International Journal of Plasticity 29 (2012) 60–76

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International Journal of Plasticity

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Twinning effects in a rod-textured AM30 Magnesium alloy

Q. Ma a,⇑, H. El Kadiri a,b, A.L. Oppedal a, J.C. Baird a, B. Li a, M.F. Horstemeyer a,b, S.C. Vogel c

a Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39762, USAb Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, USAc Los Alamos Neutron Science Center, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 October 2010Received in final revised form 27 June 2011Available online 24 August 2011

Keywords:MagnesiumTextureCrystal plasticityTwinningDislocations

0749-6419/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.ijplas.2011.08.001

⇑ Corresponding author. Tel.: +1 662 325 5457; faE-mail address: qma@cavs.msstate.edu (Q. Ma).

We experimentally and numerically investigated the effect of twinning on plasticity usingan extruded rod-textured magnesium alloy. The rod-texture is a h10 �10i-axis fiber texturethat presents a fundamentally different anisotropy correlated to twinning with respect tothe widely discussed c-axis fiber texture generated by clock rolling. We quantified a pro-fuse f10 �12gh10 �11i extension twinning along the extrusion direction (ED) that consumedthe entire parent before the inflection point in the stress–strain behavior. However, undercompression along the extrusion radial direction (ERD), the twinning model in the visco-plastic self-consistent formulation still predicts substantial extension twinning. However,in this case the stress–strain curve did not inflect, and Regime II hardening was absent.We demonstrate via EBSD analyses that the absence of Regime II hardening along theERD was due to a non-Schmid effect by multivariant ‘‘stopped’’ twinning. The intersectingvariants of stopped twins incurred twin–twin interactions that limited the twin growth.Profuse f10 �11gh10 �12i double twinning occurs both under ED and ERD but peculiarly trig-gered earlier under ERD than under ED, so the Voce model under VPSC could not capturetheir effect. The complex networks of stopped twins in the ERD clearly negate a possibleHall–Petch effect on Regime II by twin segmentation, since otherwise Regime II wouldbe more marked in the ERD. Rather, the stopped twins suggest preferential latent harden-ing within the twinned regions by parent dislocation transmutation upon their incorpora-tion in the twins. In fact, since twin–twin interactions mitigate the growth rates ofsweeping extension twin boundaries, dislocation transmutation could be limited to theextent that Regime II hardening will be eliminated.

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1. Introduction

Magnesium is historically recognized as a metal with poor formability at room temperature. The poor formability is ratio-nalized based on strong anisotropy due to the unavailability of more than five independent easy slip systems of equalstrength (on the closed packed planes) necessary to satisfy the Taylor’s criterion (Roberts, 1960). Kocks and Westlake(1967) first noted that the presence of mechanical twinning in some hexagonal close-packed (HCP) metals, such as magne-sium modifies the slip situation by essentially requiring other independent deformation modes. For instance, magnesiumexhibits a number of deformation modes: basal hai slip; f0002gh11 �20i, prismatic hai slip; f10 �10gh11 �20i, second pyrami-dal hc + ai slip; f11 �22gh11 �23i; f10 �12gh10 �11i extension twinning, f10 �11gh10 �12i contraction twinning andf10 �11g-f10 �12g double twinning (Barnett, 2007a,b; Barnett et al., 2008; Brown et al., 2005; Staroselsky and Anand,2003). All of these deformation modes have been observed using transmission electron microscopy (TEM) or/and electron

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Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76 61

backscattered diffraction (EBSD). Therefore, slip and twinning systems show a strong sensitivity to small changes in initialtexture, microstructure, temperature, chemical composition, loading path, strain rate regime, and stress state (Yoo, 1981;Thornburg and Piehler, 1975; Partridge, 1967; Christian and Mahajan, 1995). In particular, the effect of texture is currentlya focus on improving the formability and performance of magnesium alloys (Jiang et al., 2007a,b; Choi et al., 2007; Neil andAgnew, 2009).

Hardening correlated to deformation twinning is unanimously assumed in the literature to be caused by two main effects(Kalidindi, 2001; Kalidindi et al., 2003; Proust et al., 2009): (i) the increasing regional crystallographic reorientations in theparent that be from softer-to-harder or harder-to-softer orientations depending on the twinning mode, c/a ratio and texture,and (ii) initial grain refinement by twinning lamellae that is supposed to increase the Hall (1951) and Petch (1953) effect.1

The second effect is controversial with recent claims put forward by El Kadiri and Oppedal (2010) on the rate-increasing la-tent hardening in the twins by dislocation transmutations referred to as the Basinski mechanism before (Basinski et al., 1997).

Hard orientations are correlated to the higher difficulty to deform the crystal along the c-axis than any other crystal axis.In this regard, pyramidal hc + ai slip is a key deformation mode considered for magnesium and zirconium alloys above roomtemperature in polycrystal plasticity simulations (Akhtar, 1973; Jain and Agnew, 2007). In fact, it is the only slip mode as-sumed to accommodate deformation along the c-axis. Although, the pyramidal hc + aimode was recognized as a hard activeslip mode (Partridge, 1967), its critical resolved shear stress (CRSS) is sufficiently low to allow significant activity in orien-tations where the easy slip modes exhibit a very low Schmid factor (SF) (McCabe et al., 2006; Jain and Agnew, 2007). Theoperation of pyramidal hc + ai slip was demonstrated in the 1970s by Lavrentev and Pokhil (1975) and Lavrentev et al.(1977) through TEM studies, but recent studies by Agnew et al. (2001, 2005a) comparing simulated and experimental defor-mation textures confirmed the activity of this slip mode through the ‘‘double-peak’’ phenomenon.

