*2.4. Detwinning*

In the case of more complex strain paths, the detwinning mechanism should also be included in addition to twinning and slip modes. Various models address the detwinning along twinning and slip modes using crystal plasticity [22,87–89,94–97]. Here, a physicallybased Twinning-Detwinning (TDT) model is elaborated, which was developed by Wang and his coworkers [90–92] for the EVPSC framework and extended later on for the CPFE framework [9]. In this model, the twinning and detwinning mechanisms can be divided into 4 major operations as follows (Figure 9):


**Figure 9.** Deformation twinning and detwinning operations: (**a**) Initial twin-free parent grain (**b**) *Operation A*: As soon as the twin volume reaches a threshold of *f*0, twin nucleation occurs. This nucleated twin can grow according to the reduction in the volume of parent grain. (**c**) *Operation B*: The growth of the twin region can occur according to the child propagation. (**d**) *Operation C*: As the parent region grows, the twin volume reduces. (**e**) *Operation D*: Detwinning inside the twinned child leads to the twin volume reduction (After Yaghoobi et al. [9]).

The polarity of simple twinning model should be modified for these operations. In the parent grain, the crystallographic systems include *Ns* slip systems, *Nt* twin systems for *Operation A*, and *Nt* twin systems for *Operation C*. In the case of the twin child, the crystallographic systems include *Ns*−*tw* slip systems, one twin system for *Operation D*, and one twin system for *Operation B*. *Operation A* can ge<sup>t</sup> activated in parent grain for when the resolved shear in the twin system *k* is larger than the corresponding slip resistance, i.e., *τNs*+*<sup>k</sup>* > *<sup>s</sup>Ns*+*k*. *Operation B* is activated in the twin child, which is nucleated by the activation of *β*th twin variant, when *τNs* −*tw*+2 *β* < 0 and |*τNs*−*tw*+<sup>2</sup> *β* | <sup>&</sup>gt;*sNs*−*tw*+<sup>2</sup> *β* . *Operation C* initiates in the parent grain for the *k*th twin variant when *τNs*+*Nt* <sup>+</sup>*k*< 0 and *τNs*+*Nt*+*<sup>k</sup>* <sup>&</sup>gt;*sNs*+*Nt*+*k*.

*Operation D* initiates inside the twin child, which is nucleated by the activation of *β*th twin variant, when *τNs*−*tw*+<sup>1</sup> *β*> *s Ns*−*tw*+1 *β*.

In the TDT/EVPSC model developed by Wang and his coworkers [90–92], the twinned child is manifested as a new grain, and the *Operations A-D* is reflected by the change in the volume of the parent and twin grains. Wang et al. [90] used the developed TDT model to capture the cyclic response of AZ31B Mg alloy for complex loading paths. They compared the model prediction versus the experimental data provided by Lou et al. [106] (Figure 10). They also showed the variation of twin volume and how the deformation twinning and detwinning governs the twin volume and stress-strain curve shape. Furthermore, they compared the prediction of TDT/EVPSC model versus the CG results reported by Proust et al. [88] for the experimental results of AZ31B Mg alloy subjected to the case of in-plane compression followed by through-thickness compression (Figure 11). Although the CG model was able to capture some features of twinning-detwinning mechanisms, the TDT/EVPSC model considerably enhances the accuracy of the simulation results.

**Figure 10.** Comparison of predicted cyclic response of AZ31B Mg alloy versus the experimental data subjected to different loading paths of: (**a**) Uniaxial compression-tension-compression, (**b**) uniaxial tension-compression- tension. The twin volume is predicted by simulation and demonstrated at different strains (After H. Wang et al. [90]).

**Figure 11.** Comparison of predicted stress versus the absolute value of the accumulated strain in AZ31B Mg alloy versus the experimental data subjected to in-plane compression followed by compression loading through the thickness along the RD. The twin volume is predicted by simulation and demonstrated at different strains (After H. Wang et al. [90]).

