Mechanics of Materials 79 (2014) 58–72

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Mechanics of Materials

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Effects of particles and solutes on strength, work-hardeningand ductile fracture of aluminium alloys

http://dx.doi.org/10.1016/j.mechmat.2014.08.0060167-6636/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +47 73 59 46 47; fax: +47 73 59 47 01.E-mail address: tore.borvik@ntnu.no (T. Børvik).

Ida Westermann a,b, Ketill O. Pedersen a,b, Trond Furu c, Tore Børvik a,⇑, Odd Sture Hopperstad a

a Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation, Department of Structural Engineering, Norwegian University of Scienceand Technology, NO-7491 Trondheim, Norwayb SINTEF Materials & Chemistry, NO-7465 Trondheim, Norwayc Hydro Aluminium, Research and Technology Development (RTD), NO-6601 Sunndalsøra, Norway

a r t i c l e i n f o

Article history:Received 28 October 2013Received in revised form 26 June 2014Available online 6 September 2014

Keywords:Aluminium alloysFracture behaviourScanning electron microscopyFractographyGurson model

a b s t r a c t

The influence of particles and solutes on the strength, work-hardening behaviour and duc-tile fracture of four different aluminium alloys in the as-cast and homogenised condition isinvestigated in this paper. These alloys contain different types and volume fractions of par-ticles, i.e. constituent particles and dispersoids, in addition to elements in solid solution.Tensile tests on smooth and notched axisymmetric specimens are performed to determinethe work-hardening curves and the ductile fracture characteristics of the alloys. A laser-based extensometer is used to continuously measure the logarithmic strain to failure inthe minimum cross section of the specimens. Finite element simulations of the test spec-imens are used to determine the work-hardening curves to failure. Both the J2 flow theoryand the Gurson model are used to describe the stress–strain behaviour of the materials,where the latter accounts for material softening due to void growth. The microstructureof the alloys is characterised by optical and scanning electron microscopy, and fractogra-phy is performed to investigate the fracture modes. While the damage and failure mecha-nisms are similar in the four alloys, the failure strain depends markedly on the stresstriaxiality and the yield stress. The trend is that the failure strain decreases linearly withincreasing yield stress for the investigated alloys.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Aluminium alloys are increasingly used in automotivesafety components such as bumper beams and crash boxes.In design of these aluminium components, the weightshould be minimised without compromising the strengthand stiffness, the energy absorption capability and thedeformability of the final product, and thus an optimalcombination of yield strength, work-hardening and ductil-ity of the material is required. This optimisation process isnot trivial, since increased strength is usually obtained at

the cost of lowering the work-hardening and the ductility.It was shown experimentally by Lloyd (2003) that for alu-minium alloys the tensile fracture strain tends to decreaselinearly with increasing yield stress for constantmicrostructure.

The yield strength of pure aluminium is low (about10 MPa) and needs to be increased by various strengthen-ing mechanisms. The most relevant strengtheningmechanisms for aluminium alloys are solid solutionstrengthening, increasing the stress field acting upon thedislocations, and precipitation hardening due to shearingand bypassing of particles by dislocations. Yield strengthmodels for age-hardenable aluminium alloys have beenproposed by e.g. Deschamps and Brechet (1999) and

I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72 59

Myhr et al. (2001), where the effects of solute atoms andhardening precipitates are included. Another strengtheningmechanism is work-hardening due to the increase of thedislocation density with plastic deformation and the associ-ated evolution of dislocation substructures. Experimentalresults show that the work-hardening of aluminium alloysat ambient temperature depends on several microstruc-tural features, e.g. solute atoms, non-shearable precipitates,dispersoids and inclusions (Embury et al., 2006). In additionto increasing the yield stress, solute atoms decrease thedynamic recovery rate of dislocations and thus result inhigher dislocation density and increased work-hardeningrate (Ryen et al., 2006). Hard particles, such as non-sheara-ble precipitates and dispersoids, increase the initial work-hardening rate due to storage of geometrically necessarydislocations around the particles (Cheng et al., 2003;Poole et al., 2005; Zhao et al., 2013; Zhao and Holmedal,2013a). As stated by Embury et al. (2006), there are twomain approaches to the modelling of work-hardening offcc metals. In the first approach, several internal variablesare used to describe the details of the dislocation substruc-ture (e.g. Nes, 1998), while in the other approach the dislo-cation density is taken as the single internal variable (e.g.Kocks and Mecking, 2003). A work-hardening model forage-hardenable aluminium alloys based on the latterapproach was proposed by Cheng et al. (2003), while morerecently Myhr et al. (2010) proposed a two-internal-variable model using the statistically stored dislocationdensity and the geometrically necessary dislocation densityas the two internal variables.

The mechanisms controlling the evolution of damageand the ductile fracture of metallic materials are nucle-ation, growth and coalescence of microscopic voids (e.g.Benzerga and Leblond, 2010). The voids nucleate at constit-uent particles or inclusions when the stress on the particleis sufficient to induce either particle cracking or particle–matrix decohesion. Continuum models for void nucleationassume that for a given particle size and geometry, the for-mation of voids depend on the equivalent stress as well asthe hydrostatic stress acting on the particle (e.g. Anderson,2005). Based on computational cell models using the finiteelement method, it has been established that the voidgrowth and the strain to void coalescence depend on sev-eral factors including the initial void volume fraction andthe distribution and shape of the voids, the plastic anisot-ropy, work-hardening and rate sensitivity of the matrixmaterial, and the stress state (Benzerga and Leblond,2010). Micromechanics-based continuum models for voidgrowth include the Rice–Tracey model (Rice and Tracey,1969) and the Gurson model (Gurson, 1977) among others.Void coalescence occurs typically by localised plastic defor-mation and necking of the ligament between adjacentvoids. If two classes of particles of different size and spacingare present in the material, void-sheet formation may takeplace and lead to shear fracture (Teirlinck et al., 1988).Void-sheet formation is caused by nucleation, growth andcoalescence of voids at smaller and more densely spacedparticles within shear bands linking the larger voids(Anderson, 2005). A limit-load model for void coalescenceby internal necking between microvoids was proposed byThomason (Thomason, 1985a, 1985b). The Thomason

model was combined with the Gurson model by Zhangand Niemi (1995) and later with an enhanced version ofthe Gurson model by Pardoen and Hutchinson (2000).Excellent state-of-the-art reviews of models for ductiledamage and fracture based on micromechanics and contin-uum mechanics are available in Benzerga and Leblond(2010) and Besson (2010).