The importance of the prismatic hai ðf10 �10gh2 �1 �10i dislocation activity was emphasized by Koike et al. (2003) and Sty-czynski et al. (2004) in their studies on cold rolled AZ31B and AZ31 alloys. These authors demonstrated that the activity ofthis slip mode resulted in more than 20% elongation. Furthermore, the consideration of the prismatic slip in polycrystal plas-ticity simulations using Taylor’s model resulted in a better reproduction of the experimental deformation texture. Accordingto the theoretical work of Hutchinson (1977), the continuous plastic deformation in HCP polycrystal metals with randomlyoriented crystals was only possible through the activity of hai type dislocations lying on the basal and/or prismatic planes.

Polycrystal modeling of HCP metals, in addition to experimental observations, can help us to understand deformationmechanisms. Recently, continuum plasticity models incorporated in the finite element method can simulate macromechan-ical responses of magnesium alloys based on phenomenological methods (Lee et al., 2008, 2009) or on energy minimizations(Homayonifar and Mosler, 2011), but these particular models do not have a physics basis for dislocations and twinning com-pared to the crystal plasticity models. The crystal plasticity finite element method (CPFEM), using in three-dimensional grainstructures considered grain morphology, can capture stress–strain behavior and texture revolution, whereas reorientation bytwinning and their morphology were still not taken into account (Diard et al., 2005; Graff et al., 2007). The viscoplastic self-consistent (VPSC) model originally proposed by Molinari et al. (1987) and later developed and implemented by Lebensohnand Tomé (1993) into a polycrystal plasticity code offers more realistic stress estimations for HCP structures than thosebased on Taylor’s and Sachs’s assumptions. Among these available models, VSPC is an appropriate polycrystal deformationmodel, especially for HCP metals and has provided insights into deformation mechanisms in HCP metals (McCabe et al.,2006; Proust et al., 2009).

In this paper, the VPSC approach for stress estimation and an extended Voce hardening model in a polycrystal plasticityframework are used to simulate the flow curves and texture evolution of an extruded AM30 alloy under simple compressionalong two perpendicular loading directions. While the c-axis fiber textures were the most common textures analyzed incommercialized HCP metals, the contribution of this paper is to reveal the deformation mechanisms correlated to the pris-matic-axis fiber texture generated by extrusion. Another important component correlated to extrusion is the role of grainmorphology upon the loading orientation and the inhomogeneity in the extruded Mg, which changes the nucleation andgrowth mechanisms of extension twinning and results in twinning multiplicity. The axisymmetric rod-texture, grain mor-phology in an extruded AM30 give new insights into the mechanisms behind twin–slip and twin–twin interactions and pro-vide an argument for the primary role of dislocation transmutations in the hardening associated with profuse twinning. Incontrast to the c-axis fiber texture, in a rod-texture no loading exists that completely annihilates extension twinning, andwhen twinning is most reduced, easy slip seems to induce a complex structure of ‘‘stopped’’ twins that fundamentally mod-ifies the hardening mechanisms and the nature of strain path anisotropy.

2. Experimental information

A commercial extruded AM30 magnesium alloy ‘‘billet’’ with 178 mm in diameter was used in this study for mechanicaltesting and materials characterization. The chemical composition of this AM30 is listed in Table 1.

Cylindrical compression specimens illustrated in Fig. 1 with 6 mm diameter and 6 mm length were cut by the electrical dis-charge machining (EDM) method from an as-received extruded billet along the extrusion direction (ED) and along the extrusion

1 The present authors do not support a substantial effect of a Hall-Petch effect that could be induced by the presence of twin boundaries. This point is beyondthe scope of the paper.

Table 1The chemical composition (in wt.%) of AM30 bar.

Al Mn Zn Fe Si Cu Ce Ni Mg

2.54 0.40 0.018 0.003 0.008 0.011 0.025 0.005 Bal.

Fig. 1. Schematic illustration of AM30 magnesium alloy compression specimens. ED stands for loading direction parallel to the extrusion direction whereERD stands for the loading direction parallel to the extrusion radial direction.

62 Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76

radial direction (ERD) in the middle area of the AM30 billet. Here, the extrusion radial direction is defined as parallel to a radialdirection of a billet, and the third direction is termed extrusion tangent direction, ETD. All of the compression tests were per-formed at room temperature and at a strain rate of 10�3 s�1. Interrupted tests at plastic strains of 3.5%, 5.8% and 8.4% for bothloading directions were performed to analyze the evolution of deformation texture and twins. All microtexture measurementsand analyses were conducted by employing an electron back scattered diffraction (EBSD) technique from a ZEISS SUPRA-40FEG–SEM equipped with a TSL OIM data acquisition and analysis software package. For the initial texture, EBSD scans were per-formed on both sections, normal to ED and ERD, in an effort to characterize the grain morphology (Fig. 3).

