Kondori MT

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Metallurgical and MaterialsTransactions A ISSN 1073-5623Volume 45Number 8 Metall and Mat Trans A (2014)45:3292-3307DOI 10.1007/s11661-014-2211-7

Effect of Stress Triaxiality on the Flow andFracture of Mg Alloy AZ31

Babak Kondori & A. Amine Benzerga

1 23

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Effect of Stress Triaxiality on the Flow and Fractureof Mg Alloy AZ31

BABAK KONDORI and A. AMINE BENZERGA

The microscopic damage mechanisms operating in a hot-rolled magnesium alloy AZ31B areinvestigated under both uniaxial and controlled triaxial loadings. Their connection to macro-scopic fracture strains and fracture mode (normal vs shear) is elucidated using postmortemfractography, interrupted tests, and microscopic analysis. The fracture locus (strain-to-failure vsstress triaxiality) exhibits a maximum at moderate triaxiality, and the strain-to-failure is foundto be greater in notched specimens than in initially smooth ones. A transition from twinning-induced fracture under uniaxial loading to microvoid coalescence fracture under triaxial loadingis evidenced. It is argued that this transition accounts in part for the observed greater ductility innotched bars. The evolution of plastic anisotropy with stress triaxiality is also investigated. It isinferred that anisotropic plasticity at a macroscopic scale suffices to account for the observedtransition in the fracture mode from flat (triaxial loading) to shear-like (uniaxial loading).Damage is found to initiate at second-phase particles and deformation twins. Fracture surfacesof broken specimens exhibit granular morphology, coarse splits, twin-sized crack traces, as wellas shallow and deep dimples, in proportions that depend on the overall stress triaxiality andfracture mode. An important finding is that AZ31B has a greater tolerance to ductile damageaccumulation than has been believed thus far, based on the fracture behavior in uniaxialspecimens. Another finding, common to both tension and compression, is the increase in vol-umetric strain, the microscopic origins of which remain to be elucidated.

DOI: 10.1007/s11661-014-2211-7� The Minerals, Metals & Materials Society and ASM International 2014

I. INTRODUCTION

MAGNESIUM has the lowest density of all struc-tural metals (1.74 g/cm3). Mg alloys are endowed withsuperior specific stiffness and strength, and these char-acteristics make them ideal material candidates forlightweight structural applications, notably in the trans-portation industry.[1] One challenge facing their imple-mentation as wrought products in load-bearingcomponents is their relatively low ductility, which limitstheir formability at room temperature. During the lastdecade, most experimental and modeling efforts havebeen devoted to understanding the plastic flow andstrengthening of Mg alloys.[2–8] On the other hand, littleis known about the damage and fracture behavior ofthese materials. It is well established that stress statetriaxiality plays an important role in the ductile fractureof metallic alloys.[9,10] However, published studies onfracture in Mg alloys either have been restricted touniaxial loading[11–13] or consist of exploratory experi-mental studies.[14–17]

Although some differences between single- and poly-crystals are noted,[4] it is widely believed that the lowductility of pure or alloyed Mg polycrystals stems fromtheir plastic anisotropy associated with the limitednumber of deformation systems, as a result of theirhexagonal-closed-packed (hcp) crystalline structure.However, the literature remains elusive on the issue ofhow plastic anisotropy affects ductility for triaxialloading conditions. A commonly accepted understand-ing of fracture under uniaxial tensile loading perpendic-ular to the c-axis is as follows.[4] Subsequent to basalslip, anisotropic plastic flow leads to stress concentra-tions, for example, at grain boundaries (GBs), which arethen accommodated by f10�12g extension twinning.[18]

Concomitant with prismatic hai slip, the latter producesa strain transverse to the loading direction but normal tothe c-axis. While some details pertain to the hardeningbehavior that ensues, it is clearly evident that atransverse strain along the c-axis can only be producedby the so-called contraction twins and, to some extent,hcþ ai dislocations. The former concentrate large shearswhich lead to failure by strain incompatibility at thetwin boundaries or inside the twins. Clear evidence oftwin-sized microcracks parallel to f10�11g-f10�12g con-traction double twins has recently been documented indifferent alloys.[11,19]

It is emphasized that the above mechanisms pertain touniaxial loading conditions. How the plastic anisotropyplays out under more complex triaxial loading states,which are encountered during processing or in service,

BABAK KONDORI, Graduate Research Assistant, is with theDepartment of Materials Science and Engineering, Texas A&MUniversity, College Station, TX 77843-3141. A. AMINE BENZERGA,Associate Professor, is with the Department of Materials Science andEngineering and Department of Aerospace Engineering, Texas A&MUniversity. Contact e-mail: [email protected]

Manuscript submitted January 9, 2013.Article published online February 27, 2014

3292—VOLUME 45A, JULY 2014 METALLURGICAL AND MATERIALS TRANSACTIONS A

Author's personal copy

remains unexplored. In addition, in most studies,[11,19]

rectangular-prismatic tensile specimens are typicallyused, which are known to favor shear-like fracture,even in materials that are more ductile, and lessanisotropic than Mg. Under such circumstances, thematerial adjacent to the slanted fracture surface doesnot display much damage, by cavitation or otherwise. Inother words, shear failure obscures the intrinsic damagemechanisms as it becomes increasingly difficult todecouple the fracture behavior from the plastic insta-bility. Furthermore, ductility is often invoked to meanthe tensile elongation of an initially smooth bar. Inmaterials with limited post-necking deformation, thisproperty is more a measure of the hardening capacity ofthe alloy than it is a measure of the material’s resistanceto damage accumulation and cracking. To our knowl-edge, there has been no systematic study of loadtriaxiality effects on the ductile fracture of Mg alloys.

In addition to the intrinsic anisotropy of pure single-crystalline Mg, alloying and processing affect the aniso-tropic flow properties of polycrystalline Mg. Primaryprocessing, such as extrusion or rolling, generally leadsto a strong basal texture. This is the case in AZ31, whichis the alloy of interest here. Alloying not only affects thestrength and possibly the propensity for twinning, butalso leads to the formation of second-phase particles, afew of which can play a role in the ductile fractureprocess.[14,16,20] Knowledge about how texture, twin-ning, and second-phase particles affect the damageprocess across a wide range of stress states is stilllacking. The current article is a first step toward acomplete characterization of damage-initiation andprogression mechanisms in Mg alloys under multiaxialstress states. One way to circumvent the difficultiesassociated with shear failure is to use cylindricalspecimens since axisymmetric deformation states de-crease the propensity for shear localization[21]; anotherway is to introduce a notch to induce a damage processzone in the specimen. Both have been explored in thisresearch. In a recent technical brief, we reported onpreliminary experiments using smooth and sharplynotched tensile bars,[22] wherein it was noted that thestrain to failure in the notched bar was roughly the sameas that under uniaxial loading. However, since thetriaxiality range was too wide (from 1/3 to about 1.5),what happened in between those extremes was notinvestigated. In the current article, we report onadditional experiments on two other notch geometriesas well as compression specimens, all tested in the rollingdirection of a thick AZ31 plate. We also investigate theoperating damage mechanisms, the evolution of plasticanisotropy, and their dependence on stress state.

