Materials Science and Engineering A 399 (2005) 1–12

Internal strain and texture evolution duringdeformation twinning in magnesium

D.W. Browna,∗, S.R. Agnewb, M.A.M. Bourkea, T.M. Holdenc,S.C. Vogela, C.N. Tomea

a MS-H805, BLDG 622, TA-53, Los Alamos National Laboratory, Los Alamos, NM 87545, USAb Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA

c Northern Stress Technologies, Deep River, Ont., Canada K0J 1P0

Abstract

The development of a twinned microstructure in hexagonal close-packed rolled magnesium compressed in the in-plane direction has beenmonitored in situ with neutron diffraction. The continuous conversion of the parent to daughter microstructure is tracked through the variationof diffraction peak intensities corresponding to each. Approximately 80% of the parent microstructure twins by 8% compression. Elastic latticestrain measurements indicate that the stress in the newly formed twins (daughters) is relaxed relative to the stress field in the surroundingm rroundingg echanismi©

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atrix. However, since the daughters are in a plastically “hard” deformation orientation, they quickly accumulate elastic strain as surains deform plastically. Polycrystal modeling of the deformation process provides insight about the crystallographic deformation m

nvolved.2005 Published by Elsevier B.V.

eywords:Deformation twin; Neutron diffraction; Internal strain; Magnesium; hcp

. Background

.1. Deformation mechanisms of magnesium and itslloys

It has long been understood that the limited room tempera-ure formability (or generalized ductility) of magnesium andts alloys is connected with the large plastic anisotropy at theingle crystal or grain-level[1]. For instance, it is often citedhat magnesium possesses only two independent slip systemsssociated with the primary deformation mode, basal slip of/3〈1 12 0〉 or 〈a〉 type dislocations. Furthermore, it is recog-ized the incorporation of non-basal slip of〈a〉 dislocationsn prismatic{1 01 0} or even first-order pyramidal{1 01 1}lanes offers only two more independent slip modes.

Kocks and Westlake[2] first pointed out that metals canot be simply divided into two groups: ductile metals thatossess five independent easy slip modes (i.e. satisfy the

∗ Corresponding author. Tel.: +1 505 667 7904; fax: +1 505 665 2676.E-mail address:dbrown@lanl.gov (D.W. Brown).

Taylor criterion) and brittle metals that do not. They nothat there is a range of behaviors spanned by those mthat clearly do possess five independent easy slip modeductile fcc and bcc metals), those that possess nearly(e.g. hcp magnesium) and those that clearly posses(e.g. hcp beryllium and simple cubic NiAl). Additionalthey noted that the presence of mechanical twinning mfurther modifies the situation by essentially offering othedependent modes of deformation, albeit in a limited sdue to their unidirectional nature and their limited straincommodating ability (both points to be discussed in mdetail later). For magnesium alloys, which commonlyhibit mechanical twinning on{1 01 2} planes, an additionhalf of an independent slip mode was proposed. Thus, isuggested that an hcp magnesium–lithium solid solutioloy reported to slip extensively on prismatic as well as bplanes[3] should be characterized as having 41

2 independenslip modes, thus explaining the alloy’s significant duct[2].

Exactly how strong the plastic anisotropy of magnescrystals is and what deformation mechanisms are invo

921-5093/$ – see front matter © 2005 Published by Elsevier B.V.oi:10.1016/j.msea.2005.02.016

2 D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12

has been an evolving discussion as various experimentaland modeling techniques have been brought to bear on theproblem. In particular, the distinction between single crystalbehavior and the behavior of grains within a polycrystallineaggregate has become a point of consideration. Usingtransmission electron microscopy it has been demonstratedthat magnesium and its alloys do possess five independentslip modes, by the observation of second-order pyramidal{1 12 2} slip of 〈c+a〉 dislocations during compression ofc-axis aligned single crystals[4,5] and within polycrystals,particularly in connection with mechanical twins.

A more quantitative way of describing the situation is toemploy the concept of single crystal yield surface (SCYS),whereby the deformation mechanisms are characterized bytheir activation stresses, rather than as binary quantities (ac-tive or not). Polycrystal models[6–8] offer a means of con-necting the SCYS with the overall anisotropy and textureevolution within a polycrystal during deformation. Thus, ifthe SCYS is known, polycrystal behavior may be predicted.Conversely, if the polycrystal behavior is known, the SCYSmay be inferred. Using a combined polycrystal modeling andexperimental approach, Agnew et al.[9,10]were able to cor-relate the ductility enhancement of lithium alloying additions,with an increase of non-basal 1/3{1 12 3} or 〈c+a〉 slip, inaddition to the previously proposed enhancement of prismaticslip of 〈a〉 dislocations[3]. The proposed enhancement of〈 lecAs ne-s anceoa ainsw ep lt

teru n asi iump cts,fs udeda int oundi in-n ald ared orc tb sm fmi tm thatt longt ion

axis, and inactive during tension along the same axis. Thedominant dislocation slip systems, basal〈a〉, are unfavorablyoriented in both conditions. Thus, the unidirectional natureof deformation twinning results in observed high tensileand low compressive strength in worked magnesium alloys[9,10].

The current study was undertaken to further investigatethese issues; in particular, to investigate the homogeneityof plasticity by directly measuring the intergranular strainsand texture which develop during deformation. The measure-ments provide, for the first time, information regarding whatis the level of stress present in deformation twins as they form.

