Internal strain and texture evolution during deformation twinning in magnesium

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

Internal strain and texture evolution during deformation twinning in magnesium D.W. Brown a,∗ , S.R. Agnew b , M.A.M. Bourke a , T.M. Holden c , S.C. Vogel a , C.N. Tom´e a b

a MS-H805, BLDG 622, TA-53, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 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 been monitored in situ with neutron diffraction. The continuous conversion of the parent to daughter microstructure is tracked through the variation of diffraction peak intensities corresponding to each. Approximately 80% of the parent microstructure twins by 8% compression. Elastic lattice strain measurements indicate that the stress in the newly formed twins (daughters) is relaxed relative to the stress field in the surrounding matrix. However, since the daughters are in a plastically “hard” deformation orientation, they quickly accumulate elastic strain as surrounding grains deform plastically. Polycrystal modeling of the deformation process provides insight about the crystallographic deformation mechanism involved. © 2005 Published by Elsevier B.V. Keywords: Deformation twin; Neutron diffraction; Internal strain; Magnesium; hcp

1. Background 1.1. Deformation mechanisms of magnesium and its alloys It has long been understood that the limited room temperature formability (or generalized ductility) of magnesium and its alloys is connected with the large plastic anisotropy at the single crystal or grain-level [1]. For instance, it is often cited that magnesium possesses only two independent slip systems associated with the primary deformation mode, basal slip of 1/31 1 2¯ 0 or a type dislocations. Furthermore, it is recognized the incorporation of non-basal slip of a dislocations on prismatic {1 0 1¯ 0} or even first-order pyramidal {1 0 1¯ 1} planes offers only two more independent slip modes. Kocks and Westlake [2] first pointed out that metals can not be simply divided into two groups: ductile metals that possess five independent easy slip modes (i.e. satisfy the ∗

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0921-5093/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.msea.2005.02.016

Taylor criterion) and brittle metals that do not. They noted that there is a range of behaviors spanned by those metals that clearly do possess five independent easy slip modes (i.e. ductile fcc and bcc metals), those that possess nearly five (e.g. hcp magnesium) and those that clearly posses fewer (e.g. hcp beryllium and simple cubic NiAl). Additionally, they noted that the presence of mechanical twinning modes further modifies the situation by essentially offering other independent modes of deformation, albeit in a limited sense due to their unidirectional nature and their limited strain accommodating ability (both points to be discussed in more detail later). For magnesium alloys, which commonly exhibit mechanical twinning on {1 0 1¯ 2} planes, an additional half of an independent slip mode was proposed. Thus, it was suggested that an hcp magnesium–lithium solid solution alloy reported to slip extensively on prismatic as well as basal planes [3] should be characterized as having 4 21 independent slip modes, thus explaining the alloy’s significant ductility [2]. Exactly how strong the plastic anisotropy of magnesium crystals is and what deformation mechanisms are involved

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has been an evolving discussion as various experimental and modeling techniques have been brought to bear on the problem. In particular, the distinction between single crystal behavior and the behavior of grains within a polycrystalline aggregate has become a point of consideration. Using transmission electron microscopy it has been demonstrated that magnesium and its alloys do possess five independent slip modes, by the observation of second-order pyramidal {1 1 2¯ 2} slip of c + a dislocations during compression of c-axis aligned single crystals [4,5] and within polycrystals, particularly in connection with mechanical twins. A more quantitative way of describing the situation is to employ the concept of single crystal yield surface (SCYS), whereby the deformation mechanisms are characterized by their activation stresses, rather than as binary quantities (active or not). Polycrystal models [6–8] offer a means of connecting the SCYS with the overall anisotropy and texture evolution within a polycrystal during deformation. Thus, if the SCYS is known, polycrystal behavior may be predicted. Conversely, if the polycrystal behavior is known, the SCYS may be inferred. Using a combined polycrystal modeling and experimental approach, Agnew et al. [9,10] were able to correlate the ductility enhancement of lithium alloying additions, with an increase of non-basal 1/3 {1 1 2¯ 3} or c + a slip, in addition to the previously proposed enhancement of prismatic slip of a dislocations [3]. The proposed enhancement of c + a slip in Mg–Li alloys is reinforced by recent single crystal and TEM studies of Ando and Tonda [11]. Finally, Agnew and Duygulu [12] have shown that enhanced c + a slip can explain the vastly enhanced ductility of magnesium alloys at slightly elevated temperatures. The importance of secondary deformation mechanisms such as c + a slip and deformation twinning is obvious as they provide grains with a means of straining along their c-axis, which the more prevalent slip modes (e.g. basal slip of a dislocations) fail to do. In addition to offering a unique opportunity to better understand the fundamentals of plastic accommodation as it relates to ductility (or lack thereof), wrought magnesium polycrystals manifest a wide range of anisotropy effects, from strong in-plane anisotropy in the sheet form [13] to a strong tension–compression asymmetry in both the extruded and rolled conditions [1]. The latter aspect is of interest in the current study because the root of the asymmetry is found in the combination of crystallographic texture and twinning. Unlike dislocation slip, twinning is a unidirectional deformation mechanism. Twinning modes in hcp metals are distinguished by their ability to produce either tensile or compressive strain along the crystallographic c-axis, but not both. Depending upon the c/a axial ratio, hcp twinning modes may be either tensile or compressive [14]. In the case of magnesium, the dominant twinning mode√{1 0 1¯ 2}1 0 1¯ 1 is a tensile twin since c/a = 1.624 < 3. In wrought magnesium, the typical deformation textures are such that the tensile twinning mode is activated by compression along the prior working direction, e.g. parallel to the extrusion

