- Email: [email protected]
Challenges in the prediction of twin transmission at grain boundaries in a magnesium alloy
Scripta Materialia 123 (2016) 77–80
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Regular Article
Challenges in the prediction of twin transmission at grain boundaries in a magnesium alloy X. Hong, A. Godfrey ⁎, W. Liu Key Laboratory of Advanced Materials (MOE), School of Material Science and Engineering, Tsinghua University, Beijing, 100084, PR China
a r t i c l e
i n f o
Article history: Received 19 May 2016 Accepted 29 May 2016 Available online xxxx Keywords: Mg alloys Twinning Plastic deformation
a b s t r a c t An analysis of twin transmission, examining grain pairs both where twin transmission takes place, and where it does not, confirms that in general such events occur more frequently with decreasing misorientation angle. However, it is found that strain incompatibility may play a counter-balancing role in the very low angle range, and some influence of the Schmid factor on twinning is also observed. The geometric compatibility factor is found to be useful for explaining occurrences of twin transmission, but the probability of such events is rather insensitive to this parameter for many boundaries where twin transmission may be expected. © 2016 Elsevier Ltd. All rights reserved.
Twinning is an important deformation mechanism for Mg alloys and for many other hexagonal close-packed (hcp) metals because of the restricted slip system activity at room temperature in such metals [1,2]. Due to its role in plasticity, an understanding of twinning, including not just which variants form, but also to what extent, and where twinning takes place in the microstructure, is an important issue. It is already well established that grain boundaries are the most relevant sites for twin nucleation [3–5]. More recently, a specific type of grain boundary twinning event, namely twin transmission of {10−12} twins, has been widely investigated in many hexagonal metals [3,6– 9]. Twin transmission is characterized by the observation of twins in two adjacent grains, where the twins are connected to the same region of grain boundary. In this case it is assumed that either a twin in one grain promotes the nucleation of a twin in the second grain (though in ex-situ studies it cannot be excluded that they form simultaneously). Several studies of both Mg and Ti have shown that, in general, twin transmission occurs more frequently with decreasing grain boundary misorientation angle [3,5,10–12]. However, data for twin transmission in the very low misorientation angle range are less clear. For example, a detailed study by Beyerlein et al. showed that the probability of twin transmission increased continuously with decreasing grain boundary misorientation angle until an apparent maximum at approximately 10° [3]. An interesting related question is whether twin transmission becomes easier as the boundary misorientation angle decreases towards zero, or whether an additional influence of strain incompatibility across a boundary also plays a role in assisting this process. In a more general sense, it can be noted that twin transmission is important, as in addition to the role it plays in strain hardening, it also has been linked ⁎ Corresponding author. E-mail address: [email protected] (A. Godfrey).
http://dx.doi.org/10.1016/j.scriptamat.2016.05.044 1359-6462/© 2016 Elsevier Ltd. All rights reserved.
to the formation of a yield plateau and the development of localized shear bands [10,13,14]. A geometric compatibility parameter, mʹ = cosκ · cosψ, where κ and ψ are angles between the two shear directions and two twin plane normals for twin systems in two neighboring grains, has also been used for analysis of twin transmission in HCP metals [6,10,12,15]. This parameter, which has also been used to analyze strain accommodation between slip systems, and between slip and twinning systems [16,17] can be thought of as providing a measure of the efficiency of strain transfer across a grain boundary. Many studies that use this parameter to analyze twin transmission do so on a case by case basis, examining only grain pairs where twin transmission is observed. Here instead we examine data for a large number of grains, also considering grain pairs where twin transmission is not seen. In particular our interest is in not only trying to understand whether the misorientation angle and mʹ parameters can explain twin transmission observations, but also in determining to what extent they can be used to predict twin transmission events. The material used in this study was a commercially purchased hot-rolled AZ31 (Mg—3%Al—1%Zn) sheet. Samples of dimensions 5 mm × 8 mm × 10 mm (rolling direction (RD) × transverse direction (TD) × normal direction (ND)) were cut from the middle of the sheet thickness. The sample had a typical hot-rolled texture with a {0001}||ND fiber texture. Plane strain compression tests (loading direction parallel to TD, i.e. in a direction favorable for {10−12} twinning in this material) were performed at a strain rate of 10−3 s−1 to a reduction of 1.9% at room temperature, using a Gleeble 1500D system. Electron backscatter diffraction (EBSD) observations of the deformed microstructure were carried out on samples sectioned in a mid-plane parallel to the initial TD-ND plane, then mechanically ground to 2000-grit SiC paper, followed by polishing using diamond paste, and then vibratory polishing using colloidal silica. Prior to EBSD examination the sample
78
X. Hong et al. / Scripta Materialia 123 (2016) 77–80
