# Crystal Plasticity Model Analysis of the Effect of Short-Range Order on Strength-Plasticity of Medium Entropy Alloys

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Crystal Plasticity Framework

#### 2.1. Kinematics of Crystal Plasticity

#### 2.2. Dislocation Slipping

#### 2.3. Deformation Twinning

## 3. Simulations and Validation of the Constitutive Model

#### 3.1. Polycrystalline Finite Element Model

^{3}and contained a total of 55 grains (55 random points in parallel segment space generate 55 Voronoi polyhedra), so the grain size was about 13 µm.

#### 3.2. Parameter Validation

^{−3}s

^{−1}). The blue curve corresponds to the curve of low SRO (${\tau}_{SRO}^{0}$ = 10 MPa), and the purple curve is the curve of high SRO (${\tau}_{SRO}^{0}$ = 50 MPa) (Figure 1b). These calculated results and the experiments match well, which shows the applicability of the parameters to CoCrNi MEAs.

## 4. Influence of SRO on Deformation Behavior

_{1},b

_{1})). Furthermore, when observing the end of the curves, the elongation of the material rises and then falls as ${\tau}_{SRO}^{0}$ increases (Figure 2(a

_{2},b

_{2})). The observed phenomenon is plotted in Figure 2c. From Figure 2c, when ${\tau}_{SRO}^{0}\approx $ 29 MPa, the elongation takes the maximum value.

_{1},c

_{1},d

_{1})). When ${E}_{22}$ = 0.1 and ${\tau}_{SRO}^{0}=$ 0 MPa, the integration points from grain #2 are scattered in the region $\Delta {l}_{0}=0.077$ and $\Delta {\theta}_{0}=11.17\xb0$ (Figure 3(b

_{1})). Similarly, when ${E}_{22}$ = 0.1 and ${\tau}_{SRO}^{0}$ = 30 MPa, the integration points are scattered in $\Delta {l}_{30}=0.083$ and $\Delta {\theta}_{30}=12.78\xb0$ (Figure 3(c

_{1})). Moreover, when ${E}_{22}$ = 0.1 and ${\tau}_{SRO}^{0}$ = 60 MPa, $\Delta {l}_{60}=0.099$, $\Delta {\theta}_{60}=13.12\xb0$ (Figure 3(d

_{1})). The crystallographic orientation of the material changes drastically as ${\tau}_{SRO}^{0}$ increases. It can be further inferred that the increase in elongation with increasing ${\tau}_{SRO}^{0}$, when ${\tau}_{SRO}^{0}$ < 29 MPa, is due to the intense local rotation, which can provide additional macro-strain.

_{1},b

_{1},c

_{1})). From Equation (12), it is known that high local stress implies a high dislocation density; thus, quickly attaining a saturated dislocation density locally and causing the material to lose its capacity to harden (Figure 5(a

_{2},b

_{2},c

_{2})).

## 5. Conclusions

- (1)
- A set of parameters consistent with CoCrNi MEAs was determined and can be used to discuss the influence of various factors on a material’s deformation behavior.
- (2)
- Adjusting the resistance of SRO at a certain range increases both the yield strength and elongation simultaneously, but beyond this range, the yield strength increases but the elongation decreases.
- (3)
- As the resistance of SRO increases, the elongation increases and then decreases, which is attributed to the more intense local rotation with coplanar slip. Local rotation can increase the additional macro strain, while also causing a more intense stress concentration; when the resistance of SRO is low, the additional macro strain dominates the elongation increase; when the resistance is high, the stress concentration dominates the elongation decrease.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Macro stress–strain curves at different ${\tau}_{SRO}^{0}$. (

**a**) ${\tau}_{SRO}^{0}$ are equal to 0, 10~60 MPa, respectively, the local enlargements for the head (

**a**) and the end (

_{1}**a**); (

_{2}**b**) ${\tau}_{SRO}^{0}$ is equal to 25, 28, 29, 35 MPa, the values are near 30 MPa, the local enlargements for the head (

**b**) and the end (

_{1}**b**); (

_{2}**c**) the curve of yield strength (Corresponding to the blue y-axis on the right) and elongation (Corresponding to the red y-axis on the left) with ${\tau}_{SRO}^{0}$.

**Figure 3.**Pole figures at different ${\tau}_{SRO}^{0}$ and ${E}_{22}$. (

**a**) Initial grain orientation distribution. (

**b**,

_{1}**b**) when ${\tau}_{SRO}^{0}$ = 0 MPa, the orientation distribution at ${E}_{22}$ = 0.1, 0.3. (

_{2}**c**,

_{1}**c**) when ${\tau}_{SRO}^{0}$ = 30 MPa, the orientation distribution at ${E}_{22}$ = 0.1, 0.3. (

_{2}**d**,

_{1}**d**) when ${\tau}_{SRO}^{0}$ = 60 MPa, the orientation distribution at ${E}_{22}$ = 0.1, 0.3. The red part is the orientation distribution of grain #2.

