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Geometrically necessary twins in bending of a magnesium alloy
Materials Science & Engineering A 645 (2015) 298–305
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Geometrically necessary twins in bending of a magnesium alloy Z. McClelland a, B. Li a,b,n, S.J. Horstemeyer a, S. Brauer a, A.A. Adedoyin a, L.G. Hector Jrc, M.F. Horstemeyer a a
Center for Advanced Vehicular Systems, Mississippi State University, Starkville, MS 39759, USA Department of Chemical and Materials Engineering, University of Nevada, Reno, NV 89557, USA c General Motors R & D Center, 30500 Mound Road, Warren, MI 48090, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 12 May 2015 Received in revised form 5 August 2015 Accepted 6 August 2015 Available online 8 August 2015
Evidence for the formation of geometrically necessary twins (GNTs), or twins that accommodate a strain gradient in a multi-axial stress state, in quasi-static, room temperature three-point bending of a rolled ¯ } < 1011 ¯ ¯> magnesium alloy is presented. Electron backscatter diffraction analysis showed that {1012 ¯ } < 1012 ¯ ¯ > contraction twins) form in arcs in the tension zone, and extension twins (rather than {1011 that twinned grains have very low Schmid factors. The main tensile stress component in the tension zone was nearly perpendicular to the c-axis of the parent grains. The mechanism for such unusual twinning ¯ } < 1011 ¯ ¯> behavior was analyzed from the perspective of strain components that are generated by {1012 twinning. After twinning, an extension strain component along the c-axis and a contraction strain component perpendicular to the c-axis of the parent lattice are generated simultaneously due to the misfit between the parent and the twin lattice. The contraction strain component by twinning provided an extra strain accommodation for the compressive strain in the tension zone produced by the bending, ¯ } < 1011 ¯ ¯ > twinning. Thus, the despite the fact that the local stress state strongly disfavored the {1012 ¯ } < 1011 ¯ ¯ > twins in the arcs in the tension zone of the bent specimen present the characteristic of {1012 being geometrically necessary, similar to geometrically necessary dislocations and boundaries. & 2015 Elsevier B.V. All rights reserved.
Keywords: Magnesium Twinning Schmid factor Geometrically necessary twins
1. Introduction Magnesium (Mg) alloys have drawn substantial attention in recent years owing to their broad technological significance and intriguing materials science. Their high strength-to-weight ratio along with low density ( 35% lighter than Al alloys and 78% lighter than steel) make them attractive replacement materials for more mass-intensive ferrous and non-ferrous alloys currently used in transportation industries [1,2]. Magnesium alloys also offer significant potential for fuel efficiency improvements and reduced hydrocarbon emissions. However, room temperature (RT) applications of wrought Mg alloys have been limited, in part, by poor ductility. This results from the large difference in critical resolved shear stresses between basal and prismatic slip in the hexagonal 1 ¯ > basal slip close packed (HCP) lattice (two independent 3 < 1120 systems are active at RT). The stress required to plastically deform Mg along its (easy) basal slip plane is two-orders of-magnitude lower than the (hard) prismatic plane at RT, requiring elevated temperatures to enhance ductility in component forming n Corresponding author at: Department of Chemical and Materials Engineering, University of Nevada, Reno, NV 89775, USA. E-mail address: [email protected] (B. Li).
http://dx.doi.org/10.1016/j.msea.2015.08.027 0921-5093/& 2015 Elsevier B.V. All rights reserved.
processes [3,4]. Magnesium sheet formability at elevated temperatures is sensitive to the chosen material manufacturing process (e.g. direct chill, twin roll casting) [5]. The materials science of Mg alloys is quite complex. In addition to the multitude of slip systems, both active and inactive, the nearly ideal c/a ratio (γ) of Mg, promotes twinning, for which there are two systems, namely: ¯ } < 1011 ¯ ¯ > and {1011 ¯ } < 1012 ¯ ¯ >. The {1012 ¯ } < 1011 ¯ ¯ > system {1012 is the most commonly observed twinning mode in all HCP metals [6–8]. Yoo [9] plotted the twinning shear with respect to the c/a ratio of the major twinning modes in HCP metals. For γ o 3 , the slope of the twinning modes was shown to be negative but turns positive when γ 4 3 . Thus, the negative slope was as an indication that the twinning mode is c-axis extension. Hence, ¯ } < 1011 ¯ ¯ > twinning is most favorable when a tensile stress is {1012 applied along the c-axis of a Mg crystal. However, it is suppressed when a compressive stress is applied along the c-axis or a tensile stress is applied perpendicular to the c-axis. In dislocation-mediated plastic deformation, an important class of dislocations, i.e. geometrically necessary dislocations (GNDs), was first proposed by Nye [10]. The GNDs are generated when a strain gradient is present within a deformed material. For instance, in bending tests of crystalline metals, the slip planes for dislocations are curved by the deformation [10]. Thus, the curvature of the slip planes gives rise to a slip gradient on successive slip planes
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that can be compensated by GNDs [11]. Additionally, GNDs are distinguished from statistically stored dislocations (SSDs) that are created in the absence of a large strain gradient. Here, GNDs are associated with the organization of dislocations into low energy configurations, such as dislocation cells or walls that result in the development of a locally non-uniform strain field [12]. Closely related to GNDs are geometrically necessary boundaries (GNBs), including dense dislocation walls (DDWs) and microbands (MBs) that are necessitated by the accommodation of the lattice misorientation between the dislocation cells [13–17]. GNBs were proposed as a means to account for the flow stress anisotropy in deformed metals with medium to high stacking fault energies [18]. Using the GND concept in dislocation-dominated plastic deformation, Sevillano [19] first proposed the concept of “geometrically necessary twins” (GNTs). A GNT was treated as a “pseudodislocation line” at the twin edge with an effective Burgers vector in the direction of the twinning shear. The resultant hardening effect and strain accommodation by GNTs was also discussed [14]. Unfortunately, there has been a dearth of experimental evidence of GNTs since Sevillano's [19] initial proposition. Some preliminary experimental observations were reported by Li et al. [20]. In slipcontrolled deformation, dislocations interact as they intersect with each other. Dislocations also interact with secondary phases and interfaces such as grain boundaries and cell boundaries, resulting in changes in the original slip pattern via cross-slip, transmission, reflection or absorption at interfaces [21–23]. In contrast, twinning results from successive glide of twinning dislocations restricted at the twin/parent interfaces, i.e. twin boundaries (TBs) that should satisfy the invariant plane strain condition [24]. Thus, strain accommodation by shear-dominated twinning is different from dislocation slip. Alternatively, twinning is unidirectional [24] since the twinning shear is uniquely defined by the second invariant plane, i.e. the K2 plane, and the twinning direction η1. Because of these differences, GNTs that behave similarly to GNDs have received little attention in experiments. In this study, we present experimental evidence for GNTs in Mg AZ31-O, the most common of all wrought Mg alloys, deformed in quasi-static three point bending at RT. Bending was chosen, in part, because it is a significant deformation component in many practical applications in transportation industries [25]. The GNTs result from the macroscopic strain gradient produced by the bending load. We specifically characterized twins in the tension zone using electron backscatter diffraction (EBSD) and showed that they present the characteristic of being geometrically necessary.
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optical microscopy, the specimen was then mechanically re-polished followed by electrochemical polishing with an electrolyte composed of 160 mg sodium thiocyanate, 800 ml of ethanol, 80 ml of ethylene glycol monobutyl ether, and 20 ml of distilled water. Electron backscatter diffraction (EBSD) was then performed on the through-thickness cross-section using a Zeiss Supra 40 field emission gun scanning electron microscope (SEM) equipped with an EDAX Hikari EBSD detection system. The EBSD step size was 0.1 mm. When the scans were completed the data was rotated at 90° about the transverse direction (TD) using the EDAX analysis software (TSL OIM 6) so as to align it with the normal direction (ND) of the sheet specimen, which is the direction perpendicular to the tension and the compression surfaces of the bend specimen.
