Online publication date: 12 May 2010

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1 This article was downloaded by: [Los Alamos National Laboratory] On: 15 June 2010 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Philosophical Magazine Publication details, including instructions for authors and subscription information: Statistical analyses of deformation twinning in magnesium I. J. Beyerlein a ; L. Capolungo b ; P. E. Marshall b ; R. J. McCabe b ; C. N. Tomé b a Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Online publication date: 12 May 2010 To cite this Article Beyerlein, I. J., Capolungo, L., Marshall, P. E., McCabe, R. J. and Tomé, C. N.(2010) 'Statistical analyses of deformation twinning in magnesium', Philosophical Magazine, 90: 16, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Philosophical Magazine Vol. 90, No. 16, 28 May 2010, Statistical analyses of deformation twinning in magnesium I.J. Beyerlein a *, L. Capolungo b, P.E. Marshall b, R.J. McCabe b and C.N. Tome b a Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA; b Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Received 8 November 2009; final version received 17 January 2010) To extract quantitative and meaningful relationships between material microstructure and deformation twinning in magnesium, we conduct a statistical analysis on large data sets generated by electron backscattering diffraction (EBSD). The analyses show that not all grains of similar orientation and grain size form twins, and twinning does not occur exclusively in grains with high twin Schmid factors or in the relatively large grains of the sample. The number of twins per twinned grain increases with grain area, but twin thickness and the fraction of grains with at least one visible twin are independent of grain area. On the other hand, an analysis of twin pairs joined at a boundary indicates that grain boundary misorientation angle strongly influences twin nucleation and growth. These results question the use of deterministic rules for twin nucleation and Hall Petch laws for size effects on twinning. Instead, they encourage an examination of the defect structures of grain boundaries and their role in twin nucleation and growth. Keywords: statistics; hcp; grain size; twinning; magnesium alloys; polycrystalline metals 1. Introduction In polycrystalline hexagonal close packed (hcp) metals, the plastic deformation mechanisms are dislocation slip and deformation twinning. Many important characteristics of material stress strain response, such as plastic anisotropy, flow stress, hardening, and failure, are controlled by the relative contributions of slip and twinning as well as the interactions between these two mechanisms. It is well recognized that these two mechanisms are very different in nature and also believed that the differences arise primarily from the behavior of their respective dislocations [1 9]. Slip is a thermally activated process and, thus, a function of temperature and strain rate. To model plastic slip in an hcp metal, one can reasonably assign to each slip mode (e.g. basal, prismatic, pyramidal) a critical resolved shear stress (CRSS) that evolves in its own characteristic manner with strain, temperature and strain rate [10]. Deformation twinning, on the other hand, *Corresponding author. irene@lanl.gov ISSN print/issn online This material is published by permission of the Office of Basic Energy Sciences under Contract No. W-7405-ENG-36. The US Government retains for itself, and others acting on its behalf, a paid-up, non-exclusive, and irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. DOI: /

3 2162 I.J. Beyerlein et al. is less sensitive to temperature and rate effects [10 12] and the existence of a CRSS that activates twinning is sometimes not obvious from experimental evidence. Twinning has been observed to be sensitive to microstructure. Twins most often originate at grain boundaries, pile-ups, defects, slip bands, twin interfaces and cracks [13 19]. It is also recognized, as well as consistent with experimental evidence, that twinning occurs in grains whose orientation favors a high resolved shear stress on the twin system [20,21]. One of the more frequently debated microstructural effects, reported for different crystal structures, is the grain size (Hall Petch) effect on twinning [12,13]. In hcp metals, grain size effects on twinning have been studied in Zn [18,22], Zr [23 25], Ti [26], Mg [27 31], and severely deformed Mg AZ31 [32]. Many studies suggest that the grain size effect on twinning is much stronger than on slip (for a review, see [12]). The origin of the size effect is unknown and a few interpretations have been put forth. Hull [13] loosely interpreted the Hall Petch coefficient as a stress concentration factor for twin nucleation. For Mg-AZ31 Barnett [27] argued that the Hall Petch coefficient corresponds to the number of twins per unit grain boundary area. Ecob and Ralph [18] correlated twin nucleation with slip band formation and dislocation density in the neighboring grain. Hosford [33] explained the grain size effect through an energy balance between the new surfaces introduced by the twin boundaries and the mechanical work done by the twin shear. Another important characteristic of deformation twinning that distinguishes it from plastic slip is the substantial statistical variation in the incidence, morphology, numbers, spatial distribution and variant selection of twins. Such dispersion has been quantified with respect to grain size in silicon-iron by Hull [13], orientation factor in pure Zr by Reed-Hill [21], and twin thickness in silicon-iron by Priestner and Leslie [14]. Furthermore, there is significant scatter in measured CRSS values among repeated tests [7,14,34 36]. Although twins have been reported to form in samples with large grains [18,23 26], from grain boundaries [25], or in grains that are preferentially oriented [21,37], they do not necessarily occur in all large grains, at all grain boundaries, or in all grains in the same favorable orientation. Extracting meaningful and quantitative conclusions regarding relationships between microstructure and twinning from the variability in metallographic data requires a statistical analysis. Moreover, such relationships are likely to be different for twin nucleation than for twin propagation and so it is important to separate these two processes in the analysis. In the present work, we carry out a systematic statistical analysis of metallographic data on {10 12} h1011i twinning in high purity Mg. The goal is to determine the important grain-scale microstructural characteristics (such as grain size) that impact twin nucleation and growth. Collection of statistically significant sets of data on individual twins is made possible by an automated electron backscattering diffraction (EBSD) technique [38]. In analyzing these data, we investigate the relationships between the presence, thickness, orientation and number of twins as a function of grain size (area and diameter), crystal orientation, length of the grain boundary between triple junctions and neighbor misorientation angle. In conducting these analyses, we attempt to separate twin nucleation from twin growth. In spite of substantial statistical variation, many important and interesting relationships between twinning and grain size, grain orientation and grain boundary