Unlike the EVPSC framework, the TDT operations are not trivial to apply to a material point in the case of CPFE framework. Yaghoobi et al. [9] used a multiscale framework in which the untwinned and twinned regions were considered within a sub-scale model at a material point and the stresses homogenized using a Taylor-type scheme (Figure 12). This allows multiple variants to coexist at a material point, which avoids the need for the selection of predominant variants as in the PTR models. They implemented the model in an open-source CPFE software PRISMS-Plasticity [103]. The twin volume evolution of a material point for the *β*th twin variant is obtained as below:

$$\dot{f}^{\beta} = \left[1 - \sum\_{\beta=1}^{N\_t} f^{\beta}\right] \frac{\left(\dot{\gamma}^{N\_s + \beta} - \dot{\gamma}^{N\_s + N\_l + \beta}\right)}{S} - f^{\beta} \frac{\left(\dot{\tilde{\gamma}}^{N\_s - lv} - \dot{\tilde{\gamma}}^{N\_s - lv} + ^2\right)}{S} \tag{15}$$

where .*γNs*+*<sup>β</sup>* and .*γNs*+*Nt*+*<sup>β</sup>* are the shear rate inside the parent corresponds to operations A and C, and . *γ N <sup>s</sup>*−*tw*+1 *β* and . *γ Ns*−*tw*+2 *β* are the shear rates within the child nucleated due to the activation of the *β*th twin system.

**Figure 12.** A partially twinned material point in the CPFE framework (After Yaghoobi et al. [9]).

Yaghoobi et al. [9] calibrated the model using the uniaxial experimental data of extruded ZK60A Mg alloy presented by Wu [107] (Figure 13). The predictions of the modeled cyclic response of the ZK60A alloy along the extrusion direction were compared with the experimental data by Wu et al. [17] and Wu [107]. Yaghoobi et al. [9] further studied the cyclic response of ZK60A by comparing the predicted twin variation versus the normalized intensity of the {0002} diffraction peak along the longitudinal direction measured by Wu et al. [35] and Wu [107]. The result showed that not only can the multiscale TDT CPFE framework successfully capture the cyclic stress-strain response of the ZK60A (Figure 14), it can also capture the deformation twinning and detwinning during the cyclic loading (Figure 15). Furthermore, Yaghoobi et al. [9] showed the evolution of predicted basal (0001) pole figures at strains of *ε* = ±1.2% (Figure 16). It was shown that the deformation twinning resulted in the reorientation of the basal pole, which intensifies in the basal {0002} peak in the loading direction at max compressive strains of *ε* = −1.2% (Figure 16b,d). On the other hand, the detwinning mechanism removed the increased basal {0002} peak intensity at a max tensile strain of *ε* = 1.2% (Figure 16c,e).

**Figure 13.** The comparison of the simulated stress-strain response responses of ZK60A Mg alloy subjected to the uniaxial loading along the extrusion direction versus the experimental data of Wu [107] (Yaghoobi et al. [9]).

**Figure 14.** Comparison of the predicted stress-strain curve versus the experimental results of Wu et al. [17] and Wu [107] during the cyclic loading along the extrusion direction in ZK60A Mg alloy (After Yaghoobi et al. [9]).

**Figure 15.** The variation of predicted twin volume versus the strain compared to the experimentally measured change in the normalized intensity of the {0002} diffraction peak along the longitudinal direction obtained by Wu et al. [35] and Wu [107] in ZK60A Mg alloy during the cyclic loading along the extrusion direction (After Yaghoobi et al. [9]).

**Figure 16.** The evolution of basal (0001) pole figures predicted by CPFE simulation at different strains: (**a**) Initial texture, (**b**) first maximum compression (*ε* = −1.2%) in cycle 1, (**c**) first maximum tension (*ε* = 1.2%) in cycle 1, (**d**) second maximum compression (*ε* = −1.2%) in cycle 2, (**e**) second maximum tension (*ε* = 1.2%) in cycle 2 (After Yaghoobi et al. [9]).

