black speckle paint pattern. Theaverage size of a black speckle on the specimen surface is about200 lm which corresponds to about eight pixels. The softwareVIC3D (CorrelatedSolutions, Columbia, SC) is used to measure dis-placement fields, assuming an affine transformation of a 21 � 21pixel subset (24 lm/pixel). In the following, the ‘‘global’’ displace-ment and rotation, uðtÞ and hðtÞ are reported, which describe therelative motion of the points B and B0 on the specimen shoulders.As illustrated in Fig. 2(a), these points are initially positioned at avertical distance of 8 mm from the gage section center.

2.4. Measured force–displacement curves

Two series of experiments are performed: (1) proportional load-ing experiments (monotonic loading with b ¼ const: all the way tofracture), and (2) non-proportional loading experiments (twosequential proportional loading segments with different values ofb). Fig. 3 shows a summary of the loading paths in terms of the rel-ative shoulder displacement u and rotation h (left column) and theaverage normal stress versus average shear stress (right column).

The testing program for the first series of experiments(Fig. 3(a) and (b)) covered seven distinct biaxial loading angles:b ¼ 0�, b ¼ 21:9�, b ¼ 34:1�, b ¼ 45�, b ¼ 55�, b ¼ 69:5� andb ¼ 90�. The specific actuator control settings were:

cimen (1), male connector (2), sliding wedges (3), and M5 clamping screws (4), with

(b)(a)

(d)(c)

β=34.1°

β=0°

β=45°

β=90°

β=21.9°

β=55°

β=69.5°

12

3

3

β=55°

β=0° 1

2

β=55°

β=90°

12

3

β=0°

β=55°

pre-comp pre-torsion pre-tension

3

β=90°

Fig. 3. Prescribed loading paths in the experiments: axial displacement as a function of the rotation (left column), and average normal stress as function of the average shearstress in the gage section (right column) for (a) and (b) proportional loading, and (c) and (d) non-proportional loading. The end points of all curves correspond to the instantsof fracture initiation. In (d), these have been highlighted through solid dots for non-proportional loading, and compared with the results for proportional loading (crosssymbols).

462 J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474

� for b = 0�, (pure shear) the rate of rotation is (1.14�/min) whilezero axial force is imposed.� for 21.9� 6 b 6 55� (shear-dominated), the rate of rotation is

prescribed (1.14�/min) while the axial position is incrementallyadjusted to keep b constant.� for b = 69.5�, (tension-dominated), the rate of axial displace-

ment is prescribed (0.12 mm/min) while the rotational positionis incrementally adjusted to keep b constant.� for b = 90� (transverse plane strain), the rate of axial displace-

ment is prescribed (0.12 mm/min) while zero torque is imposed.

The non-proportional loading experiments (Fig. 3(c) and (d))involved two proportional loading steps:

� Pre-loading (path r in Fig. 3(d)): the specimen is loaded eitherin compression (b ¼ �90�), tension (b ¼ þ90�) or torsion(b ¼ 0�) using the same actuator settings as for proportionalloading. The pre-loading is monitored by stereo DIC, and isstopped when the equivalent plastic strain on the symmetryplane of the gage section surface reaches approximately 0.07.� Elastic unloading (path s in Fig 3(d)): the specimen is unloaded

to zero force and torque.� Re-loading (path t in Fig. 3(d)): The pre-deformed specimens

are re-loaded all the way to fracture under proportional loadingfor either torsion (b ¼ 0�), uniaxial tension ðb ¼ 55�Þ or planestrain tension (b ¼ 90�). As during the first phase of loading,the corresponding actuator setting for proportional loading isused.

In sum, seven non-proportional experiments are performed. Tohighlight the effect of the stress path on failure (i.e. fracture initi-ation) in stress space, we included cross symbols in Fig. 3(d) thatshow the average stresses at failure in the monotonic experimentsnext to solid dots that show the instant of failure afternon-proportional loading. It can already be tentatively inferredfrom the comparison of the crosses and solid dots thatpre-compression increased the material ductility while the oppo-site holds true for pre-tension.

The measured force–displacement and torque-rotation curvesare plotted as dashed lines in Fig. 4 (proportional loading) andFig. 5 (non-proportional loading). For the latter, the pre-loadingcurves (pre-compression, pre-torsion or pre-tension) are notshown as these match the corresponding recordings for propor-tional loading shown in Fig. 4.

All force–displacement curves increase monotonically, and dropabruptly with the onset of specimen fracture. No sign of pro-nounced necking prior to fracture is observed in any of the exper-iments performed. Note that fracture initiates in all experimentsbefore reaching a maximum in axial force or torque. The forceand torque measurements exhibit a Portevin–Le-Chatelier type ofserration as the gage section begins to deform plastically forb ¼ 69:5� and b ¼ 90�.

3. Hybrid experimental–numerical results

As discussed in Papasidero et al. (2014b), the direct estimationof the stress state and the strains to fracture based on surface strain

Fig. 4. Force–displacement and torque-rotation curves for proportional loading(dashed lines = experiments; solid lines = numerical simulations).

J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474 463

measurements does not always provide a good approximation ofthe loading path to fracture. As an alternative, we perform a finiteelement simulation of each experiment to obtain estimates of theloading paths to fracture.

