dynamic loading Fracture mechanism and toughness of a rolled magnesium alloy under Acta Materialia

dynamic loading

Arjun Sreedhar S

^a

, Suraj Ravindran

^b

, Gyan Shankar

^c

, S. Suwas

^c

, R. Narasimhan

^a,∗

aDepartment of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India

bGraduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125, USA

cDepartment of Materials Engineering, Indian Institute of Science, Bangalore 560012, India

a rt i c l e i nf o

Article history:

Received 8 June 2020 Revised 14 October 2020 Accepted 26 October 2020 Available online 2 November 2020 Keywords:

Dynamic fracture initiation Rolled Mg alloy

Texture Tensile twinning Crack tip constraint Fracture toughness

a b s t r a c t

Staticanddynamicfractureexperimentsareperformedusingfatiguepre-crackedthree-pointbendspec- imensofarolledAZ31 Mgalloyonaservo-hydraulicuniversal testingmachineandaHopkinsonbar setup,respectively.Theresultsareinterpretedusingin-situopticalimagingalongwithdigitalimagecor- relationanalysis.Microstructuralanalysisrevealsthatthefracturemechanismchangesfromtwin-induced quasi-brittlecrackingforstaticloadingtomicro-voidgrowthandcoalescenceunderdynamicloadingac- companiedby decreaseintensiletwinning nearthe tipwithloadingrate.By contrast,the density of twinsnear thefar-edgeofthe ligamentand associated texturechangeenhancestrongly withloading rate.Thefracturetoughnessincreasesdramaticallyathighloadingrateswhichisattributedtoenhanced workofseparationanddissipationinthebackgroundplasticzone.

© 2020ActaMaterialiaInc.PublishedbyElsevierLtd.Allrightsreserved.

1. Introduction

The mechanicalandfracturebehaviorofmagnesiumalloysare currentlythefocusofconsiderableresearchworkduetotheir at- tractiveproperties,suchaslowdensity,highspecificstrengthand good machinability.However, they havepoorcorrosionresistance aswell aslower fracturetoughnessthanAl alloyswhichhashin- deredtheirutilization[1–4].MagnesiumhasHCPcrystalstructure andplasticallydeformsatroomtemperaturebyslip,aswellasby {10¯12}and{10¯11}twins[4–6].Thesearereferredtoastensileand contraction twins(TTs, CTs), respectively(because they cause ex- tension andcontraction along c-axis).The vastly differing critical resolved shear stress (CRSS) values of various slipand twinsys- tems and strong textures make Mg alloys inherently anisotropic [3,5].

The orientation dependentstress-strainbehavioursofwrought Mg alloys under uniaxial tension and compression at different strain rates and temperatures have been studied extensively [7–

10].Ulaciaetal.[8]andDudamelletal.[9]observedthattheyield stress remains almost constant for all compression tests (up to strain rateof10³ s⁻¹),whereasfortensionalongrollingdirection (RD)itincreases.However,Zhaoetal.[10]reportedthat,whilerate dependenceduringcompressionalongrolling,transverseandnor-

∗Corresponding author.

E-mail address: narasi@iisc.ac.in (R. Narasimhan).

maldirections (RD, TD andND) forAZ31B Mgalloyis negligible up tostrain rateof10³ s⁻¹,astrongelevationofflow stressand straintofailureoccursabovestrainratesoftheorder10⁴ s⁻¹.An increase inTT activity wasalsoobserved withstrain rateforMg alloys,especiallyunderRD/TDcompression[8,9].

The fracture behaviour of Mg alloys under static loading has beenextensivelyinvestigated [2,11–22]. Somekawaetal.[2,11,12] observed an increase in K_IC and decrease in tensile twinning as grain size is refined,along with a changein fracture mechanism fromtwin-induced crack propagationto ductile void growthand coalescence. Kaushik et al. [13] conducted static experiments on notchedthree-pointbendfracturespecimensofMgsinglecrystals.

Theynoteda directcorrelation betweenevolution historiesofTT volumefractionandenergyreleaserateJeventhoughcrackexten- siontook placealong twinboundaries. Incontrastto [12],Prasad etal.[14]reportedthatfibrousfailureoccursforarolledAZ31Mg alloyfromexperimentsperformedwithnotchedcompact tension specimens.Theynotedthatasignificantcontributiontotoughness (roughly 70%) arises from profuse tensile twinning in the back- groundplasticzonewhichcorroborateswith[13].Further,astrong enhancementinJcwithnotchsizewasalsonoticedin[15]accom- panied by a brittle-ductile transition. The complex interplay be- tween stress triaxiality level (variedby changing the notch size) and plastic anisotropy has been found to influence ductility and failure mechanismin notchedcylindricalAZ31Mg alloybarsun- der tensileloading [3].Moreover, crystal plasticityfinite element

https://doi.org/10.1016/j.actamat.2020.10.059

1359-6454/© 2020 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

(CPFE)simulations[16,17]haverevealedthathighstresstriaxiality nearsharpnotchescanactivateTTs,whichinturn,promotebrittle fracture.

In automotive and aerospace applications, rapid loading of structural components may occur owing to crash or impact. Un- der suchconditions,dynamicfractureinitiationofpre-existingfa- tigue cracks atstress concentrationslike cut-outsandrivet holes becomesimportant[23].Therefore,sinceMgalloysare envisaged for such applications [1],it is imperative to investigate their dy- namic fracturebehavior. Experimental studiescarried outon var- ious steels and Al alloys [24–28] have shown that the fracture toughness enhances with loading rate, if ductile fracture takes place.Onthecontrary,itmaydecreasewithloadingrateifcleav- age fracture mechanism is dominant [24,25]. In order to ratio- nalize the above observed trends of dynamicfracture toughness withloadingrate,thestressfieldsnearadynamicallyloadedcrack tip inisotropic elastic-plasticsolidsaswell asFCC singlecrystals havebeensystematically investigated[29–33].Thesestudies have shownthatasloadingrateincreases,evenhighconstraintgeome- trieslike deeplycracked three-point bendspecimenswill experi- enceprogressivelossofconstraint(ortriaxiality). Inother words, theconstraintparameterQ[34,35]becomesmorenegativeasJ˙in- creases.Jayadevanetal.[31]andBiswasandNarasimhan[30]have applied theabove inertia-drivenconstraintlosstoexplaintheen- hancementintoughnesswithloadingrateinnearlyrateindepen- dentplasticsolids.In[30,36],itwasobservedthatmaterialinertia andstrainratesensitivityretard voidgrowthnearacracktipand enhancetheductilefracturetoughness.

IncontrasttosteelsandAlalloys,veryfewinvestigations have been conducted on dynamic fracture response of Mg alloys. Yu etal.[37]performeddynamicfracture experiments,coupledwith finite element analysis, on AZ31B Mg alloy and found an in- creasing trend in fracture toughness K_c with loading rate. Daud et al. [38] conducted dynamic fracture experiments on fatigue pre-cracked three-pointbendAZ61Mgalloyspecimensofvarying thickness using Charpy impact testing machine and noted a de- creaseintoughnesswithspecimenthickness.

However, a comprehensive study on effect of loading rateon fracture behaviourofMgalloysneedstobe performedtoaddress thefollowingimportantissues.

• How does the loading rate influence the operative fracture mechanism? Does this mechanism change with increase in loading rate akin to the effect of notch root radius in static experiments,sincetriaxialitywillbesimilarlyaffectedbyboth thesefactorsasmentionedabove?

• WhatistheeffectofloadingrateontheTTdevelopmentnear the crack tip as well as over the entire uncracked ligament?

Doesthiscausestrongchangesinthecrystallographictexture?

• Howdoesthestraindistributionintheligament(bothnearthe tipandawayfromit)changeastheloadingrateenhances?Can thisbecorrelatedwiththeTTactivityandtexturechanges?

• Whatistheinfluenceofloadingrateonthedynamiccrackini- tiation toughness, J_c? Can thistrend be rationalized fromthe role of tensile twinning in influencing the observed fracture mechanismandbackgroundplasticdissipation?