3. Results Fig. 1 shows optical microscopy of the Mg AZ31 specimen microstructure after three point bending. The image was taken on the through-thickness cross-sectional area. With the DIC illumination, two zones can be clearly distinguished, namely, a compression zone and a tension zone. In the compression zone, a very high density of deformation twins is observed. Away from the surface of the compression zone, twins are localized in bands. The tips of the bands extend near the center line, i.e. midway along the thickness of the specimen. Such localized twin bands in three point bending were first reported by Baird et al. [27]. The twin bands are aligned in two directions that are nearly perpendicular to each other. EBSD analysis showed that the twins are ¯ } < 1011 ¯ ¯ > extension twins [27]. Alternatively, no twins are {1012 observed in the tension zone of Fig. 1 at the selected magnification. After removing the DIC illumination, the twin bands present a different contrast than that in Fig. 1. As indicated by the long black arrows midway along the thickness of the specimen, the tips of the twin bands can still be seen but in weak contrast. However, another pattern of twins in the region at the vicinity of the tips of the twin bands is observed. Near the center of the field of view in Fig. 2a, where the deformation transitions from compression to tension, an arc of twins, which has a different contrast than the twin bands (indicated by the long black arrows), is observed
2. Experiments A twin-roll cast sheet of AZ31 Mg alloy [26] with thickness of 1.0 mm was cut into strips with dimensions 76 7 1 mm3. Quasi-static, three point bending was performed using an Instron 5882 testing system and an in-house three point bend fixture. The distance between the bending anvil pivots was 28 mm. The specimens were deformed at a rate of 10 mm/min and through a maximum bending angle of 132° and 30 mm displacement. The bend was parallel to the rolling direction (RD). One of the specimens was immediately cold mounted and the microstructure of the cross-section through the thickness was studied in detail using optical microscopy. This specimen was mechanically polished and then etched using an Acetic Picral solution containing 3 g picric acid, 100 ml ethanol, 10 ml distilled water, and 5 ml acetic acid. The specimen was then examined using a Zeiss Axiovert optical microscope (OM). To better reveal the deformation twins, differential interference contrast (DIC) illumination was used in imaging: this improves the contrast of the twins [27]. After
Fig. 1. Optical microscopy of localized twin bands in the compression zone of the sample after three point bending (the bending angle after springback is 132°) [20]. Differential interference contrast (DIC) illumination was used in imaging. Each band comprises a high density of extension twins. In the tension zone, where the stress disfavors extension twinning, no twins are observed at this magnification (100X). The black arrows point to the twin band tips.
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Fig. 2. Higher magnifications of the tension zone in Fig. 1 are shown [20]. (a) 500X. Near the tips of the localized twin bands (indicated by the long black arrows), arcs of twins were observed when the differential interference contrast (DIC) illumination was removed. Note that the twins in the arcs (indicated by the short black arrows) and those twins in the bands (indicated by the long black arrows) present different contrast, indicating that they may be different twin variants. (b) 500X. In the tension zone, arcs of twins follow the contour of the bent specimen. Outside the arcs, the twin density is much lower. (c) 800X. Deformation twins (indicated by the short black arrows) appear near the surface in tension with maximum tensile strain. The grain size is 5.0 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
between the red lines (indicated by the short black arrows). Lenticular twins are localized in an arc that follows the contour of the bent specimen. Away from the compression zone, which, as shown in Fig. 1 contains a high density of twins in the bands, a small arc of twins can be observed in the upper right region of the tension zone. Fig. 2b shows that close to the middle of the tension zone, a long arc of twins can be observed. This arc extends diagonally across the field of view, follows the shape of the bent specimen, and is but one of multiple arcs of twins that were captured. Fig. 2c shows the somewhat surprising result that twins (indicated by the short black arrows) appeared near the surface of the tension zone where the tensile strain is maximum in three point bending. Fig. 3a and b shows other arcs of twins at different locations inside the tension zone at higher magnifications. Because the grain size of the rolled sheet is rather small ( 5 mm), the thickness of the twin laths is only a few microns. Fig. 3a shows two neighboring arcs of twins. Fig. 3b shows another arc of twins in which more deformation twins (relative to Fig. 3a) are localized inside. All the arcs of twins follow the contour of the bent specimen. The arcs of twins in the tension zone were also observed in other Mg AZ31-O specimens after three point bending. The optical microscopy in Figs. 2 and 3 reveal arcs of twins in the tension zone with different image contrast than those twin bands in the compression zone, suggesting that the twins in the tension zone may be a different twin variant than those in the compression zone. But these images (Figs. 2 and 3) do not provide definitive crystallographic orientations of the twins in the tension ¯ } < 1011 ¯ ¯> zone in which the stress state strongly disfavors {1012 extension twins [27]. The morphology of the twins does not match ¯ } < 1012 ¯ ¯ > twins because neighboring twins the compression {1011 join together at their tips at the grain boundaries (Fig. 2a). Due to
Fig. 3. Other arcs of twins at higher magnifications inside the tension zone at different locations [20]. (a) 500X. Two neighboring arcs of twins, one between the red lines and another one between the yellow lines. (b) 800X. Another arc of twins between the red lines. All the arcs follow the contour of the bent specimen. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the repulsive force between twinning dislocations, neighboring compression twins should not join at their tips. However, it is very ¯ } < 1012 ¯ ¯ > compression twins. likely that the twins could be {1011 To identify the type of the twins in the tension zone, we performed EBSD on a bent AZ31 specimen. Because the rolled sheet was not annealed before bending, and the density of defects in the bent specimen was high, capture of twins with EBSD was very challenging. After electrochemical polishing, the arcs of twins were not observable in SEM, in other words, the arcs of twins could not be located on the well-polished surface before the EBSD scans started. To circumvent this difficulty and pinpoint the twins in the tension zone for EBSD scans, we first mechanically polished the through-thickness cross-section and then etched it to reveal the twins with optical microscopy. Then, micro-hardness indentations were performed near the arcs of twins. At each area of interest, four indentation marks forming a rectangular region were created with the twins being surrounded by the indentations. The indentation marks, each of which had a depth of 5.0 μm, served as reference points for the EBSD scans. Following indentation, the specimen was re-polished mechanically such that the indentation marks remained on the polished surface. EBSD scans were then run in the regions with the indents. After repeated scanning attempts, we captured twins in the tension zone with EBSD.
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the plotted lattices. The rotation axis is roughly parallel to the TD. One of the twinning planes was delineated by a pair of dashed lines within the hexagonal cell in P2. Fig. 5b plots the point-topoint misorientation angle between parent grain P2 and twin T2. The 88° misorientation angle again confirms that the twin is a ¯ } < 1012 ¯ ¯ > extension twin. Fig. 5c shows that the twinning {1012 process can be identified from the PF of the basal plane rotated by nearly 90° (indicated by the red arrow). Fig. 5d shows the PF of the ¯ } twinning planes. The two twinning planes in Fig. 5a were {1012 highlighted by the red circles in Fig. 5d. These two twinning planes share the same zone axis and hence both are possible twinning planes. Since the direction of the tensile stress is roughly parallel to the TD, we expect the Schmid factor of the twinning systems should be very low. Similar analysis for twin variants T1 and T3 is shown in Fig. 6a–d, where the PFs of the basal (0001) and the ¯ } twinning plane of grains P2 and P3 were plotted. The re{1012 orientation of the parent lattices by twinning is denoted by the red arrows (Fig. 6a and c). To calculate the Schmid factor of the parent grains (P1, P2 and P3 in Fig. 4b), we first recorded the Euler angles (ϕ1, Φ, ϕ2) of the individual parent grains from the IPF maps in Fig. 5a. Each of the three Euler angles corresponds to a coordinate transformation matrix, denoted as Tϕ , TΦ and Tϕ , respectively. The overall trans1 2 formation matrix from the lattice coordinate system to the specimen coordinate system (RD, TD, ND, where the ND is perpendicular to the figure plane of Fig. 5a) can be expressed as:
T = Tϕ2⋅TΦ⋅Tϕ1 Fig. 4. EBSD inverse pole figure (IPF) maps of the tension zone. As expected, a strong basal texture can be seen, typical of rolled Mg sheets [20]. (a) Twins were captured after repeated scanning efforts. (b) The pole figure of (a). (c) Magnified view of the region enclosed by the rectangle in (a). Lattices of the parent grains (P1, P2 and P3) and the corresponding twins (T1, T2 and T3) were plotted to show the orientation relationship as denoted by the hexagonal cells (highlighted view of the ̅ cells are shown in Fig. 5). Twin boundaries that satisfy 86.3 7 5° 1120 are highlighted with black lines.
Fig. 4a shows an inverse pole figure (IPF) map of a portion of the tension zone in Fig. 3b. The corresponding pole figures (PFs) are shown in Fig. 4b, which is typical of a strong basal (0001) texture of rolled Mg sheets. A few twin variants can be seen near the middle of Fig. 4a. The parent grains and the twins inside the rectangular box were cropped and magnified in Fig. 4c.