axis, and inactive during tension along the same axis. The dominant dislocation slip systems, basal a, are unfavorably oriented in both conditions. Thus, the unidirectional nature of deformation twinning results in observed high tensile and low compressive strength in worked magnesium alloys [9,10]. The current study was undertaken to further investigate these issues; in particular, to investigate the homogeneity of plasticity by directly measuring the intergranular strains and texture which develop during deformation. The measurements provide, for the first time, information regarding what is 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 strain measurements to probe the grain-level plastic anisotropy of polycrystals is a relatively new technique, with the earliest published examples attributable to MacEwen et al. [15]. In situ examinations are even more recent [16]. Magnesium is especially suitable for a basic scientific study of internal strain and texture development driven by plastic anisotropy. Notably, it is unique among hexagonal metals in that the single crystal elastic constants and coefficients of thermal expansion are nearly isotropic [17,18], while the plasticity is distinctly anisotropic. Therefore, type II (intergranular) strains have their origin in plastic anisotropy only. The use of textured polycrystal samples allows preferential probing of specific deformation mechanisms by using a variety of loading conditions. For this study, in situ compression and tensile tests have been performed on samples cut from a hot-rolled magnesium alloy AZ13B plate. Compressive loads were applied to the in-plane and through-thickness directions and tensile loads to the in-plane direction. This paper focuses on the in-plane compression (IPC) sample in which deformation twinning plays a major role. A complementary paper [10] deals in depth with the deformation of the through-thickness compression (TTC) and in-plane tension (IPT) samples, which deform almost exclusively by slip. Agnew et al. [10] used an elasto-plastic self-consistent (EPSC) model to describe the development of internal strains and the ratio of transverse plastic strains during deformation of the through-thickness compression and in-plane tension samples. The researchers concluded that, in addition to the basal and prism activity, a contribution from pyramidal slip and tensile twins must be considered to reproduces the experimental observations. Evidence is shown in this report that further supports this argument.

2. Experimental techniques 2.1. Neutron diffraction techniques Neutron diffraction measurements of internal strain and bulk texture were performed on the neutron powder

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sample through 14 different orientations for diffraction measurements, chosen to provide roughly uniform coverage of the pole figure. Rietveld analysis of the 56 diffraction patterns using eighth order spherical harmonics to describe the texture [21] was completed with the General Structure Analysis System (GSAS) software developed at LANSCE. 2.2. Sample preparation

Fig. 1. Schematic of the NPD.

diffractometer (NPD) and high intensity powder diffractometer (HIPD), respectively, at the Manuel Lujan Jr. Neutron Scattering Center, LANSCE, Los Alamos National Laboratory. Details of the diffraction instruments are similar and published elsewhere [19] and only a short description is presented here. In each instrument, the incident neutron beam impinges on the sample and is scattered onto a set of four detectors banks situated at ±90◦ and ±148◦ in the case of NPD (±153◦ , HIPD) relative to the incident beam. The timeof-flight technique utilizes the continuous energy spectrum and pulsed structure of the neutron beam to collect an entire ˚ in diffraction pattern (effective d-space range from 0.3 to 4 A) each detector panel simultaneously. The primary differences in the diffraction instrument are the lengths of the neutron flight paths, which define the resolution of the instrument and the ancillary equipment available to each instrument. The NPD has a flight path of 30 m yielding an instrumental resolution of d/d = 2.5 × 10−3 , making it suitable to measure the small shifts in lattice spacing associated with internal strains. A purpose built horizontal load frame was utilized to perform in situ measurements of the lattice strains during uniaxial compression and tension. Fig. 1 shows a schematic of the NPD indicating the sample geometry (compression) and orientation of the applied stress relative to the instrument. The load axis was oriented at 45◦ relative to the incident beam. Detectors on either side of the specimen simultaneously record data with diffraction vectors, parallel Q|| (−90◦ ) and transverse Q⊥ (+90◦ ), to the applied load [19,20]. Both detector banks have acceptance angles of ±5.5◦ in the vertical and horizontal directions. The circle in the center of the pole figures shown in Fig. 2 (and in subsequent pole figures) represents the coverage of the Q|| detector bank (−90◦ ) of the NPD relative to the pole figure of the sample. A similar sized circle located at the north and south poles would represent the coverage of the Q⊥ detector bank (+90◦ ). The high angle banks were not utilized for this measurement. The macroscopic strain was determined concurrently using an extensometer that spanned the irradiated region. In contrast, the HIPD has a 9 m flight path, which maximizes the incident neutron intensity for measurements of bulk texture. A computer-controlled goniometer rotated each

Hot-rolled magnesium alloy AZ31B plate was selected for the investigation because it represents the highest production wrought magnesium alloy and it has a well defined but simple crystallographic texture, shown in Fig. 2a. The preferred orientation of grains within the plate has basal poles parallel to the plate normal direction and exhibits a near random orientation of prism poles about that axis. Commercial magnesium alloy AZ31B has a nominal composition of 3 wt.% Al and 1 wt.% Zn, with restrictions on the transition metal impurities Fe, Ni, and Cu in order to improve the corrosion resistance 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) after 14% in-plane compression, (c) calculated texture after 14% in-plane compression (note that scale on calculated pole figures is roughly 2× larger).