For better representation of initial and deformed textures necessary for polycrystal plasticity simulations, macrotexturemeasurements were performed using a Rigaku SmartLab X-Ray Diffractometer (XRD) based on a Shultz reflection methodusing Cu target, Ka radiation, 40 kV, 30 mA measurement parameters. Six pole figuresf10 �10g; f0002g; f10 �11g; f10 �12g; f11 �20g and f10 �13g were characterized, and background and defocusing effectswere eliminated. The orientation distribution functions (ODFs) were calculated using a WIMV algorithm in the popLA package(Kallend et al., 1991) based on the five possible pole figures. The complete pole figures were recalculated based on these ODFsand were plotted by the texture software MTEX (Hielscher and Schaeben, 2008). The initial texture of AM30 measured by neu-tron diffraction on HIPPO (High-Pressure-Preferred Orientation) at LANSCE (Los Alamos Neutron Science Center) is also pre-sented in Fig. 4b for comparison. For more details of the technique and facilities, see Von Dreele (1997) and Wenk et al. (2003).

2.1. Mechanical flow behavior

The quasi-static compression true stress–strain behavior and corresponding hardening rates at room temperature and ata strain rate of 10�3 s�1 along the ED and ERD are presented in Fig. 2. The true plastic stress–strain behavior along the EDexhibits a sigmoidal-shaped flow curve in which four stages corresponding to four hardening rates arises: (i) an initialdecreasing hardening rate up to approximately 0.5% plastic strain (�140 MPa), (ii) a very low concave increasing hardeningrate, indeed an almost plateau, from approximately 0.5% to �2.5% plastic strains termed Regime I, (iii) a very high hardeningrate between approximately 2.5% and 7.5% plastic strains termed Regime II, and (iv) a decreasing hardening rate fromapproximately 7.5% plastic strain to rupture termed Regime III. These four stages are better appreciated in hardening–stress(theta–sigma) curve, which corresponds to the evolution of hardening rate as a function of stress. Note that Regime II in thehardening–stress behavior has an increasing rate.

The stress–strain behavior obtained under simple compression along the ERD seems to show exclusive concavity typicalof slip predominance. However, the hardening rate curves reveal a peculiar regime of constant slope up to 4% plastic strain,termed as Regime IIR, which followed an early slip regime up to 1.7% plastic strain. This regime is followed by a regime ofparabolic decrease with a transition marked with load drops in the hardening rate. Ductility seems to be higher under theERD compared to the ED. The saturation stress is remarkably higher under ED than under ERD, although the last stages under

(a)

(b)

Fig. 2. Measured and predicted flow behavior for an extruded AM30 Magnesium alloy loaded in compression along the extrusion direction (ED) and alongextrusion radial (ERD) direction showing results for (a) the stress–strain behaviors and (b) the hardening rates versus stress garnered from the ED and ERDstress–strain behaviors.

Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76 63

both orientations are marked by pyramidal slip activities (Fig. 8). This is indicative of a peculiar hardening behavior broughtabout by twinning.

2.2. Initial microstructure and texture

The EBSD inverse pole figure maps of initial AM30 on sections normal to the ED and ERD are presented in Fig. 3. Fig. 4ashows the recalculated pole figures for the initial state based on XRD measurements. As shown in Fig. 4, the initial texturemeasured by neutron diffraction is consistent with that measured by XRD.

The inverse pole maps reveal dual grain microstructure in which large elongated grains coexist with small equiaxedgrains. The large elongated grains correspond to large portions of the parent grains that persisted from the pre-extrusionstate without undergoing dynamic recrystallization. The small grains are the recrystallized portions of the parent grains pref-erentially nucleated at the grain boundaries of the large elongated parent grains. We expect twinning multiplicity in thisAM30 under compression along two perpendicular directions. Meanwhile, we utilize the benefit of the VPSC modeling onthe extruded Mg alloys (Agnew et al., 2005b; Yi et al., 2006) and the microstructure evolution to investigate the effect ofvarious twinning on the plasticity of the extruded AM30 alloy.

As illustrated in Fig. 4, the initial texture was nearly a rod-texture with the h10 �10i||ED component. The rod-texture wasnot completely axisymmetric as there was a slightly stronger h0001i||ERD component. These results were consistent withthe EBSD based microtexture analyses showing a preferred orientation of the large non-recrystallized portions of the parentgrains.

Fig. 3. EBSD Inverse pole figure maps of an extruded AM30 Magnesium alloy bar on (a) a section normal to the extrusion direction (ED), and (b) a sectionnormal to the extrusion radial (ERD) revealing the large aspect ratio of the grains that were not recrystallized and were elongated along the extrusiondirection. Here, the texture was rotated so the extrusion direction texture axis is still normal to the plan of view. The EBSD scan step size = 1 lm. The IPFmap of (b) was rotated 90� around ETD (the horizontal direction) for consistency with (a) in the inverse pole figure.

(a)

(b)

Fig. 4. (a) Recalculated (0002) and ð10 �10Þ pole figures of an extruded AM30 magnesium alloy based on XRD and (b) neutron diffraction.

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3. Crystal plasticity simulations

An extended Voce-type hardening rule was used to describe the evolution of the threshold stress above which the s-slip(or twin) system was activated. The extended Voce hardening model is a function of the accumulated shear strain C withinthe grain over all the slip systems and twinning systems given by the following equation,

Fig. 5. Schematic illustration of the four deformation modes used in the main simulation to model plasticity in the AM30 magnesium alloy. The doublearrow indicates slip while the single arrow reflects polarity of twinning.

Fig. 6. Evolution of the critically resolved shear stresses with the accumulated shear C for the four deformation modes considered in the simulations.