surface was etched for 10–20 s using a solution of 10 ml nitric acid, 30 ml acetic acid, 40 ml water and 120 ml ethanol. A typical example of the microstructure of a deformed sample is shown in Fig. 1a, where many instances of twin transmission can be clearly identified. In this study a total of 6886 grain boundaries, including 306 pairs of transmitted twins have been analyzed. To maximize the information available concerning twin transmission at very low angle boundaries, the misorientation angle dependence has been analyzed where necessary using 5° bins starting from the lower EBSD cut-off angle of 2° (rather than starting from 0° with a low angle cut-off at 2° as more commonly used, resulting in a first bin of reduced width). Fig. 1b plots the frequency of twin transmission as a function of grain boundary misorientation angle. For comparison the same data analyzed using bins starting at 0° are shown in the inset (top right corner of Fig. 1b), where an apparent maxima in twin transmission is seen in the 3rd bin (10–15°). The results are nevertheless in general agreement with previous studies [5,10,11]. A better parameter, however, to analyze twin transmission is the number of boundaries with a transmitted twin in a given misorientation angle range, normalized by the total number of boundaries within that range, fTT, as plotted in Fig. 1c. Here it is seen that the fraction of boundaries where twin transmission takes place increases continuously with decreasing misorientation angle. However, for misorientation angles in the range of 2–7°, the value of fTT lies under the value predicted from extrapolation of the values for higher boundary misorientation angles (see the line in Fig. 1c), suggesting that twin transmission is not increasingly easy with decreasing misorientation angle, although number of grain boundaries within this misorientation angle range is relatively small. Further insight is obtained by considering also grain pairs where a twin is present in one grain, but where twin transmission is not observed. To do this an area was examined containing 316 such grain boundaries. The fraction of grain pairs where twin transmission is seen with respect to the number of boundaries with at least one twin (i.e. including cases where a twin is present only on one side of a grain boundary), fʹTT, is shown as a function of boundary misorientation
angle in the inset of Fig. 1c. It is worth noting that the complementary fraction (1 − fʹTT) measures the probability that a boundary twin does not result in twin transmission. A general trend of increasing twin transmission fraction with decreasing misorientation angle is again seen, except for the lowest angle bin (representing 43 grain boundaries where at least one twin is present), indicating that some influence of strain incompatibility in assisting twin transmission may also play a role in this process. The data in Fig. 1 represent an experimental description of twin transmission as a function of misorientation angle. To probe the intrinsic dependence of this process on grain boundary misorientation it is also necessary to consider any possible influence of the variation in Schmid factor (SF) on twinning [18–22]. In general it is found a higher SF value gives an increased likelihood of twinning on a particular twin system, though the preferential activation of twins with lower ranked Schmid factors, or even of twins with a shear strain in the opposite sense to the applied load, has also been observed [19–21,23]. The potential influence of SF on twin transmission is highlighted in Fig. 2, which shows data for grain pairs in an undeformed sample, and plots the relationship between grain boundary misorientation angle and the maximum SF for twinning of the grains on each side of each boundary. The shape of the distribution for this typically textured sample, with increasing density of grain pairs towards the (0.5, 0.5, 0°) corner (highlighted by the projection onto the (SF1, SF2 plane)) indicates that grain pairs with low angle also tend to have high Schmid factors. The corollary of this is that a higher fraction of randomly chosen grain pairs where both grains have high SFs will have a low misorientation angle compared to grain pairs where both grains have lower SFs. As a simple test of whether this correlation has an influence on the observed misorientation angle dependence of twin transmission, the value of fTT has been recalculated considering only grain pairs where the SF values for both grains are in a certain SF range, either 0.4 b SF ≤ 0.5 or 0.3 b SF ≤ 0.4. The results are shown in Fig. 3, where it seen that the dependence of fTT on misorientation angle is noticeably different for the two cases. Note here that although the
Fig. 1. (a) EBSD map (inverse pole figure coloring) showing twin transmission events in a sample of AZ31 after compression to a strain of 1.9% (b) frequency distribution of twin transmission events as a function of grain boundary misorientation angle; (c) fraction of grain boundaries in each misorientation angle range (bins of 5° width) where twin transmission is seen, fTT (specifically, number of boundaries with twin transmission in misorientation angle range Δθ, divided by the number of grain boundaries in the same angle range). The dotted line represents a fit to the data for bins with θ N 7°. The inset shows the fraction of transmitted boundaries relative to the number of grain boundaries where at least one twin is present.
X. Hong et al. / Scripta Materialia 123 (2016) 77–80
Fig. 2. Graph showing the correlation as a result of the initial texture between grain boundary misorientation angle and the maximum Schmid factors for twinning (SF1, SF2) of the grains forming each boundary. Also shown is a projection onto the (SF1,SF2) plane highlighting the peak (0.5, 0.5, 0°).
