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Compound cross-grain boundary extension twin structure and its related twin variant selection in a deformed Mg alloy
Journal of Alloys and Compounds 716 (2017) 128e136
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Compound cross-grain boundary extension twin structure and its related twin variant selection in a deformed Mg alloy Zhang-Zhi Shi School of Materials Science and Engineering, University of Science and Technology Beijing, Xueyuan Road 30, Beijing 100083, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 January 2017 Received in revised form 2 May 2017 Accepted 3 May 2017 Available online 6 May 2017
Interrupted bi-axial compressions are applied to a hot-rolled Mg alloy with a strong basal texture in order to activate {1 0 1 2}-{1 0 1 2} double extension twinning. Compound cross-grain boundary (cross-GB) extension twin structures form after the compressions, which consist of cross-GB secondary twin pairs within their host cross-GB primary twin pairs. Shear accommodation of a twin is proven to be proportional to a geometrical m0 factor and the magnitude of its twinning shear. It is found that 60% of cross-GB twin pairs, including both the primary and the secondary ones, have a value of m0 >0.7, indicating twinning shear transmission over GB is a major mechanism. Multi-level twinning shear transmissions over GB contribute prominently to the formation of 40% of the observed compound cross-GB twin structures. About 98.3% of the twins are high Schmid factor (SF) ones, while low SF ones appear due to twinning shear transmission. About 81.7% of secondary twins have a misorientation of 〈0 14 14 1〉60 with respect to their host grains, indicating a strong confining effect of primary twins. A cross-GB twin pair could form through a mechanism of associated nucleation or isolated nucleation. High m0 value is characteristic of associated nucleation through twin-to-twin accommodation mode. Depending on the process of microstructure evolution, SF, shear accommodation and the confining effect of the matrix if necessary should be synthetically considered to predict twin variant selection related to cross-GB twin pairs. © 2017 Elsevier B.V. All rights reserved.
Keywords: Magnesium alloys EBSD Twinning Grain boundary Strain path change
1. Introduction Twinning is an important deformation mode of Mg alloys. Interaction between twins and grain boundaries (GBs) is vital for twinning in polycrystalline Mg alloys. Twin nucleation occurs preferentially at GBs, which is supported by atomistic simulation and electron back-scattering diffraction (EBSD) observation [1e3]. Schmid factor (SF) is an effective criterion for twin nucleation, which evaluates how much the macro applied stress is resolved to be the shear stress on a twinning system. Not surprisingly, twin variants with high SFs nucleate preferentially [4e7]. After nucleation, a twin usually preferentially elongates in its shear direction, which evolves shearing the matrix. It relaxes elastic energy in the matrix, but concentrates applied stress at its tip, inducing localized plasticity [8]. When it collides with a GB, the immediate neighborhood in the adjoining grain must also be sheared. Otherwise, stress will arise and accumulate in the adjoining grain along with
E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.jallcom.2017.05.020 0925-8388/© 2017 Elsevier B.V. All rights reserved.
twin growth. In addition, stress will also arise outside of a twin due to the confining effect of the surrounding matrix according to Eshelby inclusion theory [9e12]. So twin-induced stress at GB originates at least from three sources: (I) stress induced by twinning shear due to GB blocking effect, (II) applied stress concentrated at twin tip, and (III) stress resulted from the confining effect of the matrix. Twin-induced stress at GB may intrigue twin nucleation in the adjoining grain, resulting in a cross-GB twin pair [3,13e19]. Several connected cross-GB twin pairs form a long cross-GB twin band [15,16]. Intuitively, a cross-GB twin pair may form directly through twinning shear transmission over GB, which is indeed proven to be a major mechanism [13,14,16e18]. Twinning shear transmission over GB is often estimated by a geometric compatibility factor, m0 ¼ cosa,cosb, where a and b are angles between two twinning shear directions and two twinning plane normals for twinning systems in two neighboring grains, respectively [13,17,18,20,21]. When m0 ¼ 1, two twinning shears are fully compatible. This implies that elastic energy in a grain due to twin-induced stress at GB from source (I) could be fully relaxed, which is energetically
Z.-Z. Shi / Journal of Alloys and Compounds 716 (2017) 128e136
favorable. Statistical EBSD analysis reveals that 95% of 306 cross-GB primary {1 0 1 2} twin pairs in AZ31 Mg alloy have a m0 value higher than 0.7, with almost 70% having a m0 value higher than 0.9 [13]. When m0 ¼ 0, two twinning shears are independent. When m0 ¼ 1, two twinning shears are exactly opposite. By now, cross-GB primary {1 0 1 2} extension twin pairs have received the most attention [3,13,14,16e19], largely due to their high EBSD pattern indexing rates and their frequently appearance during uni-axial deformation. Recently, cross-GB primary {1 0 1 1} contraction twin pairs have been studied, in which their low EBSD pattern indexing rates have been compensated by trace analysis [15]. A minority of either cross-GB primary {1 0 1 2} or {1 0 1 1} twin pairs have been reported to have m0 values lower than 0 [15,17,22], indicating that a cross-GB primary twin pair can also form due to mechanisms other than twinning shear transmission over GB. Is it possible that two secondary twins form a cross-GB secondary twin pair within their host cross-GB primary twin pair? If they formed, is there any big difference between cross-GB secondary and primary twin pairs, since the parent crystals of the former one are twins but not ordinary grains? Although large uniaxial deformation can activate profuse {1 0 1 1}e{1 0 1 2} double twinning [23e27], difficulties arise from low EBSD pattern indexing rate and the limitation of trace analysis [15]. Fortunately, small biaxial deformation is enough to activate profuse {1 0 1 2}e{1 0 1 2} double extension twinning [5,10,28,29]. Cross-GB secondary {1 0 1 2} twin pairs have greater chances to form and to be identified by EBSD, which will provide good chances to explore the above problems.
129
system, a rectangular coordinate system (d, m, n) of it can be defined by unit vectors d//the twinning shear direction, n//the twinning plane normal, and m//the cross-product of n and d. In such a coordinate system, twinning shear tensor ‘S1’ has the simplest form [22,31], which is:
2
0 S1 ¼ 4 0 0
0 0 0
3 s 0 5; 0
(1)
2. Material and methods
where ‘s’ is the magnitude of the twinning shear. The magnitude of the twinning shear depends on crystal structure and twinning system. In Mg with c/a ¼ 1.624, s ¼ 0.129 for {1 0 1 2} twinning, while s ¼ 0.138 for {1 0 1 1} twinning [31,32]. In Ti with c/a ¼ 1.587, s ¼ 0.175 for {1 0 1 2} twinning, while s ¼ 0.218 for {1 1 2 2} twinning [32,33]. Similarly, a rectangular coordinate system (a, b, c) of any deformation system in Crystal 2 can be defined by unit vectors a//the twinning shear or slip direction, c//the twinning or slip plane normal, and b//the cross-product of c and a (Fig. 1). Twinning in Crystal 1 shears a fraction of it, which should be accommodated through deformation systems in Crystal 2 [22,31,33,34]. The shear which could be accommodated through the deformation system (a, b, c) can be calculated by rotating the twinning shear tensor S1 from coordinate systems (d, m, n) to (a, b, c) [22,31]. For calculation, the components of all the unit vectors are defined in a public coordinate system (x, y, z), in which unit vectors x, y, z can be selected to be parallel to sample directions, e.g., x//RD, y//TD, and z//ND. For rotating any second-order tensor from coordinate systems (d, m, n) to (x, y, z), the rotation matrix R1 is defined by:
2.1. Experimental procedure
R1 I ¼ ½d; m; n;
The material used was a commercial purchased hot-rolled Mg3Al-1Zn (AZ31) alloy sheet, which was annealed at 350 C for 1 h after rolling. The rolling, transverse and normal directions of the sheet were designated as RD, TD and ND, respectively. The annealed sheet exhibited a typical hot-rolled texture with most of the grains' [0 0 0 1] directions (i.e., c-axes) centering around ND, as reported previously [15]. Cubic samples of 10 mm length were cut from the sheet for compression tests. According to the initial texture and twinning geometry [10,29], compressive loading was applied first along RD to a reduction of 2.0% for producing primary {1 0 1 2} extension twins (ETWs), unloaded and then applied along TD also to a reduction of 2.0% for producing secondary ETWs, at a strain rate of 103 s1 at room temperature. The deformed samples were sectioned in a mid-plane normal to ND for EBSD observations. The sectioned faces were first ground using 2000 to 4000 grit SiC papers, and then electrolytically polished in an electrolyte of 62.5% phosphoric acid and 37.5% ethanol at 1.5 V for 1 min, at 15 C. The measured EBSD date was analyzed using Matlab™ MTEX toolbox (version 4.1.beta4, mtex-toolbox.github.io) [30].
2
d1 where matrices [d, m, n] ¼ 4 d2 d3 one can get:
R1 ¼ ½d; m; n:
m1 m2 m3
2 3 1 n1 n2 5, and I ¼ 4 0 0 n3
(2)
0 1 0
3 0 0 5. So 1
(3)
For rotating any second-order tensor from coordinate systems (x, y, z) to (a, b, c), the rotation matrix R2 is defined by:
R2 ½a; b; c ¼ I;
2
a1 where matrix [a, b, c] ¼ 4 a2 a3
(4)
b1 b2 b3
3 c1 c2 5. Similar to matrix [d, m, n], c3
2.2. Association of m0 factor with transmitted shear The following model in Fig. 1 is constructed in order to clarify the relationship between m0 factor and transmitted shear. There are two crystals in the figure, which could be grains, primary twins or secondary twins, etc. In each crystal, one deformation system is chosen to be under study, which could be a twinning or a slip system. The two deformation systems are supposed to be connected with each other at a boundary separating the two crystals, which could be a grain boundary, a twin-twin boundary, or a twingrain boundary, depending on what the two crystals are. If the deformation system in Crystal 1 (Fig. 1) is a twinning
Fig. 1. A sketch of shear transmission from one crystal into its neighboring crystal.
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matrix [a, b, c] is also a rotation matrix, so that the inverse of [a, b, c] equals the transpose of it, i.e., [a, b, c]1 ¼ [a, b, c]T. Then one can get:
2
a1 R 2 ¼ 4 b1 a3
a2 b2 b3
3 a3 b3 5 : c3
(5)
Then, for rotating the tensor S1 from coordinate systems (d, m, n) to (a, b, c), the rotation matrix R is:
2
a$d R ¼ R2 R1 ¼ 4 b$d c$d
a$m b$m c$m
3 a$n b$n 5: c$n
(6)
The resulted tensor S2 after the rotation is:
S2 ¼ R$S1 $R1 :
(7)
The tensor S2 is a second-order tensor, whose 9 elements are eij (i, j ¼ 1e3). The element e13 has a particular physical meaning [22]. According to the selection of coordinate system (a, b, c), e13 corresponds to the shear along the direction parallel to the unit vector a, on the plane normal to the unit vector c. That is to say, e13 corresponds to the shear that can be accommodated through the deformation system (a, b, c). From Eq. (1), Eq. (6) and Eq. (7), it can be deduced that:
e13 ¼ ða$dÞðc$nÞs ¼ ðcosa$cosbÞs ¼ m0 $s;
(8)
where a is the angle between two twinning shear or slip directions, while b is the angle between two twinning or slip plane normals, for two deformation systems in two neighboring crystals (Fig. 1). From Eq. (8), it is clear that m0 in fact represents how much the shear on a deformation system in a crystal can transmit over a boundary onto another deformation system in its neighboring crystal. For a transmission from a twinning system to a slip system (i.e., twin-to-slip), jm0 j should be taken into consideration, since slip could be bi-directional. However, for a transmission from a twinning system to another twinning system (i.e., twin-to-twin), m0 should be taken into consideration, since twinning is unidirectional. In twin-to-twin case, twinning shear transmission could happen when m0 >0, the magnitude of which is proportional to m0 value, according to Eq. (8). Afterwards, m0 as default refers to twin-to-twin accommodation mode in the present paper.