4. Analysis and discussion 4.1. The Schmid factor of the twinning system in the tension zone The capture of the twins in the tension zone with EBSD (Fig. 4) enables analysis of the type, orientation relationship and the Schmid factor of the twins. In Fig. 4b, three parent grains are marked as P1, P2 and P3, whereas their corresponding twins are marked as T1, T2 and T3. The grain boundaries that satisfy ¯ } < 1012 ¯ ¯ > twin orientation re¯ >, i.e. the {1011 86.3 75° <1120 lationship, are highlighted with black lines. Thus, the twins T1, T2 ¯ } < 1011 ¯ ¯ > extension twins, instead of and T3 are indeed {1012 ¯ } < 1012 ¯ ¯ > compression twins. To help identify the rotation {1011 axis of these extension twins, hexagonal cells of the parent grains and the twins are also plotted in Fig. 4b (highlighted views of the hexagonal cells are shown in Fig. 5). To further analyze the twins in the tension zone (see Figs. 2 and 3), parent grain P2 and twin T2 in Fig. 4b were cropped and ¯ } < 1011 ¯ ¯ > twinning reorients highlighted in Fig. 5a. Because {1012 ¯ >, the rotation axis the parent lattice by nearly 90° around <1120 can now be readily identified (see the green arrow in Fig. 5a) from
(1)
Applications of the transformation matrix T to the normal vector of the twinning plane (K1) and the twinning vector (η1) requires some preliminary steps. We first have to transform the Miller–Bravais indices of these vectors into the Miller indices, and then transform the Miller indices into orthogonal, cubic coordinate indices. This allows us to calculate the Schmid factor m = cos α⋅cos λ (α is the angle between the normal vector of the twinning plane and the loading direction, and λ the angle between the twinning vector and the loading direction). This transformation is denoted as B. Thus, any vector in the Miller–Bravais system a¼[uvtw] (u, v, t and w are the components in the four-index system), is transformed to a vector b ¼[xyz] in the specimen coordinate system by:
b = T ⋅B⋅a
(2)
The components of T can be computed in terms of the three Euler angles:
⎡ T ⎤ ⎡ cosϕ1 cosϕ2 − cosΦ sinϕ1 sinϕ2 ⎤ ⎥ ⎢ 11 ⎥ ⎢ ⎢ T12 ⎥ ⎢ cosϕ2 sinϕ1 + cosΦ cosϕ1 sinϕ2 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ T13 ⎥ ⎢ sinΦ sinϕ2 ⎢ T ⎥ ⎢ −sinϕ cosϕ − cosΦ sinϕ cosϕ ⎥ 21 2 1 1 2 ⎥ ⎥ ⎢ ⎢ ⎢ T22 ⎥ = ⎢ −sinϕ2 sinϕ1 + cosΦ cosϕ1 cosϕ2 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ T23 ⎥ ⎢ cosϕ2 sinΦ ⎥ ⎢ T31 ⎥ ⎢ sin sin ϕ Φ ⎥ ⎢ 1 ⎥ ⎢ T ⎥ ⎢ ⎢ 32 ⎥ −sinΦ cosϕ1 ⎥ ⎢T ⎥ ⎢ ⎣ 33 ⎦ ⎣ cosΦ ⎦
(3)
The transformation B can be computed as:
⎤ ⎡1 1 − 0⎥ ⎢ 2γ ⎥ ⎢γ ⎥ B=⎢ 3 ⎢0 0⎥ 2γ ⎥ ⎢ ⎢⎣ 0 0 1⎥⎦
(4)
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Fig. 5. Schmid factor and twin orientation relationship analysis of the deformation twins. (a) A magnified IPF map of the region containing parent P2 and twin T2 in Fig. 4b. ̅ can be readily identified from the lattices denoted by the hexagonal cells. The twinning plane is marked with dashed lines in the lower of the two The rotation axis 1120 ̅ } 101̅ 1̅ twins. (c) The PF of the hexagonal cells. (b) The misorientation angle of the twin is 88°, indicating that the twins in the tension zone are the unfavorable {1012 basal plane shows a near 90° rotation. The open circles represent the pole of the twinning plane in (a), and the red arrow represents the reorientation by twinning. (d) The PF of the twinning planes (the red circles) in grain P2 shows that the Schmid factor should be low. The tensile stress is approximately along the TD indicated in (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where γ is the c/a ratio (for Mg γ ¼ 1.624). Since the loading direction is known in the specimen coordinate system, the Schmid factor of any slip or twinning system can now be calculated. In three point bending, the stress state in the tension zone is more complex than uniaxial tension because the stresses are multi-axial. As a first-order approximation, we assumed the direction of the main component of the tensile stress is along the TD, and then calculated the Schmid factor of all the twinning systems (Figs. 5 and 6). Because there are six twin variants, i.e. three pairs of twin variants with each pair sharing a common zone axis, the calculated Schmid factors can be matched with the actual twins in the individual parent grains of P1, P2 and P3 in Fig. 4c. For all three parent grains, one of the twinning planes must have a very low Schmid factor because its pole is nearly perpendicular to the loading direction, i.e. the TD. Thus, the Schmid factors of the twin variant in the parent grains can be associated with the PFs in Figs. 5c, d and Figs. 6a to d. The results from calculations involving Eqs. (1–4) are listed in Table 1. As shown in Table 1, the Schmid factors of the twinning systems in the three parent grains have very low values, with the Schmid factor for grain P2 in Fig. 5a being closest to zero. Alternatively, Fig. 5c and the PFs in Fig. 6a and c show that the direction of the tensile stress is almost perpendicular to the c-axis [0001] for grain P2 and P3. This stress state strongly disfavors ¯ } < 1011 ¯ ¯ > twinning; however, it favors {1011 ¯ } < 1012 ¯ ¯> {1012 compression twinning [9]. The same results were noted in
analyses of additional bending tests of the AZ31 Mg sheet material. These facts raise some compelling questions: how and why does ¯ } < 1011 ¯ ¯ > extension twinning occur in the tension zone of {1012 the three point bending specimen where the Schmid factor of the twinning systems is close to zero and the tensile stress is nearly perpendicular to the c-axis? In the following, we provide an ana¯ } < 1011 ¯ ¯ > twinning. lysis of the unique nature of {1012
¯ } < 1011 ¯ ¯ > twinning in the 4.2. Strain accommodation by {1012 tension zone ¯ } < 1011 ¯ ¯ > twinning was first The non-Schmid behavior of {1012 reported by Barnett et al. [28]. It was found that the secondary ¯ } < 1011 ¯ ¯ > twins of {1011 ¯ } − {1012 ¯ } double twinning sig{1012 nificantly deviate from the Schmid-type behavior which governs shear-dominated deformation. Such behavior was attributed to the closest match between the primary and the secondary twinning plane so that the compatibility strain is minimized. In this study, ¯ } < 1011 ¯ ¯ > twins in the tension zone are not the secthe {1012 ondary twins in double twinning; rather, they are the primary twins. To understand the mechanism of the deformation twins observed in the tension zone during bending of the AZ31 Mg sheet, it is necessary to analyze the strain components that are ¯ } < 1011 ¯ ¯ > twinning. produced by the {1012 ¯ ¯ ¯ > twinning has been exThe mechanism of {1012} < 1011 tensively discussed in the literature [29–36]. According to the classical twinning theory, this twinning mode in Mg should have
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̅ } twinning planes of grain P1 (see Fig. 4). The rotations of the poles represent the Fig. 6. (a) The pole figures (PF) of the (0001) basal and (b) the corresponding {1012 ̅ } twinning planes of grain P3 are twinning process. The twinning planes are highlighted with the open red circles. (c) The PFs of (0001) basal and (d) corresponding {1012 shown. The calculated Schmid factors (see Table 1 in the text) of the twinning planes can be associated with these PFs. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Schmid factor of the possible twin variants in Fig. 4c. Euler angles (deg.)
