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Acta Materialia 56 (2008) 688–695

Twinning–detwinning behavior during the strain-controlledlow-cycle fatigue testing of a wrought magnesium alloy, ZK60A

L. Wu a,*, A. Jain b, D.W. Brown c, G.M. Stoica a, S.R. Agnew b, B. Clausen c,D.E. Fielden a, P.K. Liaw a

a Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USAb Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA

c Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 17 September 2007; received in revised form 13 October 2007; accepted 14 October 2007Available online 11 December 2007

Abstract

The twinning and detwinning behavior in a strongly textured magnesium alloy was investigated using in situ neutron diffraction dur-ing the cyclic deformation along the prior extrusion direction at the fully reversed total constant strain amplitude of 1.2% at room tem-perature. The initial preferred orientation places the c-axis in most grains perpendicular to the loading axis, and this favors extensivef1012g h1011i twinning under compressive loading. In contrast, the grains are not favorably oriented to undergo such twinning duringmonotonic tensile loading along the prior extrusion axis. This is the reason for the well-known tension–compression strength asymmetryof wrought magnesium alloys. The strength in compression is controlled by the stress required to activate twinning, while the strength intension is controlled by the harder non-basal slip mechanisms. The unique orientation relationship between the parent grains and thetwin grains favors detwinning during the subsequent loading reversal. In situ neutron-diffraction results indicate that such twinningand detwinning alternates with the cyclic loading, i.e. most of the twins formed during compression are removed when the load isreversed. However, a small volume fraction of residual twins gradually increases with increasing cycles, which may be an important fac-tor in dictating the low-cycle fatigue behavior of the magnesium alloy.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Magnesium alloy; Cyclic deformation; Twinning; In situ neutron diffraction

1. Introduction

The magnesium alloys have recently attracted a greatdeal of research interest because of their potential applica-tions for lightweight structural components [1–5]. How-ever, the number of applications for which magnesiumalloys can be used is restricted because they always exhibita high directional anisotropy and are hard to deform atroom temperature, owing to their hexagonal crystallo-graphic structure and limited deformation modes [1]. Inthis regard, mechanical twinning plays a significant rolein the magnesium deformation. At room temperature,

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.10.030

* Corresponding author.E-mail address: lwu7@utk.edu (L. Wu).

twinning may help this material to satisfy the von Misescriterion, which requires five independent deformation sys-tems for an arbitrary homogeneous straining. Aside fromthe slip of dislocations with hcþ ai Burgers vectors, whichis recognizably a very hard deformation mechanism, twin-ning is the only active deformation mode that can providestraining along the c-axis at room temperature [6].

Most common wrought magnesium alloys usually dis-play a strong in-plane tension–compression asymmetry[7]. This phenomenon has been attributed to the mechani-cal twinning on the f1012g planes along the h1011i direc-tions during compression along the prior working direction(rolling or extrusion) but not during tension along the samedirection [1]. This is primarily the result of strong textures,where a significant fraction of grains orientated with their

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L. Wu et al. / Acta Materialia 56 (2008) 688–695 689

c-axis nearly perpendicular to the prior working direction[8], and the polar nature of twinning [9]. The monotoniccompressive curve with the activation of twinning typicallyexhibits a sigmoidal (S-shaped) shape, where a low stressplateau is initially observed, followed by a higher harden-ing evolution rate [10].

Because it is such an important deformation mode ofMg alloys, mechanical twinning has been the focus of anumber of recent studies [10–18]. For magnesium alloys,with c/a � 1.624 <

p3, the f1012g h1011i tensile twin-

ning mode can accommodate extensions along the c-axisof the hexagonal lattice but not contractions along thesame direction. This tensile twinning results in a nearly90� (�86.3�) reorientation of the basal pole, as shownin Fig. 1a [1]. Thus, detwinning may occur in thetwinned areas at subsequent reversed loading with thetensile stress applied along the c-axis of the twinnedmaterials (Fig. 1b) [12,13,19]. In other words, twins candisappear or become narrower under reversed loadingor unloading, and can reappear under reloading. Brownet al. [14,15], Gharghouri et al. [11] and Oliver et al.[16] found that twinning and detwinning appear alter-nately in cyclic loading using in situ neutron scattering.Lou et al. [17] revealed twinning during in-plane com-pression and detwinning upon the subsequent tensionof an Mg alloy, AZ31B, sheet using metallography,acoustic emission and X-ray texture measurements. Thistwinning–detwinning behavior produces unusual hystere-sis loops during cyclic deformation [15–17].

