Crystallographic analysis on the primary and secondary extension twin variants in Mg alloys

{10-12}-{10-12} double twins (DTs) are observed in AZ31 Mg alloys after one path uniaxial tension. Secondary twins (STs) are produced mainly from the intersecting points of two primary twins (PTs), forming the PTing-PTed-ST grouping. The relative orientations among these variants are analyzed. Statistical analysis reveals that the two primary variants have a specific orientation relationship with the geometric compatibility factor (m’) of 0.562. The detwinning of PTed and ST also have such type of orientation. Moreover, the PTing and the selected ST have almost opposite shear directions with the angle of 151.1°. Tensor analysis conforms that these geometric orientation relationships are beneficial for minimizing the local strain incompatibility.

A few studies reported that the ET-ET type of DTs can also be formed in Mg alloys after one path strain deformation [14][15][16][17][18][19]. In some cases, the applied stress is complicated, which hinders understanding the mechanism of variant selection. Note that knowledge on the selection rule of ET-ET variants in one path deformation is still limited compared to that of CT-ET variants. In recent work [19], we found the formation of {10-12}-{10-12} DTs in AZ31 Mg alloys after simple uniaxial tension at room temperature. The crystallographic aspects of the secondary variants were analyzed and linked to the crystallographic features of the intersecting primary twins. Here, we aim to extend 1234567890''"" understanding the selection rule for these variants. Being different from the previous work, here the relative geometric relationships between the intersecting primary variants and that between one primary variant and the secondary twin are analyzed. Then the selection of secondary variants is explained from the point view of local strain accommodation. Moreover, the displacement gradient tensors associated with the primary and secondary variants are calculated and analyzed. This provides some insights into understanding the activation of these DTs.

Material and method
A hot rolled AZ31 Mg alloy plate (~70 mm in thickness) is used as the starting material. Fig. 1a is the typical EBSD map of this plate. The average grain size is determined to be ~32 μm by a linear intercept method. Some grains are larger than 100 μm. A cylindrical tensile sample is machined from the AZ31 plate with the axis direction parallel to the normal direction (ND) of the plate. Then the sample is subjected to uniaxial tension at room temperature and at a strain rate of 10 -3 s -1 . The tensile test is interrupted at ~7%. The microstructure of the deformed sample is shown in Fig. 1b. It is clear that many twins are formed in the deformed sample. At the intersecting points of some primary twins (PTs), secondary twins (STs) are produced in the thicker PTs. All the primary and secondary twins are confirmed to be of {10-12}<-1011> type. showing the deformed microstructure. Due to the hexagonal crystal symmetry, each PT has six crystallographically equivalent variants. There are three types of relative orientations among the six variants [20]. As shown in Fig. 2, these relative orientations are referred to as ortho-position (OP), meta-position (MP) and para-position (PP), respectively [21]. For OP, the angles between the two habit plane normals (ψ) and between the two shear directions (κ) are 40.0 o and 42.8 o , respectively. Thus, the geometric compatibility factor (defined as ' cos cos m In previous studies, m' has been used to evaluate the local strain accommodation between slip and twins [22][23][24][25][26]. The m' value ranges from -1 to 1. The larger the m' means the better the local strain compatibility. The ψ, κ and m' for the other types of relative orientations are also calculated, as shown in Fig. 2b. It reveals that the m' for MP and PP is very small, being 0.060 and -0.004, respectively. Previous studies indicated that the variants with OP orientation likely form in the same grain to reduce the local plastic anisotropy. As revealed in Fig. 1b and some previous work [14][15][16], STs likely initiate from the intersecting points of two PTs. Hereafter, the one containing ST is referred to as PT ed (meaning this PT is impinged by another one); another is referred to as PT ing (meaning this PT is impinging another one). These two PTs together with the ST form a PT ing -PT ed -ST grouping. In the following analysis, the ET variants 1-6 correspond to the indices of