3.1. Plasticity model

Considering the apparent isotropic initial texture and the mod-erate strains involved in the fracture experiments, an isotropicplasticity model is employed to describe the material response.Formally, the yield surface is defined in terms of the Hosford(1972) equivalent stress �rHf and a deformation resistance k,

f ðrÞ ¼ �rHf � k ¼ 0 ð2Þ

with

�rHf ¼12ððr1 � r2Þp þ ðr1 � r3Þp þ ðr2 � r3ÞpÞ

� �1=p

: ð3Þ

In the definition of the Hosford equivalent stress, r1, r2 and r3

are the ordered principal values of the Cauchy stress tensor; p isthe Hosford exponent which is a real within the interval1 6 p 61. For the two limiting cases p ¼ 1 and p ¼ 1, theTresca yield surface is retrieved, while for p ¼ 2 and p ¼ 4, theMises yield surface is retrieved (Fig. 6(a)).

Isotropic hardening is represented through a linear combinationof the Swift and Voce laws,

k½�ep� ¼ Aðeo þ �epÞn þ K þ Qð1� exp½�b�ep�Þ ð4Þ

with the Swift parameters fA; e0;ng and the Voce parametersfK;Q ; bg.

3.2. Finite element model

The experiments are simulated using Abaqus Standard 6.9EF2in conjunction with the Zmat material library (Centre deMateriaux, Ecole des Mines, France). The finite element model isshown in Fig. 2(b). The specimen portion between B and B0 is dis-cretized using four-node generalized axisymmetric elements(CGAX4), which account for twist around the z-axis. Based on astudy of the mesh-convergence with respect to the local deforma-tion fields, an element size of 25 lm is chosen within the gage sec-tion. The axial and rotational displacement of two reference nodes(N1 and N2 in Fig. 2(b)) is coupled to those of the top and bottomsurface of the model. Displacements and loads are applied to thenode N1, while node N2 is fully clamped.

One hundred implicit time steps are performed to simulate pro-portional loading. Closely mimicking the experimental procedure,the boundary conditions for proportional loading are applied:

� for b = 0�, h is imposed on N1 as measured by 3DIC, while zeroaxial load is imposed on N1;� for 21.9� 6 b 6 55�, the rotation h is imposed on N1 as measured

by 3DIC. A numerical sensor is created to get the value of thetorque, and the UAMP subroutine is used to apply the correctproportion of force on N1.� for b = 69.5�, the displacement u is imposed on N1 as measured

by 3DIC. A numerical sensor is created to get the value of theforce; the UAMP subroutine is used to apply the correct propor-tion of torque on N1.� for b = 90�, the displacement u is imposed on N1 as measured by

3DIC, while zero torque is imposed on N1.

For non-proportional loading, 100 implicit time steps are per-formed during each segment of proportional loading. Recall that thefirst phase involves pre-compression, pre-torsion or pre-tensionand then unloading, applying either u (pre-compression orpre-tension) or h (pre-torsion) as measured by 3DIC. Then, for thesecond segment, the boundary conditions are applied similarly tothose of proportional loading for b = 0�, b = 55� and b = 90�.

3.3. Identification of the plasticity model parameters

The seven plasticity model parameters fp;A; e0;n;K;Q ; bg areidentified through inverse analysis by minimizing the differencebetween the measured and computed force–displacement andmoment-rotation curves for proportional loading. The optimizationis performed by coupling the Abaqus solver with the optimizationtoolbox of Matlab. The final set of parameters corresponding tothe hardening curve shown in Fig. 6(b) is listed in Table 1. Goodagreement of the experimental curves (dashed lines) and the simu-lation results (solid lines) is observed for all biaxial loading anglesfor proportional loading (Fig. 4). Furthermore, despite the simplic-ity of the hardening model, the model also provides a reasonableapproximation of the global specimen response tonon-proportional loading (compare solid and dashed lines in Fig. 5).

3.4. Location of onset of fracture

It is assumed that a sudden drop in the measured force–dis-placement curve indicates the instant of fracture initiation. Anattempt is made to identify the location of onset of fracture basedon (1) photographs of the specimens right after the onset of frac-ture (Fig. 7), and (2) SEM pictures of the fracture surfaces (Fig. 8).In particular, abrasive marks on the latter are instrumental in iden-tifying the location of onset of fracture. For proportional loadingwithin the range 0� 6 b 6 45�, and for selected cases of

Fig. 5. Force–displacement and torque-rotation curves for non-proportional loading (dashed lines = experiments; solid lines = numerical simulations).

