# Crystal Plasticity Modeling of Anisotropic Hardening and Texture Due to Dislocation Transmutation in Twinning

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Implementation and Calibration

#### 3.1. Correspondence Method for Transmutation for the Construction of the $\alpha $ matrix

- The entirety of the dislocation density from a slip system in the volume fraction of the parent grain overtaken by a twin mode is considered to transmute to its corresponding slip system inside the twin grain. This pairing of slip systems is defined by the mappings defined by the correspondence method used by Niewczas [29] for HCP materials.
- The dislocation density of slip mode in the volume fraction of the parent grain overtaken by a twin volume is considered to be evenly distributed across the systems of that mode.
- In this simulation, the dislocation density of any parent slip system corresponding to a slip system inside the twin grain that was not part of the prismatic, basal, or 2nd order pyramidal $\langle c+a\rangle $ slip modes was assumed to contribute to debris formation inside the twin as part of ${\rho}_{deb}$.

#### 3.2. Parameters for Dissociation

- Dislocations in the prismatic, basal, and 2nd order pyramidal $\langle c+a\rangle $ modes are assumed to transmute.
- The incidence of screw type dislocations at low strain regimes is quite low. As such, for the purposes of this work, it was assumed that the contributions to dislocation density inside the twin volume fractions made by these types of dislocations were negligible.
- Relative to twin boundaries, it was assumed that dislocations with positive and negative Burgers vector occur in equal measure.

#### Parameters for Dislocation Generation and Twin Nucleation and Propagation

#### 3.3. Simulation

## 4. Results

#### 4.1. TTC Load Path

#### 4.2. IPC Load Path

## 5. Discussion

#### $\eta $ Sensitivity

## 6. Conclusions

- In place of Twin Storage Factor and Hall–Petch effects, a model for dislocation and twin boundary interactions was implemented. These methods and parameter selections were used to simulate the behavior of pure, basal textured, rolled magnesium subjected to uniaxial compression along (TTC) and perpendicular to (IPC) the dominant $\langle c\rangle $-axis of the texture. The simulation results for stress, hardening, and texture development were consistent with observed experimental results. Modal activity and twin volume growth were largely similar to simulation work performed using the TSF approach.
- The large difference between the saturation stress of the simulated IPC and TTC load paths was confirmed to be the result of transmutation effects included in the model by disabling the contributions of transmuted dislocations to the dislocation density of twin volume fractions.
- It can be stated that Hall–Petch effects cannot be assumed to be the sole or even primary source of twinning induced anisotropy in the mechanical behavior of pure magnesium.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

VPSC | Visco-Plastic Self-Consistent |

HCP | Hexagonal Close Packed |

Mg | Magnesium |

CRSS | Critical Resolved Shear Stress |

HP | Hall–Petch |

LANL | Los Alamos National Laboratory |

TSF | Twin Storage Factor |

MD | Molecular Dynamics |

CG | Composite Grain |

TTC | Through Thickness Compression |

IPC | In-Plane Compression |

## Appendix A. Parameters for Dislocation Density Based Hardening

Prismatic | Basal | Pyramidal$\langle \mathit{c}+\mathit{a}\rangle $ | |
---|---|---|---|

${k}_{1}$${}^{1}{/}_{m}$ | $2.0$ × ${10}^{9}$ | $0.25$ × ${10}^{9}$ | $2.0$ × ${10}^{9}$ |

$\dot{\u03f5}$${s}^{-1}$ | $1.0$ × ${10}^{7}$ | $1.0$ × ${10}^{7}$ | $1.0$ × ${10}^{7}$ |

${g}^{\alpha}$ | $0.0035$ | $0.0035$ | $0.003$ |

${D}_{0}^{\alpha}$ MPa | $3.4$ × ${10}^{3}$ | $10.0$ × ${10}^{3}$ | $0.08$ × ${10}^{3}$ |

${\tau}_{0}^{\alpha}$ MPa | 30 | 11 | 50 |

$\chi $ | $0.9$ | $0.9$ | $0.9$ |

$H{P}^{\alpha}$ | 0 | 0 | 0 |

$H{P}^{\alpha \beta}$ | 0 | 0 | 0 |

Tensile Twin | Compression Twin | |
---|---|---|

${\tau}_{0}^{\beta}$ MPa | ${\tau}_{crit}=15,\text{}{\tau}_{prop}=10$ | ${\tau}_{crit}=185,\text{}{\tau}_{prop}=185$ |