Schmid factor
P1 (24.9, 155.3, 335.8) P2 (336.3, 173.2, 314.5) P3 (318.4, 7.3, 171.1)
0.008–0.025 0.005–0.007 0.049–0.055
an elementary twinning dislocation with a Burgers vector of 0.024 nm, thus generating a 12.9% twinning shear [9,37,38]. Christian and Mahajan suggested that the twinning dislocation should be a twin-layer zonal dislocation [38]. Serra et al. [30,32,34] proposed a disconnection model to describe the twinning dislocations on the twin boundaries [29,33,35]. Other types of twinning dislocations were proposed by Wang et al. [31]. Li and Ma [29] showed that the maximum magnitude of the shuffles that are required for this twinning mode is about four times that of the theoretical elementary twinning dislocation. Thus, this is a shuffling dominated twinning. Indeed, the most recent in-situ transmission electron microscopy (TEM) observations of ¯ } < 1011 ¯ ¯ > twinning and detwinning in single crystal Mg {1012 showed that in both twinning and detwinning, no measurable shear strain was produced on the single crystal specimen [39–41], in sharp contrast to shear-dominated twinning in other single crystal metals [42]. Therefore, the accommodation of the shear strain by the twin arcs should not play a dominant role even in Mg
alloys such as Mg AZ31. Fig. 7 shows the crystallographic relationship between the parent lattice (Mg atoms are denoted by blue filled spheres) and the twin lattice (Mg atoms are denoted by red filled spheres). This analysis is based on the lattice correspondence in deformation twinning, i.e. a plane/direction of the parent lattice is linearly mapped to a plane/direction of the twin by a homogeneous shear ¯ } < 1011 ¯ ¯ > twinning, the basal plane of the parent is [43]. For {1012 mapped to the prismatic plane of the twin, and the prismatic plane is mapped to the basal plane of the twin. If we interpenetrate the parent and the twin lattices, as shown in Fig. 7a, such lattice correspondence can be clearly resolved. After twin¯ > ning, the parent lattice is reoriented by 90° around the <1120 axis. The theoretical misorientation angle is 86.3° for Mg, but TEM measurements have revealed that the misorientation angle for this particular twinning mode does not have a unique value and varies from 84° to 97° [44]. This was attributed to the breakdown of the invariant plane strain condition that is required for deformation twinning [45]. Irrespective of the misorientation angle, however, the lattice correspondence remains the same and the analysis in Fig. 7a and b remains accurate. ¯ } < 1011 ¯ ¯ > twinning mode arises It is now clear that the {1012 from the conversion of the parent basal plane to the twin prism plane, and the parent prism plane to the twin basal plane. In Fig. 7a, the basal plane of the parent lattice (blue atoms) is parallel to the plane of the page and the viewing direction is along the caxis of the parent (perpendicular to the plane of the page). After
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̅ } 101̅ 1̅ twinning. The parent lattice (in blue) and the twin lattice (in red) interpenetrate, with a 1120 ̅ Fig. 7. Analysis of the strain components generated by {1012 rotation axis, i.e. the horizontal axis where the two lattices coincide. (a) Lattice correspondence between the parent and the twin lattice when the viewing direction is [0001] of the parent. The double-layered prismatic plane of the parent must convert to the single-layered basal plane of the twin. After twinning, a contraction strain is generated in ̅ of the parent, an extension strain is generated along the c-axis of the parent. The yellow the vertical direction due to the misfit (Δd). (b) When the viewing direction is 1120 arrows denote the atomic shuffles required to accomplish the basal-to-prism and prism-to-basal conversions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
twinning, the basal plane (single-layered and perpendicular to the plane of the page) of the twin lattice (red atoms) becomes parallel to the prismatic plane of the parent which is double-layered. Hence, to convert the prismatic plane of the parent to the basal plane of the twin, the parent atoms on the prismatic plane (double-layered) must shuffle and adjust their positions to arrive at the basal plane of the twin (single-layered) [33] (the shuffles are denoted by the yellow arrows). This conversion creates a misfit strain due to the dimensional difference Δd ( =c − 3 a , where a and c ¯ > direction, i.e. the vertical are lattice parameters) in the <1100 direction of the parent lattice. This misfit strain equals γ− 3 3
≈ − 6.24%. Hence, a negative compressive strain is generated
by twinning in the direction perpendicular to the c-axis of the parent. In the direction of the c-axis of the parent, however, a positive tensile strain is generated, as shown in Fig. 7b. In this plot, the ¯ >, and the basal plane of the parent is viewing direction is <1120 parallel to the prismatic plane of the twin. Similarly, the parent atoms on the basal plane (single-layered) must shuffle and adjust their positions to arrive at the prismatic plane of the twin (doublelayered; the shuffles are denoted by the yellow arrows). As such, a misfit strain, which equals
3 −γ γ
≈ 6.65% , is generated. This is a
positive tensile strain along the c-axis of the parent lattice. Thus, ¯ } < 1011 ¯ ¯ > twinning is most favorable if a when γ < 3 , {1012 tensile load is applied along the c-axis of the grains. This analysis also explains how “tension twinning” got its name [9]. The arcs of twins in the tension zone of the three point bending specimen (see Figs. 1–3) can now be accounted for by the misfit strains generated by the mechanism shown in Fig. 7a and b. For those twins in the tension zone, the major component of the stress is a tensile stress. This stress component introduces a compressive strain component in the direction along the normal of the sheet specimen (see “ND” in Fig. 1). Without the extension twins, the strain would have to be accommodated mainly by non-basal dislocations with a limited number of independent slip systems and ¯ } < 1012 ¯ ¯ > compression twinning high critical stresses. The {1011 may be activated, but experiments suggest that this type of twinning has very low volume fraction due to the high critical ¯ } < 1011 ¯ ¯ > extension stress as well [9]. The presence of the {1012 twins, however, provides an extra mechanism for strain
accommodation. The stress state in three point bending strongly disfavors the extension twinning, i.e. as evidenced by the near zero Schmid factor (see Table 1) and a tensile stress perpendicular to the c-axis. However, the high local strain generated in three point bending at large bending angles ( 132° in this study) is able to activate the extension twinning, such that the compressive strain component can be effectively accommodated by the contraction strain (see Fig. 7a and b) generated by the extension twinning in the direction perpendicular to the c-axis of the parent grains. From ¯ } < 1011 ¯ ¯ > twinning mode is strainthis standpoint, the {1012 controlled rather than stress-controlled. Hence, these stress-unfavorable twins can be characterized as GNTs as originally proposed by Sevillano [19], despite the fact that the actual stress state in the tension zone of a three-point bending specimen is multiaxial and complex. Note that the extension strain component (Fig. 7b) generated by the extension twinning along the c-axis has to be accommodated by dislocation mechanisms near the twin boundaries. This may lead to high local stresses at the twin boundaries. The resultant twin-slip interactions in the tension zone may affect the deformation behavior and sheet forming. Whether the presence of the extension twins in the tension zone during sheet bending facilitates crack nucleation and propagation, or actually suppresses crack nucleation and improves the bendability of the AZ31 Mg alloy is worth of further studies. In our three point bending tests of the same Mg AZ31 material, when we bent the specimens to larger bending angles until the specimens fractured, cracks were always observed near the surface with the maximum tensile strain, but no cracks were observed in the compression zone where high density twins are present. In rolled Mg sheets with a strong basal texture, dislocation slip, especially the non-basal slip, dominates the plastic deformation in the tension zone. As the tensile strain builds up in the tension zone, the activation of the GNTs may be able to locally relax the strain. From this standpoint, the GNTs could delay the nucleation of cracks in the tension zone, suggesting the possibility of designing Mg alloys wherein GNT formation is increased. Therefore, it will be of great interest to investigate how the GNTs affect Mg sheet forming at RT. The volume fraction of the GNTs will increase with increasing bending angle, and hence increasing strain. Thus, further investigation is needed to establish a quantitative connection between the density of twins in the arcs to the
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local strain gradient in three point bending.
5. Conclusions We performed three point bending tests on a twin-roll cast Mg AZ31 sheet alloy. Microstructure examinations revealed high density, localized twin bands in the compression zone where the ¯ } < 1011 ¯ ¯ > twinning in this highly stress state strongly favors {1012 textured material. However, in the tension zone, where the stress ¯ } < 1011 ¯ ¯ > twinning, arcs of state strongly disfavors {1012 ¯ } < 1011 ¯ ¯ > twins were also observed. These twins follow the {1012 contour of the bent specimen. EBSD analysis on the misorientation angle and the rotation axis confirmed that the twins in the tension zone are indeed ¯ } < 1011 ¯ ¯ > extension twins, not {1011 ¯ } < 1012 ¯ ¯ > compression {1012 twins, although the stress state strongly favors compression twinning. Schmid factor calculations show that these twins have very low Schmid factor values, one in particular being very close to zero, and the main tensile stress component is nearly perpendicular to the c-axis of the individual grains. ¯ } < 1011 ¯ ¯ > extension The presence of the unfavorable {1012 twins can be understood from the strain components produced by this twinning mode. Due to the misfit strains after reorientation of ¯ > axis, an extension the parent lattice by 90° around the <1120 strain is generated along the c-axis of the parent grain; however, a contraction strain is also generated in the direction perpendicular to the c-axis of the parent grain. In the tension zone of the bent specimen, the compressive strain along the normal direction of the ¯ } < 1011 ¯ ¯ > twins, such that the sheet specimen activates the {1012 compressive strain is accommodated by the contraction strain component generated by the twinning. Hence, the stress-un¯ } < 1011 ¯ ¯ > twins have the characteristic of geofavorable {1012 metrically necessary twins, similar to GNDs and GNBs that have been extensively observed in cubic metals.
Acknowledgments Z. McClelland, B. Li, S.J. Horstemeyer, S. Brauer, A.A. Adedoyin and M.F. Horstemeyer gratefully appreciate the support from Center for Advanced Vehicular Systems, Mississippi State University.