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common for magnesium alloys revealed a grain size of approximately 50 ␮m [22]. The compression samples (10 mm in diameter and 22 mm in height) used in the study were electro-discharge machined from near the midplane of the plate. 2.3. Experimental programme The sample was initially oriented within the load frame so that the basal poles of the main texture component were in the diffraction plane (horizontal), but transverse to the loading direction. Fig. 3 shows a schematic of typical grains within the polycrystal relative to the load axis in the neutron diffraction instrument (NPD). The orientation of a typical grain prior to twinning, shown schematically in the left hand grain in the figure, results in strong (0 0 0 2) peak intensity in the transverse detector, Q⊥ and absence of (0 0 0 2) intensity in the parallel detector Q|| . The prism poles, {1 0 1¯ 0}, {1 1 2¯ 0}, {2 1 3¯ 0}, are initially distributed in a vertical plane containing the load axis and diffract solely into the parallel detector. Throughout this paper this will be termed the parent microstructure or parent grain orientation. During in-plane compression, the stress axis is initially perpendicular to the basal poles, making basal slip difficult to activate, but facilitating tensile twinning on the ¯ system. With a c/a ratio of 1.624, the twin {1 0 1¯ 2} 1 0 1¯ 1 plane normal is 43.3◦ from the basal pole. During a twinning event, the symmetry axis of the crystal structure rotates 180◦ about the twin plane normal [23,24] resulting in a 86.6◦ reorientation of the basal pole from perpendicular to the stress axis to nearly parallel. Given the sample orientation, the nearly 90◦ reorientation of the basal pole during twinning results in a transfer of (0 0 0 2) diffraction intensity from the Q⊥ to the Q|| detector, as shown schematically in the right hand grain in Fig. 3. Of course, the prism plane diffraction intensity concomitantly transfers to the Q⊥ detector. The “newborn” twinned orientations will be referred to as the daughter

Fig. 3. Schematic of magnesium grains in-plane compression sample relative to the diffractometer. The orientations of the grains represent the bulk texture of the IPC sample. The left and right grains represent before and after a twinning event, respectively.

microstructure or daughter grain orientation. The transfer of intensity between the two detector banks is a direct measure of the twinned volume fraction and allows us to monitor the development of the twinned microstructure. By taking advantage of the initial texture and the characteristic twin reorientation, this experimental arrangement provides a unique opportunity to simultaneously monitor the development of the internal strains in both the parent and daughter grains. Since (0 0 0 2) diffraction intensity is initially absent from the Q|| detector, the position in d-space of the initial (0 0 0 2) diffracted intensity in this detector provides unobstructed information about the initial strain state of daughter grains as they appear. Likewise, the prism pole intensity is initially absent from the Q⊥ detector, and the appearance of {1 0 1¯ 0} and {1 1 2¯ 0} intensity provides information about the initial strain state of the daughter grains perpendicular to the applied stress. In contrast, the strain state of the parent grains as they are consumed may be gleaned from the d-space of diminishing (0 0 0 2) peaks in the Q⊥ detector and {1 0 1¯ 0} and {1 1 2¯ 0} peaks in the Q|| detector. As far as the authors are aware, this represents the first study of the stress state of twins as they form during in situ loading.

3. Self-consistent modeling In the past, visco-plastic self-consistent (VPSC) [8] and elasto-plastic self-consistent (EPSC) [25,26] polycrystal models have been used to simulate the mechanical response of Mg alloys [9,10]. In these models each grain is treated as (visco-plastic or elasto-plastic) ellipsoidal inclusion embedded in (and interacting with) the homogeneous effective medium that represents the aggregate. The properties of the grain and of the medium are anisotropic, the crystallographic orientation of the grain with respect to the medium is accounted for. Stress and strain in each grain are a consequence of the grain-medium interaction, and differ from grain to grain. The VPSC model allows to tackle large strains and to account for hardening, texture evolution, and twin reorientation. However, the VPSC model has the disadvantage of not accounting for elasticity and, as a consequence, it does not address the initial elastic–plastic transition, or the evolution of internal strain. The EPSC model, on the other hand, does account for elasticity but does not currently incorporate the crystal rotation associated with slip, much less the large-scale reorientation associated with twinning. As a consequence, it is not applicable for simulating large deformation or texture evolution due to twinning. Agnew et al. [9] applied the VPSC model to predict the in-plane and through-thickness mechanical response of the same Mg plate examined in this study to compressive strains up to 50% strain. The EPSC model was employed to describe the internal strains developed during deformation to 10% of in-plane tension and through-thickness compression samples [10], which do not exhibit significant texture development.

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As discussed above, neither the EPSC nor the VPSC models are particularly well suited to this study, since the sample undergoes large textural reorientations and the region of interest is relatively small strains, roughly 10%. However, in the current study the VPSC model is used to lend understanding regarding the contribution of twinning to texture evolution, and to obtain information about the activity of other deformation modes. A detailed description of the VPSC formalism as it applies to the present calculations can be found in Ref. [9]. The VPSC simulations utilized in this paper have maintained the critical resolved shear stresses and hardening parameters used in Ref. [9] to aid in interpretation of the experimental data. In what follows, we compare experimental and predicted stress/strain curves and textures, associated with in-plane compression. At present, it is observed that twinning evolves faster in the simulation and, as a consequence, so does the texture and the hardening. From a qualitative point of view, however, the conclusions concerning system activity appear to be valid since the final deformation texture is correctly predicted. For a description of the polycrystal model and predominant twin reorientation scheme used in the simulations, the reader is referred to Ref. [27]. 4. Results 4.1. Macroscopic mechanical behavior The mechanical response of this material was described in our earlier publications [9,10] and only an overview will be given here. The measured and calculated in-plane stress/strain curves are shown in Fig. 4. The flow curve of a through-thickness sample is also shown for comparison. The difference between the flow curves of the two sample orientations is due to the difference in loading direction relative to the basal fiber. The load on the through-thickness sample is parallel to the basal fiber, whereas the load on the in-plane sample is normal to the basal fiber. The stress/strain curve of the through-thickness sample is typical for a ductile

Fig. 4. Solid lines represent measured stress/strain curves for throughthickness (TTC) and in-plane (IPC) compression of AZ31B magnesium alloy. Dashed line represents calculated stress/strain curves for IPC. Solid circles represent points during IPC at which neutron diffraction measurements were taken.

