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Mechanics of Materials

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In-situ quantification of intra and intergranular deformation in puremagnesium using full-field measurements at low and high strain rates

Peter Malchow, Behrad Koohbor, Suraj Ravindran, Addis Kidane⁎

Department of Mechanical Engineering, University of South Carolina, 300 Main St, Columbia, SC 29208, USA

A R T I C L E I N F O

Keywords:MagnesiumDuctilityGrain boundaryHopkinson barDigital image correlation

A B S T R A C T

Due to the presence of weak grain boundaries as well as significantly coarse grain structure with minimaldeformability, grain boundary cracking is almost an inevitable source of failure when a cast magnesium-basedalloy is deformed at low homologous temperatures. The main objective in this work is to quantitatively in-vestigate the contribution of inter- and intragranular deformation to the macroscale deformability and ductilityof nominally pure as-cast magnesium subjected to quasi-static and dynamic loading. The proposed mesoscalefull-field measurement approach is first presented and verified through measurements carried out to investigatedeformation and grain boundary cracking in cast magnesium subjected to quasi-static loading. The method isthen extended into the analysis of mesoscale deformation and failure under dynamic loading conditions using anexperimental setup consisting of a split Hopkinson pressure bar (SHPB) and a high-speed imaging system. In bothcases, the effect of the initial grain configuration on the local deformation response of Mg is investigated. Theresults indicate that the contribution of grain boundary region deformation to the total deformation exerted onthe Mg samples is significant and depends on the initial grain configuration. Also, the strain rate sensitivity of thematerial is found to be dominated by the material deformation in the vicinity of grain boundaries.

1. Introduction

The effects of grain boundaries on the properties of crystallinematerials ranging from deformation resistance to electrical conductivityhave been studied for decades. From the mechanical behavior per-spective, the presence of large fractions of grain boundaries (equivalentto having finer grains) is generally associated with higher strength,higher fracture toughness and improved ductility (Dieter, 1986).However, certain metals and alloys may still show relatively lowstrength and brittle failure response even with fine grain structures.Low strength grain boundaries and the delayed activation of slip sys-tems in these polycrystalline metals are documented as being theprincipal sources for such mechanical behaviors (Hughes et al., 2007).Magnesium and its alloys are examples for such relatively brittle me-tallic systems. Despite having a high specific strength which makesthem appealing to automotive and aerospace applications, magnesiumand its alloys are among a group of materials which show low ductilityat room temperature. This behavior is mainly due to limited activationof deformation systems at low homologous temperatures, whereas theactivation of non-basal slip systems at elevated temperatures can sig-nificantly increase the degree of intragranular deformation, enhancingthe ductility (Mordike et al., 2001).

Grain refinement through severe plastic deformation followed bythermomechanical processing has been widely used to increase theductility of Mg-based alloys at low working temperatures. (Arab et al.,Arab and Akbarzadeh 2013; Li et al., 2009; Yamashita et al., 2001).Producing a uniform and fine equiaxed grain structure with dispersedthermally stable particles has been identified as a method for processingof Mg alloys with superior forming ability (Xie et al., 2017). A refinedequiaxed microstructure enables grain boundary sliding, a major me-chanism for superplastic deformation in Mg alloys, even at low homo-logous temperatures. In as-cast conditions, Mg alloys generally exhibitpoor deformability due to both composition and grain structure het-erogeneities. Forming ability issues exacerbate in the case of cast pureMg, as no thermal stabilizing particle would be present to hinder graingrowth during processing. Therefore, due to the presence of weak grainboundaries, as well as significantly coarse grains with minimal de-formability, grain boundary cracking is almost an inevitable cause offailure when cast Mg is deformed at low homologous temperatures.

Efforts have been devoted to explor the characteristics of grainboundary cracking and understand intergranular fracture and nomin-ally brittle behavior in various alloy systems. Metals with hexagonalclose-packed (hcp) crystal structure have been given special attentiondue to their susceptibility to grain boundary cracking. Among studies

https://doi.org/10.1016/j.mechmat.2018.07.007Received 3 May 2018; Received in revised form 5 July 2018; Accepted 12 July 2018

⁎ Corresponding author.E-mail address: kidanea@cec.sc.edu (A. Kidane).

Mechanics of Materials 126 (2018) 36–46

Available online 27 July 20180167-6636/ © 2018 Elsevier Ltd. All rights reserved.

T

that explore the origin of grain boundary cracking in hcp metals is thework of Hughes et al. (2007), in which a pseudo-three-dimensionalmodel consisting of hexagonal arrays was used to model cleavagecracking in polycrystalline zinc. Although such modeling approachescan facilitate a more in-depth study of possible mechanisms governingdeformation and failure in the examined materials systems, in-situ ex-perimental measurements to verify modeling results are still veryscarce. The lack of systematic experimental studies in this area may bedue to challenges which arise with the implementation of high re-solution in-situ measurement protocols that enable quantitative-basedinvestigations, especially at grain scales.