3. Experimental results and analysis Fig. 2a shows a compound cross-GB twin structure measured by EBSD after the two-step compressions. In the figure, boundaries with misorientations larger than 5 are outlined in black, while those with a misorientation of 〈1 2 1 0〉86.3 (±7 offset) are outlined in white, indicating an {1 0 1 2}〈1 0 1 1〉 extension twinning relationship. For convenience, host grain, primary twin, and secondary twin are designated by G, G-P, and G-P-S, respectively. Such a designation clearly points out the parent-twin relationship in a complex microstructure, the advantage of which will be shown afterwards. The six possible ETW variants (EVs) are defined as follows [22,34]: EV1: (1 0 1 2)[1 0 1 1], EV2: (0 1 1 2) [0 1 1 1], EV3: (1 1 0 2)[1 1 0 1], EV4: (1 0 1 2)[1 0 1 1], EV5: (0 1 1 2)[0 1 1 1], and EV6: (1 1 0 2)[1 1 0 1]. Usually, the SFs of the EVs are different, which can be ranked from the highest one to the lowest one, i.e., rank 1 corresponds to the highest one while rank 6 corresponds to the lowest one. Note that secondary ETWs and their host primary twins form respectively during the second
TD and the first RD compressions, so that SFs of them should be calculated respectively with loadings along TD and RD [10,29]. In addition, SF ratio of a certain twin variant EVi (i ¼ 1e6) can be calculated by SFratio¼(SF of EVi)/(SF of rank 1). SF, SF rank and SF ratio together clearly indicate whether a certain twin variant EVi is favored by macro applied stress, compared with its competitive twin variants EVj (j ¼ 1e6, jsi) in the same host crystal, which could be a grain or a primary twin. In Fig. 2a, primary twins G1-P1 and G2-P1 form a cross-GB primary twin pair during the first RD compression. SFs, SF ranks and SF ratios of G1-P1 and G2-P1 are calculated to be 0.494/2/0.988 and 0.489/2/0.986, respectively. Although both of them are high SF twin variants, those with the highest SFs in grains G1 and G2 are absent. In fact, when SF ratio is larger than 0.98, the SF difference between the highest and the second highest ones in a grain should be smaller than 0.01, which is too small to be distinguished for twin variant selection. In this case, additional criteria ought to be adopted. In Table 1, SFs of the six possible EVs in grain G2 and their m0 values with respect to G1-P1 have been listed. It can be seen that G2-EV3 corresponding to G2-P1 in Fig. 2a has the largest m0 value close to 1, indicating almost complete twinning shear transmission over GB. Although G2-EV6 has the largest SF, its m0 value is close to 0. During the second TD compression, secondary twins G1-P1-S1 and G2-P1-S1 form a cross-GB secondary twin pair within their host cross-GB primary twins, resulting in a compound cross-GB twin structure. SFs, SF ranks and SF ratios of G1-P1-S1 and G2P1-S1 are calculated to be 0.461/1/1 and 0.396/2/0.983, respectively. The appearance of G1-P1-S1 is anticipated, since its SF is the highest with a SF rank of 1. However, according solely to SF, it is unclear why G2-P1-S1 with a SF rank of 2 is the most preferred. In Table 1, SFs of the six possible EVs in primary twin G2-P1 and their m0 values with respect to G1-P1-S1 are also listed. It can be seen that variant G2-EV3-EV6 corresponding to twin G2-P1-S1 in Fig. 2a has the largest m0 value close to 1. Therefore, multi-level twinning shear transmissions happen over the GB, in which the primary and the secondary twinning shear transmissions are related to the primary and the secondary twins, respectively. Thirty compound cross-GB twin structures similar to that in Fig. 2a have been detected by several EBSD measurements. Fig. 2b shows the 60 m0 values associated with them. Twelve of the 30 (i.e., 40%) compound cross-GB twin structures have both m0 values of cross-GB primary and secondary twin pairs larger than 0.7. Therefore, multi-level twinning shear transmissions over GB prominently contribute to their formations. The threshold for ‘prominent’ is taken as m0 0.7 in this study. Note that twinning shear transmission can happen as long as m0 >0, according to Eq. (8). The SFs and SF ratios of their related 48 (¼12 4) twins are shown in subfigure (I) in Fig. 2c. It can be seen that the majority (i.e., ~95.8%) of the twins have SF ratios larger than 0.8, indicating that they are high SF twins compared with their competitive twin variants in their host crystals. Twinning shear transmission over GB can activate low SF twin variants in compound cross-GB twin structures. Such a microstructure is shown in Fig. 3a. Primary twins G3-P1 (SF/SF rank/SF ratio ¼ 0.458/1/1) and G4-P1 (SF/SF rank/SF ratio ¼ 0.395/1/1) with the highest SFs form a cross-GB primary twin pair with the highest m0 value, as can be seen in Table 2. Although G4-EV4 has a high m0 value larger than 0.7 (Table 2), its SF is close to 0. Although G4-EV2 has the second highest SF (Table 2), its m0 value with G3-P1 is close to 0. Therefore, twin variant G4-EV5 with both the highest SF and m0 values is highly preferred than its competitors. Due to the second applied TD stress, secondary twins G3-P1-S1 (SF/SF rank/SF ratio ¼ 0.328/2/0.912) and G4-P1-S1 (SF/SF rank/SF ratio ¼ 0.178/4/ 0.598) form a cross-GB secondary twin pair in their host cross-GB
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Table 1 For prominent multi-level twinning shear transmissions over GB in Fig. 2a, SFs of the six possible EVs in grain G2 (i.e., G2-EVi, i ¼ 1e6) and their m0 values with G1-P1, the related a and b angles, and those for the six possible EVs in primary twin G2-P1 (i.e., G2-EV3-EVj, j ¼ 1e6) with G1-P1-S1.
G2-EVi Observed a ( ) b ( ) m0 value SF G2-EV3-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 87.21 67.36 0.019 0.163
e 52.51 34.87 0.499 0.087
G2-P1 9.74 5.40 0.981 0.489
e 36.62 42.68 0.590 0.159
e 75.96 72.56 0.073 0.090
e 97.72 82.85 0.017 0.496
e 44.50 36.10 0.576 0.337
e 76.29 73.04 0.069 0.001
e 87.71 92.80 0.002 0.403
e 71.38 82.70 0.041 0.344
e 37.30 49.95 0.512 0
G2-P1-S1 7.05 10.29 0.977 0.396
Candidates of preferred twin variants are highlighted in bold.
primary twins, resulting in a compound cross-GB twin structure (Fig. 3a). Although the SF of G4-P1-S1 is low, it has the highest m0 value with G3-P1-S1, which is close to 1. High SF secondary twin G3-P1-S1, whose parent is primary twin G3-P1 with a much larger size than primary twin G4-P1, is likely to be first activated and then intrigues secondary twinning in G4-P1. Since the highest possible SF of G4-EV5-EVj (j ¼ 1e6) is smaller than 0.3 (Table 2), neither of them could largely release the elastic energy aroused by the applied TD stress in G4-EV5. In this case, it is more requisite to release the increased elastic energy in G4-EV5 due to twin shearing of G3-P1S1, which favors G4-EV5-EV1 with a very high m0 value close to 1 the most. Six of the 30 (i.e., 20%) compound cross-GB twin structures have m0 values of cross-GB primary twin pairs larger than 0.7, but those of cross-GB secondary twin pairs smaller than 0.2, as shown in Fig. 2b. They are compound cross-GB twin structures solely featured with prominent primary twinning shear transmission over GB. A representative microstructure of them is shown in Fig. 3b, in which Euler angles of each microstructure component in concern are also listed in order to make clear parent-twin relationship by calculation. In grain G6, primary twins G6-P1(a~d) all correspond to twin variant G6-EV3, which are separated by secondary twins G6P1-S1 and G6-P1-S2. This indicates that G6-P1(a~d) are remnants of a primary twin G6-P1. High SF primary twins G6-P1 (SF/SF rank/ SF ratio ¼ 0.365/2/0.909) and G5-P1 (SF/SF rank/SF ratio ¼ 0.420/2/ 0.939) form a cross-GB primary twin pair. It can be seen in Table 3 that G6-P1 (i.e., G6-EV3) has the highest possible m0 value of 0.874 with G5-P1. Although G6-EV6 has the highest SF (Table 3), its m0 value is close to 0. Although G6-EV2 has a high m0 >0.7 (Table 3), its SF is lower than 0.15. However, prominent secondary twinning shear transmission does not happen in Fig. 3b. High SF secondary twins G5-P1-S1 (SF/ SF rank/SF ratio ¼ 0.496/2/0.995) and G6-P1-S1 (SF/SF rank/SF ratio ¼ 0.343/1/1) form a cross-GB secondary twin pair. It can be seen in Table 3 that none of the possible secondary twin variants G6-EV3-EVj (j ¼ 1e6) has a m0 value larger than 0.7 with G5-P1-S1.
Fig. 2. (a) A compound cross-GB twin structure measured by EBSD. Boundaries with misorientations larger than 5 are outlined in black, while those with a misorientation of 〈1 2 1 0〉86.3 (±7 offset) are outlined in white, indicating an {1 0 1 2}〈1 0 1 1〉 extension twinning relationship. The measured Euler angles of the crystals are listed in the down part of the figure. (b) 30 compound cross-GB twin structures and their corresponding m0 values, in which symbols ‘o’ and ‘*’ refer to cross-GB priamry and secondary twin pairs, respectively. (c) SFs and SF ratios of 120 twins associated with the 30 compound cross-GB twin structures, in which symbols ‘o’ and ‘*’ refer to priamry and secondary twins, respectively.
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Fig. 3. Compound cross-GB twin structures with: (a) proniment multi-level twinning shear transmissions, resulting in the formation of a low SF twin G4-P1-S1; (b) solely prominent primary twinning shear transmission; (c) solely prominent secondary twinning shear transmission; (d) nearly none twinning shear transmission over GB. The measured Euler angles of the crystals are listed in the down parts of the figures.
In this case, secondary twin variants with high SFs should be preferred. The observed G6-P1-S1 corresponds to twin variant G6EV3-EV1 with the highest SF (Table 3). Another observed one is G6P1-S2 in some distance away from G5-P1-S1, which corresponds to twin variant G6-EV3-EV4 with the second highest SF (Table 3). Six of the 30 (i.e., 20%) compound cross-GB twin structures have m0 values of cross-GB primary twin pairs smaller than 0.7, but those of cross-GB secondary twin pairs larger than 0.7, as shown in Fig. 2b. They are compound cross-GB twin structures solely featured with prominent secondary twinning shear transmission over GB. A representative microstructure of them is shown in
Fig. 3c. High SF primary twins G7-P1 (SF/SF rank/SF ratio ¼ 0.471/2/ 0.964) and G8-P1 (SF/SF rank/SF ratio ¼ 0.497/2/0.994) form a cross-GB primary twin pair. Although G8-EV5 has a very high m0 value close to 1 with G7-P1 (Table 4), its SF is lower than 0.15. Although G8-EV3 has the highest SF (Table 4), its m0 value with G7P1 is close to 0. Twin variant G8-EV6 with the second highest SF and a medium m0 value is the most preferred, which corresponds to the observed twin G8-P1. Secondary twins G7-P1-S1 (SF/SF rank/SF ratio ¼ 0.378/1/1) and G8-P1-S1 (SF/SF rank/SF ratio ¼ 0.336/3/ 0.819) form a cross-GB secondary twin pair. It is curious why possible high SF secondary twin variants in G8-P1 with SF ranks of
Z.-Z. Shi / Journal of Alloys and Compounds 716 (2017) 128e136 Table 2 For prominent multi-level twinning shear transmissions over GB in Fig. 3a, SFs of the six possible EVs in grain G4 (i.e., G4-EVi, i ¼ 1e6) and their m0 values with G3-P1, the related a and b angles, and those for the six possible EVs in primary twin G4-P1 (i.e., G4-EV5-EVj, j ¼ 1e6) with G3-P1-S1.