1.2. In situ internal stress assessments

The use of neutron diffraction-based internal strainmeasurements to probe the grain-level plastic anisotropy ofpolycrystals is a relatively new technique, with the earliestpublished examples attributable to MacEwen et al.[15]. Insitu examinations are even more recent[16]. Magnesiumis especially suitable for a basic scientific study of internalstrain and texture development driven by plastic anisotropy.Notably, it is unique among hexagonal metals in that thesingle crystal elastic constants and coefficients of thermalexpansion are nearly isotropic[17,18], while the plasticityis distinctly anisotropic. Therefore, type II (intergranular)s

ren-t g av res-s s cutf res-s nessd Thisp ple inw le-m het sion( Ag-n SC)m andt n oft sam-p basala ndt peri-m thatf

2

2

raina der

c+a〉 slip in Mg–Li alloys is reinforced by recent singrystal and TEM studies of Ando and Tonda[11]. Finally,gnew and Duygulu[12] have shown that enhanced〈c+a〉lip can explain the vastly enhanced ductility of magium alloys at slightly elevated temperatures. The importf secondary deformation mechanisms such as〈c+a〉 slipnd deformation twinning is obvious as they provide grith a means of straining along theirc-axis, which the morrevalent slip modes (e.g. basal slip of〈a〉 dislocations) fai

o do.In addition to offering a unique opportunity to bet

nderstand the fundamentals of plastic accommodatiot relates to ductility (or lack thereof), wrought magnesolycrystals manifest a wide range of anisotropy effe

rom strong in-plane anisotropy in the sheet form[13] to atrong tension–compression asymmetry in both the extrnd rolled conditions[1]. The latter aspect is of interest

he current study because the root of the asymmetry is fn the combination of crystallographic texture and twing. Unlike dislocation slip, twinning is a unidirectioneformation mechanism. Twinning modes in hcp metalsistinguished by their ability to produce either tensileompressive strain along the crystallographicc-axis, but nooth. Depending upon thec/aaxial ratio, hcp twinning modeay be either tensile or compressive[14]. In the case oagnesium, the dominant twinning mode{1 01 2}〈1 01 1〉

s a tensile twin sincec/a = 1.624<√

3. In wroughagnesium, the typical deformation textures are such

he tensile twinning mode is activated by compression ahe prior working direction, e.g. parallel to the extrus

trains have their origin in plastic anisotropy only.The use of textured polycrystal samples allows prefe

ial probing of specific deformation mechanisms by usinariety of loading conditions. For this study, in situ compion and tensile tests have been performed on samplerom a hot-rolled magnesium alloy AZ13B plate. Compive loads were applied to the in-plane and through-thickirections and tensile loads to the in-plane direction.aper focuses on the in-plane compression (IPC) samhich deformation twinning plays a major role. A compentary paper[10] deals in depth with the deformation of t

hrough-thickness compression (TTC) and in-plane tenIPT) samples, which deform almost exclusively by slip.ew et al.[10] used an elasto-plastic self-consistent (EPodel to describe the development of internal strains

he ratio of transverse plastic strains during deformatiohe through-thickness compression and in-plane tensionles. The researchers concluded that, in addition to thend prism activity, a contribution from pyramidal slip a

ensile twins must be considered to reproduces the exental observations. Evidence is shown in this report

urther supports this argument.

. Experimental techniques

.1. Neutron diffraction techniques

Neutron diffraction measurements of internal stnd bulk texture were performed on the neutron pow

D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12 3

Fig. 1. Schematic of the NPD.

diffractometer (NPD) and high intensity powder diffrac-tometer (HIPD), respectively, at the Manuel Lujan Jr.Neutron Scattering Center, LANSCE, Los Alamos NationalLaboratory. Details of the diffraction instruments are similarand published elsewhere[19] and only a short description ispresented here. In each instrument, the incident neutron beamimpinges on the sample and is scattered onto a set of fourdetectors banks situated at±90◦ and±148◦ in the case ofNPD (±153◦, HIPD) relative to the incident beam. The time-of-flight technique utilizes the continuous energy spectrumand pulsed structure of the neutron beam to collect an entirediffraction pattern (effectived-space range from 0.3 to 4A) ineach detector panel simultaneously. The primary differencesin the diffraction instrument are the lengths of the neutronflight paths, which define the resolution of the instrumentand the ancillary equipment available to each instrument.

The NPD has a flight path of 30 m yielding an instrumentalresolution of�d/d= 2.5× 10−3, making it suitable to mea-sure the small shifts in lattice spacing associated with internalstrains. A purpose built horizontal load frame was utilized toperform in situ measurements of the lattice strains during uni-axial compression and tension.Fig. 1 shows a schematic ofthe NPD indicating the sample geometry (compression) andorientation of the applied stress relative to the instrument. Theload axis was oriented at 45◦ relative to the incident beam. De-tectors on either side of the specimen simultaneously recordd -v rb r-i ress s thect teda ge oft otu wasd nnedt

ax-i ts ofb ach

sample through 14 different orientations for diffraction mea-surements, chosen to provide roughly uniform coverage ofthe pole figure. Rietveld analysis of the 56 diffraction pat-terns using eighth order spherical harmonics to describe thetexture[21] was completed with the General Structure Anal-ysis System (GSAS) software developed at LANSCE.

2.2. Sample preparation

Hot-rolled magnesium alloy AZ31B plate was selected forthe investigation because it represents the highest productionwrought magnesium alloy and it has a well defined but simplecrystallographic texture, shown inFig. 2a. The preferred ori-entation of grains within the plate has basal poles parallel tothe plate normal direction and exhibits a near random orienta-tion of prism poles about that axis. Commercial magnesiumalloy AZ31B has a nominal composition of 3 wt.% Al and1 wt.% Zn, with restrictions on the transition metal impuri-ties Fe, Ni, and Cu in order to improve the corrosion resis-tance of the alloy. The alloy was obtained as a 1′′ (24.5 mm)hot-rolled plate in the soft-annealed condition (O-temper).Conventional metallography, including a picral-acetic etch

Fig. 2. Measured texture of rolled magnesium plate (a) as-received, (b) after14% in-plane compression, (c) calculated texture after 14% in-plane com-pression (note that scale on calculated pole figures is roughly 2× larger).

ata with diffraction vectors, parallelQ|| (−90◦) and transerseQ⊥ (+90◦), to the applied load[19,20]. Both detectoanks have acceptance angles of±5.5◦ in the vertical and ho

zontal directions. The circle in the center of the pole figuhown inFig. 2(and in subsequent pole figures) representoverage of theQ|| detector bank (−90◦) of the NPD relativeo the pole figure of the sample. A similar sized circle locat the north and south poles would represent the covera

heQ⊥ detector bank (+90◦). The high angle banks were ntilized for this measurement. The macroscopic strainetermined concurrently using an extensometer that spa

he irradiated region.In contrast, the HIPD has a 9 m flight path, which m

mizes the incident neutron intensity for measuremenulk texture. A computer-controlled goniometer rotated e

4 D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12

common for magnesium alloys revealed a grain size of ap-proximately 50�m [22]. The compression samples (10 mmin diameter and 22 mm in height) used in the study wereelectro-discharge machined from near the midplane of theplate.