Recent advances in multiscale full-field measurements, in particularthe development of digital image correlation (DIC), have facilitatedaccurate quantification of mesoscale deformation in a variety of ma-terial systems, from metals to polymer and ceramic composites(Efstathiou et al., 2010; Koohbor et al., 2015; Raabe et al., 2003; Tracyet al., 2015; Zhao et al., 2008). Constant improvement of spatial re-solution in imaging devices has also led to the enhancement of full-fielddeformation measurements at grain and sub-grain levels. Such ad-vancements currently allow for studying grain-level in-situ strain lo-calization and damage (Tasan et al., 2014), slip band formation (DiGioacchino and Quinta da Fonseca2013), micro-strain evolution inultra-fine grained alloys (Zhang et al., 2014), in-situ phase transfor-mation (Das et al., 2016), active slip system identification in poly-crystalline metals (Chen et al., 2017; Di Gioacchino et al., 2015; Gueryet al., 2016a), and parameter identification for crystal plasticity mod-eling and simulations (Bertin et al., 2016; Guery et al., 2016b). Despiteconstant improvements in spatial resolution of in-situ measurementscarried out under slow deformation rates, research works dealing within-situ characterization of mesoscale deformation at high strain ratesare still very scarce. In fact, to the best of our knowledge, there existonly a few studies that take into account in-situ dynamic deformationmeasurements in metallic systems (Bodelot et al., 2015). Recent trendsin the literature have seen growing interest in the applications of full-field measurements and the in-situ characterization of local deforma-tion and failure phenomena in a number of different materials systemssubjected to dynamic loading conditions (Koohbor et al., 2017a;Ravindran et al., 2016a; Ravindran et al., 2016b). Advances in high-speed and ultra-high-speed cameras, in terms of both spatial and tem-poral resolutions, have paved the way for such studies (Pierron et al.,2011).

The main objective in this work is to present the latest findings inmesoscale full-field measurements conducted on nominally pure castmagnesium. The methodology introduced here provides quantitativeinformation on the contributions of inter and intragranular strains tothe macroscale forming ability of the material. In the following, detailsregarding material preparation and the proposed experimental ap-proach are provided. Discussions regarding the quasi-static and thedynamic testing protocols are provided. Next, results obtained throughoptical-based digital image correlation measurements carried out toinvestigate deformation and grain boundary cracking under quasi-staticloading conditions are presented. The approach is then extended intothe analysis of mesoscale deformation and failure in high strain rateconditions. Roles of initial grain configuration on the deformation re-sponse of cast Mg under different strain rates is compared and discussedin detail. Finally, the contribution of inter-granular strains to the overalldeformation of samples with various grain orientations is quantified.

2. Experimental

2.1. Material and specimen geometry

The material examined in this work was nominally pure (99.9%) as-cast magnesium. The reason for choosing the as-cast condition wassupported by the facts that (1) grain structure in a cast magnesium ingotis coarse enough to enable optical-based mesoscale full-field

measurements with proper resolution, (2) the grain boundaries aremechanically weak, escalating the probability of grain boundarycracking and intergranular fracture, and (3) a single ingot of castmagnesium offers a wide variety of grain shapes and orientations. Thelatter point is discussed in more detail in the following sections.

Cubic samples (10×10×10 mm3) were cut from a single castingot using a band saw and machined using a milling apparatus.Samples for quasi-static loading were extracted from various locationsof the ingot to provide a range of different grain shape/orientations.Fig. 1 schematically shows the locations from which quasi-static sam-ples were extracted. It should be noted that an as-cast magnesium ingotcontains a variety of local grain structures. Due to the nature of soli-dification processes in pure metals, columnar grains are formed on theinner mold wall and are elongated towards the center of the ingot. Acentral zone containing equiaxed grains is formed at the ingot center.Depending on the purpose, specimen location and loading directionscan be selected such that a variety of different loading conditions andmechanical responses can be studied concerning the initial grain con-figuration. The samples were extracted such that the influence of grainorientation and loading direction would be reflected in the measure-ments. As such, for simplicity, samples were named as “H”, “V” and “E”,for horizontal, vertical and equiaxed grain configurations, respectively(see Fig. 1).

Common surface preparation practice, including stepwise grindingand polishing with an aqueous alumina powder mixture, was applied toeach sample. The relatively soft mechanical nature of cast magnesiummade it challenging to achieve a mirror surface finish. However, thiswas not considered to be a significant issue in the present work becausethe sample surface was intentionally speckled to enable image corre-lation. The primary intention with polishing was to reveal the grainboundaries without the obstruction of major scratches or defects on thesurface. The locations of these boundaries were later used to distinguishbetween inter and intragranular deformation domains. Specimens wereetched for 10 minutes in an etchant solution consisting of picric acid,acetic acid, water, and ethanol. The typical microstructure of a mag-nesium sample in this work is depicted in Fig. 2a. It should be statedthat no notable impurity content was detected in the samples. Thecoarse millimeter-sized grain structure of the cast specimen is clearlyshown in Fig. 2a.

One surface of each cubic sample was speckled for full-field DICmeasurements. Speckle pattern production was achieved by firstcoating the sample surface with a thin layer of flat white paint.Immediately afterward, a thin coat of black carbon powder was sprayedon the partially-dried white paint to produce a high contrast and denseblack and white pattern suitable for DIC measurements. The averagepowder particle size in this work was 10 µm, small enough to enable theselection of a small correlation window (subset size) that allows for

Fig. 1. Schematic representation of the locations in a single Mg ingot fromwhich samples were extracted. Loading directions are marked with bold arrows.Chill zones on the mold interior walls are not illustrated.

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detection and study of highly localized deformation patterns developedin the vicinity of grain boundaries. Note that for our purposes, thecarbon powder offers a much finer speckle configuration at highermagnifications than traditional black paint. Fig. 2b shows a typicalpattern applied to samples in this work.

2.2. Experimental setup

Quasi-static experiments were carried out in a conventional com-pression testing machine with a 22 kN load-cell capacity. The mea-surement accuracy of the load-cell was ± 10 N, equivalentto ± 0.1MPa margin of error in stress measurements. The specimenwas inserted between hardened stainless steel plates and compressed ata constant rate of 1mm/min which is equivalent to a nominal strainrate of 1.7× 10−3/s.