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Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Geometrically necessary twins in bending of a magnesium alloy Z. McClelland a, B. Li a,b,n, S.J. Horstemeyer a, S. Brauer a, A.A. Adedoyin a, L.G. Hector Jrc, M.F. Horstemeyer a a
Center for Advanced Vehicular Systems, Mississippi State University, Starkville, MS 39759, USA Department of Chemical and Materials Engineering, University of Nevada, Reno, NV 89557, USA c General Motors R & D Center, 30500 Mound Road, Warren, MI 48090, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 12 May 2015 Received in revised form 5 August 2015 Accepted 6 August 2015 Available online 8 August 2015
Evidence for the formation of geometrically necessary twins (GNTs), or twins that accommodate a strain gradient in a multi-axial stress state, in quasi-static, room temperature three-point bending of a rolled ¯ } < 1011 ¯ ¯> magnesium alloy is presented. Electron backscatter diffraction analysis showed that {1012 ¯ } < 1012 ¯ ¯ > contraction twins) form in arcs in the tension zone, and extension twins (rather than {1011 that twinned grains have very low Schmid factors. The main tensile stress component in the tension zone was nearly perpendicular to the c-axis of the parent grains. The mechanism for such unusual twinning ¯ } < 1011 ¯ ¯> behavior was analyzed from the perspective of strain components that are generated by {1012 twinning. After twinning, an extension strain component along the c-axis and a contraction strain component perpendicular to the c-axis of the parent lattice are generated simultaneously due to the misfit between the parent and the twin lattice. The contraction strain component by twinning provided an extra strain accommodation for the compressive strain in the tension zone produced by the bending, ¯ } < 1011 ¯ ¯ > twinning. Thus, the despite the fact that the local stress state strongly disfavored the {1012 ¯ } < 1011 ¯ ¯ > twins in the arcs in the tension zone of the bent specimen present the characteristic of {1012 being geometrically necessary, similar to geometrically necessary dislocations and boundaries. & 2015 Elsevier B.V. All rights reserved.
Keywords: Magnesium Twinning Schmid factor Geometrically necessary twins
1. Introduction Magnesium (Mg) alloys have drawn substantial attention in recent years owing to their broad technological significance and intriguing materials science. Their high strength-to-weight ratio along with low density ( 35% lighter than Al alloys and 78% lighter than steel) make them attractive replacement materials for more mass-intensive ferrous and non-ferrous alloys currently used in transportation industries [1,2]. Magnesium alloys also offer significant potential for fuel efficiency improvements and reduced hydrocarbon emissions. However, room temperature (RT) applications of wrought Mg alloys have been limited, in part, by poor ductility. This results from the large difference in critical resolved shear stresses between basal and prismatic slip in the hexagonal 1 ¯ > basal slip close packed (HCP) lattice (two independent 3 < 1120 systems are active at RT). The stress required to plastically deform Mg along its (easy) basal slip plane is two-orders of-magnitude lower than the (hard) prismatic plane at RT, requiring elevated temperatures to enhance ductility in component forming n Corresponding author at: Department of Chemical and Materials Engineering, University of Nevada, Reno, NV 89775, USA. E-mail address: [email protected] (B. Li).
http://dx.doi.org/10.1016/j.msea.2015.08.027 0921-5093/& 2015 Elsevier B.V. All rights reserved.
processes [3,4]. Magnesium sheet formability at elevated temperatures is sensitive to the chosen material manufacturing process (e.g. direct chill, twin roll casting) [5]. The materials science of Mg alloys is quite complex. In addition to the multitude of slip systems, both active and inactive, the nearly ideal c/a ratio (γ) of Mg, promotes twinning, for which there are two systems, namely: ¯ } < 1011 ¯ ¯ > and {1011 ¯ } < 1012 ¯ ¯ >. The {1012 ¯ } < 1011 ¯ ¯ > system {1012 is the most commonly observed twinning mode in all HCP metals [6–8]. Yoo [9] plotted the twinning shear with respect to the c/a ratio of the major twinning modes in HCP metals. For γ o 3 , the slope of the twinning modes was shown to be negative but turns positive when γ 4 3 . Thus, the negative slope was as an indication that the twinning mode is c-axis extension. Hence, ¯ } < 1011 ¯ ¯ > twinning is most favorable when a tensile stress is {1012 applied along the c-axis of a Mg crystal. However, it is suppressed when a compressive stress is applied along the c-axis or a tensile stress is applied perpendicular to the c-axis. In dislocation-mediated plastic deformation, an important class of dislocations, i.e. geometrically necessary dislocations (GNDs), was first proposed by Nye [10]. The GNDs are generated when a strain gradient is present within a deformed material. For instance, in bending tests of crystalline metals, the slip planes for dislocations are curved by the deformation [10]. Thus, the curvature of the slip planes gives rise to a slip gradient on successive slip planes
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that can be compensated by GNDs [11]. Additionally, GNDs are distinguished from statistically stored dislocations (SSDs) that are created in the absence of a large strain gradient. Here, GNDs are associated with the organization of dislocations into low energy configurations, such as dislocation cells or walls that result in the development of a locally non-uniform strain field [12]. Closely related to GNDs are geometrically necessary boundaries (GNBs), including dense dislocation walls (DDWs) and microbands (MBs) that are necessitated by the accommodation of the lattice misorientation between the dislocation cells [13–17]. GNBs were proposed as a means to account for the flow stress anisotropy in deformed metals with medium to high stacking fault energies [18]. Using the GND concept in dislocation-dominated plastic deformation, Sevillano [19] first proposed the concept of “geometrically necessary twins” (GNTs). A GNT was treated as a “pseudodislocation line” at the twin edge with an effective Burgers vector in the direction of the twinning shear. The resultant hardening effect and strain accommodation by GNTs was also discussed [14]. Unfortunately, there has been a dearth of experimental evidence of GNTs since Sevillano's [19] initial proposition. Some preliminary experimental observations were reported by Li et al. [20]. In slipcontrolled deformation, dislocations interact as they intersect with each other. Dislocations also interact with secondary phases and interfaces such as grain boundaries and cell boundaries, resulting in changes in the original slip pattern via cross-slip, transmission, reflection or absorption at interfaces [21–23]. In contrast, twinning results from successive glide of twinning dislocations restricted at the twin/parent interfaces, i.e. twin boundaries (TBs) that should satisfy the invariant plane strain condition [24]. Thus, strain accommodation by shear-dominated twinning is different from dislocation slip. Alternatively, twinning is unidirectional [24] since the twinning shear is uniquely defined by the second invariant plane, i.e. the K2 plane, and the twinning direction η1. Because of these differences, GNTs that behave similarly to GNDs have received little attention in experiments. In this study, we present experimental evidence for GNTs in Mg AZ31-O, the most common of all wrought Mg alloys, deformed in quasi-static three point bending at RT. Bending was chosen, in part, because it is a significant deformation component in many practical applications in transportation industries [25]. The GNTs result from the macroscopic strain gradient produced by the bending load. We specifically characterized twins in the tension zone using electron backscatter diffraction (EBSD) and showed that they present the characteristic of being geometrically necessary.
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optical microscopy, the specimen was then mechanically re-polished followed by electrochemical polishing with an electrolyte composed of 160 mg sodium thiocyanate, 800 ml of ethanol, 80 ml of ethylene glycol monobutyl ether, and 20 ml of distilled water. Electron backscatter diffraction (EBSD) was then performed on the through-thickness cross-section using a Zeiss Supra 40 field emission gun scanning electron microscope (SEM) equipped with an EDAX Hikari EBSD detection system. The EBSD step size was 0.1 mm. When the scans were completed the data was rotated at 90° about the transverse direction (TD) using the EDAX analysis software (TSL OIM 6) so as to align it with the normal direction (ND) of the sheet specimen, which is the direction perpendicular to the tension and the compression surfaces of the bend specimen.
