# Crystal Plasticity Simulation of Magnesium and Its Alloys: A Review of Recent Advances

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## Abstract

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## 1. Introduction

## 2. Crystal Plasticity Models

#### 2.1. Overview

#### 2.2. Twinning

#### 2.3. Stress Relaxation

#### 2.4. Detwinning

- Operations A: Twin nucleation and growth due to parent grain reduction.
- Operations B: Twin growth due to the twinned child propagation.
- Operations C: Twin shrinkage due to the parent propagation.
- Operations D: Detwinning in the twinned child.

## 3. In-situ Experiments

#### 3.1. In-Situ DIC Experiments

#### 3.1.1. Microscale Deformation Mechanisms

#### 3.1.2. Effects of Heat Treatment

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

#### 3.2.1. Micromechanics of Twinning

#### 3.2.2. Detwinning

#### 3.3. In-Situ Neutron Diffraction

#### Micromechanics of Twinning

## 4. Conclusions and Future Works

- Real-time crystal plasticity simulation coupled to in-situ experiments to guide identification of outliers that can in-turn improve crystal plasticity theories.
- A general map to include the effect of alloying for a variety of Mg alloys using crystal plasticity models along with synchrotron X-ray techniques in a consistent framework.
- Using machine learning techniques to learn the crystal plasticity models and generate surrogate models which can be used to design specific Mg alloy loading paths to achieve target properties.
- Developing a crystal plasticity model with a physically based twinning and detwinning model, which include the correct isotropic and kinematic hardenings to capture the appropriate cyclic response of Mg alloy. This is extremely important in the prediction of fatigue simulation using crystal plasticity simulation.
- Developing an integrated framework of crystal plasticity models and phase field simulation to better capture the twin morphology in Mg alloys.
- The interaction of slip modes and twinning and detwinning mechanisms, which is typically reflected as latent hardening in crystal plasticity models.
- Improved modeling of slip/twin and grain boundary interactions: Effects of grain size in different Mg alloys using crystal plasticity models, specifically via micro-Hall Petch models whose parameters can be inferred through experiments [98,102] and including the effect of grain boundary on twin nucleation and growth in crystal plasticity models.
- Integrating the crystal plasticity models of Mg and its alloys with the PRISMS-Fatigue framework [109] to investigate the effects of texture, grain morphology, sample size, multiaxial strain, and strain amplitude on their fatigue response.
- Coupling the crystal plasticity models with phase-field simulations to address the effect of deformation mechanisms such as plastic slip and twinning on the dynamic recrystallization of Mg alloys.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The schematics of the transformation of the untwinned parent unit cell (blue segments and red and blue atoms) into the twinned children unit cell (cyan segments and magenta and cyan atoms for the first two layers close to the twin boundary) during $\left(10\overline{1}2\right)\left[\overline{1}011\right]$ twinning (After Namakian and Voyiadjis [40]).

**Figure 2.**Minimum energy paths of four deformation mechanisms associated with π_1 plane, i.e., homogeneous CTW nucleation, heterogeneous nucleation mechanism of pyramidal-I 〈c + a〉 slip, i.e., $1/3\left[1\overline{1}\overline{2}3\right]\left(101\overline{1}\right)$ slip to CTW $\left[1\overline{0}12\right]\left(101\overline{1}\right)$, heterogeneous nucleation mechanism of pyramidal-I 〈a〉 slip $1/3\left[1\overline{2}1\overline{0}\right]\left(101\overline{1}\right)$ to CTW $\left[101\overline{2}\overline{}\right]$_(π1_D), and pyramidal-I 〈c + a〉 slip $1/3\left[1\overline{1}\overline{2}3\right]\left(101\overline{1}\right)$ (

**a**) Liu et al. [42] EAM potential (

**b**)Wu et al. [43] MEAM potential (After Namakian et al. [41]).

**Figure 3.**Different configurations of the body in finite strain framework (After Yaghoobi et al. [103]).

**Figure 4.**(

**a**) The crystallography of the extension twinning $\left\{10\overline{1}2\right\}\u27e8\overline{1}011\u27e9$ in Mg and its alloys. (

**b**) Description of deformation systems in the orthonormal system $\left\{{e}_{i}^{c}|i=1,2,3\right\}$ (After Staroselsky and Anand [64]).

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

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

**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 $\alpha $ 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]).

**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]).

**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]).

**Figure 12.**A partially twinned material point in the CPFE framework (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 $\left\{0002\right\}$ 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 $\left(0001\right)$ pole figures predicted by CPFE simulation at different strains: (

**a**) Initial texture, (

**b**) first maximum compression $\left(\epsilon =-1.2\%\right)$ in cycle 1, (

**c**) first maximum tension $\left(\epsilon =1.2\%\right)$ in cycle 1, (

**d**) second maximum compression $\left(\epsilon =-1.2\%\right)$ in cycle 2, (

**e**) second maximum tension $\left(\epsilon =1.2\%\right)$ in cycle 2 (After Yaghoobi et al. [9]).

**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 $\epsilon =1.23\%,$ (

**b**) the map of normalized maximum principal strain maps at $\epsilon =4.86\%,$ (

**c**) the inverse pole figure map, (

**d**) a probability distribution of the strain at different applied tensile strains of $\epsilon =1.23\%,2.91\%,\mathrm{and}4.86\%$ (After Githens et al. [25]).