Traditionally, studies of ductile fracture are based onmechanical testing and microscopy. In recent years, X-raytomography has made it possible to study the nucleation,growth and coalescence of voids in more details. Maireet al. (2005) performed tomography experiments to inves-tigate the formation and growth of voids in an aluminiummatrix containing spherical ceramic particles and used theresults to validate an extended version of the Rice-Traceymodel. Weck et al. (2008) studied the nucleation, growthand coalescence of voids in a model material with a core/shell design. The core consisted of a pure aluminiummatrix with particles, while the shell was made of parti-cle-free aluminium. The results were applied to develop amodified version of the Brown–Embury model (Brownand Embury, 1973) for coalescence. Maire et al. (2011)quantified the damage in three different aluminium alloysusing in situ tensile tests in synchrotron X-ray tomogra-phy. The results were used to fit the parameters in a mod-ified version of the Rice-Tracey model for void growth andto adapt an existing model for void nucleation. Thuillieret al. (2012) studied the ductile damage in thin sheets ofaluminium alloy AA6016-T4 by X-ray micro-tomographyand used the Gurson model to analyse the results.

In this paper, the effects of particles and solutes on thestrength, work-hardening and ductile fracture are studiedexperimentally for four aluminium alloys. Two pure Alalloys with different iron content, an AlMn alloy and anAlMgSi alloy were investigated, all in the as-cast andhomogenised condition, i.e. without any further heat treat-ment after homogenisation. The selected alloys contain dif-ferent types and volume fractions of particles, i.e.constituent particles and dispersoids, and, in addition, theAlMn alloy contains Mn and the AlMgSi alloy containsMg and Si in solid solution. The solubility of iron in alumin-ium is low and assumed to have little or no effect on thework-hardening of the alloys investigated. Tensile testson smooth and notched axisymmetric specimens are car-ried out to determine the stress–strain behaviour and theductile fracture characteristics of the alloys. Optical andscanning electron microscopy is used together with frac-tography to characterise the microstructure of the alloysand to study the damage and fracture mechanisms as func-tion of the stress state. Finite element analysis is combinedwith the experimental results to determine the work-hard-ening curves of the materials to failure. In the simulations,both the J2 flow theory and the Gurson model, whichaccounts for material softening due to void growth, areused to describe the stress–strain behaviour.

2. Materials

Four aluminium alloys were investigated in this work.The materials were provided as DC-cast extrusion ingots

60 I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72

with a diameter of 100 mm produced at the laboratorycasting facilities at Hydro Aluminium R&D Sunndal. Toease the reading, the four alloys will be designated Al0.2Fe,Al0.8Fe, Al1.2Mn, and AlMgSi throughout this work. Theexact composition is, however, given in Table 1. TiB hasbeen added as grain refiner to control the grain size andavoid abnormal grains during casting. The ingots were sub-jected to the homogenisation procedures listed in Table 2,using a laboratory furnace. The temperature–time cyclesare similar to what is used in industrial practice and con-sist of a soaking treatment followed by a predeterminedcooling rate.

3. Experimental procedures

3.1. Mechanical testing

Tensile tests were performed with smooth and notchedaxisymmetric samples. The geometry of the samples isillustrated in Fig. 1. The tensile axis of the specimen wasoriented along the longitudinal axis of the cast ingot. Threeparallel tests were carried out at room temperature foreach of the alloys. The cross-head velocity of the universaltensile testing machine was 1.2 mm/min, which corre-sponds to an average strain rate of 5 � 10�4 s�1 before neck-ing for the smooth specimens.

The force and the diameter at the minimum cross sec-tion of the specimen were measured continuously untilfracture. The latter was made possible using an in-housemeasuring rig with two perpendicular lasers that accu-rately measured the specimen diameter. Each laser pro-jected a light beam with dimension 13 � 0.1 mm2

towards a detector located on the opposite side of the spec-imen. Thus, the two orthogonal lasers created a box of laserlight of 13 � 13 � 0.1 mm3 around the minimum cross sec-tion of the sample. As the specimen was deformed, thecontinuous change in diameters was observed by thedetectors. The system consists of a high-speed, contact-lessAEROEL XLS13XY laser gauge with 1 lm resolution, whichwas installed on a mobile frame to ensure that the diame-ters always were measured at minimum cross section.During elongation, the sample was scanned at a frequencyof 1200 Hz and the measured data were transferred by thebuilt-in electronics to a remote computer via fast Ethernet.The measured diameters are denoted D1 and D2 in thisstudy.

The Cauchy stress and the logarithmic strain were cal-culated as

r ¼ FA

and e ¼ lnA0

Að1Þ

Table 1Composition in wt% of the four alloys. Grain refiner (TiB) has been added tothe alloys to obtain a homogeneous grain structure.