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metal; linear elasticity followed by a smooth elastic–plastic transition. In contrast, the flow curve of the in-plane sample displays abrupt yielding at 60 MPa followed by a low hardening rate plateau that is frequently associated with deformation twinning [28] or a stress induced phase transformation [29]. After a plastic strain of roughly 5%, an inflection is apparent in the flow curve as the work-hardening begins to increase with increased strain. Finally, beyond 8% deformation the flow stress of the in-plane sample has matched or surpassed that of the through-thickness sample and the work-hardening begins to decrease for a second time. 4.2. Ex situ measurement and modeling of texture development The experimentally determined and calculated (0 0 0 2) and {1 0 1¯ 0} pole figures of the in-plane compressed samples are shown in Fig. 1b and c, respectively, after −14% deformation. For ease of discussion, we have superposed a polar axis on the first pole figure with the azimuthal angle, Θ, zero at the center of the pole figure, and the polar angle, Ψ , zero to the right and increasing counter-clockwise [30]. Again, the circle in the center of the pole figures represents the coverage of the Q|| detector of the NPD relative to the pole figures. Although not shown, the complete orientation distribution functions (ODF’s) were also calculated using spherical harmonic and WIMV methods [31]. The volume fraction of the sample that has re-oriented during deformation was then calculated from the integrated difference in the initial and final orientation distribution functions [30]. Initially, the samples possess a fiber texture with the basal poles predominately parallel to the plate’s rolling normal direction. After the deformation, the sample showed a strong textural reorientation that can be associated with the activity of the {1 0 1¯ 2} tensile twinning model. The post-deformation basal fiber is aligned with the load direction, and is slightly elliptical with the major axis at Ψ = 90◦ (along the rolling normal direction). Assuming that large-scale reorientation is associated solely with twinning, the twinned volume fraction in the in-plane compressed sample is estimated to be 0.80 after 14% deformation of the sample. This estimate of the twinned volume fraction ignores reorientation due to other mechanisms such as slip, which are likely small corrections at such small macroscopic deformations. The VPSC calculation reproduced the deformation texture satisfactorily. In general, the features are sharper in the calculated pole figure than observed. Also, the basal fiber in the calculated pole figure is split into lobes in the Ψ = 0–180◦ direction, which is not observed in the measured pole figure. Activity of the pyramidal slip system causes rotation of the basal poles away from the load direction [9]. Over-activity of the pyramidal slip system in the calculation results in the spurious splitting of the basal fiber. The calculation predicts the twin volume fraction reaches a maximum of 0.85 by 5% at which point twinning saturates and it subsequently remains constant. The eccentricity of the measured basal pole figure

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

is reproduced by the model and is linked with the remainder of the parent microstructure at Ψ = 90 and 270◦ and θ = 45◦ which did not twin. 4.3. In situ measurement of texture development The evolution of the diffraction patterns during in situ deformation of the in-plane compression sample is shown in Fig. 5. Corresponding to the pole figures shown in Fig. 2 there is initially no diffracted intensity from the basal planes, (0 0 0 2), in the Q|| detector bank while the diffraction peaks from prism planes, {1 0 1¯ 0} and {1 1 2¯ 0}, are strong. The situation is reversed in the Q⊥ detector. Because the NPD detectors have an acceptance of ±5.5◦ , {1 0 1¯ 2} twinning manifests as an exchange of (0 0 0 2) intensity from the Q⊥ to the Q|| detector (and vice versa for the prism planes) during deformation. Therefore, the diffracted intensity of the (0 0 0 2) in the Q|| detector bank may be used to monitor the development of the twinned (daughter) microstructure during deformation. Figs. 6a and b show the integrated intensities (normalized to the background counts) of several grain orientations in the Q|| detector as a function of applied stress. In order to put all of the intensities on one scale, the peaks that disappear

Fig. 6. Normalized intensities of several diffraction peaks {h k i l} in the parallel detector.

(appear) have been normalized to 100 at their initial (final) point. Uncertainties in the diffracted intensities have been intentionally omitted for clarity, but in general, they are roughly 10%. The figure includes three grain orientations within the basal plane (Fig. 6a), as well as five grain orientations that are in the plane containing the {1 0 1¯ 0} and (0 0 0 2) poles (Fig. 6b), also called the plane of shear of the {1 0 1¯ 2} twin system [24]. The {1 0 1¯ 0} normal is in the plane of shear and in the basal plane. The {2 1 3¯ 0} and {1 1 2¯ 0} plane normal are also in the basal plane but are 19.1 and 30◦ from the plane of shear, respectively. The {1 0 1¯ 1}, {1 0 1¯ 2}, {1 0 1¯ 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 changes in peaks that are initially weak or completely absent, e.g. (0 0 0 2) in the parallel detector bank. However, both the increase in some peak intensities and the decrease in others are indicative of twinning activity. The (0 0 0 2), {1 0 1¯ 2}, and {1 0 1¯ 3} peaks exhibit a marked increases in intensity between −50 and −70 MPa signaling the onset of twinning. The (0 0 0 2) intensity increases more rapidly and reaches a maximum that is not observed in the {1 0 1¯ 2} or {1 0 1¯ 3} curves. The initial intensity changes are less obvious in the peaks that start strong, such as the {1 0 1¯ 0}, {2 1 3¯ 0}, {1 1 2¯ 0} and {1 0 1¯ 1} peaks, and it is difficult to discern the initiation of twinning from them. However, it is clear that the {1 0 1¯ 0} diminishes most rapidly, followed in order by the {2 1 3¯ 0} and {1 1 2¯ 0}. The {1 0 1¯ 1} intensity shows an initial increase in intensity that is outside of errors bars, the source of which is unknown, and subsequently diminishes at larger stresses than the three prism planes. Recall that the (0 0 0 2) and {1 0 1¯ 0} have special significance in this measurement because they are initially absent from the parallel and transverse detectors, respectively. As such they provide clean information pertaining to the development of the daughter and parent microstructures. The transfer of diffracted intensity in the parallel detector from parent to daughter grains is shown quantitatively as a function of imposed plastic strain in Fig. 7. Diffracted intensity