4 Philosophical Magazine 2163 Table 1. Chemical specification of the high-purity magnesium (weight ppm) used in this study. Metal Al Ca Mn Zr Zn Sn Si Pb ppm misorientation can be extracted, some of which are shown to conflict with conventional thought. Furthermore, some microstructural features that influence twin nucleation are found not to influence twin growth and some features affecting growth are found not to affect nucleation. 2. Material details and testing 2.1. Starting texture and deformation mechanisms The material used in this study is 99.96% Mg commercially available hot-rolled plate. The material compositional specification is given in Table 1. Compression test specimens with a rectangular geometry (7 8 9 mm) were machined from the mid-thickness of the plate. These were heat-treated at 200 C for 30 min in vacuo, resulting in equi-axed grains with a large grain size distribution. At room temperature, the flow stress and hardening rate of Mg are controlled by both slip and twinning. By far, the easiest slip mode is basal slip, 1/3[1120](0001). Prismatic slip, 1/3[1120]{10 10}, and pyramidal or hc þ ai slip, 1/3h112 3i{11 22}, are secondary slip systems. Compared to the non-basal slip systems, {1012} twinning is easy to activate for suitably oriented crystals [20,39 41]. The texture of the sample before compression was measured by neutron diffraction and is displayed in Figure 1 as (0001) and (1010) pole figures in the 3-plane, where the 3-axis is parallel to the through-thickness direction of the plate (TT). These indicate a strong non-axisymmetric initial texture, in which the basal poles are tilted away from the TT-direction, toward the RD-direction. Compression tests were performed at room temperature at a strain rate of 10 3 /s in the in-plane transverse direction (TD). Figure 2 shows the corresponding true stress true strain curve up to 13% [42]. The first 7 8% strain interval, where the hardening rate increases with strain, is dominated by twin nucleation and growth. For the material studied here, the sample was unloaded at 3% strain, where the macroscopic flow stress was approximately 50 MPa and twin formation is still in a relatively early stage. In this work, geometric Schmid factors m are calculated with respect to the sense and direction of the macroscopic loading, using the orientation of the glide plane and slip direction in the case of slip, or the twin plane (K 1 ) and shear direction ( 1 ) in the case of twinning. Although the stress state in the grain is likely to deviate from the macroscopic stress, it is not possible to know the local stress states generated before unloading and sectioning the sample. For each grain, the six possible twin variants v (i) are classified in order of decreasing Schmid factor, m (i), i ¼ 1,..., 6: variant v (1) has the highest Schmid factor m (1), variant v (2) the second highest m (2), and the sixth variant v (6) the lowest m (6). Due to the symmetry of the hcp crystal structure, variants

5 2164 I.J. Beyerlein et al. Figure 1. Basal (0002) and prism (10 10) pole figures of the starting texture measured by neutron diffraction of the high purity Mg specimen analyzed in this work. The through-thickness (TT) axis is normal to the pole figure. Figure 2. True stress true strain response of high purity Mg. The specimen used for this study was unloaded after 3% true compressive strain (unpublished data courtesy of Mr. Andrew Oppedal). v (3) and v (4) cannot have Schmid factors greater than and variants v (5) and v (6) cannot have Schmid factors greater than for a compressive loading state. Combining a Schmid factor analysis with estimates of the critical stresses for slip and twinning indicates a priori which mechanisms are likely to be activated under the applied loading condition. Single crystal constitutive modeling of the compression response of pure Mg single crystals under various loading directions [20] estimates that the initial CRSS values for basal slip and {1012} twinning are three to ten times less than that for prismatic slip. For the population studied here, approximately 52% of the grains are oriented with their highest twin Schmid factor m twin larger than the highest Schmid factor for basal slip m basal. While a smaller fraction (11%) is oriented with their highest twin Schmid factor larger than that for prismatic slip m prism, the difference between m prism and m twin is less than 0.1 in most of the grains, which is too small to compensate for the significantly higher CRSS for prismatic slip compared to that for {1012} twinning. In summary, considering the initial texture, the ease at which {1012} twinning is activated, and the fact that {1012} twinning can

6 Philosophical Magazine 2165 Figure 3. EBSD scans of Mg strained to 3% at room temperature in the in-plane direction (TD-direction in Figure 1). The TD direction lies normal to the plane of the scan. TT-direction orientations are shown. The black arrow indicates one of many twin crossings within this scanned area. accommodate extension along the c-axis, while basal and prismatic slip cannot, copious amounts of {10 12} twinning should be activated under the current loading state, TD-compression EBSD analysis of deformed microstructure The compression specimen after 3% strain was sectioned on a plane normal to the compression axis roughly in the center of the sample. It was prepared for EBSD by mechanically grinding and polishing to 1 mm diamond in propylene glycol and then chemically polishing in 10% nitric and water for 10 s. The EBSD data was collected in an FEI XL30 FEG-SEM at 20 kv using a 1-mm step size. Forty-two distinct mm scans were collected from different locations on the sample. The EBSD scans were then analyzed using an automated twin characterization method [38]. After 3% strain, the total area twin fraction of {1012} twins A tw is 42%. We did not observe any other twin types and secondary twinning was negligible. Through-thickness (TT) direction orientation maps of six EBSD scans of the deformed sample sectioned normal to the compression direction (TD-direction) after 3% strain are displayed in Figure 3. As shown, the grain microstructure, occurrence of twinning and twin morphologies vary non-homogeneously across the sample. Table 2 lists the EBSD data sets and population sizes used in the present statistical analysis: the total number of grains, N g ; twinned grains, N tg ; twins, N tw ; and grain boundaries, N 1 gb. The upper case N signifies quantities calculated by summing over all EBSD scans. A twinned grain is defined as a grain that contained at least one visible twin. A grain boundary is considered crossed if there is at least one adjoining twin pair; that is, two twins belonging to different grains and meeting at the boundary at the same location within some tolerance.