data for 0.3 b SF ≤ 0.4 show some scatter, these results are taken from an area covering more than 650 grain pairs in this category. The differences are expected to be smaller when comparing more similar SF ranges, but nevertheless highlight the need to consider such an influence in any detailed analysis of the intrinsic dependence on boundary angle of twin transmission. An alternative parameter that has been used for analysis of twin transmission events is the geometric compatibility factor, mʹ, representing the degree of alignment between the twinning systems in adjacent grains. Fig. 4 shows the value of mʹ for each of 306 twin pairs taken from the investigated sample. In each case the twin system was first identified in each grain [24], and then the geometric compatibility factor for the twin pair, mʹTT, was calculated. More than 95% of the twin pairs have a value of mʹTT N 0.7 (Fig. 4a), with almost 70% having mʹTT N 0.9. In many cases, however mʹTT is not the maximum possible value for the grain pair (mʹmax, calculated by considering the 6 possible twin variants for each grain) as illustrated in Fig. 4b. This difference reflects the fact that it is possible for two twin systems to be well aligned (giving high mʹmax), but for neither to be well oriented for twinning (i.e. have a low SF). To further understand the twin transmission events, we calculate the parameter mʹ(1,2), defined as the maximum value of mʹ based on the observed twin on one side, i.e. assuming that the twin first forms in one grain (grain 1 or 2) and is then transmitted to the other (grain 2 or 1) on one of the possible 6 twin systems. Note that as the sequence is not known we consider therefore both possibilities and use both values if they are different.
79
For the current data more than 98% twin transmission events are found to have mʹ values equal to mʹ(1,2). This result agrees with the analysis in previous investigations [12,15,17] that the transmitted twin variant is always the one with the highest mʹ value correlating with the first formed twin. These data, however, do not provide information about the ability of this parameter to predict occurrences of twin transmission. For this, it is necessary to consider also grain pairs in which twin transmission is not seen (i.e. as previously considering also grain pairs where a twin is present only in one grain). Such an analysis is given in Fig. 4b, where for the same EBSD map (316 grain pairs with one boundary twin and 116 grain pairs where twin transmission is seen) mʹ values are plotted as a function of boundary misorientation angle for all grain boundaries where twin transmission is observed (mʹTT), and for all grain boundaries where a twin is present in one grain but is not transmitted (mʹ1T, based on the relationship of the observed twin to the 6 possible variants in the neighbor grain). Consistent with the earlier analysis based on misorientation angle, the number of grains where twin transmission does not take place increases with increasing boundary misorientation angle. A more important observation is that for misorientation angles less than ≈30° the mʹ1T values are not systematically lower compared to the mʹTT values, but cover a similar range, i.e. grain boundary pairs exist where a boundary is twinned on one side but not the other even though a twin system is well aligned in the neighboring grain (i.e. has a high mʹ value). The dependence of twin transmission on the geometric compatibility parameter is probed further by calculation of the complementary cumulative distribution function (CCDF) for twin transmission, defined here as f CCDF ðxÞ ¼
N m0TT Nx N m0TT Nx þ N m01T Nx
ð1Þ
where N(mʹ N x) is the number of grain boundary pairs with mʹ N x for cases either of twin transmission (TT) or a single boundary twin (1T). This value represents the fraction of grain boundaries where twin transmission is seen relative to the number of all grain pairs with at least one boundary twin (i.e., including boundaries where twin transmission could have taken place but did not), for all boundaries with the geometric compatibility factor (mʹTT for twin transmission pairs and mʹ1T for single twin boundaries) greater than a given value. The results are shown in Fig. 4c. A positive correlation with fCCDF is seen with increasing x for low values (mʹ b 0.9), indicating that in this range larger mʹ values for a twinned boundary correspond to a higher probability of twin transmission taking place. However, in the range 0.9 b x b 0.95 the value of fCCDF is approximately constant at ≈0.5. In this range therefore the fraction of possible twin transmission events is rather insensitive to the value of mʹ. It can be noted also that this range of mʹ values corresponds in the present sample to grain boundary
Fig. 3. Fraction of grain boundaries where twin transmission is seen as a function of grain boundary misorientation angle (analyzed in bins of 5° width) for grain pairs where both grains have maximum Schmid factors in the range (a) 0.40–0.50, or (b) 0.30–0.40.
80
X. Hong et al. / Scripta Materialia 123 (2016) 77–80
Fig. 4. (a) Normalized frequency distribution (bin width = 0.1) of the geometric compatibility factor (mʹ) for the observed 306 twin transmission events; (b) plot showing mʹ values for each observed twin transmission pair (mʹTT, red squares), and maximum mʹ value for grain boundaries with twin on only one side of the boundary (mʹ1T, green triangles). The small blue circles show the maximum possible mʹmax values for all grain pairs in the sample; (c) complementary cumulative distribution function showing the fraction of grain boundary pairs with at least one twin where twin transmission takes place.