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

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

from the twins begins to appear at strains of about 1%, concurrent with the initial decrease in slope of the macroscopic stress/strain curve (see Fig. 4), and increases linearly with plastic strain to roughly 6% deformation. Twinning appears to saturate after 8% plastic deformation. The total twinned volume fraction after 14% imposed strain determined from the 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, the twinned volume fraction is approximately proportional to the intensity of the (0 0 0 2) diffraction peak in the Q|| detector. The rate of increase of the twinned volume fraction can then be correlated with the slope of the (0 0 0 2) intensity curve and is 0.145 per unit plastic strain. The figure also shows the (0 0 0 2) intensity as a function of plastic strain calculated by the model. Clearly, twinning occurs much too rapidly in the model, saturating at 4% strain. Subsequently, the calculation predicts a maximum and sharp decrease in the (0 0 0 2) peak intensity that may be associated with the onset of pyramidal slip activity in daughter grains in the model. The rotation of the basal poles away from the load direction that results in the split lobes shown in the calculated pole figure (Fig. 2) removes the poles from the coverage of the NPD detector designated schematically by the circle in the center of the pole figures. The observed (0 0 0 2) intensity displays a similar, albeit considerably weaker, maximum and ensuing decrease. Concurrent with the saturation of twinning, there is an increased hardening rate observed in the macroscopic flow curve, which through the link with the polycrystalline model, may be associated with the onset of pyramidal slip in daughter grains. 4.4. In situ measurement of internal strain development Lattice strains for the grain orientations which contribute to a given (hkl) diffraction peak are calculated from the measured interplanar spacing using Eq. (1) ε=

d hkl − d0hkl , d0hkl

(1)

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where d0 represents the unstrained interatomic spacing. The initial interplanar spacing is good approximations of the unstrained spacings (d0hkl ) because the material is fully annealed and has a nearly isotropic coefficient of thermal expansion [18]. The uncertainty in the strain measurements represents the combination (error propagation) of the statistical uncertainty in the peak fitting, δdhkl , based upon the estimated standard deviations (esd’s) of the peak positions output from GSAS, as well as the uncertainty of the calculated unstrained lattice parameters, δd0hkl , 
δd0hkl δd hkl (2) + hkl δε = ε d hkl d0 Again due to the initial texture, the diffracted intensity of the (0 0 0 2) peak in the Q|| detector is initially absent. Based on the arguments presented above, this peak is, therefore, uniquely poised to monitor the volume fraction of twins and the strains within the twins as they form. The only problem with such an approach is the lack of an unstrained lattice parameter measurement to use in the above strain calculations. Using the unstrained lattice parameter from the other detector bank circumvented this problem. Due to the possibility that sample position results in different Bragg peak positions within the two detector banks, an estimate of the error incurred was made. Small displacements of the sample relative to the center of the instrument cause a slight discrepancy in the peak positions and, therefore, the determined lattice parameters between the Q|| and Q⊥ detectors. This effect could skew the strain results if not taken into account. The average ratio of the d-spaces between the Q|| and Q⊥ detectors was found for six diffraction peaks (few peaks were strong in both banks) to be 1.0011 with a standard deviation of 0.00012 (120 ␮␧) and the unstrained lattice parameters were corrected for this shift. To check for repeatability, this process was done on other “standard” texture free samples where roughly 20 independent peaks were available in each bank and was shown to be repeatable. The internal strains in the parent grains parallel and perpendicular to the load axis are shown as a function of applied load in Fig. 8a. It is important to note explicitly that parents grains are aligned such that the {1 0 1¯ 0} poles are parallel to the stress axis and diffract into the Q|| detector, while the (0 0 0 2) poles are perpendicular to the load axis and diffract into the Q⊥ detector. The dashed lines on the figure represent the theoretical linear elastic response based on the single crystal moduli [17]. The parents exhibit linear elastic behavior until 70 MPa. Beyond the elastic limit, the evolution of elastic strain (i.e. stress) in the parents saturates. At higher aggregate stresses, other grain orientations, including the twins themselves, are shown to sustain higher than average stresses. This sort of composite load sharing behavior is observed in all types of polycrystals [16]. The internal strains in the daughter grains are also shown as a function of applied load in Fig. 8b. Once again, dashed

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

lines represent the calculated elastic response. The lattice strains within the twins, whose basal poles are parallel to the applied stress field, are first measurable at a macroscopic load of −70 MPa, after a small amount of twin reorientation has taken place. The elastic strain in the first daughter grains observed is roughly −700 ␮␧, considerably less than then the parent and surrounding grains. As the macroscopic load continues to increase from −70 to −220 MPa, the slope of the lattice stress/strain curve, 32 GPa, is less than the calculated elastic response of grains with basal poles parallel to the applied load, 48 GPa. Above 220 MPa (macroscopic strain of 0.08), an inflection is observed in the daughter applied stress versus elastic strain curve and the slope increase to greater than the linear elastic response. The initial transverse strains in the daughter grains are closer to the calculated linear elastic response. The uncertainty is larger because the counting statistics are not as good for the {1 0 1¯ 0} peak as the (0 0 0 2) peak. Upon further loading, and subsequent unloading, the response of the daughter grains perpendicular to the applied load closely follows that calculated from the elastic modulus and Poisson’s ratio.