In order to answer the above questions, static and dynamic three-point bend fracture experiments are conductedon a rolled (basal-textured)AZ31Mgalloyusinganuniversaltestingmachine (UTM) and Hopkinson pressure bar, respectively. In-situ optical imagingalongwithdigitalimagecorrelation(DIC)analysisisper- formed to extract the displacement and strain fields. Also, mi- crostructural investigations including optical microscopy, fractog- raphy and electron back scattered diffraction (EBSD) are carried out on the fracturedsamples to get key insights on TT develop- ment, texturechanges andoperativefracture mechanism. An im-

portant aspect ofthe present work isthat thedisplacement and straindata obtainedfromDIC mapsareusedalongwiththepro- cedure suggested by Nakamura et al. [39] to compute the time history of J. This obviates recourse to complementary finite ele- mentanalysisorusinghandbookformula,asin[37,38].Theresults showastrongenhancementinfracturetoughness,Jc,alongwitha changeinfracturemechanismfromtwinning-inducedquasi-brittle fracturetoductilevoidgrowthandcoalescenceasloadingrateen- hances.Theintensityoftwinningneartheloading-endofthelig- ament and associated texture change enhance strongly with im- pactvelocity.Thus,tensiletwinningplays adualroleondynamic fracture ofMgalloys.It affectsnear-tip fractureprocesses(which canbedetrimental)andtheplasticworkexpendedintheligament (whichisbeneficial).

2. Materialsandmethods

2.1. Materialandspecimenpreparation

Single edge notched specimens are machined using wire-cut electricdischargemachining(EDM) fromanas-received12.7 mm thick rolled AZ31 Mg alloy plate. The chemical composition by weightpercentageofthisalloyispresentedinTable1.Fig.1shows aschematicofaspecimenwhichhasalength L=90mm,width, W= 20mmandthickness, B= 6.3mm.The specimensare ma- chinedsuchthat RDisalongitslengthandTDisalongits width.

Earlierstudies [14,20] haveshownthat fracture specimenstested under quasi-static loading with the crack along RD and TD dis- playnegligibledifferencesinresponse.Astarternotchofdiameter about350μmisfirstcut along thewidthusingEDM toa length of9.3mm.Itisthensubjectedtofatigueloadingunderfour-point bend configuration witha small load amplitudeof 1.4 kN in an UTM, INSTRON 8502, at a frequency of 3 Hz till a crack length ofabout0.7mmgrowsfromthestarternotch.Thus,theeffective cracklengthtowidthratioofthespecimena/W=0.5.

Optical metallographyandEBSD afterpre-cracking are carried out following the procedure explained inSection 2.3 to examine theinitialmicrostructureofthealloyandtoascertainifanytwin- ninghasoccurredduringthespecimenpreparation.Fig.2(a)shows an opticalmicrographwhileFigs. 2(b)and(c)display theinverse polefigure(IPF)mapand(0001)/(11¯20)polefigures(PFs),respec- tively,nearthefatiguepre-cracktip.Abimodalgrainsizedistribu- tion can be observedwith an average grainsize of 10.6± 5 μm whichwasestimatedbythearea fractionmethod.Also, itcan be seenfromFigs.2(b)and(c)thatthetextureisnear-basal.Figs.2(a) and(b)confirm thatno twinninghasbeeninducedduringspeci- menpreparation,especiallynearthefatiguepre-cracktip.

Table 1

Composition of AZ31 Mg alloy by weight%.

Element Mg Al Zn Mn

Weight% 95.88 2.75 1.12 0.14

Fig. 1. Schematic of the specimen employed in both the static and dynamic tests along with a detailed view of the initial pre-crack tip that is grown by fatigue load- ing from an EDM-cut notch. All dimensions are in mm.

Fig. 2. (a) Optical micrograph near fatigue pre-crack tip (which is indicated by the red vertical arrow), (b) IPF map near the fatigue pre-crack tip and (c) corresponding (0 0 01) and (11 ¯2 0) pole figures. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2.2. Experimentalprocedure

2.2.1. Staticfracturetest

Static, mode I fracture test is carried out under three-point bendconfigurationusingtheabovespecimen(Fig.1) witha span of80mmandover-hangsof5mmateitherendontheUTM,with custom-made grips. A displacement rateof 0.001 mm/sec is ap- pliedinthisexperimentwhichisperformedthreetimestoobtain repeatable results. In addition to the loadand actuatordisplace- ment, images of the deformed specimen are captured at regular intervalswitha digitalcamera,Nikon D7000,havingaresolution of16.9megapixels.Aspecklepatternofveryfineblackdotsovera whitebackgroundisappliedonthespecimensurfaceusingacrylic paint withthe help ofan air-brush. The displacement andstrain fields aresubsequently obtainedby analyzingtheseimagesusing Vic-2DDICsoftwaretoobtainthedisplacementandstrainfields.

Thespecimenrotation,

θ

^{,aboutthehingepoint,locatedahead}

of the crack tip on the ligament is ascertained from the crack mouthopeningdisplacement

δ

^{m,determinedfromDICanalysis,as}

θ

⁼

δ

^m

(

^a+rb

)

^{. (1)}

In theabove equation,r (0< r< 1) isa factorthatgives the location of thehinge point,which isinferred from normalstrain variation on the ligamentasabout 0.4. The specimenrotation

θ

^,

isdividedintoelasticandplasticparts,

θ

eand

θ

p.Theplasticpart

θ

^{pisemployedalongwiththemomentMonthecrackplane,com-}

putedfromtheload,toestimatetheenergyreleaserateJas J= Je+Jp, where,

Je= K²

E , Jp=

η

Bb θ^p

∫

0

Md

θ

p, (2)

areelasticandplasticcontributionstoJandthefactor

η

^istakenas

2[40,41].TrialanderrorcalculationsshowedthatneglectingJeand computingJdirectlyfromtheexpressiongivenaboveforJp,byem- ployingthetotalrotation

θ

^{,makesnegligibledifferenceespecially}

atlaterstagesbecauseofprofuseplasticityintheligament.

2.2.2. Dynamicfracturetests

ThedynamicfractureexperimentsarecarriedoutusingaHop- kinsonpressurebarsetupwithaluminiumbarsofdiameter19mm (seesupplementarymaterial).Asinglebarconfigurationwithouta

transmittedbarisemployed withthespecimensupportedagainst two cylindrical anvils which are fixed to a rigid frame. The load versus time history for the dynamictests are obtained with the help ofstrain gaugesfixed atthe mid-lengthofthe incident bar, whichis 1.82m long.Atup ofdiameter6.35 mm isused atthe endofthisbar, andnoplasticdeformation isobserved init dur- ingtheexperiments.Also,norotationconstraintisimposedatthe loadingpoint.The strikerbarisoflength 45.7cm.Dynamictests are carried out at four loading rates by changing the striker bar velocity V₀ which,in turn, isachieved by adjusting the pressure inthegasgun.ThevaluesofV₀ correspondingtothesefourload- ing rates are about 9m/s, 13 m/s, 17 m/sand 20m/s, although thereare slight variationsamongst themultiple testscarriedout foreach case.Thesignalsfromthestraingaugesarepassedtoan oscilloscopethroughanamplifier.