G4-EVi Observed a ( ) b ( ) m0 value SF G4-EV5-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 97.94 90.66 0.002 0.016
e 82.47 97.10 0.016 0.370
e 99.44 127.25 0.099 0.272
e 43.25 12.90 0.710 0.013
G4-P1 4.01 30.39 0.861 0.395
e 45.37 67.61 0.268 0.293
G4-P1-S1 11.142 15.274 0.947 0.178
e 46.508 26.696 0.615 0.069
e 85.958 57.424 0.038 0.298
e 104.736 75.243 0.065 0.216
e 87.792 70.975 0.013 0.074
e 48.983 47.081 0.447 0.256
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Table 5 For nearly none twinning shear transmission over GB in Fig. 3d, SFs of the six possible EVs in grain G10 (i.e., G10-EVi, i ¼ 1e6) and their m0 values with G9-P1, the related a and b angles, and those for the six possible EVs in primary twin G10-P1 (i.e., G10-EV2-EVj, j ¼ 1e6) with G9-P1-S1.
G10-EVi Observed a ( ) b ( ) m0 value SF G10-EV2-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 96.37 83.27 0.013 0.107
G10-P1 71.85 69.87 0.107 0.486
e 31.61 38.29 0.668 0.125
e 14.25 3.10 0.968 0.098
e 56.86 39.51 0.422 0.468
e 89.62 70.68 0.002 0.115
e 113.10 67.47 0.150 0.338
e 92.28 46.73 0.027 0.003
e 51.12 19.38 0.592 0.406
e 19.48 25.97 0.848 0.337
e 53.82 53.36 0.352 0.003
G10-P1-S1 94.42 70.20 0.026 0.408
Candidates of preferred twin variants are highlighted in bold.
Candidates of preferred twin variants are highlighted in bold.
Table 3 For solely prominent primary twinning shear transmission over GB in Fig. 3b, SFs of the six possible EVs in grain G6 (i.e., G6-EVi, i ¼ 1e6) and their m0 values with G5-P1, the related a and b angles, and those for the six possible EVs in primary twin G6-P1 (i.e., G6-EV3-EVj, j ¼ 1e6) with G5-P1-S1.
twin structures are featured without any prominent twinning shear transmission over GB. A microstructure with nearly none twinning shear transmission over GB is shown in Fig. 3d. High SF primary twins G9-P1 (SF/SF rank/SF ratio ¼ 0.471/1/1) and G10-P1 (SF/SF rank/SF ratio ¼ 0.486/1/1) form a cross-GB primary twin pair. Each of them corresponds to the twin variant with the highest possible SF in its host grain. It can be seen in Table 5 that m0 value of them is close to 0. Comparatively, G10-EV5 in Table 5 not only has the second highest SF, but also has a medium m0 value of 0.422. If twinning shear transmission played an important role, G10-EV5 should be more preferred, just as the case of G8-EV6 in Table 4. In Fig. 3d, high SF secondary twins G9-P1-S1 (SF/SF rank/SF ratio ¼ 0.424/2/0.951) and G10-P1-S1 (SF/SF rank/SF ratio ¼ 0.408/ 1/1) form a cross-GB secondary twin pair in their host cross-GB primary twin pair. Once again, their m0 value is close to 0. Similarly, G10-EV2-EV3 in Table 5 should be more preferred, if twinning shear transmission played an important role. The case in Fig. 3d indicates that a compound cross-GB twin structure can form due to mechanisms other than twinning shear transmission over GB.
G6-EVi Observed a ( ) b ( ) m0 value SF G6-EV3-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 78.03 49.43 0.135 0
e 37.47 13.76 0.771 0.147
G6-P1 12.89 26.35 0.874 0.365
e 55.28 60.01 0.285 0.012
e 90.95 79.70 0.003 0.172
e 101.41 75.28 0.050 0.402
G6-P1-S1 29.48 105.18 0.228 0.343
e 41.05 66.48 0.301 0.044
e 57.57 53.92 0.316 0.217
G6-P1-S2 64.22 83.24 0.051 0.306
e 56.54 122.90 0.300 0.049
e 39.72 137.14 0.564 0.249
Candidates of preferred twin variants are highlighted in bold.
1 and 2 are absent. As shown in Table 4, G8-EV6-EV3 has the second highest SF (i.e., SF rank 2), but its m0 value with G7-P1-S1 is close to 0. However, unless a reliable calculation of free energy change due to the two-step compressions is made, the preference of G8-EV6EV1 over G8-EV6-EV6 cannot be predicted unambiguously, since the latter one has the highest SF (i.e., SF rank 1) over 0.4 and a medium m0 value over 0.5. Six of the 30 (i.e., 20%) compound cross-GB twin structures have both m0 values of cross-GB primary and secondary twin pairs smaller than 0.7, as shown in Fig. 2b. These compound cross-GB Table 4 For solely prominent secondary twinning shear transmission over GB in Fig. 3c, SFs of the six possible EVs in grain G8 (i.e., G8-EVi, i ¼ 1e6) and their m0 values with G7P1, the related a and b angles, and those for the six possible EVs in primary twin G8P1 (i.e., G8-EV6-EVj, j ¼ 1e6) with G7-P1-S1.
G8-EVi Observed a ( ) b ( ) m0 value SF G8-EV6-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 89.05 63.99 0.007 0.136
e 78.09 101.47 0.041 0.113
e 80.48 69.27 0.059 0.499
e 40.55 41.54 0.569 0.138
e 10.69 8.81 0.971 0.111
G8-P1 52.34 33.57 0.509 0.497
G8-P1-S1 7.09 6.97 0.985 0.336
e 36.47 46.16 0.557 0.003
e 71.28 79.61 0.058 0.407
e 88.52 91.84 0.001 0.333
e 77.64 74.47 0.057 0.003
e 45.79 38.72 0.544 0.410
Candidates of preferred twin variants are highlighted in bold.
4. Discussions 4.1. Parent crystals in a compound cross-GB twin structure The observed 30 compound cross-GB twin structures consist of 60 cross-GB twin pairs, of which 60% have a m0 >0.7 (Fig. 2b). Twinning shear transmission over GB prominently contributes to their formations. Parent crystals in these structures refer to host grains for primary twins and host primary twins for secondary twins. Misorientation between two parent crystals can be expressed by a (rotation axis)-(minimum rotation angle) pair. Intuitively, twinning shear transmission is more likely to happen over the boundary between two parent crystals with a small misorientation. Fig. 4a shows the relationship between misorientation angle q (i.e., minimum rotation angle) and twinning shear transmission in the observed 30 compound cross-GB twin structures. There exist 30 host grains and 30 host primary twins. About 93.3% of the host grains and about 86.7% of the host primary twins have q values smaller than 35 . For those in the region of m0 >0.7 and q < 35 in Fig. 4a, both the cross-GB primary and secondary twin pairs have a tendency of m0 value increasing with q value decreasing. However, about 35.7% of the host grains with q < 35 and about 30.8% of the host primary twins with q < 35 produce cross-GB twin pairs with m0 values smaller than 0.7. Twin pairs with high m0 values are absent possibly due to their low SFs, such as the case of
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Fig. 4. (a) Effect of misorientation between two neighboring parent crystals on twinning shear transmission in the observed 30 compound cross-GB twin structures. Symbol ‘o’ represents m0 value of a cross-GB primary twin pair and misorientation angle between their host grains, while symbol ‘*’ represents m0 value of a cross-GB secondary twin pair and misorientation angle between their host primary twins. (b) Misorientations of 60 secondary twins in the 30 compound cross-GB twin structures with respect to their host grains. (c) A sketch of mechanisms of cross-GB twin pair formation. ‘C’ and ‘T’ denote parent crystal and twin, respectively. (d) A sketch of two possible processes of associated twin nucleation. In (c) and (d), the parent crystal ‘C’ refers to a host grain for a primary twin or a primary twin for a secondary twin.
G8-EV5 and G7-P1 (Table 4). About 98.3% of the 120 observed twins have SF ratios larger than 0.7 (Fig. 1c), indicating that they are twin variants with relatively high SFs compared with other possible twin variants in their host crystals. Thus, low SF restricts twinning shear transmission over GB. Some twin pairs with low m0 values appear, since the geometry of their host crystals does not allow a high m0 value, such as the case of G6-EV3-EVj (j ¼ 1e6) and G5-P1-S1 (Table 3). The 36 possible double extension twin variants can be classified into four groups according to their misorientations with respect to their host grains: (I) 0 (i.e., detwinning), (II) 〈1 2 1 0〉7.4 , (III) 〈0 14 14 1〉60 , and (IV) 〈1 7 8 0〉60.4 [10]. As shown in Fig. 4b, about 81.7% of the 60 secondary twins in the 30 compound crossGB twin structures have a misorientation of 〈0 14 14 1〉60 , while all the rest have a misorientation of 〈1 7 8 0〉60.4 . The obvious preference of secondary twins with 〈0 14 14 1〉60 is resulted from the confining effect of their host primary twins [10]. Such a confining effect also works for {1 0 1 1}-{1 0 1 2} double twins [11,24,35]. 4.2. Mechanisms of cross-GB twin pair formation Formation of a cross-GB twin pair is the basis of formation of a compound cross-GB twin structure. As sketched in Fig. 4c, there are two possible mechanisms of cross-GB twin pair formation: (I) associated nucleation and (II) isolated nucleation. The former one refers to conditions when the influence of one twin on the
nucleation of another twin cannot be neglected, of which two possible processes of microstructure evolution (i.e., C1-T1 and C2T1, and C1-T2 and C2-T2) are sketched in Fig. 4c. The latter one refers to conditions when the influence of one twin on the nucleation of another twin is negligible, of which three possible processes of microstructure evolution (i.e., C3-T1 and C4-T1, C3-T2 and C4-T1, and C3-T3 and C4-T2) are also sketched in Fig. 4c. For twin pairs formed due to isolated nucleation, it is not surprising that they have high SFs but low m0 values. When plastic strain is small, the possibility of formation of a twin pair is low. However, when plastic strain is large enough, C3-T1 and C4-T1 in Fig. 4c for example would consume the majority or eventually all of C3 and C4, respectively. Then, the possibility of them to form a twin pair is very high. For twin pairs formed due to associated nucleation, two possible processes of twin nucleation are drawn in Fig. 4d. The internal stress near the boundary between C1 and C2 in Fig. 4d is a combination of the applied macro stress, twin-induced stress at GB as mentioned in introduction, and boundary stress due to different plastic deformation behaviors in differently oriented crystals. When plastic strain is small, it is reasonable to neglect boundary stress in a single-phase material with a strong texture, as in the present study. The internal stress should be relaxed in C2 through plastic deformation, including twinning and dislocation slipping. One possibility as path (I) in Fig. 4d is through twinning (i.e., twinto-twin accommodation), resulting in the formation of a cross-GB twin pair with a high m0 value when it is allowed by geometry.
Z.-Z. Shi / Journal of Alloys and Compounds 716 (2017) 128e136 Table 6 For twin variant selection of G10-P1-S1 in Fig. 3d, SFs of the six possible EVs in primary twin G10-P1 (i.e., G10-EV2-EVj, j ¼ 1e6) and their m0 BS values. G10-EV2-EVj
EV1
EV2
EV3
EV4
EV5
EV6
Observed m0 BS SF
e 0.824 0.338
e 0.751 0.003
e 0.482 0.406
e 0.188 0.337
e 0.209 0.003
G10-P1-S1 0.523 0.408
Candidates of preferred twin variants are highlighted in bold.