464 J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474

non-proportional loading (b = 0� after pre-compression, b = 0� afterpre-tension, b = 55� after pre-torsion), the upper and lower parts ofthe specimen remained partially attached despite the abrupt loaddrop. For all other tests, the specimens separated immediately(within less than 1 s) into two parts at the instant of fracture initi-ation. With regards to the location of onset of fracture, the follow-ing observations are made:

� Fracture initiated near the (toroidal) transition fillet between thecylindrical gage section and shoulder regions for 0� 6 b 6 34.1�(Fig. 7(a)). Furthermore, SEM observations provide evidence thatthe crack propagated earlier near the outer surface, where clearabrasive marks can be seen; the inner surface was the last to fail,as shear dimples can still be seen near the inner radius(Fig. 8(a)). Experimental evidence for crack initiation near thefillet radius is also found for b = 0� after pre-tension (Fig. 7(b))and after pre-compression, as well as for b = 55� afterpre-tension (Fig. 7(c)) and after pre-compression.� For proportional loading within the range 45� 6 b 6 69.5�, the

crack meanders within a narrow band around the symmetryplane (Fig. 7(d)). Abrasive marks as well as dimples are visibleon many regions of the fracture surface for proportional loadingwith b = 45� and 55�; however, their apparent random distribu-tion does not allow for the determination of the radial coordi-nate of the location of onset of fracture (Fig. 8(b)). Almost noabrasive marks are seen for b = 69.5�.� For proportional loading with b = 90�, the crack meanders

within a narrow band near the upper fillet (Fig. 7(e)). Almostno abrasive marks are visible on the fracture surface (Fig. 8(c)).

� For loading with b = 90� after pre-compression and afterpre-torsion, the crack meanders within the entire gage section(i.e. between the upper and lower fillet, see Fig. 7(f)) with no vis-ible abrasive marks on the SEM pictures of the fracture surface.� For b = 55� loading after pre-torsion, the crack meanders

between the lower fillet and the symmetry plane (Fig. 7(g)).

The corresponding spatial distributions of the equivalent plasticstrain within the specimen gage section as estimated through finiteelement analysis are shown in Fig. 9 for all experiments performed.Observe that the locations of onset of fracture identified experi-mentally (which was possible for b = 0� proportional andnon-proportional, and for b = 55� after pre-tension andpre-compression) coincide with the locations of maximum equiva-lent plastic strain in the simulations. For each experiment, theloading path to fracture is therefore extracted at the integrationpoint which featured the highest equivalent plastic strain at theinstant of onset of fracture. The corresponding points are high-lighted in Fig. 9 using solid black dots.

3.5. Surface strain fields obtained from stereo DIC

The position of the optical system had been chosen for the pur-pose of measuring the relative motion of the specimen shoulders.However, owing to the high resolution of the camera system, thedisplacement fields within the gage section can still be determinedwith a spatial resolution of about 25 lm using the same images.This resolution is not sufficient to obtain reliable strain

Fig. 6. (a) Yield surfaces in the P-plane; (b) identified stress–strain curve foraluminum 2024-T351.

J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474 465

measurements at the grain level, but it provides rough estimates ofthe strain variations from grain-to-grain.

Based on the displacement field provided by the VIC3D algo-rithm (Version 2010, Correlated Solutions), we computed the sur-face deformation gradient using a finite difference algorithmwithout any artificial smoothening. Furthermore, we computedthe logarithmic surface strains (ezz, ehh and ehz) and the effectivevon Mises strain through the direct application of the von Misesequivalent strain definition to the logarithmic strain tensor,

�eeff ¼2ffiffiffi3p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

zz þ e2hh þ ezzehh þ e2

hz

q: ð5Þ

The effective strain field within a central area of 1 mm � 1 mmof the gage section is shown in Fig. 10 as a function of the initialsurface position coordinates fXh ¼ 2HR; Zg for three selected pro-portional loading experiments:

� Torsion (b ¼ 0�, g ffi 0): the effective strain as averaged over the1 mm � 1 mm area is h�eeff i ¼ 0:30 at the instant of specimenfracture. The deformation map shows a very heterogeneousdeformation field with a standard deviation of 0.075; the

Table 1Plasticity model parameters identified.

p (–) A (MPa) eo (–) n (–) K (MPa) Q (MPa) b (–)

6.6 357.8 0.00795 0.189 149.5 105.2 18.9

corresponding coefficient of variation (standard deviation nor-malized by the mean) is 24%. The deformation appears to local-ize in individual grains (see representative grain structure inFig. 10(d) for reference) with effective strains as high as 0.85in the most highly strained grains.� Uniaxial tension (b ¼ 55�, g ffi 0:33): the average effective strain

to fracture is 0.23 for this type of loading. As for torsion, weobserve a locally heterogeneous strain field, but with a signifi-cantly smaller coefficient of variation of 15%. The maximumlocal strain observed in the area shown in Fig. 10(b) is 0.40.� Plane strain tension (b ¼ 90�, g ffi 0:58): here, the average sur-

face strain within the central gage section area is 0.08 at theinstant of specimen fracture. The surface strain fields exhibitsa normalized standard deviation of 12% with a maximum valueof 0.10 within the 1 mm � 1 mm area shown in Fig. 10(c).

Visual inspection revealed that the intensity of surface wrin-kling associated with the orange skin of the deformed specimensseems to coincide with the locations of strain localization identi-fied from 3DIC. The measurement of highly heterogeneous strainfields is in qualitative agreement with the microscopic observa-tions of others for aluminum alloys (e.g. Khan and Liu, 2012;Ghahremaninezhad and Ravi-Chandar, 2012; Haltom et al.,2013). A more detailed discussion of microscopic observations foraluminum 2014-T351 can be found in Papasidero et al. (2014a)

Recall that according to our observations and analysis, fractureinitiated either near the toroidal fillet or on the inner specimensurface (Fig. 9, the only exception is b ¼ 45�). The measured surfacestrains at the gage section center must therefore be interpreted aslower bounds, i.e. it can only be concluded that fracture initiatesunder proportional loading at a strain that is greater than the mea-sured surface strain (for the corresponding stress state). A directcomparison of the measured average effective von Mises strainsat the onset of specimen fracture and those computed on the outersurface at the specimen center is provided by Table 2. The remark-able agreement may be seen as another validation of the hybridexperimental–numerical procedure used in this study.