II. EXPERIMENTAL PROCEDURE

A. Material

The material used in the current study is from a 1.25¢¢(32 mm)-thick, hot-rolled AZ31B plate (2.5–3.5 wt pctAl, 0.7–1.3 wt pct Zn, and 0.2–1.0 wt pct Mn) providedby Magnesium Elektron company in the H24 condition

(strain hardened and partially annealed). The plate isreported to three principal directions, rolling or longi-tudinal (L), transverse (T) and short-transverse orthrough-thickness (S); see Figure 1(b). Metallographicsamples form different planes were cut using a diamondsaw, ground with SiC paper and fine polished using1-, 0.3-, and 0.05-lm alumina suspensions. Water wasused during grinding only. Isopropyl alcohol was usedfor rinsing and sometimes acetone as ultrasonic cleanser.For etching, acetic picral solution (4.2 g picric acid, 10mL acetic acid, 70 mL ethanol, and 10 mL water) wasused for 5 seconds. Optical microscopy (OM) andscanning electron microscopy (SEM) were both used inmicrostructural observations. Grain-size distributionsand average figures were determined using the lineintercept method.[23] Compositional variations in themicrostructure and the identification of second-phaseparticles were determined using energy dispersion spec-troscopy (EDS) and wavelength dispersion spectroscopy(WDS) in SEM. Crystallographic texture measurementswere carried out using a Bruker-AXS D8 X-ray diffrac-tometer (XRD) with Cu-Ka radiation on a sample fromthe plate’s mid-section to get (0002) and (10�10) polefigures using a 5º grid size and an 85º sample tilt.

B. Mechanical Testing

Compression, tension, and notched specimens were cutout along the rolling direction of the plate (Figure 1(b))and deformed either to crack initiation or completefracture. The S orientation was systematically markedon both ends of each specimen. Cylindrical specimenswere used exclusively to deconvolute, as far as possible,structural effects from intrinsic properties. Their geom-etry is sketched in Figure 1(a). The same experimentsenabled us measure (i) the yield and flow behaviors ofthe material; (ii) the evolution of plastic anisotropieswith plastic strain (in a single specimen) and with stresstriaxiality (from one specimen to another); and (iii) thestrains to failure. The macroscopic fracture locus soobtained spans a wide range of stress state triaxialities.At least two specimens were used for each type of test.To increase confidence when scatter was large, up to twoadditional tests were carried out.Compression tests were carried out on a servo-

hydraulic MTS machine (Model 318.25) with a loadcell capacity of 250 kN at a strain rate of 10�3 s�1. Apure nickel antiseize lubricant was used to prevent earlybarreling. Each test was interrupted at regular strainintervals, and the specimen was unloaded to enable themeasurement of its current height, H, and diameters, UT

and US along the initial principal directions, namely, Tand S, respectively. True axial and lateral strains weredefined as

eL ¼ lnH

H0

� �; eX ¼ ln

UX

U0

� �; ½1�

where X stands for either T or S, and the superscript 0refers to initial values. The accuracy on lateral strains(eX) is 0.1 pct. The evolution of anisotropic plasticityis quantified in terms of an anisotropy ratio defined as

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 45A, JULY 2014—3293


RL ¼ eTeS: ½2�

All compression tests were continued until the pinsfailed in shear. A distinct load drop occurred beforethe specimen split in two pieces. The value of axialstrain, eL, at the load drop is taken as a measure ofstrain-to-failure initiation, ei: In addition, a strain tocomplete fracture, ef; was defined on the basis ofcross-sectional area variation:

ef ¼ lnAf

A0

� �� eT

��fþ eS

��f; ½3�

where the area of the fractured specimen, Af, wasmeasured postmortem in OM. Also, eT

��fand eS

��fare

the lateral strains in Eq. [1]2 taken at failure. Theidentification in Eq. [3]2 assumes an elliptical shape forthe fractured cross section. An alternative measurementof Af assuming an oval shape led to small differences inevaluating ef:

Uniaxial tension experiments were carried out at aninitial strain rate of 10�3 s�1 on a servo-hydraulicMTS machine (Model 380.50) equipped with a 250-kNload cell. True axial strain was measured beforenecking using a laser extensometer over a gaugelength of 30 mm. Also, a radial extensometer (detailsof which will follow) was used to measure the diameterreduction in real time. The measurement was madealong the S direction in some cases and along the Tdirection in others. In addition, each test was pausedat regular strain intervals to measure the lateraldiameters using a caliper (accuracy better than0.005 mm). This method expedited measurements ofplastic anisotropy compared with the unloading–reloading method used in the compression tests. Trueaxial and lateral strains were defined similar to Eq. [1]substituting the gauge length l for H in (Eq. [1])1. Alltensile specimens failed in shear. However, this failureoccurred after necking. Although postnecking defor-mation is small, it is generally important to distinguishbetween failure initiation and complete fracture as in

(a)

T

S

c

L

(b) (c)

Fig. 1—(a) Geometry and (b) orientation of round specimens used. (c) Custom-made knives for a radial extensometer, here kept mounted on theRN2 specimen.



the case of compression. Ideally, the strain-to-failureinitiation is defined as

ei ¼ lnAi

A0

� �� jeTj

��iþ jeSj

��i: ½4�

A robust estimate of ei thus requires a measurement ofAi, the area of the cross section at the neck at crackinitiation. Interrupting a tensile test at incipient crackingis quite challenging. Therefore, in most tests, the lastvalue recorded for the axial strain eL was used as alower-bound estimate of ei: In general, this identificationcan lead to errors because of necking. Exceptionally, onetest was interrupted successfully at the load drop, i.e., atcrack initiation. Although the orientation of the tensilespecimen in this test was off the L direction by about45 deg in the rolling plane, this case enabled us estimatethe error made by identifying ei with the last recordedvalue of eL at about 4 pct.