Material Fe Mn Mg Si Al

Al0.2Fe 0.2 – – 0.05 Bal.Al0.8Fe 0.8 – – 0.05 Bal.Al1.2Mn 0.2 1.2 – 0.05 Bal.AlMgSi 0.2 – 0.5 0.4 Bal.

where F is the force, A0 ¼ p4 D2

0 is the initial cross-sectionarea, and D0 is the initial diameter of the gauge section.The current area of the cross section was assumed to beelliptical and obtained as

A ¼ p4

D1D2 ð2Þ

The plastic strain is obtained as ep = e � r/E, where E isYoung’s modulus. The stress and strain measures in Eq.(1) represent average values over the minimum cross sec-tion for the notched samples and for the smooth samplesafter incipient necking. Plastic incompressibility wasassumed to obtain the logarithmic strain in Eq. (1). At largestrains, volume changes caused by void formation andgrowth may invalidate this assumption and induce inaccu-racy in the estimated average logarithmic strain.

3.2. Microstructure characterisation

Samples of the four materials in the as-cast and homog-enised state were mechanically ground and polished fol-lowed by electro polishing. Subsequently, the size anddistribution of the particles were obtained by image pro-cessing of back scattered electron (BSE) micrographs takenin a Hitachi SU-6600 FESEM operated at 5.0 kV. The pol-ished specimens were also anodized at room temperaturefor 2 min using HBF4 to reveal the grain structure. Fracturesurfaces of the failed tensile tests were investigated in aZeiss Gemini Supra 55 VP FESEM operated at 10 kV. Thiswas done for both the smooth and the notched testgeometries.

To examine the microstructure evolution and the frac-ture mechanisms, the specimens were sliced and the crosssection in the longitudinal direction of the failed tensilespecimens was polished and investigated for the variousalloys in the scanning electron microscope.

Alstruc (Dons et al., 1999; Dons, 2001; Røyset et al.,2004) was used to calculate the volume fraction of constit-uent particles after solidification, the wt% of elements insolid solution and b0-Mg2Si after homogenisation. The pro-gram is based on standard solidification and diffusion the-ory and consists of three modules, i.e. solidification,homogenisation and an Mg2Si module. It is assumed thatthe metal solidifies one part at a time with the concentra-tion in the recent layer of solid-state aluminium almostproportional to the concentration in the liquid. The ‘‘con-stants’’ of proportionality are called distribution coeffi-cients, and the values are found in the phase diagram.The microstructural input parameters are the composition,the dendrite arm spacing and the grain size. The main out-put parameters are the temperature as a function of thesolid fraction, the concentration profile in the solid-statefrom the dendrite centre to the dendrite boundary, the vol-ume fraction of each type of particle, the temperatureinterval in which they form, and tentative particle sizes.

4. Numerical simulations

Finite element simulations of the tensile tests onsmooth and notched samples were conducted to deter-mine the work-hardening behaviour of the materials after

Table 2Homogenisation procedure for the four alloys.

Material Heating rate [�C/h] Holding temperature [�C] Holding time [h] Cooling rate to RT [�C/h]

Al0.2Fe 100 580 5 200Al0.8Fe 100 580 5 200Al1.2Mn 100 600 5 200AlMgSi 100 585 2.5 300

904030

20 5

R=7.25

Ø6

±0.

01

M10

90

10 6

R

M10

(a)

(b)

Fig. 1. Test geometries of (a) the smooth specimen, and (b) the notchedspecimen where two values of the radius R (2.0 and 0.8 mm) have beentested.

0 1 2 3 4Diameter reduction [mm]

0

1

2

3

4

5

Forc

e[k

N]

Experimental resultsNumerical simulations

Al0.8Fe

Al0.2Fe

Al1.2Mn

AlMgSi

Fig. 2. Force-diameter reduction curves for the four materials from testsand simulations of smooth samples subjected to tensile loading.

I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72 61

necking. This information is not available directly from theexperiments, since the stress and strain fields are inhomo-geneous in the notched specimens and in the smooth spec-imens after incipient necking.

The numerical simulations were carried out using theexplicit solver of the finite element code LS-DYNA(www.lstc.com, 2013). Axisymmetric elements were usedto mesh the smooth and notched tensile samples. Theelements were four-node quadrilaterals with one-pointquadrature and stiffness-based hourglass control to avoidzero-energy modes. Mass scaling was used to reduce thecomputation time, and it was checked that the kineticenergy remained negligible compared with the internalenergy of the samples during the deformation process. 20elements were used across the radius for all three samplegeometries, which gives a characteristic element size of0.15 mm.

The materials were first modelled using the J2 flowtheory, i.e. the von Mises yield criterion, the associated flowrule and isotropic hardening were adopted. The work-hard-ening curve was modelled using a two-term Voce rule, i.e.

req ¼ r0 þX2

i¼1

RiðeeqÞ ¼ r0 þX2

i¼1

Q ið1� expð�CieeqÞÞ ð3Þ

where req ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 r0 : r0

qis the von Mises equivalent stress, r0

being the deviatoric stress tensor, and eeq ¼R t

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23 Dp : Dp

qdt

is the von Mises equivalent plastic strain. The plastic rate-of-deformation tensor Dp is defined by the associated flow

rule. Further, r0 is the yield stress, Ri are the hardeningvariables, and Qi and Ci are hardening constants. Thework-hardening rate h is obtained from Eq. (3) by differen-tiation as

hðeeqÞ ¼dreq

deeq¼X2

i¼1

CiQ i expð�CieeqÞ ð4Þ

Note that hi � CiQi, i = 1, 2, represent the contributionsto the initial work-hardening rate from the two work-hardening terms R1 and R2, while Qi, i = 1, 2, are the satura-tion values of these two terms. The saturation stressobtained at large strains is thus rsat = r0 + Q1 + Q2. We willhere arrange the terms so that C1 P C2, which implies thatthe first term R1 saturates at lower strain than the secondterm R2. Accordingly, the parameters Q1 and C1 are impor-tant for the work-hardening at small and moderate strains(less than about 0.1 for the actual materials), while theparameters Q2 and C2 determine the work-hardening atlarger strains.

