# Studying Grain Boundary Strengthening by Dislocation-Based Strain Gradient Crystal Plasticity Coupled with a Multi-Phase-Field Model

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. The Multi Phase Field Model

#### 2.2. Elasticity

**C**${}_{\alpha}$ is the 4th order elastic stiffness tensor, ${\sigma}_{\alpha}$ is the 2nd Piola-Kirchhoff stress, “:” represents double contraction in the form (a${}_{ij}$):(b${}_{ij}$) = ${\sum}_{i}{\sum}_{j}$ a${}_{ij}$ b${}_{ji}$ and ${\mathit{\u03f5}}_{\alpha}^{\mathrm{el}}$ is the Lagrangian strain. Now, as the stiffness tensor and the elastic strains are known for each phase field, the evaluation of the driving force is simple. The continuum mechanical homogenization sets several rules and evaluates effective values of mechanical properties with the help of phase fraction and the parameters related to the phase. The resulting total strain $\mathit{\u03f5}$ should be weighted as the average of strains associated with a phase field as

#### 2.3. Plasticity

**d**and

**l**refer to the slip direction vector and to the line direction vector used to evaluate the edge (first term in this equation,

**l**is normal to

**d**) and screw components (second term, where

**l**is parallel to

**d**) of GND. These vectors are given in Table 1 for each slip system.

## 3. Simulation Setup

## 4. Results and Discussion

#### 4.1. Effect of the Grain Size on the Distribution of Dislocation Density

#### 4.2. Averaged Stress and Dislocation Density Under the Influence of Model Parameters

## 5. Conclusions

- Our work shows that by applying a dislocation-based strain gradient crystal plasticity model, we can capture many aspects of grain boundary strengthening as it is observed in experiments. This conforms to the Hall-Petch model in which the introduction of special properties for grain boundaries is not necessary.
- The model introduced in our work is capable of recapturing the Hall-Petch relation with an exponent of −0.5 for the grain size dependence. Furthermore our model is consistent with the experimental observations of the evolution of the Hall-Petch coefficient with progressing plastic deformation and the initial state of the material with respect to dislocation density.
- The value of the Hall-Petch coefficient predicted by our model is significantly smaller than those observed through experiments and the strain gradient plasticity is unable to explain the grain boundary strengthening at the onset of plastic yielding. This has been discussed in light of the initial state of the material in particular with respect to the initial GND density prior to mechanical testing.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

CP | Crystal plasticity |

MPF | Multi phase field |

GND | Geometrically necessary dislocations |

SSD | Statistically stored dislocations |

RVE | Representative volume element |

${\tau}^{\mathrm{s}}$ | Resolved shear stress on s slip system |

${\tau}_{\mathrm{c}}^{\mathrm{s}}$ | Critical resolved shear stress on s slip system |

${\nu}^{\mathrm{s}}$ | Dislocation velocity on s slip system |

${\nu}_{0}$ | Initial dislocation velocity |

${\u03f5}^{\left(\mathrm{p}\right)}$ | Equivalent plastic strain |

$\dot{\gamma}$ | Shear strain rate |

c${}_{1}$ | Geometrical constant |

G | Shear modulus |

s | Arbitrary slip system |

$\Omega $ | Domain/system size |

$\varphi $ | Order parameter/phase field parameter |

$\alpha $ | Arbitrary name for a phase phase/grain |

F | Total free energy of the system |

${f}^{\mathrm{int}}$ | Interfacial free energy |

${f}^{\mathrm{el}}$ | Elastic or mechanical free energy |

N | Total number of slip systems or phases |

$\eta $ | Interfacial width |

${\rho}_{\mathrm{total}}$ | Density of dislocations |

${\rho}_{\mathrm{SSD}}$ | Density of SSD |

${\rho}_{\mathrm{GND}}$ | Density of GND |

k${}_{1}$ | SSD storage parameter |

k${}_{2}$ | SSD annihilation parameter |

m | Strain rate sensitivity parameter |

${\sigma}_{\alpha \beta}$ | Interfacial energy between arbitrary phase or grain $\alpha $ and $\beta $ |