In order to study the effect of stress triaxiality on thedeformation and fracture of AZ31, round-notched (RN)specimens with three different notch geometries wereused, as in a previous study.[24] Inside the notch, thestress state is triaxial. In addition to the major axialstress, R; there are two equal minor (principal) stresses,each denoted by r<R: Let h � r=R: Each notched bar isessentially characterized by a certain level of h. Thestress triaxiality, T, is defined as the ratio of thehydrostatic stress to the von Mises effective stress. Foraxisymmetric loading, T is related to h through

T � 1

3signR

2hþ 1

j1� hj : ½5�

Alternatively, each bar is labeled based on the notchseverity parameter, f; equal to ten times the notchradius to specimen diameter at the notch root. Threevalues of f were explored, and the correspondingspecimens are denoted by RNf (Figure 1(a)). There isa direct relation between notch severity and stress tri-axiality. The lower the value of f the higher the levelsof stress triaxiality, as inferred from finite-element cal-culations.[25] Taking the notch height as a gaugelength, a nominal strain rate of 3 9 10�4 s�1 wasimposed in all the cases. In the notched bars, the useof an axial extensometer would be pointless unless thegauge is restricted to the height of the notch, which isdifficult given the size of our specimens. Instead, theinstantaneous diameter along the S direction was con-tinuously measured thanks to a custom-made radialextensometer. The latter consists of two knives(Figure 1(c)) made of a superalloy material mountedon a MTS clip-on displacement gage 632.02E�20 (notshown), which is commonly used in standard fracture-toughness tests. As in the compression of pins, a fewnotched bar experiments were interrupted, and thespecimens were unloaded several times to measure theanisotropy ratios, especially at incipient macroscopiccrack formation. Unlike in the case of initially smoothtensile bars or compression pins, the plastic strains arespatially nonuniform in the gauge section of a notchedbar. Hence, the following definitions are typicallyadopted; see Reference 24:

�eX ¼ lnUX

U0

� �; RL ¼ �eT

�eS; �ef ¼ �eT

��fþ �eS

��f;

�ei ¼ �eT��iþ �eS

��i;

½6�

where the bar stands for spatial averaging over theminimum-diameter section (absolute values taken whereappropriate). These definitions are the counterpart ofEqs. [1]1 and [2] to [4] in uniaxial bars. Most tests wereinterrupted at crack initiation, but a few were continueduntil final fracture. Crack initiation is detected by asudden drop in the load–displacement curve (see, Figure2.9 in Reference 10 for further background on thisidentification). For each notch geometry, one specimenwas taken to complete fracture, and then subsequentones were interrupted at crack initiation.

C. Fractography

In the literature, two methods are typically used toidentify damage-initiation sites.[10] The first, morestraightforward, consists of observing fracture surfacesin SEM. If void formation from second-phase particlesoccurs, and there is residual void–particle contactpostmortem, as would prevail in low-triaxiality fracture,particles would be observed at the bottom of dimples.This is the standard method for identifying second-phase particles inside dimples by means of dispersionspectroscopy. The second, more tedious, consists ofinterrupting mechanical tests, and subsequently section-ing the specimens and searching (in OM or SEM) forsome evidence of void formation.[24] Each method hasits own advantages. The disadvantage of the first is thatmere observation of particles on the fracture surfacedoes not necessarily inform on the extent to which theywere involved in the fracture process. The disadvantageof the second is that it only samples through two-dimensional sections. In general, a combination of bothis necessary to obtain a full picture. This has beenaccomplished in the current study. In material systemswhere particles are larger than, say 1 lm, X-ray com-puted microtomography[26] and laminography[17] pres-ent obvious advantages over the destructive methodsused here. It would be difficult, however, to rely on thesemethods for identifying damage-initiation sites at lowerlength scales, such as twinning-induced cracks. Toprevent oxidation, the fracture surfaces of brokenspecimens were sprayed immediately after testing witha silicone mold release spray then placed and held in avacuum desiccator before being examined in SEM. It isworth noting that, even with extreme care, oxidation issuch a major problem in magnesium that fracturesurfaces can only be observed once. For this reason,the testing campaign has been paced to accommodateSEM observations of oxide-free fracture surfaces. Occa-sionally, EDS and WDS spectra were recorded.In addition to the SEM fractographs, the tensile

(smooth and notched) specimens that were deformeduntil failure initiation (see above) were cut longitudi-nally using wire electro-discharge machining (EDM).Observations were then made in OM and SEM in L–S



planes, i.e., sections that contain the loading direction(L) and the short-transverse direction (S). Samplescontaining the damage process zone were subsequentlymounted in an epoxy resin to prepare metallographicsections. The method described for microstructuralobservations (Section II–A) was followed. Magnesiumand its alloys are among the most difficult metallicsamples to prepare for microstructural examination.This is particularly true in heavily deformed specimenswith features such as microcracks, microvoids andeventually macroscopic cracks. Extra care was thustaken to prepare the sections for damage observations.

III. RESULTS

A. Microstructure of Undeformed Material

Figure 2(a) depicts the microstructure of the materialviewed in the L–S plane. The material exhibits a dualgrain size distribution with small (~3 to 4 lm) and large(~12 lm) grains. The contrast bands along the rollingdirection are attributed to variations in Al content,based on WDS micro-analysis. Observations at highermagnification (Figure 2(b)) reveal second-phase parti-cles and scarce deformation twins. Pole figures show astrong basal texture typical of wrought AZ31B products(Figure 2(c)). The c-axis of most grains is oriented alongthe through-thickness direction S.

Three types of second-phases particles were observedin undeformed samples. The first type consists ofintermetallic Al-Mn rich, probably Al8Mn5 disper-soids.[27] They appear as stringers elongated in therolling direction or as isolated particles. Their compo-sition was ascertained using EDS. The second type ofparticles consists of Mg17Al12 intermetallics with a cubicstructure. They are present in two forms: fine, oftensubmicron lamellar precipitates, and larger equiaxedparticles. The fine precipitates form most likely duringcooling form hot-rolling temperatures. They appearmostly in regions of high Al content and decorate grainboundaries. On the other hand, the particles formduring solidification. While larger than the precipitates,their number frequency is much smaller. They can befound at grain boundaries or in the grain interior. Thepresence of both Al-Mn and Mg17Al12 intermetallics isexpected in Mg-Al-Zn-Mn alloy systems.[28] A third typeof particles was occasionally observed. They appeared asequiaxed and darker in SEM. The presence of an O peakin the EDS spectra suggests that these particles might beMgO oxides mixed with Mg-Al particles. The presenceof oxide/intermetallic particles has recently been reportedin Reference 16 and will be further ascertained onfracture surfaces.

B. Flow Behavior and Anisotropy

Typical nominal stress vs nominal strain responses intension and compression are shown in Figure 3(a)illustrating the unloading parts for the measurement ofanisotropy ratios. The corresponding true stress–straincurves, depicted in Figure 3(b), exhibit the usual

tension–compression asymmetry. True-stress correc-tions were made to keep the same hardening rate asbefore necking. Better corrections are possible[29] butwere not pursued here because the post-necking defor-mation is small. The compressive yield strength isconsiderably lower than in tension but the ultimatestrengths are comparable.Figure 4 shows typical results of plastic anisotropy

evolution. In compression, the relatively easy activation

Fig. 2—(a) Initial microstructure of hot-rolled AZ31B plate in L–Splane exhibiting a dual grain size distribution. (b) Second-phase par-ticles and some deformation twins in as-rolled material. (c) XRDpole figures corresponding to (0002) and ð10�10Þ planes.



of extension twinning renders the material significantlymore deformable along S (i.e., parallel to the c-axis)than along T (Figure 4(a)). The lack of deformabilityalong T up to a strain of ~5 pct is a strong indicator ofthe limited number of deformation systems that areactive post-yield. In fact, the formation of extensiontwins at the onset of plastic deformation also results inan unusual lateral contraction along the T direction;notice the initial decrease in eT in the inset of Figure 4(a).This phenomenon is consistent with recent observa-tions.[30] In subsequent stages of plastic flow, accumu-lated activity on other deformation systems, such asbasal and prismatic hai slip, induces enough plasticstrain along T to overcome the negative strain producedby extension twins and thus results in a net positivestrain along this direction.