$H{P}^{\beta}$ | 0 | 0 |

$H{P}_{TW}^{\beta \beta}$ | 0 | 0 |

${C}^{\beta 1}$ | 0 | 0 |

${C}^{\beta 2}$ | 0 | 0 |

${C}^{\beta 3}$ | 0 | 0 |

## Appendix B. Correspondence Method; Summary of Equations

## Appendix C. Parameters for Transmutation and Dissociation: Alpha Matrix Composition

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**Figure 3.**Simulated and experimental mechanical response under TTC compression. (

**a**) Stress-strain. (

**b**) Hardening rate vs. strain.

**Figure 4.**Simulated slip and twinning activity under TTC compression compared to simulated results from Oppedal et al. [14]. (

**a**) Simulated modal contributions vs. strain. (

**b**) Simulated twin volume growth vs. strain.

**Figure 5.**Comparison of simulated and experimental textures under TTC load conditions at $\u03f5$ = 0.09. (

**a**) Simulated {0002} and {10$\overline{1}$0} pole figures. (

**b**) Experimental {0002} and {10$\overline{1}$0} pole figures.

**Figure 6.**Simulated and experimental mechanical response under IPC compression. (

**a**) Stress-strain. (

**b**) Hardening rate vs. strain.

**Figure 7.**Simulated parent slip and primary twinning activity under IPC compression compared to simulated results from Oppedal et al. [14]. (

**a**) Simulated modal contributions vs. strain. (

**b**) Simulated twin volume growth vs. strain.

**Figure 8.**Simulated twin slip and secondary twinning activity under IPC compression compared to simulated results from Oppedal et al. [14]. (

**a**) Simulated modal contributions to strain vs. strain. (

**b**) Simulated twin volume growth vs. strain.

**Figure 9.**Texture comparison of simulated and experimental textures under IPC load conditions at $\u03f5$ = 0.12. (

**a**) Simulated {0002} and {10$\overline{1}$0} pole figures. (

**b**) Experimental {0002} and {10$\overline{1}$0} pole figures.

**Figure 10.**Simulated stress-strain curves with a dissociation parameter of $\eta $ = 0.5 and dissociation parameter $\eta $ = 1.0. These values correspond to a state of active dislocation transmutation and a state of no transmutation, respectively.

**Figure 11.**Simulated dislocation density evolution. (

**a**) Modal dislocation density vs. strain. (

**b**) Dislocation density for parent and primary twin volume fractions vs. primary twin volume fraction.

Al | Ca | Mn | Zr | Zn | Sn | Si | Pb | Mg |
---|---|---|---|---|---|---|---|---|

30 | 10 | 40 | 10 | 130 | 10 | 40 | 10 | Balance |

NMSE | |
---|---|

Transmutation (TTC) | $5.7613$% |

TSF (TTC) | $16.0268$% |

Transmutation (IPC) | $21.0105$% |

TSF (IPC) | $16.7439$% |

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**MDPI and ACS Style**

Allen, R.M.; Toth, L.S.; Oppedal, A.L.; El Kadiri, H.
Crystal Plasticity Modeling of Anisotropic Hardening and Texture Due to Dislocation Transmutation in Twinning. *Materials* **2018**, *11*, 1855.
https://doi.org/10.3390/ma11101855

**AMA Style**

Allen RM, Toth LS, Oppedal AL, El Kadiri H.
Crystal Plasticity Modeling of Anisotropic Hardening and Texture Due to Dislocation Transmutation in Twinning. *Materials*. 2018; 11(10):1855.
https://doi.org/10.3390/ma11101855

**Chicago/Turabian Style**

Allen, Robert M., Laszlo S. Toth, Andrew L. Oppedal, and Haitham El Kadiri.
2018. "Crystal Plasticity Modeling of Anisotropic Hardening and Texture Due to Dislocation Transmutation in Twinning" *Materials* 11, no. 10: 1855.
https://doi.org/10.3390/ma11101855