2.3. Experimental programme

The sample was initially oriented within the load frameso that the basal poles of the main texture component werein the diffraction plane (horizontal), but transverse to theloading direction.Fig. 3shows a schematic of typical grainswithin the polycrystal relative to the load axis in the neutrondiffraction instrument (NPD). The orientation of a typicalgrain prior to twinning, shown schematically in the left handgrain in the figure, results in strong (0 0 0 2)peak intensityin the transverse detector,Q⊥ and absence of (0 0 0 2)inten-sity in the parallel detectorQ||. The prism poles,{1 01 0},{1 12 0}, {2 13 0}, are initially distributed in a vertical planecontaining the load axis and diffract solely into the paralleldetector. Throughout this paper this will be termed the parentmicrostructure or parent grain orientation.

During in-plane compression, the stress axis is initiallyperpendicular to the basal poles, making basal slip dif-ficult to activate, but facilitating tensile twinning on the{ ¯ ¯¯

p ge 180ae s axist arly9 ultsit andg n-s -b hter

F e rela-t bulkt d aftera

microstructure or daughter grain orientation. The transfer ofintensity between the two detector banks is a direct measureof the twinned volume fraction and allows us to monitor thedevelopment of the twinned microstructure.

By taking advantage of the initial texture and the char-acteristic twin reorientation, this experimental arrangementprovides a unique opportunity to simultaneously monitor thedevelopment of the internal strains in both the parent anddaughter grains. Since (0 0 0 2) diffraction intensity is ini-tially absent from theQ|| detector, the position ind-space ofthe initial (0 0 0 2) diffracted intensity in this detector pro-vides unobstructed information about the initial strain stateof daughter grains as they appear. Likewise, the prism poleintensity is initially absent from theQ⊥ detector, and theappearance of{1 01 0} and{1 12 0} intensity provides infor-mation about the initial strain state of the daughter grains per-pendicular to the applied stress. In contrast, the strain state ofthe parent grains as they are consumed may be gleaned fromthed-space of diminishing (0 0 0 2)peaks in theQ⊥ detectorand{1 01 0} and{1 12 0} peaks in theQ|| detector. As far asthe authors are aware, this represents the first study of thestress state of twins as they form during in situ loading.

3. Self-consistent modeling

at al re-s ist clu-s eouse prop-e , thec them in area ifferf

d toa nta-t f nota s nota tiono doesa thec scaler ce, iti turee

hei f thes rainsu cribet % ofi mples[ t.

1 01 2} 〈1 011〉 system. With ac/a ratio of 1.624, the twinlane normal is 43.3◦ from the basal pole. During a twinninvent, the symmetry axis of the crystal structure rotates◦bout the twin plane normal[23,24]resulting in a 86.6◦ reori-ntation of the basal pole from perpendicular to the stres

o nearly parallel. Given the sample orientation, the ne0◦ reorientation of the basal pole during twinning res

n a transfer of (0 0 0 2) diffraction intensity from theQ⊥ toheQ|| detector, as shown schematically in the right hrain inFig. 3. Of course, the prism plane diffraction inteity concomitantly transfers to theQ⊥ detector. The “neworn” twinned orientations will be referred to as the daug

ig. 3. Schematic of magnesium grains in-plane compression samplive to the diffractometer. The orientations of the grains represent theexture of the IPC sample. The left and right grains represent before antwinning event, respectively.

In the past, visco-plastic self-consistent (VPSC)[8]nd elasto-plastic self-consistent (EPSC)[25,26] polycrys-

al models have been used to simulate the mechanicponse of Mg alloys[9,10]. In these models each grainreated as (visco-plastic or elasto-plastic) ellipsoidal inion embedded in (and interacting with) the homogenffective medium that represents the aggregate. Therties of the grain and of the medium are anisotropicrystallographic orientation of the grain with respect toedium is accounted for. Stress and strain in each graconsequence of the grain-medium interaction, and d

rom grain to grain.The VPSC model allows to tackle large strains an

ccount for hardening, texture evolution, and twin reorieion. However, the VPSC model has the disadvantage occounting for elasticity and, as a consequence, it doeddress the initial elastic–plastic transition, or the evoluf internal strain. The EPSC model, on the other hand,ccount for elasticity but does not currently incorporaterystal rotation associated with slip, much less the large-eorientation associated with twinning. As a consequens not applicable for simulating large deformation or texvolution due to twinning.

Agnew et al.[9] applied the VPSC model to predict tn-plane and through-thickness mechanical response oame Mg plate examined in this study to compressive stp to 50% strain. The EPSC model was employed to des

he internal strains developed during deformation to 10n-plane tension and through-thickness compression sa10], which do not exhibit significant texture developmen

D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12 5

As discussed above, neither the EPSC nor the VPSC mod-els are particularly well suited to this study, since the sampleundergoes large textural reorientations and the region of in-terest is relatively small strains, roughly 10%. However, in thecurrent study the VPSC model is used to lend understandingregarding the contribution of twinning to texture evolution,and to obtain information about the activity of other deforma-tion modes. A detailed description of the VPSC formalism asit applies to the present calculations can be found in Ref.[9].

The VPSC simulations utilized in this paper have main-tained the critical resolved shear stresses and hardening pa-rameters used in Ref.[9] to aid in interpretation of the exper-imental data. In what follows, we compare experimental andpredicted stress/strain curves and textures, associated within-plane compression. At present, it is observed that twin-ning evolves faster in the simulation and, as a consequence,so does the texture and the hardening. From a qualitativepoint of view, however, the conclusions concerning systemactivity appear to be valid since the final deformation textureis correctly predicted. For a description of the polycrystalmodel and predominant twin reorientation scheme used inthe simulations, the reader is referred to Ref.[27].

4. Results

4

ribediw lanes fa ison.T pleo rel-a nesss n thei strainc ctile

F ough-t siuma . Solidc sure-m

metal; linear elasticity followed by a smooth elastic–plastictransition. In contrast, the flow curve of the in-plane sampledisplays abrupt yielding at 60 MPa followed by a low harden-ing rate plateau that is frequently associated with deformationtwinning [28] or a stress induced phase transformation[29].After a plastic strain of roughly 5%, an inflection is apparentin the flow curve as the work-hardening begins to increasewith increased strain. Finally, beyond 8% deformation theflow stress of the in-plane sample has matched or surpassedthat of the through-thickness sample and the work-hardeningbegins to decrease for a second time.