In tension, on the other hand, the maximum lateralstrain is along T (Figure 4(b)). This change is rational-ized as follows. Under tensile loading, the polarity ofextension twinning is such that it is not activated aseasily as in compression. Instead, basal and prismaticslips are active from the outset of plastic deformation.These mechanisms, however, cannot accommodate the

lateral strain along the S direction, only along T. Toaccommodate the lateral straining along S, i.e., thecontraction along the c-axis, contraction twinning and/

(a)

0

100

200

300

400

500

0 4 8 12 16 20

Nom

inal

str

ess

(MP

a)

Nominal strain (%)

compression

tension

(b)

0

100

200

300

400

0 4 8 12 16 20

Tru

e st

ress

(M

Pa)

True strain (%)

compression

tension

Fig. 3—(a) Typical nominal stress vs nominal strain curves depictingfull and partial unloadings in compression and tension, respectively,for R-value measurements. (b) True stress–strain curves. The arrowindicates the onset of necking. Post-necking corrections were appliedkeeping a similar hardening rate.

(a)

-3

0

3

6

9

12

15

0 4 8 12 16 20

Late

ral S

trai

n (%

)

True strain (%)

T

S

compression

0

1

2

0 1 2

(b)

-10

-8

-6

-4

-2

0

0 4 8 12 16 20

Late

ral S

trai

n (%

)

True strain (%)

T

S

tension

(c)

0

2

4

6

8

10

0 4 8 12 16 20

1/(R

L )

True strain (%)

compression

tension

-10

0

10

0 1 2 3

Fig. 4—Lateral strain (along T or S as labeled) vs axial strain alongL in (a) compression and (b) tension. Dotted lines point to a refer-ence isotropic response. Inset in (a) shows the negative (contraction)strain along T in the early stages of compression. The necking strainis indicated by the vertical dashed line in (b). (c) Anisotropy ratio vsaxial strain showing the transient and steady states. A detail of thetransient (negative strain ratio) is depicted in the inset. Values of1/RL larger than unity mean a greater ability to deform along S(parallel to the c-axis).



or pyramidal slip must occur. The fact that the criticalresolved shear stresses of these systems are relativelyhigh explains the delay in their activation so that the Tdirection experiences more strain than the S direction. Itis important to note that all deformation systems (slipand twinning) must be active in the early stages ofplastic flow because some net deformation along S ismeasured from the outset.

The evolution of the anisotropy ratio RL with strain isshown in Figure 4(c), where the inverse of RL is plottedfor convenience. The error in R ratio measurement islarge at strains below 0.1 pct (of order of R itself), butdecreases rapidly (0.2R at eL ¼ 0:005 and 0.02 R ateL ¼ 0:05). For this reason, the ordinate axis is trun-cated for clarity. Also the inset shows the negative strainratio shortly after yielding. Figure 4(c) reveals twoessential aspects of plastic flow anisotropy in AZ31B:(i) the existence of a steady state after a transient regime;and (ii) the extreme behavior during the transient. Forinstance, in compression values of 1/RL in excess of 10are amenable to the lack of deformability along T(Figure 4(a)). In tension, the S direction is actually moredeformable (RL < 1 as in compression) during thetransient. In steady state, however, the anisotropy ratiois about 1.8 in tension and 0.35 in compression. The factthat RL < 1 in compression and RL > 1 in tension isconsistent with the rationales provided above in whatconcerns the evolution of lateral strains.

Interestingly, the evolution of anisotropic deforma-tion is accompanied by a non-negligible dilatation.Figure 5 depicts the volumetric strain ekk (the sum of thethree logarithmic strains eL þ eT þ eS) vs the axial straineL: A positive value of ekk indicates dilatation whereas anegative value indicates volume shrinkage up to exper-imental errors. Similar strain measurements were doneon three different specimens in both tension andcompression. The results were reproducible with littlescatter. The error bars become larger in compressionafter a 0.1 strain because of barreling (not visible to thenaked eye but quantified using top, center and bottomdiameter measurements). The development of dilata-tional strains in tension was already evident in

Figure 4(b) as the T-strain follows approximately theisotropic reference line while the S-strain is smaller. In aplastically incompressible material, the curves corre-sponding to the two lateral strains would be oneither side of the dotted line so that the constraint�ðeT þ eSÞ ¼ eL is satisfied. Figure 5 shows that there isa net increase in volume in both compression (2 pct) andtension (up to 6 pct after necking).All of the above applies to uniaxial loading only.

C. Mechanical Response Under Triaxial Loading

Notched bars are usually employed in fracture stud-ies.[24,25] In strongly anisotropic materials such as Mg,the introduction of a notch plays a dual role since it alsoallows to investigate the effect of triaxial loading on themechanical anisotropy. Figure 6 shows the mechanicalresponse of the three types of RN specimens. Eachresponse is given in terms of the applied load vs diameterreduction along the S direction (Figure 1(b)). Someexpected scatter was observed on the maximum strainsbefore load drop but there was less scatter on the limitloads. For reference, the response of the smooth bar isalso provided in Figure 6. In that case, continuousmeasurement of DUS was acquired in addition to thediscrete set of values made available on the basis ofthe anisotropy ratio measurements. However, since theradial extensometer is usually not located at the neckedsection the change in diameter recorded post-neckingconstitutes a lower bound.In a plastically isotropic material the axial limit load

increases with increasing the acuity of the notch as aresult of notch-enhanced stress triaxiality.[9] In AZ31Bthis trend is observed in Figure 6; it is however weaker.Interestingly, there is barely any variation of the limitload between the RN4 and RN2 specimens. Thisobservation hints at the interplay of plastic flowanisotropy and the achievable stress state triaxialitiesinside notches. This point will be discussed further inSection IV.

-2

0

2

4

6

8

0 4 8 12 16 20

Vol

ume

chan

ge (

%)

True strain (%)

compression

tension

Fig. 5—Relative volume change vs axial strain in compression andtension. Vertical dashed line indicates necking in tension.

0

100

200

300

400

0 0.03 0.06 0.09 0.12 0.15

F/A

0 (M

Pa)

ΔΦS/Φ0

RN2RN4

RN10tension

Fig. 6—Force, divided by the initial cross-sectional area, vs relativereduction in diameter along S for the tensile (smooth and notched)specimens. Filled symbols indicate the values used to define nominalstrains at failure initiation (abrupt load drop).