7 2166 I.J. Beyerlein et al. Table 2. Statistical information obtained from EBSD scans of deformed Mg at 3% strain. Twins refer to the {10 12} type. Number of grains N g 2339 Number of twinned grains N tg 1534 Number of twins N tw 8550 Number of grain boundaries N gb 6343 Number of crossed grain boundaries 1160 (8%) a Note: a The 8% is based on the total population of grain boundaries used for determining crossed grain boundaries statistics, which was larger than N gb, the population used for the other grain boundary statistics. Several other quantities are calculated for each grain, twin, and grain boundary, and are identified using lower case letters. For each grain, the number of twins (n g ) it contains is determined. For each twin, its twin variant v (i) among the six {10 12} twin variants in hexagonal metals is determined. Also, the thickness of each twin is calculated. As shown in Figure 3, most {10 12} twins are irregular in shape and therefore the measured twin thickness w is defined as the minor axis of an ellipse fitted to each twin. To calculate the true twin thickness (w t ), the measured w from the two-dimensional (2D) scan is corrected by multiplying by the cosine of the angle between the twin plane and the normal to the sample [38]. Each grain boundary is characterized by its length (l gb ), number of twins intersecting it (n tgb ) from one grain (one side), the thickness of the twin that intersects the boundary (w), and the misorientation angle across it (). These quantities will be defined more precisely later Grain area and grain boundary length distribution For each grain, its area a g, which includes any twinned area, is measured. Grains less than mm 2 in area (25 points) could not be confidently defined as grains using the EBSD scan data and were excluded from the analysis. Figure 4a shows the distribution of grain areas for this sample for the range mm 2. Based on the apparent exponential character of the grain area distribution, common Gaussian-based mean and variance estimates do not apply. Alternatively, the distribution is defined by an exponential distribution, which has the following cumulative distribution function: Pða g 5 AÞ ¼1 expð AÞ, where P(a g 5A) is the probability that the area a g is less than or equal to A and the parameter is mm 2. The corresponding mean (1/) and variance (1/ 2 )is 2941 mm 2 and mm 4, respectively. An effective diameter d g can be estimated for each grain by associating its area with a circular cross-section a g ¼ dg 2 /4. The equivalent diameter (d g ) distribution derived from the measured a g is shown in Figure 4b. In a 2D scan, the perimeter of each grain ( p g ) is the sum of a number of grain boundary segments, the trace of the grain boundary between two adjacent triple points. In an inhomogeneous grain microstructure such as this, the lengths of the grain boundary segments bounding a single grain will vary. Two grains with the

8 Philosophical Magazine 2167 (a) (b) Figure 4. (a) Grain area and (b) circle equivalent grain diameter distribution across the sample. Grain areas smaller than mm 2 (corresponding to diameter smaller than 5.25 mm) were discarded. The first column in the grain area plot represents grains between and 250 mm 2 and the first column in the diameter plots diameters between 5.25 and 8 mm. same p g will not necessarily be bounded by the same number of segments with the same lengths. Likewise, the lengths of these segments l gb will vary across the microstructure. Figure 5 shows the frequency distribution of all l gb in the microstructure. Similar to grain area, we find that the distribution of l gb cannot be analyzed using common Gaussian statistics. The shape of this distribution in Figure 5 indicates that it can also be characterized by an exponential distribution: Pðl gb 5 LÞ ¼1 expð LÞ, ð2þ where P(l gb 5L) is the probability that the length l gb is less than or equal to L. For l gb, the exponential parameter is ¼ mm 1 and its mean is 1/ ¼ 28.6 mm.

9 2168 I.J. Beyerlein et al. Figure 5. Distribution of grain boundary lengths across the sample. The criterion of discarding grains smaller than mm 2 affects the minimum possible observed boundary length. It is understood that the l gb and a g measured in a 2D scan correspond to sections of 2D grain boundary facets and three-dimensional (3D) grains, respectively. As a consequence, a segment l gb is a chord taken at random on the grain boundary plane formed between two adjoining grains and an area a g is a random cross-section of the grain. Because the grains were equi-axed and the sample size significant, the trends found in the following analysis are expected to be similar in 3D [43]. 3. Results The present analysis aims to examine deformation twinning from a statistical viewpoint, utilizing large, statistically relevant data sets that can be readily obtained within a reasonable time period from EBSD and in doing so, attempts to extract separate relationships for twin nucleation and twin growth. Characterization of the initial stages of twin nucleation and propagation requires transmission electron microscopy (TEM) or high-resolution TEM (HRTEM) and is beyond the scope of the EBSD work presented here. Recent TEM studies [15,44,45] have observed {1012} twins in hcp metals, several tens of nanometers in width. Classical TEM work on Zn crystals by Price [15] has shown that after embryo formation, the twin tip advances inside the grain, propagates the twin lamella and creates two twin boundaries. Once the twin has made contact with the opposing grain boundary, the two twin boundaries separate, thereby thickening the twin. Here, each observed twin lamella is associated with a successful nucleation event, one that has led to an observable amount of propagation following nucleation, and hence, a contribution to plastic deformation. Unsuccessful nucleation events, involving twins that nucleate but do not propagate or propagate a little, cannot be detected with the resolution of EBSD, and do not contribute to overall plastic deformation. Regarding twin growth, the present 2D EBSD sections show that most twins

10 Philosophical Magazine 2169 Figure 6. Gray bars: fraction of grains in which at least one twin can be identified plotted as a function of the highest twin Schmid factor m (1) in the grain. White bars: fraction of grains versus their highest twin Schmid factor m (1). traversed the entire grain and only very few terminated within the grain interior. As a consequence, the processes involved in twin propagation cannot be distinguished from those involved in twin thickening. Rather, we refer to twin growth in a general sense as including all those processes that follow nucleation and are involved in the expansion of the twin domain. In summary, in the following EBSD analysis, observance of at least one micron-scale twin in a grain will be interpreted as an occurrence of twin nucleation. Corresponding nucleation-related statistics will be calculated with respect to the entire sample set of grains N g. Twin growth, on the other hand, will be related to twin thickness (either its measured w or corrected true value w t ) and the corresponding growth-related statistics will consider only those grains containing twins (N tg ) or grain boundaries connected to twins (N tgb.) Crystallographic grain orientation effects In this section, crystallographic effects on twinning are analyzed in two ways. The first considers twinning with respect to m (1), the highest twin Schmid factor of the grain. The factor m (1) is used solely as a measure of the suitability of the grain crystallographic orientation for twinning. Two grains with the same m (1), however, do not necessarily have the same orientation. The second is an analysis of the Schmid factor of each individual twin lamella m to determine which variant was activated. Figure 6 plots the fraction of grains (gray bars) in which at least one twin could be identified as a function of m (1) of the grain. 2 For comparison, the fraction of all grains (white bars) for a given m (1) is also included. As shown, the analysis finds that the incidence of twinning increases with increasing m (1) or, conversely, the frequency of untwinned grains in which no twins formed (difference between the two bin