The parameters were determined using numerical opti-misation. To this end, the code LS-OPT (www.lstc.com,2013), which is an optimisation tool that interacts withLS-DYNA, was used. The measured force-diameter reduc-tion curves from the tests on the smooth samples (seeFig. 2) were used as target curves, and 4 � 20 series of 10simulations, i.e. a total of 800 simulations, were run tooptimise the parameter sets. The work-hardening parame-ters of the four alloys are compiled in Table 3, which alsogives the 0.2% proof stress r0.2 of the alloys obtained fromthe tensile tests. The resulting numerical force-diameter

62 I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72

reduction curves are plotted in Fig. 2. It is seen that theagreement with the experimental data is good.

The influence of mesh size was investigated by runningsimulations with three mesh densities. The number of ele-ments across the radius was 10, 20 and 40. Simulationswere conducted for the smooth sample and notched sam-ple with 0.8 mm notch radius using the work-hardeningcurve identified for Al0.8Fe. The results showed that therewas no significant influence of mesh size for the tworefined meshes on the predicted stress–strain curves.

As will be shown later, it turned out that significantsoftening occurred in the tests of the notched samples,owing to void growth in the materials. In an attempt toaccount for this softening, additional simulations wereconducted with the Gurson model (Gurson, 1977). TheGurson yield criterion is here used in the form proposedby Tvergaard (1981) as

U ¼r2

eq

r2M

þ 2fq1 coshq2 trr2rM

� �� ð1þ q3f 2Þ ¼ 0 ð5Þ

where f is the void volume fraction. The parameters q1, q2

and q3 are here given the values q1 = 1.5, q2 = 1 and q3 ¼ q21,

as suggested by Tvergaard (1981). The flow stress of thematrix material, rM, is defined as (cf. Eq. (3))

rM ¼ r0 þX2

i¼1

RiðeMÞ ¼ r0 þX2

i¼1

Q ið1� expð�CieMÞÞ ð6Þ

where eM is the plastic strain of the matrix material,defined from the plastic power as

r : Dp ¼ ð1� f ÞrM _eM ð7Þ

As for the J2 flow theory, Dp is defined by the associatedflow rule. It is noted that in the absence of voids (f = 0), theGurson model reduces to the J2 flow theory, with yieldfunction U ¼ r2

eq r2M

�� 1 ¼ 0. In this study, void nucleation

is neglected in the model, and the void growth is expressedas

_f ¼ ð1� f ÞtrDp ð8Þ

The initial value of the void volume fraction is denotedf0 and is the only additional parameter in the Gursonmodel compared with the J2 flow theory. It is here assumedthat the voids have minor influence on the work-hardeningat stress triaxialities occurring during the uniaxial tensiletest (Xue et al., 2010). This means that the yield stressand hardening parameters obtained for the J2 flow theorycan be used for the matrix material, and f0 is the only addi-tional parameter that needs to be calibrated. It will bedemonstrated below that this is a reasonable assumptionfor sufficient small values of f0. All simulations with theGurson model were carried out using the same meshes

Table 3Yield strength and work-hardening parameters of the four alloys.

Material r0 [MPa] Q1 [MPa] C1

Al0.2Fe 24.37 31.46 32.5Al0.8Fe 30.68 37.93 35.5Al1.2Mn 39.51 51.40 35.6AlMgSi 66.26 62.00 32.3

as for the simulations with the J2 flow theory. To avoidnumerical problems at large plastic strains, the elementswere eroded as the void volume fraction f reached 0.4 inthe integration point. Simulations with higher criticalvalues of f were also performed, but the resulting work-hardening curves were not noticeably changed when thisvalue was altered.

5. Results and discussion

5.1. Initial microstructure

The four alloys, all in the as-cast and homogenised con-dition, consist of an equiaxed grain structure with a grainsize of approximately 60 lm. Optical micrographs of thegrain structures are shown in Fig. 3. Approximately 1000grains were measured using the linear intercept method.From the optical micrographs, the constituent particlesare mainly observed on the grain boundaries and in theinter-dendritic regions. Further investigations in the SEMusing back-scattered electrons (BSE), Fig. 4, show the sizeand distribution of constituent particles in more detail.Particle fraction and length obtained by image processingof BSE images containing sufficient numbers of constituentparticles (ca. 1000) are listed in Table 4. Several pictures,which constitute a total area of approximately0.135 mm2, were investigated to determine the area frac-tion of particles.

The constituent particles are of the type AlmFe for theAl0.2Fe and Al0.8Fe alloys, where m is between 3 and 6(Dons et al., 1999). In both these alloys, larger constituentparticles are observed along with clusters of smaller parti-cles, cf. Fig. 4(a) and (b). The inhomogeneous particle dis-tribution is formed because iron segregates to the grainboundaries and in-between the dendrite arms duringsolidification. The particles are of irregular shape withrounded edges. Particles having a maximum length ofapproximately 9.2 and 8.3 lm for the Al0.2Fe and Al0.8Fe,respectively, were measured. In this investigation we havenot distinguished between the different types of constitu-ent particles formed during processing. Based on Alstrucsimulations, both alloys were found to contain a smallamount of Fe and Si (less than 0.1 wt%) in solid solution,see Table 5.