_{2}**Figure 4.**(

**a**) Model of single crystal tensile simulation; the parallel section for the developed model, grain orientation as shown by the tetrahedron enclosed by the slip plane, $\left\{111\right\}$ plane is $45\xb0$ to the tensile direction, $\langle 1\overline{1}0\rangle $ direction is perpendicular to the paper surface. (

**b**) Comparison curves of macro-strain and average of local strain using single crystal tensile simulation at different ${\tau}_{SRO}^{0}$ = 30 MPa. (mesh size ~ 4 μm). (

**c**) Comparison of mesh deformation under different ${\tau}_{SRO}^{0}$ for ${E}_{22}$ = 0.1.

**Figure 5.**Mises stress and dislocation density distribution next to (

**a**,

_{1}**a**) ${\tau}_{SRO}^{0}$ = 0 MPa, (

_{2}**b**,

_{1}**b**) ${\tau}_{SRO}^{0}$= 30 MPa, (

_{2}**c**,

_{1}**c**) ${\tau}_{SRO}^{0}$ = 60 MPa, with the ${E}_{22}$ = 0.5.

_{2}Symbol | Physical Mean | Value |
---|---|---|

${C}_{11}$$,\text{}{C}_{12}$$,\text{}{C}_{44}$ | Elastic constants | 249, 156, 142 GPa |

${\tau}_{sol}$ | Solid solution strength | 200 MPa |

${N}_{s}$ | Total number of slip systems | 12 |

$b$ | Burgers vector | 0.2522 nm |

${N}^{*}$ | Saturated number of piled-up dislocation | 39 |

$l$ | Mean spacing between slip bands | 223 nm |

${\sigma}_{0}$$,\text{}{k}_{HP}$ | Hall–Petch coefficient (Converted to resolved shear stress) | $20\text{}\mathrm{MPa}$$,\text{}88\text{}\mathrm{MPa}\cdot {\mathsf{\mu}\mathrm{m}}^{1/2}$ |

$k$ | Forest dislocation hardening constant | 0.0488 |

${\rho}_{0}^{\alpha}$ | The initial dislocation density of the slip system | $4\text{}{\mathsf{\mu}\mathrm{m}}^{-2}$ |

${v}_{0}$ | Reference velocity for dislocation slip | $2\times {10}^{-4}\mathrm{m}/\mathrm{s}$ |

${Q}_{s}$ | The activation energy for dislocation slip | 0.27 eV |

$p,q$ | The exponent in slip velocity | 0.75, 2.5 |

${d}_{anni}$ | Annihilation distance for dislocations | 1.1 $b$ |

${\xi}_{\alpha {\alpha}^{\prime}}$ | Interaction coefficient between slip systems | 0.122, 0.122, 0.625, 0.07, 0.137, 0.122 |

${N}_{tw}$ | Total number of twin systems | 12 |

$h$, $t$ | The width and thickness of twin lamellas | 10 μm, 0.01 μm |

${f}_{\mathrm{max}}^{\beta}$ | Maximum twin fraction of twin system | 0.01 |

${V}_{cs}$ | Cross-slip volume | $0.0469\text{}{\mathrm{nm}}^{3}$ |

$A$ | Twinning transition profile width exponent | 5 |

${\dot{N}}_{0}$ | reference twin nucleation rate | 2 s^{−1} |

${\xi}_{\alpha \beta}$ | Interaction coefficient between slip and twin systems | 0.0 (coplanar) 0.042 (cross-slip) |

${\xi}_{\beta {\beta}^{\prime}}$ | Interaction coefficient between twin systems | 0.0 (coplanar) 0.468 (non-coplanar) |

${\mathrm{\Gamma}}_{sf}$ | Stacking fault energy | $22{\text{}\mathrm{mJ}/\mathrm{m}}^{2}$ |

${\tau}_{SRO}^{0}$ | Dislocation resistance of SRO | 10, 30, 50 MPa |

${\gamma}_{0}$ | reference strain | 0.25 |

${\xi}_{\alpha \eta}$ | Interaction coefficient between slip and degree of SRO | 3 (coplanar) −1 (non-coplanar) |

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**MDPI and ACS Style**

Li, C.; Cao, F.; Chen, Y.; Wang, H.; Dai, L.
Crystal Plasticity Model Analysis of the Effect of Short-Range Order on Strength-Plasticity of Medium Entropy Alloys. *Metals* **2022**, *12*, 1757.
https://doi.org/10.3390/met12101757

**AMA Style**

Li C, Cao F, Chen Y, Wang H, Dai L.
Crystal Plasticity Model Analysis of the Effect of Short-Range Order on Strength-Plasticity of Medium Entropy Alloys. *Metals*. 2022; 12(10):1757.
https://doi.org/10.3390/met12101757

**Chicago/Turabian Style**

Li, Chen, Fuhua Cao, Yan Chen, Haiying Wang, and Lanhong Dai.
2022. "Crystal Plasticity Model Analysis of the Effect of Short-Range Order on Strength-Plasticity of Medium Entropy Alloys" *Metals* 12, no. 10: 1757.
https://doi.org/10.3390/met12101757