Theincidentandreflectedpulsesrecordedbythestraingauges areusedtoobtaintheappliedloadPas:

P=AbEb

( ε

I+

ε

R

)

^{. (3)}

Here,

ε

I and

ε

R are incident and reflected strain signals, respectively, which are suitably shifted to account for time lag causedbywave propagationintheincident barandcorrectedfor dispersion, A_b is the area of cross-section of the bar andE_b, its Young’s modulus. The deformation of the specimen is recorded atthe rateof0.5million frames per second witha resolutionof 400×250pixelsusingaShimadzuHPV-X2highspeedcamera(so asto capturethe stress-wave dominantloading). In order toen- able DIC analysis forobtaining displacements andstrain fields, a specklepattern isfirst applied onthe specimensurfacewhich is viewedbythehigh-speedcamera.Theenergyreleaseratehistory, J(t)iscalculatedusingEq.(2)asinthestaticanalysis.Tothisend, thespecimenrotationaboutthehingepointontheligament,

θ

^(t),

isestimatedfromCMODhistory

δ

^{m(t),whichisextractedfromthe}

DICanalysis,alongwithEq.(1).However,thecontactloadP(t)ob- tained through Eq.(3) isnot likely to be equilibrated withreac- tions impartedbythesupports upto acertain timeowing toin- ertialeffects.Thelatterweremeasured bypiezoelectric loadcells mountedonthefixedsupportswhichconfirmthattheendcondi- tionsareestablishedduringthecourseofthetest.

The time over which inertialeffects are expected to be dom- inant has been estimated by Nakamura et al. [39] and Ireland [42]as2

τ

^,where

τ

^{~ 23W/co,cobeingtheelasticbarwavespeed,}

iscalledastransitiontime[39] orinertialoscillationtime period

Table 2

Test velocity, time to fracture, fracture mechanism, fracture toughness, average loading rate, critical CTOD, and twin area fraction near crack tip for static and dynamic tests.

Test Velocity, V 0(m/s)

Time to fracture, t f(μs)

Fracture Mechanism

Fracture Toughness, J c(N/mm)

Average loading rate, j GN/ms

Critical CTOD, δtc(μm)

Twin area fraction near crack tip (%)

Static Brittle 32 ±0.5 0.13 11.95 ±1.13

9 133 ±5 Ductile 60 ±1.5 0.45 0.24 7.58 ±0.3

13 103 ±7 Ductile 77 ±2.0 0.75 0.34 7.29 ±1.07

17 78 ±4 Ductile 81 ±1.5 1.03 0.39 5.00 ±0.4

20 65 ±1 Ductile 88 ±2.0 1.35 3.88 ±0.39

Note: Critical CTOD for V 0 = 20 m/s could not be reliably estimated due to loss in correlation in the images.

[42] ofthe specimen.Fortime t <

τ

^{,the ratioof kinetic energy}

to strain energy of the specimen exceeds 1 and it decays to a valuelessthan0.2onlybeyondatimeof2

τ

^{.Aswillbeseensub-}

sequently, thetime corresponding tofracture initiation, t_f, forall dynamictestsconductedfallswithin the above notedtime range of 2

τ

^{, which is estimated to be 160 μs for the present Mg al-}

loyspecimens.Hence,themeasured contactloadPisnotdirectly used to compute the bending moment M aboutthe hinge point onthecrackplanebutinsteadanalternateprocedureproposed in [39] based on distribution of strains on a remote cross-section is employed. Thisis explainedbelow, andmoredetails aregiven in supplementarymaterial.

Since the regionaround the ligamentundergoes heavy plastic deformation, which makes precise evaluation of the strains near the cracktip andimpactedge difficult,thebendingmoment and shear force, M andV over a cross-section located at a distancel (greaterthan0.5b)fromthecrackplaneisfirstestimated.Forthis purpose,thestrain variationonthespecimensurfaceoversucha cross-section, deduced fromthe DIC maps, is used along with a numericalalgorithm tocompute thenormal(bending) andtrans- verse (shear) stress distributions on the cross-section. This algo- rithm,whichisbasedonpolycrystalplasticityassumingTaylorho- mogenization,isgiveninthesupplementarymaterial.Itmakesuse ofsinglecrystalplasticityconstitutiveequationsatthelevelofin- dividualgrains,specifictoMg, whichtakesintoaccountbothdis- location slip on various systems and deformation twinning. The phenomenological hardening laws proposed by Zhang and Joshi [43] accounting forslip-slip,twin-twin andtwin-slip interactions are incorporatedintheseconstitutiveequations.Theserateequa- tionsarenumericallyintegratedwithrespecttotimeusingtherate tangent modulusalgorithmofPeirceetal.[44].Thesinglecrystal propertiesinthemodelaretakentobethesameasthosegivenin [17,45]. Also, aset ofdominantlattice orientation extractedfrom the initial EBSD maps (Fig. 2(b)) are assigned to the grains that constitute therepresentativevolume element(RVE)oftheunder- lying microstructure in the homogenized polycrystal model (see [45] for the pole figures constructed from these extracted orien- tations).

The bending moment M about the hinge point on the crack plane is then obtained by the method explained in the supple- mentary material, which is based on a simple dynamic analysis of the beam-type specimen. In order to get a more robust esti- mate ofM,a few cross-sectionson eitherside ofthecrackplane withl >0.5b areconsidered andthe averageistaken. Theabove method isfirstappliedto thestatictestdatato evaluateM.This, along withthebendingmomentcomputeddirectlyfromtheload for this test, are plotted (after smoothing) against the specimen rotation inFig.3(a). Thetwo M-

θ

^{variationsdiffer bylessthan}

10%,whichmaybeduetopossiblethicknessdependenceofstrains andstressesthatcannotbe estimatedfromDICanalysis.Thecrit- ical value ofenergy releaserate, denoted as J_C, is ascertained as thevalue ofJcorrespondingtothetime atwhichthecrackstarts to grow in the static anddynamic tests. This stage is identified

throughcarefulvisualinspectionoftheimagescapturedduringthe test.

Eachof theabove experiments (staticand dynamic)wascon- ducteda few timesinorder toverifythe repeatabilityof there- sults.Inthestaticcaseandthreeofthedynamictests(V₀=9m/s, 17m/s and20 m/s), atleast three experimentswere performed, whileforthecasewithV₀=13m/srepeatabilityofthedatawas checked with two tests. The energy release rate histories ascer- tainedfrom allthe experimentsconducted arepresented insup- plementarymaterial andcanbe seen tobe quite consistentwith theJ_cvaluesforeachloadingcasevaryingtowithin5%(referalso Table2).

2.3. Microscopicexamination

Microstructuralstudieswerecarriedoutonthefracturedspec- imenswhichwere keptina desiccatortoprevent oxidation.Frac- tography was performed on a TESCAN VEGA 3 scanning elec- tron microscope (SEM),withsecondary electron imaging. Surface preparationfor EBSDincluded mechanical polishingusing silicon carbide abrasive paper and cloth polishing with diamond paste followedbyelectro-polishing.Theelectrolyte employed wascom- posedofa3:5solutionoforthophosphoricacidandethanol.EBSD analysis was conducted on a TESCAN MIRA 3 SEM with a stage tilt of 70°, accelerating voltage of 25 kV, beam intensity 12 and astepsizeof0.6μm.Surfacepreparationforopticalmicrography includedetchinginadditiontoabove mentionedpolishingproce- dure.Etchingwasdoneusinganacetic-picraletchant,for5s,tore- vealthegrainboundariesandtwins.Theoptical micrographywas carriedoutonLEICA DM2700Mmicroscopeatdifferentmagnifi- cationlevels.

3. Results

3.1. Macroscopicresponse

Figs. 3(b) to (d) show Mversus

θ

^{variationsfor dynamicex-}

perimentsconductedatthethreeloadingrates.Itcanbeobserved fromtheseplotsandthecorrespondingvariationforthestaticcase (Fig.3(a))thatMincreasessteeplywith

θ

^{initiallyandthentends}

tosaturateatsome levelfor

θ

> 0.04rad,owingtodevelopment oflarge scale yielding in thespecimen (see Sec.3.2).On compar- ingthe M-

θ

^{plotspertaining tothestaticanddynamictests, itis}

evidentthat thesaturationmoment valuesare higherforthelat- ter. Thus, the saturation value of M enhances fromabout 30 N- m for the static case (Fig. 3(a)) to around 45 N-m for the dy- namictest withV₀ =20m/s(Fig.3(d)).This enhancementisat- tributed to lattice reorientation in many grains caused by more pronouncedtensiletwinningneartheloading-endoftheligament inthe dynamictests aswill be seen inSec.3.5. Thus, asthe lat- ticeinthesegrainsreorientswhenTTsenvelopthem,hardsystems likepyramidal

^c+a

^{slipandCTsareactivatedwhichsharplyin-}

creasestheflowstressleadingtoacharacteristicsigmoidalstress-

Fig. 3. Moment versus specimen rotation plots for (a) static test and the dynamic tests corresponding to (b) V 0 = 9 m/s, (c) V 0 = 13 m/s, and (d) V 0 = 20 m/s. The moment extracted from both DIC analyses as explained in Sec.2.2 and directly from load for the static case are shown in (a).