However, the cyclic twinning–detwinning behavior ofmagnesium alloys is far from fully understood. In the cur-rent research, the twinning–detwinning behavior ofwrought magnesium alloy, ZK60A, was investigated atthe Los Alamos National Scattering Center (LANSCE)using in situ neutron diffraction. This is the first detailedreport of twinning–detwinning behavior of the alloyZK60A, and the results serve to highlight the similaritiesand distinctions between two of the most common wroughtmagnesium alloys, AZ31B and ZK60A. Implications for

Fig. 1. Schematic of the f1012g h1011i tensile twin system in magne-sium: (a) 86.3� reorientation of the twin grain relative to the parent grain[1]; (b) applied loading directions with respect to the c-axis [19]. The solidarrows indicate the applied loading directions favorable for the tensiletwinning, and the open arrows indicate the applied loading directionsunfavorable for the tensile twinning.

the low-cycle fatigue behavior of wrought magnesiumalloys in general are discussed.

2. Experimental details

The magnesium sample used in this study was machinedfrom a commercial extruded magnesium plate with T5 tem-per (solution-treated at 535 �C for 2 h, quenched in hotwater and aged at 185 �C for 24 h), which has a nominalcomposition of 6.0% Zn, 0.5% Zr (wt.%) and Mg as bal-ance. The initial texture exhibits the combined features ofthe typical rolling and extrusion textures (Fig. 2). Thereare two major texture components, one with the basal polesperpendicular to the plate normal (ND) and the other withthe basal poles parallel to the transverse direction. There isa greater angular spread in the pole density from the NDtoward the extrusion direction (ED) than toward the trans-verse direction (TD). In this study, those grains with theirinitial c-axis parallel to the plate normal (Fig. 2) wereextensively twinned under the compressive loadings anddetwinned during the subsequent reversed loadings, andtheir behavior was readily captured using in situ neutrondiffraction.

In order to analyze the microstructures by opticalmicroscopy, the cylindrical specimens were cut in theED–ND plane. Specimens were prepared using standardmetallographic techniques, finishing with 1 lm diamondpaste in methanol. Acetic picral solution (5 g picric acid,10 ml water, 5 ml acetic acid and 90 ml ethanol) wasemployed to etch the specimens for 5–10 s, which revealedthe grains and/or twins.

In situ neutron-diffraction measurements were con-ducted on the Spectrometer for MAterials Research atTemperature and Stress (SMARTS) (Fig. 3) at the Man-uel Lujan Jr. Neutron Scattering Center, LANSCE.Details of the SMARTS have been published elsewhere[10,14] and only a short description is presented here.The threaded-end sample, having a 19.05 mm gage sec-tion with a 6.35 mm diameter, was mounted in the hor-izontal load frame, and the prior plate normal wascarefully aligned in the horizontal direction so that alarge volume fraction of grains have their basal polesin the diffraction plane (horizontal), but transverse tothe loading direction. The loading axis is oriented at45� relative to the incident beam (Fig. 3) and the twodetector banks situated at ±90� simultaneously recordtwo complete diffraction patterns with diffraction vectorsparallel (Q||) and perpendicular (Q\) to the applied load.The cyclic testing was performed using the Instron-cus-tomized load frame under fully reversed cyclic loadingconditions at a constant total strain amplitude of 1.2%with a triangular waveform at room temperature. Themacroscopic strain was measured using an extensometerattached to the sample, and the imposed cyclic frequencywas 0.5 Hz. The initial loading was compressive, whichresults in significant extension twinning during the firstquarter cycle.

Fig. 2. Initial texture of the as-extruded magnesium alloy, ZK60A. The scale at right indicates the relative diffraction intensity (1.0 = random). ED,extrusion direction; ND, normal direction; and TD, transverse direction.

Fig. 3. Schematic of the SMARTS configuration at LANSCE.