Another possibility as path (II) in Fig. 4d is through dislocation slipping (i.e., twin-to-slip accommodation). In Mg and its alloys at room temperature, basal slip has a critical resolved shear stress (CRSS) much lower than those of other slip modes [36,37], so that it is more likely to be activated. It is experimentally evidenced that profuse basal slip is activated to relax twin-induced stress at GB in pure Mg, which can be sufficiently extensive to allow complete blunting of a twin [8]. Similarly in pure Ti, prismatic and pyramidal slip bands are observed to be correlated with deformation twins at GBs [38]. Then, C1, C1-T1 and C2 form GB triple junctions (Fig. 4d), which have an effect of stress concentration. Thus, C2-T1 will nucleate preferentially near the triple junctions, which is supported by EBSD observations (e.g., Fig. 2 in Ref. [22]). In this case, m0 value of C1-T1 and C2-T1 is irrelevant to the formation of C2-T1, which of course could be low. When twin-induced stress at GB from C2-T1 is important, path (II) in Fig. 4d can lead to the formation of C2-T1 with low positive or even negative SF [22]. Under this condition, twin variant selection of C2-T1 depends more on shear accommodation in C1-T1 or in C1 (Fig. 4d). Such a formalism may explain why G10-EV2-EV6 is more preferred than G10-EV2-EV3, although their SFs are comparatively high (Table 5). Here, G9, G9-P1, and G10-P1-S1 in Fig. 3d correspond to C1, C1-T1 and C2-T1 in Fig. 4d, respectively. According to Section 2.2, the largest absolute m0 value among those of G10-P1-S1 and the three possible basal slip systems (i.e., {0 0 0 1}〈2 1 1 0〉) in G9-P1 can be taken to measure the ability of G9-P1 to accommodate the shear of G10-P1-S1 through basal slip, which is designated as m0 BS. As shown in Table 6, G10-EV2-EV6 has a higher m0 BS value than G10-EV2-EV3, indicating that basal slip in G9-P1 can accommodate more twinning shear of G10-EV2-EV6. 5. Conclusions After a strain path change, 30 compound cross-GB extension twin structures are detected by EBSD in a deformed AZ31 Mg alloy, which consist of cross-GB secondary extension twin pairs within cross-GB primary extension twin pairs. The study of these compound cross-GB extension twin structures leads to the following conclusions: 1. It is proved that the shear that can be accommodated through a deformation system is proportional to m0 factor and the magnitude of the original shear. When m0 >0, twinning shear transmission could happen. When m0 is high, which is set to be m0 0.7 in this study, twinning shear transmission is considered to be prominently contributed to formation of a cross-GB twin pair. 2. Associated nucleation and isolated nucleation are two possible mechanisms of cross-GB twin pair formation. As to the former one, shear accommodation is possible to be achieved through twin-to-twin (i.e., twinning shear transmission) or twin-to-slip modes. Twin-to-twin accommodation mode usually leads to a high m0 value as long as it is allowed by geometry. However, twin-to-slip accommodation mode or isolated nucleation could lead to a low m0 value.
135
3. There are 60 cross-GB twin pairs related to the 30 compound cross-GB twin structures, of which 60% have a m0 >0.7, indicating that twinning shear transmission over GB is a major mechanism of cross-GB twin pair formation. 40% of the compound cross-GB twin structures form due to multi-level twinning shear transmissions over GB, the primary and the secondary levels of which refer to twinning shear transmission between primary and secondary twins, respectively. 40% of them form due solely to primary or secondary twinning shear transmission over GB. The rest 20% form without any prominent twinning shear transmission over GB. 4. About 98.3% of the 120 twins in the 30 compound cross-GB twin structures are high SF twins with a SFratio 0.7. Low SF twins are preferred due to twinning shear transmission. Over 85% of the parent crystals have a misorientation angle smaller than 35 . About 81.7% of the 60 secondary twins in the 30 compound cross-GB twin structures have a misorientation of 〈0 14 14 1〉 60 with respect to their host grains, indicating that the confining effect of primary twin on variant selection of secondary twin is strong. 5. For a cross-GB twin pair formed due to twin-to-twin accommodation mode, SF and m0 factor should be considered to predict twin variant selection. For variant selection of secondary twins, its misorientation with respect to its host grain should also be taken into consideration. However, a cross-GB twin pair could form through several possible processes of microstructure evolution, which makes prediction of its formation complicated. Acknowledgement This work was supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 51601010), and Fundamental Research Funds for the Central Universities (Project: Deformation behavior and microstructure evolution of difficult-to-deform metals under complex loadings). References [1] C.D. Barrett, H.E. Kadiri, The roles of grain boundary dislocations and disclinations in the nucleation of {10-12} twinning, Acta Mater. 63 (2014) 1e15. , An atomic and probabilistic perspective on [2] J. Wang, I.J. Beyerlein, C.N. Tome twin nucleation in Mg, Scr. Mater. 63 (2010) 741e746. [3] A. Khosravani, D.T. Fullwood, B.L. Adams, T.M. Rampton, M.P. Miles, R.K. Mishra, Nucleation and propagation of twins in AZ31 magnesium alloy, Acta Mater. 100 (2015) 202e214. [4] S. Godet, L. Jiang, A.A. Luo, J.J. Jonas, Use of Schmid factors to select extension twin variants in extruded magnesium alloy tubes, Scr. Mater. 55 (2006) 1055e1058. [5] S.H. Park, S.-G. Hong, J.H. Lee, Y.-H. Huh, Texture evolution of rolled Mg-3Al1Zn alloy undergoing a {10-12} twinning dominant strain path change, J. Alloys Compd. 646 (2015) 573e579. [6] G.-S. Song, S.-H. Zhang, L. Zheng, L. Ruan, Twinning, grain orientation and texture variation of AZ31 Mg alloy during compression by EBSD tracing, J. Alloys Compd. 509 (2011) 6481e6488. [7] L. Jiang, J.J. Jonas, R.K. Mishra, A.A. Luo, A.K. Sachdev, S. Godet, Twinning and texture development in two Mg alloys subjected to loading along three different strain paths, Acta Mater. 55 (2007) 3899e3910. [8] M.R. Barnett, N. Stanford, A. Ghaderi, F. Siska, Plastic relaxation of the internal stress induced by twinning, Acta Mater. 61 (2013) 7859e7867. [9] J.D. Eshelby, The elastic field outside an ellipsoidal inclusion, Proceedings of the Royal Society of London, Ser. A 252 (1959) 561e569. [10] Z.-Z. Shi, Y.-D. Zhang, F. Wagner, T. Richeton, P.-A. Juan, J.-S. Lecomte, L. Capolungo, S. Berbenni, Sequential double extension twinning in a magnesium alloy: combined statistical and micromechanical analyses, Acta Mater. 96 (2015) 333e343. [11] P.-A. Juan, S. Berbenni, L. Capolungo, Prediction of internal stresses during growth of first- and second-generation twins in Mg and Mg alloys, Acta Mater. 60 (2012) 476e486. , D.W. Brown, [12] C.C. Aydıner, J.V. Bernier, B. Clausen, U. Lienert, C.N. Tome Evolution of stress in individual grains and twins in a magnesium alloy aggregate, Phys. Rev. B 80 (2009), 024113-1-024113-6. [13] X. Hong, A. Godfrey, W. Liu, Challenges in the prediction of twin transmission at grain boundaries in a magnesium alloy, Scr. Mater. 123 (2016) 77e80.
136
Z.-Z. Shi / Journal of Alloys and Compounds 716 (2017) 128e136
[14] H.H. Yu, Y.C. Xin, A. Chapuis, X.X. Huang, R.L. Xin, Q. Liu, The different effects of twin boundary and grain boundary on reducing tension-compression yield asymmetry of Mg alloys, Sci. Rep. 6 (2016), 29283e1-29283-8. [15] Z.-Z. Shi, X.-F. Liu, Characteristics of cross grain boundary contraction twin pairs and bands in a deformed Mg alloy, J. Alloys Compd. 692 (2017) 274e279. [16] R.L. Xin, C.F. Guo, Z.R. Xu, G.D. Liu, X.X. Huang, Q. Liu, Characteristics of long {10-12} twin bands in sheet rolling of a magnesium alloy, Scr. Mater. 74 (2014) 96e99. [17] R.L. Xin, Y.C. Liang, C.H. Ding, C.F. Guo, B.S. Wang, Q. Liu, Geometrical compatibility factor analysis of paired extension twins in extruded Mg-3Al1Zn alloys, Mater. Des. 86 (2015) 656e663. [18] M.R. Barnett, M.D. Nave, A. Ghaderi, Yield point elongation due to twinning in a magnesium alloy, Acta Mater. 60 (2012) 1433e1443. [19] H.C. Chen, T.M. Liu, D.W. Hou, D.F. Shi, Study on the paired twinning behavior in a hot rolled AZ31 magnesium alloy via interrupted in situ compression, Mater. Sci. Eng. A 667 (2016) 402e408. [20] L. Wang, P. Eisenlohr, Y. Yang, T.R. Bieler, M.A. Crimp, Nucleation of paired twins at grain boundaries in titanium, Scr. Mater. 63 (2010) 827e830. [21] J. Luster, M.A. Morris, Compatibility of dependence on deformation in twophase Ti-AI alloys: microstructure and orientation relationships, Metall. Mater. Trans. A 26 (1995) 1745e1756. [22] Z.-Z. Shi, Y.-D. Zhang, F. Wagner, P.-A. Juan, S. Berbenni, L. Capolungo, J.S. Lecomte, T. Richeton, On the selection of extension twins with low Schmid factors in a deformed Mg alloy, Acta Mater. 83 (2015) 17e28. [23] S. Niknejad, S. Esmaeili, N.Y. Zhou, The role of double twinning on transgranular fracture in magnesium AZ61 in a localized stress field, Acta Mater. 102 (2016) 1e16. [24] E. Martin, L. Capolungo, L. Jiang, J.J. Jonas, Variant selection during secondary twinning in Mge3%Al, Acta Mater. 58 (2010) 3970e3983. [25] M.R. Barnett, Twinning and the ductility of magnesium alloys Part II. “Contraction” twins, Mater. Sci. Eng. A 464 (2007) 8e16. [26] M.R. Barnett, M.D. Nave, C.J. Bettles, Deformation microstructures and textures of some cold rolled Mg alloys, Mater. Sci. Eng. A 386 (2004) 205e211.
[27] M. Lentz, M. Risse, N. Schaefer, W. Reimers, I.J. Beyerlein, Strength and ductility with {10-11}-{10-12} double twinning in a magnesium alloy, Nat. Commun. 7 (2016), 11068e1-11068-7. [28] S.H. Park, S.G. Hong, C.S. Lee, In-plane anisotropic deformation behavior of rolled Mg-3Al-1Zn alloy by initial {10-12} twins, Mater. Sci. Eng. A 570 (2013) 149e163. [29] S.H. Park, S.G. Hong, J.H. Lee, C.S. Lee, Multiple twinning modes in rolled Mge3Ale1Zn alloy and their selection mechanism, Mater. Sci. Eng. A 532 (2012) 401e406. [30] F. Bachmann, R. Hielscher, H. Schaeben, Texture analysis with mtex e free and open source software toolbox, Solid State Phenom. 160 (2010) 63e68. [31] J.J. Jonas, S.-J. Mu, T. Al-Samman, G. Gottstein, L. Jiang, E. Martin, The role of strain accommodation during the variant selection of primary twins in magnesium, Acta Mater. 59 (2011) 2046e2056. [32] J.W. Christian, S. Mahajan, Deformation twinning, Prog. Mater Sci. 39 (1995) 1e157. [33] H. Qin, J.J. Jonas, H. Yu, N. Brodusch, R. Gauvin, X. Zhang, Initiation and accommodation of primary twins in high-purity titanium, Acta Mater. 71 (2014) 293e305. [34] S.J. Mu, J.J. Jonas, G. Gottstein, Variant selection of primary, secondary and tertiary twins in a deformed Mg alloy, Acta Mater. 60 (2012) 2043e2053. [35] M.R. Barnett, Z. Keshavarz, A.G. Beer, X. Ma, Non-Schmid behaviour during secondary twinning in a polycrystalline magnesium alloy, Acta Mater. 56 (2008) 5e15. [36] W.B. Hutchinson, M.R. Barnett, Effective values of critical resolved shear stress for slip in polycrystalline magnesium and other hcp metals, Scr. Mater. 63 (2010) 737e740. [37] J. Koike, Enhanced deformation mechanisms by anisotropic plasticity in polycrystalline Mg alloys at room temperature, Metall. Mater. Trans. A 36 (2005) 1689e1696. [38] L. Wang, Y. Yang, P. Eisenlohr, T.R. Bieler, M.A. Crimp, D.E. Mason, Twin nucleation by slip transfer across grain boundaries in commercial purity titanium, Metall. Mater. Trans. A 41 (2010) 421e430.