Fig. 4. Energy release rate histories for (a) static test as a function of load and (b) dynamic tests with different V 0as functions of time.

strain curve under TD/RD compression (see, for example, [6,10]).

Thus, it is expected that the magnitude of compressive normal stress on the ligament near the loading edge will be signifi- cantly higher in the dynamic tests at later stages, giving rise to muchlargersaturationbendingmomentascomparedtothestatic tests.

Figs. 4(a)and(b)presenttheevolution historiesofenergyre- leaserate(J) forthestaticanddynamicexperiments,respectively.

Inthesefigures,Jisplottedagainstloadforthecaseofthestatic test andagainsttime forthedynamicexperiments.Afilled circle (•) on all the curves shown in these figures indicates the crack initiation stage.It should firstbe notedfromFig.4(a) that under static loading, J evolvesslowly up to loadof 1000 N and there- after rises sharplyowingto significant plasticdeformation inthe ligament.Fig.4(b)indicatesthatJincreasesappreciablyfasterwith respecttotimeastheimpactspeedV₀ increases.

For example, corresponding to t = 50 μs, the value of J en- hancesfrom15to53N/mm asV₀ changes from9to 20m/s.On furtherexaminingFig.4(b),itcanbenotedthatfracturetoughness Jc (refer to the stages marked by ‘•’ symbols) increases strongly withV₀,whilethetimetofracture,t_f,reduces.ThevaluesofJcand t_f pertainingtoalltestsconductedarepresentedinTable2along withthe averageloadingrate,J˙= J_c/t_f.It can benotedfromthis tablethatasV₀changesfrom9to20m/s,Jcenhancesfrom60to 88N/mm,whereast_freducesfrom133to65μs.Thus,thehighest average loadingrate generatedin thesedynamic fracture testsis J˙= 1.35GN/ms.Bycontrast,thestaticfracture toughnessisonly 32 N/mm, which is abouthalf of that pertaining to V₀ = 9 m/s case. A qualitativelysimilar trend invariation of Kc withloading ratewasreportedbyYuetal.[37]fromtheirdynamicexperiments conductedon AZ31B Mg alloyusing three-point bend specimens withcracklinealongNDandspanalongRD.

Fig. 5. CTOD measured at a distance of 150 μm behind the crack tip for different tests plotted as functions of (a) time and (b) energy release rate. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Normal strain ( ε22) contours at different time instants for a dynamic test with V 0 = 20 m/s corresponding to (a) t = 0 μs, (b) t = 32 μs, (c) t = 48 μs, (d) t = 54 μs, (e) t = 60 μs and (f) t = 64 μs. The crack tip location is indicated by vertical white arrow in (c)-(f), and the hinge point as H in (e). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The evolution of crack tip opening displacement (CTOD),

δ

^t,

measured at150 μm behindthe pre-crack tip, withtime ispre- sentedinFig.5(a).AsinthecaseofFig.4,afilledcircle(•) onall thecurvesshowninthisfigureindicatesthecrackinitiationstage.

ItcanbeobservedthatrateofincreaseofCTODwithtimeaswell astheCTODatcrackinitiationincreasewithloadingrate.Forex- ample, att =50μs,

δ

^{t enhancesfrom0.04to 0.15μm,while the}

critical CTOD atcrack initiation,

δ

^{tc, increases from 0.24 to 0.39}

μm asV₀ changes from9to 17m/s(seealso Table2,where the latter values are summarized). It is important to note by com- paring Figs. 5(a)and 4(b)that time histories of CTODand Jper- tainingtodifferentV₀ arequalitativelysimilar. HenceinFig.5(b), CTODisplottedagainstJ,correspondingtothestaticanddynamic tests, inorderto examinethe correlationbetweenthesetwo im- portant fracturecharacterizingparameters [35,46–48]. Indeed,the plots displayed in Fig. 5(b) do show a reasonable linear depen- dence of

δ

^{t (measured directly using DIC) andJ} (determined by applying Eq.(2))forall tests. Thiscorroborateswith thework of

Shih[47]based onHRRsolution[35,48]andthereforelendscon- fidenceto themethodology adoptedinthiswork toestimate the latter.

3.2. Normalstraindistributiononspecimensurface

Fig. 6(a) to (f) show contours of normal strain

ε

22, deduced fromDICanalysis,onthespecimensurface,correspondingtoase- quenceoftimeinstantsfrombeginningofloadinguptocrackiniti- ationstageforatypicaldynamictestconductedwithstrikerveloc- ityV₀=20m/s.ThesestraincontoursconfirmthattheDICanaly- sisoftheimagesacquiredusingthehighspeedcameracanindeed delineate the stress wave dominant loading experienced by the specimensincethe time rangeoverwhichtheseplots areshown fallswithin

τ

^{.OnexaminingFig.6(b),itisevidentthatacompres-}

sivestrainfieldbeginstodevelopneartheloading-edgeofthelig- amentfollowingimpactbytheincidentbar.Att=48μs,Fig.6(c) showstwo distinct symmetriclobes ofstrain haveformed atthe

Fig. 7. Normal strain ( ε22) contours at crack initiation stage for: (a) static test, and dynamic tests with (b) V 0 = 9 m/s, (c) V 0 = 13 m/s, (d) V 0 = 17 m/s, (e) V 0 = 20 m/s.

(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

far-edge, whereas a perceptible tensile strain field also starts to envelopthecracktip(indicatedbywhiteverticalarrow).Withfur- ther increasein time,thecompressive strain field atthefar-edge ofthe ligamentandthetensilestrain fieldnearthecrack tipen- hanceinmagnitudeandspreadoverlargerregions(Fig.6(d),(e)).

Also,thetwostrainlobesatthefar-edgemergetogethertoforma roughlysemi-circularshape(Fig.6(e)).Thus,Fig.6(e)showsawell developedbendingstrainfieldintheligamentwiththehingepoint (where reversal insign ofnormal strain on the ligamentoccurs) located atH (which is about0.4bfromthe cracktip). The strain contoursnearthecracktipareroundedinshape,whichistypical oftheplanestressconditionthatisexpectedonthespecimensur- face[49].Fig.6(f)depictsthestagejustaftercrackinitiation,with a significantlyintensifiedstrain field nearthe cracktipandsome loss ofcorrelationthat hasoccurredintheDIC strain mapowing tocrackextensionfromtheinitialcracktip(seeverticalarrow).

Thenormalstrain(

ε

22)contoursatthecrackinitiationstagefor allthetestsarepresentedinFigs.7(a)to(e).Theseimagesindicate thatthestraindistributiononthespecimensurfaceisqualitatively similar forthe staticandalldynamictests. However, itis impor- tant to note that the magnitude oftensile strain prevailing near the crack tipat theinitiation stage enhancesstrongly withload- ing rate(compare Fig.7(e)with7(a) and(b)).Thisimpliesstrain (and stress)intensification closeto cracktipathighloading rates (specifically, forV₀ = 17 and20 m/s).Concurrently, examination ofthe loading-edgeofthe ligamentshowsthatthe magnitudeof compressivestraininthisregionalsoenhancesappreciablyathigh V₀ (seeFig.7(e)) andthesize scaleover whichitprevailsis also larger. This has an important bearing on the dissipation due to plasticworkoccurringinthefar-fieldregionwithincreaseinload- ingrate,whichaswillbeseenbelow,iscausedbyprofusetensile twinning.