misorientation angles from 13° to 25°, i.e. the range where one third of twin transmission events are seen. At larger values (mʹ N 0.95) a strong positive correlation with mʹ is again seen, though for very high mʹ values (corresponding to very low angle misorientations) a flattening of the curve is found, in agreement with the trends also seen in Fig. 1c. It is worth noting that although misorientation angle and mʹ are inter-related, the two parameters show some differences with regard to prediction of twin transmission events (i.e. when considering all grain pairs where at least one twin is present). In particular the relative insensitivity of the probability of twin transmission to mʹ for the range of boundary misorientation angles where such events are seen is likely to reflect the fact that any specific twin transmission event (including nucleation of twin pairs at a grain boundary) may be influenced by a combination of factors, including the local stress state, the presence of pre-existing twins, and the local dislocation configuration [9]. The observations therefore highlight the advantage of modelling twin transmission in a similar probabilistic (rather than deterministic) manner to that already proposed for single twinning events [25]. In summary, twin transmission during plane strain compression of an AZ31 magnesium sample has been investigated with a view to understand better how such events can be predicted. It is found in agreement with previous studies that in general twin transmission probability increases with decreasing boundary misorientation angle, though some influence of the Schmid factor on twinning is also found and may need to be considered when analyzing the intrinsic ease of twin transmission as a function of boundary misorientation. Although the geometric compatibility factor, mʹ, is found to be useful for explaining occurrences of twin transmission, this parameter is not so efficient for prediction of such events, as the probability of twin transmission is insensitive to mʹ for a range of values over which 1/3 of the observed twin transmission events are observed. For both the misorientation angle and the mʹ parameter the experimental data show that twin transmission does not become increasingly easy with decreasing misorientation angle in the very low misorientation angle regime, suggesting that strain incompatibility may play some role also in nucleation of twin transmission events. Such very low angle boundaries are, however, relatively infrequent for samples with a typical texture.
Acknowledgements Financial support for this work from the National Key Basic Research Program of China (grant number 2013CB632204) is gratefully acknowledged.
References [1] J.W. Christian, S. Mahajan, Prog. Mater. Sci. 39 (1995) 1–157. [2] M.H. Yoo, Metall. Trans. A 12 (1981) 409–418. [3] I.J. Beyerlein, L. Capolungo, P.E. Marshall, R.J. McCabe, C.N. Tomé, Philos. Mag. 90 (2010) 2161–2190. [4] J. Wang, I.J. Beyerlein, C.N. Tomé, Scr. Mater. 63 (2010) 741–746. [5] I.J. Beyerlein, R.J. McCabe, C.N. Tomé, J. Mech. Phys. Solids. 59 (2011) 988–1003. [6] L. Wang, P. Eisenlohr, Y. Yang, T.R. Bieler, M.A. Crimp, Scr. Mater. 63 (2010) 827– 830. [7] R.J. McCabe, G. Proust, E.K. Cerreta, A. Misra, Int. J. Plast. 25 (2009) 454–472. [8] J. Kacher, A.M. Minor, Acta Mater. 81 (2014) 1–8. [9] A. Khosravani, D.T. Fullwood, B.L. Adams, T.M. Rampton, M.P. Miles, R.K. Mishra, Acta Mater. 100 (2015) 202–214. [10] M.R. Barnett, M.D. Nave, A. Ghaderi, Acta Mater. 60 (2012) 1433–1443. [11] A. Fernández, A. Jérusalem, I. Gutiérrez-Urrutia, M.T. Pérez-Prado, Acta Mater. 61 (2013) 7679–7692. [12] R.L. Xin, C.F. Guo, Z.R. Xu, G.D. Liu, X.X. Huang, Q. Liu, Scr. Mater. 74 (2014) 96–99. [13] S. Sandlöbes, S. Zaefferer, I. Schestakow, S. Yi, R. Gonzalez-Martinez, Acta Mater. 59 (2011) 429–439. [14] H.E. Kadiri, J. Kapil, A.L. Oppedal, L.G. Hector Jr., S.R. Agnew, M. Cherkaoui, S.C. Vogel, Acta Mater. 61 (2013) 3549–3563. [15] C.F. Guo, R.L. Xin, C.H. Ding, B. Song, Q. Liu, Mater. Sci. Eng. A 609 (2014) 92–101. [16] J. Luster, M.A. Morris, Metall. Mater. Trans. A 26A (1995) 1745–1756. [17] L. Wang, Y. Yang, P. Eisenlohr, T.R. Bieler, M.A. Crimp, D.E. Mason, Metall. Mater. Trans. A 41 (A) (2010) 421–429. [18] S.-G. Hong, S.H. Park, C.S. Lee, Acta Mater. 58 (2010) 5873–5885. [19] M.R. Barnett, A. Ghaderi, J.Q. da Fonseca, J.D. Robson, Acta Mater. 80 (2014) 380–391. [20] L. Capolungo, P.E. Marshall, R.J. McCabe, I.J. Beyerlein, C.N. Tomé, Acta Mater. 57 (2009) 6047–6056. [21] A. Chapuis, Y.C. Xin, X.J. Zhou, Q. Liu, Mater. Sci. Eng. A 612 (2014) 431–439. [22] Y. Pei, A. Godfrey, J. Jiang, Y. Zhang, W. Liu, Q. Liu, Mater. Sci. Eng. A 550 (2010) 138–145. [23] Z.-Z. Shi, Y.D. Zhang, F. Wagner, P.-A. Juan, S. Berbenni, L. Capolungo, J.-S. Lecomte, T. Richeton, Acta Mater. 83 (2015) 17–28. [24] J. Jiang, A. Godfrey, W. Liu, Q. Liu, Scr. Mater. 58 (2008) 122–125. [25] I.J. Beyerlein, C.N. Tome, Proc. R. Soc. A 466 (2010) 2517–2544.