3. Results Fig. 1 shows optical microscopy of the Mg AZ31 specimen microstructure after three point bending. The image was taken on the through-thickness cross-sectional area. With the DIC illumination, two zones can be clearly distinguished, namely, a compression zone and a tension zone. In the compression zone, a very high density of deformation twins is observed. Away from the surface of the compression zone, twins are localized in bands. The tips of the bands extend near the center line, i.e. midway along the thickness of the specimen. Such localized twin bands in three point bending were first reported by Baird et al. [27]. The twin bands are aligned in two directions that are nearly perpendicular to each other. EBSD analysis showed that the twins are ¯ } < 1011 ¯ ¯ > extension twins [27]. Alternatively, no twins are {1012 observed in the tension zone of Fig. 1 at the selected magnification. After removing the DIC illumination, the twin bands present a different contrast than that in Fig. 1. As indicated by the long black arrows midway along the thickness of the specimen, the tips of the twin bands can still be seen but in weak contrast. However, another pattern of twins in the region at the vicinity of the tips of the twin bands is observed. Near the center of the field of view in Fig. 2a, where the deformation transitions from compression to tension, an arc of twins, which has a different contrast than the twin bands (indicated by the long black arrows), is observed
2. Experiments A twin-roll cast sheet of AZ31 Mg alloy [26] with thickness of 1.0 mm was cut into strips with dimensions 76 7 1 mm3. Quasi-static, three point bending was performed using an Instron 5882 testing system and an in-house three point bend fixture. The distance between the bending anvil pivots was 28 mm. The specimens were deformed at a rate of 10 mm/min and through a maximum bending angle of 132° and 30 mm displacement. The bend was parallel to the rolling direction (RD). One of the specimens was immediately cold mounted and the microstructure of the cross-section through the thickness was studied in detail using optical microscopy. This specimen was mechanically polished and then etched using an Acetic Picral solution containing 3 g picric acid, 100 ml ethanol, 10 ml distilled water, and 5 ml acetic acid. The specimen was then examined using a Zeiss Axiovert optical microscope (OM). To better reveal the deformation twins, differential interference contrast (DIC) illumination was used in imaging: this improves the contrast of the twins [27]. After
Fig. 1. Optical microscopy of localized twin bands in the compression zone of the sample after three point bending (the bending angle after springback is 132°) [20]. Differential interference contrast (DIC) illumination was used in imaging. Each band comprises a high density of extension twins. In the tension zone, where the stress disfavors extension twinning, no twins are observed at this magnification (100X). The black arrows point to the twin band tips.
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Fig. 2. Higher magnifications of the tension zone in Fig. 1 are shown [20]. (a) 500X. Near the tips of the localized twin bands (indicated by the long black arrows), arcs of twins were observed when the differential interference contrast (DIC) illumination was removed. Note that the twins in the arcs (indicated by the short black arrows) and those twins in the bands (indicated by the long black arrows) present different contrast, indicating that they may be different twin variants. (b) 500X. In the tension zone, arcs of twins follow the contour of the bent specimen. Outside the arcs, the twin density is much lower. (c) 800X. Deformation twins (indicated by the short black arrows) appear near the surface in tension with maximum tensile strain. The grain size is 5.0 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
between the red lines (indicated by the short black arrows). Lenticular twins are localized in an arc that follows the contour of the bent specimen. Away from the compression zone, which, as shown in Fig. 1 contains a high density of twins in the bands, a small arc of twins can be observed in the upper right region of the tension zone. Fig. 2b shows that close to the middle of the tension zone, a long arc of twins can be observed. This arc extends diagonally across the field of view, follows the shape of the bent specimen, and is but one of multiple arcs of twins that were captured. Fig. 2c shows the somewhat surprising result that twins (indicated by the short black arrows) appeared near the surface of the tension zone where the tensile strain is maximum in three point bending. Fig. 3a and b shows other arcs of twins at different locations inside the tension zone at higher magnifications. Because the grain size of the rolled sheet is rather small ( 5 mm), the thickness of the twin laths is only a few microns. Fig. 3a shows two neighboring arcs of twins. Fig. 3b shows another arc of twins in which more deformation twins (relative to Fig. 3a) are localized inside. All the arcs of twins follow the contour of the bent specimen. The arcs of twins in the tension zone were also observed in other Mg AZ31-O specimens after three point bending. The optical microscopy in Figs. 2 and 3 reveal arcs of twins in the tension zone with different image contrast than those twin bands in the compression zone, suggesting that the twins in the tension zone may be a different twin variant than those in the compression zone. But these images (Figs. 2 and 3) do not provide definitive crystallographic orientations of the twins in the tension ¯ } < 1011 ¯ ¯> zone in which the stress state strongly disfavors {1012 extension twins [27]. The morphology of the twins does not match ¯ } < 1012 ¯ ¯ > twins because neighboring twins the compression {1011 join together at their tips at the grain boundaries (Fig. 2a). Due to
Fig. 3. Other arcs of twins at higher magnifications inside the tension zone at different locations [20]. (a) 500X. Two neighboring arcs of twins, one between the red lines and another one between the yellow lines. (b) 800X. Another arc of twins between the red lines. All the arcs follow the contour of the bent specimen. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the repulsive force between twinning dislocations, neighboring compression twins should not join at their tips. However, it is very ¯ } < 1012 ¯ ¯ > compression twins. likely that the twins could be {1011 To identify the type of the twins in the tension zone, we performed EBSD on a bent AZ31 specimen. Because the rolled sheet was not annealed before bending, and the density of defects in the bent specimen was high, capture of twins with EBSD was very challenging. After electrochemical polishing, the arcs of twins were not observable in SEM, in other words, the arcs of twins could not be located on the well-polished surface before the EBSD scans started. To circumvent this difficulty and pinpoint the twins in the tension zone for EBSD scans, we first mechanically polished the through-thickness cross-section and then etched it to reveal the twins with optical microscopy. Then, micro-hardness indentations were performed near the arcs of twins. At each area of interest, four indentation marks forming a rectangular region were created with the twins being surrounded by the indentations. The indentation marks, each of which had a depth of 5.0 μm, served as reference points for the EBSD scans. Following indentation, the specimen was re-polished mechanically such that the indentation marks remained on the polished surface. EBSD scans were then run in the regions with the indents. After repeated scanning attempts, we captured twins in the tension zone with EBSD.
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the plotted lattices. The rotation axis is roughly parallel to the TD. One of the twinning planes was delineated by a pair of dashed lines within the hexagonal cell in P2. Fig. 5b plots the point-topoint misorientation angle between parent grain P2 and twin T2. The 88° misorientation angle again confirms that the twin is a ¯ } < 1012 ¯ ¯ > extension twin. Fig. 5c shows that the twinning {1012 process can be identified from the PF of the basal plane rotated by nearly 90° (indicated by the red arrow). Fig. 5d shows the PF of the ¯ } twinning planes. The two twinning planes in Fig. 5a were {1012 highlighted by the red circles in Fig. 5d. These two twinning planes share the same zone axis and hence both are possible twinning planes. Since the direction of the tensile stress is roughly parallel to the TD, we expect the Schmid factor of the twinning systems should be very low. Similar analysis for twin variants T1 and T3 is shown in Fig. 6a–d, where the PFs of the basal (0001) and the ¯ } twinning plane of grains P2 and P3 were plotted. The re{1012 orientation of the parent lattices by twinning is denoted by the red arrows (Fig. 6a and c). To calculate the Schmid factor of the parent grains (P1, P2 and P3 in Fig. 4b), we first recorded the Euler angles (ϕ1, Φ, ϕ2) of the individual parent grains from the IPF maps in Fig. 5a. Each of the three Euler angles corresponds to a coordinate transformation matrix, denoted as Tϕ , TΦ and Tϕ , respectively. The overall trans1 2 formation matrix from the lattice coordinate system to the specimen coordinate system (RD, TD, ND, where the ND is perpendicular to the figure plane of Fig. 5a) can be expressed as:
T = Tϕ2⋅TΦ⋅Tϕ1 Fig. 4. EBSD inverse pole figure (IPF) maps of the tension zone. As expected, a strong basal texture can be seen, typical of rolled Mg sheets [20]. (a) Twins were captured after repeated scanning efforts. (b) The pole figure of (a). (c) Magnified view of the region enclosed by the rectangle in (a). Lattices of the parent grains (P1, P2 and P3) and the corresponding twins (T1, T2 and T3) were plotted to show the orientation relationship as denoted by the hexagonal cells (highlighted view of the ̅ cells are shown in Fig. 5). Twin boundaries that satisfy 86.3 7 5° 1120 are highlighted with black lines.
Fig. 4a shows an inverse pole figure (IPF) map of a portion of the tension zone in Fig. 3b. The corresponding pole figures (PFs) are shown in Fig. 4b, which is typical of a strong basal (0001) texture of rolled Mg sheets. A few twin variants can be seen near the middle of Fig. 4a. The parent grains and the twins inside the rectangular box were cropped and magnified in Fig. 4c.