It is emphasized that the fracture strains identified in the fol-lowing subsections are macroscopic strains, i.e. average strainsover a population of grains.

3.6. Loading paths to fracture

The loading path to fracture describes the evolution of theequivalent plastic strain as a function of the stress state at the loca-tion where fracture initiates (or is assumed to initiate). For an iso-tropic material, the stress state can be conveniently characterizedby the stress triaxiality g and the Lode angle parameter �h,

g ¼ I1

3ffiffiffiffiffiffiffi3J2

p ð6Þ

and

�h ¼ 1� 2p

acos3ffiffiffi3p

2J3

ðJ2Þ3=2

" #ð7Þ

with the stress tensor invariants

I1 ¼ tr½r�; J2 ¼12

s : s and J3 ¼ det½s�

and s denoting the deviatoric part of the Cauchy stress tensor.As mentioned in the previous subsection, the evolutions of g, �h

and �ep are extracted at the integration point where the Hosfordequivalent plastic strain is maximal. Fig. 11 depicts the corre-sponding loading paths to fracture for proportional loading in theðg; �epÞ- and ð�h;�epÞ-planes. Observe that the stress state remains

(a) proportional β=0°

(c) β=55° after pre-tensi

(e) proportional β=90°

(g) β=55° with pre-torsi

(b) β=0° after pre-compr

on (d) proportional β=5

(f) β=90° with pre-tors

on

ession

5°

ion

Fig. 7. Photographs of selected specimens at the instant of onset of fracture.

466 J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474

remarkably constant throughout the experiments. The highestductility (i.e. strain to fracture) is achieved for b = 0� (�ef ¼ 0:29).The lowest ductility is observed for b ¼ 90� ð�ef ¼ 0:13Þ. Betweenthe limit stress states of pure shear ðb ¼ 0�Þ and plane strain ten-sion ðb ¼ 90�Þ, the present results suggest a local maximum in duc-tility for uniaxial tension (b ¼ 55�, �ef ¼ 0:24).

The loading paths for non-proportional loading (Fig. 12) eluci-date the effects of pre-compression, pre-torsion and pre-tensionon the strain to fracture for pure shear ðb ¼ 0�Þ, uniaxial tensionðb ¼ 55�Þ, and plane strain tension ðb ¼ 90�Þ:

� Applying pre-compression increases the material ductility in allcases. For b = 0�, the strain to fracture increases from 0.29 (pro-portional loading) to 0.32 (after pre-compression). For b ¼ 55�,it increases from 0.24 to 0.27. The most significant increase isobserved for b ¼ 90�: the ductility increases from 0.13 to 0.24,i.e. by more than 80%.� Applying pre-torsion also appears to increase the ductility: from

0.24 to 0.29 for b = 55�, and from 0.13 to 0.18 for b = 90�.� Applying pre-tension seems to decrease the ductility. The strain

to fracture decreases from 0.29 to 0.25 for b = 0�, and from 0.24to 0.19 for b = 55�.

The loading path to fracture for b ¼ 55� after pre-torsion isshown as a dashed line in Fig. 12(b) to indicate a lack of certaintyin this hybrid experimental–numerical result. Fig. 5(c) shows a

substantial difference in the hardening of the measured and com-puted force–displacement curves (compare solid and dashed redlines). In particular, the simulation curve exhibits a maximum priorto the onset of fracture which is an indicator of localization withinthe gage section. Consequently, the strain to fracture reported inFig. 12(b) for this loading case is expected to be too large. Anotherparticular feature of this experiment is that the maximum equiva-lent plastic strain is reached away from the specimen surface, andnot near the fillet, like for all other tests. The loading path to frac-ture therefore was extracted at a point where the pre-loading hadnot yet reached an equivalent plastic strain of 0.07.

Repetitions of the proportional loading experiments for b = 0�,34.1�, 55� and 90� revealed only little experimental scatter. Forb = 34.1� and b = 90�, the same strain to fracture was found, whilefor b = 0� and b = 55� the strains to fracture in the repeated exper-iments were 16% higher than those shown in Fig. 11. A summary ofhybrid experimental–numerical results is given by Tables 3 and 4for proportional and non-proportional loading, respectively.

4. Hosford–Coulomb fracture initiation model

The Hosford–Coulomb fracture initiation model was proposedby Mohr and Marcadet (2015) to predict the onset of fracture inadvanced high strength steel sheets (DP590, DP780, TRIP780)under proportional loading. It is based on the assumption that duc-tile fracture is imminent with the shear/normal localization of

(a)

(b)

(c)

200μm

200μm

shear dimples

abrasive marks

10μm

Fig. 8. SEM view of fracture surfaces: (a) dimpled surface as observed for b = 90�,(b) abrasive marks as observed for b = 55�, (c) shear dimples and abrasive marksobserved for b = 0�.