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Regular Article
Challenges in the prediction of twin transmission at grain boundaries in a magnesium alloy X. Hong, A. Godfrey ⁎, W. Liu Key Laboratory of Advanced Materials (MOE), School of Material Science and Engineering, Tsinghua University, Beijing, 100084, PR China
a r t i c l e
i n f o
Article history: Received 19 May 2016 Accepted 29 May 2016 Available online xxxx Keywords: Mg alloys Twinning Plastic deformation
a b s t r a c t An analysis of twin transmission, examining grain pairs both where twin transmission takes place, and where it does not, confirms that in general such events occur more frequently with decreasing misorientation angle. However, it is found that strain incompatibility may play a counter-balancing role in the very low angle range, and some influence of the Schmid factor on twinning is also observed. The geometric compatibility factor is found to be useful for explaining occurrences of twin transmission, but the probability of such events is rather insensitive to this parameter for many boundaries where twin transmission may be expected. © 2016 Elsevier Ltd. All rights reserved.
Twinning is an important deformation mechanism for Mg alloys and for many other hexagonal close-packed (hcp) metals because of the restricted slip system activity at room temperature in such metals [1,2]. Due to its role in plasticity, an understanding of twinning, including not just which variants form, but also to what extent, and where twinning takes place in the microstructure, is an important issue. It is already well established that grain boundaries are the most relevant sites for twin nucleation [3–5]. More recently, a specific type of grain boundary twinning event, namely twin transmission of {10−12} twins, has been widely investigated in many hexagonal metals [3,6– 9]. Twin transmission is characterized by the observation of twins in two adjacent grains, where the twins are connected to the same region of grain boundary. In this case it is assumed that either a twin in one grain promotes the nucleation of a twin in the second grain (though in ex-situ studies it cannot be excluded that they form simultaneously). Several studies of both Mg and Ti have shown that, in general, twin transmission occurs more frequently with decreasing grain boundary misorientation angle [3,5,10–12]. However, data for twin transmission in the very low misorientation angle range are less clear. For example, a detailed study by Beyerlein et al. showed that the probability of twin transmission increased continuously with decreasing grain boundary misorientation angle until an apparent maximum at approximately 10° [3]. An interesting related question is whether twin transmission becomes easier as the boundary misorientation angle decreases towards zero, or whether an additional influence of strain incompatibility across a boundary also plays a role in assisting this process. In a more general sense, it can be noted that twin transmission is important, as in addition to the role it plays in strain hardening, it also has been linked ⁎ Corresponding author. E-mail address: [email protected] (A. Godfrey).
http://dx.doi.org/10.1016/j.scriptamat.2016.05.044 1359-6462/© 2016 Elsevier Ltd. All rights reserved.
to the formation of a yield plateau and the development of localized shear bands [10,13,14]. A geometric compatibility parameter, mʹ = cosκ · cosψ, where κ and ψ are angles between the two shear directions and two twin plane normals for twin systems in two neighboring grains, has also been used for analysis of twin transmission in HCP metals [6,10,12,15]. This parameter, which has also been used to analyze strain accommodation between slip systems, and between slip and twinning systems [16,17] can be thought of as providing a measure of the efficiency of strain transfer across a grain boundary. Many studies that use this parameter to analyze twin transmission do so on a case by case basis, examining only grain pairs where twin transmission is observed. Here instead we examine data for a large number of grains, also considering grain pairs where twin transmission is not seen. In particular our interest is in not only trying to understand whether the misorientation angle and mʹ parameters can explain twin transmission observations, but also in determining to what extent they can be used to predict twin transmission events. The material used in this study was a commercially purchased hot-rolled AZ31 (Mg—3%Al—1%Zn) sheet. Samples of dimensions 5 mm × 8 mm × 10 mm (rolling direction (RD) × transverse direction (TD) × normal direction (ND)) were cut from the middle of the sheet thickness. The sample had a typical hot-rolled texture with a {0001}||ND fiber texture. Plane strain compression tests (loading direction parallel to TD, i.e. in a direction favorable for {10−12} twinning in this material) were performed at a strain rate of 10−3 s−1 to a reduction of 1.9% at room temperature, using a Gleeble 1500D system. Electron backscatter diffraction (EBSD) observations of the deformed microstructure were carried out on samples sectioned in a mid-plane parallel to the initial TD-ND plane, then mechanically ground to 2000-grit SiC paper, followed by polishing using diamond paste, and then vibratory polishing using colloidal silica. Prior to EBSD examination the sample
78
X. Hong et al. / Scripta Materialia 123 (2016) 77–80