D. Effect of Triaxiality on Plastic Anisotropy

Themeasured steady-state values of the anisotropy ratioRL are documented in Figure 7. The data for uniaxialtension bars and compression pins include those analyzedindetail in Section III–B andFigure 4. The values reportedfor the round notched specimens are according to defini-tions (Eq. [6]) at the abrupt loaddrop.The data inFigure 7shows a clear trend of a decreasing anisotropy ratio withincreasing stress triaxiality for positive (tensile) values ofthe latter.Moreprecisely, the lateral direction ofmaximumdeformability changes fromT, under uniaxial loading, to Sunder triaxial loading. This holds for any amount ofsuperposed lateral stress. As a consequence, RL< 1 in allnotched bars. In addition, the anisotropy is stronger in theRN2 specimen (with the sharpest notch) than in the RN10specimen (with the shallowest notch). For reference, thevalue of RL in uniaxial tension averaged over all realiza-tions is about 1.75.

The anisotropy ratios can also be measured post-mortem for specimens taken to complete rupture. In thiscase, the notation Rf

L is used. Figure 7(b) documents thevalues for all specimen types. Two observations arenoteworthy. First, the postmortem anisotropy ratio isgreater than the steady-state value, i.e., Rf

L >RL, irre-spective of specimen type. Second, in the notched barsthe above trend is such that Rf

L > 1 whereas RL < 1.This, in particular, implies that if measurements weretaken only after fracture, then one would have missedthe fact that the deformability in notched bars is actuallyhigher along the c-axis than that perpendicular to it.Also, it must be emphasized that while RL is represen-tative of the material’s deformation, Rf

L also measuresthe signature of the overall dilation associated with theanisotropic crack growth process.

E. Fracture Loci

The effect of stress triaxiality on fracture is bestrepresented in a fracture locus. The latter depicts somemeasure of fracture against stress triaxiality. Figure 8shows two fracture loci using either the strain-to-failureinitiation,�ei of Eq. [6]4, or the strain to complete fracture,�efof Eq. [6]3. Since the history of triaxiality evolution in thenotched specimens is not known exactly, the failure strainsare plotted against the specimen type. In uniaxial com-pression and tension, the triaxiality values T (Eq. [5]) are�1/3 and1/3, respectively, before barrelingornecking.Fornotchedbars, there aremoredatapoints for�ei (Figure 8(a))than for�ef (Figure 8(b)) sincemost testswere interruptedatcrack initiation to enable observating damagemechanismsand ascertaining the presence and location of the macro-scopic crack. In uniaxial specimens, there are more datapoints for �ef since all of them fractured. Recall that thevalues of �ei in Figure 8(a) for uniaxial tension are lower-bound estimates. Upper-bound estimates for �ei are obvi-ously the strains to complete fracture, �ef: In fact, a tighterupper-bound estimate was obtained using the smooth-bartest interrupted at the incipient formation of shear failureand then taking measurements inside the neck. Thus, thevalues provided for �ei in Figure 8(a) for tension are at themost 4 pct less than the actual values.

Details aside, the salient features from Figure 8 are asfollows. First, the fracture loci exhibit a maximum atmoderate stress triaxiality. The maximum ductility isachieved in the RN10 specimen (shallow notch). Second,the strain to complete fracture �ef is found to be greater innotched specimens than that in the initially smooth ones(Figure 8(b)). For �ei; the trend is similar, albeit weaker.Third, in any given specimen, there is a differencebetween �ef and �ei: This difference is large in the notchedbars and small in the initially smooth bars (the fact thatthe �ei values in the latter are lower bounds confirms theobservation.) Finally, the strain-to-failure initiationdoes not vary between the RN4 and RN2 specimens(Figure 8(a)). As already observed in Figure 6, thediameter reduction (at the abrupt load drop) in thesetwo specimens is close to that of the smooth bars. Notethat the reported strain-to-failure initiations account foranisotropy as per Figure 7 (i.e., DUS 6¼ DUT).

F. Fracture Modes

Side views of broken specimens (not shown forbrevity) showed that the macroscopic fracture path isslanted in uniaxial loading (compression and tension)

(a)

0

0.4

0.8

1.2

1.6

2

-2 0 2 4 6

RL

Specimen Type

tensioncompression

RN10 RN4 RN2

(b)

0

0.5

1

1.5

2

2.5

3

-2 0 2 4 6

RL

f

Specimen Type

tension RN10 RN4 RN2compression

Fig. 7—Anisotropy ratios RL for various specimens measured (a) insteady-state, before crack initiation; and (b) postmortem. For an iso-tropic material RL = 1 (horizontal dashed line).



and nominally flat in triaxial loading. All compressionpins broke in two pieces. The normal to the fractureplane is contained in the L–S plane at �±45 deg fromthe loading axis. Other ductile metals do not break incompression before some significant barreling. Thelatter usually leads to the development of tensile stresseson the outer boundary, but, since the stress state insideremains compressive, the main crack does not usuallylead to the breaking of the specimen in fully separatepieces. Shear failure also prevails in the round tensilespecimens. As noted above, this is rarely observed inmetals. The fracture surface is rougher than in com-pression. The normal to the fracture plane may becontained in the L–S plane as in compression, or mayvary about that orientation leading to a somewhatconical fracture surface. The strong propensity for shearlocalization in the round tensile specimens is indicativeof a few destabilizing effects. This point will be discussedlater in Section IV. The macroscopically flat fracturesurfaces of notched specimens are rougher. Also, somesecondary cracks are observed on the outer surfacebelow and above the main crack. Detailed SEMmicrographs will be analyzed after reporting on prefer-ential and competing sites for damage initiation.

G. Damage-Initiation Sites

Using the methods described in Section II–C, twofundamental sites were identified for potential damageinitiation: twins and second-phase particles. An attemptwas made to observe each site in longitudinal sections aswell as on the fracture surface.Figure 9 shows two examples of fully developed twin-

sized voids in the vicinity of a nascent macroscopic crack(not shown). These observations were made in the onlyuniaxial tension test that was successfully interruptedbefore complete (shear) failure. The first microcrack(Figure 9(a)) is located about 30 lm ahead of themacrocrack and appears to be arrested at the top-and bottom-grain boundaries. The second microcrack(Figure 9(b)) is farther away, at a distance of about

(a)

0

0.05

0.1

0.15

0.2

0.25

-2 0 2 4 6

- ε i

Specimen Typetensioncom

pressionRN10 RN4 RN2

(b)

0

0.1

0.2

0.3

0.4

4 6 8 10 12

- ε f

Specimen Type

tensioncompression

RN10 RN4 RN2

Fig. 8—(a) Strain-to-failure initiation (nominally defined at the loaddrop) for all specimens tested. (b) Strains to complete fracture mea-sured postmortem using top-view micropgaphs of failed specimens.