Fig. 7. Misorientation profile within a parent grain before testing showing the substantial misorientation peaks around 1� typical of a high dislocationdensity that developed after extrusion. The misorientation peaks at approximately 5� could correspond to dislocation substructures.

Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76 65

s ¼ ss0 þ ðss

1 þ hs1CÞ 1� exp �C

hs0

ss1

��������

� �� �; ð1Þ

where ss0 and ss

1 are the initial and back-extrapolated critical resolved shear stresses of the active s-slip or twin system, hs0

and hs1 are the initial and asymptotic hardening rates, respectively. These four parameters are different for each s-slip defor-

(a)

(b)

Fig. 8. Simulated relative activity of deformation modes under simple compression along (a) the extrusion direction (ED) and (b) the extrusion radialdirection (ERD).

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mation mode and constitute parameters to be correlated to reproduce the stress–strain behavior for both loading orienta-tions. In addition, we allow for ‘‘self’’ and ‘‘latent’’ hardening to be operational by defining coupling coefficients hss0 , shownin Eq. (2), between two arbitrary deformation modes. The latent hardening between crystallographic slip systems and twinscan be justified by the obstacles that dislocations encounter at the twin boundaries, which could underlie a Hall–Petch phe-nomenon specific to twin interfaces as argued by Tomé et al. (1991). A predominant twin reorientation (PTR) scheme devel-oped by Tomé et al. (1991) was used to account for twinning reorientation in terms of the twin volume fraction evolution.The stress–strain behaviors could not be modeled without the twin–slip latent hardening. Eventually, the increment inthreshold stress due to shear activity in the grain from all deformation modes is given by:

Ds ¼ dsdC

Xhss0Dc0; ð2Þ

where Dc0 is the accumulated shear by deformation mode s0.For active deformation modes, we disregarded f10 �11gh10 �12i contraction twinning since their introduction did not pro-

vide a benefit for better model correlation. This stems from their high critical resolved shear stress. In addition to the basal-hai ðf0002gh2 �1�10iÞ and the pyramidal-hc + ai ðf11 �22gh11 �23iÞ slip systems, the prismatic-hai slip f10 �10gh2 �1 �10i modewas also selected. As such, only extension twinning and hc + ai slip could accommodate plastic strain along the c-axis inthe simulations. These deformation modes are illustrated in Fig. 5.

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4. Results and discussion

4.1. Simulation hardening parameters and effects of twinning

The extended Voce hardening parameters of the four deformation modes were adjusted to a best fit of the experimentalstress–strain curves, and the associated VPSC parameters for the simulation are listed in Table 2. A different simulation thatincluded the contraction twinning f10 �11gh10 �12i is not listed in Table 2. The correlated stress–strain curves were not re-ported here since they gave similar predictions for the ERD type loading. In fact, the value of the CRSS of the contraction twinnecessary for a good fit upon ED, was set too high to be reached upon loading in the ERD. EBSD results reveal that contractiontwinning appears in fact at a higher CRSS in the ED than in the ERD although in grains with similar Schmid factors.

This peculiar effect may stem from a dependence of the contraction twinning CRSS to prior dislocation types and struc-tures, which is different when comparing ERD to ED in this rod-textured AM30 billet. Since the Voce model formulation can-not capture this effect, we did not use the contraction twinning in the simulations. CRSS variation of the four adopteddeformation modes with respect to the accumulated strain C based on the best fitting parameters are presented in Fig. 6.

The CRSS of the second order pyramidal hc + ai slip reached about 700 MPa at the saturation stage, but the prismatic haislip and the basal hai slip exhibited little increase in their CRSS, and then the saturation stage was quickly reached. This couldprobably be explained by their high initial dislocation densities in the extruded billet before testing as shown by Fig. 7. Infact, Fig. 7 revealed within a single parent grain misorientation peaks around 5� typical of subgrains, and an important pop-ulation of misorientation peaks around 1� typical of dislocations. The retarded saturation of pyramidal slip could be due totheir lower dislocation density in the initial state compared to basal and prismatic. This lower density of pyramidal disloca-tion could be explained by the faster tendency to annihilate pyramidal dislocations at the die exit due to their relatively high-er stacking fault energy.

Note that the CRSS evolution curves presented in Fig. 6 did not reflect any latent hardening effects as expressed by Eq. (2).The initial CRSS of the basal hai, the prismatic hai, and the pyramidal hc + ai slip modes were found to be 28 MPa, 88 MPa, and90 MPa, respectively. These values are close to those obtained by Agnew et al. (2006) for an AZ31 magnesium alloy based onan elastoplastic self-consistent polycrystal model. However, the CRSS of extension twinning (=65 MPa) is higher compared tothose reported in previous results by Agnew et al. (2006) (=30 MPa) and by Brown et al. (2005) (25–35 MPa), for an AZ31alloy, respectively.

Nevertheless, the 65 MPa CRSS value of extension twinning falls reasonably well in the range of the CRSS values reportedby Gharghouri et al. (1999) (65–75 MPa) in a 7.7% Al containing magnesium alloy. The discrepancy between these resultsmight be due to the differences in grain size, grain shape, solid solution, and defect structures that can influence the twinPeierls stress (Christian and Mahajan, 1995).