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Compound cross-grain boundary extension twin structure and its related twin variant selection in a deformed Mg alloy Zhang-Zhi Shi School of Materials Science and Engineering, University of Science and Technology Beijing, Xueyuan Road 30, Beijing 100083, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 January 2017 Received in revised form 2 May 2017 Accepted 3 May 2017 Available online 6 May 2017
Interrupted bi-axial compressions are applied to a hot-rolled Mg alloy with a strong basal texture in order to activate {1 0 1 2}-{1 0 1 2} double extension twinning. Compound cross-grain boundary (cross-GB) extension twin structures form after the compressions, which consist of cross-GB secondary twin pairs within their host cross-GB primary twin pairs. Shear accommodation of a twin is proven to be proportional to a geometrical m0 factor and the magnitude of its twinning shear. It is found that 60% of cross-GB twin pairs, including both the primary and the secondary ones, have a value of m0 >0.7, indicating twinning shear transmission over GB is a major mechanism. Multi-level twinning shear transmissions over GB contribute prominently to the formation of 40% of the observed compound cross-GB twin structures. About 98.3% of the twins are high Schmid factor (SF) ones, while low SF ones appear due to twinning shear transmission. About 81.7% of secondary twins have a misorientation of 〈0 14 14 1〉60 with respect to their host grains, indicating a strong confining effect of primary twins. A cross-GB twin pair could form through a mechanism of associated nucleation or isolated nucleation. High m0 value is characteristic of associated nucleation through twin-to-twin accommodation mode. Depending on the process of microstructure evolution, SF, shear accommodation and the confining effect of the matrix if necessary should be synthetically considered to predict twin variant selection related to cross-GB twin pairs. © 2017 Elsevier B.V. All rights reserved.
Keywords: Magnesium alloys EBSD Twinning Grain boundary Strain path change
1. Introduction Twinning is an important deformation mode of Mg alloys. Interaction between twins and grain boundaries (GBs) is vital for twinning in polycrystalline Mg alloys. Twin nucleation occurs preferentially at GBs, which is supported by atomistic simulation and electron back-scattering diffraction (EBSD) observation [1e3]. Schmid factor (SF) is an effective criterion for twin nucleation, which evaluates how much the macro applied stress is resolved to be the shear stress on a twinning system. Not surprisingly, twin variants with high SFs nucleate preferentially [4e7]. After nucleation, a twin usually preferentially elongates in its shear direction, which evolves shearing the matrix. It relaxes elastic energy in the matrix, but concentrates applied stress at its tip, inducing localized plasticity [8]. When it collides with a GB, the immediate neighborhood in the adjoining grain must also be sheared. Otherwise, stress will arise and accumulate in the adjoining grain along with
E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.jallcom.2017.05.020 0925-8388/© 2017 Elsevier B.V. All rights reserved.
twin growth. In addition, stress will also arise outside of a twin due to the confining effect of the surrounding matrix according to Eshelby inclusion theory [9e12]. So twin-induced stress at GB originates at least from three sources: (I) stress induced by twinning shear due to GB blocking effect, (II) applied stress concentrated at twin tip, and (III) stress resulted from the confining effect of the matrix. Twin-induced stress at GB may intrigue twin nucleation in the adjoining grain, resulting in a cross-GB twin pair [3,13e19]. Several connected cross-GB twin pairs form a long cross-GB twin band [15,16]. Intuitively, a cross-GB twin pair may form directly through twinning shear transmission over GB, which is indeed proven to be a major mechanism [13,14,16e18]. Twinning shear transmission over GB is often estimated by a geometric compatibility factor, m0 ¼ cosa,cosb, where a and b are angles between two twinning shear directions and two twinning plane normals for twinning systems in two neighboring grains, respectively [13,17,18,20,21]. When m0 ¼ 1, two twinning shears are fully compatible. This implies that elastic energy in a grain due to twin-induced stress at GB from source (I) could be fully relaxed, which is energetically
Z.-Z. Shi / Journal of Alloys and Compounds 716 (2017) 128e136
favorable. Statistical EBSD analysis reveals that 95% of 306 cross-GB primary {1 0 1 2} twin pairs in AZ31 Mg alloy have a m0 value higher than 0.7, with almost 70% having a m0 value higher than 0.9 [13]. When m0 ¼ 0, two twinning shears are independent. When m0 ¼ 1, two twinning shears are exactly opposite. By now, cross-GB primary {1 0 1 2} extension twin pairs have received the most attention [3,13,14,16e19], largely due to their high EBSD pattern indexing rates and their frequently appearance during uni-axial deformation. Recently, cross-GB primary {1 0 1 1} contraction twin pairs have been studied, in which their low EBSD pattern indexing rates have been compensated by trace analysis [15]. A minority of either cross-GB primary {1 0 1 2} or {1 0 1 1} twin pairs have been reported to have m0 values lower than 0 [15,17,22], indicating that a cross-GB primary twin pair can also form due to mechanisms other than twinning shear transmission over GB. Is it possible that two secondary twins form a cross-GB secondary twin pair within their host cross-GB primary twin pair? If they formed, is there any big difference between cross-GB secondary and primary twin pairs, since the parent crystals of the former one are twins but not ordinary grains? Although large uniaxial deformation can activate profuse {1 0 1 1}e{1 0 1 2} double twinning [23e27], difficulties arise from low EBSD pattern indexing rate and the limitation of trace analysis [15]. Fortunately, small biaxial deformation is enough to activate profuse {1 0 1 2}e{1 0 1 2} double extension twinning [5,10,28,29]. Cross-GB secondary {1 0 1 2} twin pairs have greater chances to form and to be identified by EBSD, which will provide good chances to explore the above problems.
129
system, a rectangular coordinate system (d, m, n) of it can be defined by unit vectors d//the twinning shear direction, n//the twinning plane normal, and m//the cross-product of n and d. In such a coordinate system, twinning shear tensor ‘S1’ has the simplest form [22,31], which is:
2
0 S1 ¼ 4 0 0
0 0 0
3 s 0 5; 0
(1)
2. Material and methods
where ‘s’ is the magnitude of the twinning shear. The magnitude of the twinning shear depends on crystal structure and twinning system. In Mg with c/a ¼ 1.624, s ¼ 0.129 for {1 0 1 2} twinning, while s ¼ 0.138 for {1 0 1 1} twinning [31,32]. In Ti with c/a ¼ 1.587, s ¼ 0.175 for {1 0 1 2} twinning, while s ¼ 0.218 for {1 1 2 2} twinning [32,33]. Similarly, a rectangular coordinate system (a, b, c) of any deformation system in Crystal 2 can be defined by unit vectors a//the twinning shear or slip direction, c//the twinning or slip plane normal, and b//the cross-product of c and a (Fig. 1). Twinning in Crystal 1 shears a fraction of it, which should be accommodated through deformation systems in Crystal 2 [22,31,33,34]. The shear which could be accommodated through the deformation system (a, b, c) can be calculated by rotating the twinning shear tensor S1 from coordinate systems (d, m, n) to (a, b, c) [22,31]. For calculation, the components of all the unit vectors are defined in a public coordinate system (x, y, z), in which unit vectors x, y, z can be selected to be parallel to sample directions, e.g., x//RD, y//TD, and z//ND. For rotating any second-order tensor from coordinate systems (d, m, n) to (x, y, z), the rotation matrix R1 is defined by:
2.1. Experimental procedure
R1 I ¼ ½d; m; n;
The material used was a commercial purchased hot-rolled Mg3Al-1Zn (AZ31) alloy sheet, which was annealed at 350 C for 1 h after rolling. The rolling, transverse and normal directions of the sheet were designated as RD, TD and ND, respectively. The annealed sheet exhibited a typical hot-rolled texture with most of the grains' [0 0 0 1] directions (i.e., c-axes) centering around ND, as reported previously [15]. Cubic samples of 10 mm length were cut from the sheet for compression tests. According to the initial texture and twinning geometry [10,29], compressive loading was applied first along RD to a reduction of 2.0% for producing primary {1 0 1 2} extension twins (ETWs), unloaded and then applied along TD also to a reduction of 2.0% for producing secondary ETWs, at a strain rate of 103 s1 at room temperature. The deformed samples were sectioned in a mid-plane normal to ND for EBSD observations. The sectioned faces were first ground using 2000 to 4000 grit SiC papers, and then electrolytically polished in an electrolyte of 62.5% phosphoric acid and 37.5% ethanol at 1.5 V for 1 min, at 15 C. The measured EBSD date was analyzed using Matlab™ MTEX toolbox (version 4.1.beta4, mtex-toolbox.github.io) [30].
2
d1 where matrices [d, m, n] ¼ 4 d2 d3 one can get:
R1 ¼ ½d; m; n:
m1 m2 m3
2 3 1 n1 n2 5, and I ¼ 4 0 0 n3
(2)
0 1 0
3 0 0 5. So 1
(3)
For rotating any second-order tensor from coordinate systems (x, y, z) to (a, b, c), the rotation matrix R2 is defined by:
R2 ½a; b; c ¼ I;
2
a1 where matrix [a, b, c] ¼ 4 a2 a3
(4)
b1 b2 b3
3 c1 c2 5. Similar to matrix [d, m, n], c3
2.2. Association of m0 factor with transmitted shear The following model in Fig. 1 is constructed in order to clarify the relationship between m0 factor and transmitted shear. There are two crystals in the figure, which could be grains, primary twins or secondary twins, etc. In each crystal, one deformation system is chosen to be under study, which could be a twinning or a slip system. The two deformation systems are supposed to be connected with each other at a boundary separating the two crystals, which could be a grain boundary, a twin-twin boundary, or a twingrain boundary, depending on what the two crystals are. If the deformation system in Crystal 1 (Fig. 1) is a twinning
Fig. 1. A sketch of shear transmission from one crystal into its neighboring crystal.
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matrix [a, b, c] is also a rotation matrix, so that the inverse of [a, b, c] equals the transpose of it, i.e., [a, b, c]1 ¼ [a, b, c]T. Then one can get:
2
a1 R 2 ¼ 4 b1 a3
a2 b2 b3
3 a3 b3 5 : c3
(5)
Then, for rotating the tensor S1 from coordinate systems (d, m, n) to (a, b, c), the rotation matrix R is:
2
a$d R ¼ R2 R1 ¼ 4 b$d c$d
a$m b$m c$m
3 a$n b$n 5: c$n
(6)
The resulted tensor S2 after the rotation is:
S2 ¼ R$S1 $R1 :
(7)
The tensor S2 is a second-order tensor, whose 9 elements are eij (i, j ¼ 1e3). The element e13 has a particular physical meaning [22]. According to the selection of coordinate system (a, b, c), e13 corresponds to the shear along the direction parallel to the unit vector a, on the plane normal to the unit vector c. That is to say, e13 corresponds to the shear that can be accommodated through the deformation system (a, b, c). From Eq. (1), Eq. (6) and Eq. (7), it can be deduced that:
e13 ¼ ða$dÞðc$nÞs ¼ ðcosa$cosbÞs ¼ m0 $s;
(8)
where a is the angle between two twinning shear or slip directions, while b is the angle between two twinning or slip plane normals, for two deformation systems in two neighboring crystals (Fig. 1). From Eq. (8), it is clear that m0 in fact represents how much the shear on a deformation system in a crystal can transmit over a boundary onto another deformation system in its neighboring crystal. For a transmission from a twinning system to a slip system (i.e., twin-to-slip), jm0 j should be taken into consideration, since slip could be bi-directional. However, for a transmission from a twinning system to another twinning system (i.e., twin-to-twin), m0 should be taken into consideration, since twinning is unidirectional. In twin-to-twin case, twinning shear transmission could happen when m0 >0, the magnitude of which is proportional to m0 value, according to Eq. (8). Afterwards, m0 as default refers to twin-to-twin accommodation mode in the present paper.