Fig. 9—SEM micrograph of longitudinal section of a uniaxial tensilebar interrupted at macroscopic crack formation showing (a) a twin-sized void near the tip of the macroscopic slanted crack, and (b)other twin related microcracks with different sizes near the macro-scopic crack.



90 lm from the main crack, and seems to extend overtwo grains probably because of the coalescence of twomicrocracks. As will be subsequently shown, observa-tions on the fracture surface suggest that these twin-sized voids are flat microcracks with a noncircular base.These features are reminiscent of those reported post-fracture by Barnett[11] who indicated that the crack-likevoids either consumed a region that appeared to havebeen occupied by a twin or, in rare cases, could be seenforming in the twin interior. The genesis of the twin-sized voids is more difficult to ascertain. They may resultfrom plastic strain accumulation inside the twin by basaland prismatic slip[4,11] or by twin-boundary failure. Theobservation of void embryos in regions with large twinfrequency (white arrows in Figure 9(b)) suggests acorrelation between twinning and void formation. Onthe other hand, since voids are not observed at all thetwin sites, concomitant factors may be necessary.Therefore, an alternative rationale for void formationin twins is the stress concentration resulting from twinintersections (Figure 9(a)) or twin–GB intersections.Note that the presence of crack-like voids of differentsizes in a single cross section indicates that voidformation is a continuous process with nucleation,growth, and coalescence phases.

The second site for damage initiation consists of asubgroup of the second-phase particles as categorizedin Section III–A. There were enough particle residualsinside dimples to allow for a limited, yet conclusivestatistical analysis of the type of particles involved.Specifically, out of about 20 EDS analyses, it wasdetermined that about 25 pct were Al-Mn, 25 pctMgO, and 50 pct pure Mn particles. Hence, these datashow that most particles leading to void formation areMn particles; recall that none was identified onmetallographic sections of undeformed material. Also,we found no evidence that the Mg17Al12 intermetallicparticles were active nucleation sites. Figure 10 sum-marizes the three types of particles involved in voidnucleation in AZ31B. Figure 10(a) illustrates voidinitiation at a stringer of Al-Mn. The micrographwas taken from the uniaxial tensile specimen deformedup to crack initiation. The slanted macrocrack shownis about 450 lm long (the twin-sized void of Figure 9(a)is just ahead of its upper tip, while the free surface is atthe bottom). The fact that the crack opening is somuch larger at the location of the particle than it isnear the free surface suggests that the particle mighthave been the initiation point and then, the shear lipformed. In any case, the EDS spectrogram is actuallynot associated with this specific particle but withanother Al-Mn particle inside a dimple (not shownfor brevity). The micrograph in Figure 10(b) and itsassociated EDS spectrum unequivocally show that theMn particles are involved in void nucleation. Theassertion that these are pure Mn particles is based onthe phase diagram of a binary Mg-Mn system.[28] Thus,the Mg peak must be the effect of the surroundingmatrix. Finally, although oxidation of fracture surfacescan be an issue, as indicated earlier, Figure 10(c) clearlyshows the presence of an MgO particle (and not a film)inside the dimple.

H. Damage Mechanisms

There is a clear transition in the fundamental damagemechanism with the increasing stress triaxiality. This isillustrated in Figure 11. Under predominately uniaxialloading, the fracture surface exhibits mainly quasi-brittle features. Figure 11(a) shows, for instance, a deepcrack in the small process zone of the smooth bar. Thecrack appears to have initiated at a deformation twin,such as that shown in Figure 9(b). On the other hand, ata moderately low triaxiality (RN10 specimen), micro-void growth to coalescence becomes the dominantfeature on the fracture surface, Figure 11(b). Thisexplains the higher strain to failure attained in RN10specimens (Figure 8). On further increasing the stress

Fig. 10—Three types of particles observed inside dimples and theirEDS spectrograms: (a) Al-Mn particles (here in a smooth bar); (b)pure Mn particles (in a RN10 specimen); and (c) Oxide particle(same RN10).



triaxiality (RN4 and RN2 specimens), the dimplesbecome shallower (Figure 11(c)). As mentioned above,another common observation in RN4 and RN2 speci-mens concerns the flat facets that appear to be traces oftwin-sized cracks (Figure 11(c)). Here, there are twocoalescing cracks each being confined to a grain. Thepresence of such cracks is consistent with our observa-tion of the twin-sized voids normal to the major loadingaxis (Figure 9(a)).

For completeness, the fractographs corresponding tothe smooth, RN10, and RN2 specimens are nowanalyzed at higher resolutions. Figure 12 shows thesalient features in the smooth bar. The overview in (a)shows an essentially slanted surface with a thin processzone (flat), (b) depicts a split in the thin process zone,and (c) some deep and diamond-like shallow dimples;note the shear zone to the right of the figure. In a fewinstances, second-phase particles are observed at thecenter of dimples (d) suggesting some contribution ofthese particles to damage under uniaxial loading.However, this potential contribution is frustrated bythe shear-like fracture and the predominance of twin-ning-induced damage (e). At a finer scale, smallerdimples are seen, which appear to be the result of grainpull-out giving a granular morphology to the surface (f).Since fine particles are observed, it is hypothesized thatthese are the Mg-Al precipitates which may have causedlocal intergranular fracture. Such features have alsobeen observed in Reference 13.

The characteristic features of the fracture surface ofthe RN10 specimen are shown in Figure 13. Theoverview in (a) shows some coarse splits, one of whichis shown in (b) at a higher magnification. Splits eitherresult from the coalescence along T of multiple voids orfrom the joining of two shear cracks. Although the flatfacets are less frequent in RN10 specimens, they are alsoobserved (c) surrounded by dimples. The predominatelydimpled character of the surface is evident in parts (d)through (f). In this region, most particles inside dimplesare pure Mn particles.

Interestingly, the area frequency of deep dimplesdecreases in the RN2 specimen (Figure 14). Twofeatures emerge: splits of shorter length (a), and flatfacets (b). With the introduction of a sharp notch, theratio h of lateral stress to axial stress increases. Oneprincipal lateral stress is along S, which would favorthe activation of softer deformation systems, notablyextension twinning. Since flat facets are less frequentlyobserved at lower triaxialities, it is possible that thesefeatures originated from the coalescence of cracksgrowing along the boundaries of deformation twins,mostly extension twins, the formation of which isfacilitated at higher triaxialities.[22] Other features onthe fracture surface are similar to those observed inthe process zone of the fracture surface of smoothbars. These include deep cracks with sizes close to thesize of large grains (Figures 14(c) and (d)), elongateddimples (Figures 14(d) through (f)), and the second-phase particles at the center of dimples (Figures 14(e)and (f)).