**Figure 18.**Identification of active deformation mode using SEM-DIC experiment and CPFE simulation at a tensile strain of $\epsilon =2.91\%$(

**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]).

**Figure 19.**Comparison of active deformation modes versus corresponding Schmid factor at a uniaxial tensile strain of $\epsilon =4.86\%$ (

**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 $\epsilon =-4.2\%$ obtained using the SEM-DIC experiment (After Githens et al. [25]).

**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]).

**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 $\epsilon =-3.23\%$ 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 ${\epsilon}_{1}/\langle {\epsilon}_{1}\rangle $ for WE43 Mg alloy subjected to different heat treatment conditions after solution treatment at the uniaxial tensile strain of $\epsilon =-3.23\%$ along the rolling direction (After Ganesan et al. [8]).

**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**) $\epsilon =0.76\%$ (SEM-DIC), (

**b**) $\epsilon =0.76\%$ (Simulation), (

**c**) $\epsilon =4.83\%$ (SEM-DIC), (

**d**) $\epsilon =4.83\%$ (Simulation), (

**e**) $\epsilon =8.15\%$ (SEM-DIC), (

**f**) $\epsilon =8.15\%$ (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]).

**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 $\epsilon =0\%,0.2\%,0.4\%,\mathrm{and}1.4\%$ at which the measurements of diffraction were performed. (

**b**) is a magnified version of (

**a**) in the strain range of $\epsilon =0\%-2\%$ (After Abdolvand et al. [27]).

**Figure 28.**Effect of twinning on texture: (

**a**) The $\left\{0002\right\}$ pole figures at a tensile strain of $\epsilon =0\%$ (Step-1), (

**b**) the $\left\{0002\right\}$ pole figures at a tensile strain of $\epsilon =0.4\%$ (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 $\epsilon =0.4\%$ (After Abdolvand et al. [27]).

**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**) ${\sigma}_{33}$, (

**b**) ${\sigma}_{cp}$, (

**c**) ${\sigma}_{n},$ (

**d**) ${\tau}_{rs}$. The results for a single untwinned parent grain and two twin variants nucleated from that grain (

**e**) ${\sigma}_{33},$ (

**f**) ${\sigma}_{cp,}$ (

**g**) ${\sigma}_{n},$ (

**h**) ${\tau}_{rs}$. (

**i**) The schematics of the stress components presented in (

**a**–

**h**) (After Abdolvand et al. [27]).

**Figure 30.**The cyclic response of pure Mg samples during the cyclic loading with a strain amplitude of 0.75% along the extrusion direction, including the stress-strain loops and variations of basal $\left\{0002\right\}$ peak intensity along with loading and normal directions during the cyclic loading (

**a**–

**c**) Cycle 1 (

**d**–

**f**) Cycle 2, (

**g**–

**i**) Cycle 200 (

**j**–

**l**) Cycle 500 (After Murphy-Leonard et al. [7]).

**Figure 31.**The variation of (

**a**) twin initiation stress and (

**b**) detwinning exhaustion stress versus the number of cycles in the case of cyclic loadings with the strain amplitude of 0.75% and 0.52% (After Murphy-Leonard et al. [7]).

**Figure 32.**The variation of twin intensity at maximum tensile strain versus the number of cycles in the case of cyclic loadings with the strain amplitude of 0.75% and 0.52% (After Murphy-Leonard et al. [7]).

**Figure 33.**The stress-strain response of pure Mg sample during cyclic loading along the extrusion direction: CPFE simulation results for cycle 2 (Δ symbol) are compared versus the experimental results (After Aagesen et al. [108]).

**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]).

**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.**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.

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**MDPI and ACS Style**

Yaghoobi, M.; Voyiadjis, G.Z.; Sundararaghavan, V.
Crystal Plasticity Simulation of Magnesium and Its Alloys: A Review of Recent Advances. *Crystals* **2021**, *11*, 435.
https://doi.org/10.3390/cryst11040435

**AMA Style**

Yaghoobi M, Voyiadjis GZ, Sundararaghavan V.
Crystal Plasticity Simulation of Magnesium and Its Alloys: A Review of Recent Advances. *Crystals*. 2021; 11(4):435.
https://doi.org/10.3390/cryst11040435

**Chicago/Turabian Style**

Yaghoobi, Mohammadreza, George Z. Voyiadjis, and Veera Sundararaghavan.
2021. "Crystal Plasticity Simulation of Magnesium and Its Alloys: A Review of Recent Advances" *Crystals* 11, no. 4: 435.
https://doi.org/10.3390/cryst11040435