*2.2. Twinning*

In the case of Mg and its alloys, extension twinning 10121011 is very important, along with the plastic slip (Figure 4). The simplest modification to include twinning into the CP formulation is to consider twinning as pseudo-slip systems (See, e.g., Staroselsky and Anand [64]), and their contribution to the plastic velocity gradient tensor can be described as follows:

$$\mathbf{L}^{\mathbf{P}} = \sum\_{a=1}^{N\_t + N\_t} \dot{\gamma}^a \mathbf{S}^a \tag{8}$$

where *Nt* is the number of twinning systems. Figure 5 shows the kinetics of slip and twinning defined in Equation (8). The relation between the plastic slip of twinning systems and their corresponding twin volume fraction can be described as follows:

$$
\dot{f}^{\beta} = \frac{\dot{\gamma}^{\beta}}{S} \tag{9}
$$

where . *f β* is the rate of change in twin volume fraction of twin pseudo-slip system *β* and *S* is the characteristic twin shear strain, which defines the amount of shear associated with twinning. The value of *S* depends on the *c*/*a* ratio, which can be defined as follows [104]:

$$S = \frac{\sqrt{3}}{c/a} - \frac{c/a}{\sqrt{3}} \tag{10}$$

where *a* and *c* are depicted in Figure 4. In the case of Mg and its alloys, the *S* = 0.129.

**Figure 4.** (**a**) The crystallography of the extension twinning 1012-1011 in Mg and its alloys. (**b**) Description of deformation systems in the orthonormal system *eci i* = 1, 2, 3 (After Staroselsky and Anand [64]).

**Figure 5.** The deformation modes kinematics in finite strain framework (After Yaghoobi et al. [103]).

Although the twin pseudo-slip system's contribution to plastic velocity gradient tensor is similar to that of the slip systems, they are not precisely treated similarly due to the polar nature of twinning. In other words, only the tensile component of stress along the c-axis can trigger the extension twinning. This is reflected in the CP formulation for the twinning pseudo-slip systems as follows:

$$\begin{cases} \quad \text{if } \sigma: \mathbf{S}^{\beta} \le 0 \to \tau^{\beta} = 0\\ \quad \text{if } \sigma: \mathbf{S}^{\beta} > 0 \to \tau^{\beta} = \sigma: \mathbf{S}^{\beta} \end{cases} \tag{11}$$

Kalidindi [62] further enhanced the effect of twinning on crystal plasticity formulation. The work considers the contribution of stress in both untwinned and twinned region as follows:

$$\boldsymbol{\sigma} = \left(1 - \sum\_{\beta=1}^{N\_l} f^{\beta} \right) \boldsymbol{\sigma}^{\mathrm{mt}} + \sum\_{\beta=1}^{N\_l} f^{\beta} \boldsymbol{\sigma}^{\mathrm{tw}, \emptyset} \tag{12}$$

where *f β* is the reoriented volume fraction of grain according to the twin system *β*. Kalidindi [62] then presented the modified macroscopic plastic velocity gradient tensor **L**p, which includes the contribution of multiple twinned systems as below:

$$\mathbf{L}^{\mathcal{P}} = \left(1 - \sum\_{\beta=1}^{N\_l} f^{\beta}\right) \sum\_{a=1}^{N\_s} \dot{\gamma}^a \mathbf{S}\_{\text{sl}}^a + \sum\_{a=1}^{N\_l} \dot{S} \dot{f}^{\beta} \mathbf{S}\_{\text{tw}}^{\beta} + \sum\_{\beta=1}^{N\_l} f^{\beta} \left(\sum\_{a=1}^{N\_{s-\text{tw}}} \dot{\gamma}^a \mathbf{S}\_{\text{sl}-\text{tw}}^a\right) \tag{13}$$

where **<sup>S</sup>***<sup>α</sup>*sl, **<sup>S</sup>***β*tw, and **<sup>S</sup>***<sup>α</sup>*sl−tw denote the Schmid tensors for the slip systems in parent grain, twin systems in parent grain, and slip systems in twinned children, respectively. *Ns*−*tw* denotes the number of slip systems in the twinned region. Equation (10) describes three mechanisms contributing to the plastic velocity gradient **L**p. The first term defines the contribution of slip inside the parent grain, the second term defines the contribution of twin systems in parent grain, and the third term is the summation of the contributions of slip systems in all twinned children.