IV. DISCUSSION

A key finding in the current study is that AZ31B has agreater tolerance to ductile damage accumulation thanhas been believed so far on the basis of the fracturebehavior in uniaxial specimens, clouded by the occur-rence of shear failure in the latter. Effective strains tofailure of about 0.35 were measured (Figure 8) undermoderately triaxial loading (radial-to-axial stress ratio h

Fig. 11—Salient features of the fracture surface under (a) uniaxialtension, (b) moderately triaxial tension (RN10), and (c) triaxial ten-sion (RN2). The qualitative change in features illustrates a transitionin microscopic damage mechanisms.



Fig. 12—(a) Fracture surface of a smooth bar exhibiting (b) few splits, (c) diamond-like and sheared dimples, (d) a dimple with a particle inside,(e) quasi-brittle facets, and (f) granular morphology.

Fig. 13—(a) Fracture surface of a shallow notched bar (RN10) exhibiting (b) a few splits, (c) rarely observed facets, and (d) through (f) com-monly observed dimples.



~0.30 in RN10 bars). This figure is larger than what hasbeen achieved through microstructural manipulations,e.g., by grain refinement,[8] and close to the valuereported by Mukai et al.[31] under uniaxial loading aftersevere plastic deformation. Two measures of ductilitywere reported in Figure 8. Although �ei is a betterquantity to work with,* there is some arbitrariness in

defining crack initiation. As indicated in Section II–B,initiation is identified with a sudden load drop. Incompression, a small drop was noticed in most tests; theexample shown in Figure 3(a) is consistent with obser-vations by Barnett.[11] In tension, the abrupt load droppost-necking is more visible (Figure 3(a)). In notchedbars, the force decreases after the limit load but crackinitiation is associated with the abrupt load drop beyondthe limit load (filled symbols in Figure 6).

An important contribution to the damage tolerancerests on the transition from twinning-induced domi-nated fracture to micro-void coalescence dominatedfracture (Figure 11), which manifests in a substantialincrease in strain-to-failure under moderately triaxialloadings (Figure 8). Postmortem fractography analysisrevealed quite rich features on the fracture surfaces suchas granular morphology, coarse splits along the Tdirection, twin-sized crack traces with different orienta-tions as well as shallow and deep dimples. These featureswere observed in all specimens but in proportions thatclearly depend on the overall stress triaxiality and

fracture mode. At the fine scale, damage was found toinitiate at three kinds of second-phase particles and atthe interfaces between matrix and deformation twins.Cracking in Mg alloys due to contraction twins and

double twins has been observed[11] and rationalized.[4,18]

Our own observations of twinning-induced crack-likevoids (Figure 9) corroborate this finding. Twinning-induced cracking is likely a continuous process, assuggested by the observation of crack embryos (whitearrows in Figure 9(b)). Fully formed cracks are arrestedat grain boundaries (Figure 9(a)). Their orientationsdepend on local twin orientations, and therefore, is notnecessarily normal to the major load. However, theinvestigation of twinning-induced voids has been so farlimited to uniaxial loading perpendicular to the c-axis.How the propensity for this phenomenon is affected bythe stress state has remained unexplored to date.Fundamentally speaking, this is best explored in puresingle- or polycrystalline Mg. When there are competingdamage-initiation sites (twin-induced vs particle-inducedvoids), the effect of twinning-induced cracks is no longercontrolling at moderate triaxialities (as would occur inthe shallow RN10 specimens). On further increasing thetriaxiality, however, the propensity for the formation oftwin-sized cracks on the fracture surface increases asobserved in the RN2 specimens (Figure 11(c)). If theobserved flat facets are indeed traces of twin boundarycracks, then the area fraction of twin cracks perpendic-ular to the loading axis is much higher in the RN2specimens than in the initially smooth specimens.Further study is needed to elucidate the effects ofstress triaxiality on twinning-induced cracking in thismaterial.

Fig. 14—(a) Fracture surface of sharp notched bar (RN2) showing (b) large flat facets, (c) a magnified view of a split, and (d) through (f) otherbrittle-like features, including shallow dimples.

*Spatial nonuniformity, notably of triaxiality, is greater after crackinitiation.



Second-phase particles play an important role in thefracture of AZ31B. Their role has been underestimatedin the literature because of the focus on uniaxialspecimens where void growth is hindered by the onsetof shear failure. Most previous investigations havecentered on the role of precipitates in twin suppression,e.g., Reference 15, but not as damage-initiation sites. Afew studies, however, have discussed the role of particlesin the low-temperature fracture of Mg alloys.[14,16,17,20]

Lugo et al.[16] have identified cracked particles onpolished longitudinal sections of deformed tensile spec-imens. In an investigation of heat treatment effects onthe fracture properties of AZ31, Marya et al.[14] haveshown evidence of dimpled fracture and identifiedseveral kinds of particles inside dimples. They used atapered specimen, hence generating a low triaxiality, yetlarger than 1/3. Their fracture surface resembles that ofthe RN10 specimen. However, their observations wererestricted to postmortem fractography of rupturedsurfaces. In Section III–A, three types of particles werereported: two are intermetallic (Al-Mn and Mg-Al) andoxides (MgO). There is yet another type of intermetallicparticles that can possibly form in AZ31B. Hort et al.[28]

indicate that in a binary Mg-Mn system, pure Mnparticles are able to form. Although the maximumsolubility of Mn in Mg is 2.2 wt pct, which is higherthan the Mn content of AZ31, high solidification ratesmight lead to the formation of Mn particles in this alloy.Mn particles have indeed been observed on fracturesurfaces (Figure 10(b)). Thus, in addition to oxides, twotypes of intermetallic particles (Al-Mn and pure Mn)were found to participate actively in the ductile damageprocess. Al-Mn particles, often present as stringers, wereseen in longitudinal sections of specimens deformed tocrack initiation (Figure 10(a)) and inside shallow dim-ples. On the other hand, the Mg-Al precipitates, whichare much smaller in size, were not found to be involvedin any significant way in damage initiation althoughthey may affect the plastic flow properties of the matrix.

Shear-like fracture is frequently observed in othermaterials such as aluminum alloys[32] or steel,[33] butonly when the tensile specimens are sheet (plane stress)or thick (plane strain) specimens. However, it is rarelyobserved in round tensile specimens. Some high-strengthaluminum alloys[34] constitute an exception. A rationalefor this behavior is found in the classical localizationanalysis of Rice,[21] which shows the propensity forshear band formation under plane-strain or plane-stressstates if at least one factor destabilizing the plastic flowis present. This may include strength-differential effects,pressure-sensitivity, or damage-induced softeningamong other factors. Another destabilizing factor thathas not received the same attention in the literature isplastic anisotropy. Rice’s localization analysis has shownthat axisymmetric deformation states are extremely stiffagainst shear band formation. The fact that our roundsmooth bars fail in a slanted mode, just like thick or thinspecimens, hints at the strong effect of anisotropy incausing shear failure. It is emphasized that an averageeffect of anisotropy suffices to cause shear failure, asopposed to specific grain-level deformation mechanisms.