The Al1.2Mn alloy contains the largest area fraction ofconstituent particles of the four alloys, see Table 4, dueto the high Mn content. The constituent particles aremainly rounded Al6Mn particles because of the low Si con-tent (Li and Arnberg, 2003; Huang and Ou, 2009) and thelargest particles observed have a length of 12.7 lm. Onlya small fraction of the particles were observed to be of

Q2 [MPa] C2 r0.2 [MPa]

5 97.26 1.90 26.728 95.24 2.15 33.699 106.22 2.16 43.516 126.46 4.21 71.21

Fig. 3. Microstructure of the four alloys. (a) Al0.2Fe, (b) Al0.8Fe, (c) Al1.2Mn, and (d) AlMgSi.

Fig. 4. SEM back scatter images of the four materials showing the size and distribution of constituent particles. (a) Al0.2Fe, (b) Al0.8Fe, (c) Al1.2Mn, and (d)AlMgSi.

I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72 63

Table 4Area fraction, fp, maximum observed length, and mean and standard deviation of maximum length of constituent particles in the four materials.

Material fp [%] Maximum observed particle length [lm] Mean maximum length [lm] Standard deviation [lm]

Al0.2Fe 0.3 9.2 0.64 0.92Al0.8Fe 1.5 8.3 0.47 0.72Al1.2Mn 2.4 12.7 1.10 1.14AlMgSi 0.5 26.5 2.12 2.91

Table 5Alstruc simulations of volume fraction of particles in % and elements insolid solution in wt%.

Material Volume fraction of particles [%] Elements insolid solution[wt%]

AlmFea Fe Si

Al0.2Fe 0.285 0.055 0.047Al0.8Fe 2.005 0.081 0.046

Volume fraction of particles [%] Elements in solidsolution [wt%]

prim-Al6Mn

sec-Al6Mn

a-AlFeSi

Fe Si Mn

Al1.2Mn 1.694 0.474 0.203 0.006 0.038 0.591

Volume fraction ofparticles [%]

Elements in solid solution[wt%]

a-AlFeSi b0-Mg2Si Fe Si Mg

AlMgSi 0.452 0.130 0.015 0.272 0.407

a m = 3–6.

0 0.4 0.8 1.2 1.6εeq

0

50

100

150

200

250

300

σ eq ,θ

[MPa

]

Al0.2FeAl0.8FeAl1.2MnAlMgSi

Fig. 5. Equivalent stress, req, and hardening modulus, h, versus theequivalent plastic strain, eeq. All curves are stopped at failure in therespective uniaxial tensile tests, assumed to occur at maximum truestress.

64 I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72

the more irregular shaped a-AlFeSi particles. In addition,Al1.2Mn contains a high density of small 0.1–1 lm disper-soids and 0.59 wt% Mn in solid solution (Table 5).

The AlMgSi alloy contains a rather low area fraction ofconstituent particles, but this alloy contains the largestparticles among the four alloys. The maximum observedparticle length is 26.5 lm (Table 4). The area fraction ofconstituent particles in the lower size range is similar tothat observed for the Al0.2Fe alloy. Two categories of con-stituent particles are expected in the AlMgSi alloy aftersolidification, namely a-AlFeSi particles and primary b-Mg2Si particles. However, at sufficient high homogenisa-tion temperature and long soaking time, the latter particleswill dissolve, and during cooling from the homogenisationtemperature finer b0-Mg2Si precipitates form. In addition,Alstruc predicts 0.41 wt% Mg and 0.27 wt% Si in solid solu-tion after homogenisation (Table 5).

The experimental measurements of the area fraction ofparticles and the simulations using Alstruc show goodagreement, see Tables 4 and 5. Note that when comparingthe results of the Alstruc simulations and the measuredarea fraction of particles, we are not distinguishingbetween various types of constituent particles, and the vol-ume fraction is assumed to be equal to the area fraction.The largest deviation is seen for the Al0.8Fe alloy wherethe measurements show 1.5 vol% of constituent particles,while the Alstruc simulation predict 2 vol%. The valuesfor the elements in solid solution of the different alloysare solemnly based on the Alstruc simulations. This is also

the case for the volume fraction of secondary Al6Mnparticles.

5.2. Strength and work-hardening

Fig. 5 presents the equivalent stress req and the harden-ing modulus h as functions of the equivalent plastic straineeq for the four materials, as obtained with the combinedexperimental and numerical method discussed in Section 4.These results are based on Eqs. (3) and (4) using theparameter sets compiled in Table 3. The curves in Fig. 5are stopped at incipient failure (or void coalescence) inthe respective tests, assumed to occur at maximum truestress in the tensile tests on smooth samples (see alsoFig. 8). It is seen that the four alloys represent a large spanwith respect to strength and work-hardening. The AlMgSialloy exhibits the highest strength, followed by Al1.2Mn,Al0.8Fe and Al0.2Fe, in that order. The four alloys have dif-ferent work-hardening rates for small strains, say less thanabout 0.15, while for larger strains the Al0.2Fe, Al0.8Fe andAl1.2Mn alloys exhibit rather similar work-hardening. Atthis strain R1 has virtually reached the saturation valueQ1 for the actual values of C1, see Table 3, and the work-hardening is determined by Q2 and C2, which are seen tobe similar for these three alloys. The hardening rate forthe AlMgSi alloy is different. A higher value of C2 impliesthat the work-hardening saturates at a lower strain level,whereas the higher value of Q2 means that the saturationvalue of R2 is greater for this alloy.

0 40 80 120 160 200σeq – σ0 [MPa]

0

500

1000

1500

2000

2500

3000θ

[MPa

]Al0.2FeAl0.8FeAl1.2MnAlMgSi

Fig. 6. Hardening modulus, h, versus work-hardening, req � r0. All curvesare stopped at failure in the respective uniaxial tensile tests, assumed tooccur at maximum true stress.

0 0.4 0.8 1.2 1.6εeq

0

0.2

0.4

0.6

0.8

1

θ / σ

eq

Al0.2FeAl0.8FeAl1.2MnAlMgSi

Fig. 7. Normalised hardening modulus, h/req, versus equivalent plasticstrain eeq. All curves are stopped at failure in the respective uniaxialtensile tests, assumed to occur at maximum true stress.