Images of the speckled surface during loading were acquired using a5 MP CCD camera (Grasshopper GS3-U3-51S5M-C, FLIR) at a rate of 1image per second and at a full-field resolution of 2448× 2048 pixel2.Imaging rate was synchronized with the load-cell sampling rate throughthe use of a data acquisition box, such that for each and every imageacquired during loading, a corresponding load data point was alsoobtained. The camera was equipped with a 100mm macro lens(Tokina) to enable high magnification imaging with minimal imagedistortion. Uniform lighting was provided by high-intensity white LEDlights.

Loading and imaging were continued until complete specimenfailure. Images acquired during loading were then imported into thecorrelation software Vic-2D (Correlated Solutions, Inc.) for post-pro-cessing. In this software, subset and step sizes of 33 pixels and 3 pixelswere used, respectively. The strain filter size was selected to be 5, thesmallest filter allowable in the software. Note that selecting these imagecorrelation parameters was based on an extensive optimization process,details of which are beyond the scope of this work, but can be foundelsewhere (Koohbor et al., 2017b; Ravindran et al., 2017). In addition,the DIC parameters used in this work allow for investigating strongdeformation gradients formed across grain boundaries.

Dynamic experiments were conducted in a conventional SHPB ap-paratus consisting of 25mm diameter stainless steel bars. Loading wasinitiated by a steel striker bar propelled by an inert helium gas gun. Thelength of both the incident and transmitted bars was 1.82m, while thelength of the striker bar was 0.45m. The necessary striker bar velocitythat causes yielding of the specimen was calculated to be approximately11m/s based on well-established SHPB equations (Chen andSong, 2011). The nominal strain rate applied on the specimen in theSHPB experiments was 1100 /s.

A single high-speed CCD camera (Fastcam SA-X2, Photron) was usedto capture images of the dynamically deformed specimen during anapproximately 500 μs period. The frame rate used was 100,000 frames

per second. The spatial resolution of the camera at the utilized framerate was 384× 260 pixel2. For each experiment, the ends of the samplewere lubricated, and the specimen was held firmly between the incidentand transmitted bars while illuminated by a Lumen 200 metal arc lamp.Lighting was kept only for several seconds to minimize heating effects.Strain gauges on both the incident and transmitted bars were used todetect the impact signal. These strain gauges were connected to a signalamplifier which outputted to an oscilloscope that captured the impactwaveform. This impact signal also initiated the triggering of thecamera.

Similar to our quasi-static experiments, images captured via high-speed camera were processed using a subset size of 13× 13 pixel2 and astep size of 1 pixel with a strain filter size of 5. These DIC parametersallow for studying localized deformed domains with sufficient resolu-tion.

Measurement performance, i.e. displacement and strain noise floors,for both quasi-static and dynamic camera systems were determinedthrough a series of baseline measurements similar to the approachfollowed in Ref. (Koohbor et al., 2016). Strain noise floors for quasi-static and dynamic experiments were found to be 945 µε and 1530 µε,respectively. Due to negligible out-of-plane displacements as well as asufficiently high depth of field in the utilized imaging system, the in-fluence of defocusing which resulted from out-of-plane motion wasminimal in all measurements.

Finally, one must note that adiabatic heating of the sample in dy-namic deformation conditions may have some influence on the mobilityof grain boundaries (Rittel et al., 2012). Since in-situ full-field tem-perature measurement was not possible at high strain rate conditions,temperature effects were not considered in the present study.

3. Results and discussion

3.1. Global quasi-static stress-strain response

As previously described, a single ingot of cast magnesium maycontain different grain structures. This is a consequence of the solidi-fication process, a brief discussion of which was presented earlier. Thepresence of different grain shapes and orientations was taken advantageof in this work to explain the possible deformation and failure me-chanisms that occurred in coarse-grained cast magnesium. Quasi-staticuniaxial compression was applied on specimens with different initialgrain shapes and orientations. Fig. 3 shows the stress-strain curvesobtained for three cubic specimens with different initial grain struc-tures. Note that the global strain measurement in our quasi-static ex-periments was performed using the built-in ‘virtual extensometer' toolof the utilized DIC software, which provides highly accurate strainmeasurements with sub-pixel resolution. This being far more precisethan the conventional compressive strain measurement based on

Fig. 2. (a) Optical micrograph showing the grain structure of a typical cast Mg specimen (sample ‘E’ in Fig. 1). (b) speckle patterned surface prepared for in-situ DIC.

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platens displacement, as confirmed in the literature (Ravindran et al.,2016b; Sutton et al., 2009).

In all cases, specimens show a nominally linear initial stress-strainresponse, with nonlinear deformation initiating at approximately35MPa. This region is designated as ‘Region 1’, in which the slightdifferences between the curves are expected to be due to the initialorientation and size of grains. It should be emphasized here that de-pending on the crystallographic orientation of individual grains in eachsample, some grains may start to plastically deform before the others(Zhao et al., 2008); however, the global yielding occurs when all grainsstart to deform plastically.

After nominal yielding and upon entering Region 2, all curvesconverge and an orientation-independent response is observed. Notethat it is anticipated that the samples with different grain structuresshow a different plastic deformation response. However, the deforma-tion mechanisms for these samples might be such that a unified ap-parent global mechanical response is obtained for our three samples, atleast after the initial yielding. This behavior is explained through mi-crostructural deformation measurements, discussed in detail in theforthcoming sections.