It is of considerable importance to note that our simulations predicted remarkably well the hardening–stress (theta–sig-ma) curve in contrast to the simulations reported in the literature, where the hardening–stress curve is hardly shown (Proustet al., 2009). Although in Proust et al. (2009), the model correlates seemingly well the stress–strain behavior, the correlationto the theta-sigma is unrealistic. Moreover, the model and experimental hardening rates within Regime II in the theta–sigmacurve are of opposite signs. This underlies that the model in Proust et al. (2009) overlooked some fundamental aspects of thehardening mechanisms related to twinning.

The predicted evolution of the relative activity of slip and twinning evolving as a function of strain of all four active defor-mation modes is presented in Fig. 8.

4.1.1. Behavior in the extrusion direction (ED)Fig. 8 shows four subsequent regimes of deformation mode activities for compression along the ED direction. From Fig. 8a,

one can see that in the strain range below 0.5% plastic strain, basal slip is slightly predominant over extension twinning. Thisexplains the initial stage clearly seen in the theta–sigma curve where the hardening rate is rapidly decreasing. Between 0.5%and 2.5% plastic strain, extension twinning rapidly increased up to a maximum relative activity of approximately 45%. Thisstage corresponded to a deformation mostly accommodated by twinning before twin coalescence began (Jiang et al., 2007b;Kalidindi et al., 2003). The rapid onset of lengthwise thickening of the twins nearly led to a plateau characteristic of Regime I.Texture hardening effect by twinning from soft to hard orientations should exist in this case, but it was not sufficient to trig-ger a rapid hardening as in Regime II. In the plastic strain range between 2.5% and 7.5%, the activity of the extension twinningmarkedly decreased. This corresponds to a mixed stage of twin growth and coalescence where the twinned lamellae grew

Table 2Best fitting Voce hardening parameters for VPSC simulations.

Modes s0 (MPa) s1 (MPa) h0 MPa) h1 (MPa) hss0 (pri) hss0 (bas) hss0 (pyr) hss0 (etw)

pria 88 40 32 0 2 2 3 8bas 28 17 654 0 3 1 3 9pyr 90 701 7701 0 0.1 0.1 0.8 1etw 65 0 0 0 1 1 1 1

a pri – prismatic hai slip; bas – basal hai slip; pyr – pyramidal second hc + ai slip; etw – extension twinning.

68 Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76

larger than their relative aspect ratio and merged into each other. Therefore, at least the upper-half of Regime II should becharacterized by slip dominance within the twins. This can be verified by the dominance of slip over twinning, which startedwell before 4.5% plastic strain.

With slip predominated, Regime II exhibits a substantial and peculiar increase in the hardening rate. When the strain ex-ceeded approximately 7.5%, slip activity exhibits the usual exhibited decrease in the hardening rate. The transition wasmarked by a remarkable inflection into Regime III. A sudden rupture occurred at 12% plastic strain correlated to double twin-

Fig. 9. Inverse pole figure maps of an AM30 magnesium alloy compressed along ED (out of the plane of the paper) up to three plastic strain levels (a) 3.5%,(b) 5.8%, and (c) 12% showing an important amount of extension twinning (red indicating basal texture), which practically invaded the entire parent grains(blue indicating the initial prismatic texture). The micrograph in (d) is an image quality of the IPF map in (c) which exhibits better the double twinsf10 �11g-f10 �12g that profusely affect the extension twins. The EBSD scan step size was set equal to 0.2 lm. (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this article.)

Fig. 10. Texture analyses using EBSD of an ED sample deformed to 12% plastic strain revealing double twinning f1 0 �11g-f1 0 �12g (DT) (a) affecting theprimary f10 �1 2g extension twins (PTT). The colors used here correspond to the colors of the IPF map in Fig. 9c. The occurrence of this double twinning isconfirmed (b) by the common h1 �210i misorientation axis, and (c) by the peak misorientation that is approximately 38�.

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ning of the extension twins (Fig. 9d). The double twins under ED was identified by the typical misorientation of 38� h1 �210iin Mg (Jiang et al., 2007b) shown in Fig. 10.

EBSD analyses in Fig. 9 revealed that both parent and recrystallized grains exhibited segmentation by twinning. This grainrefinement seems to peak at 3.5% plastic strain. At higher strains, twin lamella quickly began to coalesce. At approximately6% plastic strain, the matrix was nearly entirely twinned as shown in Fig. 9b. The highest grain refinement by twinning atlower strains is inconsistent with a possible Hall–Petch effect on hardening in Regime II. Regime II expanded beyond 6% plas-tic strain, which was beyond twin coalescence regime. This phenomenon questions also the validity of the classical latenthardening approach (Tomé et al., 1991) of twinning on slip used in hardening models implemented within the VPSC code,not just the Voce hardening model.

4.1.2. Behavior in the extrusion radial direction (ERD)Fig. 8b indicates that the basal slip was largely responsible for accommodating strain under compression along the ERD

(up to 65% relative activity at 2% strain) until material rupture took place. However, the relative activity of extension twin-ning was substantial (up to 20% at 2% strain) and reached over half of that recorded along the ED under the same strain.

The IPF map in Fig. 11a reveals that the extension twins are actually mostly stopped twins. Fig. 11b and c show the ori-entation relationship of the primary ‘‘stopped’’ twins, the secondary extension twins and the twinning sequence in the ERDsample. Residual twins occurred rarely from edge to edge in ERD, while under ED we scarcely observed stopped twins. Thelenticular shape and a wedged tip totally embedded within the parent matrix are the main characteristics of a stopped twin.Stopped twinning occurs when the shape change by the twin is accommodated in the matrix before the first twinning slipdislocations reach the grain boundary, that is, during the lengthwise thickening stage. The twin accommodation is thenachieved by slip and not by any type of dislocation-free kinking. Elastic strain could not have accommodated the twin sheareither, since otherwise, the twin would have disappeared upon stress removal, which was not the case here.