4.2. Ex situ measurement and modeling of texturedevelopment

The experimentally determined and calculated (0 0 0 2)and{1 01 0} pole figures of the in-plane compressed samplesare shown inFig. 1b and c, respectively, after−14% defor-mation. For ease of discussion, we have superposed a polaraxis on the first pole figure with the azimuthal angle,Θ, zeroat the center of the pole figure, and the polar angle,Ψ , zero tothe right and increasing counter-clockwise[30]. Again, thecircle in the center of the pole figures represents the coverageof theQ|| detector of the NPD relative to the pole figures.Although not shown, the complete orientation distributionfunctions (ODF’s) were also calculated using spherical har-m es n cal-c nalo

asalp l di-r rongt tivityo ionb htlyen n isa tioni 0.80a f thet therm tionsa

tex-t n thec er intd ure.A thebo thes ictst 5%a ainsc gure

.1. Macroscopic mechanical behavior

The mechanical response of this material was descn our earlier publications[9,10] and only an overviewill be given here. The measured and calculated in-ptress/strain curves are shown inFig. 4. The flow curve othrough-thickness sample is also shown for compar

he difference between the flow curves of the two samrientations is due to the difference in loading directiontive to the basal fiber. The load on the through-thickample is parallel to the basal fiber, whereas the load on-plane sample is normal to the basal fiber. The stress/urve of the through-thickness sample is typical for a du

ig. 4. Solid lines represent measured stress/strain curves for thrhickness (TTC) and in-plane (IPC) compression of AZ31B magnelloy. Dashed line represents calculated stress/strain curves for IPCircles represent points during IPC at which neutron diffraction meaents were taken.

onic and WIMV methods[31]. The volume fraction of thample that has re-oriented during deformation was theulated from the integrated difference in the initial and firientation distribution functions[30].

Initially, the samples possess a fiber texture with the boles predominately parallel to the plate’s rolling normaection. After the deformation, the sample showed a stextural reorientation that can be associated with the acf the{1 01 2} tensile twinning model. The post-deformatasal fiber is aligned with the load direction, and is sliglliptical with the major axis atΨ = 90◦ (along the rollingormal direction). Assuming that large-scale reorientatiossociated solely with twinning, the twinned volume frac

n the in-plane compressed sample is estimated to befter 14% deformation of the sample. This estimate o

winned volume fraction ignores reorientation due to oechanisms such as slip, which are likely small correct such small macroscopic deformations.

The VPSC calculation reproduced the deformationure satisfactorily. In general, the features are sharper ialculated pole figure than observed. Also, the basal fibhe calculated pole figure is split into lobes in theΨ = 0–180◦irection, which is not observed in the measured pole figctivity of the pyramidal slip system causes rotation ofasal poles away from the load direction[9]. Over-activityf the pyramidal slip system in the calculation results inpurious splitting of the basal fiber. The calculation predhe twin volume fraction reaches a maximum of 0.85 byt which point twinning saturates and it subsequently remonstant. The eccentricity of the measured basal pole fi

6 D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12

Fig. 5. Diffraction patterns during in-plane compression of magnesium in parallel and perpendicular detector banks as a function of applied stress.The patternshave been offset vertically for clarity.

is reproduced by the model and is linked with the remainderof the parent microstructure atΨ = 90 and 270◦ andθ = 45◦which did not twin.

4.3. In situ measurement of texture development

The evolution of the diffraction patterns during in situ de-formation of the in-plane compression sample is shown inFig. 5. Corresponding to the pole figures shown inFig. 2there is initially no diffracted intensity from the basal planes,(0 0 0 2), in theQ|| detector bank while the diffraction peaksfrom prism planes,{1 01 0} and{1 12 0}, are strong. The sit-uation is reversed in theQ⊥ detector. Because the NPD detec-tors have an acceptance of±5.5◦, {1 01 2} twinning manifestsas an exchange of (0 0 0 2)intensity from theQ⊥ to theQ||detector (and vice versa for the prism planes) during deforma-tion. Therefore, the diffracted intensity of the (0 0 0 2) in theQ|| detector bank may be used to monitor the development ofthe twinned (daughter) microstructure during deformation.

Figs. 6a and b show the integrated intensities (normalizedto the background counts) of several grain orientations in theQ|| detector as a function of applied stress. In order to putall of the intensities on one scale, the peaks that disappear

Fp

(appear) have been normalized to 100 at their initial (final)point. Uncertainties in the diffracted intensities have been in-tentionally omitted for clarity, but in general, they are roughly10%. The figure includes three grain orientations within thebasal plane (Fig. 6a), as well as five grain orientations thatare in the plane containing the{1 01 0} and (0 0 0 2)poles(Fig. 6b), also called the plane of shear of the{1 01 2} twinsystem[24]. The{1 01 0} normal is in the plane of shear andin the basal plane. The{2 13 0} and{1 12 0} plane normal arealso in the basal plane but are 19.1 and 30◦ from the planeof shear, respectively. The{1 01 1}, {1 01 2}, {1 01 3} and(0 0 0 2)poles are all within the plane of shear and are 28.1,46.9, 74.6 and 90.0◦ from the basal plane, respectively.

The measurements are most sensitive to intensity changesin peaks that are initially weak or completely absent, e.g.(0 0 0 2) in theparallel detector bank. However, both the in-crease in some peak intensities and the decrease in others areindicative of twinning activity. The (0 0 0 2), {1 01 2}, and{1 01 3} peaks exhibit a marked increases in intensity be-tween−50 and−70 MPa signaling the onset of twinning.The (0 0 0 2)intensity increases more rapidly and reaches amaximum that is not observed in the{1 01 2} or {1 01 3}curves.