4.5. Peak broadening In addition to changes in peak position, the data also reveal that there are changes in peak width. The ideal instrumental resolution of the NPD, FWHM, is d/d = 2.5 × 10−3 . However, in the present configuration, with a long horizontal sample, the instrumental resolution is likely to be degraded, but has not been explicitly measured. If we assume that the peak shapes are well described by Gaussian functions, the sample broadening is determined by subtracting instrumental broadening from the measured peak width in quadrature. In the current measurements, the (0 0 0 2) diffraction peak from the twins broadens considerably from its inception at −70 MPa until twinning completes at −220 MPa. Fig. 9 shows the broadening of the parent {1 0 1¯ 0} and daughter (0 0 0 2) reflections parallel to the load as a function of applied stress; the area in which twinning is active is shaded. ¯ peak is near In the elastic regime, the FWHM of the {1 0 10} the instrumental resolution and constant to within uncertain¯ ties. With the activation of deformation twinning, the {1 0 10} peak begins to broaden significantly and continues to do so as the parents are consumed by the twins. When the daughter grains first appear, the (0 0 0 2) peak width is similar to ¯ the {1 0 10}, but immediately increase with subsequent deformation. The broadening saturates as twinning completes and remains constant beyond −220 MPa or 8% deformation as well as on unloading.

5. Discussion

Fig. 9. Broadening of parent and daughter diffraction peaks as a function of applied stress. The instrumental resolution has been approximately removed. Closed (open) symbols represent data taken on loading (unloading).

For the purposes of discussion, the macroscopic flow curve of the in-plane compression sample will be broken into three distinct regimes: (1) the elastic–plastic transition, where deformation twinning initiates, (2) the low stress plateau where twinning dominates the deformation, and (3) the subsequent increased hardening, where twinning has saturated and other deformation mechanisms must activate. The microscopic

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phenomena observed through the development of the diffraction pattern will be related to each region. 5.1. Formation of twins Macroscopically, rolled magnesium compressed in the inplane direction linearly deforms in the elastic region up to −50 MPa. Between −50 and −70 MPa the flow curve departs from linearity concomitant with a significant change in diffraction peak intensity manifesting the onset of deformation twinning. In general, it has been asserted that, like slip, deformation twinning is primarily controlled by the resolved shear stress on the twin system [24], that is twinning obeys a Schmid law, τt = σc cos λ cos χ

(3)

where λ and χ are the angles between the load axis and the twin plane normal and twin direction. Specifically, Gharghouri et al. [18] confirmed this in the case of {1 0 1¯ 2} tensile twinning in magnesium reporting a CRSS of 65–75 MPa in Mg 7.7 at.% Al alloys. The initial sample texture and alignment employed in this experiment strictly define λ and χ, and thus the Schmid factor. The basal poles in the parent microstructure are initially nearly horizontal and transverse to the applied load. In order for the daughter basal pole to appear in the Q|| detector, the plane of shear, both the {1 0 1¯ 0} and {1 0 1¯ 2} poles must also be horizontal. Moreover, a pole of the {1 0 1¯ 0} family must be parallel to the load axis. If a prism pole of the {1 1 2¯ 0} or {2 1 3¯ 0} families were horizontal, the basal pole of the daughter grain would rotate out of the horizontal plane during twinning and would not be detected. The initiation of twinning was detected between 50 and 70 MPa by the appearance of (0 0 0 2) intensity parallel to the load. In the orientation defined by the experimental geometry, the maximum Schmid factor, m = cos λ cos χ, is 0.499 making the critical resolved shear stress of the twin system between 25 and 35 MPa. The stress step size between diffraction measurements was not fine enough to determine the CRSS to better accuracy. The visco-plastic model employed in Ref. [9] assumed a CRSS of 15 MPa, while the elasto-plastic model of Ref. [10] uses an initial value of 30 MPa. In both cases twins harden linearly because of slip activity with a gradient of 30 MPa. The diminishing intensity in parent grains ({1 0 1¯ 0}, {1 1 2¯ 0}, and {2 1 3¯ 0} poles parallel to the load) also signals the onset of twinning. Unfortunately, because the initial changes in intensity are small relative to the large original intensity, we cannot utilize the parent grains to accurately ascertain the CRSS. However, the stresses at which the intensity of each has been halved, σ1/2 , can be interpolated from the data in Fig. 6 with reasonable accuracy and are shown in Table 1. The maximum Schmid factor on these families of parent grain orientations {h k i l} are also summarized in Table 1. Grains with a {1 0 1¯ 0} pole along the compression

9

Table 1 Schmid factor and stress at which intensity is halved for several grain orientations {hkil} Grain orientation

Maximum Schmid factor

σ1/2 ± 2 (MPa)