The formation of stopped twinning in the ERD but not in the ED in grains although having a similar Schmid factor to thatunder ED suggests an effect of grain morphology. The stress state for a load acting normal to the elongated grain boundary inthe ERD was to induce easier/earlier nucleation of lattice slip dislocations. This would increase the extent of twin–slip inter-

ERD a

ERD

ED ED ED

b

DREDRE

c PST SET

Fig. 11. EBSD analyses on ERD sample deformed up to 5.8% plastic strain showing (a) an IPF map exhibiting a complex network of multivariant primary‘‘stopped’’ twins (PST) neighboring other grains that accommodated strain mostly by slip. The stopped twins induced in the primary extension twinssecondary extension twins (SET) at the intersection vertices of the intersecting twins. This twinning sequence is proven by (b) the common h1 �210i axis, thecommon ð10 �12Þ K1 plane, and (c) the characteristic misorientation (87�). The EBSD scan step size was set equal to 0.2 lm. The black lines in IPF map (a)correspond to boundaries with misorientation greater than 15�.

70 Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76

a

b

ED

ERD ERD

ED

c d

µ

Fig. 12. EBSD analyses on an ERD sample deformed up to 12% plastic strain showing (a) an IPF map exhibiting profuse f10 �11g-f10 �1 2g double twinning(DT). The double twinning sequence is identified by (b) the characteristic misorientation (38�) around (d) the common h1�210i axis. The rotation of the c-axisby double twinning is shown in the (0001) pole figure of (c). The EBSD scan step size was set equal to 0.2 lm.

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actions during the lengthwise thickening of a twin. The same effect could be promoted by slip dislocations that cross-slippedfrom neighboring grains deforming mostly by slip, which is a characteristic misorientation situation encountered in a rod-texture loaded in the ERD. Accordingly, stopped twin occurrence benefits from this inhomogeneous extruded Mg.

In either case, the twinning dislocations ahead of the lengthening twin would interact with the substantial lattice slipdislocations to cause a partial recombination of the twinning dislocations ahead of the tapering boundary. Recombinationof twinning dislocations was in fact observed at the end of stopped twins either at the edge of a single crystal or at the ter-minating grain boundary (Hull, 1961). Here dislocation interactions occur before the twin reaches the other side of the grainboundary.

This situation is similar to that of the emissary dislocation mechanism suggested by Sleeswyk (1962). Emissary disloca-tions, however, form as a homogeneous dissociation in every third twinning disconnection, so they are energetically unfa-vored. Furthermore, the dissociation suggested by Sleeswyk is appropriate for bcc structures where the twin compositionplane is an actual slip plane of the structure. That is, the emissary dislocations move away from the twin along K1 to the par-ent lattice. However, it is interesting to see that a recombination of twinning dislocations whether promoted or not by otherslip dislocations could result in a zero net Burgers vector of the twin boundary even if the parent structure is converted intothe twin structure. Activation of dislocation sources within the grain ahead of the lengthening tapering boundary of the twinis unlikely here, since the same effect would have occurred in the ED.

Another phenomenon from the IPF map of Fig. 11a is the abundant formation of multivariant twins not encountered inthe ED. The onset of multivariant twinning could have expanded to 6% plastic strain as observed in the load drops in hard-ening–stress curve of Fig. 2b. The most common multivariant twinning identified by EBSD corresponded to 7� misorientationrelationship between the two twin variants (Jiang et al., 2007b), and these twins were again mostly stopped twins. Themultivariant twinning in the ERD not exhibited in the ED although in grains with similar Schmid factor could also be ex-plained by the normal stresses acting in the elongated grains. In fact, as tacitly implied by the schematic of Reed-Hill andAbbaschian (1992) for necessary accommodating kinks, a high number of surface tilts would be geometrically needed toaccommodate strain in an elongated grain than in an equiaxed grain when the stress is acting normal to its main axis. This

Measured Predicted

(a) Measured, 3.5% strain (b) Predicted, 3.5% strain

(c) Measured, 5.8% strain (d) Predicted, 5.8% strain

(e) Measured, 8.4% strain (f) Predicted, 8.4% strain

Fig. 13. (a), (c), and (e) measured and (b), (d), and (f) predicted pole figures along the extrusion direction (ED) under compression at three strain levels(3.5%, 5.8%, and 8.4%).

72 Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76

mandates a higher number fraction of twins and as such multivariant twinning. Here again, multivariant twins are promotedalso by the heterogeneity in this AM30.

An important outcome from the network of intersecting twins is the twin–twin interactions. Further growth of theincoming twin impinging another twin requires accommodation of the incident shear to propagate in the obstacle twin. Gen-erally, it was observed that only a fraction of the incident shear could be transmitted to the obstacle twin so propagation ofthe incident twin is rendered difficult (Remy, 1981). An important conclusion by Mahajan and Chin (1973) is that the accom-modation of the incident shear by slip in the obstacle twin is much more unlikely to occur than secondary twinning. This isconsistent with the occurrence of secondary twinning highlighted in the IPF maps of Fig. 11; only impinging twins with sec-ondary twins at the intersection vertices could noticeably grow by edgewise thickening.