surface was etched for 10–20 s using a solution of 10 ml nitric acid, 30 ml acetic acid, 40 ml water and 120 ml ethanol. A typical example of the microstructure of a deformed sample is shown in Fig. 1a, where many instances of twin transmission can be clearly identified. In this study a total of 6886 grain boundaries, including 306 pairs of transmitted twins have been analyzed. To maximize the information available concerning twin transmission at very low angle boundaries, the misorientation angle dependence has been analyzed where necessary using 5° bins starting from the lower EBSD cut-off angle of 2° (rather than starting from 0° with a low angle cut-off at 2° as more commonly used, resulting in a first bin of reduced width). Fig. 1b plots the frequency of twin transmission as a function of grain boundary misorientation angle. For comparison the same data analyzed using bins starting at 0° are shown in the inset (top right corner of Fig. 1b), where an apparent maxima in twin transmission is seen in the 3rd bin (10–15°). The results are nevertheless in general agreement with previous studies [5,10,11]. A better parameter, however, to analyze twin transmission is the number of boundaries with a transmitted twin in a given misorientation angle range, normalized by the total number of boundaries within that range, fTT, as plotted in Fig. 1c. Here it is seen that the fraction of boundaries where twin transmission takes place increases continuously with decreasing misorientation angle. However, for misorientation angles in the range of 2–7°, the value of fTT lies under the value predicted from extrapolation of the values for higher boundary misorientation angles (see the line in Fig. 1c), suggesting that twin transmission is not increasingly easy with decreasing misorientation angle, although number of grain boundaries within this misorientation angle range is relatively small. Further insight is obtained by considering also grain pairs where a twin is present in one grain, but where twin transmission is not observed. To do this an area was examined containing 316 such grain boundaries. The fraction of grain pairs where twin transmission is seen with respect to the number of boundaries with at least one twin (i.e. including cases where a twin is present only on one side of a grain boundary), fʹTT, is shown as a function of boundary misorientation
angle in the inset of Fig. 1c. It is worth noting that the complementary fraction (1 − fʹTT) measures the probability that a boundary twin does not result in twin transmission. A general trend of increasing twin transmission fraction with decreasing misorientation angle is again seen, except for the lowest angle bin (representing 43 grain boundaries where at least one twin is present), indicating that some influence of strain incompatibility in assisting twin transmission may also play a role in this process. The data in Fig. 1 represent an experimental description of twin transmission as a function of misorientation angle. To probe the intrinsic dependence of this process on grain boundary misorientation it is also necessary to consider any possible influence of the variation in Schmid factor (SF) on twinning [18–22]. In general it is found a higher SF value gives an increased likelihood of twinning on a particular twin system, though the preferential activation of twins with lower ranked Schmid factors, or even of twins with a shear strain in the opposite sense to the applied load, has also been observed [19–21,23]. The potential influence of SF on twin transmission is highlighted in Fig. 2, which shows data for grain pairs in an undeformed sample, and plots the relationship between grain boundary misorientation angle and the maximum SF for twinning of the grains on each side of each boundary. The shape of the distribution for this typically textured sample, with increasing density of grain pairs towards the (0.5, 0.5, 0°) corner (highlighted by the projection onto the (SF1, SF2 plane)) indicates that grain pairs with low angle also tend to have high Schmid factors. The corollary of this is that a higher fraction of randomly chosen grain pairs where both grains have high SFs will have a low misorientation angle compared to grain pairs where both grains have lower SFs. As a simple test of whether this correlation has an influence on the observed misorientation angle dependence of twin transmission, the value of fTT has been recalculated considering only grain pairs where the SF values for both grains are in a certain SF range, either 0.4 b SF ≤ 0.5 or 0.3 b SF ≤ 0.4. The results are shown in Fig. 3, where it seen that the dependence of fTT on misorientation angle is noticeably different for the two cases. Note here that although the
Fig. 1. (a) EBSD map (inverse pole figure coloring) showing twin transmission events in a sample of AZ31 after compression to a strain of 1.9% (b) frequency distribution of twin transmission events as a function of grain boundary misorientation angle; (c) fraction of grain boundaries in each misorientation angle range (bins of 5° width) where twin transmission is seen, fTT (specifically, number of boundaries with twin transmission in misorientation angle range Δθ, divided by the number of grain boundaries in the same angle range). The dotted line represents a fit to the data for bins with θ N 7°. The inset shows the fraction of transmitted boundaries relative to the number of grain boundaries where at least one twin is present.
X. Hong et al. / Scripta Materialia 123 (2016) 77–80
Fig. 2. Graph showing the correlation as a result of the initial texture between grain boundary misorientation angle and the maximum Schmid factors for twinning (SF1, SF2) of the grains forming each boundary. Also shown is a projection onto the (SF1,SF2) plane highlighting the peak (0.5, 0.5, 0°).