J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474 467

deformation within a narrow band (Rice, 1977; Mear andHutchinson, 1985; Nahshon and Hutchinson, 2008; Barsoum andFaleskog, 2011; Dunand and Mohr, 2014). In the present work,the phenomenological Hosford–Coulomb model is extended fur-ther to account for the effect of severe loading path changes.

4.1. Modeling of the effect of stress state

The backbone of the Hosford–Coulomb model is a shear andnormal localization criterion of the form

12ððr1 � r2Þa þ ðr2 � r3Þa þ ðr3 � r1ÞaÞ

� �1a

þ cðr1 þ r3Þ

¼ rc ð8Þ

with 0 < a 6 2 denoting the Hosford exponent (of the fractureinitiation model), the friction coefficient c P 0, and the cohesionrc > 0. Mohr and Marcadet (2015) transformed the above criterionfrom the ordered principal stress space fr1;r2;r3g to the mixedstress–strain space fg; �h; �epg for a von Mises material with isotropichardening. Here, the same transformation is performed for a Levy–Hosford material (i.e. the material’s plastic response is defined bythe constitutive equations given in Section 3.1).

Using the Lode angle parameter dependent functions

f 1½�h� ¼23

cosp6ð1� �hÞ

h i; f 2½�h� ¼

23

cosp6ð3þ �hÞ

h iand f 3½�h� ¼ �

23

cosp6ð1þ �hÞ

h i; ð9Þ

the principal stresses ri may be expressed in terms of the modifiedHaigh–Westergaard coordinates fg; �h; �rg,

ri ¼ ðf i½�h� þ gÞ�r: ð10Þ

According to the Hosford–Coulomb model, the von Mises equiva-lent stress at the onset of fracture reads

�r ¼ rc

12 ððf 1 � f 1Þ

a þ ðf 2 � f 1Þa þ ðf 3 � f 1Þ

a� �1

a þ cðf 1 þ f 3 þ 2gÞ:

ð11Þ

Furthermore, the Hosford equivalent stress (as defined in the plas-ticity model) at the onset of fracture reads

�rHf ¼rc

12 ððf 1 � f 1Þ

p þ ðf 2 � f 1Þp þ ðf 3 � f 1Þ

p� �1

p

12 ððf 1 � f 1Þ

a þ ðf 2 � f 1Þa þ ðf 3 � f 1Þ

a� �1

a þ cðf 1 þ f 3 þ 2gÞ:

ð12Þ

Using the isotropic hardening law, the resulting expression of theHosford equivalent strain to fracture for proportional loading, �epr

f

is obtained:

�eprp ½g;�h� ¼ k�1 k½b�ð1þ cÞ 1

2ððf 1� f 1Þpþðf 2� f 1Þ

pþðf 3� f 1ÞpÞ

� �1p

12ððf 1� f 1Þ

aþðf 2� f 1Þaþðf 3� f 1Þ

a� �1

aþ cðf 1þ f 3þ2gÞ

24

35 ð13Þ

Note that k�1 is the inverse of the hardening function given byEq. (4). In the above operation, without changing the mathematicalform of the model, we substituted the model parameter rc throughk½b�ð1þ cÞ in a way that the new parameter b is equal to the strainto fracture for uniaxial tension. In sum, the above isotropic fractureinitiation model for proportional loading features three freeparameters: a, b and c. Fig. 13 shows a plot of the ‘‘fracture surface’’defined by Eq. (13). The parameter a basically controls the Lodeangle sensitivity of the strain to fracture, with the limiting caseof Lode angle independence for a ¼ 2. The dependence on thestress triaxiality is controlled by the friction parameter c P 0 withstress triaxiality independence for c ¼ 0. The parameter b (strain tofracture for uniaxial tension) controls the overall amplitude of thestrain to fracture.

The corresponding criterion for plane stress conditions is shownin Fig. 14(a). Recall that the Lode angle parameter is a function ofthe stress triaxiality in the case of plane stress. Irrespective ofthe choice of parameters (except for a ¼ 2), the model features a

Fig. 9. Equivalent plastic strain fields from FEA at the onset of fracture. The solid black dots indicate the location of maximum equivalent plastic strain at the onset of fracture.The open black dot indicate the location where D first reaches unity (only visible if different from the location of maximum equivalent plastic strain).

468 J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474

ductility valley for biaxial tension ð1=3 6 g 6 2=3Þ, i.e. the strain tofracture is the lowest for plane strain tension ðg ¼ 1=

ffiffiffi3pÞ, and the

highest for uniaxial ðg ¼ 1=3Þ and equi-biaxial tension ðg ¼ 2=3Þ.

4.2. Modeling of the effect of loading path

Fracture initiation models (or coalescence criteria) that dependon the microstructural state predicted by porous plasticity models(see review of Benzerga and Leblond, 2010) are naturally loadingpath dependent due to the loading path dependence inheritedfrom the porous plasticity model. However, when using a

non-porous plasticity model with an ‘‘uncoupled’’ fracture initia-tion model, it is necessary to incorporate the loading path depen-dence of the strain to fracture directly into the fracture initiationmodel. Here, this is achieved in a phenomenological manner usinga damage indicator modeling framework (e.g. Fischer et al., 1995).