The initial intensity changes are less obvious in the peaksthat start strong, such as the{1 01 0}, {2 13 0}, {1 12 0} and{1 01 1} peaks, and it is difficult to discern the initiation oftda sei hichi essest

-c bsentf y. Ass de-v . Thet omp func-t

ig. 6. Normalized intensities of several diffraction peaks{h k i l} in thearallel detector.

winning from them. However, it is clear that the{1 01 0}iminishes most rapidly, followed in order by the{2 13 0}nd{1 12 0}. The{1 01 1} intensity shows an initial increa

n intensity that is outside of errors bars, the source of ws unknown, and subsequently diminishes at larger strhan the three prism planes.

Recall that the (0 0 0 2) and{1 01 0} have special signifiance in this measurement because they are initially arom the parallel and transverse detectors, respectiveluch they provide clean information pertaining to theelopment of the daughter and parent microstructuresransfer of diffracted intensity in the parallel detector frarent to daughter grains is shown quantitatively as a

ion of imposed plastic strain inFig. 7. Diffracted intensity

D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12 7

Fig. 7. Measured intensity from parent (open circle) and daughter (closecircle) grain orientations as a function of plastic strain. Dashed line representscalculated intensity from daughter grains.

from the twins begins to appear at strains of about 1%, con-current with the initial decrease in slope of the macroscopicstress/strain curve (seeFig. 4), and increases linearly withplastic strain to roughly 6% deformation. Twinning appearsto saturate after 8% plastic deformation. The total twinnedvolume fraction after 14% imposed strain determined fromthe ODF’s measured before and after deformation is 0.80,but we will assume that was achieved by 8% deformation.Whilst twinning is the dominant deformation mechanism, thetwinned volume fraction is approximately proportional to theintensity of the (0 0 0 2) diffraction peak in theQ|| detector.The rate of increase of the twinned volume fraction can thenbe correlated with the slope of the (0 0 0 2) intensity curveand is 0.145 per unit plastic strain. The figure also shows the(0 0 0 2)intensity as a function of plastic strain calculated bythe model. Clearly, twinning occurs much too rapidly in themodel, saturating at 4% strain.

Subsequently, the calculation predicts a maximum andsharp decrease in the (0 0 0 2)peak intensity that may be as-sociated with the onset of pyramidal slip activity in daughtergrains in the model. The rotation of the basal poles away fromthe load direction that results in the split lobes shown in thecalculated pole figure (Fig. 2) removes the poles from the cov-erage of the NPD detector designated schematically by thecircle in the center of the pole figures. The observed (0 0 0 2)intensity displays a similar, albeit considerably weaker, max-i ationo ed int thep et ofp

4 t

utet ea-s

ε

whered0 represents the unstrained interatomic spacing. Theinitial interplanar spacing is good approximations of the un-strained spacings (dhkl

0 ) because the material is fully annealedand has a nearly isotropic coefficient of thermal expansion[18]. The uncertainty in the strain measurements representsthe combination (error propagation) of the statistical uncer-tainty in the peak fitting,δdhkl, based upon the estimatedstandard deviations (esd’s) of the peak positions output fromGSAS, as well as the uncertainty of the calculated unstrainedlattice parameters,δdhkl

0 ,

δε = ε

(δdhkl

dhkl+ δdhkl

0

dhkl0

)(2)

Again due to the initial texture, the diffracted intensity ofthe (0 0 0 2)peak in theQ|| detector is initially absent. Basedon the arguments presented above, this peak is, therefore,uniquely poised to monitor the volume fraction of twins andthe strains within the twins as they form. The only problemwith such an approach is the lack of an unstrained lattice pa-rameter measurement to use in the above strain calculations.Using the unstrained lattice parameter from the other detec-tor bank circumvented this problem. Due to the possibilitythat sample position results in different Bragg peak positionswithin the two detector banks, an estimate of the error in-curred was made.

ntero k po-s s be-t hes io oft rs nks)ta r thiss e ono 0 in-d hownt

per-p pliedl tsg lt e( racti pre-s singlec av-i ofe er ag-g twinst esses.T d ina

owna ed

mum and ensuing decrease. Concurrent with the saturf twinning, there is an increased hardening rate observ

he macroscopic flow curve, which through the link witholycrystalline model, may be associated with the onsyramidal slip in daughter grains.

.4. In situ measurement of internal strain developmen

Lattice strains for the grain orientations which contribo a given (hkl) diffraction peak are calculated from the mured interplanar spacing using Eq.(1)

= dhkl − dhkl0

dhkl0

, (1)

Small displacements of the sample relative to the cef the instrument cause a slight discrepancy in the peaitions and, therefore, the determined lattice parameterween theQ|| andQ⊥ detectors. This effect could skew ttrain results if not taken into account. The average rathed-spaces between theQ|| andQ⊥ detectors was found foix diffraction peaks (few peaks were strong in both bao be 1.0011 with a standard deviation of 0.00012 (120��)nd the unstrained lattice parameters were corrected fohift. To check for repeatability, this process was donther “standard” texture free samples where roughly 2ependent peaks were available in each bank and was s

o be repeatable.The internal strains in the parent grains parallel and

endicular to the load axis are shown as a function of apoad inFig. 8a. It is important to note explicitly that parenrains are aligned such that the{1 01 0} poles are paralle

o the stress axis and diffract into theQ|| detector, while th0 0 0 2)poles are perpendicular to the load axis and diffnto theQ⊥ detector. The dashed lines on the figure reent the theoretical linear elastic response based on therystal moduli[17]. The parents exhibit linear elastic behor until 70 MPa. Beyond the elastic limit, the evolutionlastic strain (i.e. stress) in the parents saturates. At highregate stresses, other grain orientations, including the

hemselves, are shown to sustain higher than average strhis sort of composite load sharing behavior is observell types of polycrystals[16].

The internal strains in the daughter grains are also shs a function of applied load inFig. 8b. Once again, dash

8 D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12

Fig. 8. Internal strains in (a) parent and (b) daughter grains parallel and perpendicular to the load axis. Closed (open) symbols represent data takenon loading(unloading). Dashed lines represent elastic behavior.

lines represent the calculated elastic response. The latticestrains within the twins, whose basal poles are parallel to theapplied stress field, are first measurable at a macroscopic loadof −70 MPa, after a small amount of twin reorientation hastaken place. The elastic strain in the first daughter grains ob-served is roughly−700��, considerably less than then theparent and surrounding grains. As the macroscopic load con-tinues to increase from−70 to−220 MPa, the slope of thelattice stress/strain curve, 32 GPa, is less than the calculatedelastic response of grains with basal poles parallel to the ap-plied load, 48 GPa. Above 220 MPa (macroscopic strain of0.08), an inflection is observed in the daughter applied stressversus elastic strain curve and the slope increase to greaterthan the linear elastic response.