3. Experimental results and analysis Fig. 2a shows a compound cross-GB twin structure measured by EBSD after the two-step compressions. In the figure, boundaries with misorientations larger than 5 are outlined in black, while those with a misorientation of 〈1 2 1 0〉86.3 (±7 offset) are outlined in white, indicating an {1 0 1 2}〈1 0 1 1〉 extension twinning relationship. For convenience, host grain, primary twin, and secondary twin are designated by G, G-P, and G-P-S, respectively. Such a designation clearly points out the parent-twin relationship in a complex microstructure, the advantage of which will be shown afterwards. The six possible ETW variants (EVs) are defined as follows [22,34]: EV1: (1 0 1 2)[1 0 1 1], EV2: (0 1 1 2) [0 1 1 1], EV3: (1 1 0 2)[1 1 0 1], EV4: (1 0 1 2)[1 0 1 1], EV5: (0 1 1 2)[0 1 1 1], and EV6: (1 1 0 2)[1 1 0 1]. Usually, the SFs of the EVs are different, which can be ranked from the highest one to the lowest one, i.e., rank 1 corresponds to the highest one while rank 6 corresponds to the lowest one. Note that secondary ETWs and their host primary twins form respectively during the second
TD and the first RD compressions, so that SFs of them should be calculated respectively with loadings along TD and RD [10,29]. In addition, SF ratio of a certain twin variant EVi (i ¼ 1e6) can be calculated by SFratio¼(SF of EVi)/(SF of rank 1). SF, SF rank and SF ratio together clearly indicate whether a certain twin variant EVi is favored by macro applied stress, compared with its competitive twin variants EVj (j ¼ 1e6, jsi) in the same host crystal, which could be a grain or a primary twin. In Fig. 2a, primary twins G1-P1 and G2-P1 form a cross-GB primary twin pair during the first RD compression. SFs, SF ranks and SF ratios of G1-P1 and G2-P1 are calculated to be 0.494/2/0.988 and 0.489/2/0.986, respectively. Although both of them are high SF twin variants, those with the highest SFs in grains G1 and G2 are absent. In fact, when SF ratio is larger than 0.98, the SF difference between the highest and the second highest ones in a grain should be smaller than 0.01, which is too small to be distinguished for twin variant selection. In this case, additional criteria ought to be adopted. In Table 1, SFs of the six possible EVs in grain G2 and their m0 values with respect to G1-P1 have been listed. It can be seen that G2-EV3 corresponding to G2-P1 in Fig. 2a has the largest m0 value close to 1, indicating almost complete twinning shear transmission over GB. Although G2-EV6 has the largest SF, its m0 value is close to 0. During the second TD compression, secondary twins G1-P1-S1 and G2-P1-S1 form a cross-GB secondary twin pair within their host cross-GB primary twins, resulting in a compound cross-GB twin structure. SFs, SF ranks and SF ratios of G1-P1-S1 and G2P1-S1 are calculated to be 0.461/1/1 and 0.396/2/0.983, respectively. The appearance of G1-P1-S1 is anticipated, since its SF is the highest with a SF rank of 1. However, according solely to SF, it is unclear why G2-P1-S1 with a SF rank of 2 is the most preferred. In Table 1, SFs of the six possible EVs in primary twin G2-P1 and their m0 values with respect to G1-P1-S1 are also listed. It can be seen that variant G2-EV3-EV6 corresponding to twin G2-P1-S1 in Fig. 2a has the largest m0 value close to 1. Therefore, multi-level twinning shear transmissions happen over the GB, in which the primary and the secondary twinning shear transmissions are related to the primary and the secondary twins, respectively. Thirty compound cross-GB twin structures similar to that in Fig. 2a have been detected by several EBSD measurements. Fig. 2b shows the 60 m0 values associated with them. Twelve of the 30 (i.e., 40%) compound cross-GB twin structures have both m0 values of cross-GB primary and secondary twin pairs larger than 0.7. Therefore, multi-level twinning shear transmissions over GB prominently contribute to their formations. The threshold for ‘prominent’ is taken as m0 0.7 in this study. Note that twinning shear transmission can happen as long as m0 >0, according to Eq. (8). The SFs and SF ratios of their related 48 (¼12 4) twins are shown in subfigure (I) in Fig. 2c. It can be seen that the majority (i.e., ~95.8%) of the twins have SF ratios larger than 0.8, indicating that they are high SF twins compared with their competitive twin variants in their host crystals. Twinning shear transmission over GB can activate low SF twin variants in compound cross-GB twin structures. Such a microstructure is shown in Fig. 3a. Primary twins G3-P1 (SF/SF rank/SF ratio ¼ 0.458/1/1) and G4-P1 (SF/SF rank/SF ratio ¼ 0.395/1/1) with the highest SFs form a cross-GB primary twin pair with the highest m0 value, as can be seen in Table 2. Although G4-EV4 has a high m0 value larger than 0.7 (Table 2), its SF is close to 0. Although G4-EV2 has the second highest SF (Table 2), its m0 value with G3-P1 is close to 0. Therefore, twin variant G4-EV5 with both the highest SF and m0 values is highly preferred than its competitors. Due to the second applied TD stress, secondary twins G3-P1-S1 (SF/SF rank/SF ratio ¼ 0.328/2/0.912) and G4-P1-S1 (SF/SF rank/SF ratio ¼ 0.178/4/ 0.598) form a cross-GB secondary twin pair in their host cross-GB
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Table 1 For prominent multi-level twinning shear transmissions over GB in Fig. 2a, SFs of the six possible EVs in grain G2 (i.e., G2-EVi, i ¼ 1e6) and their m0 values with G1-P1, the related a and b angles, and those for the six possible EVs in primary twin G2-P1 (i.e., G2-EV3-EVj, j ¼ 1e6) with G1-P1-S1.
G2-EVi Observed a ( ) b ( ) m0 value SF G2-EV3-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 87.21 67.36 0.019 0.163
e 52.51 34.87 0.499 0.087
G2-P1 9.74 5.40 0.981 0.489
e 36.62 42.68 0.590 0.159
e 75.96 72.56 0.073 0.090
e 97.72 82.85 0.017 0.496
e 44.50 36.10 0.576 0.337
e 76.29 73.04 0.069 0.001
e 87.71 92.80 0.002 0.403
e 71.38 82.70 0.041 0.344
e 37.30 49.95 0.512 0
G2-P1-S1 7.05 10.29 0.977 0.396
Candidates of preferred twin variants are highlighted in bold.
primary twins, resulting in a compound cross-GB twin structure (Fig. 3a). Although the SF of G4-P1-S1 is low, it has the highest m0 value with G3-P1-S1, which is close to 1. High SF secondary twin G3-P1-S1, whose parent is primary twin G3-P1 with a much larger size than primary twin G4-P1, is likely to be first activated and then intrigues secondary twinning in G4-P1. Since the highest possible SF of G4-EV5-EVj (j ¼ 1e6) is smaller than 0.3 (Table 2), neither of them could largely release the elastic energy aroused by the applied TD stress in G4-EV5. In this case, it is more requisite to release the increased elastic energy in G4-EV5 due to twin shearing of G3-P1S1, which favors G4-EV5-EV1 with a very high m0 value close to 1 the most. Six of the 30 (i.e., 20%) compound cross-GB twin structures have m0 values of cross-GB primary twin pairs larger than 0.7, but those of cross-GB secondary twin pairs smaller than 0.2, as shown in Fig. 2b. They are compound cross-GB twin structures solely featured with prominent primary twinning shear transmission over GB. A representative microstructure of them is shown in Fig. 3b, in which Euler angles of each microstructure component in concern are also listed in order to make clear parent-twin relationship by calculation. In grain G6, primary twins G6-P1(a~d) all correspond to twin variant G6-EV3, which are separated by secondary twins G6P1-S1 and G6-P1-S2. This indicates that G6-P1(a~d) are remnants of a primary twin G6-P1. High SF primary twins G6-P1 (SF/SF rank/ SF ratio ¼ 0.365/2/0.909) and G5-P1 (SF/SF rank/SF ratio ¼ 0.420/2/ 0.939) form a cross-GB primary twin pair. It can be seen in Table 3 that G6-P1 (i.e., G6-EV3) has the highest possible m0 value of 0.874 with G5-P1. Although G6-EV6 has the highest SF (Table 3), its m0 value is close to 0. Although G6-EV2 has a high m0 >0.7 (Table 3), its SF is lower than 0.15. However, prominent secondary twinning shear transmission does not happen in Fig. 3b. High SF secondary twins G5-P1-S1 (SF/ SF rank/SF ratio ¼ 0.496/2/0.995) and G6-P1-S1 (SF/SF rank/SF ratio ¼ 0.343/1/1) form a cross-GB secondary twin pair. It can be seen in Table 3 that none of the possible secondary twin variants G6-EV3-EVj (j ¼ 1e6) has a m0 value larger than 0.7 with G5-P1-S1.
Fig. 2. (a) A compound cross-GB twin structure measured by EBSD. Boundaries with misorientations larger than 5 are outlined in black, while those with a misorientation of 〈1 2 1 0〉86.3 (±7 offset) are outlined in white, indicating an {1 0 1 2}〈1 0 1 1〉 extension twinning relationship. The measured Euler angles of the crystals are listed in the down part of the figure. (b) 30 compound cross-GB twin structures and their corresponding m0 values, in which symbols ‘o’ and ‘*’ refer to cross-GB priamry and secondary twin pairs, respectively. (c) SFs and SF ratios of 120 twins associated with the 30 compound cross-GB twin structures, in which symbols ‘o’ and ‘*’ refer to priamry and secondary twins, respectively.
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Fig. 3. Compound cross-GB twin structures with: (a) proniment multi-level twinning shear transmissions, resulting in the formation of a low SF twin G4-P1-S1; (b) solely prominent primary twinning shear transmission; (c) solely prominent secondary twinning shear transmission; (d) nearly none twinning shear transmission over GB. The measured Euler angles of the crystals are listed in the down parts of the figures.
In this case, secondary twin variants with high SFs should be preferred. The observed G6-P1-S1 corresponds to twin variant G6EV3-EV1 with the highest SF (Table 3). Another observed one is G6P1-S2 in some distance away from G5-P1-S1, which corresponds to twin variant G6-EV3-EV4 with the second highest SF (Table 3). Six of the 30 (i.e., 20%) compound cross-GB twin structures have m0 values of cross-GB primary twin pairs smaller than 0.7, but those of cross-GB secondary twin pairs larger than 0.7, as shown in Fig. 2b. They are compound cross-GB twin structures solely featured with prominent secondary twinning shear transmission over GB. A representative microstructure of them is shown in
Fig. 3c. High SF primary twins G7-P1 (SF/SF rank/SF ratio ¼ 0.471/2/ 0.964) and G8-P1 (SF/SF rank/SF ratio ¼ 0.497/2/0.994) form a cross-GB primary twin pair. Although G8-EV5 has a very high m0 value close to 1 with G7-P1 (Table 4), its SF is lower than 0.15. Although G8-EV3 has the highest SF (Table 4), its m0 value with G7P1 is close to 0. Twin variant G8-EV6 with the second highest SF and a medium m0 value is the most preferred, which corresponds to the observed twin G8-P1. Secondary twins G7-P1-S1 (SF/SF rank/SF ratio ¼ 0.378/1/1) and G8-P1-S1 (SF/SF rank/SF ratio ¼ 0.336/3/ 0.819) form a cross-GB secondary twin pair. It is curious why possible high SF secondary twin variants in G8-P1 with SF ranks of
Z.-Z. Shi / Journal of Alloys and Compounds 716 (2017) 128e136 Table 2 For prominent multi-level twinning shear transmissions over GB in Fig. 3a, SFs of the six possible EVs in grain G4 (i.e., G4-EVi, i ¼ 1e6) and their m0 values with G3-P1, the related a and b angles, and those for the six possible EVs in primary twin G4-P1 (i.e., G4-EV5-EVj, j ¼ 1e6) with G3-P1-S1.