3.3. Fractography

The fracturedsurfaces ofthe broken specimensare examined using a SEM to gain insights on the operative fracture mecha- nism. Figs. 8(a)-(c) display fractographs taken with 1000x mag- nification closeto the fatiguepre-crack front nearthe mid-plane of the specimens corresponding to the static and dynamic tests withV₀ = 9m/sand 17m/s. The fatiguepre-crack front isnear thebottomedgeoftheseimages(seewhitedashedlines)andthe crackpropagatesupwardasindicatedbytheblackarrowattheex- tremeleftofthefigure.Adistinctdifferenceinthefracturesurface morphologyneartheinitialcrackfrontcanbe perceivedbetween thestaticanddynamictests.Thus, thefracturesurfacepertaining to the former (Fig. 8(a)) is decorated predominantly with quasi- brittle features like elongated groovesor flutessuch asthose la- beled asA,B andC, whichare believedtobe induced bytensile twins [3,12,13,15]. These features, which are quite different from theshallowstriation-likeimpressionsseeninthefatiguepre-crack region(belowthewhitedashed lineinFig.8(a)), werelabeledas Cleavage3 regime inthe fracture mechanismmapspertaining to MgalloysbyGandhiandAshby[50].Kaushiketal.[13]conducted experimentsusingnotchedthree-pointbendspecimensofMgsin- glecrystalandnotedthatthe crackgrowsalong twinboundaries with deflections at twin-twin intersections. They reported elon- gatedstripesorgroovesonthefracturesurfaceandconcludedthat these form by intersection of twins (of another variant) on the fracture surface. Similar groove-like features on the fracture sur- facewerealsonotedin[12,15]fromstaticfracture testsusingfa- tiguepre-crackedspecimensandbyKondoriandBenzerga[3]from cylindricaltensile specimenswith finecircumferential notches of AZ31Mgalloy.Bycontrast,farfewdimpleswereobservedonthe fracturesurfacesofthestaticallyloadedspecimen.

Fig. 8. SEM fractographs near fatigue pre-crack (FPC) front (white dashed lines) taken with 10 0 0x magnification corresponding to: (a) static test and dynamic tests with (b) V 0= 9 m/s, (c) V 0= 17 m/s. Crack propagation direction is vertical as indicated by the arrow on the left.

On the other hand, the fracture surfaces pertaining to the dynamically loaded specimens (Figs. 8(b) and (c)) display no such twin-induced brittle features but instead show a distribu- tion of micro-voids. In order to quantify the difference between the fracturemorphology seen inthesefigures andthestaticcase (Fig.8(a)), surfaceroughness(Rq) measurements weremadeover square regions ofsize 500μm usingan opticalprofilometer with a resolution of50 nm.However, the Rq values were foundto be about the same (~ 8.5 μm) for the static anddynamic test with V₀ = 9m/s, whereas it wasmarkedlylower (~3.8 μm) fora test corresponding toV₀ =13m/s,whichonlyindicatesthatthedim- plesareshallowforthehigherimpactspeedcase.Asanalternative distinguishing measure ofthe fractographic features, their aspect ratiowasquantifiedbyidentifying100suchfeaturesforeachload- ing caseandtracing themwithImage-Jsoftware(seesupplemen- tary material). The averageaspect ratios were found to be 6, 1.6 and1.3forthespecimentestedunderstaticloadinganddynamic loadingwithV₀=9and17m/s,respectively.Thesevaluesconfirm that thefractographicfeatures arehighlyelongatedgrooves/flutes forthestaticcase,whereastheyaremoreroundedinthedynami- callyloadedspecimen,whichalongwiththevisualexaminationof Figs.8(b)and(c),atteststhattheyaredimples.

Furtherconfirmation that(the almost)equi-axedfeatures such asthoselabeledasD,E,F(Fig.8(b)),IandH(Fig.8(c))are voids stemsfromthefactthey seemtohavebeennucleatedbyfracture / debonding of inclusions (see, for example, E in Fig. 8(b)). EDS analysis conductedhereandinprevious studies[3,20]show that thesevoidnucleating particlesinAZ31Mgalloysarepure Mn or Al-Mn(about25%Al,70%Mnand5%Mgbyweight),Mg₁₇Al₁₂in- termetallicsandoxides(MgO).Mostofthesevoidshavecoalesced withneighboringonesbycompleteinternal ligamentnecking,re- sultinginknife-edgelikefeatures attheir periphery(suchasE,H andI). Afew tear-ridgescharacteristicof abruptruptureofinter- void ligaments are occasionally noticed (like G in Fig. 8(b)). Al- though there is a large scatterin the voidsizesforeach loading case, their average diameterisfound toincrease from6 to9μm asV₀ changesfrom9m/sto20m/s.

Figs. 9(a),(b)and(c)show fractographs(1000xmagnification) takenatadistanceof1.5mmaheadofthefatiguepre-crackfront nearthespecimenmid-planecorrespondingtostaticanddynamic tests with V₀ = 9 m/s and 17 m/s. It can be observed by com- paringFig.8(a)and9(a)thatthefracturesurfacefeaturespertain- ingtothestatictesthavechangedwithcrackgrowth.Thus,unlike Fig.8(a),manymicro-voids(suchasthoseindicatedasK,LandM) can benoticed inFig.9(a)withevidenceoffracturedparticles in some cases(see M)along witha fewbrittle featureslikethat in- dicated asJ.Thelatterarefar morenumerousclosetotheinitial crackfront(Fig.8(a)).Thus,abrittletoductiletransitionseemsto

haveoccurredwithcrackextensioninthestaticexperiment,which wasalsoreportedbyPrasadetal.[15]intheirstaticfracturetests withfatiguepre-crackedfour-pointspecimens.Ontheotherhand, thefractographspertainingtothedynamictestscontinuetoshow dimplesfaraheadoftheinitialcrackfrontaswell(Fig.9(b),(c)).In fact,somelargevoidswithfracturedparticles insidethem(likeN andO)canbenoticedinFig.9(b).Asinthecaseoftheregionnear thepre-crackfront (Fig.8(b),(c)),onlyafew tearridges(suchas PinFig.9(b))canbeseenandmostvoidshavecoalescedbycom- pleteinternalligamentnecking.

3.4. Opticalmicrographsofdeformedspecimens

InFigs.10(a)and(b),lowmagnificationopticalimagesofade- formedspecimencorrespondingtoastatictestandthetwohalves ofa fracturedspecimen obtainedaftera dynamictest conducted withVo =17m/sareshown. Specimensare polishedandetched followingtheprocedureoutlinedinSection2.3,inordertoreveal thenatureofinelasticdeformation.

A large,roughly semi-circularbright region can be noticed in both specimens spreading downwards from the (upper) loading edge (Figs. 10(a) and (b)). Further, a curious pattern of curved bands resembling fingers can be observed near the bottom por- tion of the above bright region. The static specimen (Fig. 10(a)) alsoshowsabrightdeformedregionenvelopingthecracktipnear theloweredge.Inordertounderstandhowtheabovenotedbands formneartheloadingedge,oneofthestatictestswasinterrupted andthespecimenwasunloaded. Itwasthenoptically imagedaf- terfollowingtheusualpolishing/etchingprotocol.Thisdeformed specimen,whichisdisplayedinFig.10(c),revealsasetofparallel whitebandsthatareroughlyequallyspacedextendingdownwards fromthetopedge.Withcontinuation ofloading,thesebandsap- parentlywidenandmergewitheachother,exceptnearthebottom portionwherethey retaintheirdiscrete shape,formingthebright semi-circularzoneobservedinFigs.10(a)and(b).