${\mathbf{P}}^{s}$ | Symmetric part of Schmidt tensor on slip system s |

${\mathit{\sigma}}_{\mathrm{vM}}$ | von Mises equivalent stress |

$\mathit{\u03f5}$ | Total strain tensor |

${\mathit{\u03f5}}^{\mathrm{el}}$ | Elastic strain tensor |

${\mathit{\u03f5}}^{*}$ | Eigen strain tensor |

${\mathit{\u03f5}}^{\left(\mathrm{p}\right)}$ | Plastic strain tensor |

${\mathit{\sigma}}_{\mathrm{ij}}$ | Stress tensor |

$\Lambda $ | Nye’s dislocation tensor |

$\mathbf{C}$ | Stiffness tensor |

$\mathbf{F}$ | Deformation gradient |

$\mathbf{b}$ | Burgers vector |

$\mathbf{d}$ | Slip direction vector |

$\mathbf{l}$ | Slip plane tangent vector |

$\mathbf{n}$ | Slip plane normal vector |

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**Figure 1.**(

**a**) Orientation distribution, (

**b**) flow stress for a polycrystal with grain diameter of 0.4 $\mathsf{\mu}$m and an offset of 0.2% of global plastic strain to define the onset of plastic yielding

**Figure 2.**Distribution of (

**a**) equivalent stress and (

**b**) equivalent plastic strain corresponding to the onset of plastic deformation, which is defined here by a global plastic strain of 0.2%.

**Figure 3.**Distribution of geometrically necessary dislocation density ${\rho}_{\mathrm{GND}}$ in the deformed RVEs with a grain diameter D of (

**a**) 16 $\mathsf{\mu}$m (

**b**) 1.6 $\mathsf{\mu}$m (

**c**) 0.8 $\mathsf{\mu}$m (

**d**) 0.4 $\mathsf{\mu}$m at a global plastic strain of 5%.

**Figure 4.**Distribution of statistically stored dislocation density ${\rho}_{\mathrm{SSD}}$ in the deformed RVEs with the grain diameter D of (

**a**) 16 $\mathsf{\mu}$m (

**b**) 1.6 $\mathsf{\mu}$m (

**c**) 0.8 $\mathsf{\mu}$m (

**d**) 0.4 $\mathsf{\mu}$m at a global plastic strain of 5%.

**Figure 5.**Distribution of total dislocation density ${\rho}_{\mathrm{total}}$ in the deformed RVEs with a grain diameter D of (

**a**) 16 $\mathsf{\mu}$m (

**b**) 1.6 $\mathsf{\mu}$m (

**c**) 0.8 $\mathsf{\mu}$m (

**d**) 0.4 $\mathsf{\mu}$m at a global plastic strain of 5%.

**Figure 6.**Effect of grain size on the evolution of the global (

**a**) geometrically necessary dislocation density, (

**b**) statistically stored dislocation density, (

**c**) total dislocation density, (

**d**) flow stress.

**Figure 7.**(

**a**) Hall-Petch coefficient calculated at an offset of 0.2% and 0.5% of total strain, (

**b**) evolution of ${\rho}_{\mathrm{SSD}}$ and ${\rho}_{\mathrm{GND}}$ at 0.5% and 5% of total strain with variation in grain size.

**Figure 8.**Effect of variation of initial total dislocation density on the global (

**a**) geometrically necessary dislocation density, (

**b**) statistically stored dislocation density, and (

**c**) total dislocation density (

**d**) flow stress.

**Figure 9.**Influence of variation of SSD storage on the global (

**a**) geometrically necessary dislocation density, (

**b**) statistically stored dislocation density, (

**c**) total dislocation density, (

**d**) flow stress.

**Figure 10.**Effect of variation of dislocation annihilation on the global (

**a**) geometrically necessary dislocation density, (

**b**) statistically stored dislocation density, (

**c**) total dislocation density, (

**d**) flow stress.

**Figure 11.**Effect of dislocation annihilation on ${\overline{\rho}}_{\mathrm{SSD}}$ and ${\overline{\rho}}_{\mathrm{GND}}$ at (