4.2.1. Linear damage accumulation lawThe direct application of (13) as fracture initiation criterion is

only valid for monotonic proportional loading (constant stress tri-axiality and Lode angle parameter throughout loading). For mildlynon-proportional loading, Mohr and Marcadet (2015) made use ofa linear ‘‘damage’’ accumulation law: Let D denote a damage

(a) β=0 )b( β=55

(c) β=90 (d)

0.30

>0.45

<0.15

effε

0.23

>0.35

<0.11

effε

0.08

>0.12

<0.04

effε

200μm

Fig. 10. Effective von Mises strain as a function of the initial surface coordinates within a central area of 1 mm � 1 mm as determined through stereo DIC at the instant ofspecimen fracture: (a) torsion, (b) uniaxial tension, (c) plane strain tension. The colormaps are chosen such that the central value (yellow) is equal to the average h�eeff i over theentire area. The extreme values (red and blue) correspond to 1:5h�eeff i and 0:5h�eeff i, respectively; (d) representative EBSD image showing the undeformed polycrystallinemicrostructure within a 1 mm � 1 mm area of the ðeh; ezÞ-plane. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version ofthis article.)

Table 2Comparison between the average effective von Mises strain at the onset of fracture onthe specimen surface.

b (�) �eeff (FEA) �eeff (DIC)

0 0.28 0.355 0.24 0.2390 0.08 0.08

J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474 469

indicator of initial value D ¼ 0; fracture is then assumed to initiatewhen D ¼ 1 after accumulating damage according to the evolutionequation

dD ¼ d�ep

�eprf ½g; �h�

ð14Þ

The term ‘‘damage’’ is used in quotes as D only partly representsphysical damage due to progressive void nucleation (which triggersthe localization of deformation at the microscale, and whose effecton the plastic material response is neglected). The above evolutionis linear in the sense that D is a linear function of the equivalentplastic strain for proportional loading.

To illustrate the effect of the loading path on the strain to frac-ture, we computed the fracture envelope for proportional loadingall the way to fracture after three different pre-loadings up to astrain of 0.1 (Fig. 14(b)):

1. Pre-compression: the strain to fracture for proportional loadingat g ¼ �1=

ffiffiffi3p

is much higher than that for proportional loadingwithin the range 0 6 g 6 2=3 (Fig. 14(a)). As a result, less dam-age is accumulated during pre-loading as compared to directloading within the range 0 6 g 6 2=3. The fracture envelopefor two-step loading with pre-compression (red curve) there-fore lies above the envelope for single-step proportional loading(black curve). In other words, the total equivalent plastic strainthat can be accumulated prior to the onset of fracture increasesafter pre-compression.

2. Pre-tension: plane strain tension is the most critical stress state.As a result, more damage is accumulated during pre-tension ascompared to loading at any other plane stress state. The frac-ture envelope for two-step loading with pre-tension (bluecurve) therefore lies below the envelope for single-step propor-tional loading (black curve). In other words, pre-loading underplane strain tension reduces the overall ductility of thematerial.

3. Pre-torsion: pre-loading under torsion increases the ductilityfor subsequent proportional loading at stress triaxialitiesabove 0.

The above model response is in remarkable qualitative agree-ment with the experimental observations for two-step loading.The comparison with the experimental results shows that themain effect of pre-loading on the subsequent fracture responseunder proportional loading is captured by the model, i.e.

Fig. 11. Loading paths to fracture for proportional loading (black solid lines) andmodel predictions of fracture initiation (blue solid dots): (a) Hosford equivalentplastic strain versus stress triaxiality, (b) Hosford equivalent plastic strain as afunction of the Lode angle parameter.

470 J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474

pre-compression and pre-torsion increase the ductility, whereaspre-tension decreases the ductility.

Fig. 12. Loading paths to fracture for non-proportional loading (black solid lines)and model prediction of fracture initiation (blue solid dots) for different types ofpre-loading: (a) pre-compression, (b) pre-torsion, and (c) pre-tension. (For inter-pretation of the references to color in this figure legend, the reader is referred to theweb version of this article.)

4.3. Non-linear damage accumulation law

A non-linear damage accumulation law is employed to improvethe model predictions from a quantitative point of view. The readeris referred to Xue (2007) and Bai (2008) for an overview onnon-linear damage accumulation laws. Instead of assuming a lin-ear relationship between the damage indicator and the equivalentplastic strain for proportional loading (black curve in Fig. 15(a)), anon-linear relationship as shown by one of the dashed curves inFig. 15(a) is used. In differential form, the corresponding evolutionequation reads

dD ¼ m�ep

�eprf ½g; �h�

!m�1d�ep

�eprf ½g; �h�

ð15Þ

with the exponent m > 0. Fig. 15(b) shows the damage evolution fora torsion experiment (i) after pre-loading under compression up to�ep ¼ 0:07, (ii) pre-loading under tension up to �ep ¼ 0:07, and (iii)without any pre-loading.