data for 0.3 b SF ≤ 0.4 show some scatter, these results are taken from an area covering more than 650 grain pairs in this category. The differences are expected to be smaller when comparing more similar SF ranges, but nevertheless highlight the need to consider such an influence in any detailed analysis of the intrinsic dependence on boundary angle of twin transmission. An alternative parameter that has been used for analysis of twin transmission events is the geometric compatibility factor, mʹ, representing the degree of alignment between the twinning systems in adjacent grains. Fig. 4 shows the value of mʹ for each of 306 twin pairs taken from the investigated sample. In each case the twin system was first identified in each grain [24], and then the geometric compatibility factor for the twin pair, mʹTT, was calculated. More than 95% of the twin pairs have a value of mʹTT N 0.7 (Fig. 4a), with almost 70% having mʹTT N 0.9. In many cases, however mʹTT is not the maximum possible value for the grain pair (mʹmax, calculated by considering the 6 possible twin variants for each grain) as illustrated in Fig. 4b. This difference reflects the fact that it is possible for two twin systems to be well aligned (giving high mʹmax), but for neither to be well oriented for twinning (i.e. have a low SF). To further understand the twin transmission events, we calculate the parameter mʹ(1,2), defined as the maximum value of mʹ based on the observed twin on one side, i.e. assuming that the twin first forms in one grain (grain 1 or 2) and is then transmitted to the other (grain 2 or 1) on one of the possible 6 twin systems. Note that as the sequence is not known we consider therefore both possibilities and use both values if they are different.
79
For the current data more than 98% twin transmission events are found to have mʹ values equal to mʹ(1,2). This result agrees with the analysis in previous investigations [12,15,17] that the transmitted twin variant is always the one with the highest mʹ value correlating with the first formed twin. These data, however, do not provide information about the ability of this parameter to predict occurrences of twin transmission. For this, it is necessary to consider also grain pairs in which twin transmission is not seen (i.e. as previously considering also grain pairs where a twin is present only in one grain). Such an analysis is given in Fig. 4b, where for the same EBSD map (316 grain pairs with one boundary twin and 116 grain pairs where twin transmission is seen) mʹ values are plotted as a function of boundary misorientation angle for all grain boundaries where twin transmission is observed (mʹTT), and for all grain boundaries where a twin is present in one grain but is not transmitted (mʹ1T, based on the relationship of the observed twin to the 6 possible variants in the neighbor grain). Consistent with the earlier analysis based on misorientation angle, the number of grains where twin transmission does not take place increases with increasing boundary misorientation angle. A more important observation is that for misorientation angles less than ≈30° the mʹ1T values are not systematically lower compared to the mʹTT values, but cover a similar range, i.e. grain boundary pairs exist where a boundary is twinned on one side but not the other even though a twin system is well aligned in the neighboring grain (i.e. has a high mʹ value). The dependence of twin transmission on the geometric compatibility parameter is probed further by calculation of the complementary cumulative distribution function (CCDF) for twin transmission, defined here as f CCDF ðxÞ ¼
N m0TT Nx N m0TT Nx þ N m01T Nx
ð1Þ
where N(mʹ N x) is the number of grain boundary pairs with mʹ N x for cases either of twin transmission (TT) or a single boundary twin (1T). This value represents the fraction of grain boundaries where twin transmission is seen relative to the number of all grain pairs with at least one boundary twin (i.e., including boundaries where twin transmission could have taken place but did not), for all boundaries with the geometric compatibility factor (mʹTT for twin transmission pairs and mʹ1T for single twin boundaries) greater than a given value. The results are shown in Fig. 4c. A positive correlation with fCCDF is seen with increasing x for low values (mʹ b 0.9), indicating that in this range larger mʹ values for a twinned boundary correspond to a higher probability of twin transmission taking place. However, in the range 0.9 b x b 0.95 the value of fCCDF is approximately constant at ≈0.5. In this range therefore the fraction of possible twin transmission events is rather insensitive to the value of mʹ. It can be noted also that this range of mʹ values corresponds in the present sample to grain boundary
Fig. 3. Fraction of grain boundaries where twin transmission is seen as a function of grain boundary misorientation angle (analyzed in bins of 5° width) for grain pairs where both grains have maximum Schmid factors in the range (a) 0.40–0.50, or (b) 0.30–0.40.
80
X. Hong et al. / Scripta Materialia 123 (2016) 77–80
Fig. 4. (a) Normalized frequency distribution (bin width = 0.1) of the geometric compatibility factor (mʹ) for the observed 306 twin transmission events; (b) plot showing mʹ values for each observed twin transmission pair (mʹTT, red squares), and maximum mʹ value for grain boundaries with twin on only one side of the boundary (mʹ1T, green triangles). The small blue circles show the maximum possible mʹmax values for all grain pairs in the sample; (c) complementary cumulative distribution function showing the fraction of grain boundary pairs with at least one twin where twin transmission takes place.