## **3. In-situ Experiments**

#### *3.1. In-Situ DIC Experiments*

#### 3.1.1. Microscale Deformation Mechanisms

Githens et al. [25] used the in-situ DIC under scanning electron microscopy (SEM-DIC) to study the deformation mechanisms of WE43-T5 Mg alloy with weak basal texture subjected to uniaxial tension and compression along the rolling direction. The SEM-DIC technique provided the full-field strain maps. They first investigated the response of the WE43 alloy during uniaxial tension (Figure 17). They showed that the heterogeneous pattern of strain does not vary after the yield point. They also studied the strain probability distribution, which confirms that the strain map does not vary after yielding (*ε* = 2.91% and 4.86%). However, this pattern is different from the one before yielding (*ε* = 1.23%). They simulated the sample using CPFE simulation open-source software of PRISMS-Plasticity [103] and compared the predicted strain map with the experimental observations, which agrees well (Figure 18).

**Figure 17.** Using SEM-DIC technique to investigate the underlying deformation mechanisms in WE43-T5 Mg alloy during tensile loading along the rolling direction: (**a**) The map of normalized maximum principal strain maps at *ε* = 1.23%, (**b**) the map of normalized maximum principal strain maps at *ε* = 4.86%, (**c**) the inverse pole figure map, (**d**) a probability distribution of the strain at different applied tensile strains of *ε* = 1.23%, 2.91%, and 4.86% (After Githens et al. [25]).

**Figure 18.** Identification of active deformation mode using SEM-DIC experiment and CPFE simulation at a tensile strain of *ε* = 2.91% (**a**) SEM-DIC strain map, (**b**) slip traces for different deformation mode along with the basal Schmid factor map, (**c**) CPFE simulation strain map, (**d**) the activity of basal slip system obtained from CPFE simulation (After Githens et al. [25]).

They not only verified the CPFE framework using the SEM-DIC results, but they also resolved and categorized the strain from individual slip traces for the first time in any Mg alloy, as shown in Figure 18a. They used this information to categorize the active slip/twinning modes with respect to their nominal Schmid factors in both SEM-DIC and CPFE results (Figure 19). The results show that the basal modes sustain the most plastic deformation during uniaxial tension. The non-basal slip modes are less active. The least active is the extension twinning, which is because of the tested sample texture and loading direction. Githens et al. [25] also studied the distribution of active slip/twin modes during the compression loading (Figure 20). The activity of extension twinning is considerably increased during compression compared to the tensile loading (Figure 19). In addition, the non-basal slip modes contribution to the deformation is very limited during compression (Figure 20).

**Figure 19.** Comparison of active deformation modes versus corresponding Schmid factor at a uniaxial tensile strain of *ε* = 4.86% (**a**) SEM-DIC experimental data, (**b**) CPFE simulation results (After Githens et al. [25]).

**Figure 20.** Active deformation modes versus corresponding Schmid factor at uniaxial compression strain of *ε* = −4.2% obtained using the SEM-DIC experiment (After Githens et al. [25]).

3.1.2. Effects of Heat Treatment

Ganesan et al. [8] incorporated the in-situ DIC along the SEM technique to investigate the effect of heat treatment on the response of WE43 Mg alloy with the weak basal texture. They started with the WE43-T5 sample and applied different heat treatment conditions of the solution treated (ST) and aging times of 15 min, 2 h, 4 h, and 16 h (T6 condition). They investigated the effect of heat treatment on the macro responses of WE43 Mg (Figure 21). The results showed that the T5 condition had the highest yield stress, which can be attributed to its finer grain size. Within the heat-treated samples, the T6 condition had the maximum yield strength, which was due to the role of precipitates. Each of the samples was then subjected to the in-situ uniaxial tension loading along the rolling direction, where the SEM-DIC technique was used to gather the deformation maps. Figure 22 shows the normalized maximum principal strain maps at *ε* =3.23% in samples with different aging times. They showed that the strain map of the sample with T6 condition had fewer localized strains. In order to quantitatively investigate their observation, Ganesan et al. [8] studied the strain probability distribution of the tested samples (Figure 23). They showed that it is ∼ 3.84 more probable to find a localized strain of 4 × (Average applied strain) in underaged samples compared to the T6 condition.

**Figure 21.** The stress-strain response of Mg alloy WE43 with different heat treatment conditions during uniaxial tension along the rolling direction (After Ganesan et al. [8]).