4.4. Model calibration

In a first step, the fracture initiation model parameters fa; b; cgare calibrated based on the proportional experiments for b ¼ 0�,

Table 4Summary of results for non-proportional loading.

hgistepl h�eistepl hgistep h�eistep2 �epre-loading �efinal gfinal�hfinal

0� pre-compression �0.58 �0.07 0.04 0.12 0.071 0.32 0.01 0.020� pre-tension 0.57 0.13 �0.04 �0.11 0.074 0.25 0.00 0.0055� pre-compression �0.58 �0.07 0.36 0.94 0.071 0.27 0.38 0.8955� pre-torsion 0.00 0.01 0.34 0.99 0.042 0.29 0.36 0.9855� pre-tension 0.57 0.13 0.34 0.94 0.075 0.19 0.37 0.9190� pre-compression �0.58 �0.07 0.58 0.11 0.071 0.24 0.58 0.0890� pre-torsion 0.00 0.00 0.58 0.08 0.071 0.18 0.58 0.08

Table 3Summary of results for proportional loading.

hgi h�hi �e gfinal�hfinal

0� 0.003 0.01 0.29 0.004 0.0121.9� 0.14 0.39 0.27 0.14 0.434.1� 0.22 0.62 0.24 0.22 0.6345� 0.25 0.74 0.22 0.25 0.7355� 0.33 0.98 0.24 0.35 0.9869.5� 0.45 0.57 0.17 0.44 0.690� 0.57 0.11 0.13 0.58 0.09

J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474 471

b ¼ 55� and b ¼ 90� assuming m ¼ 1. Recall that the parameter b isequal to the strain to fracture for uniaxial tension. Consequently,the strain to fracture for b ¼ 55� provides a good starting valuefor b. A quick optimization showed that the parameters a ¼ 1:13,b ¼ 0:24 and c ¼ 0:085 provide an exact prediction of the strainto fracture for all three proportional experiments.

In a second step, all four model parameters fa; b; c;mg are iden-tified through inverse analysis using the loading paths fornon-proportional loading shown in Fig. 12. The exact integral crite-rion (Eq. (14)) is evaluated for each calibration experiment. The dif-ference between the experimental and predicted strains to fractureis then minimized using a derivative-free simplex algorithm. The‘‘optimal’’ model parameters are a ¼ 1:21, b ¼ 0:21, c ¼ 0:076 andm ¼ 0:71.

Fig. 13. Hosford–Coulomb fracture envelope identified for aluminum 2024-T351.

4.5. Model validation

The comparison of the experimental and predicted strains tofracture allows for a first validation of the fracture initiation model.Recall that the end points of the loading paths shown in Figs. 11and 12 correspond to the measured instants of onset of fracture.The corresponding model predictions are shown as solid blue dots.Overall the model predictions agree well with the experimentalresults. The largest differences are observed for b ¼ 21:9�

(�eexpf ¼ 0:29 vs. �emod

f ¼ 0:24) under proportional loading (Fig. 11),

and for b ¼ 0� after pre-tension (�eexpf ¼ 0:29 vs. �emod

f ¼ 0:24) fornon-proportional loading (Fig. 12(c)).

The numerical simulations of all experiments are also repeatedwith the fracture initiation model active at each integration point.The predicted instants of onset of fracture are highlighted by solidblack dots in Figs. 4 and 5. The overall agreement is good for allloading cases. For proportional loading, the displacement/rotationto fracture is slightly overestimated for b ¼ 90� and b ¼ 0�, andunderestimated for all other cases. This result is consistent withthe ‘‘local’’ validation shown in Fig. 11, where the strain to fractureis overestimated for b ¼ 90� and b ¼ 0�. This consistency may beseen as a partial validation of the assumed location of onset of frac-ture made during model calibration. It is validated more accuratelyby comparing the assumed location of onset of fracture (solid dotsin Fig. 9) with the locations at which the damage indicator reachedunity first (open dots in Fig. 9). These locations coincide for allloading cases except for b ¼ 55� where the model predicts fracture

initiation on the inner specimen surface while the maximumequivalent plastic strain is reached within the bulk of the gagesection.

5. Discussion

As outlined in the introduction, the experimental results of Baoand Wierzbicki (2004) on aluminum 2024-T351 served frequentlyas the basis for the validation of fracture initiation models. It istherefore worth confronting the present results on aluminum2024-T351 with those from Bao and Wierzbicki (2004).

Following the processing procedure proposed by Bao andWierzbicki (2004), we calculated the average stress triaxialitythroughout loading,

hgi ¼ 1�ef

Z �ef

0gðd�epÞ ð16Þ

and computed the von Mises equivalent plastic strain to fracture,

�e fvm ¼

Z �ef

0

�rHf

�r

� �d�ep: ð17Þ

Fig. 15. Non-linear damage accumulation law: (a) Effect of the exponent mincluding the special case of linear damage accumulation for m = 1, (b) Evolution ofdamage in a torsion experiment after pre-loading under tension (blue) andcompression (red), as well as for proportional loading (black); the dashed linesshow the predictions for linear damage accumulation (m = 1), the solid lines depictthe results for non-linear damage accumulation (m = 0.8). (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

Fig. 14. (a) Hosford-Coulomb fracture envelope for proportional plane stressloading; (b) Fracture envelope for g P 0 after pre-loading up to �ep ¼ 0:07 undercompression (g ¼ �0:58, red curve), tension (g ¼ 0:58, blue) and torsion (g ¼ 0,magenta). (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

472 J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474

The corresponding data points fhgi; �e fvmg from our proportional

loading experiments (blue dots) are plotted next to the Bao–Wierzbicki (B&W) data (red dots) for the same range of averagestress triaxialities in Fig. 16. We also included the calibratedHosford–Coulomb envelope for plane stress (blue line) as well asthe criterion suggested by B&W1 (red line) in Fig. 16. The quantita-tive comparison reveals that the strains reported by B&W are oftenmuch higher (up to 0.46) than the present results (up to 0.30).This might be possibly due differences in the microstructures ofthe tested aluminum 2024-T351 alloys. More importantly, theB&W data suggests a dramatic decrease of the strain to fracturebetween uniaxial tension and pure shear, while our data suggestsan increase in ductility instead. It is worth noting that the determi-nation of the stress state and strain to fracture is affected by several(possibly significant) uncertainties in the assumed location of onsetof fracture, and in the mechanical model used for hybridnumerical-experimental analysis.