Therefore, the limited growth of twins in the ERD was due to a combined effect of twin–twin interactions and multivar-iant twinning leading to stopped twins which stem from the heterogeneity of the extruded AM30. These mechanisms arebeyond the reach of the twinning models in the VPSC using a Voce type law. As such, there was a systemic overestimationof twin activity in the ERD.

This overestimation of twinning activity explains the absence of Regime II in the experimental ERD stress–strain behavior,which underlies a non-Schmid effect of twinning. This could be rationalized by a dislocation transmutation effect on RegimeII put forward by El Kadiri and Oppedal (2010). The limited extent of twin growth would dramatically mitigate transmuta-tion of parent dislocations upon their incorporation in the twins, since the twin boundaries can only sweep a limited volumeof the parent. As such, a competition takes place between hardening by transmutation-induced latent hardening and soft-ening due to stoppage of twins. That is, the ERD stress–strain behavior is not purely parabolic but shows a tendency of adopt-

Measured Predicted

(a) Measured, 3.5% strain (b) Predicted, 3.5% strain

(c)Measured, 5.8% strain (d) Predicted, 5.8% strain

(e) Measured, 8.4% strain (f) Predicted, 8.4% strain

Fig. 14. (a), (c), and (e) measured and (b), (d), and (f) predicted pole figures along the extrusion radial direction (ERD) under compression at three strainlevels (3.5%, 5.8%, and 8.4%).

Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76 73

ing a linear trend with load drops due to the onset of multivariant twinning before a regime of slip predominance that couldtake place. The overestimation of twinning caused a substantial misprediction of the ERD hardening–stress curve. As such,although the stress–strain curve seems to be well predicted, none of the stages along ERD were satisfactorily predicted in therate form.

According to the Hall–Petch effect, the network of multivariant stopped twins should have greatly contributed to induce aRegime II hardening rate in the ERD with probably a similar magnitude as that in the ED. In fact, segmentation in the ERDwas finer since growth was limited when compared to that recorded in the ED, and the opposite would be expected if theHall–Petch effect were in operation. This would induce the twin boundaries to act as more effective barriers thus increasingthe ERD hardening rate. However, there was no type of Regime II in the ERD, and as such a Hall–Petch effect cannot operateby twin segmentation unless the twin boundaries lose coherency with the matrix.

The competition between extension twinning and their impediment by twin–twin interaction and stopped twinning isconsistent with the plateau observed in the ERD hardening–stress curve between 1.5% and 4% plastic strain. After this pla-teau, secondary load drops appeared in the hardening–stress curve. Fig. 12 reveals profuse f10 �11g-f10 �12g double twin-ning. The position of the f10 �11g-f10 �12g double twins in the basal pole figure shown in Fig. 12b and c is consistent withthe measured macrotexture result by XRD in Fig. 14a, c and e. It is important to note that double twinning occurred at a con-siderably lower stress in the ERD than in the ED, which suggests a non-Schmid effect correlated to double twinning. Thisobservation reveals the sensitivity of double twinning to the type and structure of prior dislocations. This is not accountedfor in the present formulation of the hardening law in the VPSC code.

If the double twinning affects the stopped twins, then Regime II hardening would be reduced according to the mechanismof preferential latent hardening in the twin by dislocation transmutation. In fact, the f10 �11g contraction twins, a componentof the double twins, are known to have significantly smaller growth rates compared to extension twins due to their relativelyhigh characteristic shear (Yoo, 1981). As such, the contraction twins could not have resulted in a high hardening rate basedon the dislocation transmutation effect of twinning (El Kadiri and Oppedal, 2010).

Fig. 15. Twin volume fraction progression as a function of strain for simple compression along the extrusion direction (ED) obtained by incrementalintegrations of recalculated pole figure intensities with an angle tolerance of 38�. The lower-limit and upper-limit twin fractions were calculated based onthe a angle tolerance of 29� and 48�, respectively.

74 Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76

Given all of the observations regarding the texture, twins, CRSS, and the grain morphology, the best modeling correlationsto the Regime II hardening could not be realized without resorting to high values of nontraditional latent hardening. Anyattempt to increase the pyramidal slip or latent hardening slip/slip leads systemically to an overestimation of the ERDstress–strain behavior. Otherwise, Regime II is substantially underestimated for the ED. It is of considerable importanceto note that the high values of latent hardening twin/slip were not only necessary to capture the high extent of Regime IIin the ED, but also to capture the higher saturation stress under the ED. The latent hardening of twin/slip does not reflecthere a Hall–Petch effect at the twin interfaces as it is the case for grain boundaries (Hall–Petch effect), rather it postulatedhere to reflect latent hardening in the twins increased by the multiplicity of dislocations which increased upon dislocationtransmutation by the twinning shear (El Kadiri and Oppedal, 2010). Incorporating the dislocation transmutation effect oftwinning, instead of the Hall–Petch effect of twin boundaries, into latent hardening of twin/slip in VPSC model will simulatemore accurate texture and stress–strain data of magnesium.

4.2. Texture evolution

The predicted and the measured {0002} and f10 �10g pole figures for strains equal to 3.5%, 5.8%, and 8.4% for simple com-pression in the ED and ERD are presented in Figs. 13 and 14, respectively.