11 2170 I.J. Beyerlein et al. heights) increases as m (1) decreases. Two other important conclusions can be reached from Figure 6, which were less anticipated. First, twins appeared in grains over a wide range of m (1), not only in those grains oriented most favorably for twinning (m (1) 0.375). Second, in the range 0.5 m (1) 0.375, we find that the fraction of untwinned grains is non-zero. Most (90%), but not all favorably oriented grains, twinned. Therefore, twins cannot be expected to nucleate in all grains favorably oriented for twinning or to never nucleate in those that are less favorably oriented. Note that this analysis does not mean that the twin variants formed in the twinned grains were v (1). Basal slip is the easiest slip mode in Mg and it is possible that how well a grain is oriented for basal slip can affect the likelihood of {1012} twinning. The analysis in Figure 7, which locates the (a) twinned and (b) untwinned grains on a m basal versus map, demonstrates that this is not strictly true. While most of the twinned m twin grains lie in the range m basal 5m twin, where they are more favorably oriented for twinning over basal slip, the preference is not strong. As shown, twinned grains can be found over the entire range where deformation twinning is permitted, even in the region where m basal 4m twin 40. Only in the neighborhood where m twin is very close to zero and m basal is at its maximum of 0.5 are grains less likely to twin in comparison with basal slip. Figure 8 examines the distribution of Schmid factors on all N tw observed twins and shows an increasingly larger number of twins with higher Schmid factors. Although the trend that more twins form with higher Schmid factors is expected, it is notable that the number of twins with Schmid factor less than is still significant. The apparent scatter in m suggests that while grain orientation is important, other factors are at play. In addition, the analysis in Figure 8 does not reveal the variant associated with the observed twin. The relative frequency of each variant is analyzed in Figure 9, which shows the number fraction of each twin variant v (i) observed (relative to the total number of twins) as a function of the Schmid factor of the twin. As expected, twins with m are exclusively v (1) or v (2) but, interestingly, the variant observed over the entire range is not always v (1). In the upper range m40.375, the frequency of the second variant v (2) is significant. In the interval m50.375, twins of even lower rank variants form. For bins spanning up to m ¼ 0.35, v (3) and v (4) occur more often than v (1). It is evident from Figure 9 that variability in twin variant selection is substantial. In interpreting results in Figure 9, a few things must be kept in mind. First, some grains contain more than one set of twins and thus a lower rank variant may have been nucleated in addition to a higher ranked variant in a single grain. All twins are included in the analysis. Second, differences among m (i), i ¼ 1,..., 6, decreases as m (1) decreases below In this range, the preference for v (1) over the other variants is reduced, and local stress deviations may have an influence comparable to that of the macroscopic stress state. The fractions of each variant occurring over the entire population of twins (N tw ) are summarized in Table 3. The highest twin variant v (1) occurs most often. Even so, the probabilities of forming v (2) and v (3) (instead of v (1) in most cases) are still significant. This result is unexpected since, by definition, v (1) has a higher geometric Schmid factor than v (2) and v (3).

12 Philosophical Magazine 2171 (a) (b) Figure 7. Value of the maximum basal slip Schmid factor m basal plotted versus the highest twin Schmid factor m twin for (a) twinned and (b) untwinned grains measured in the EBSD scans. The solid line represents m twin ¼ m basal. The grains were separated into two populations: (a) the ones with at least one twin, and (b) the ones without twins. Observe that not all grains favorably oriented for twinning did twin, and many grains non-favorably oriented for twinning did twin. To investigate orientation effects on twin growth, in Figure 10 we plot for each twin its true thickness w t versus its Schmid factor. The w t is found to increase with Schmid factor, indicating that higher resolved shear stresses favor twin growth. The assumption that twin growth is governed by resolved shear stresses based on long-range stress fields appears to be well suited for Mg, at least for those twins that have nucleated and are well developed. The same result was found in the case of Zr [24], where we speculated that if back stresses induced by the neighboring grain in reaction to the localized twin shear are independent of orientation, then a higher

13 f 2172 I.J. Beyerlein et al. Figure 8. Fraction of twins within a given twin Schmid factor interval. Figure 9. Fraction of each twin variant v (i) (i ¼ 1,..., 6) relative to the total number of twins in the sample plotted versus their Schmid factor. Table 3. Fraction of {10 12} twins of each variant found in the 3% strained samples analyzed here. Strain level v (1) v (2) v (3) v (4) v (5) v (6) 3%

14 Philosophical Magazine 2173 Figure 10. True thickness of individual twins as a function of their Schmid factor. resolved shear stress should help to overcome this reaction shear and give a growth advantage to those twins with a higher Schmid factor. Other quantities that may be considered indicators of twin growth are the twinned area per grain and twin fraction per grain. These measures are properties of the grain rather than the individual twin. They depend on the number of twins in the grain (n g ), the true thickness of each (w c ), and the grain area a g (or diameter d g ). These properties w t and n g are examined next as a function of grain area a g Grain size effects The microstructure of this material is found to be highly inhomogeneous, considering that the entire sample set (N g ) spans three orders of magnitude in grain area a g (two orders in d g ). Such variability in a g provides a way of probing the grain size effect among grains that experienced the same macroscopic stress and strain history. In the present analysis, based on 2D sections of the microstructure, the size of equiaxed grains can be characterized either by grain diameter d g (a 1D measure) or area a g (a 2D measure). To investigate grain size effects on the incidence of twinning, we show in Figure 11a the fraction of grains of a given a g containing at least one twin (gray bars) compared to the total fraction of grains of a given a g (white bars). For an alternative view, the fraction of twinned grains of a given a g (ratio of the gray to white bars for each area interval) is plotted as a function of grain area. Figure 11b plots the same statistics versus grain diameter d g. In contrast to conventional thought, Figure 11 shows that, for the most part, grain area and diameter have no obvious effect on whether or not the grain twinned, since the fraction of twinned grains versus area a g (and d g ) is more or less constant. A large grain is just as likely to have no twins as a