Experimental data for other aluminum alloys also suggest ahigher strain to fracture for pure shear than for uniaxial tension.

1 We encountered a small discrepancy between the curves shown in Fig. 20 of Baoand Wierzbicki (2004) and the corresponding criteria given by Eqs. (8) and (9) of theirpaper. To ensure the continuity of the failure envelope at a stress triaxiality of 0.4, weadjusted the parameter of the criterion given by Eq. (9).

For example, Mohr and Henn (2004, 2007) observed an increasein ductility from uniaxial tension to pure shear in their results frombutterfly specimens for a cast aluminum alloy. The experimentalresults from Bruenig et al. (2008) for an unspecified aluminumalloy also show a higher ductility for pure shear than for uniaxialtension. A more comprehensive experimental study has been per-formed for aluminum 6061-T6 by Haltom et al. (2013). Their ten-sion–torsion tests generated ten data points between pure shearand uniaxial tension that suggest a decrease in ductility as functionof stress triaxiality in that range.

Despite multiple arguments challenging the validity of the Baoand Wierzbicki (2004) data, it is noted that the discussion as towhether the strain to fracture for pure shear is higher or lower thanthat for uniaxial tension is probably only of minor importance froma modeling point of view. Our current understanding of fractureinitiation at low stress triaxialities suggests that there is a compe-tition of Lode and stress triaxiality effects. If the stress triaxialityeffect is weak and the Lode effect dominates (which might havebeen the case in the Bao–Wierzbicki material), the strain to frac-ture for pure shear will be lower than that for uniaxial tension.

Fig. 16. Comparison of present experimental results (blue data points) withexperimental results of Bao and Wierzbicki (2004) for the same aluminum alloy(but possibly different microstructure). Different from the previous figures, the vonMises equivalent plastic strain is shown on the ordinate. The solid (dashed) linesshow the fracture envelopes as fitted to data for plane stress (axisymmetric)loading.

J. Papasidero et al. / International Journal of Solids and Structures 69–70 (2015) 459–474 473

Small changes to the parameters of the proposed Hosford–Coulomb model will already change the trends (e.g. reducing theexponent a and/or the friction coefficient c). A comparison of thestrain to fracture for pure shear with that for plane strain tensionmight be a better indicator of possible inconsistencies in experi-mental data sets. These two stress states feature the same Lodeparameter, and the effect of stress triaxiality should always resultin a lower strain to fracture for plane strain tension. A data pointfor plane strain tension has been added to the B&W data byWierzbicki et al. (2005), featuring the same strain to fracture asfor pure shear. Our data on the other hand suggest that the strainto fracture for pure shear is more than twice as high as that forplane strain tension, which is probably the main qualitative con-tradiction with the B&W data.

6. Conclusions

The effect of the stress state and loading path on the onset ofductile fracture in aluminum 2024-T351 is determined throughtension–torsion experiments on tubular specimens with a stockygage section of uniform thickness. The experimental program forproportional loading covers stress states ranging from pure shear(zero stress triaxiality) to plane strain tension (stress triaxialityof 0.58). The plastic response of the aluminum alloy is modeledusing a non-quadratic plasticity model with isotropic Swift–Vocehardening, while a Hosford–Coulomb fracture initiation model isemployed to predict the instant of onset of ductile fracture. Thecalibrated fracture initiation model for aluminum 2024-T351 sug-gests that the increase in ductility associated with a decrease instress triaxiality is stronger than the decrease in ductility associ-ated with a change of the Lode angle parameter from axisymmetrictension to pure shear. In particular, a higher strain to fracture isobserved for pure shear than for uniaxial tension. This result isnot only obtained from hybrid experimental–numerical analysisof the experimental data, but also confirmed by direct surfacestrain field measurements using stereo digital image correlation.

Experiments with two distinct proportional loading steps eluci-dated the strong effect of the loading path on the equivalent plasticstrain at the onset of fracture. It is found that pre-compressionincreases the strain to fracture for pure shear, uniaxial tensionand plane strain tension, whereas pre-tension decreases the corre-sponding strains to fracture. It is shown that a simple non-linear

damage accumulation rule can capture the observed effect ofnon-linear loading paths on the strain to fracture.

Acknowledgements

The financial support of Jessica Papasidero through a MongeFellowship from Ecole Polytechnique is gratefully acknowledged.This work was also supported by the Sésame 2006 grants fromthe Région Ile-de-France. The partial financial support throughthe French National Research Agency (Grant ANR-11-BS09-0008,LOTERIE) is gratefully acknowledged. The authors are grateful toProfessor Tomasz Wierzbicki (MIT) for valuable discussions andcomments on our manuscript.

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