misorientation angles from 13° to 25°, i.e. the range where one third of twin transmission events are seen. At larger values (mʹ N 0.95) a strong positive correlation with mʹ is again seen, though for very high mʹ values (corresponding to very low angle misorientations) a flattening of the curve is found, in agreement with the trends also seen in Fig. 1c. It is worth noting that although misorientation angle and mʹ are inter-related, the two parameters show some differences with regard to prediction of twin transmission events (i.e. when considering all grain pairs where at least one twin is present). In particular the relative insensitivity of the probability of twin transmission to mʹ for the range of boundary misorientation angles where such events are seen is likely to reflect the fact that any specific twin transmission event (including nucleation of twin pairs at a grain boundary) may be influenced by a combination of factors, including the local stress state, the presence of pre-existing twins, and the local dislocation configuration [9]. The observations therefore highlight the advantage of modelling twin transmission in a similar probabilistic (rather than deterministic) manner to that already proposed for single twinning events [25]. In summary, twin transmission during plane strain compression of an AZ31 magnesium sample has been investigated with a view to understand better how such events can be predicted. It is found in agreement with previous studies that in general twin transmission probability increases with decreasing boundary misorientation angle, though some influence of the Schmid factor on twinning is also found and may need to be considered when analyzing the intrinsic ease of twin transmission as a function of boundary misorientation. Although the geometric compatibility factor, mʹ, is found to be useful for explaining occurrences of twin transmission, this parameter is not so efficient for prediction of such events, as the probability of twin transmission is insensitive to mʹ for a range of values over which 1/3 of the observed twin transmission events are observed. For both the misorientation angle and the mʹ parameter the experimental data show that twin transmission does not become increasingly easy with decreasing misorientation angle in the very low misorientation angle regime, suggesting that strain incompatibility may play some role also in nucleation of twin transmission events. Such very low angle boundaries are, however, relatively infrequent for samples with a typical texture.
Acknowledgements Financial support for this work from the National Key Basic Research Program of China (grant number 2013CB632204) is gratefully acknowledged.
References [1] J.W. Christian, S. Mahajan, Prog. Mater. Sci. 39 (1995) 1–157. [2] M.H. Yoo, Metall. Trans. A 12 (1981) 409–418. [3] I.J. Beyerlein, L. Capolungo, P.E. Marshall, R.J. McCabe, C.N. Tomé, Philos. Mag. 90 (2010) 2161–2190. [4] J. Wang, I.J. Beyerlein, C.N. Tomé, Scr. Mater. 63 (2010) 741–746. [5] I.J. Beyerlein, R.J. McCabe, C.N. Tomé, J. Mech. Phys. Solids. 59 (2011) 988–1003. [6] L. Wang, P. Eisenlohr, Y. Yang, T.R. Bieler, M.A. Crimp, Scr. Mater. 63 (2010) 827– 830. [7] R.J. McCabe, G. Proust, E.K. Cerreta, A. Misra, Int. J. Plast. 25 (2009) 454–472. [8] J. Kacher, A.M. Minor, Acta Mater. 81 (2014) 1–8. [9] A. Khosravani, D.T. Fullwood, B.L. Adams, T.M. Rampton, M.P. Miles, R.K. Mishra, Acta Mater. 100 (2015) 202–214. [10] M.R. Barnett, M.D. Nave, A. Ghaderi, Acta Mater. 60 (2012) 1433–1443. [11] A. Fernández, A. Jérusalem, I. Gutiérrez-Urrutia, M.T. Pérez-Prado, Acta Mater. 61 (2013) 7679–7692. [12] R.L. Xin, C.F. Guo, Z.R. Xu, G.D. Liu, X.X. Huang, Q. Liu, Scr. Mater. 74 (2014) 96–99. [13] S. Sandlöbes, S. Zaefferer, I. Schestakow, S. Yi, R. Gonzalez-Martinez, Acta Mater. 59 (2011) 429–439. [14] H.E. Kadiri, J. Kapil, A.L. Oppedal, L.G. Hector Jr., S.R. Agnew, M. Cherkaoui, S.C. Vogel, Acta Mater. 61 (2013) 3549–3563. [15] C.F. Guo, R.L. Xin, C.H. Ding, B. Song, Q. Liu, Mater. Sci. Eng. A 609 (2014) 92–101. [16] J. Luster, M.A. Morris, Metall. Mater. Trans. A 26A (1995) 1745–1756. [17] L. Wang, Y. Yang, P. Eisenlohr, T.R. Bieler, M.A. Crimp, D.E. Mason, Metall. Mater. Trans. A 41 (A) (2010) 421–429. [18] S.-G. Hong, S.H. Park, C.S. Lee, Acta Mater. 58 (2010) 5873–5885. [19] M.R. Barnett, A. Ghaderi, J.Q. da Fonseca, J.D. Robson, Acta Mater. 80 (2014) 380–391. [20] L. Capolungo, P.E. Marshall, R.J. McCabe, I.J. Beyerlein, C.N. Tomé, Acta Mater. 57 (2009) 6047–6056. [21] A. Chapuis, Y.C. Xin, X.J. Zhou, Q. Liu, Mater. Sci. Eng. A 612 (2014) 431–439. [22] Y. Pei, A. Godfrey, J. Jiang, Y. Zhang, W. Liu, Q. Liu, Mater. Sci. Eng. A 550 (2010) 138–145. [23] Z.-Z. Shi, Y.D. Zhang, F. Wagner, P.-A. Juan, S. Berbenni, L. Capolungo, J.-S. Lecomte, T. Richeton, Acta Mater. 83 (2015) 17–28. [24] J. Jiang, A. Godfrey, W. Liu, Q. Liu, Scr. Mater. 58 (2008) 122–125. [25] I.J. Beyerlein, C.N. Tome, Proc. R. Soc. A 466 (2010) 2517–2544.