15 2174 I.J. Beyerlein et al. Figure 11. Total fraction of grains of a given a g (white bars) and fraction of grains that contain at least one twin (gray bars) plotted as a function of (a) grain area and (b) circle equivalent grain diameter. Lineþsymbols: Twin frequency, given by the ratio of gray and white bars, representing the fraction of grains of a given area that contain at least one twin. small grain. An exception may be observed among the smallest grains in the sample (a g mm 2 ), where the fraction of twinned grains appears to decrease with decreasing grain size. This decrease may be an artifact of random EBSD sectioning: smaller area sections of a grain are less likely to intersect any twins it may contain. The analyses in the remainder of this section will consider size

16 Philosophical Magazine 2175 Figure 12. (a) Number of twins per grain as a function of grain area a g and (b) true twin thickness as a function of grain area. effects with respect to grain area a g since it is directly measured, as opposed to grain diameter d g, which is estimated from a g via an equal area circle approximation. The most notable size effect is found in Figure 12a, which presents the number of twins per grain n g as a function of area a g. As shown, n g increases as a g increases. In contrast, a g has little effect on twin thickness w t, as shown in Figure 12b. Therefore, provided that nucleation has taken place, larger twinned grains contain more twins than smaller twinned grains, while the thicknesses of their twins were more or less the same. Consequently, if we take n g ka g, where k is a proportionality constant, the twinned area per grain (a t n g w t d g kw t d 3 g ) and twin area fraction (a t /a g kw t d g ) will increase with grain size. Although not shown, we find this second important size effect to indeed be the case.

17 2176 I.J. Beyerlein et al. Figure 13. Number of twins per grain as a function of grain area and highest twin Schmid factor Effects of orientation and grain size In the last two sections, the effects of crystallographic orientation and grain size were analyzed separately. The map presented in Figure 13 examines the combined effects of grain orientation and grain size on the number of twins per grain. The crystallographic orientation is represented by m (1), the highest Schmid factor among the six variants in the grain and grain size is represented by grain area a g. We observe in Figure 13 that the number of twins per grain increases with both m (1) and a g. Mapping the data in this way shows that, even for the less suitably oriented grains (with m (1) 50.3), the number of twins per grain increases as a g increases. Conclusions from the 2D map in Figure 13 are consistent with those from the separate analyses on orientation and size effects. This consistency occurs because grain size and grain crystal orientation were found to be uncorrelated Grain boundary effects The previous sections examine twin formation and growth with respect to a g, which is related to the volume taken up by the grain. In this section, we investigate the effects of the properties of individual boundaries between a grain and its neighbors.

18 Philosophical Magazine 2177 Correlations for grain boundary segments will be different than those for grain diameter d g or grain area a g. Grain diameter d g is related to the grain perimeter, d g (or grain boundary surface area, dg 2 ), whereas a grain boundary segment (or a grain boundary facet), which is bounded by triple points (or triple lines), constitutes only a part of the grain boundary perimeter (or grain boundary surface area). Under the assumption that twins nucleate, propagate and thicken from the grain boundaries, it follows that some parts of the grain boundary (e.g. particular facets) may have properties that make them preferred sites for nucleation and growth more so than others. Such dependencies cannot be elucidated by considering measures of the entire grain size, such as d g or a g, and for this reason, we shift our focus to the role of individual grain boundaries with a given segment length l gb and misorientation angle. Ideally, one would like to utilize in the statistical analysis only those boundaries that can be recognized as being responsible for nucleating the observed twins. Unfortunately, in most cases, EBSD analysis cannot identify the boundary from which a twin nucleated. In a 2D scan, twins that traverse the entire grain (which is most of them) intersect two grain boundary segments. However, in actuality, the 3D twin intersects many more boundaries, say, p, where p can be five or larger. Therefore, the likelihood that the twin originated in one of the two boundary segments observed in the scan is 2/p. Moreover, a single twin that traverses the entire grain contributes to the statistics of both boundaries it intersects. Since these two grain boundaries have different misorientation and length l gb, attributing two separate intersections with the same twin will add some dispersion to the statistic. Nevertheless, any systematic correlation between twin nucleation and grain boundary characteristics could still be apparent in the statistics, albeit weakened, above the statistical noise. There is one situation in which it is possible to identify the boundary from which a twin originated. Occasionally, a twin appears to cross a grain boundary at the same position (see black arrow in Figure 3). In this case, the automated EBSD analysis identifies the event as two different twins, one belonging to each of the two adjoining grains and associates them with the same boundary. Twin pairs joined at a grain boundary signify one of two possible twin nucleation-related events. One, it could indicate sequential twinning, where one twin, upon terminating at a grain boundary, triggers another twin in the adjoining crystal [18]. Two, it could be the result of simultaneous twin nucleation into the two adjoining grains. In either case, these crossed twin pairs are nucleation events that the EBSD analysis can identify and associate with a particular grain boundary segment. In our analysis, these crossed twin pairs were characterized in EBSD as two adjoining twins coinciding spatially at a grain boundary and having similar thicknesses. The criterion used is that the shared thickness w AB over the total thickness spanned by both twins A and B i.e. (w A þ w B w AB ), is greater than 50%. Defined in this way, this event occurred across 8% of the grain boundaries in the present Mg sample. 3 The 50%-shared thickness criterion is rough but it allows for cases in which one twin grows thicker because its grain may be more favorably oriented for twinning than the other grain. For comparison, we analyze the influence of grain boundary characteristics on first, the entire population of twins and next, just the adjoining twin pairs. In the following, for each grain boundary segment, the number of twin/grain boundary