G4-EVi Observed a ( ) b ( ) m0 value SF G4-EV5-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 97.94 90.66 0.002 0.016
e 82.47 97.10 0.016 0.370
e 99.44 127.25 0.099 0.272
e 43.25 12.90 0.710 0.013
G4-P1 4.01 30.39 0.861 0.395
e 45.37 67.61 0.268 0.293
G4-P1-S1 11.142 15.274 0.947 0.178
e 46.508 26.696 0.615 0.069
e 85.958 57.424 0.038 0.298
e 104.736 75.243 0.065 0.216
e 87.792 70.975 0.013 0.074
e 48.983 47.081 0.447 0.256
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Table 5 For nearly none twinning shear transmission over GB in Fig. 3d, SFs of the six possible EVs in grain G10 (i.e., G10-EVi, i ¼ 1e6) and their m0 values with G9-P1, the related a and b angles, and those for the six possible EVs in primary twin G10-P1 (i.e., G10-EV2-EVj, j ¼ 1e6) with G9-P1-S1.
G10-EVi Observed a ( ) b ( ) m0 value SF G10-EV2-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 96.37 83.27 0.013 0.107
G10-P1 71.85 69.87 0.107 0.486
e 31.61 38.29 0.668 0.125
e 14.25 3.10 0.968 0.098
e 56.86 39.51 0.422 0.468
e 89.62 70.68 0.002 0.115
e 113.10 67.47 0.150 0.338
e 92.28 46.73 0.027 0.003
e 51.12 19.38 0.592 0.406
e 19.48 25.97 0.848 0.337
e 53.82 53.36 0.352 0.003
G10-P1-S1 94.42 70.20 0.026 0.408
Candidates of preferred twin variants are highlighted in bold.
Candidates of preferred twin variants are highlighted in bold.
Table 3 For solely prominent primary twinning shear transmission over GB in Fig. 3b, SFs of the six possible EVs in grain G6 (i.e., G6-EVi, i ¼ 1e6) and their m0 values with G5-P1, the related a and b angles, and those for the six possible EVs in primary twin G6-P1 (i.e., G6-EV3-EVj, j ¼ 1e6) with G5-P1-S1.
twin structures are featured without any prominent twinning shear transmission over GB. A microstructure with nearly none twinning shear transmission over GB is shown in Fig. 3d. High SF primary twins G9-P1 (SF/SF rank/SF ratio ¼ 0.471/1/1) and G10-P1 (SF/SF rank/SF ratio ¼ 0.486/1/1) form a cross-GB primary twin pair. Each of them corresponds to the twin variant with the highest possible SF in its host grain. It can be seen in Table 5 that m0 value of them is close to 0. Comparatively, G10-EV5 in Table 5 not only has the second highest SF, but also has a medium m0 value of 0.422. If twinning shear transmission played an important role, G10-EV5 should be more preferred, just as the case of G8-EV6 in Table 4. In Fig. 3d, high SF secondary twins G9-P1-S1 (SF/SF rank/SF ratio ¼ 0.424/2/0.951) and G10-P1-S1 (SF/SF rank/SF ratio ¼ 0.408/ 1/1) form a cross-GB secondary twin pair in their host cross-GB primary twin pair. Once again, their m0 value is close to 0. Similarly, G10-EV2-EV3 in Table 5 should be more preferred, if twinning shear transmission played an important role. The case in Fig. 3d indicates that a compound cross-GB twin structure can form due to mechanisms other than twinning shear transmission over GB.
G6-EVi Observed a ( ) b ( ) m0 value SF G6-EV3-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 78.03 49.43 0.135 0
e 37.47 13.76 0.771 0.147
G6-P1 12.89 26.35 0.874 0.365
e 55.28 60.01 0.285 0.012
e 90.95 79.70 0.003 0.172
e 101.41 75.28 0.050 0.402
G6-P1-S1 29.48 105.18 0.228 0.343
e 41.05 66.48 0.301 0.044
e 57.57 53.92 0.316 0.217
G6-P1-S2 64.22 83.24 0.051 0.306
e 56.54 122.90 0.300 0.049
e 39.72 137.14 0.564 0.249
Candidates of preferred twin variants are highlighted in bold.
1 and 2 are absent. As shown in Table 4, G8-EV6-EV3 has the second highest SF (i.e., SF rank 2), but its m0 value with G7-P1-S1 is close to 0. However, unless a reliable calculation of free energy change due to the two-step compressions is made, the preference of G8-EV6EV1 over G8-EV6-EV6 cannot be predicted unambiguously, since the latter one has the highest SF (i.e., SF rank 1) over 0.4 and a medium m0 value over 0.5. Six of the 30 (i.e., 20%) compound cross-GB twin structures have both m0 values of cross-GB primary and secondary twin pairs smaller than 0.7, as shown in Fig. 2b. These compound cross-GB Table 4 For solely prominent secondary twinning shear transmission over GB in Fig. 3c, SFs of the six possible EVs in grain G8 (i.e., G8-EVi, i ¼ 1e6) and their m0 values with G7P1, the related a and b angles, and those for the six possible EVs in primary twin G8P1 (i.e., G8-EV6-EVj, j ¼ 1e6) with G7-P1-S1.
G8-EVi Observed a ( ) b ( ) m0 value SF G8-EV6-EVj Observed a ( ) b ( ) m0 value SF
EV1
EV2
EV3
EV4
EV5
EV6
e 89.05 63.99 0.007 0.136
e 78.09 101.47 0.041 0.113
e 80.48 69.27 0.059 0.499
e 40.55 41.54 0.569 0.138
e 10.69 8.81 0.971 0.111
G8-P1 52.34 33.57 0.509 0.497
G8-P1-S1 7.09 6.97 0.985 0.336
e 36.47 46.16 0.557 0.003
e 71.28 79.61 0.058 0.407
e 88.52 91.84 0.001 0.333
e 77.64 74.47 0.057 0.003
e 45.79 38.72 0.544 0.410
Candidates of preferred twin variants are highlighted in bold.
4. Discussions 4.1. Parent crystals in a compound cross-GB twin structure The observed 30 compound cross-GB twin structures consist of 60 cross-GB twin pairs, of which 60% have a m0 >0.7 (Fig. 2b). Twinning shear transmission over GB prominently contributes to their formations. Parent crystals in these structures refer to host grains for primary twins and host primary twins for secondary twins. Misorientation between two parent crystals can be expressed by a (rotation axis)-(minimum rotation angle) pair. Intuitively, twinning shear transmission is more likely to happen over the boundary between two parent crystals with a small misorientation. Fig. 4a shows the relationship between misorientation angle q (i.e., minimum rotation angle) and twinning shear transmission in the observed 30 compound cross-GB twin structures. There exist 30 host grains and 30 host primary twins. About 93.3% of the host grains and about 86.7% of the host primary twins have q values smaller than 35 . For those in the region of m0 >0.7 and q < 35 in Fig. 4a, both the cross-GB primary and secondary twin pairs have a tendency of m0 value increasing with q value decreasing. However, about 35.7% of the host grains with q < 35 and about 30.8% of the host primary twins with q < 35 produce cross-GB twin pairs with m0 values smaller than 0.7. Twin pairs with high m0 values are absent possibly due to their low SFs, such as the case of
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Fig. 4. (a) Effect of misorientation between two neighboring parent crystals on twinning shear transmission in the observed 30 compound cross-GB twin structures. Symbol ‘o’ represents m0 value of a cross-GB primary twin pair and misorientation angle between their host grains, while symbol ‘*’ represents m0 value of a cross-GB secondary twin pair and misorientation angle between their host primary twins. (b) Misorientations of 60 secondary twins in the 30 compound cross-GB twin structures with respect to their host grains. (c) A sketch of mechanisms of cross-GB twin pair formation. ‘C’ and ‘T’ denote parent crystal and twin, respectively. (d) A sketch of two possible processes of associated twin nucleation. In (c) and (d), the parent crystal ‘C’ refers to a host grain for a primary twin or a primary twin for a secondary twin.
G8-EV5 and G7-P1 (Table 4). About 98.3% of the 120 observed twins have SF ratios larger than 0.7 (Fig. 1c), indicating that they are twin variants with relatively high SFs compared with other possible twin variants in their host crystals. Thus, low SF restricts twinning shear transmission over GB. Some twin pairs with low m0 values appear, since the geometry of their host crystals does not allow a high m0 value, such as the case of G6-EV3-EVj (j ¼ 1e6) and G5-P1-S1 (Table 3). The 36 possible double extension twin variants can be classified into four groups according to their misorientations with respect to their host grains: (I) 0 (i.e., detwinning), (II) 〈1 2 1 0〉7.4 , (III) 〈0 14 14 1〉60 , and (IV) 〈1 7 8 0〉60.4 [10]. As shown in Fig. 4b, about 81.7% of the 60 secondary twins in the 30 compound crossGB twin structures have a misorientation of 〈0 14 14 1〉60 , while all the rest have a misorientation of 〈1 7 8 0〉60.4 . The obvious preference of secondary twins with 〈0 14 14 1〉60 is resulted from the confining effect of their host primary twins [10]. Such a confining effect also works for {1 0 1 1}-{1 0 1 2} double twins [11,24,35]. 4.2. Mechanisms of cross-GB twin pair formation Formation of a cross-GB twin pair is the basis of formation of a compound cross-GB twin structure. As sketched in Fig. 4c, there are two possible mechanisms of cross-GB twin pair formation: (I) associated nucleation and (II) isolated nucleation. The former one refers to conditions when the influence of one twin on the
nucleation of another twin cannot be neglected, of which two possible processes of microstructure evolution (i.e., C1-T1 and C2T1, and C1-T2 and C2-T2) are sketched in Fig. 4c. The latter one refers to conditions when the influence of one twin on the nucleation of another twin is negligible, of which three possible processes of microstructure evolution (i.e., C3-T1 and C4-T1, C3-T2 and C4-T1, and C3-T3 and C4-T2) are also sketched in Fig. 4c. For twin pairs formed due to isolated nucleation, it is not surprising that they have high SFs but low m0 values. When plastic strain is small, the possibility of formation of a twin pair is low. However, when plastic strain is large enough, C3-T1 and C4-T1 in Fig. 4c for example would consume the majority or eventually all of C3 and C4, respectively. Then, the possibility of them to form a twin pair is very high. For twin pairs formed due to associated nucleation, two possible processes of twin nucleation are drawn in Fig. 4d. The internal stress near the boundary between C1 and C2 in Fig. 4d is a combination of the applied macro stress, twin-induced stress at GB as mentioned in introduction, and boundary stress due to different plastic deformation behaviors in differently oriented crystals. When plastic strain is small, it is reasonable to neglect boundary stress in a single-phase material with a strong texture, as in the present study. The internal stress should be relaxed in C2 through plastic deformation, including twinning and dislocation slipping. One possibility as path (I) in Fig. 4d is through twinning (i.e., twinto-twin accommodation), resulting in the formation of a cross-GB twin pair with a high m0 value when it is allowed by geometry.
Z.-Z. Shi / Journal of Alloys and Compounds 716 (2017) 128e136 Table 6 For twin variant selection of G10-P1-S1 in Fig. 3d, SFs of the six possible EVs in primary twin G10-P1 (i.e., G10-EV2-EVj, j ¼ 1e6) and their m0 BS values. G10-EV2-EVj
EV1
EV2
EV3
EV4
EV5
EV6
Observed m0 BS SF
e 0.824 0.338
e 0.751 0.003
e 0.482 0.406
e 0.188 0.337
e 0.209 0.003
G10-P1-S1 0.523 0.408
Candidates of preferred twin variants are highlighted in bold.