690 L. Wu et al. / Acta Materialia 56 (2008) 688–695

Owing to the high penetration of neutrons into the bulkmaterials, the entire volume of the specimen can be probedby the neutron beam regardless of the specimen orientationduring the measurements. Twinning takes place through-out the volume of the sample; the full penetration of thesamples by neutrons provides reliable volumetric informa-tion. It is thus quite easy to detect twinning by neutron dif-fraction, which results in a change in the peak intensity.The transfer of the f0002g diffraction intensity betweenthe two detector banks is a direct measure of the twinnedvolume fraction and allows us to monitor the evolutionof the twinned microstructures. In particular, the uniqueorientation relationship between the parent grains andthe twin grains coupled with the SMARTS configurationfacilitates the study of the cyclic twinning–detwinningbehavior.

3. Results and discussion

Fig. 4 clearly shows that the hysteresis loops loadedalong the extrusion direction (Fig. 4a) are distinct fromthose loaded along the transverse direction (Fig. 4b). InFig. 4b, the hysteresis loops are symmetric between thecompression and tension cycle, which are usually the resultof the dislocation slip-dominated deformation in mostmaterials [20], while in the current Mg alloy, the cyclic

deformation behavior along the transverse direction withthe normal-shaped loops (Fig. 4b) are to be studied usingin-situ neutron and synchrotron diffractions, and theresults will be reported shortly. In contrast, the hysteresisloops in Fig. 4a are asymmetric, with a sigmoidal shapecharacteristic of mechanical twinning [15]. Upon the initialcompressive yielding at �150 MPa, the sample showed lit-tle hardening and the maximum compressive stress at 1.2%was �170 MPa. This type of strain hardening plateau istypical of materials which deform by twinning, includinghexagonal close-packed Mg alloys [10,13,21] and Zr alloys[22,23], as well as low-symmetry martensite shape memoryalloys such as NiTi [24].

Upon reversal, a significant Bauschinger effect wasobserved, such that the material yielded even before thestress became positive. The reverse yielding is much moregradual than the abrupt elasto-plastic transition observedduring the initial compressive loading. During the reversal,at �0% strain the loop shows an inflection, and beyond thispoint, the hardening rate rapidly increases. It will be shownbelow that this increase in the hardening rate correlateswith the exhaustion of the detwinning mechanism. Oncethe grains are completely detwinned, the resulting orienta-tion is hard with respect to tensile deformation by basalslip. This demands activation of the harder non-basal dis-location slip [6] or f1011g h1012i compression twinningmechanisms [21,25]. Thus, a high maximum tensile stressof �310 MPa was reached.

On the unloading from +1.2% strain, the deviation fromlinear elastic unloading occurs roughly at a stress of�100 MPa and a strain of +0.2%. Because the yieldingduring the second compression cycle started early, morecompressive plastic strain was induced prior to reversinga second time. Interestingly, the maximum compressivestress during the second compressive cycle was similar tothe first, and this is an additional testimony to the verylow hardening rate which accompanies twinning domi-nated deformation in Mg alloys. Furthermore, it suggeststhat more twinning occurs during the second cycle thanthe first, and the in situ neutron diffraction data discussedbelow corroborate this conclusion. The maximum tensile

Fig. 4. Hysteresis loops loaded along the prior extrusion direction (a) and along the transverse direction (b).

L. Wu et al. / Acta Materialia 56 (2008) 688–695 691

stress during the second cycle was comparable to that ofthe first as well; however, the inflection of the tensile curveappeared later, at a strain of �0.5%. Again, it is under-stood that more twinning took place during the secondcompressive cycle, and more strain is required to detwinthe material during the second tensile cycle than duringthe first. The in situ diffraction data enable us to make firmconclusions regarding these points. The subsequent hyster-esis loops closely follow the shape of the second cycle. Nev-ertheless, in the compressive quarter cycles, the materialshowed an ever increasing hardening rate during the latercycles (see, for example, the hysteresis loop of the cycle200 in Fig. 4a); in addition, the maximum compressiveand tensile stresses developed distinctly with increasingcycles (Fig. 5).

Fig. 5. Stress–response curves showing the variation of tensile andcompressive stresses with the fatigue cycles at the total strain amplitudeof 1.2%.