I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72 65

The work-hardening rate is more clearly visualised inFig. 6 showing the hardening modulus h plotted againstthe work-hardening req � r0. From this plot it is clearlyseen that the work-hardening rate for a given microstruc-ture, as represented here by the work-hardening, is consis-tently highest for the AlMgSi alloy, followed by, in turn,Al1.2Mn, Al0.8Fe and Al0.2Fe. This classification holdsfrom initial yielding until failure. The plot also shows thesimilarity in work-hardening between the Al0.2Fe, Al0.8Feand Al1.2Mn alloys in the saturation phase of the work-hardening (i.e. for large strains). A similar approach wasused by Mondal et al. (2013) to display the work-harden-ing behaviour of an AA7010 aluminium alloy.

The differences in strength and work-hardening behav-iour of the four alloys are assumed to be related mainly tothe differences in solute content. The constituent particlesplay an important role in the creation of voids, but haveless effect on the yield strength and the work-hardeningrate. The Alstruc simulations indicate that the Al0.2Feand Al0.8Fe alloys have about the same solute contentwhile the volume fraction of constituent particles is severaltimes higher in the latter (see Table 5). The influence of thislarge difference in the volume fraction of constituent par-ticles is a small, but still significant, increase in the yieldstrength and the initial hardening rate. The large differ-ences observed in the yield strength and the work-harden-ing rate between the four alloys are assumed mainly due toelements in solid solution (Ryen et al., 2006; Zhao andHolmedal, 2013b), but also due to clusters/co-clusters ofelements (Starink et al., 2012) and small precipitates(Myhr et al., 2010). The highest yield stress and work-hard-ening rate are observed for the AlMgSi alloy, which has thehighest amount of alloying elements in solid solution and,in addition, contains hardening precipitates due to theslow cooling rate from the homogenisation temperaturefor these materials.

The ratio h/req equals unity at diffuse necking in a uni-axial tension test, according to Considère’s criterion, while

after necking this ratio decreases continuously towardsfailure. This behaviour is displayed in Fig. 7, which showsh/req versus eeq. It is also interesting to observe that h/req

equals 0.05–0.1 at incipient failure for all alloys.The stress–strain curves obtained numerically by the

parameter sets listed in Table 3 are compared with exper-iments in Fig. 8. Results are given for the J2 flow theory andthe Gurson model. The plots give the average Cauchystress, r = F/A, versus the average logarithmic strain,e = ln (A0/A), in the minimum cross section of the tensilespecimens. The numerical results obtained with the J2 flowtheory are discussed first. Since the results of the smoothspecimens were used in the calibration procedure, theagreement is good between the numerical and experimen-tal stress–strain curves to fracture. For the notched speci-mens the numerical stress–strain curves consistentlyoverestimate the stress level and the discrepancy tendsto increase towards fracture. The reason for this is believedto be damage evolution in the specimens caused by voidgrowth. Since the notched specimens have higher stresstriaxiality than the smooth specimens used in the calibra-tion, the damage is expected to be more severe in thenotched specimens inducing increased damage softeningdue to accelerated void growth. Experimental evidencefor this will be given in Section 5.3. Damage softening isnot included in these numerical simulations, and the over-estimation of the stress level in the notched specimensseems reasonable. This conclusion is strengthened by theresults obtained with the Gurson model. In these simula-tions, the initial void volume fraction f0 was calibrated bytrial and error to the available experimental data, givingf0 equal to 0.001 for the Al0.2Fe alloy, 0.0025 for theAl0.8Fe alloy and 0.005 for the Al1.2Mn and AlMgSi alloys.It was not attempted here to relate f0 to the measured vol-ume fraction of constituent particles. With the adoptedvalues of f0, good agreement was obtained for both testswith smooth and notched specimens, cf. Fig. 8, showingthat the Gurson model is capable of describing the

0 0.4 0.8 1.2 1.6 2ε

0

50

100

150

200

250

σ [M

Pa]

Experimental resultsvon Mises

Gurson (f0 = 1 x10-3)

Al0.2Fe

Smooth

R2.0

R0.8

0 0.4 0.8 1.2 1.6ε

0

50

100

150

200

250

σ [M

Pa]

Experimental resultsvon MisesGurson (f0 = 2.5 x10-3)

Al0.8Fe

Smooth

R2.0R0.8

0 0.4 0.8 1.2 1.6ε

0

50

100

150

200

250

300

σ [M

Pa]

Experimental resultsvon MisesGurson (f0 = 5 x10-3)

Al1.2Mn

SmoothR2.0

R0.8

0 0.2 0.4 0.6 0.8ε

0

50

100

150

200

250

300

350

σ [M

Pa]

Experimental resultsvon MisesGurson (f0 = 5 x10-3)

AlMgSi

Smooth

R2.0

R0.8

(a) (b)

(c) (d)

Fig. 8. Plots of Cauchy stress, r = F/A, versus logarithmic strain, e = ln (A0/A), from tests and simulations of smooth and notched specimens of the fourmaterials.

66 I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72

observed stress triaxiality dependent material softening. Itis also seen that the softening is limited for the smoothtests, thus validating the adopted calibration procedurebased on the J2 flow theory (see also Xue et al., 2010).

In addition to damage evolution, there are two otherimportant sources of error in the experimental–numericalapproach adopted to determine the work-hardeningcurves.