The initial transverse strains in the daughter grains arecloser to the calculated linear elastic response. The uncer-tainty is larger because the counting statistics are not asgood for the{1 01 0} peak as the (0 0 0 2)peak. Upon fur-ther loading, and subsequent unloading, the response of thedaughter grains perpendicular to the applied load closely fol-lows that calculated from the elastic modulus and Poisson’sratio.

F ion ofa oved.C

4.5. Peak broadening

In addition to changes in peak position, the data also re-veal that there are changes in peak width. The ideal instru-mental resolution of the NPD, FWHM, is�d/d= 2.5× 10−3.However, in the present configuration, with a long horizon-tal sample, the instrumental resolution is likely to be de-graded, but has not been explicitly measured. If we assumethat the peak shapes are well described by Gaussian func-tions, the sample broadening is determined by subtractinginstrumental broadening from the measured peak width inquadrature.

In the current measurements, the (0 0 0 2) diffraction peakfrom the twins broadens considerably from its inception at−70 MPa until twinning completes at−220 MPa.Fig. 9shows the broadening of the parent{1 01 0} and daughter(0 0 0 2)reflections parallel to the load as a function of ap-plied stress; the area in which twinning is active is shaded.In the elastic regime, the FWHM of the{1 010} peak is nearthe instrumental resolution and constant to within uncertain-ties. With the activation of deformation twinning, the{1 010}peak begins to broaden significantly and continues to do soas the parents are consumed by the twins. When the daugh-ter grains first appear, the (0 0 0 2)peak width is similar tothe {1 010}, but immediately increase with subsequent de-formation. The broadening saturates as twinning completesa na

5

urveo hreed de-f eret uenti otherd opic

ig. 9. Broadening of parent and daughter diffraction peaks as a functpplied stress. The instrumental resolution has been approximately remlosed (open) symbols represent data taken on loading (unloading).

nd remains constant beyond−220 MPa or 8% deformatios well as on unloading.

. Discussion

For the purposes of discussion, the macroscopic flow cf the in-plane compression sample will be broken into tistinct regimes: (1) the elastic–plastic transition, where

ormation twinning initiates, (2) the low stress plateau whwinning dominates the deformation, and (3) the subseqncreased hardening, where twinning has saturated andeformation mechanisms must activate. The microsc

D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12 9

phenomena observed through the development of the diffrac-tion pattern will be related to each region.

5.1. Formation of twins

Macroscopically, rolled magnesium compressed in the in-plane direction linearly deforms in the elastic region up to−50 MPa. Between−50 and−70 MPa the flow curve de-parts from linearity concomitant with a significant change indiffraction peak intensity manifesting the onset of deforma-tion twinning. In general, it has been asserted that, like slip,deformation twinning is primarily controlled by the resolvedshear stress on the twin system[24], that is twinning obeys aSchmid law,

τt = σc cosλ cosχ (3)

where λ and χ are the angles between the load axisand the twin plane normal and twin direction. Specifi-cally, Gharghouri et al.[18] confirmed this in the case of{1 01 2} tensile twinning in magnesium reporting a CRSS of65–75 MPa in Mg 7.7 at.% Al alloys.

The initial sample texture and alignment employed in thisexperiment strictly defineλ andχ, and thus the Schmid fac-tor. The basal poles in the parent microstructure are initiallynearly horizontal and transverse to the applied load. In orderf ep obbo thed dur-i

and7 el me-tm be-t ctionm SS tob f.a odelo sest iento

{ ig-n itialc inali y as-c nten-s mt owni lieso inT on

Table 1Schmid factor and stress at which intensity is halved for several grain orien-tations{hkil}Grain orientation Maximum Schmid factor σ1/2 ± 2 (MPa)

{1 01 0} 0.499 −101{2 13 0} 0.445 −107{1 12 0} 0.374 −111{1 01 1} 0.305 −197

axis have a high Schmid factor ofm= 0.499 and reach halfintensity at the lowest stress. On the other hand, in grainswith {2 13 0} or {1 12 0} pole parallel to the stress axis, themaximum Schmid factor ism= 0.445 or 0.374, respectively.Correspondingly,σ1/2 increases with decreasing Schmid fac-tor. In fact,σ1/2 linearly decreases with Schmid factor overthis small data set. Finally, in a grain with a{1 01 1} polealigned with the compression axis, the highest Schmid fac-tor ism= 0.305 andσ1/2 =−197 MPa. This is much largerthat would be expected based on the three prism poles, butmay be obscured by the initial increase in the{1 01 1} diffrac-tion intensity. There is an intermediate orientation of{1 01 1}and we do not presently understand the source of the initialincrease in intensity. While the Schmid law cannot be quanti-tatively tested nor a CRSS determined, the data is consistentwith the conclusion of Gharghouri et al.[18] that twinningin Mg is controlled by a Schmid law.

The experimentally determined textures indicate thatroughly 80% of the original parent microstructure hastwinned during deformation to 14%. Through comparisonwith the calculated pole figure (Fig. 2) the remainder of theparent microstructure is associated with grains oriented withbasal poles atΨ = 90 and 270◦ andθ = 45◦, which result inthe observed eccentricity of the basal fiber. The Schmid factorfor twinning is relatively low in these grain orientations, be-tween 0.15 and 0.3 depending on orientation about the basalp antlyo andt slipm

5

s atw tals twin-n ter onc n thed stals ugh-t ita -p in inte ghterg at 50a

or the daughter basal pole to appear in theQ|| detector, thlane of shear, both the{1 01 0} and{1 01 2} poles must alse horizontal. Moreover, a pole of the{1 01 0} family muste parallel to the load axis. If a prism pole of the{1 12 0}r {2 13 0} families were horizontal, the basal pole ofaughter grain would rotate out of the horizontal plane

ng twinning and would not be detected.The initiation of twinning was detected between 50

0 MPa by the appearance of (0 0 0 2)intensity parallel to thoad. In the orientation defined by the experimental geory, the maximum Schmid factor,m = cosλ cosχ, is 0.499aking the critical resolved shear stress of the twin system

ween 25 and 35 MPa. The stress step size between diffraeasurements was not fine enough to determine the CRetter accuracy. The visco-plastic model employed in Re[9]ssumed a CRSS of 15 MPa, while the elasto-plastic mf Ref. [10] uses an initial value of 30 MPa. In both ca

wins harden linearly because of slip activity with a gradf 30 MPa.