Asmallregionencompassing thebandsinthe bottomportion ofthebrightsemi-circularregion,withinthebluerectangularbox indicated inFig.10(b), isexamined undera highresolutionopti- cal microscope.This opticalmicrograph ispresentedin Fig.10(d) andshows two parallel bands, whose boundariesare marked by reddashedlines,whichareseveralgrainsizesinwidthcontaining a highdensityof tensiletwins.The region betweenthesebands, whichhasbothsmallandbiggrainsandisalsoseveralgrainsizes inwidth, iscompletelyfreeoftwins.Acarefulexaminationofthe twotwinbandsinFig.10(d)revealsthatthetwinsextendroughly parallel to the orientation of the band and ascontinuous chains spreadingacross grain boundaries.Possible reasons fortheorigin

Fig. 9. SEM fractographs about 1.5 mm away from original crack front taken with 10 0 0x magnification corresponding to: (a) static test and dynamic tests with (b) V 0= 9 m/s, (c) V 0= 17 m/s. Crack propagation direction is vertical as indicated by the arrow on the left.

Fig. 10. Optical images showing deformed specimen corresponding to (a) static test, (b) dynamic test with V 0= 17 m/s (both fractured halves) and (c) a static test after interrupting and unloading. (d) A higher magnification optical micrograph corresponding to a small region within the blue rectangular box marked in (b) showing two twin bands. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

ofthesetensiletwinbandsandtheirinfluenceonthefracturere- sistanceoftheMgalloywillbediscussedinSec.4.

InFig.11(a),anopticalmicrographtakennearthecracktipaf- ter a shortextension isshown. Inthismicrograph,the crackline can be seen atthebottom left corner,withthe currentcracktip indicated by a red horizontal line. Considerable twinning can be perceivedingrainssurroundingthetipspreadinguptoadistance of about100 μm ahead of the tip and 150 μm in the direction normaltothecrackline.AnalysisofEBSDmaps(seeSec.3.5)con- firms thattheseare tensiletwins.Infact, some grainshavemul- tipleintersectingtwins(seegrainmarkedasQ)andseveralparal- leltwins,possiblyofthesamevariant(grainidentifiedasR).Also,

someofthetwinsextendacrossgrainboundaries(seelocationin- dicated asS)similar toFig. 10(d).In Fig.11(b),an optical micro- graph ofthe region nearthe initial crack tip, obtainedfrom one ofthebrokenhalvesofaspecimentestedunderdynamicloading withV₀ = 17m/s, is displayed. Here, thecrack lineis along the rightverticalboundaryoftheimageandtheapproximatelocation oftheinitialcracktipismarkedbyaredhorizontalarrow.Incon- trast to Fig. 11(a), very few twins may be noticed in the grains above and to the side ofthe initial crack tip. This suggests that thereisareductioninamountoftwinningaroundtheoriginaltip withenhancementinloadingrate.Inordertoconfirmthis,rectan- gulardomains ofsize200μm ×170 μmcloseto theinitialcrack

Fig. 11. Optical micrographs near crack tip corresponding to (a) static test and (b) dynamic test with V 0= 17 m/s. Optical micrographs near the far-end of the ligament close to loading edge corresponding to (c) static test and (d) dynamic test with V 0= 17 m/s.

tip(liketheopticalimagesshowninFig.11(a),(b))wereanalyzed using the point count method (ASTM E562–19 [51]) andthe av- eragetwinarea fractionwasestimated. Thesevaluesare givenin Table2,andcanbeobservedtodecreasefromabout12%forstatic loadingto4%forthedynamictestwithV₀=20m/s.

Figs.11(c)and(d)showopticalmicrographsobtainedfromthe loading-end of the ligament (refer to the top edge of brightre- gion in Figs. 10(a), (b)),corresponding tothe static anddynamic test withV₀ = 17m/s,respectively. Boththesemicrographsindi- cate massivetwinning at the far-endof the ligamentwith many grainshavingmultipleparallel twins(see,forinstance,grain Tin Fig.11(d))andintersectingtwins(grainUinFig.11(d)).Twinsex- tendingacrossboundariesofmultiplegrains(seetheonesadjacent to grain TinFig.11(d)) canalso beperceived. Again,EBSD maps confirm that these are tensile twins, andas will be seen below, resultinstrongchangesinthetextureinthisregion.

3.5. EBSDmapsnearcracktipandspecimenfaredge

InFig.12(a),theIPFmapneartheextendedcracktipalongwith (0001) and (1120) PFs corresponding to a static test are shown.

The crackthat hasextendedin theverticaldirectioncan beseen in the IPFimage. Similar IPFmaps andPFs pertaining to the re- gion near the initial crack tip, extractedfrom one of the broken halvesofspecimenssubjectedtodynamicloadingwithV₀=9and 17 m/s are presented in Fig. 12(b) and(c), respectively. In these IPF maps,the initial crackis along the lower edge ofthe image.

TheIPFmapcorrespondingtothestatictest(Fig.12(a))showsthat mostgrainscontinuetoremainintheinitial(almostbasal)orien- tation. However, thegrainspresenton thesidesofthecrack line and up toa distance ofabout100 μm ahead ofit are populated with deformation twins. Analysis ofthe angle change ofthe lat- ticewithin thesetwins,withrespecttotheparentgrainindicates

that it is about86°, which confirms that these are TTs. By con- trast,fewertwinsarenoticedinthegrainsneartheinitialcracktip fromthe IPFspertaining tothedynamictests(seeFigs.12(b)and (c)).Infact,hardly anytwinsarevisibleinthe IPFcorresponding to V₀ = 17m/s (Fig. 12(c)).This corroborateswiththe reduction in average twin area fraction estimated from the optical micro- graphstakennearthecracktip.OncomparingthePFsdisplayedin Fig.12withthosepertainingtotheundeformedspecimen(Fig.2), it can be noticed that for the statically testedspecimen anddy- namiccasewithV0 = 9m/s, some ofthebasal poleshave tilted fromnearNDtowardsRD(see,inparticular,(0001)PFsshownin Figs.12(a),(b)). However,theoverall texturechangesaresmallfor thespecimentestedwithV₀=17m/s(Fig.12(c)).

TheIPFmapsandPFsobtainedfromtheregionadjacenttothe loading-endoftheligamentforastatictestanddynamictestsper- formedwithV₀=9and17m/saredisplayedinFigs.13(a)-(c),re- spectively.ItcanbenotedfromFig.13(a)that mostofthegrains, whichwereinitially innear-basalorientation,havemultiplepar- allelorintersecting twinsasnotedfromtheopticalmetallograph presentedinFig.11(c). Ascomparedto the(0001)PF takenfrom theregionclosetothecracktip(Fig.12(a)),manybasalpoleshave shiftedtotheperiphery inthePFshowninFig.13(a),althougha strongconcentrationofthesepolesatthecenter(i.e.,closetoND) isstillpresent. Perceptiblechangeinthe(1120)PFincomparison to theinitial one (Fig. 2) can alsobe noticed fromFig. 13(a).On examiningthe IPFforthe caseV₀ = 9m/s(Fig.13(b)), itcan be seen that the population ofTTs is higher compared tothe static case.Infact,thelatticeinsomegrains,suchasthosemarkedbyV andW,havealmostcompletelyre-orientedduetotensiletwinning (althoughasmallportionofthelattergrain stillappears tobein theinitial near-basal orientation).The PFs pertainingto thiscase showmoreprofoundchangesfromtheinitialonesascomparedto the statically testedspecimen(Fig.13(a)).Thus, the (0001) PFin

Fig. 12. IPF maps and PFs extracted near the initial crack tip corresponding to (a) static test and dynamic test with (b) V 0= 9 m/s, (c) V 0= 17 m/s. (d) Legends for IPFs and PFs. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 13(b)indicates thatthe intensityofpoles attheperiphery is moreappreciablecomparedtoFig.13(a)andthatatthecenterhas decreased. The(1120) PFalsoexhibitsstronger changes(compare withFig.2).