For the ED, the predicted pole figures are in good agreement with the measured ones. The goodness of the predictions isnot limited to the location and position of the texture component but most importantly to the magnitude of intensity andintensity distribution. This point is of considerable importance, since a prediction discrepancy of texture intensity wouldhave a direct effect on the hardening rate as twinning reorients the matrix to hard orientations. A good correlation in thiscase would have been achieved only by multiple discrepancies that artificially annihilate each other. We emphasize here thatmost of the literature only compares final textures and not its evolution upon strain, which has led to discrepancies in fittingparameters. Following this fact, the reliability of our fitted parameters is substantiated by the good reproduction of the twinvolume fraction evolution as depicted in Fig. 15. Here, the extension twin volume fraction was measured by intensity inte-gration around the basal pole within approximately 40� tolerance for a angle (Bunge pole figure coordinate parameters (Bun-ge, 1982)), and 29�, 48� tolerance as the lower- and upper-bound twin volume fraction. There was, however, a slightoverestimation at the last stages of plastic deformation before rupture.

With regard to compression along the ERD, there were some discrepancies between the measured and predicted pole fig-ures shown in Fig. 14, although they had basically similar aspects of the strongest texture components. The predicted tex-tures overestimated the intensities along the ERD||<0001> component. This is due to the overestimation of the growth ofextension twinning as mentioned earlier. Furthermore, the measured textures show a double peak phenomenon aroundthe ERD||<0001> component not captured by the simulations. As shown in Fig. 12c, this double peak is due to the spreadcaused by the profuse f10 �11g-f10 �12g double twinning.

The VPSC simulations were not able to reproduce this texture even when contraction twinning and double twinning wereadmitted by allowing both extension and contraction twins to retwin. In fact, the model was not able to capture the differ-ence in the critical resolved shear stress of contraction twinning from ED to ERD without violating the appropriate values forthe parameters of other modes.

Q. Ma et al. / International Journal of Plasticity 29 (2012) 60–76 75

5. Conclusions

The detailed mechanisms relating the texture, twinning and dislocation formation, grain morphology, and stress–strainbehavior were examined for an AM30 magnesium alloy that experienced extrusion. One important point was that an AM30magnesium alloy with a rod-texture could not be compressed without a noticeable contribution of extension twinning.Along the extrusion direction, extension twinning was profuse, and the extended Voce hardening model with apparent latenthardening from twin–slip could predict fairly well the hardening–stress (theta-sigma) curve without contradicting the mod-el correlation with the twin volume fraction. Along the extrusion radial direction, the model substantially overestimates theactivity of extension twinning. The reason lies behind the formation of stopped twins and multivariant twinning. The forma-tion of stopped twins, only observed under the extrusion radial direction, was attributed to an interaction between earlytwinning dislocations and lattice slip dislocations enhanced by a grain morphology effect and cross slip from neighboringgrains deforming mostly by easy slip. This situation is similar to that of emissary slip proposed by Sleeswyk (1962). Onlytwins with secondary twinning at the intersection vertices grew in a noticeable fashion. Multivariant twinning dependedupon the specific types of accommodation effects needed at the grain boundary when an elongated grain was loaded normalto its axis.

Both stopped twinning and multivariant twinning led to a limited twin growth primarily through twin–twin interactions.These phenomena seem to be consistent with latent hardening by dislocation transmutation and incompatible with a Hall–Petch effect by twin segmentation, when explaining the absence of Regime II hardening under the extrusion radial direction.

The f10 �11gh10 �12i double twinning was profuse under both the extrusion direction and the extrusion radial direction.However, the critical resolved shear stress of the contraction twins was unexpectedly smaller under the extrusion radialdirection than under the extrusion direction. This critical resolved shear stress difference between longitudinal and trans-verse directions points to a dislocation type and grain morphology effect that cannot be captured through the Voce typehardening formulation in the VPSC code used in this paper.

All of these phenomena are important for performance extrusion textures since real-world conditions would correspondto intermediate loading between the extrusion direction and the extrusion radial direction.

The simulated crystal plasticity texture results showed good agreement with measured textures for the extrusion direc-tion and an important discrepancy from measured textures along the extrusion radial direction. This discrepancy is attrib-uted mainly to a double-peak spread by profuse double twinning.

Acknowledgements

The authors are grateful to the financial support from the Center for Advanced Vehicular Systems (CAVS) at MississippiState University. The authors are also grateful to Stephen Horstemeyer and Arsalan Adil for their work in mechanical testing.This study was also supported by the Department of Energy and the National Energy Technology Laboratory under AwardNumber No. DE-FC26-02OR22910, and the US Army TACOM Life Cycle Command under Contract No. W56HZV-08-C-0236through a subcontract with Mississippi State University, and was performed for the Simulation Based Reliability and Safety(SimBRS) research program. The authors acknowledge Alan Luo (General Motors Company), and Joy Hines Forsmark andJohn Allison (Ford Motor Company) for their leadership and encouragement of the larger USAMP/DOE Integrated Computa-tional Materials Engineering for Magnesium Program. This report was prepared as an account of work sponsored by anagency of the United States Government. Neither the United States Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, com-pleteness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would notinfringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name,trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favor-ing by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not nec-essarily state or reflect those of the United States Government or any agency thereof. Such support does not constitute anendorsement by the Department of Energy of the work or the views expressed herein. UNCLASSIFIED: Dist A. Approvedfor public release.

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