19 2178 I.J. Beyerlein et al. Figure 14. Number of grain boundary/twin intersections per grain boundary segment and per grain as a function of grain boundary length. All grain boundaries are considered in the analysis, not just those with twins connected to them. intersections n tgb and measured twin thickness w (thickness of the twin along the boundary) are studied as a function of grain boundary length l gb (the distance along the grain boundary between two triple points) and misorientation angle. The number n tgb counts the number of twins intersecting each grain boundary between its two bounding triple points from each grain sharing the boundary. The misorientation angle across a grain boundary adjoining two grains A and B is calculated from the following: ¼ min cos 1 trðr A R T B Þ 1, ð3þ 2 where R A and R B are the rotation matrices that transform from the crystal axes of grains A and B, respectively, to the sample axes. Due to crystal symmetry, R is not unique and the same orientation can be represented by more than one R. Equation (3) defines by the minimum of the angle between all possible equivalent symmetries Effects of grain boundary properties on the entire twin population When considering the general population of twins, we find that over 90% of all grain boundaries had at least one twin connected to it. As shown in Figure 14, n tgb increases linearly with l gb. While interesting, this result is, in part, due to the increasing chance of twin/grain boundary intersections as l gb becomes longer. Figure 15 shows the variation in thickness w with l gb. Unlike the relationship between a g and w, a positive correlation is seen between l gb and w; however, this result can also be an artifact of 2D sectioning. The measured thickness of the twin w that intersects the grain boundary is limited by the grain boundary length l gb to which it is connected. Since w can be no longer than the l gb, w can also increase artificially with l gb.

20 Philosophical Magazine 2179 Figure 15. Twin thickness w as a function of grain boundary length. The twin thickness in this plot is not the true twin thickness, but rather the length of intersection between the twin and the grain boundary measured in the plane of the EBSD scan. All grain boundaries are considered in the analysis, not just those with twins connected to them. Grain boundary fraction Figure 16. Distribution of grain boundary misorientation angles (Equation (3)) across the 3% strained sample. Next, we examine the effects of grain boundary misorientation angle. Figure 16 shows the distribution of across all grain boundary segments considered in the deformed sample. Due to the strong texture in this material, the angles are heavily weighted within A minimum misorientation angle of 5 is used to define grain boundaries, and as a result, the first bin in Figure 16, i.e ,is empty. For the remaining histograms, is binned in 5 increments starting at 5.

21 2180 I.J. Beyerlein et al. (a) (b) Figure 17. (a) Number of grain boundary/twin intersections per grain boundary segment and per grain as a function of grain boundary misorientation angle. (b) Thickness of each twin as a function of the misorientation angle across the grain boundary to which it is connected. A misorientation angle between two grains can generate stress gradients across the boundary due to elastic and plastic anisotropy, which can potentially influence twin nucleation and growth. Because Mg is almost isotropic elastically [46], elastic stress gradients are not expected to change with. However, Mg crystals are plastically anisotropic, and the likelihood of a grain to plastically accommodate the twinning shear taking place in its neighbor would depend on their relative misorientation. The analysis in Figure 17 suggests that twinning is preferred from low grain boundary angles. Figure 17a shows that the number of twin/grain boundary intersections decreases as increases. The correlation is noticeable despite the

22 Philosophical Magazine 2181 Figure 18. Number of grain boundary/twin intersections per grain boundary segment and per grain as a function of grain boundary misorientation angle between the c-axes. additional uncertainty that arises in the analysis from associating each twin with both its bounding grain boundary segments. The chance that the twin nucleated from one of these segments is not high, i.e. 2/p, where p is the number of boundaries the twin intersects in 3D. Regarding growth, w also tends to decrease with, as shown in Figure 17b. The observation of more and thicker twins at smaller motivates probing just the misorientation angle between the c-axes. Misorientation angles 530 can result either from non-parallel or parallel c-axes, two cases that can lead to vastly different single crystal deformation responses in pure Mg [47] and would produce different stress gradients across the boundary. Figure 18 shows the number of twin intersections per grain boundary as a function of c-axis angle misorientation. (Similar to, is also the minimum among all possible equivalent symmetries.) In Figure 18, it is seen that the reduction in the number of twins with is more evident than with. In particular, the preference for smaller misorientations, 520, is more apparent. Qualitatively, however, results found for were very similar to those found for in Figure Effects of grain boundary properties on the crossed twin population The grain boundary analyses above are repeated for just the subset of adjoining twin pairs (Table 2). In Figure 19a, we plot, using dark-colored bars, the distribution of the minimum misorientation angle twin between all adjoining twin pairs found in the scans. Figure 19a shows that adjoining twin pairs are more likely to have lower twin. To understand the nature of this distribution, Figure 19a also plots, using light-colored bars, the minimum misorientation possible twin,min between the 36 possible pairs of twin variants for the two neighboring grains associated with each grain boundary. The similarity between the measured twin distribution and theoretical twin,min distribution is striking. Small deviations of the measured twin

23 2182 I.J. Beyerlein et al. (a) (b) Figure 19. (a) Dark bars: frequency distribution of adjoined twin pairs ( crossed frequency ) as a function of misorientation angle between the two adjoining twin. Light bars: distribution of the misorientation for grain boundaries containing at least one adjoined twin pair. (b) Shared length between the adjoining twin pairs as a function of grain boundary misorientation angle. from twin,min can be easily attributed to slip within the matrix and/or twin that changes the matrix/twin orientation relationship from the ideal. Furthermore, it can be shown that twin,min is equal to the corresponding grain boundary misorientation angle. Figure 19a also shows that the frequency of twin boundary crossings decreases as grain boundary increases. The favorable comparison in Figure 19 between twin and twin,min ¼ indicates that the nucleation events leading to twin pairs joined at a boundary: (1) produces twins in each crystal of like variant (with similar orientation), (2) is facilitated across low angle grain boundaries, and (3) can

