*3.3. In-Situ Neutron Diffraction* MicromechanicsofTwinning

Brown et al. [34] incorporated the in-situ neutron diffraction to investigate the evolution of internal strain and texture during extension twinning in the AZ31B Mg alloy. They applied different loading paths that include in-plane compression (IPC), in-plane tension (IPT), and through-thickness compression (TTC), where the c-axis of most of the grains were perpendicular to the loading direction. Among these three loading paths, the extension twinning was the governing deformation mechanism in the IPC, which was the main focus of their study [34]. They used the VPSC framework to simulate the sample response during in-plane compression. Figure 34 presents the stress-strain responses of AZ31B Mg alloy during TTC and IPC. The circles on the IPC curve represent the strains at which the in-situ neutron diffraction measurements were conducted. The VPSC results were also presented as the dashed line for the IPC loading. The stress-strain response of IPC loading yields at the stress value of ∼60 MPa, which was followed by a plateau with a small hardening rate. This plateau is commonly attributed to the role of extension twinning [34]. The hardening rate starts increasing at the inflection point that occurs at the strain of ∼5%. For strain values larger than ∼8%, the stress during IPC becomes equal or larger than that of the TTC loading, and the rate of work hardening decreases again.

**Figure 34.** The stress-strain responses of AZ31B Mg alloy during thorough-thickness compression (TTC) and in-plane compression (IPC). The circles on IPC curve represent the strains at which the in-situ neutron diffraction measurements were conducted. The VPSC results are also presented as the dashed line for the IPC loading (After Brown et al. [34]).

Brown et al. [34] investigated the diffraction pattern of the AZ31B Mg alloy subjected to the IPC loading path in parallel and perpendicular detector banks at different stress values, as shown in Figure 35. Initially, the parallel detector banker does not detect any diffraction peaks from {0002}, while the transverse detector bank shows very strong diffraction peaks from {0002}. As the applied compression increases, the diffraction peaks intensity detected by the parallel detector banker increases, while the one detected by the transverse detector bank decreases. Brown et al. [34] stated that the increase in the diffraction peaks intensity of {0002} detected by the parallel detector banker manifests the development of twinned grains. Accordingly, they measured the twinning variation and showed that the twinning initiates at the strain ∼1%. It was observed that the twinning volume linearly increases with the applied strain up to a strain ∼6%. The twinning is then saturated at the strain ∼8%, with the maximum twinning volume fraction of 80%. Brown et al. [34] mentioned that the variation of twin volume fraction is almost proportional to the increase in the diffraction peaks intensity of {0002} detected by the parallel detector banker, with the slope of 0.145 per unit plastic strain. The VPSC predicted faster twin

formation, which saturates at the strain ∼4%. The increased hardening rate at larger strain observed in Figure 34 can be attributed to the twin saturation and initiation of pyramidal slip in the twinned regions [34].

**Figure 35.** The diffraction pattern of the AZ31B Mg alloy subjected to in-plane compression loading in parallel and perpendicular detector banks at different stress values (After Brown et al. [34]).

Figure 36 shows the variation of lattice strain parallel and perpendicular to the loading axis in untwinned parent grains and daughter twinned grains versus the applied load [34]. Brown et al. [34] obtained the lattice strain for a grain orientation contributing to a given (*hkl*) diffraction peak as follows:

$$\varepsilon = \frac{d^{\text{lkl}} - d\_0^{\text{lkl}}}{d\_0^{\text{lkl}}} \tag{17}$$

where *d*0 denotes the unstrained interatomic spacing, which can be approximated by the initial interplanar spacing. Figure 36a shows the internal strain in the untwinned parent grain parallel and normal to the loading axis versus the applied load. The theoretical linear elastic relation is shown as dashed lines. Prior to the applied compressive stress of 70 MPa, the untwinned parent grains behave according to the linear elasticity. Afterward, the lattice strain becomes less than the elastic prediction as other grains with different orientations, including the daughter grains, which sustain stresses larger than average applied stress.

**Figure 36.** The variation of lattice strain parallel and perpendicular to the loading axis versus the applied load: (**a**) Untwinned Parent grains, (**b**) daughter twinned grains (After Brown et al. [34]). The dashed line represents the theoretical linear elastic relation. Closed markers denote the loading path, while the open markers represent the unloading path.