The variation of the stress response with the number offatigue cycles is an important behavior of the low-cyclefatigue process. Such a cyclic-stress response dependsmainly on the mechanical and/or cyclic stability of theintrinsic microstructural features, particularly as a resultof the competitive processes between the hardening fromthe multiplication of dislocations [26] and the twin bound-aries as barriers to dislocation slip [27], and the softeningdue to the annihilation and rearrangement of dislocations[26,28,29]. The peak compressive stress exhibits continuouscyclic hardening until the final fracture. The neutron dif-fraction data suggest that this is due to an increasing vol-ume fraction of twins which form during the cycling. Onthe other hand, the material exhibits a tensile hardeningduring the initial 10 cycles and then the cyclic tensile soft-ening dominates until a final abrupt stress drop. The over-all shape of the hysteresis loops and the neutron diffractiondata again provide an explanation for this more compli-cated observation.

As stated in the experimental section of this paper, com-plete diffraction spectra were collected using the time-of-flight technique in detector banks located at +90 and 90�from the incident beam. Due to the 45� orientation of thesample stress axis, these detector banks collect informationfrom grains with their diffracting plane normal vectors ori-ented parallel and transverse to the stress axis, respectively.The diffraction data reveal a number of different aspectsrelated to the structure and stress state of the material.For example, the lattice spacings derived from Braggslaw can be used to determine the state of internal latticestrain (and stress via Hooke’s law) [10]. The appearanceor disappearance of peaks can signify a phase transforma-tion or, in the present case, mechanical twinning. Due tothe �90� reorientation of the basal poles which occurs withf1012g h10 11i twinning and the initial texture, thechanges in the basal f0002g peak intensities in the paralleland transverse detector banks can be directly related to the

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degree of twinning that has occurred. This fact wasexploited to advantage in previous in situ studies of twin-ning in Mg [11,15].

In short, the initially high basal f0002g peak intensityin the transverse bank (Bank 1) is indicative of the largevolume fraction of ‘‘parent” grains which are favorably ori-ented for twinning during the compression cycle, and canbe correlated with the high intensity in the basal f0002gpole figure parallel to the ND of the initial texture(Fig. 2). The initially low intensity in the parallel bank(Bank 2) correlates with the initially low intensity in thef0002g basal pole figure along the ED (Fig. 2).

Fig. 6 shows the normalized-diffraction intensity evolu-tions of the basal f0002g peak during the cyclic deforma-tion along the extrusion direction corresponding to thehysteresis loops presented in Fig. 4a. The intensity data isplotted as a function of the ‘‘run number” (which was usedfor the experimental bookkeeping to identify the diffractiondata), and the actual cycle number is also indicated in theplots. An increase in the f0 002g peak intensity withinthe parallel bank indicates an increase in the volume frac-tion of twins and may be correlated with a concurrentdecrease in the corresponding intensity diffracting fromthe parent grains in the transverse bank. A decrease inthe parallel bank intensity is, in the present set of experi-ments, correlated with detwinning events (or the shrinkingof twins and reversion to the parent orientation).

A detailed view of the intensity evolution is provided forthe first few cycles in Fig. 7. At the initial compression, theintensity of the basal f000 2g poles remained unchangedbecause the deformation is elastic and the initial micro-structure of ZK60A with T5 temper is free of twins(Fig. 8a). Upon yielding, at a strain of �0.2% (Fig. 4a),the intensity begins to gradually increase, which indicatesthat the twinning was activated, and a lot of needle-liketwin bands were observed in some favorably orientedgrains at a strain of 1.2% (Fig. 8b). Previous in situ diffrac-tion experiments focused on the monotonic deformation

Fig. 6. Normalized intensity evolutions of basal f

have shown that as much as 80% of the volume fractionof favorably oriented grains can undergo twinning duringthe first 8% strain [10,30]. There is a direct relationshipbetween the volume fraction of twins, ft, the characteristictwinning shear, ct (for tensile twins in Mg alloys, ct = 0.130[31]), the Schmid factor of the twin variant and grain orien-tation, mg,t, and the macroscopic strain, et, accommodatedby twinning:

et ¼ f tmg;tct ð1ÞIn the present case, the Schmid factor for twinning in theparent grains with basal poles parallel to ND or TD canbe approximated as �0.5. Determining the volume fractionof twins from the increase in the f0 002g intensity withinthe parallel bank confirms that the plastic deformation dur-ing the compression stroke is dominated by twinning. Be-cause twinning is dominating the strain accommodation,and the overall strain hardening rate is low, it can be in-ferred that massive twinning leads to a low strain-harden-ing rate (Fig. 4a) [19].