Upon further deformation and entering Region 3, intergranularfailure initiates in the form of cracks at grain boundaries and triplejunctions, finally leading to failure. Fig. 4 shows a typical evolution ofintergranular cracking and final failure in sample ‘E’. The strain mapshown in Fig. 4e indicates that failure initiates from boundaries onwhich highest strain magnitudes are developed. Interestingly, there arealso regions on the sample surface that show very small local strains.Formation of such low strain regions may be associated with locallydeveloped basal textures which limit the deformability (Liu andWang, 2016).

Fig. 4 shows that intergranular failure initiates from triple junctions.Failure occurs at global compressive strains of approximately 8%.Specimen failure is revealed through the formation of small visiblecracks on the surface, as well as a drop in the global stress magnitude.Note that depending on the initial grain structure, failure stress andstrains differ for specimens tested in this work. The lowest failure stresswas obtained for sample ‘E’, i.e. the one with relatively small equiaxed

grains (i.e. highest volume fraction of grain boundaries1). Since failureis governed by grain boundary cracking in this material, it is indeedexpected that a specimen with high grain boundary fractions would failat comparably lower global stress and strain, as clearly observed in thiswork. Conversely, sample ‘H’, which is characterized by having co-lumnar grains with shortest grain axis aligned with the loading direc-tion, fails at comparably larger strains. This indicates that the failure inthe examined samples is essentially controlled by the amount of localstrain developed at the boundaries. The orientation of grains in sample‘H’ limits the amount of shear strain developed at grain boundary re-gions, therefore pushing global failure strains to larger values. In con-trast, for sample ‘V’, in which columnar grains are mostly aligned withthe direction of compressive load, shear deformation on boundaries isexpected to be higher and therefore, failure takes place at relativelylower strains.

Regardless of the orientation of the grains with respect to theloading direction, it was evidently shown that in quasi-static loading,failure always takes place by grain boundary fracture. This is indeed awell-understood phenomenon particularly for hcp metals and is relatedto the role of grain boundaries in transferring in-grain cleavage fromone grain to another (Hughes et al., 2007). In hcp metals, owing toinsufficient active slip systems at low homologous temperatures, clea-vage may occur in grains over some well-defined planes. Cleavageplanes in neighboring grains do not necessarily meet in a line in theircommon grain boundary. Therefore, the two cleavage planes withinadjacent grains intercept at the grain boundary at some angle. De-pending on the fracture resistance of the boundary, the mismatchedstrain resulting from the cleavage planes may be partially accom-modated at the boundary. Previous studies have suggested that thedeformation accommodation at grain boundaries significantly con-tributes to the overall failure strain of the material (Crocker et al., 2005;Hughes et al., 2007). However, the amount of such contributions hasnot been verified through in-situ experimental measurements. Oneparticular objective of this work, as presented and discussed in detail in

Fig. 3. Quasi-static stress-strain curves obtained for three different samples shown on the right side of the plot.

1 Quantification of effective grain boundary fraction will be discussed inSection 3.4.

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the following, is to quantify the contribution of such grain boundaryaccommodation in the overall deformation of the examined material.

As a final remark in the study of global stress-strain response, ourspecimens in this work were deliberately machined to dimensions suchthat the study of grain-scale deformations and grain interactions usingan optical-based approach would be possible. One may argue thatmaterial with finer grains would have been readily used in high mag-nification optical (or electron) microscopes to conduct the same type ofcharacterization. Although this would be a valid argument, our primaryintention in this work was to implement a systematic approach that canbe used to facilitate studies in quasi-static as well as dynamic loadings.In dynamic loading, as discussed earlier in the introduction, highmagnification full-field measurements are limited to optical-basedtechniques. Our attempt in the present work was to implement a sys-tematic approach that enables obtaining directly comparable resultsacquired from quasi-static and dynamic experiments with the purposeof explaining possible deformation and failure mechanisms that lead tolow ductility in cast magnesium.

3.2. Global dynamic stress-strain response

Global dynamic stress-strain curves were obtained for the Mg spe-cimens using a conventional SHPB apparatus. Fig. 5 shows dynamicstress-strain curves obtained from two independent tests. The curvesrepresent the constitutive response of samples with an initial grainstructure similar to samples ‘H’ and ‘V’ in Fig. 3. However, a conclusivecomparison between the dynamic stress-strain curves and those pre-viously reported in Fig. 3 may not be made here. This is because (1) themeasurement accuracy in SHPB experiments is comparably lower thanthat of quasi-static experiments. This is because in quasi-static experi-ments, the applied load is directly measured by high accuracy load-cells, while in SHPB experiments the stress is estimated from signalsmeasured by the strain gauges and based on one-dimensional waveequations. (2) Depending on the material examined, the transient stressstate may expand over several microseconds at the early stages of de-formation (Koohbor et al., 2016; Pierron et al., 2014). Such a behavior

invalidates the stress measurements within the linearly elastic regime.Therefore, no conclusive remarks similar to those made for quasi-staticexperiments could be made here. Instead, a direct comparison betweenthe stress-strain curves obtained in quasi-static and dynamic conditionshas been made. Comparing quasi-static and dynamic constitutivecurves, the following remarks are highlighted:

1 Considering the general shape of the curves obtained from dynamicexperiments, a downward concave shape is exhibited. This shape isgenerally considered as an indication for higher work hardeningrates, which in the particular case of Mg and its alloys is related to

Fig. 4. Evolution of intergranular cracking in quasi-static compression applied on a magnesium sample with microstructure shown in (a). Global compressive strainapplied on the specimen is 6%, 8% and 12% in (b), (c) and (d), respectively. Location of the first visible cracks is magnified in (c). (e) DIC contour plot of the εyy straincomponent taken at approximately 8% global strain, where the mechanical failure is initiated. The highly localized deformation behavior is concentrated at theboundaries and in particular at the triple junctions (Scale bar= 2mm).