Firstly, rate-independent plasticity was assumed in thesimulations, while the strain rate in the minimum crosssection of the smooth tensile sample increases with strain-ing after the onset of necking. The average strain rateinside the neck was found to increase almost linearly afterthe onset of necking, and depending on the ductility of thealloy, the value at failure was between one and two magni-tudes higher than the initial one. If the alloys exhibit signif-icant rate sensitivity, the influence of neglecting theincreased strain rate inside the neck is an over-estimationof the work-hardening compared to what would have beenthe case if the strain rate was kept constant. The rate

sensitivity of the actual alloys has not been determinedexperimentally and the error introduced by disregardingthe strain rate effect cannot be quantified. A study on thestrain rate sensitivity of AA1200 and AA3103 performedby Lademo et al. (2010) indicates that the rate sensitivitycould be significant and thus this issue should be furtherinvestigated.

Secondly, the alloys considered in this work were testedin the as-cast and homogenised condition, and were there-fore assumed to have isotropic mechanical properties.However, during tensile testing to large strains, the grainsbecome elongated and crystallographic texture develops inthe neck area that may lead to plastic anisotropy. In thesimulations, this is not accounted for and the materialswere assumed to remain isotropic to failure.

5.3. Damage and fracture

Fig. 9 shows the fracture surfaces of the smooth tensiletest specimens. The macroscopic appearance of the

Fig. 9. Overview of the fracture surface of the smooth tensile specimens. (a) Al0.2Fe, (b) Al0.8Fe, (c) Al1.2Mn, and (d) AlMgSi.

Fig. 10. Higher magnifications of the fracture surfaces in the smooth test geometries. (a) Al0.2Fe, (b) Al0.8Fe, (c) Al1.2Mn, and (d) AlMgSi. All materialsshow a ductile fracture with some large deep dimples and smaller dimples in between. Constituent particles are observed in the fracture surface.

I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72 67

Fig. 11. Fracture surfaces of alloy Al0.8Fe: notched specimens with (a) notch radius R = 2.0 mm and (b) notch radius R = 0.8 mm. Both geometries show afracture surface with a mixture of few wide and shallow dimples with smaller dimples in between.

Fig. 12. Initial void adjacent to a particle cluster in the undeformedAl1.2Mn alloy.

68 I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72

fracture is similar for all the alloys and specimens investi-gated independent of the geometry. However, as seen inFig. 9, the cross-sectional area at fracture significantlyincreases with increasing strength due to the reductionin ductility. The main global fracture appearance is cup-and-cone implying that the fracture starts from the centreand grows outward, but in a rather rough serrated mode.

From the fractographs of the smooth specimens inFig. 10 it is observed that the fracture surface is coveredwith two categories of dimples, i.e. a low density of coarsedimples and a higher density of small dimples. Thenotched specimens, only represented here for the Al0.8Fe

(a)(a)

Fig. 13. Void growth and fracture mechanism. (a) SEM micrograph in the longschematic illustration of void growth and coalescence.

alloy in Fig. 11, have more shallow dimples compared tothe smooth specimens. This is attributed to higher triaxial-ity in the notched specimens. Coarse constituent particlesare seen at the bottom of the dimples. Even though parti-cles are not observed at the bottom of all the dimples itcannot be excluded that particles have been present. Theparticles may have fallen out during fracture or they maybe present in the opposite fracture surface.

It is apparent that the size and number density of theconstituent particles influence the size and density of thedimples. The Al0.2Fe alloy has fewer but coarser dimplescompared to the Al0.8Fe alloy, cf. Fig. 10(a) and (b). BothAl0.8Fe and Al1.2Mn seem to have approximately the samedensity of coarse dimples which is consistent with the areafraction of primary particles. However, it is worth to noticethat the constituent particles are different in chemicalcomposition and shape for the different alloys.

The high density of voids and the constituent particlesobserved at the bottom of several of the voids in the frac-ture surfaces indicate that void initiation and growth is amajor mechanism for fracture. Back-scatter scanning elec-tron microscope investigation of longitudinal cross sec-tions, Fig. 12, also reveals that small voids are alreadypresent in the material after casting, but before tensiletesting. These are located adjacent to or in-between theconstituent particles. The void in Fig. 12 was not visibleat an accelerating voltage of 5–10 kV. However, it appeared

(b)(b)

itudinal direction of the Al1.2Mn alloy close to the fracture surface, (b)

Fig. 14. Longitudinal section from the centre of the fractured tensile tests of alloy Al0.8Fe. (a) Smooth test geometry showing alignment of particles due toelongated grains, and (b) test geometry with notch radius R = 2.0 mm showing void formation and small deformation of the grains in the material due tohigher triaxiality.

0 0.4 0.8 1.2 1.6 2εM

0

0.04

0.08

0.12

0.16

0.2

Voi

dvo

lum

ef r

act io

n(f )

R0.8R2.0Smooth

Al0.2Fe

0 0.4 0.8 1.2 1.6 2εM

0

0.04

0.08

0.12

0.16

0.2

Voi

dvo

lum

ef r

act io

n(f )

R0.8R2.0Smooth

Al0.8Fe

0 0.4 0.8 1.2 1.6 2εM

0

0.04

0.08

0.12

0.16

0.2

Voi

dvo

lum

ef r

act io

n(f )

R0.8R2.0Smooth

Al1.2Mn

0 0.4 0.8 1.2 1.6 2εM

0

0.04

0.08

0.12

0.16

0.2

Voi

dvo

lum

ef r

act io

n(f )

R0.8R2.0Smooth

AlMgSi

(a) (b)

(c) (d)

Fig. 15. Void growth in at the centre of the minimum cross section until failure as obtained from FE simulations of smooth and notched specimens of thefour materials. Failure is defined by the diameter reduction at maximum true stress in the tensile tests.

I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72 69

(a) Smooth (b) R2.0 (c) R0.8

Fig. 16. Colour plots of void volume fraction f in the minimum cross section of Al0.8Fe at the same diameter reduction as giving failure in the experimentsobtained from FE simulations of smooth and notched specimens with the Gurson model.