Finally, turningto Fig.13(c)pertaining tothe specimentested with V₀ = 17 m/s, it can be observed from the IPF that in al- mostallgrains,thelatticehascompletelyre-orientedsothatNDis along[1120]or[1010]direction,duetotensiletwinsfullyenvelop- ingthesegrains.The(0001)PFconfirmsthisobservationwithcon- centrationof polesseen attheperiphery, whereas thereisnegli- gibleintensityatthecenter.The(1120)PFalsoindicatesa strong intensity atthecenter incontrast tothe initial one (referFig.2) where this was negligible.Thus, Fig. 13 shows that thedensity of tensile twinsand theassociatedchanges intexture neartheloading edgeforthedynamicallytestedspecimensaremoreprofoundthanthe static onesand,infact,enhancewithimpactvelocity.Thisisconsis- tent with thehigher (compressive) normalstrain levels observed

intheaboveregionatcrackinitiationastheloadingrateincreases (referFig.7).

4. Discussion

4.1. Effectofloadingrateonfracturemechanism

As noted in Sec.3.3, the SEM fractograph near the fatigue pre-crack front corresponding to the static test (Fig. 8(a)) shows predominantly quasi-brittle features like elongated grooves or slits which have been traced to twin-induced brittle cracks [3,12,13,15,50]. Indeed,the optical micrographs and IPFmaps for thestatictestsexhibitafairlyhighaverageareafractionofTTsin the neighborhood of the crack tip (Figs. 11(a) and 12(a)), which reinforces the inference based on the fractographs. Onthe other hand,inthedynamictests, theTT areafractionaround thecrack tip diminishes (see Table 2). Concurrently, the fractographs (Fig.

Fig. 13. IPF maps and PFs near loading edge corresponding to (a) static test and dynamic tests conducted with (b) V 0= 9 m/s, (c) V 0= 17 m/s. (d) Legends for IPFs and PFs. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

8(b)-(c))display nearly equi-axeddimples, whichestablishes that a quasi-brittletoductiletransitioninfracture mechanismsoccurs asloading changes fromstaticto dynamicinthe presentexperi- ments.

In ordertorationalizethesetrends, itisimportant tofirstre- call that previous static tests [3,15] have shown similar fracture mechanism transition asnotch acuity (or near-tip stress triaxial- ity)isreducedfromhightomoderatelevels.Further,itwasnoted in [15] that the average twinarea fraction in grains around the tip drops from13.5% for a fatiguepre-cracked bend specimento 6.8%foranotchedspecimenwithrootradiusof60μm.CPFEcom-

putations [17] of theexperiments reportedin [3]have alsoindi- cated similar reduction in TT volume fractionbetween a sharply notchedcylindricalRN-2specimenandmoderatelynotchedRN-10 specimen(which display above notedfracture mechanismtransi- tion). At a more fundamental level, it has been postulated that TTsare triggered by pile-upofbasal dislocationsatgrain bound- aries [45,52]. Thus, it may be concluded that as stress triaxial- ity near a notch or crack tip decreases from high to moderate levels: (a) twin density near the tip diminishes and (b) quasi- brittleto ductilefracture mechanismtransitionoccursinthisMg alloy.

or higher (see Table 2). From the elastic-plastic finite element analysis reported in [30] for a three point bend specimen with a/W = 0.5, the drop in near-tip hydrostatic stress, with respect to static case, is found to be around 0.6

σ

0 corresponding to dy- namicloadingwithJ˙of0.5GN/(ms), where

σ

0 isthetensileyield strength. This confirms that stress triaxiality will be lowered by an amount similar tothat between theRN-2 andRN-10 notched cylindricalspecimenstestedin[3]whichtriggeredfracturemech- anismtransitioninthisalloy.Thisexplainstheoriginoftheabove transition seen between the static anddynamically loaded spec- imens (even withV₀ = 9 m/s) mentioned above. These features areillustratedschematicallyinFig.14(a)and(b).Forthestatically loaded specimen, high stress triaxiality caused by the J-dominant HRRfield[48]prevailingoutsidethefractureprocesszone(FPZ)close to thecracktip may be responsiblefor activating theTTs andcon- sequent quasi-brittle failure (Fig. 14(a)). On the other hand, the two-parameterJ-Qfield[34,35]existsnearthetipinthedynamically loadedspecimens,withthetriaxialityparameterQbecomingincreas- ingly negative as J˙ enhances[29–33],effecting thechange to slow voidgrowthandcoalescenceintheFPZ(Fig.14(b)).

4.2. Dependenceoffracturetoughnessonloadingrate

A strong enhancement in fracture toughness J_c, with loading rate is observed for the present Mg alloy. Thus, there is an al- most200%increase inJc correspondingtothedynamictestswith V0 = 20m/sascompared to staticloading.This enhancementis higher than in many other engineering alloys [24–28]. The pro- nouncedincreaseinJcwithJ˙forthepresentAZ31Mgalloycanbe rationalized bynotingthat thecontributionto fracturetoughness arises from two sources, viz., intrinsic work of separation inthe fracture process zone, denoted here asJ_c^fp,and from background plasticdissipation,indicatedasJ^bp_c .Aswillbediscussedbelow,the evidencepresentedinSec.3demonstratesthatboththesecompo- nentsincreasewithloadingrate.

4.2.1. Contributiontotoughnessfromintrinsicworkofseparation J_c^{f p}

The contributionfromintrinsicworkofseparationdependson the operativefracture mechanism. AsnotedinSec.4.1,thismech- anism changes from quasi-brittle to ductile as loading changes fromstatictodynamic.Moreover,inthecontextofisotropicplas- ticsolids,ithasbeennotedthatductilefractureprocesseswillbe impeded asloadingrateenhancesowing tohigherlevels ofneg- ative Q caused by inertial effects [30,36]. This corroborates with retardationinvoidgrowthastriaxialitydropswhichhasbeenob- served forbothisotropicplasticsolids[54]andMgsinglecrystals [21,22].Anaddedfactorisstrainratesensitivitywhichwasshown in[36]tofurtherslowdownvoidgrowthunderdynamicloading.

Theabovenoteddropinnear-tipstresstriaxialitywouldtrans- lateintermsofelevationinJ_c^{f p}asshownbyPrasadetal.[15]from theirstaticfracturetestswithincreaseinnotchrootradius.Anap- proximate estimate ofthis enhancementcan be madeby consid-

25N/mmmaybeexpectedowingdropinnear-tipstress triaxial- ity.

The above features anticipated in the variation of J_c^{f p} with Q areschematicallyillustratedinFig.15.Alsoindicatedinthisfigure are thespecific valuesof J_c^{f p} pertainingto thestatic case(where Q = Q_s) and underdynamic loading with increasing J˙ (where Q becomesprogressivelymorenegativesothatQ₃ <Q₂<Q₁).Asis clearfromthisschematic, J_c^{f p}willenhance withJ˙ andthe brittle- ductiletransitionwouldfurtheraccentuatethecontributiontothe dynamicfracturetoughnessfromthenear-tipfractureprocessesas comparedtoJ_cs^{f p}.

4.2.2. Contributiontotoughnessfrombackgroundplasticdissipation Amajorcontributiontofracturetoughness,J_c^bp,arisesfromdis- sipation in the background plastic zone, which includes the in- elasticdeformationprocessesofslipandtwinning.TheDICstrain maps presented in Fig. 6 and 7 indicate that strong plastic de- formation occurs in the region around the crack tip as well as closetotheloadingedge.The opticalimages,Fig.10(a),(b)show a large bright semi-circular zone near the loading-edge of the ligament,which underhighresolution microscopy revealstensile twinsforminginbands(Fig.10(d)).Thisintriguingpatternoftwin bands wasalso observed by Baird et al. [56] in thin, unnotched AZ31Mgalloysheetspecimensunderthree-pointbending.Under thisloadingcondition,thefar-fieldregionissubjectedtocompres- sionnormaltotheligament(notefromFig.7that

ε

22 isnegative inthisregion).Duetothenear-basaltextureofthisalloy,withc- axisalignedclosetoND,theabovecompressionnormaltothelig- amentfavoursformationofTTsinthisregiontoaccommodatethe expected out-of-plane bulging of the specimen. Here, it is to be notedthat,inadditiontovariationinlatticeorientationandthere- fore the Schmid factor for TTs, the critical resolved shear stress (CRSS) associated with twinning is also expected to vary due to the variationin grain size (bimodalgrain size distributionof the presentalloy)withthelargergrainshavinglowerCRSSvalues[57]. Thus, theaverageinitial spacingbetweenthetwinbandsseen in Fig.10(c)wouldbe determinedby the spatialfluctuation inratio ofSchmidfactortoCRSSfortensiletwinning.