Fig. 5. Dynamic stress-strain curves obtained from two independent measure-ments. Test 1 and 2 are performed on samples similar to ‘H’ and ‘V’, respec-tively. (see Fig. 1).

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the occurrence of higher amounts of twinning during deformation(Barnett et al., 2004; Li et al., 2009). Increased amounts of twinninglead to higher levels of work hardening, resulting in a noticeableconcave shape in the stress-strain curve. Interestingly, the samplecorresponding to Test 2 in Fig. 5 had a similar structure to sample‘H’ in Fig. 3. For this sample, as most of the grains were orientedperpendicular to the loading direction, minimal shear strain wasexpected to develop on grain boundaries and the applied globaldeformation was mostly consumed in deforming the grains ratherthan shearing the boundaries.

2 Nominal failure strains achieved in dynamic loading are slightlyhigher than those in quasi-static experiments. This observation isconsistent with previous ones (Li et al., 2009), and is explainedthrough heightened twinning and dislocation activities in highstrain rate compression of magnesium (Dixit et al., 2015; Li et al.,2012).

Note that the formation of twins could not be observed and studiedin this work due to limitations in the spatial resolution of our full-fieldmeasurements. In particular, an in-situ study of twinning in dynamicloading requires substantially higher resolutions and could not beachieved using the utilized imaging system. The failed specimen char-acteristics in dynamic condition were similar to those observed inquasi-static experiments. The fracture, in the case of dynamic condition,was also seen to initiate from the boundaries and triple junctions in thematerial, leading to total failure.

Fig. 6 compares the constitutive response of the material under twoloading conditions. Curves representing quasi-static and dynamic re-sponse were the averages of those previously shown in Figs. 3 and 5.The strain rate sensitivity parameter, m, was determined from thesecurves using the following simple definition:

=m ln σ σln

(ɛ) ( / )(ɛ̇ /ɛ̇ )

d s

d s

ɛ

(1)

where subscripts, ‘d' and ‘s' represents dynamic and quasi-static, re-spectively. Strain rate sensitivity was determined as a function of strainand was shown to undergo an initial increase with strain. The initialincrease in the strain rate hardening is consistent with the data pre-viously reported by Dixit et al., 2015), and is due to a rapid increase oftwins in the structure which promote the overall work-hardening of thematerial subjected to dynamic deformation. Strain rate sensitivity thenreaches a maximum at a strain magnitude of about 2% and decreasesafterward. The rate of decreasing m with strain accelerates at

strains> 6%, within a region that corresponds to Region 3 previouslyshown in Fig. 3. The trend observed for the parameter m confirms thatstrain rate sensitivity of the material is indeed related to the micro-structural aspects. Specifically, as the contribution of intergranulardeformation becomes more prominent at higher global strains, thematerial is expected to become less strain rate sensitive. This furtherindicates that the two competing mechanisms described earlier, i.e.intra and intergranular deformations might show different strain ratesensitivities (Lin et al., 2008; Song et al., 2009). Further elaborationsand quantification in this regard are provided in the forthcoming sec-tions where local strain maps are presented and discussed.

3.3. Local strain mapping and possible deformation mechanisms

The global stress–strain behavior of the samples indicated that theremust be two significant mechanisms governing the overall deformationand failure of the cast magnesium samples. These two are intra andintergranular deformation, the former was shown to be dominant atlower global strains. Intergranular deformation, on the other hand, wasdescribed as the major deformation mechanism controlling deformationand failure at grain boundaries, becoming more critical at largernominal strains and characterized as being less sensitive to the appliedstrain rate.

To provide more quantitative evidence on these competing me-chanisms, focus is given here to the full-field measurement data. Fig. 7ashows typical local strain maps obtained for sample ‘H' (see Figs. 1 and3) under quasi-static loading conditions. Initial grain boundaries areoverlaid on strain maps to give an insight on the location of high andlow strain domains. In Fig. 7a, the columns represent contours maps atapproximately 1, 2, 3 and 4% global strain. From these images, thestrain localization between the grain boundaries is evidenced by thelarge contrasts along and across the grain boundary overlay. This isespecially clear when analyzing the εyy and εxx maps. As the evolutionof the strain progresses, the horizontal orientation of the grains and theinitialization of grains sliding over each other is evident. Substantialvariability in the normal components of local strain is accommodatedby the formation of strong shear deformations in the material, as shownin Fig. 7a.

It is clearly observed that in-grain and grain boundary deformationsshow very different values than the global strains applied. The visiblywavy patterns are a result of the highly heterogeneous strain behavior.For instance, when examining the εyy component, the large bands of redindicate little localized deformation, in contrast to the purple and pinkregions which represent high magnitude strain behavior. A similar re-sponse was consistently seen in all specimens in this work.