20 30 40 50 60 70 80

σ0.2 [MPa]

0

0.4

0.8

1.2

1.6

2

ε f

Failure strain - SmoothFailure strain - R2.0Failure strain - R0.8Linear best fit

Al1.2Mn

AlMgSi

Al0.8Fe

Al0.2Fe

Smooth

R2.0

R0.8

Fig. 17. Plots of measured failure strain versus yield stress for allmaterials and tests.

70 I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72

when increasing the accelerating voltage to 20 kV as thepenetration depth of the electron beam increased. As men-tioned above, the constituent particles are not homoge-neously distributed and are mainly located at the grainboundary due to the solidification conditions. Large areaswithout particles are therefore present in the grain interiorand between the dendrites.

Investigations of longitudinal sections close to the frac-ture surface in Fig. 13(a) show that large voids origin fromconstituent particles and expand into the particle-free area.Since the particles are located to the grain boundaries, voidsare observed to expand into both grains, i.e. on both sides ofthe particle, resulting in two large voids as schematicallyillustrated in Fig. 13(b). The large voids can meet anotherlarge void, coalesce and leave a sharp edge, or it can meetclusters of particles creating a row of small voids, as seene.g. in Figs. 10(d) and 11. On the fracture surface this is seenas large dimples with a particle at the bottom and severalsmall dimples containing particles. However, also smalldimples without particles are observed, indicating thatother mechanisms are possible when linking the large voidstogether, i.e. interacting slip planes creating voids in thejunction points (Pedersen et al., 2008).

Scanning electron microscope investigations of the frac-ture surfaces unveil that the large voids contain circularslip traces on the surface stretching from the bottom tothe top, see Figs. 10 and 11. This could imply that voidslocated in grains with favourable slip systems will expandand, depending on the triaxiality, they can either expand todevelop elongated voids in the tensile test direction or theycan expand to become more rounded voids. The formerrepresents the voids formed in smooth tensile specimenswhile the latter also result in more shallow voids formedin specimens with a pre-machined notch. Fig. 14 showslongitudinal sections from the centre of fractured samplesof alloy Al0.8Fe. In the smooth sample, the particles arealigned due to elongation of the grains at large strainsand the voids are elongated due to the relatively low stresstriaxiality. The sample with notch radius 2.0 mm exhibits ahigh density of equiaxed voids due to the higher stress tri-axiality and less deformation of the grains due to the lowerfailure strain.

The predicted evolution of the void volume fraction asobtained with the Gurson model for the four materials is

shown in Fig. 15. The results are from the centre of thespecimens and the curves are stopped at the diameterreduction leading to failure in the experiments. It isobserved that the void volume fraction is consistentlyhigher at failure in the notched specimens than in thesmooth one. The highest void volume fraction at failureis obtained for the notch with smallest radius and thuswith the highest stress triaxiality. This is in qualitativeagreement with the experimental findings, see Fig. 14,but the predicted void volume fractions are high. However,the Gurson model has been used here as a phenomenolog-ical damage model to simulate the softening observed inthe notched samples, and this should be kept in mindwhen considering these results. Colour plots of the voidvolume fraction at failure are provided in Fig. 16 for theAl0.8Fe material.

5.4. Failure strain

In view of the results presented by Lloyd (2003), it isinteresting to evaluate the influence of the yield stress onthe failure strain. The results are presented in Fig. 17 where

I. Westermann et al. / Mechanics of Materials 79 (2014) 58–72 71

the failure strain ef obtained in the tensile tests usingsmooth and notched samples is plotted against the yieldstress r0.2, which is here taken to represent the strengthof the materials. It is seen that there is a linear decreaseof ef with increasing r0.2, in agreement with Lloyd’s results.Linear relationships between the reduction of area at frac-ture and the yield strength were also found for AlCuMg andAlMgSi alloys by Liu et al. (2011).

6. Concluding remarks

Smooth and notched tensile tests have been performedfor four aluminium alloys with different alloying content,namely two AlFe alloys, an AlMn alloy and an AlMgSi alloy,all in the as-cast and homogenised condition. The AlFealloys contain different volume fractions of iron-rich con-stituent particles and, in addition, small amounts of Feand Si in solid solution. In addition to the constituent par-ticles, the AlMn alloy contains dispersoids and Mn in solidsolution whereas the AlMgSi contains Mg and Si in solidsolution or as clusters in addition to b0-Mg2Si precipitates.Using a laser-based measuring system in combination withnonlinear finite element analysis, the work-hardeningcurves have been determined to incipient failure. A weak-ness of the method is that the work-hardening curvesmight be influenced by material damage at large plasticstrains and increased strain rate after necking.

At a given microstructure, as defined by the work-hard-ening, the AlMgSi alloy consistently exhibits the highestwork-hardening rate, followed by Al1.2Mn, Al0.8Fe andAl0.2Fe, in that order. The latter three alloys display similarwork-hardening rate at large plastic strain. The damageand failure mechanisms are the same in the four alloys,namely nucleation, growth and coalescence of voids, butthe strain to failure depends markedly on stress triaxialityand on the yield stress. It is found that the strain to failuredecreases linearly with increasing yield stress for thesealloys. The fracture surfaces display both large and smalldimples which are associated with the constituent parti-cles. The Gurson model is capable of describing with goodaccuracy the softening of the materials due to void growthobserved, particularly for the notched specimens wherethe highest stress triaxialities occur.

Acknowledgements

The financial support of this work from the StructuralImpact Laboratory (SIMLab), Centre for Research-basedInnovation (CRI) at the Norwegian University of Scienceand Technology (NTNU), is gratefully acknowledged.

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