The diminishing intensity in parent grains ({1 01 0},1 12 0}, and {2 13 0} poles parallel to the load) also sals the onset of twinning. Unfortunately, because the inhanges in intensity are small relative to the large origntensity, we cannot utilize the parent grains to accuratelertain the CRSS. However, the stresses at which the iity of each has been halved,σ1/2, can be interpolated frohe data inFig. 6 with reasonable accuracy and are shn Table 1. The maximum Schmid factor on these famif parent grain orientations{h k i l} are also summarizedable 1. Grains with a{1 01 0} pole along the compressi

ole. Also, in these grains the basal poles are significffset from the nominal orientation transverse to the load

herefore, they are more likely to deform through basalodes, on which the Schmid factor is nearly 0.5.

.2. Internal strains in parent and daughter grains

While the initial parent orientation defines the streshich a given grain twins, it is the rotation of the crysymmetry and the associated shear achieved during theing event that determines the stress state of the daughreation and throughout subsequent deformation. Giveiffraction elastic constants derived from the single crytiffnesses of magnesium, isolated grains having the daer orientation ((0 0 0 2)parallel to the load) would exhib

lattice strain of−1460�� in a −70 MPa uniaxial comressive stress field. However, the initial observed stra

he daughter grains is roughly−700��, roughly half of thexpected lattice strain. It should be noted that the daurains likely formed between the measurement pointsnd 70 MPa. Thus the initial observation of−700�� on the

10 D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12

daughter orientation represents an upper bound on the strainin the daughter grains at formation. The relaxed state of thedaughter grains relative to the surrounding matrix can be un-derstood in terms of the schematic shown inFig. 3. Becausethe c/a ratio of magnesium is less than

√3, the twinning

shear geometrically “shortens” the grain along the load axis(lengthens it along the initialc-axis), and a fraction of theelastic compressive strain is replaced by this plastic compo-nent. The constraint of the matrix on the grain then places thedaughter in tension relative to the macroscopic stress field,that is the applied compressive load.

While deformation twinning is the dominant deformationmechanism, from 1 to 8% strain, the slope of the daughterstress/strain curve, 32± 3 GPa (seeFig. 8), is considerablyless than the (0 0 0 2)specific elastic modulus, 48 GPa. Thisslope represents the averaged response of existing and newlyformed daughter grains. Since the results discussed aboveestablish the fact that the newborn daughters form relaxedrelative to the matrix, and then the existing daughter grainsmust accumulate strain more rapidly than the aggregate as awhole. This makes sense, since the daughters themselves arein a hard orientation, poorly oriented for all but the hardestslip and twinning modes and will accept a greater proportionof the elastic strain as surrounding grains deform inelastically.As a result, lattice strain builds up faster than linear elasticitydue to load sharing with the parents and surrounding grains,w ices xt liedl untilt le.

overt n-t ot cursi etrys rval,s ture( sed.

5p

to0 tive.C ena,n on-n truc-t ins.T at lit-t thisr pidly.A ed toa idals low),t

Daughters that formed early at−70 MPa continue to ac-cumulate elastic strain during subsequent deformation withan effective modulus,Eeff, of 32 GPa, or a total strain ofε = 0.0047 by−220 MPa. The development of elastic strainin the parent grains saturated once twinning began at roughly−70 MPa (Fig. 8). Therefore, it is likely that daughters thatform near the completion of twinning do so in a state similarto that of the initial daughters, that is relatively unstrained.This would lead to a spread of the interplanar spacings of the(0 0 0 2)planes between the first and last daughters of 0.008Aconsistent with the observed increase in the FWHM of thedaughter peak after−220 MPa and is thus likely the sourceof the broadening.

5.4. Role of twinning in plastic deformation

Obviously, twinning is playing an important role in thedeformation from 0.5 to 8% deformation. It is interesting toconsider how much of the plastic deformation is accommo-dated by twinning in this regime. For a given grain orienta-tion,g the strain achieved parallel to the applied load,ε11, isrelated to the characteristic shear of the twin system,S, bythe Scmidt factorm, specific to that grain orientation, and thevolume fraction of the grain which is twinned,ν,

ε11(g) = m(g)γtw = m(g)ν(g)S. (4)

da

ε

w andt sim-p nota soa re-c hmidfob totalp mayt

a volves l-o stics , int t.U twini asticd mo-d ingb ing

hich are in a plastically “soft orientation”. The elastic latttrains in the parent grains, shown inFig. 8a, continue to relahroughout this region. In fact, perpendicular to the appoad, the elastic strain in the parent grains remains zerohe corresponding diffraction peak is no longer resolvab

The transfer to the daughter microstructure occurshe range from−70 to−220 MPa as manifested by the coinuous increase in (0 0 0 2) diffraction intensity parallel the applied load. The spread over which twinning ocs not a grain orientation effect, as the diffraction geomtrictly defines the orientation. Rather, at each stress inteufficient twinning occurs to relax the parent microstrucFig. 8a) and then ceases until further deformation is impo

.3. Variability in intergranular stresses indicated byeak broadening

Significant peak broadening, FWHM from 0.002.006A, was observed in the regime where twinning is aconvoluted within the peak broadening are two phenomamely the type III intragranular strains (most frequently cected with the stress fields of dislocations and their s

ures) as well as grain-to-grain variations or the type II strahe slope of the daughter stress/strain curve indicates th

le plastic relaxation of the daughter grains is occurring inegime since the elastic strain/stress accumulates so ras such, the increase in broadening cannot be attributn increase in dislocation density. Indeed, when pyramlip does activate in the daughter grains (discussed behe sample broadening is no longer increasing.