(**e**) 16 h aged (T6 condition)

**Figure 22.** The effect of heat treatment condition on normalized maximum principal strain map subjected to a uniaxial tension along the rolling direction at a strain of *ε* = −3.23% for different heat treatment conditions of: (**a**) Solution treated, (**b**) 15 min aged, (**c**) 2 h aged, (**d**) 4 h aged, (**e**) 16 h aged (T6 condition) (After Ganesan et al. [8]).

**Figure 23.** The probability distribution of normalized maximum principal strain *ε*1/*<sup>ε</sup>*1 for WE43 Mg alloy subjected to different heat treatment conditions after solution treatment at the uniaxial tensile strain of *ε* = −3.23% along the rolling direction (After Ganesan et al. [8]).

They further investigated the effect of heat treatment using the CPFE simulation. They validated the CPFE framework and showed that the simulation can capture the stress-strain response of samples with different heat treatment conditions (Figure 24). They further validated the simulation by predicting the strain map of WE43-T6 at different tensile strains (Figure 25). The results showed that not only can the CPFE capture the macro responses, but it can also accurately model the local field maps. After validation of the CPFE framework, Ganesan et al. [8] used the CPFE simulation to investigate the deformation mechanisms involved in the response of T5 and T6 conditions. Accordingly, they compared the contribution of each deformation mode for each of these heat treatment conditions of T5 and T6 during uniaxial tension and compression (Figure 26). They showed that the heat treatment condition alters the contribution of different deformation modes. A sample with a T5 condition has finer grains, which alter the CRSS of deformation modes. For instance, they obtained the ratio of *CRSS*prismatic/*CRSS*pyramidal*a* = 1.02 in the T5 condition, while this ratio was 0.88 for the T6 condition. They stated that this was the underlying mechanism for more active pyramidal *a* in the T5 condition compared to the activity of prismatic, while pyramidal *a* slip mode was nearly inactive in the T6 condition. Finally, Ganesan et al. [8] used the CPFE simulation results to capture the Hall–Petch constant for different deformation modes in WE43 Mg alloy, which were 4.53, 11.33, 8.9, 11.18, 5.1 MPa × mm1/2 for different deformation modes of basal, prismatic, pyramidal *<sup>a</sup>*, pyramidal c + *<sup>a</sup>*, and extension twinning, respectively.

**Figure 24.** Comparison of the stress-strain curves of WE43 with different heat treatment conditions of ST, T5, and T6 subjected to uniaxial tension along the rolling direction: CPFE simulation versus the experimental results (After Ganesan et al. [8]).

**Figure 25.** The strain maps obtained by the SEM-DIC experiment compared to the CPFE prediction in WE-43 T6 sample during uniaxial tension at different strain values: (**a**) *ε* = 0.76% (SEM-DIC), (**b**) *ε* = 0.76% (Simulation), (**c**) *ε* = 4.83% (SEM-DIC), (**d**) *ε* = 4.83% (Simulation), (**e**) *ε* = 8.15% (SEM-DIC), (**f**) *ε* = 8.15% (Simulation) (After Ganesan et al. [8]).

**Figure 26.** The comparison of deformation modes relative activity in T5 and T6 heat treatment conditions of WE43 Mg alloy during uniaxial loadings along the rolling direction obtained by the CPFE simulation: (**a**) Uniaxial tension, (**b**) uniaxial compression (After Ganesan et al. [8]).

#### *3.2. In-Situ Synchrotron X-ray Techniques*

#### 3.2.1. Micromechanics of Twinning

Abdolvand et al. [27] used a combination of CPFE and in-situ 3DXRD experiments to investigate deformation twinning in the AZ31B Mg alloy. They applied the uniaxial tension on the sample with the texture favored for twinning during the loading and used the in-situ 3DXRD to study the microstructural evolution. They calibrated the CPFE model to capture the stress-strain response, as shown in Figure 27. They also studied the effect of twinning on texture. Accordingly, they measured the {0002} pole figures at a tensile strain of *ε* = 0% and *ε* = 0.4%. In the case of *ε* = 0%, as shown in Figure 28a, basal poles were observed to be aligned almost parallel to the normal direction. However, in the case of *ε* = 0.4%, as shown in Figure 28b, the deformation twinning reorients the {0002} poles, which were observed towards the rolling direction and transverse direction on the outer edges of the pole figure.