{1 0 1¯ 0} {2 1 3¯ 0} {1 1 2¯ 0} {1 0 1¯ 1}

0.499 0.445 0.374 0.305

−101 −107 −111 −197

axis have a high Schmid factor of m = 0.499 and reach half intensity at the lowest stress. On the other hand, in grains with {2 1 3¯ 0} or {1 1 2¯ 0} pole parallel to the stress axis, the maximum Schmid factor is m = 0.445 or 0.374, respectively. Correspondingly, σ1/2 increases with decreasing Schmid factor. In fact, σ1/2 linearly decreases with Schmid factor over this small data set. Finally, in a grain with a {1 0 1¯ 1} pole aligned with the compression axis, the highest Schmid factor is m = 0.305 and σ1/2 = −197 MPa. This is much larger that would be expected based on the three prism poles, but may be obscured by the initial increase in the {1 0 1¯ 1} diffraction intensity. There is an intermediate orientation of {1 0 1¯ 1} and we do not presently understand the source of the initial increase in intensity. While the Schmid law cannot be quantitatively tested nor a CRSS determined, the data is consistent with the conclusion of Gharghouri et al. [18] that twinning in Mg is controlled by a Schmid law. The experimentally determined textures indicate that roughly 80% of the original parent microstructure has twinned during deformation to 14%. Through comparison with the calculated pole figure (Fig. 2) the remainder of the parent microstructure is associated with grains oriented with basal poles at Ψ = 90 and 270◦ and θ = 45◦ , which result in the observed eccentricity of the basal fiber. The Schmid factor for twinning is relatively low in these grain orientations, between 0.15 and 0.3 depending on orientation about the basal pole. Also, in these grains the basal poles are significantly offset from the nominal orientation transverse to the load and therefore, they are more likely to deform through basal slip modes, on which the Schmid factor is nearly 0.5. 5.2. Internal strains in parent and daughter grains While the initial parent orientation defines the stress at which a given grain twins, it is the rotation of the crystal symmetry and the associated shear achieved during the twinning event that determines the stress state of the daughter on creation and throughout subsequent deformation. Given the diffraction elastic constants derived from the single crystal stiffnesses of magnesium, isolated grains having the daughter orientation ((0 0 0 2) parallel to the load) would exhibit a lattice strain of −1460 ␮␧ in a −70 MPa uniaxial compressive stress field. However, the initial observed strain in the daughter grains is roughly −700 ␮␧, roughly half of the expected lattice strain. It should be noted that the daughter grains likely formed between the measurement points at 50 and 70 MPa. Thus the initial observation of −700 ␮␧ on the

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daughter orientation represents an upper bound on the strain in the daughter grains at formation. The relaxed state of the daughter grains relative to the surrounding matrix can be understood in terms of the schematic shown in √ Fig. 3. Because the c/a ratio of magnesium is less than 3, the twinning shear geometrically “shortens” the grain along the load axis (lengthens it along the initial c-axis), and a fraction of the elastic compressive strain is replaced by this plastic component. The constraint of the matrix on the grain then places the daughter in tension relative to the macroscopic stress field, that is the applied compressive load. While deformation twinning is the dominant deformation mechanism, from 1 to 8% strain, the slope of the daughter stress/strain curve, 32 ± 3 GPa (see Fig. 8), is considerably less than the (0 0 0 2) specific elastic modulus, 48 GPa. This slope represents the averaged response of existing and newly formed daughter grains. Since the results discussed above establish the fact that the newborn daughters form relaxed relative to the matrix, and then the existing daughter grains must accumulate strain more rapidly than the aggregate as a whole. This makes sense, since the daughters themselves are in a hard orientation, poorly oriented for all but the hardest slip and twinning modes and will accept a greater proportion of the elastic strain as surrounding grains deform inelastically. As a result, lattice strain builds up faster than linear elasticity due to load sharing with the parents and surrounding grains, which are in a plastically “soft orientation”. The elastic lattice strains in the parent grains, shown in Fig. 8a, continue to relax throughout this region. In fact, perpendicular to the applied load, the elastic strain in the parent grains remains zero until the corresponding diffraction peak is no longer resolvable. The transfer to the daughter microstructure occurs over the range from −70 to −220 MPa as manifested by the continuous increase in (0 0 0 2) diffraction intensity parallel to the applied load. The spread over which twinning occurs is not a grain orientation effect, as the diffraction geometry strictly defines the orientation. Rather, at each stress interval, sufficient twinning occurs to relax the parent microstructure (Fig. 8a) and then ceases until further deformation is imposed. 5.3. Variability in intergranular stresses indicated by peak broadening Significant peak broadening, FWHM from 0.002 to ˚ was observed in the regime where twinning is active. 0.006 A, Convoluted within the peak broadening are two phenomena, namely the type III intragranular strains (most frequently connected with the stress fields of dislocations and their structures) as well as grain-to-grain variations or the type II strains. The slope of the daughter stress/strain curve indicates that little plastic relaxation of the daughter grains is occurring in this regime since the elastic strain/stress accumulates so rapidly. As such, the increase in broadening cannot be attributed to an increase in dislocation density. Indeed, when pyramidal slip does activate in the daughter grains (discussed below), the sample broadening is no longer increasing.

Daughters that formed early at −70 MPa continue to accumulate elastic strain during subsequent deformation with an effective modulus, Eeff , of 32 GPa, or a total strain of ε = 0.0047 by −220 MPa. The development of elastic strain in the parent grains saturated once twinning began at roughly −70 MPa (Fig. 8). Therefore, it is likely that daughters that form near the completion of twinning do so in a state similar to 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 0.008 A consistent with the observed increase in the FWHM of the daughter peak after −220 MPa and is thus likely the source of the broadening. 5.4. Role of twinning in plastic deformation Obviously, twinning is playing an important role in the deformation from 0.5 to 8% deformation. It is interesting to consider how much of the plastic deformation is accommodated by twinning in this regime. For a given grain orientation, g the strain achieved parallel to the applied load, ε11 , is related to the characteristic shear of the twin system, S, by the Scmidt factor m, specific to that grain orientation, and the volume fraction of the grain which is twinned, ν, ε11 (g) = m(g)γtw = m(g)ν(g)S.