Figure 36b shows the internal strain in the daughter twinned grains parallel and normal to the loading axis versus the applied load. The lattice strain in daughter grains can be first identified at the compressive applied stress of 70 MPa, where twin reorientation occurs. Brown et al. [34] reported an interesting observation in which the lattice strain of newly twinned daughters was −700 *με*, which was considerably less than the untwinned parent and the surrounding grains. In other words, the daughter twinned grains were in a relaxed state compared to the surrounding grains. In the region of applied compressive loading of 70 MPa − 220 MPa, the increase in lattice strain was much larger than the elastic prediction. The stress/lattice strain slope was reported as 32 GPa, which was smaller than the elastic slope of 48 GPa calculated for grains with basal pole parallel to the loading axis. This can be attributed to the fact that the daughter twinned grains were hard orientations and sustain more elastic strain compared to the untwinned parent and surrounding grains, which have soft orientations and deform inelastically [34]. For applied stress greater than 220 MPa, one can see an inflection point in the variation of lattice strain versus the applied loading as the variation of strain sustained by the daughter grain is less than the elastic prediction.

Brown et al. [34] calculated the CRSS required for twin formation using the in-situ neutron diffraction experimental data. They observed that the twin formation occurs at the stress in the region of 50 MPa − 70 MPa. In the case of the tested sample, the maximum Schmid factor is 0.499, which leads to the twinning CRSS of 25 to 35 MPa for AZ31B Mg alloy. In the last step, Brown et al. [34] tried to derive how much strain is sustained by twinning. They derived the contribution of twin to the total plastic strain as follows:

$$\frac{\Delta\varepsilon\_{\text{twin}}}{\Delta\varepsilon\_{\text{plastic}}} = 0.32S \frac{\Delta\upsilon}{\Delta\varepsilon\_{\text{plastic}}}\tag{18}$$

where the coefficient 0.32 is obtained based on the texture of tested AZ31B Mg alloy, *S* is the characteristic twin shear strain defined in Equation (10), and *υ* is the twin volume fraction. They estimated that 61% of the plastic deformation is sustained by twinning mechanisms at the strain of 5%. Afterward, the twinning rate versus the strain decreases in which it saturates at the strain of 8%. At this strain, 42% of the plastic deformation is sustained by the twinning mechanism.

#### **4. Conclusions and Future Works**

In the present study, the crystal plasticity models, which have been incorporated for Mg and its alloys, are reviewed. These models include different deformation mechanisms such as plastic slip, twinning, and detwinning. The recent experimental frameworks, such as in-situ neutron diffraction, 3D high energy synchrotron X-ray techniques, and digital image correlation under scanning electron microscopy, which have been incorporated along crystal plasticity models to investigate the properties of Mg and its alloys, are also reviewed. Although many studies have tried to model the behavior of Mg and its alloys using crystal plasticity, which was often coupled by the advanced in-situ techniques, there are still some critical challenges that need to be addressed. A summary is presented for the future challenges as below:


**Author Contributions:** Conceptualization, M.Y., G.Z.V., and V.S.; methodology, M.Y., G.Z.V., and V.S.; formal analysis, M.Y.; investigation, M.Y., G.Z.V., and V.S.; resources, G.Z.V.; data curation, M.Y.; writing—original draft preparation, M.Y.; writing—review and editing, M.Y., G.Z.V., and V.S.; visualization, M.Y.; supervision, G.Z.V. and V.S.; project administration, M.Y., G.Z.V., and V.S.; funding acquisition, G.Z.V., M.Y., and V.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The current work is partially funded by the NSF EPSCoR CIMM project under award #OIA-1541079. In addition, this work is partially supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award#DE-SC0008637 as part of the Center for Predictive Integrated Structural Materials Science (PRISMS Center) at the University of Michigan. We also acknowledge the financial cost-share support of the University of Michigan College of Engineering and Office of the Vice President for Research.

**Acknowledgments:** George Z. Voyiadjis acknowledges the funding from NSF EPSCoR CIMM project under award #OIA-1541079. Mohammadreza Yaghoobi and Veera Sundararaghavan acknowledge the funding from U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award#DE-SC0008637 as part of the Center for Predictive Integrated Structural Materials Science (PRISMS Center) at the University of Michigan. Mohammadreza Yaghoobi and Veera Sundararaghavan also acknowledge the financial cost-share support of the University of Michigan College of Engineering and Office of the Vice President for Research.

**Conflicts of Interest:** The authors declare no conflict of interest.
