## 3 Discussion
### Evolution of Slip Systems
#### 3.3.2 Analysis of Schmid Factors  
According to the Schmid's law [35; 36], it was believed that when the CRSS was reached, some specific crystal planes would slip along a specific direction, that is, when the shear stress on the slip system reached a critical value, the plastic deformation began. In this study, we adopted the uniaxial tensile simulation, and the SFs were analyzed. For grains with different orientations, the SFs were calculated in the crystal coordinate system and the primary uniaxial tensile direction should be transferred from sample coordinate system to the crystal coordinate system by rotation matrix \(g\) (see Equation (1)). After that, the SFs were calculated by the slip system and the transferred loading direction. Therefore, the activation of slip systems in different grains was determined by the SFs. The distribution of SFs, corresponding to the macro strains of 0.7%, 0.9%, 2.4%, 4%, 7.8%, and 15.5%, is shown in Figure 11. According to the data of \(\alpha\) phase, it was shown that the prismatic slip systems with the SFs in a range of 0.1-0.2 activated when the macro strain = 0.7% and 0.9%. When the macro strain reached 2.4%, the SFs lower than 0.1 of pyramidal\(<\)a\(>\) slip systems activated. In comparison, the SFs of \(\beta\) phase were much larger, especially at the macro strain = 0.9% and 2.4%, it was shown that all the SFs were in the range of 0.4-0.5 when the slip systems activated. When the macro strain reached 4%, there existed SFs lower than 0.1 of \(\left\{110\right\}<\)111\(>\) and SFs in the range of 0.1-0.2 of \(\left\{112\right\}<\)111\(>\). By comparing the difference of the SFs of the \(\alpha\) and \(\beta\) phases, it was found that the SFs of the \(\alpha\) phase were lower than the SFs of the \(\beta\) phase on the whole, and there was a situation that the slip systems, with lower SFs in the range of 0.1-0.2, became activated preferentially, which was also described in the literature [10]. However, according to the Schmid's law, the slip system with the higher SF can be activated easily, contrary to the result above. On this basis, an assumption was proposed: the grains of \(\alpha\) phase were deviated from uniaxial stress due to grain interactions, which meant the local realistic loading direction of grains of \(\alpha\) phase, deviated from the predominant loading direction. Generally, a resolved shear stress is determined from the full Schmid tensors unless the stress state is uniaxial, what is unlikely for a local stress state. In this study, we analyze the major principal stress to verify the assumption and prove the influence of grain interactions on the SFs quantitatively. For example, when the macro strain reached 0.7%, the prismatic slip systems of grain No. 3 and 11 were activated, while the SF of \(\left(01\overline{1}0\right)\left[\overline{2}110\right]\) in grain No. 3 was 0.136 and the SF of \(\left(01\overline{1}0\right)\left[\overline{2}110\right]\) in grain No. 11 was 0.321. It was found that the \(\left(01\overline{1}0\right)\left[\overline{2}110\right]\) in grain No. 3 should not be the primary slip system. Therefore, as shown in Figure 12a, the macro deformation direction was parallel to X axis, and the direction of major principal stress \(\delta_{1}\) deviated from the X-axis and the deviation angle was \(\theta\) (ignoring the deviation angle between the direction of \(\delta_{1}\) and Z-axis), which was used to fit with the Euler angle and recalculate the SFs. As shown in Figure 12b, the \(\theta\) of grain No. 3 was \(30^{\circ}\) and the \(\theta\) of grain No. 11 was close to \(0^{\circ}\). After recalculation, the SF of \(\left(01\overline{1}0\right)\left[\overline{2}110\right]\) in grain No. 3 was 0.333, which meant that the \(\left(01\overline{1}0\right)\left[\overline{2}110\right]\) slip system was easily activated. Figure 11: Distribution of Schmid factors (SFs) at macro strains = 0.7%, 0.9%, 2.4%, 4.0%, 7.8%, 15.5%.