4. Analysis and discussion 4.1. The Schmid factor of the twinning system in the tension zone The capture of the twins in the tension zone with EBSD (Fig. 4) enables analysis of the type, orientation relationship and the Schmid factor of the twins. In Fig. 4b, three parent grains are marked as P1, P2 and P3, whereas their corresponding twins are marked as T1, T2 and T3. The grain boundaries that satisfy ¯ } < 1012 ¯ ¯ > twin orientation re¯ >, i.e. the {1011 86.3 75° <1120 lationship, are highlighted with black lines. Thus, the twins T1, T2 ¯ } < 1011 ¯ ¯ > extension twins, instead of and T3 are indeed {1012 ¯ } < 1012 ¯ ¯ > compression twins. To help identify the rotation {1011 axis of these extension twins, hexagonal cells of the parent grains and the twins are also plotted in Fig. 4b (highlighted views of the hexagonal cells are shown in Fig. 5). To further analyze the twins in the tension zone (see Figs. 2 and 3), parent grain P2 and twin T2 in Fig. 4b were cropped and ¯ } < 1011 ¯ ¯ > twinning reorients highlighted in Fig. 5a. Because {1012 ¯ >, the rotation axis the parent lattice by nearly 90° around <1120 can now be readily identified (see the green arrow in Fig. 5a) from
(1)
Applications of the transformation matrix T to the normal vector of the twinning plane (K1) and the twinning vector (η1) requires some preliminary steps. We first have to transform the Miller–Bravais indices of these vectors into the Miller indices, and then transform the Miller indices into orthogonal, cubic coordinate indices. This allows us to calculate the Schmid factor m = cos α⋅cos λ (α is the angle between the normal vector of the twinning plane and the loading direction, and λ the angle between the twinning vector and the loading direction). This transformation is denoted as B. Thus, any vector in the Miller–Bravais system a¼[uvtw] (u, v, t and w are the components in the four-index system), is transformed to a vector b ¼[xyz] in the specimen coordinate system by:
b = T ⋅B⋅a
(2)
The components of T can be computed in terms of the three Euler angles:
⎡ T ⎤ ⎡ cosϕ1 cosϕ2 − cosΦ sinϕ1 sinϕ2 ⎤ ⎥ ⎢ 11 ⎥ ⎢ ⎢ T12 ⎥ ⎢ cosϕ2 sinϕ1 + cosΦ cosϕ1 sinϕ2 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ T13 ⎥ ⎢ sinΦ sinϕ2 ⎢ T ⎥ ⎢ −sinϕ cosϕ − cosΦ sinϕ cosϕ ⎥ 21 2 1 1 2 ⎥ ⎥ ⎢ ⎢ ⎢ T22 ⎥ = ⎢ −sinϕ2 sinϕ1 + cosΦ cosϕ1 cosϕ2 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ T23 ⎥ ⎢ cosϕ2 sinΦ ⎥ ⎢ T31 ⎥ ⎢ sin sin ϕ Φ ⎥ ⎢ 1 ⎥ ⎢ T ⎥ ⎢ ⎢ 32 ⎥ −sinΦ cosϕ1 ⎥ ⎢T ⎥ ⎢ ⎣ 33 ⎦ ⎣ cosΦ ⎦
(3)
The transformation B can be computed as:
⎤ ⎡1 1 − 0⎥ ⎢ 2γ ⎥ ⎢γ ⎥ B=⎢ 3 ⎢0 0⎥ 2γ ⎥ ⎢ ⎢⎣ 0 0 1⎥⎦
(4)
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Fig. 5. Schmid factor and twin orientation relationship analysis of the deformation twins. (a) A magnified IPF map of the region containing parent P2 and twin T2 in Fig. 4b. ̅ can be readily identified from the lattices denoted by the hexagonal cells. The twinning plane is marked with dashed lines in the lower of the two The rotation axis 1120 ̅ } 101̅ 1̅ twins. (c) The PF of the hexagonal cells. (b) The misorientation angle of the twin is 88°, indicating that the twins in the tension zone are the unfavorable {1012 basal plane shows a near 90° rotation. The open circles represent the pole of the twinning plane in (a), and the red arrow represents the reorientation by twinning. (d) The PF of the twinning planes (the red circles) in grain P2 shows that the Schmid factor should be low. The tensile stress is approximately along the TD indicated in (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where γ is the c/a ratio (for Mg γ ¼ 1.624). Since the loading direction is known in the specimen coordinate system, the Schmid factor of any slip or twinning system can now be calculated. In three point bending, the stress state in the tension zone is more complex than uniaxial tension because the stresses are multi-axial. As a first-order approximation, we assumed the direction of the main component of the tensile stress is along the TD, and then calculated the Schmid factor of all the twinning systems (Figs. 5 and 6). Because there are six twin variants, i.e. three pairs of twin variants with each pair sharing a common zone axis, the calculated Schmid factors can be matched with the actual twins in the individual parent grains of P1, P2 and P3 in Fig. 4c. For all three parent grains, one of the twinning planes must have a very low Schmid factor because its pole is nearly perpendicular to the loading direction, i.e. the TD. Thus, the Schmid factors of the twin variant in the parent grains can be associated with the PFs in Figs. 5c, d and Figs. 6a to d. The results from calculations involving Eqs. (1–4) are listed in Table 1. As shown in Table 1, the Schmid factors of the twinning systems in the three parent grains have very low values, with the Schmid factor for grain P2 in Fig. 5a being closest to zero. Alternatively, Fig. 5c and the PFs in Fig. 6a and c show that the direction of the tensile stress is almost perpendicular to the c-axis [0001] for grain P2 and P3. This stress state strongly disfavors ¯ } < 1011 ¯ ¯ > twinning; however, it favors {1011 ¯ } < 1012 ¯ ¯> {1012 compression twinning [9]. The same results were noted in
analyses of additional bending tests of the AZ31 Mg sheet material. These facts raise some compelling questions: how and why does ¯ } < 1011 ¯ ¯ > extension twinning occur in the tension zone of {1012 the three point bending specimen where the Schmid factor of the twinning systems is close to zero and the tensile stress is nearly perpendicular to the c-axis? In the following, we provide an ana¯ } < 1011 ¯ ¯ > twinning. lysis of the unique nature of {1012
¯ } < 1011 ¯ ¯ > twinning in the 4.2. Strain accommodation by {1012 tension zone ¯ } < 1011 ¯ ¯ > twinning was first The non-Schmid behavior of {1012 reported by Barnett et al. [28]. It was found that the secondary ¯ } < 1011 ¯ ¯ > twins of {1011 ¯ } − {1012 ¯ } double twinning sig{1012 nificantly deviate from the Schmid-type behavior which governs shear-dominated deformation. Such behavior was attributed to the closest match between the primary and the secondary twinning plane so that the compatibility strain is minimized. In this study, ¯ } < 1011 ¯ ¯ > twins in the tension zone are not the secthe {1012 ondary twins in double twinning; rather, they are the primary twins. To understand the mechanism of the deformation twins observed in the tension zone during bending of the AZ31 Mg sheet, it is necessary to analyze the strain components that are ¯ } < 1011 ¯ ¯ > twinning. produced by the {1012 ¯ ¯ ¯ > twinning has been exThe mechanism of {1012} < 1011 tensively discussed in the literature [29–36]. According to the classical twinning theory, this twinning mode in Mg should have
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̅ } twinning planes of grain P1 (see Fig. 4). The rotations of the poles represent the Fig. 6. (a) The pole figures (PF) of the (0001) basal and (b) the corresponding {1012 ̅ } twinning planes of grain P3 are twinning process. The twinning planes are highlighted with the open red circles. (c) The PFs of (0001) basal and (d) corresponding {1012 shown. The calculated Schmid factors (see Table 1 in the text) of the twinning planes can be associated with these PFs. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Schmid factor of the possible twin variants in Fig. 4c. Euler angles (deg.)