24 Philosophical Magazine 2183 Figure 20. Dark bars: shared length between the adjoining twin pairs as a function of grain boundary length. For comparison, the light bars are the thicknesses of twins connected to the grain boundary (from Figure 15). influence the choice of twin variant formed in a neighbor. Both the proposed sequential-twinning and dual-nucleation mechanisms described earlier imply a co-dependence in which the twin variant nucleated in the first grain dictates the choice of variant in the second one. Furthermore, such strong correlations between twin crossing on likely influence the statistical analyses performed on the entire twin population. In particular, the larger number of twin/grain boundary intersections for grain boundary angles smaller than 45, seen above in Figure 17a, could be related to sequential twinning or dual twin nucleation into two neighboring grains. Recall also from Figure 17b that twins connected to low grain boundaries grew thicker than those connected to higher grain boundaries. From the adjoining twin pair sample set, Figure 19b plots the grain boundary length w AB shared by the two adjoining twins as a function of twin. Evidentally, w AB tends to increase with decreasing twin. Similarities in the observed trends between the adjoining twin pairs (Figure 19) and all twins connected to grain boundaries (Figures 17 and 18) indicate that lower angle grain boundaries increase the propensity of twinning and promote twin growth. The effect of grain boundary length l gb on the growth of these twin pairs is examined in Figure 20. Figure 20 plots the shared boundary length w AB between each pair as a function of l gb (dark-colored bars). For comparison, Figure 20 also includes the thickness w of all twins intersecting boundaries of the same length (light-colored bars) taken from Figure 15. Remarkably, w AB is significantly larger than w despite the fact that, by definition, w AB is less than or equal to the thinnest of the two twins in each pair. It is likely that when these joined twin pairs grow concurrently, the backstress, which would otherwise impede their growth, is lowered. The above analyses were repeated using a more stringent sequential twinning criterion in which the thickness of the adjoining twins must be within 90%

25 2184 I.J. Beyerlein et al. (instead of 50%). While the fraction of grain boundaries with at least one crossed twin pair dropped, the same trends were observed, namely the same peak value and same consistency between twin,min based on the minimum angle criterion and the observed twin. 4. Discussion 4.1. Orientation effects Grain orientation greatly influences the propensity for twin nucleation and growth with the more favorably oriented grains being more likely to form at least one visible twin and containing thicker twins. However, the statistical variability in these observations was too great to consider these processes determined by orientation alone. Twins did not form in all grains of similar favorable orientation (Figure 6) and the observed twin did not always correspond to the variant with the highest Schmid factor (Figure 9). The variation was likely not due to variations in grain size. The incidence of twinning, whether or not at least one twin was observed in a given grain, was not particularly sensitive to grain size (for grains with areas larger than 1000 mm 2 ) and we found no correlation between grain size and orientation. Some of the variability found in these single-point statistical analyses can be attributed to the effect of paired statistics the misorientation angle between two neighboring grains. More twins and thicker twins were connected to grain boundary segments with lower misorientation angles, 545, than to those with higher angles. The fact that these correlations are strong is notable, since the analysis associated each fully traversed twin with the two boundaries it connects (not knowing from which boundaries it originated) and hence with two values of, a procedure which tends to increase the dispersion and weaken trends. This uncertainty is removed in the case of twin pairs joined at a grain boundary at the same position. With adjoined twin pairs it is possible to associate the parent boundary with its twin because they signify that a nucleation event took place at the shared boundary. Adjoined twin pairs may have occurred, for instance, by sequential twinning, when one twin terminating at the boundary nucleates another one in the neighboring grain, 4 or by dual twinning, when both nucleated at the same site. Like the individual twins, the frequency of adjoined twin pair observations and their thickness (shared grain boundary) also were found to increase as decreases. Even more profound was the observation that the adjoining twins grew considerably thicker than single twins for the same. This last observation deserves further discussion. It may be explained by first assuming that the thickness of a twin at a grain boundary is a measure of how well the neighboring grain accommodates the shear strain provided by the twin, since the amount of strain requiring accommodation is proportional to its thickness. For instance when the neighboring grain (or boundary) is rigid, a twin might be expected to become narrow when approaching the grain boundary, as the adjoining grain cannot accommodate a twin shear. A reaction stress from the neighboring grain is generated that hinders the twin from thickening. Examples of this case can be seen in Figure 3 where twins intersecting boundaries with grains in (0001) compression (red grains, not oriented well for twinning or slip) often come to a

26 Philosophical Magazine 2185 point at the grain boundary. The opposite effect of an accommodating neighbor grain is most evident in the present study in the enhanced growth of two twins in neighboring grains located at approximately the same point on the grain boundary (i.e. adjoining twin pairs). As shown by Figure 19, these twins occur on similarly oriented twinning systems and hence are less likely to induce inhibiting backstresses on each other. As a result, the pair will grow thicker at the grain boundary than a single twin that does not overlap with other twins in the neighboring grain. Occurrence of adjoined twin pairs, particularly in the upper range of 430, will contribute to the dispersion in twin variant selection with respect to the one expected considering only crystallographic orientation. Although the preference to occur across low misorientation angle grain boundaries is strong, adjoining twin pairs are observed over the entire range of. For all values of, the misorientation between the pair twin was nearly equal to its grain boundary. In summary, results from examining adjoined twin pairs indicate that local neighbor affects twinning since a twin growing in grain A can: (1) nucleate a twin in a neighboring grain B that is not as favorably oriented for twinning, (2) select a variant for this twin that is not necessarily the most geometrically favored one in grain B, and (3) enhance its growth more so than a twin which would have nucleated in grain B independent of twins in neighboring grains (compare Figures 17a and 19b). Statistical variation with respect to grain orientation may also indicate a strong influence of the local resolved shear stress in the region of the twin. The actual resolved shear stress on the twin plane is expected to be different than the macroscopic stress multiplied by the (geometric) Schmid factor that is calculated here and used as a measure of orientation. At the grain scale, average grain stresses will deviate from grain to grain and from the macroscopic applied stress state due to differences in anisotropic deformation between a grain and the polycrystal. At the subgrain scale, stress gradients across the grain are expected due to interactions with its individual neighborhood and characteristics specific to the neighborhood (number of neighbors, their slip and twin activity and orientation, etc). At even lower length-scales, variations in defect content in the grain boundaries will lead to even finer stress fluctuations. Local stress fluctuations will need to be introduced in models for determining when and where twins will nucleate and which variant will be selected. In this regard, efforts are currently underway towards developing a physically based probabilistic twin nucleation model [48] and understanding the role of non-uniform stresses at grain boundaries, produced, for instance, by a dislocation pile up [1] Origin of the grain size effect The effect of grain area a g or diameter d g on whether or not at least one twin appeared in the grain or on twin thickness w was small or negligible. These results are in conflict with associating a Hall Petch effect to the incidence of deformation twinning in hcp metals. The same results would be obtained if we considered instead a length scale associated with the grain boundary, such as grain perimeter d g, since d g differs from the grain perimeter d g by only a constant. In other words, our statistical analysis does not support the idea that at least one twin is more likely to