The aggregate strain response,ε11, is the texture weighteverage of the single grain response

11 = S∑

g

m11νf = Sm11ν, (5)

heref (g) represents the orientation distribution functionhe sum is over the grains that twin. We have made thelifying assumption that the twinned volume fraction isfunction of grain orientation, i.e. all grains that twin dot some average rate.Fig. 6demonstrates that this is not pisely true, but it is a reasonable approximation. The Scactor on a grain with optimal orientation ism= 1/2. Averagedver the initial texture of the rolled magnesium, ¯m is found toe approximately 0.32. The incremental fraction of thelastic strain accommodated by deformation twinning

hen be written as

�εtwin

�εplastic= 0.32S

�ν

�εplastic, (6)

ssuming that the average orientation factor does not eubstantially during the deformation.Fig. 7shows the devepment of the twinned volume fraction as a function of platrain. The slope of the curve or the rate of twin formationhe linear region of the curve (εp < 5%) is 0.145/strain unising Eq.(6), and the characteristic shear of the tensile

n magnesium, 0.131, we estimate that from 0 to 5% pleformation, 61% of the plastic deformation is accomated by twinning. After 5% plastic deformation, twinnegins to slow down and by 8% deformation, where twinn

D.W. Brown et al. / Materials Science and Engineering A 399 (2005) 1–12 11

has saturated, 42% of the plastic deformation has been ac-complished by twinning. Reed-Hill performed similar anal-yses on rolled zirconium, in which the basal poles were ori-ented in the rolling plane, under applied in-plane tension. Heestimated the texture weighted Schmid factor to be 0.5, andreported that only 15% of the plastic deformation of rolledzirconium under applied in-plane tension was attributable totwinning[32]. A similarly low value, 12%, has recently beenfound for extruded zirconium compressed perpendicular tothe basal texture, and an intermediate ratio, 21%, in rolledberyllium compressed perpendicular to the basal fiber[33].

5.5. Mechanisms of c-axis compression (〈c + a〉 slip and{1 01 1} twinning)

Beyond−220 MPa or 8% strain, the macroscopic flowcurve again begins to decrease in slope. At this point, the slopeof the daughter applied stress versus elastic strain curve in-creases to greater than the linear elastic response, indicatingactivity of a plastic deformation mechanism in the daugh-ter grains that allows for relaxation in thec direction. Theinflection in the macroscopic and lattice stress/strain curvescoincides with the maximum observed in the (0 0 0 2)inten-sity parallel to the load axis (seeFig. 7). Ghargouri et al.[18] speculated that{1 01 1} 〈1 01 2〉 compression twinningmight explain some of their observations of small intensityc d int ag-n ther them tudies[ es along〈 easei ouldb , thec thert

ultsi the{o orre-sa n-s thatt rebytc al in-t ringu

a-t tureo com-p ilart ugh-t

stress/strain curve in likewise oriented grains[10] and the be-havior was satisfactorily reproduced with the EPSC model byassuming pyramidal activity. The through-thickness samplesalso showed a comparable reduction of (0 0 0 2) diffractionintensity parallel to the load when pyramidal slip activated.Beyond 10% strain, the in-plane compression sample is actu-ally stronger than its through-thickness counterpart, despitethe similar textures. It is speculated that the increased strengthis due to the remaining twin boundaries acting as barriers todislocation motion, similar to reducing the grain size[9].

6. Conclusions

Deformation twinning is active in textured magnesiumwhen loaded in compression transverse to the basal poles.The activity of deformation twinning is associated with thelow stress flow regime of the stress/strain curve from 70 to200 MPa (0.1–8% strain). The onset of deformation twinningon the{1 01 2} tensile twinning in magnesium is consistentwith a Schmid law, as concluded by Ghargouri et al.[18],and the critical resolved shear stress of the{1 01 2} 〈1 01 1〉in ally AZ31B is estimated to be between 25 and 35 MPa.Unobstructed diffraction from newly formed daughter grainsindicates that the twins are initially relaxed relative to them Sim-i xedd uentl loada rd”d nars reasei finald∼ in-n oft s arep tiono oda-t

A

t LosA 05-E atedb 25.T ionalu

R

ork,

hanges at large strain, which could not be understooerms of the more common slip or twinning modes of mesium. The monotonic strain level that was achieved ineported tests was insufficient to yield a final conclusion inatter. The present results, and those of other recent s

10], lead to the conclusion that〈c+a〉 slip provides a moratisfactory explanation for the observations. The stress0 0 0 1〉 could be relaxed by either mechanism. The decrn the basal fiber texture along the compression axis ce understood in terms of either mechanism. Howeverhange in overall texture points to the slip mechanism rahan the twin.

As mentioned earlier, any twinning mechanism resn a characteristic finite reorientation of the crystal. For1 01 1} compression twin, there is approximately a 56◦ re-rientation of the basal poles. Hence, there must be a cponding increase in intensity within the (0 0 0 2)pole figuret about 56◦ from the compression axis if twinning is respoible for the texture change. It was previously suggestedhis might be a pseudo-elastic deformation mode, whewinning under load “untwinned” during the unload[18]. Theurrent data clarifies this point since the decrease in basensity, which occurs at the highest loads, remains dunloading.

It should be noted that at this point in the deformion (subsequent to the tensile twinning regime), the texf the in-plane compression sample has been almostletely reoriented (Fig. 2b) and the situation becomes sim

o the through-thickness compression experiment. Throhickness compression produced a similar (0 0 0 2) lattice

acroscopic applied stress along the loading direction.larly, the elastic lattice stain in the parent grains is relauring the formation of the daughters. During subseq

oading, the daughter grains absorb a large fraction of thes it is shed from the “soft” parent orientation to the “haaughter orientation. A significant variation in interplapacing within the daughter grains, manifested as an incn diffraction peak breadth, arises between the initial andaughters to form. Throughout the twinning regime,εp < 8%,42% of the plastic deformation is accommodated by twing. Deformation twinning continues until roughly 80%

he microstructure is reoriented such that the basal polereferentially aligned with the applied load at the complef the test. After twinning has been exhausted accomm

ion of c-axis compression is achieved by〈c+a〉 slip.

cknowledgements

Research sponsored by the U.S. Dept. of Energy alamos National Laboratory under contract number W-74NG-36 and at the Oak Ridge National Laboratory opery UT-Battelle, LLC, under contract DE-AC05-00OR227he Manuel Lujan Jr. Neutron Scattering Center is a natser facility funded in part by the U.S. DOE.

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