The TTs thus nucleated adjacent to the loading edge, extend across thegrainsbefore gettingarrested at their boundaries. The stress concentration created at the arrested twin tips nucleate twinsintheadjacentgrainsandtheprocessrepeatsformingcon- tinuousbandsoftwinsspreading acrossseveralgrainsasseenin Fig.10(c)and(d).Thesebandswidenandmergewithfurtherload- ingenvelopingtheentireregionintheupperpartofthespecimen (abovethe plastichinge point). Thisfar-field plasticregion, char- acterizedbyprofuse twinning,is indicatedinthe schematicsdis- playedinFig.14(a)and(b).

Fig. 7 shows that at crack initiation, the intensity of normal strain both in the crack tip plastic zone and far-field region, as well asthe extentover which they prevailenhance withloading rate.Moreover,theIPFmaps(Fig.13)indicateprogressivelyhigher

Fig. 14. Schematic representation of fracture process zone (FPZ) and far-field plastic region corresponding to (a) static test with high triaxiality near crack tip and (b) dynamic test with moderately reduced levels of triaxiality near crack tip.

Fig. 15. Schematic showing variation of J c^{f p}with crack tip constraint parameter Q . The specific values of J c^{f p}for the static case ( Q = Q s) and those pertaining to increas- ingly negative Q levels ( Q 1, Q 2, Q 3) attained with enhancement in J are indicated. ˙

densityofTTsnearthespecimenfar-edgewithmanygrainsexpe- riencingcompletelattice re-orientationathighV₀ leadingtopro- nounced texturechanges (Fig.13(c)). Thiscorroborates well with theobservationbyDudamelletal.[9]forAZ31Mgalloythatunder RDcompressionTTpropagationisenhancedathighstrainrates.

As suggested by Prasad et al.[14], the contribution from TTs to thebackgroundplasticdissipation intheligamentcanbe esti- matedas

U_tw^p = W_tw^pVtw, (4)

where, W_tw^p =

τ

tw

γ

tw is the specific plastic work due to tensile twinning and Vtw = BAtw favg is the volume of twinned region in the ligament. Here,

τ

^{tw is the average CRSS for tensile twin-}

ning (which isassumedas40 MPain thefollowing computation [14,58])and

γ

^{twisthetwinningshearwhichis0.13[59].Also,Atw}

isthetotal(in-plane)areaoftheregionadjacenttotheuncracked ligament wheretwinning is observed (semi-circular whiteregion inFig.10(a),(b),whichhasaradiusofabout6mm),Bisthespec- imenthicknessandfavgistheaveragetwinvolumefractioninthis region.

Toobtainapproximateupperboundstof_avg,EBSDmapsonthe specimen surface near the far-edge such as those shown in Fig.

13 were used to evaluate thetwinned area fraction (see supple- mentarymaterial). Assumingthatthesetwinspersistthroughthe specimenthicknessasnotedin[13],favgisthenestimatedas0.21, 0.32and0.64forthestaticcaseanddynamicloadingwithV₀ =9 and17m/s,respectively.Further,inviewofEq.(2),thecontribution fromtensiletwinninginbackgroundplasticzonetothetoughness can be writtenasJ^bp_c ^,tw=

η

^Utw^p/(^Bb)^.On using thedata indicated

above,thisisfoundtobeabout12.5N/mmforthestaticcaseand about19and38N/mmforthedynamicloadingwithV₀ = 9and 17m/s, respectively.Theseroughestimatesrangebetween32and 47%ofthe corresponding totalJ_c valuesgiven inTable2 andare thereforesubstantial.Moreimportantly,itmustbenotedthat the contributiontothetoughnessfromdissipationduetotwinningin theregionadjacent totheligamentincreasesby afactorofthree asloadingchangesfromstatictodynamicwithV₀=17m/s.

Interestingly,theoverall enhancementintoughness asloading changes from static to dynamicwith V₀ = 9 m/s, by combining the above two approximate estimates for J_c^{f p}and J_c^bp,tw is about 31N/mmwhichiscomparabletotheactualvalueof28N/mmin- dicated inTable1. Thissuggeststhat theanalysispresentedhere providesarationalbasisforinterpretingtheresults.

5. Concludingremarks

Staticanddynamicfractureexperiments(correspondingtodif- ferentloading rates)havebeen conductedinthiswork, usingfa- tiguepre-crackedthree-pointbendspecimensofarolledAZ31Mg alloy.The focusofthe work hasbeenon understandinghow TTs influencethefracturebehaviorathighloadingratescausedbyim- pact(resultinginJ˙greaterthan0.5GN/(ms)).Thekeyobservation is that while the density ofTTs nearthe loading edge enhances stronglywithloadingratecausingthelatticeinmanygrainstore- orient,therebyimpartingpronouncedtexturechanges,itdecreases nearthecracktip.Theformer,whichisconsistentwithhighstrain rate compression experiments performed along RD [9], leads to strongtexturehardening aswell astoelevationinthe saturation valueofthe bendingmomentontheligament.Moreimportantly, itenhancesthecontributiontotheoveralltoughnessduetodissi- pationinthebackgroundplasticzone(whichisroughlyestimated tobemorethanafactorof3forV₀=17m/sascomparedtostatic loading).

ThedecreaseinTTareafractionnearthetipontheotherhand, isattributed todrop instress triaxiality causedby local material inertiawhichhasbeenshowninpreviouselastic-plasticcomputa- tionalstudies[29–33]toenhancewithJ˙.Infact,evenforthelow- estJ˙ofabout0.5GN/(ms),thisconstraintloss(|Q

σ

0|)isexpected tobearound0.6

σ

0[30],whichiscomparabletothedropinstress triaxialitywithincreaseinnotch rootradiusbetweenthesharply grooved RN2andmoderatelynotchedRN10 cylindricalspecimens testedunderstaticloading by KondoriandBenzerga [3]. Thisre- sultsinaquasi-brittle toductiletransition(asin[3])whenload- ing changes from static to dynamic (even for the lowest V₀ of 9 m/s) with the fracture morphology comprising ofhighly elon- gatedgrooves(havingaspect ratioofabout6) fortheformerand nearly equi-axeddimples(having aspect ratioaround1.5) forthe

ing components (such as in automotive or aircraft applications) wherepossibleimpactloadingmayoccurtriggeringdynamicfrac- tureatstressconcentrations.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingfinan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

R.NarasimhanwouldliketogratefullyacknowledgetheScience and EngineeringResearch Board(Government ofIndia) forfinan- cialsupportthroughtheJ.C.BoseFellowshipresearchgrant.Theau- thorswouldliketothankProfessorG.Ravichandranforproviding experimental facilities andDr. Zev Lovinger forhelping withthe dynamictests.

Supplementarymaterials

Supplementary material associated with this article can be found,intheonlineversion,atdoi:10.1016/j.actamat.2020.10.059.

References

[1] B.L. Mordike , T. Ebert , Magnesium Properties - applications - potential, Mater.

Sci. Eng. A. 302 (2001) 37–45 .

[2] H. Somekawa , K. Nakajima , A. Singh , T. Mukai , Ductile fracture mechanism in fine-grained magnesium alloy, Philos. Mag. Lett. 90 (2010) 831–839 . [3] B. Kondori , A .A . Benzerga , Effect of stress triaxiality on the flow and fracture

of Mg alloy AZ31, Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 45 (2014) 3292–3307 .

[4] M.R. Barnett , Twinning and the ductility of magnesium alloys. Part I: “Tension”

twins, Mater. Sci. Eng. A. 464 (2007) 1–7 .

[5] S.R. Agnew , Ö. Duygulu , Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B, Int. J. Plast. 21 (20