27 2186 I.J. Beyerlein et al. Table 4. Comparison of {10 12} twinning data between pure Mg and pure Zr. Material (compression strain, temperature) Grain area a (mm 2 ) Average grain boundary length (mm) Fraction of twinned grains (%) Fraction of grains oriented for twinning over slip b (%) Average true twin thickness (mm) Mg (3%, 300 K) (basal) (prismatic) Zr (5%, 10%, 77 K) (prismatic) Notes: a Grain areas in Mg were highly variable and could not be described by a simple mean. b This calculation does not consider the stresses required to activate each mechanism. appear in larger grains than in smaller grains, at least for the grain area range studied here, i.e. a g 421 mm 2. The only notable effects of grain area a g observed were the increase in number of twins per twinned grain (n g ) and grain twinned fraction with grain area. The origin of these positive grain size effects are likely related to a length scale of the grain boundary. Grain boundaries are regions of interfacial defects, such as ledges, stacking faults and grain boundary dislocations [1], that may serve as nucleation sites for twinning. In addition to containing twin nuclei sources, they are also the likely locations for the relatively high internal stresses required to convert defects into twin nuclei. Furthermore, for a given grain, some grain boundary facets shared between two adjoining grains may be more favorable for twinning than the others, such as those with lower misorientation angle, as seen in recent atomistic simulations [1]. Therefore, it may not be the entire grain volume that directly increases the number of twins, but the larger areas of preferential grain boundaries that carry with them proportionally more defect sites for nucleation Comparison between {1012} twinning in Mg and Zr Although not as comprehensive, another statistical study was carried out for {1012} twinning in high-purity Zr having a similar rolling texture as the Mg studied here [24]. The Zr material was also loaded in compression in an in-plane direction to induce {1012} twinning. Unlike the present study, the previous study on Zr did not evaluate each individual twin lamella. Instead, it evaluated each grain and considered the average properties of the twins within them. In this section, the results for Zr and Mg are compared to gain further insight on the mesoscopic microstructural effects on twinning. Table 4 compares some data between the two materials. Generally, twinning occurred more readily in Mg than Zr. As shown in Table 4, a higher fraction of grains formed twins in Mg than Zr, although the Mg was deformed to a lower strain and at a higher temperature. The Mg sample was strained to 3% with a flow stress of 60 MPa at room temperature, whereas the Zr sample was strained to 5 and 10% with flow stresses of 270 and 370 MPa, respectively, at 77 K. Also, while in Zr twinning is delayed and some amount of plastic flow occurs prior to twinning [10,21], in Mg

28 Philosophical Magazine 2187 Figure 21. Compression direction inverse pole figures for an hcp crystal of (a) the difference between the highest Schmid factor for twinning and the highest Schmid factor for basal slip (m twin m basal ) and (b) the difference between the highest Schmid factor for twinning and the highest Schmid factor for prismatic slip (m twin m prism ). micron-scale twins appear without delay at about the same strain as macroscopic yielding [39 41,49]. We envision two possible explanations for the differences between Zr and Mg. The first one is based on grain orientation and the preferred slip mode of these two materials: basal slip for Mg and prismatic slip for Zr. In general, hcp grains that are oriented for {101 2} twinning are also oriented relatively well for prismatic slip, and not very well for basal slip. To demonstrate, Figure 21 shows inverse pole figures of the difference (a) between the highest Schmid factors m (1) for twinning and basal slip, (m twin m basal ) and (b) between twinning and prismatic slip, (m twin m prism ) for an hcp crystal in uniaxial compression. The maximum value for (m twin m prism ) is less than 0.1, whereas it can be as high as 0.5 for (m twin m basal ). Therefore, for Zr, prismatic slip and {1012} twinning can be activated concurrently with their relative roles depending on temperature and strain rate [10]. For Mg, however, basal slip and {1012} twinning are less likely to be activated concurrently. As a result, for the present texture and loading direction, {1012} twinning is more prevalent in Mg than Zr because in Mg it does not have to compete with the easy slip mode, whereas in Zr, it does. A second, more speculative explanation involves the local stress required for nucleating twins at grain boundaries. If it is much larger in Zr than in Mg, then the nucleation will be delayed in Zr until strain hardening from dislocation storage (macroscopic effect) and pile ups at grain boundaries (localized effect) can raise the total local stresses at the grain boundaries to a sufficient level. Moreover, analytical stability calculations based on dislocation theory [8] indicate that pile ups of basal slip dislocations (the easy slip mode in Mg) can aid slip dislocation dissociation reactions into twinning partials but pile ups of prismatic slip dislocations (the easy slip mode in Zr) cannot. 5. Conclusions In this work, a systematic, statistical analysis on large sets of metallographic EBSD data was conducted to identify correlations between microstructural features, such as grain orientation, grain size, and grain boundary misorientation and the nucleation and growth of {1012} deformation twins in Mg. The following are the key results