(4)

The aggregate strain response, ε¯ 11 , is the texture weighted average of the single grain response  ε¯ 11 = S (5) m11 νf = S m ¯ 11 ν¯ , g

where f (g) represents the orientation distribution function and the sum is over the grains that twin. We have made the simplifying assumption that the twinned volume fraction is not a function of grain orientation, i.e. all grains that twin do so at some average rate. Fig. 6 demonstrates that this is not precisely true, but it is a reasonable approximation. The Schmid factor on a grain with optimal orientation is m = 1/2. Averaged over the initial texture of the rolled magnesium, m ¯ is found to be approximately 0.32. The incremental fraction of the total plastic strain accommodated by deformation twinning may then be written as εtwin ν = 0.32S , εplastic εplastic

(6)

assuming that the average orientation factor does not evolve substantially during the deformation. Fig. 7 shows the development of the twinned volume fraction as a function of plastic strain. The slope of the curve or the rate of twin formation, in the linear region of the curve (εp < 5%) is 0.145/strain unit. Using Eq. (6), and the characteristic shear of the tensile twin in magnesium, 0.131, we estimate that from 0 to 5% plastic deformation, 61% of the plastic deformation is accommodated by twinning. After 5% plastic deformation, twinning begins to slow down and by 8% deformation, where twinning

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

11

has saturated, 42% of the plastic deformation has been accomplished by twinning. Reed-Hill performed similar analyses on rolled zirconium, in which the basal poles were oriented in the rolling plane, under applied in-plane tension. He estimated the texture weighted Schmid factor to be 0.5, and reported that only 15% of the plastic deformation of rolled zirconium under applied in-plane tension was attributable to twinning [32]. A similarly low value, 12%, has recently been found for extruded zirconium compressed perpendicular to the basal texture, and an intermediate ratio, 21%, in rolled beryllium compressed perpendicular to the basal fiber [33].

stress/strain curve in likewise oriented grains [10] and the behavior was satisfactorily reproduced with the EPSC model by assuming pyramidal activity. The through-thickness samples also showed a comparable reduction of (0 0 0 2) diffraction intensity parallel to the load when pyramidal slip activated. Beyond 10% strain, the in-plane compression sample is actually stronger than its through-thickness counterpart, despite the similar textures. It is speculated that the increased strength is due to the remaining twin boundaries acting as barriers to dislocation motion, similar to reducing the grain size [9].

5.5. Mechanisms of c-axis compression (c + a slip and {1 0 1¯ 1} twinning)

6. Conclusions

Beyond −220 MPa or 8% strain, the macroscopic flow curve again begins to decrease in slope. At this point, the slope of the daughter applied stress versus elastic strain curve increases to greater than the linear elastic response, indicating activity of a plastic deformation mechanism in the daughter grains that allows for relaxation in the c direction. The inflection in the macroscopic and lattice stress/strain curves coincides with the maximum observed in the (0 0 0 2) intensity parallel to the load axis (see Fig. 7). Ghargouri et al. ¯ compression twinning [18] speculated that {1 0 1¯ 1} 1 0 1¯ 2 might explain some of their observations of small intensity changes at large strain, which could not be understood in terms of the more common slip or twinning modes of magnesium. The monotonic strain level that was achieved in the reported tests was insufficient to yield a final conclusion in the matter. The present results, and those of other recent studies [10], lead to the conclusion that c + a slip provides a more satisfactory explanation for the observations. The stress along 0 0 0 1 could be relaxed by either mechanism. The decrease in the basal fiber texture along the compression axis could be understood in terms of either mechanism. However, the change in overall texture points to the slip mechanism rather than the twin. As mentioned earlier, any twinning mechanism results in a characteristic finite reorientation of the crystal. For the {1 0 1¯ 1} compression twin, there is approximately a 56◦ reorientation of the basal poles. Hence, there must be a corresponding increase in intensity within the (0 0 0 2) pole figure at about 56◦ from the compression axis if twinning is responsible for the texture change. It was previously suggested that this might be a pseudo-elastic deformation mode, whereby twinning under load “untwinned” during the unload [18]. The current data clarifies this point since the decrease in basal intensity, which occurs at the highest loads, remains during unloading. It should be noted that at this point in the deformation (subsequent to the tensile twinning regime), the texture of the in-plane compression sample has been almost completely reoriented (Fig. 2b) and the situation becomes similar to the through-thickness compression experiment. Throughthickness compression produced a similar (0 0 0 2) lattice

Deformation twinning is active in textured magnesium when loaded in compression transverse to the basal poles. The activity of deformation twinning is associated with the low stress flow regime of the stress/strain curve from 70 to 200 MPa (0.1–8% strain). The onset of deformation twinning on the {1 0 1¯ 2} tensile twinning in magnesium is consistent with a Schmid law, as concluded by Ghargouri et al. [18], ¯ and the critical resolved shear stress of the {1 0 1¯ 2} 1 0 1¯ 1 in ally AZ31B is estimated to be between 25 and 35 MPa. Unobstructed diffraction from newly formed daughter grains indicates that the twins are initially relaxed relative to the macroscopic applied stress along the loading direction. Similarly, the elastic lattice stain in the parent grains is relaxed during the formation of the daughters. During subsequent loading, the daughter grains absorb a large fraction of the load as it is shed from the “soft” parent orientation to the “hard” daughter orientation. A significant variation in interplanar spacing within the daughter grains, manifested as an increase in diffraction peak breadth, arises between the initial and final daughters to form. Throughout the twinning regime, εp < 8%, ∼42% of the plastic deformation is accommodated by twinning. Deformation twinning continues until roughly 80% of the microstructure is reoriented such that the basal poles are preferentially aligned with the applied load at the completion of the test. After twinning has been exhausted accommodation of c-axis compression is achieved by c + a slip.

Acknowledgements Research sponsored by the U.S. Dept. of Energy at Los Alamos National Laboratory under contract number W-7405ENG-36 and at the Oak Ridge National Laboratory operated by UT-Battelle, LLC, under contract DE-AC05-00OR22725. The Manuel Lujan Jr. Neutron Scattering Center is a national user facility funded in part by the U.S. DOE.

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