Schmid factor
P1 (24.9, 155.3, 335.8) P2 (336.3, 173.2, 314.5) P3 (318.4, 7.3, 171.1)
0.008–0.025 0.005–0.007 0.049–0.055
an elementary twinning dislocation with a Burgers vector of 0.024 nm, thus generating a 12.9% twinning shear [9,37,38]. Christian and Mahajan suggested that the twinning dislocation should be a twin-layer zonal dislocation [38]. Serra et al. [30,32,34] proposed a disconnection model to describe the twinning dislocations on the twin boundaries [29,33,35]. Other types of twinning dislocations were proposed by Wang et al. [31]. Li and Ma [29] showed that the maximum magnitude of the shuffles that are required for this twinning mode is about four times that of the theoretical elementary twinning dislocation. Thus, this is a shuffling dominated twinning. Indeed, the most recent in-situ transmission electron microscopy (TEM) observations of ¯ } < 1011 ¯ ¯ > twinning and detwinning in single crystal Mg {1012 showed that in both twinning and detwinning, no measurable shear strain was produced on the single crystal specimen [39–41], in sharp contrast to shear-dominated twinning in other single crystal metals [42]. Therefore, the accommodation of the shear strain by the twin arcs should not play a dominant role even in Mg
alloys such as Mg AZ31. Fig. 7 shows the crystallographic relationship between the parent lattice (Mg atoms are denoted by blue filled spheres) and the twin lattice (Mg atoms are denoted by red filled spheres). This analysis is based on the lattice correspondence in deformation twinning, i.e. a plane/direction of the parent lattice is linearly mapped to a plane/direction of the twin by a homogeneous shear ¯ } < 1011 ¯ ¯ > twinning, the basal plane of the parent is [43]. For {1012 mapped to the prismatic plane of the twin, and the prismatic plane is mapped to the basal plane of the twin. If we interpenetrate the parent and the twin lattices, as shown in Fig. 7a, such lattice correspondence can be clearly resolved. After twin¯ > ning, the parent lattice is reoriented by 90° around the <1120 axis. The theoretical misorientation angle is 86.3° for Mg, but TEM measurements have revealed that the misorientation angle for this particular twinning mode does not have a unique value and varies from 84° to 97° [44]. This was attributed to the breakdown of the invariant plane strain condition that is required for deformation twinning [45]. Irrespective of the misorientation angle, however, the lattice correspondence remains the same and the analysis in Fig. 7a and b remains accurate. ¯ } < 1011 ¯ ¯ > twinning mode arises It is now clear that the {1012 from the conversion of the parent basal plane to the twin prism plane, and the parent prism plane to the twin basal plane. In Fig. 7a, the basal plane of the parent lattice (blue atoms) is parallel to the plane of the page and the viewing direction is along the caxis of the parent (perpendicular to the plane of the page). After
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̅ } 101̅ 1̅ twinning. The parent lattice (in blue) and the twin lattice (in red) interpenetrate, with a 1120 ̅ Fig. 7. Analysis of the strain components generated by {1012 rotation axis, i.e. the horizontal axis where the two lattices coincide. (a) Lattice correspondence between the parent and the twin lattice when the viewing direction is [0001] of the parent. The double-layered prismatic plane of the parent must convert to the single-layered basal plane of the twin. After twinning, a contraction strain is generated in ̅ of the parent, an extension strain is generated along the c-axis of the parent. The yellow the vertical direction due to the misfit (Δd). (b) When the viewing direction is 1120 arrows denote the atomic shuffles required to accomplish the basal-to-prism and prism-to-basal conversions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
twinning, the basal plane (single-layered and perpendicular to the plane of the page) of the twin lattice (red atoms) becomes parallel to the prismatic plane of the parent which is double-layered. Hence, to convert the prismatic plane of the parent to the basal plane of the twin, the parent atoms on the prismatic plane (double-layered) must shuffle and adjust their positions to arrive at the basal plane of the twin (single-layered) [33] (the shuffles are denoted by the yellow arrows). This conversion creates a misfit strain due to the dimensional difference Δd ( =c − 3 a , where a and c ¯ > direction, i.e. the vertical are lattice parameters) in the <1100 direction of the parent lattice. This misfit strain equals γ− 3 3
≈ − 6.24%. Hence, a negative compressive strain is generated
by twinning in the direction perpendicular to the c-axis of the parent. In the direction of the c-axis of the parent, however, a positive tensile strain is generated, as shown in Fig. 7b. In this plot, the ¯ >, and the basal plane of the parent is viewing direction is <1120 parallel to the prismatic plane of the twin. Similarly, the parent atoms on the basal plane (single-layered) must shuffle and adjust their positions to arrive at the prismatic plane of the twin (doublelayered; the shuffles are denoted by the yellow arrows). As such, a misfit strain, which equals
3 −γ γ
≈ 6.65% , is generated. This is a
positive tensile strain along the c-axis of the parent lattice. Thus, ¯ } < 1011 ¯ ¯ > twinning is most favorable if a when γ < 3 , {1012 tensile load is applied along the c-axis of the grains. This analysis also explains how “tension twinning” got its name [9]. The arcs of twins in the tension zone of the three point bending specimen (see Figs. 1–3) can now be accounted for by the misfit strains generated by the mechanism shown in Fig. 7a and b. For those twins in the tension zone, the major component of the stress is a tensile stress. This stress component introduces a compressive strain component in the direction along the normal of the sheet specimen (see “ND” in Fig. 1). Without the extension twins, the strain would have to be accommodated mainly by non-basal dislocations with a limited number of independent slip systems and ¯ } < 1012 ¯ ¯ > compression twinning high critical stresses. The {1011 may be activated, but experiments suggest that this type of twinning has very low volume fraction due to the high critical ¯ } < 1011 ¯ ¯ > extension stress as well [9]. The presence of the {1012 twins, however, provides an extra mechanism for strain
accommodation. The stress state in three point bending strongly disfavors the extension twinning, i.e. as evidenced by the near zero Schmid factor (see Table 1) and a tensile stress perpendicular to the c-axis. However, the high local strain generated in three point bending at large bending angles ( 132° in this study) is able to activate the extension twinning, such that the compressive strain component can be effectively accommodated by the contraction strain (see Fig. 7a and b) generated by the extension twinning in the direction perpendicular to the c-axis of the parent grains. From ¯ } < 1011 ¯ ¯ > twinning mode is strainthis standpoint, the {1012 controlled rather than stress-controlled. Hence, these stress-unfavorable twins can be characterized as GNTs as originally proposed by Sevillano [19], despite the fact that the actual stress state in the tension zone of a three-point bending specimen is multiaxial and complex. Note that the extension strain component (Fig. 7b) generated by the extension twinning along the c-axis has to be accommodated by dislocation mechanisms near the twin boundaries. This may lead to high local stresses at the twin boundaries. The resultant twin-slip interactions in the tension zone may affect the deformation behavior and sheet forming. Whether the presence of the extension twins in the tension zone during sheet bending facilitates crack nucleation and propagation, or actually suppresses crack nucleation and improves the bendability of the AZ31 Mg alloy is worth of further studies. In our three point bending tests of the same Mg AZ31 material, when we bent the specimens to larger bending angles until the specimens fractured, cracks were always observed near the surface with the maximum tensile strain, but no cracks were observed in the compression zone where high density twins are present. In rolled Mg sheets with a strong basal texture, dislocation slip, especially the non-basal slip, dominates the plastic deformation in the tension zone. As the tensile strain builds up in the tension zone, the activation of the GNTs may be able to locally relax the strain. From this standpoint, the GNTs could delay the nucleation of cracks in the tension zone, suggesting the possibility of designing Mg alloys wherein GNT formation is increased. Therefore, it will be of great interest to investigate how the GNTs affect Mg sheet forming at RT. The volume fraction of the GNTs will increase with increasing bending angle, and hence increasing strain. Thus, further investigation is needed to establish a quantitative connection between the density of twins in the arcs to the
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local strain gradient in three point bending.
5. Conclusions We performed three point bending tests on a twin-roll cast Mg AZ31 sheet alloy. Microstructure examinations revealed high density, localized twin bands in the compression zone where the ¯ } < 1011 ¯ ¯ > twinning in this highly stress state strongly favors {1012 textured material. However, in the tension zone, where the stress ¯ } < 1011 ¯ ¯ > twinning, arcs of state strongly disfavors {1012 ¯ } < 1011 ¯ ¯ > twins were also observed. These twins follow the {1012 contour of the bent specimen. EBSD analysis on the misorientation angle and the rotation axis confirmed that the twins in the tension zone are indeed ¯ } < 1011 ¯ ¯ > extension twins, not {1011 ¯ } < 1012 ¯ ¯ > compression {1012 twins, although the stress state strongly favors compression twinning. Schmid factor calculations show that these twins have very low Schmid factor values, one in particular being very close to zero, and the main tensile stress component is nearly perpendicular to the c-axis of the individual grains. ¯ } < 1011 ¯ ¯ > extension The presence of the unfavorable {1012 twins can be understood from the strain components produced by this twinning mode. Due to the misfit strains after reorientation of ¯ > axis, an extension the parent lattice by 90° around the <1120 strain is generated along the c-axis of the parent grain; however, a contraction strain is also generated in the direction perpendicular to the c-axis of the parent grain. In the tension zone of the bent specimen, the compressive strain along the normal direction of the ¯ } < 1011 ¯ ¯ > twins, such that the sheet specimen activates the {1012 compressive strain is accommodated by the contraction strain component generated by the twinning. Hence, the stress-un¯ } < 1011 ¯ ¯ > twins have the characteristic of geofavorable {1012 metrically necessary twins, similar to GNDs and GNBs that have been extensively observed in cubic metals.
Acknowledgments Z. McClelland, B. Li, S.J. Horstemeyer, S. Brauer, A.A. Adedoyin and M.F. Horstemeyer gratefully appreciate the support from Center for Advanced Vehicular Systems, Mississippi State University.
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