**Figure 27.** Stress-strain curves for AZ31B Mg alloy during uniaxial tension along normal direction obtained by continuous deformation, in-situ 3DXRD experiment, and CPFE simulation. The steps of 1 to 4 correspond to the strain values of *ε* = 0%, 0.2%, 0.4%, and 1.4% at which the measurements of diffraction were performed. (**b**) is a magnified version of (**a**) in the strain range of *ε* = 0% − 2% (After Abdolvand et al. [27]).

**Figure 28.** Effect of twinning on texture: (**a**) The {0002} pole figures at a tensile strain of *ε* = 0% (Step-1), (**b**) the {0002} pole figures at a tensile strain of *ε* = 0.4% (Step-3), (**c**) twin volume computed by CPFE versus the measured twin fraction, (**d**) variation of predicted twin volume versus misorientation between normal to the basal plane of the grain and loading direction predicted by CPFE at a tensile strain of *ε* = 0.4% (After Abdolvand et al. [27]).

Abdolvand et al. [27] also predicted the twin volume fraction using the CPFE simulation, which agrees well with the measured twin volume fraction (Figure 28c). They further investigated the CPFE simulation results by studying the variation of predicted twin volume versus misorientation angle (*θ*) between normal to the basal plane of the grain and the loading direction predicted by CPFE at a tensile strain of *ε* = 0.4% (Figure 28d). In the case of grains with *θ* > 40◦ , no twinning was predicted by CPFE. Furthermore, the scatter of twin volume for grains with similar *θ* showed that global *θ* was not the only governing factor of twinning volume as the effect of the local neighborhood and load sharing considerably varied the predicted twin volume fraction.

Another important subject Abdolvand et al. [27] addressed was the stress inside the twinned children and parent grains using the in-situ 3DXRD results (Figure 29). They selected 15 parent grains along 20 twinned children nucleated within those parent grains. Figure 29i shows different components of the stress presented in the results summary. Different stress components were the normal stress *σ*33, stress along the c-axis of the untwinned parent *<sup>σ</sup>cp*, and normal and shear stress according to the twinned child plane, i.e., *σn* and *τrs*, respectively. The mean value of each stress was represented by a line. Abdolvand et al. [27] observed that the mean normal stress *σ*33 and *<sup>σ</sup>cp* almost followed the applied stress trend. Because of the texture (c-axis is almost aligned with the loading direction), the values of *σ*33 and *<sup>σ</sup>cp* were close. The values of *σn* in untwinned parents and twinned children grains were close, which were in the region of 23 −30 MPa, which did not vary considerably during the loading. Interestingly, Abdolvand et al. [27] observed that the shear stress, which was resolved on the twinning plane, i.e., *τrs*, was considerably different in untwinned parent and twinned children grains. They reported that the average value of *τrs* in untwinned parent grains was 22 MPa higher than that of the twinned children grains. To further analyze the experimental results, Abdolvand et al. [27] focused on an untwinned parent grain with its two twinned children grains, as shown in Figure 29e–h, which were nucleated due to the activation of the twin variants 1 and 6. The twin variant 1 was nucleated at the normal stress of *σn* ∼20 MPa, which was close to that of the parent, as shown in Figure 29a–d. The twin variant 6 appeared at the strain value of 0.4%, in which the shear stress *τrs* is ∼19 MPa in the untwinned parent grain and ∼ −10 MPa for the twinned grain. The results showed that the shear stress, which was resolved on the twinning plane, i.e., *τrs*, was smaller for both twin variants compared to the parent grains, which was in line with the results shown in Figure 29a–d.

**Figure 29.** Stress evolution inside the untwinned parent and twinned children grains: The results for 15 parent grains and 20 twinned children nucleated within those parent grains (**a**) *σ*33, (**b**) *<sup>σ</sup>cp*, (**c**) *σn*, (**d**) *τrs*. The results for a single untwinned parent grain and two twin variants nucleated from that grain (**e**) *σ*33, (**f**) *<sup>σ</sup>cp*, (**g**) *σn*, (**h**) *τrs*. (**i**) The schematics of the stress components presented in (**<sup>a</sup>**–**h**) (After Abdolvand et al. [27]).