The stress–strain response departs from linear elasticcompressive unloading almost immediately (Fig. 4a), indi-cating the activity of some plastic dissipation. Correspond-ingly, as soon as the straining is reversed, the f00 02gintensity began decreasing. This indicates that detwinningrequires much lower absolute stresses to occur in alloyZK60A relative to the stress required to activate the twinsthemselves. This places the present results in agreementwith previous twinning–detwinning observations of alloyAZ31B [15,17]. In fact, the detwinning begins so quicklythat it is suggested that the twins and parents must containsignificant internal stresses that drive the detwinning event.This hypothesis is presently being explored in detail usingthe collected in situ lattice strain data, and will be reportedshortly.

The f0002g intensity in the parallel bank, indicative ofthe twin volume fraction, continued to decrease all the wayback to the background intensity until the plastic compres-

0002g poles as a function of the run number.

Fig. 7. Normalized intensity evolutions of basal f0002g poles in the Bank 2 during the first few cycles.

Fig. 8. Optical microstructures at various strain states: (a) initial state (0% strain), free of twins; (b) first maximum compression (�1.2% strain), a lot oftwins in some favorably oriented grains; (c) first maximum tension (+1.2% strain), disappearance of twins.

L. Wu et al. / Acta Materialia 56 (2008) 688–695 693

sive strain was recovered at 0%. As discussed and suggestedin the discussion of the hysteresis loops above, this resultconfirmed that the twin grains formed during the compres-sive deformation were entirely removed by detwinning,which was corroborated by the observed disappearanceof twins at the maximum tensile strain of +1.2% inFig. 8c. Again, this trend explains why the stress–straincurve becomes concave up at 0% (Fig. 4a), and beyond thispoint, the deformation was accommodated by the harderdeformation mechanisms within the parent grains, suchas prismatic <a> and pyramidal hcþ ai dislocation slip[32] along with possible f1011g h1012i compression twin-ning [21,25]. These mechanisms result in a higher strainhardening rate up to the maximum tensile stress(Fig. 4a), but they do not result in a rapid texture evolu-tion, thus the f0002g peak intensities remain unchangedduring this portion of the hysteresis loop (compare Figs4a and 7).

During the unloading from the maximum tensile strain,the f0002g pole intensity was equal to the backgroundintensity through the zero stress until the plastic deforma-tion begins again during the second compression stroke

at a strain of +0.5%, at which point the intensity curveincreased once again (Fig. 7). Actually, the plastic strainingbegan at +0.2% (Fig. 4a); however, we did not collectin situ diffraction data at this point. Because the secondcompressive straining induces more plastic strains thanthat introduced in the first cycle (Fig. 4a), a large volumefraction of twins were formed and, consequently, the max-imum intensity is higher at the end of the second compres-sive cycle than that of the first cycle.

During the second tensile reversal, the inflection point inthe hysteresis loop appeared at �+0.5%, which is consider-ably larger than the strain required to reach the inflectionpoint during the first tensile reversal, but is comparableto the starting point of the second compression (Fig. 4a)because a larger volume fraction of twins is available tobe detwinned and, concurrently, more tensile strain canbe accommodated by detwinning. Notably, the strainaccommodated by detwinning is the same as that of twin-ning, only opposite in sign (see Eq. (1)). This hypothesisis confirmed by the fact that the f0002g peak intensityin the parallel bank does not return close to thebackground level at 0% strain as in the first cycle.

Fig. 9. Typical fractography of the fatigue sample at the total strainamplitude of 1.2%.

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Unfortunately, diffraction data were not collected at levelsintermediate to +0 and +1.2% to confirm the exact strainlevel at which the detwinning mechanism is exhausted;however, it can be inferred to be �0.5% based on the hys-teresis-loop shapes of the first and second cycles (Fig. 4a).