Locally developed strains can take significantly higher values thanthose applied globally. For instance, the ɛyy contour map shown in theupper right corner of Fig. 7a, corresponding to a 4% global strain ex-hibits areas that contain strains up to twice as high as those appliedglobally. It is interesting to note that the nominal failure strain of thematerial in both quasi-static and dynamic loadings is below 10%, whilelocal strain maps indicate substantially higher in-grain strains. A reasonfor such phenomenon might be that in spite of the hexagonal closedpacked crystal structure of pure Mg, relatively small (c/a) ratios= 1.622, give rise to the activation of slip modes other than merely thebasal slip system. In particular, prismatic and pyramidal slip modes aswell as < >{0112} 0111 twinning systems, have also been identified to beactive deformation modes in pure magnesium (Styczynski et al., 2004).These modes can all contribute to the relatively large in-gain straindevelopment in the material, also proving that low global failure strainis essentially controlled by limited deformation and failure resistance ofgrain boundaries. Of course, this explanation may not be generalized toall Mg alloys, since alloying elements are established to be capable ofaltering the c/a ratio in magnesium (Minarik et al., 2016). With regardto observations made in the current work, there is smaller deformationactivity occurring within the grains in comparison to the boundaries

Fig. 6. Comparing global constitutive responses in quasi-static and dynamicloadings. Strain rate sensitivity, m, is determined as a function of strain usingthese curves.

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(see Fig. 7a). The high concentration of strain occurring preferentiallyat the boundaries in our experiments indicates that these particulargrains’ orientations lend themselves to be more resistant to twinningactivation (Wang et al., 2014). Regardless, strong strain gradientsacross grain boundaries lead to significant sliding of grains, the evi-dence for which is presented in Figs. 7b and 7c.

3.4. Quantification of in-grain and grain boundary deformation

Although strong deformation heterogeneities are evident in straincontours, full-field DIC measurements do not make a distinction be-tween data obtained inside of grains and over the boundaries.Distinguishing in-grain deformation from that developed over the grainboundaries is essential in the present work since we intend to quantifythe contribution of each to the total strain applied to the samples.Therefore, a simple approach is proposed where a mechanically re-levant ‘effective grain boundary thickness (EGBT)’ parameter is in-troduced to distinguish between deformation of the grain boundaryregions and deformation developed at grain interiors. The proposedidea of EGBT is schematically shown in Fig. 8. Note that the

mechanically relevant effective boundary thickness introduced here isdifferent from grain boundary width used for diffusion studies(Prokoshkina et al., 2013). The concept used in diffusion studies con-siders a very narrow region typically with dimensions of a few nan-ometers utilized to characterize mass transfer through grain bound-aries. The notion of EGBT proposed here is significantly larger in size,several orders of magnitudes larger than those used in diffusion studies.In addition, EGBT is not an inherent material property, rather it de-pends mainly on the dimensions of the area of interest in a DIC-basedmeasurement, and is highly sensitive to image correlation parametersused in a DIC analysis. In particular, image correlation parameters thatfacilitate measuring more localized deformation patterns at higherspatial resolutions allow for selecting thinner EGBTs. For more detaileddiscussions regarding the influence of image correlation parameters inmeasuring mesoscale deformation at highly localized regions refer to(Koohbor et al., 2017b; Ravindran et al., 2017).

In order to quantify the EGBT in this work, local strain profilesacross grain boundaries were considered. Fig. 9 shows a typical evo-lution of in-plane strain components across a short representative line‘L’. Local strain data over the representative line segment ‘L’ wereplotted and fitted with proper Gaussian-type curves, as shown inFig. 9c. Gaussian curve fitting in this work was conducted in MATLAB®.Once the experimental data were fitted with the proper Gaussianfunction, the arithmetic mean and the standard deviation was identi-fied, and the span of one standard deviation of the data was taken as thethickness of the effective grain boundary (δGB). Due to higher accuracyand better spatial resolution of quasi-static measurements in this work,EGBT measurements were carried out for quasi-static specimens and theobtained values were used for the case of dynamic loading conditions aswell. The approach mentioned above was carried out several times formultiple line segments at various specimen locations. The average of allδGB measurements was obtained as 200 µm. This value was taken as theeffective boundary thickness. The ratio of the surface area of effectivegrain boundaries to the sample surface, referred to as grain boundaryfraction (ξGB), was calculated for each sample using the following ex-pression:

= ×ξ δ ddGB

GB GB2 (2)

Fig. 7. (a) A typical contour maps are depicted, showing the evolution of in-plane strain components at different global strain magnitudes for sample ‘H’. Loading wasapplied in (-)ydirection. Grain boundary outline is overlaid on contour maps. Images of initial and post-failure conditions of sample ‘H’ are shown in (b) and (c),respectively.

Fig. 8. Schematic representation of mechanically relevant effective grainboundary thickness (EGBT) and grain interior (GI) regions in this work.Effective boundary thickness is denoted by δGB.

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where dGB is the total length of grain boundaries visible on the samplesurface, and d denotes the in-plane sample dimensions (= 10mm). ξGBvalues for samples E, H, and V were determined as 68%, 42% and 48%,respectively.

Next, strain evolutions over grain boundaries and grain interiors areconsidered and analyzed in order to estimate the contribution of eachmechanism to the total strain. Our methodology will be analogous tothat proposed by Langdon (2006). In this approach, the total strainapplied to the specimen, ɛT, may be expressed as:

= +ɛ ɛ ɛT GI GB (3)

where ɛGI, is the strain associated with just the intragranular (in-grain)deformation, and ɛGB is the strain developed over the entire network ofgrain boundaries. Notice that the latter, i.e. ‘grain boundary strain’hereafter refers to the strain developed over narrow effective grainboundaries, the thickness of which was determined as δGB= 200 µm.The contribution of grain boundary strains to the total strain, η, is thenexpressed as:

=η ɛɛGB

T (4)

whereas ( − η1 ) indicates the contribution of in-grain strains to the totalstrain applied. Furthermore, due to the complex in-plane strain pat-terns, to enable a direct comparison of local and global strains, thedefinition of equivalent von Mises strain, ɛeq, is utilized to determineboth in-grain and grain boundary strains. Equivalent strain in this re-gard is calculated as:

= ⎡⎣

+ + ⎤⎦

′ ′′ ′iɛ 23

(ɛ ɛ 2ɛ ) ( : GI or GB )ieq

xx yy xy2 2 2

1/2

(5)

From a practical point of view, separation of in-grain and grainboundary regions within a whole-field area of interest are cumbersome.To accomplish this task with higher accuracy, we carried out two se-parate image correlation runs for each sample. As shown in Fig. 10,first, the whole speckled area was selected as the area of interest and allstrain components were determined. Next, image correlation was

performed again, this time with the initial area of interest only con-taining the grain interiors, from which the effective grains boundarieswere separated from the beginning. Finally, the strain data for just thegrain boundaries was calculated using Eq. (3). Digital image correlationfor strictly the grain boundary regions was not possible due to the areasof interest being far too narrow. Image correlation parameters (subset,step and strain filter) were chosen to be the same for all DIC runs.Again, this approach is not rigorous and provides just a broad gen-eralization about deformation activity occurring in the regions of thegrain boundaries. Finally, all strain components were spatially averagedover the corresponding area of interest and further analyses were per-formed.

Fig. 11 shows the variation of the parameter η with respect to totalstrain applied on our quasi-static samples. All samples show significantvariations in the value of η at strain levels lower than 0.02. This strainrange is consistent with Region 1 discussed earlier in Fig. 3, and isassociated with the initiation of plastic deformation in the material, theoccurrence of which is highly heterogeneous due to the initial Schmidtfactor of the grains. After plastic yielding, the η parameter takes onvalues close to 55% for sample ‘H’. This indicates that the contributionof in-grain deformation and deformation of the grain boundary regionsis almost the same in the deformation of the sample with grains mostlyoriented perpendicular to the loading direction. In sample ‘E’, the ηfactor displays higher values than those of sample ‘H’. This behavior canbe related to the comparably larger fraction of grain boundaries in thissample, which can accommodate higher local deformations in the areasmore closely associated with grain boundaries. The η parameter for thissample takes on values close to 70%. Finally, in sample ‘V’, i.e. the onewith grain boundaries mostly aligned with the direction of the far fieldload, the highest values of the η parameter is measured. This observa-tion is indeed consistent with our previous discussions and can be at-tributed to the occurrence of large local shear strains within the grainboundary network. For this sample, the contribution of grain bound-aries strains to the overall deformation is estimated to be above 80%.

Interestingly, for all samples, a slight change takes place at totalstrain values close to 0.05. This strain value is approximately the

Fig. 9. Typical contour maps are depicted, showing a highly heterogeneous distribution of (a) ɛyy and (b) ɛxx with a representative line segment ‘L’ used to determineEGBT. Variation of local strain components along L is shown in (c) and is used to determine δGB.

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transition strain between Regions 2 and 3 in Fig. 3. Due to the un-availability of deformation data within the bulk of the samples, nofurther discussion can be provided on whether or not any internal da-mage in the form of grain boundary cracking occurred in the material.However, this can be a subject of interest for future computationalstudies or those facilitated by the use of X-ray tomography and volu-metric image correlation (Langdon, 1970; Lenoir et al., 2007). Re-gardless, the result indicate that the role of grain boundaries in ac-commodating the deformation applied quasi-statically on puremagnesium is substantial. The contribution of grain boundaries in thisregard was quantified to be at least 50%. This value is higher than thatpreviously obtained through computational approaches (Hughes et al.,2007), and can be verified through more accurate crystal plasticity andmicromechanics models in the future. Of course, one should bear inmind that the measurements in this work were limited to a single ma-terial with weak grain boundaries. However, a similar approach can beimplemented to study other HCP metals with more regular grainstructures and mechanically stronger grain boundaries.

Similar to the approach used to identify the η parameter in quasi-static loading, this parameter was obtained for dynamic experiments aswell. Fig. 12 shows the variation of this parameter with respect to totalstrain applied to the samples in the SHPB experiments. Note that thestrain rate, in this case, is approximately 1100 /s and that the number

of data points is significantly smaller due to the limited temporal re-solution of the imaging in our dynamic experiments. The general trendsof the η parameter in dynamic experiments are similar to the quasi-static data with the only difference being the value of the parameter η,which is significantly smaller in the dynamic compared with the quasi-static tests. Results presented in Fig. 12 indicate that in high strain rateexperiments, the maximum contribution of grain boundaries to theoverall deformation of the samples is approximately 40%, roughly halfof the values obtained for quasi-static conditions.

To enable a quantitative assessment of the role of initial grain or-ientation on the contribution of grain boundary strain to the total de-formation, quasi-static and dynamic η values were directly compared.Note that the direct comparison of the η data for quasi-static and dy-namic loading conditions requires the same number of data points, andthis was indeed a limiting point in our experiments because of theconsiderably different experiment and interframe times. Therefore, ηcurves obtained for quasi-static and dynamic cases were first fitted with6th degree polynomials, as shown in Fig. 13. The curves were re-generated using the fitted polynomials and comparisons were made.