Generally speaking, the occurrence of shear failureobscures the fundamental damage mechanisms at thegoverning length scales. Damage processes are betterunderstood in the absence of shear failure. To suppressthe latter, our first attempt has been to use cylindricalspecimens for uniaxial loading, either in tension orcompression. All these specimens failed in shear. Asindicated above, this is likely the signature of stronganisotropic plasticity. On the other hand, the introduc-tion of notches has been effective in creating a damageprocess zone (macroscopically flat fracture). In turn, theuse of notched bars has allowed us investigate the effectof stress triaxiality on microscopic and macroscopicaspects of fracture in AZ31B.The concave shape of the hardening curve in com-

pression (Figure 3(b)) is associated with the activationof extension-twinning and accommodation effects.[35–38]

In tension along L, on the other hand, basal andprismatic slips as well as limited extension twinning(compared with compression)[2] are all activated inaddition to contraction twinning, which is necessary toaccommodate the transverse strain along the c-axis.[4] Afew pyramidal slips may also be activated as evidencedby the observation of hcþ ai dislocations.[2,39] Volumet-ric strain measurements in Mg alloys have not receiveddue attention in the literature. In fact, plastic incom-pressibility is often a priori assumed to infer Lankfordstrain ratios from limited measurements, e.g., Reference2. The relative increase in volume (dilatation) reportedin Figure 5 is another finding that warrants moredetailed investigation, particularly in relation to damageprocesses.The role of plastic anisotropy in the fracture of Mg

alloys, and more generally in other materials, is ratherpoorly understood. In the case of AZ31B, we havediscussed in what precedes two manifestations of plasticanisotropy: (i) the propensity for twin-induced micro-cracks at the microscale, and (ii) the onset of shearfailure under uniaxial loading at the macroscale. Thesetwo phenomena are by no means fully descriptive of theextent to which plastic anisotropy affects damage andfailure.In wrought polycrystalline AZ31, the common basal

texture imparts a certain type of macroscopic yieldingand flow anisotropy. How the latter evolves with stresstriaxiality has not been studied to date. Our measure-ments of the anisotropy ratio RL have shown a cleartransition from a situation of lower deformability alongthe c-axis (uniaxial tension perpendicular to c-axis) tosituations of higher deformability along the c-axis atmoderate-to-high triaxialities, as indicated by values ofRL < 1 in Figure 7. The fact RL < 1 under triaxialloading may be explained as follows. In the notchedbars, there are two minor lateral stresses, r, in additionto the major axial stress R (along L). In particular, thelateral stress along the S direction presumably favors theformation of extension twinning. If the material wereisotropic, for example, obeying a von Mises yieldcriterion, then the yield condition would be written as

R� r ¼ Y; ½7�



with Y being the yield stress. In terms of the measuredaxial stress, R; this yield condition is

R ¼ Yð1� hÞ�1:

where h is the lateral-to-axial stress ratio, which isrelated to the stress triaxiality T through Eq. [5]; seeSection II–B. Taking Y = 130 MPa as an estimate forthe yield stress in simple tension and T = 1.45 asrepresentative of the triaxiality in the RN2 notched bar,one gets h = 0.53 by inverting Eq. [5] in the branch0 £ h £ 1, so that the expected apparent yield stress inthe notched bar would be R

��y¼ 276 MPa. Actual stress

triaxiality levels vary pointwise within the specimen anddepend on details of the plastic flow properties of thematrix. However, the value of 1.45 is representative, seee.g., Reference 25. The value of 276 MPa, estimated forR��y; is greater than the measured value of ~190 MPa

(Figure 6). Of course, the material is not isotropic andEq. [7] does not apply. However, the fact that theapparent axial stress at yield in the RN2 specimen ismuch smaller than expected for an isotropic materialcan only be explained by the activation of softerdeformation systems. One such system consists ofextension twinning, which is activated because of theapplication of a lateral stress r along S, i.e., parallel tothe c-axis. Under such circumstances, extension twin-ning may settle at stresses of ~80 MPa or lower.Therefore, since the lateral stress at yield ishR��y� 100MPa, the activation of twinning is indeed

plausible. The three-dimensional picture is quite com-plex, but it is clear that more deformation systems areactivated under triaxial loading, which in turn mayexplain the decrease in the net deformation anisotropyof the notched specimens, as observed in Figure 7.

V. CONCLUSIONS

The macroscopic flow characteristics and microscopicdamage mechanisms have been investigated at roomtemperature in a magnesium alloy using various spec-imen geometries, resulting in a wide range of stress statetriaxialities and a major normal stress along the rollingdirection.

1. Dilatations were measured during plastic flow,apparently for the first time. Dilatations of up to 4and 2 pct were recorded in tension (before necking)and compression, respectively. Further study is requiredto reveal their fundamental origins.

2. On the basis of data gathered for AZ31B, Mgalloys containing void-forming second-phase parti-cles may have a better tolerance to ductile damageaccumulation than that commonly reported basedon the fracture behavior under uniaxial loading.

3. In the as-rolled AZ31B, three factors affect theincrease in ductility under moderately triaxial load-ing compared with uniaxial tensile loading: (i) acti-vation of more deformation systems under triaxialloading; (ii) transition from twinning-controlledfracture to microvoid coalescence fracture (at the

microscale); and (iii) transition from slanted (shear)failure to macroscopically flat fracture (at the mac-roscale). The effect of plastic flow anisotropy ismanifest in all.

4. Under triaxial loading, the tensile stress parallel tothe c-axis causes the activation of softer deformationsystems, such as extension twinning. The macro-scopic manifestation of this is lower-than-expectedaxial loads in notched bars and a saturation of theaxial load when the lateral-to-axial stress ratioexceeds ~0.40. Further study is needed to provide amicroscopic evidence for this phenomenon.

5. Shear failure in smooth round tensile bars is a rarephenomenon, and its occurrence is indicative of astrong destabilizing effect of anisotropic plasticity.Shear failure most likely occurs subsequent to flowlocalization. Under such circumstances, the damageprocess zone is too small to allow inferring any con-clusive evidence of intrinsic damage mechanisms.On the other hand, the notched bars are well suitedfor investigating damage and fracture.

6. Past its maximum, the strain to failure decreaseswith the increasing stress triaxiality, which suggeststhat void growth processes are active. However, thepresence of predominately shallow dimples on thefracture surfaces at high triaxialities suggests anearly coalescence of the nucleated microcracks,regardless of whether the latter initiate on second-phase particles or inside deformation twins.

ACKNOWLEDGMENTS

This research was supported by the NPRP GrantNo 4-1411-2-555 from the Qatar National ResearchFund (a member of Qatar Foundation). The state-ments made herein are solely the responsibility of theauthors.

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