**a**) 0.5% plastic strain and (

**b**) 5% plastic strain.

Slip System | Plane Normal | Slip Direction | Line Direction |
---|---|---|---|

s | n | d | l |

1 | [111] | [1$\overline{1}$0] | [11$\overline{2}$] |

2 | [111] | [10$\overline{1}$] | [$\overline{1}$2$\overline{1}$] |

3 | [111] | [01$\overline{1}$] | [$\overline{2}$11] |

4 | [$\overline{1}$11] | [$\overline{1}\overline{1}$0] | [1$\overline{1}$2] |

5 | [$\overline{1}$11] | [101] | [12$\overline{1}$] |

6 | [$\overline{1}$11] | [0$\overline{1}$1] | [211] |

7 | [1$\overline{1}$1] | [110] | [$\overline{1}$12] |

8 | [1$\overline{1}$1] | [10$\overline{1}$] | [121] |

9 | [1$\overline{1}$1] | [0$\overline{1}\overline{1}$] | [21$\overline{1}$] |

10 | [1$1\overline{1}$] | [1$\overline{1}$0] | [$\overline{1}\overline{1}\overline{2}$] |

11 | [11$\overline{1}$] | [101] | [1$\overline{2}\overline{1}$] |

12 | [11$\overline{1}$] | [011] | [2$\overline{1}$1] |

Parameters | Symbol | Value | Unit | Ref. |
---|---|---|---|---|

Anisotropic elastic constant | C${}_{11}$ | 108.2 | GPa | [43] |

Anisotropic elastic constant | C${}_{12}$ | 61.3 | GPa | [43] |

Shear Modulus | C${}_{44}$ = G | 28.5 | GPa | [43] |

Strain rate sensitivity | m | 0.025 | - | |

Lattice friction stress | ${\tau}_{o}$ | 80 | MPa | |

SSD storage parameter | k${}_{1}$ | 2 × ${10}^{9}$ | - | [40] |

SSD annihilation parameter | k${}_{2}$ | 10 | - | [40] |

Initial total dislocation density | ${\rho}_{\mathrm{total}\left(\mathrm{i}\right)}$ | 1 × ${10}^{13}$ | m${}^{-2}$ | |

Geometrical factor for flow stress | c${}_{1}$ | 0.3 | - | [20,44] |

Referential dislocation velocity | ${\nu}_{0}$ | 1 × ${10}^{-3}$ | ms${}^{-1}$ | |

Interfacial energy | ${\sigma}_{\alpha \beta}$ | 0.24 | Jm${}^{-2}$ | [28] |

Spacediscretization | $\Delta $x | 0.1 | $\mathsf{\mu}$m | |

Timediscretization | $\Delta $t | 1 | $\mathsf{\mu}$s | |

Interfacial width | $\eta $ | 4.5 | $\Delta $x | |

Domain size | $\Omega $ | 128 × 128 | $\Delta $x | |

Length of Burger’s vector | b | 0.286 | nm |

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**MDPI and ACS Style**

Amin, W.; Ali, M.A.; Vajragupta, N.; Hartmaier, A.
Studying Grain Boundary Strengthening by Dislocation-Based Strain Gradient Crystal Plasticity Coupled with a Multi-Phase-Field Model. *Materials* **2019**, *12*, 2977.
https://doi.org/10.3390/ma12182977

**AMA Style**

Amin W, Ali MA, Vajragupta N, Hartmaier A.
Studying Grain Boundary Strengthening by Dislocation-Based Strain Gradient Crystal Plasticity Coupled with a Multi-Phase-Field Model. *Materials*. 2019; 12(18):2977.
https://doi.org/10.3390/ma12182977

**Chicago/Turabian Style**

Amin, Waseem, Muhammad Adil Ali, Napat Vajragupta, and Alexander Hartmaier.
2019. "Studying Grain Boundary Strengthening by Dislocation-Based Strain Gradient Crystal Plasticity Coupled with a Multi-Phase-Field Model" *Materials* 12, no. 18: 2977.
https://doi.org/10.3390/ma12182977