Analysis and discussion
The EBSD map in Fig. 3a shows   Fig. 4a shows the (0002) stereographic projections for a-d and the six potential PTs. Those for b, e and the six potential STs are shown in Fig. 4b. Here, the potential variant nearest to the observed twin is considered as the active one. In this way, the active twin systems are determined, and their habit planes and shear directions are highlighted in the unit cells as shown in Fig. 3b. It indicates that twins b and c have OP orientation, meaning that their co-activations can minimize the local strain anisotropy. The detwinning of b can be considered as the "twin" a formed in the "matrix" b. Crystallographic calculations indicate that ψ and κ between detwin b and twin e is 80 o and 151.1 o , respectively. Therefore, the detwin b also has OP orientation with e. Moreover, as indicated in Fig. 3b, twins c and e have close shear directions. Their angle is calculated to be ~151 o . It is worth to mention that the orientation relationship between e and c (or d) is of 49.7°<-909-4>. According to the previous work, this geometric relationship favors the activation of STs in terms of basal slip glide [19].  Fig. 5b. It is seen that twin b has OP orientation with c (or d). Detwin b and twin e also have such kind of orientation. The relationships between the first and second generations of variants are summarized in Fig. 5c. Similar crystallographic analyses are performed on more than 20 PT ing -PT ed -ST groupings. It consistently confirms that the PT ing and PT ed have OP orientation with m'=0.562. The detwinning of PT ed and ST also have such relative orientation. These analyses imply that the relative orientation between PT ing and PT ed and that between PT ed detwin and ST determine the selection of secondary variants. Such kind of grouping is beneficial for minimizing the local strain incompatibility.  As illustrated in Table 1, six potential STs can be formed in PT1 (the PT ed ). There are 36 relative orientations between the potential ST and PT1 detwin or the other PTs (i.e. PT2-PT6). The ψ, κ and m' for the 36 combinations are calculated and listed in Table 1. In our experiment, more than 90% of the observed PT ing -PT ed -ST groupings have ψ, κ and m' of 80.0 o , 151.1 o and -0.15, respectively. This corresponds to the combinations of PT2-ST12 or PT6-ST16 in Table 1. According to the indices of twin variants, PT2 (or PT6) has OP orientation with PT1. Similarly, ST12 (or ST16) has OP relationship with PT1 detwin. In addition, Table 1 indicates that the angles between the shear directions of ST12 and PT2 (or of ST16 and PT6) is close to 180 o . This implies that the formation of ST and PT ing pairs is helpful to minimize the strain incompatibility at PT ed boundary. On the other hand, the angle between the habit plane normals of ST and PT ing is less important for determining the secondary variants. Table 1. The theoretical orientation relationships (ψ, κ and m') between PTi and PT1j.
The strain accommodation between twinning and dislocation glide has been assessed by tensor analysis. As reported in literature [4], the displacement gradient tensor for ET in Mg alloy is 0 0 0.129 0 0 0 . This is expressed in the coordinate system (CS) formed by the shear direction (SD), shear plane normal (SPN) and habit plane normal (HPN). Knowing the twin variant and the orientations of twins and the matrix, this tensor can be transformed to any twin or matrix CS. Then the required strain accommodation can be analyzed. For the PT ing -PT ed -ST grouping shown in Fig. 3a, the tensors associated with the six potential PTi are transformed to the CS of PT ed (i.e. twin b). Similarly, those associated with the six potential STj are transformed in this CS as well. The calculation results are listed in Table 2. The calculated SFs for various variants are also given. In Table 2, c and d correspond to v1 and e corresponds to v4. It is clear that twin e has a small SF of -0.059, implying its formation is not favored by the applied stress. According to literature [4], the components of e13 and e23 in the tensor of 11  values of e13 + e23 are close for c (or d) and e, and their signs are of opposite. This implies that the coactivation of PT ing and ST can partly cancel the accommodation strain required from basal slip. This is in agreement with the m' analysis reported in previous work [19]. Table 2. The displacement gradient tensors associated with PT ing and ST expressed in the PT ed crystal frame for the grouping shown in Figure 3b.

Conclusions
Statistical analysis is performed on the PT ing -PT ed -ST groupings formed in AZ31 Mg alloys after tensile deformation. The following major conclusions can be drawn: (1) The two primary variants (PT ing and PT ed ) in the groupings likely form OP orientation with ψ and κ being 40.0 o and 42.8 o , respectively. This corresponds to a m' of 0.562, the largest one among the three possible values.
(2) The detwining of PT ed and ST also have OP orientation, indicating the groupings of these variants are beneficial for minimizing the local strain anisotropy.
(3) PT ing and the selected ST almost have opposite shear directions (with the angle of 151.1 o ), implying that the local strain generated by these two variants can be partly cancelled. The extra accommodation strain might be provided by basal slip.