In the twinning models, there is a key part that should be defined as the reorientation. Figure 6 shows the crystallography of the deformation twinning, which includes the untwinned region (parent grain) and twinned region (child). While Equation (9) defines the amount of twin volume required for a certain amount of shear strain, it does not define the reorientation in the model. Different criteria have been introduced for the reorientation. Van Houtte [60] was the first researcher to propose a method for reorientation. In this method, reorientation of the grain is decided depending on the volume fractions calculated at every incremental step along with using a random criterion. Tomé et al. [61] modified the model presented by Van Houtte [60] using the predominant twin reorientation scheme (PTR) in which the grain is reoriented according to its predominant twin variant as the integration of twin volume throughout the time satisfy specific criteria.

**Figure 6.** The crystallographic description of extension twinning (After Yaghoobi et al. [103]).

#### *2.3. Stress Relaxation*

The in-situ neutron diffraction experiment on the Mg alloy showed that the stress relaxation occurs in the twinned region compared to the untwinned region of grain [34]. Different crystal plasticity models were incorporated to capture the stress relaxation during twinning. The simplest model is to consider larger CRSSs for twin nucleation than growth [12,93]. Wu et al. [105] presented a comprehensive physically-based twin model implemented in the EVPSC framework. They presented this twin model, which was called 'TNPG,' to capture twin nucleation, propagation, and growth for magnesium alloys (Figure 7). In Figure 7a, the twinning process initiates with twin nucleation. It is assumed that the grain boundary is the preferred nucleation site. As the twin propagates according to Figure 7b, stress relaxation occurs. Different notation is used throughout the current review for twinning resistance as *s<sup>α</sup>* compared to the one used by Wu et al. [105], which used *τ*<sup>ˆ</sup>*<sup>α</sup>* (Figure 7). The twinning resistance *s<sup>α</sup>* decreases during the propagation phase until it reaches its minimum value of *sαg* when the twin volume is *f α* = *f αg* . Afterward, the twinning resistance increases due to the hardening occurring in the twin thickening process. Wu et al. [105] captured the stress relaxation by modifying the twin systems resistances *s<sup>α</sup>* as follows:

$$s^{a} = \begin{cases} & s\_0^a - \frac{s\_0^a - s\_\mathcal{S}^a}{f\_\mathcal{S}^a} f^a & \left( \text{if } f^a \le f\_\mathcal{S}^a \right) \\ & s\_\mathcal{S}^a + \left( s\_1^a + h\_1^a \Gamma^a \right) \left( 1 - \exp \left( -\frac{h\_0^a}{s\_1^a} \Gamma^a \right) \right) & \left( \text{if } f^a > f\_\mathcal{S}^a \right) \end{cases} \tag{14}$$

where Γ*<sup>α</sup>* is the accumulated shear of *α*th twinning system, which is calculated after its corresponding twin volume *f α* surpass *f αg* (see Figure 7). They used this model to capture the stress-strain response of AZ31B Mg alloy obtained by Lou et al. [106]. They also compared the predicted twin volume versus the experimental results. Figure 8 shows that the model developed by Wu et al. [105] can successfully capture both the stress-strain and twin volume during uniaxial compression and tension. Qiao et al. [73] reviewed the available scheme to model stress relaxation in the CPFE framework. They used the model developed by Wu et al. [105] for EVPSC and incorporated it in the CPFE model to capture the stress relaxation in Mg single crystal during twinning.

**Figure 7.** Extension twinning in Mg and its alloys. (**a**) The schematics of extension twin nucleation, propagation, and growth, (**b**) variation of slip resistance for extension twinning system *α* at different stages of deformation twinning (After Wu et al. [105]).

**Figure 8.** Comparison of the experimental and simulated response of the AZ31B Mg alloy during uniaxial loadings along the rolling direction. (**a**) Stress-strain curves of uniaxial tension and compression, (**b**) twin volume fraction of uniaxial compression (After Wu et al. [105]).