The cyclic hardening and softening behavior can also beexplained by the variations in the loop shapes, when con-sidered in light of the corresponding in situ neutron diffrac-tion data. The continuously increasing basal f0002g peakintensity in the parallel bank and the correspondingdecreasing of the f0002g peak intensity in the transversedetector bank at the point of maximum compressive strainindicate the presence of an ever increasing volume fractionof twins (Fig. 6). This coincides with the compressive hard-ening of the material with the increasing number of cycles(Fig. 5). Similarly, the inflection point on the tensile side ofthe hysteresis loops is delayed to an increasing strain level(Fig. 4a), and thus the exhaustion of detwinning is contin-ually delayed to higher strain levels. The strain at which thehysteresis loop inflects is important because it is onlybeyond this point that the strain hardening rapidlyincreases, and the cyclic softening observed on the tensileside of the hysteresis loops at the later cycles (Fig 5) is mostprobably related to the decreasing amount of the post-det-winning dislocation-based flow in the parent grains. Fromapproximately 25 cycles onwards, the basal f0 002g peakintensity in the parallel bank never returned to the back-ground level (Fig. 6), suggesting that there is a residualtwin content that remained throughout the entire cycle.With the increasing number of cycles, the volume fractionof residual twins increases (Fig. 6), resulting in the ever-increasing hardening rate observed in the compressivequarter cycles of the hysteresis loops (Fig. 4a). It is sug-gested that an increasing amount of dislocation debriswithin the material makes twinning–detwinning more diffi-cult with cycling, which could simultaneously explain boththe cyclic hardening on the compression side and the cyclicsoftening on the tensile side of the hysteresis loops.

Fig. 9 shows the typical fractography of the magnesiumsample fatigued at a total constant strain amplitude of1.2% at room temperature. There exist a great number ofparallel traces, and secondary cracks propagated in a trans-granular mode along specific directions, which are presum-ably related to twin boundaries. The connection betweenfatigue cracking and deformation twinning will be furtherinvestigated by sectioning through the fracture surfaceand performing electron backscattered diffraction, in con-junction with fractographic stereology, as has recently beenapplied to aluminum alloys [33]. During the cyclic defor-mation, there are presumably strong interactions betweenthe slip and twinning [9,22,27]. This is particularly impor-tant because the newly formed twins are in a plasticallyhard orientation with their basal poles parallel to the com-pressive straining direction [15]. Furthermore, the increas-ing volume fraction of residual twins with cycles mayserve as barriers to the dislocation motion and vice versa[20,21]. Such complicated deformation interactions may

be important damage accumulation mechanisms that ulti-mately lead to the massive shear-band formation and crackinitiation, once slip and twinning cannot accommodate fur-ther plastic straining [9,19].

In summary, the intensity evolutions of f000 2g basalpoles observed via in situ neutron diffraction measurements(see Figs. 6 and 7) can be correlated with the activation ofthe f1012g h1011i twinning in this material (see Figs. 1–3), and these results may be used to explain the strangelyshaped hysteresis loops (Fig. 4a).

4. Summary and conclusions

The cyclic deformation of the magnesium alloy, ZK60A,was investigated using in situ neutron scattering to examinethe twinning–detwinning behavior. The major conclusionsare summarized as follows:

(i) the unique orientation relationship between the par-ent and the twin grains in the magnesium alloys facil-itates the investigation of the twinning–detwinningbehavior using the two-detector in situ neutron dif-fraction system of the SMARTS spectrometer;

(ii) the intensity transfer of f000 2g basal poles in thetwo detector banks can be reasonably related to theactivation of f1012g h101 1i twinning in this mate-rial, which are characterized by the strangely shapedhysteresis loops;

(iii) the fact that twinning and detwinning alternates withthe cyclic loading was confirmed using in situ neutronscattering, i.e. most twins formed during compressionare removed via detwinning when the load isreversed;

(iv) a small volume fraction of residual twins wasdetected, which gradually increases with increasingcycles, and may be an important factor in under-standing the low-cycle fatigue behavior of the magne-sium alloy;

L. Wu et al. / Acta Materialia 56 (2008) 688–695 695

(v) overall, the observed cyclic twinning–detwinningbehavior in ZK60A alloy was similar to the phenom-enon reported for AZ31B alloy, although their initialtextures are significantly different.

Acknowledgements

The authors are grateful to the financial support fromthe National Science Foundation – Combined ResearchCurriculum Development Program (CRCD) under EEC-9527527, with Ms. M. Poats as the Program Director,and the National Science Foundation – InternationalMaterials Institutes (IMI) Program under DMR-0231320,with Dr. C. Huber as the Program Director. The Los Ala-mos National Scattering Center (LANSCE) is a nationaluser facility funded by the United States Department ofEnergy, Office of Basic Energy Science – Materials Scienceunder the contract number of W-7405-ENG-36 with theUniversity of California.

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