Another possibility as path (II) in Fig. 4d is through dislocation slipping (i.e., twin-to-slip accommodation). In Mg and its alloys at room temperature, basal slip has a critical resolved shear stress (CRSS) much lower than those of other slip modes [36,37], so that it is more likely to be activated. It is experimentally evidenced that profuse basal slip is activated to relax twin-induced stress at GB in pure Mg, which can be sufficiently extensive to allow complete blunting of a twin [8]. Similarly in pure Ti, prismatic and pyramidal slip bands are observed to be correlated with deformation twins at GBs [38]. Then, C1, C1-T1 and C2 form GB triple junctions (Fig. 4d), which have an effect of stress concentration. Thus, C2-T1 will nucleate preferentially near the triple junctions, which is supported by EBSD observations (e.g., Fig. 2 in Ref. [22]). In this case, m0 value of C1-T1 and C2-T1 is irrelevant to the formation of C2-T1, which of course could be low. When twin-induced stress at GB from C2-T1 is important, path (II) in Fig. 4d can lead to the formation of C2-T1 with low positive or even negative SF [22]. Under this condition, twin variant selection of C2-T1 depends more on shear accommodation in C1-T1 or in C1 (Fig. 4d). Such a formalism may explain why G10-EV2-EV6 is more preferred than G10-EV2-EV3, although their SFs are comparatively high (Table 5). Here, G9, G9-P1, and G10-P1-S1 in Fig. 3d correspond to C1, C1-T1 and C2-T1 in Fig. 4d, respectively. According to Section 2.2, the largest absolute m0 value among those of G10-P1-S1 and the three possible basal slip systems (i.e., {0 0 0 1}〈2 1 1 0〉) in G9-P1 can be taken to measure the ability of G9-P1 to accommodate the shear of G10-P1-S1 through basal slip, which is designated as m0 BS. As shown in Table 6, G10-EV2-EV6 has a higher m0 BS value than G10-EV2-EV3, indicating that basal slip in G9-P1 can accommodate more twinning shear of G10-EV2-EV6. 5. Conclusions After a strain path change, 30 compound cross-GB extension twin structures are detected by EBSD in a deformed AZ31 Mg alloy, which consist of cross-GB secondary extension twin pairs within cross-GB primary extension twin pairs. The study of these compound cross-GB extension twin structures leads to the following conclusions: 1. It is proved that the shear that can be accommodated through a deformation system is proportional to m0 factor and the magnitude of the original shear. When m0 >0, twinning shear transmission could happen. When m0 is high, which is set to be m0 0.7 in this study, twinning shear transmission is considered to be prominently contributed to formation of a cross-GB twin pair. 2. Associated nucleation and isolated nucleation are two possible mechanisms of cross-GB twin pair formation. As to the former one, shear accommodation is possible to be achieved through twin-to-twin (i.e., twinning shear transmission) or twin-to-slip modes. Twin-to-twin accommodation mode usually leads to a high m0 value as long as it is allowed by geometry. However, twin-to-slip accommodation mode or isolated nucleation could lead to a low m0 value.
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3. There are 60 cross-GB twin pairs related to the 30 compound cross-GB twin structures, of which 60% have a m0 >0.7, indicating that twinning shear transmission over GB is a major mechanism of cross-GB twin pair formation. 40% of the compound cross-GB twin structures form due to multi-level twinning shear transmissions over GB, the primary and the secondary levels of which refer to twinning shear transmission between primary and secondary twins, respectively. 40% of them form due solely to primary or secondary twinning shear transmission over GB. The rest 20% form without any prominent twinning shear transmission over GB. 4. About 98.3% of the 120 twins in the 30 compound cross-GB twin structures are high SF twins with a SFratio 0.7. Low SF twins are preferred due to twinning shear transmission. Over 85% of the parent crystals have a misorientation angle smaller than 35 . About 81.7% of the 60 secondary twins in the 30 compound cross-GB twin structures have a misorientation of 〈0 14 14 1〉 60 with respect to their host grains, indicating that the confining effect of primary twin on variant selection of secondary twin is strong. 5. For a cross-GB twin pair formed due to twin-to-twin accommodation mode, SF and m0 factor should be considered to predict twin variant selection. For variant selection of secondary twins, its misorientation with respect to its host grain should also be taken into consideration. However, a cross-GB twin pair could form through several possible processes of microstructure evolution, which makes prediction of its formation complicated. Acknowledgement This work was supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 51601010), and Fundamental Research Funds for the Central Universities (Project: Deformation behavior and microstructure evolution of difficult-to-deform metals under complex loadings). References [1] C.D. Barrett, H.E. Kadiri, The roles of grain boundary dislocations and disclinations in the nucleation of {10-12} twinning, Acta Mater. 63 (2014) 1e15. , An atomic and probabilistic perspective on [2] J. Wang, I.J. Beyerlein, C.N. Tome twin nucleation in Mg, Scr. Mater. 63 (2010) 741e746. [3] A. Khosravani, D.T. Fullwood, B.L. Adams, T.M. Rampton, M.P. Miles, R.K. Mishra, Nucleation and propagation of twins in AZ31 magnesium alloy, Acta Mater. 100 (2015) 202e214. [4] S. Godet, L. Jiang, A.A. Luo, J.J. Jonas, Use of Schmid factors to select extension twin variants in extruded magnesium alloy tubes, Scr. Mater. 55 (2006) 1055e1058. [5] S.H. Park, S.-G. Hong, J.H. Lee, Y.-H. Huh, Texture evolution of rolled Mg-3Al1Zn alloy undergoing a {10-12} twinning dominant strain path change, J. Alloys Compd. 646 (2015) 573e579. [6] G.-S. Song, S.-H. Zhang, L. Zheng, L. Ruan, Twinning, grain orientation and texture variation of AZ31 Mg alloy during compression by EBSD tracing, J. Alloys Compd. 509 (2011) 6481e6488. [7] L. Jiang, J.J. Jonas, R.K. Mishra, A.A. Luo, A.K. Sachdev, S. Godet, Twinning and texture development in two Mg alloys subjected to loading along three different strain paths, Acta Mater. 55 (2007) 3899e3910. [8] M.R. Barnett, N. Stanford, A. Ghaderi, F. Siska, Plastic relaxation of the internal stress induced by twinning, Acta Mater. 61 (2013) 7859e7867. [9] J.D. Eshelby, The elastic field outside an ellipsoidal inclusion, Proceedings of the Royal Society of London, Ser. A 252 (1959) 561e569. [10] Z.-Z. Shi, Y.-D. Zhang, F. Wagner, T. Richeton, P.-A. Juan, J.-S. Lecomte, L. Capolungo, S. Berbenni, Sequential double extension twinning in a magnesium alloy: combined statistical and micromechanical analyses, Acta Mater. 96 (2015) 333e343. [11] P.-A. Juan, S. Berbenni, L. Capolungo, Prediction of internal stresses during growth of first- and second-generation twins in Mg and Mg alloys, Acta Mater. 60 (2012) 476e486. , D.W. Brown, [12] C.C. Aydıner, J.V. Bernier, B. Clausen, U. Lienert, C.N. Tome Evolution of stress in individual grains and twins in a magnesium alloy aggregate, Phys. Rev. B 80 (2009), 024113-1-024113-6. [13] X. Hong, A. Godfrey, W. Liu, Challenges in the prediction of twin transmission at grain boundaries in a magnesium alloy, Scr. Mater. 123 (2016) 77e80.
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[14] H.H. Yu, Y.C. Xin, A. Chapuis, X.X. Huang, R.L. Xin, Q. Liu, The different effects of twin boundary and grain boundary on reducing tension-compression yield asymmetry of Mg alloys, Sci. Rep. 6 (2016), 29283e1-29283-8. [15] Z.-Z. Shi, X.-F. Liu, Characteristics of cross grain boundary contraction twin pairs and bands in a deformed Mg alloy, J. Alloys Compd. 692 (2017) 274e279. [16] R.L. Xin, C.F. Guo, Z.R. Xu, G.D. Liu, X.X. Huang, Q. Liu, Characteristics of long {10-12} twin bands in sheet rolling of a magnesium alloy, Scr. Mater. 74 (2014) 96e99. [17] R.L. Xin, Y.C. Liang, C.H. Ding, C.F. Guo, B.S. Wang, Q. Liu, Geometrical compatibility factor analysis of paired extension twins in extruded Mg-3Al1Zn alloys, Mater. Des. 86 (2015) 656e663. [18] M.R. Barnett, M.D. Nave, A. Ghaderi, Yield point elongation due to twinning in a magnesium alloy, Acta Mater. 60 (2012) 1433e1443. [19] H.C. Chen, T.M. Liu, D.W. Hou, D.F. Shi, Study on the paired twinning behavior in a hot rolled AZ31 magnesium alloy via interrupted in situ compression, Mater. Sci. Eng. A 667 (2016) 402e408. [20] L. Wang, P. Eisenlohr, Y. Yang, T.R. Bieler, M.A. Crimp, Nucleation of paired twins at grain boundaries in titanium, Scr. Mater. 63 (2010) 827e830. [21] J. Luster, M.A. Morris, Compatibility of dependence on deformation in twophase Ti-AI alloys: microstructure and orientation relationships, Metall. Mater. Trans. A 26 (1995) 1745e1756. [22] Z.-Z. Shi, Y.-D. Zhang, F. Wagner, P.-A. Juan, S. Berbenni, L. Capolungo, J.S. Lecomte, T. Richeton, On the selection of extension twins with low Schmid factors in a deformed Mg alloy, Acta Mater. 83 (2015) 17e28. [23] S. Niknejad, S. Esmaeili, N.Y. Zhou, The role of double twinning on transgranular fracture in magnesium AZ61 in a localized stress field, Acta Mater. 102 (2016) 1e16. [24] E. Martin, L. Capolungo, L. Jiang, J.J. Jonas, Variant selection during secondary twinning in Mge3%Al, Acta Mater. 58 (2010) 3970e3983. [25] M.R. Barnett, Twinning and the ductility of magnesium alloys Part II. “Contraction” twins, Mater. Sci. Eng. A 464 (2007) 8e16. [26] M.R. Barnett, M.D. Nave, C.J. Bettles, Deformation microstructures and textures of some cold rolled Mg alloys, Mater. Sci. Eng. A 386 (2004) 205e211.
[27] M. Lentz, M. Risse, N. Schaefer, W. Reimers, I.J. Beyerlein, Strength and ductility with {10-11}-{10-12} double twinning in a magnesium alloy, Nat. Commun. 7 (2016), 11068e1-11068-7. [28] S.H. Park, S.G. Hong, C.S. Lee, In-plane anisotropic deformation behavior of rolled Mg-3Al-1Zn alloy by initial {10-12} twins, Mater. Sci. Eng. A 570 (2013) 149e163. [29] S.H. Park, S.G. Hong, J.H. Lee, C.S. Lee, Multiple twinning modes in rolled Mge3Ale1Zn alloy and their selection mechanism, Mater. Sci. Eng. A 532 (2012) 401e406. [30] F. Bachmann, R. Hielscher, H. Schaeben, Texture analysis with mtex e free and open source software toolbox, Solid State Phenom. 160 (2010) 63e68. [31] J.J. Jonas, S.-J. Mu, T. Al-Samman, G. Gottstein, L. Jiang, E. Martin, The role of strain accommodation during the variant selection of primary twins in magnesium, Acta Mater. 59 (2011) 2046e2056. [32] J.W. Christian, S. Mahajan, Deformation twinning, Prog. Mater Sci. 39 (1995) 1e157. [33] H. Qin, J.J. Jonas, H. Yu, N. Brodusch, R. Gauvin, X. Zhang, Initiation and accommodation of primary twins in high-purity titanium, Acta Mater. 71 (2014) 293e305. [34] S.J. Mu, J.J. Jonas, G. Gottstein, Variant selection of primary, secondary and tertiary twins in a deformed Mg alloy, Acta Mater. 60 (2012) 2043e2053. [35] M.R. Barnett, Z. Keshavarz, A.G. Beer, X. Ma, Non-Schmid behaviour during secondary twinning in a polycrystalline magnesium alloy, Acta Mater. 56 (2008) 5e15. [36] W.B. Hutchinson, M.R. Barnett, Effective values of critical resolved shear stress for slip in polycrystalline magnesium and other hcp metals, Scr. Mater. 63 (2010) 737e740. [37] J. Koike, Enhanced deformation mechanisms by anisotropic plasticity in polycrystalline Mg alloys at room temperature, Metall. Mater. Trans. A 36 (2005) 1689e1696. [38] L. Wang, Y. Yang, P. Eisenlohr, T.R. Bieler, M.A. Crimp, D.E. Mason, Twin nucleation by slip transfer across grain boundaries in commercial purity titanium, Metall. Mater. Trans. A 41 (2010) 421e430.