Fig. 14 compares the η parameter by showing the ratio of thisparameter for dynamic loading over that of quasi-static experiments.Results are shown for both samples ‘H’ and ‘V’, i.e. those with grains

Fig. 10. Illustration of the approach utilized to separate in-grain deformation from grain boundary region deformation. Solid green regions in each image indicate theDIC area of interest. Showing from left to right: Full-field DIC area of interest, in-grain DIC area of interest, GI, and grain boundary area of interest, GB. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Variation of the parameter η with total strain for quasi-static samples.Symbols ‘H’, ‘V’ and ‘E’ indicate ‘Horizontal’, ‘Vertical’ and ‘Equiaxed’ grainorientations, respectively. See Fig. 1 for more details.

Fig. 12. Variation of the parameter η with total strain for dynamic samples.Symbols ‘H’ and ‘V’ indicate ‘Horizontal’ and ‘Vertical’ grain orientations, re-spectively. See Fig. 1 for more details.

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elongated perpendicular and parallel to the loading direction, respec-tively. The following remarks must be noted: (1) Contribution of thedeformation of grain boundary regions to the total applied strain ismore prominent for the sample with grains elongated parallel to theglobal loading direction, i.e. sample ‘V’. This is in fact anticipated andattributed to the more substantial shear deformation process. (2) Afteryielding, i.e. in Region 2, the relative difference between the η valuesconstantly decreases with applied strain. This means that the con-tribution of grain boundary region deformation in dynamic loadinggradually decreases compared with quasi-static loading conditions. Bycomparing the grain boundary strain with the strain rate sensitivity (m)data shown earlier in Fig. 6, a consistent trend between the η-ratio andm can be found. Form this data along with the data shown in Fig. 13, it

can be concluded that the deformation of the grain boundary regionsplays a more significant role in the overall strain rate sensitivity of thematerial. At large strain magnitudes applied dynamically, the overalldeformation of the material is controlled to a larger extent by the in-grain deformation mechanisms rather than the shear-dominated grainboundary deformation mechanisms. On the other hand, a relativelyhigher strain rate sensitivity of the material observed at smaller totalstrains can be associated with a grain boundary deformation me-chanism.

The concluding statements above may be of particular significancein modeling works that tend to study deformation and failure responseof brittle polycrystalline systems. In this regard, appropriate modelsaccounting for the strain rate dependence of grain boundary deforma-tion seem to be essential to obtain more accurate and realistic materialresponse. It must be emphasized here that crystal plasticity constitutivemodels incorporating strain rate sensitivity in intragranular deforma-tions are well-established (Barlat et al., 2002; Salem et al., 2005). Theobservations in this work confirm that similar models are required forintergranular deformation studies, as well. In addition, the results foundin this work can be used as guidelines in the development of grainboundary engineering (GBE) strategies in the processing of materialwith desirable mechanical properties (Cao et al., 2017; Randle, 2004;Wantanabe and Tsurekawa, 2004). The approach introduced in thiswork can be applied in studies carried out in the area of GBE, mainlyshedding light on the mechanical characteristics of grain boundaries ina material subjected to high strain rate loading conditions.

Last but not least, experimental results such as those obtained in thiswork can complement multiscale modeling data obtained via variouscomputational tools. For instance, recent molecular dynamics (MD)simulations suggest that 30° tilt and twist grain boundaries in pure Mgcan be highly mobile as they correspond to energy minima (Liu andWang, 2016). Results from similar modeling efforts combined experi-mental findings reported in this work can provide the roadmap for theprocessing of advanced engineered Mg alloys with superior strengthand deformability that will be appealing to aerospace and automotiveapplications.

4. Conclusions

Mesoscale full-field deformation characteristics of nominally purecast magnesium were investigated utilizing digital image correlationunder both quasi-static and dynamic loading conditions. The quasi-static mechanical response of Mg as a function of initial grain or-ientation was studied along with the evolution of intergranular crackingunder uniaxial compression. The contributions of in-grain and inter-grain deformations to the total deformation of the samples werequantified. Initial grain configuration was confirmed to play a sig-nificant role in both the local and global deformation response ofsamples at quasi-static deformation conditions.

An approach facilitated by the use of a split Hopkinson bar appa-ratus and a high-speed camera was also proposed to capture local dy-namic deformation behavior of the samples in-situ. The contributions ofboth inter and intragranular strain in dynamic deformation were alsoquantified, the former of which was found to be grain orientation-de-pendent, also playing a substantial role in the macroscale deformation.Strain rate sensitivity of samples was measured and found to be relatedto the amount of deformation occurring in the immediate area of grainboundaries. The key findings of the present can be summarized as:

(1) The initial grain orientation with respect to the loading directioncan alter the ratio of grain boundary deformation to the total de-formation.

(2) Irrespective of the grain orientation direction, the ratio of grainboundary deformation to the total deformation is higher in thequasi-static loading than dynamic loading condition.

Fig. 13. η parameter data used for direct comparing of quasi-static and dynamicresults. The data shown in this figure is for a sample with grains aligned per-pendicular to the loading direction, i.e. sample ‘H’.

Fig. 14. Ratio of η parameter for dynamic and quasi-static conditions versustotal strain. Subscripts ‘dyn’ and ‘QS’ denote dynamic and quasi-static loading,respectively.

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The results from this work can be used to provide insight into thedominant deformation mechanisms in magnesium, which in turn can beapplied to the modeling and crafting of systems comprised of other lowductility polycrystalline metals.

Acknowledgments

This work was made possible following a technique developedthrough a program supported by the Air Force Office of ScientificResearch (AFOSR) under Grant No. FA9550-14-1-0209 and FA9550-16-1-0623. The support of AFOSR is gratefully acknowledged.

Supplementary materials

Supplementary material associated with this article can be found, inthe online version, at doi:10.1016/j.mechmat.2018.07.007.

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