# Grain Structure Rearrangement by Means the Advanced Statistical Model Modified for Describing Dynamic Recrystallization

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## Abstract

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## 1. Introduction

## 2. Establishment of the Correspondence of the Grain Structure Rearranged during the Process of Recrystallization

**.**We suppose that the grains which were recrystallized according to the Bailey-Hirsch criterion have the misorientation coherency with the parent grain [5,54]. The new grain is separated from the parent by the high-angle boundary and has the initial misorientation with respect to the parent grain within the range from 10° to 15° with a random axis of rotation [5,54]. The DRX grain orientation with respect to the neighboring (absorbed) grain is arbitrary. The index «+» in Figure 1 denotes the corresponding internal variables which describe the appearance of newly recrystallized grains.

**Σ**) of the representative volume of the material is minimum, and the conditions for recrystallization are kept unchanged during subsequent deformation.

## 3. Selection and Justification of the Method for Finding a Solution to the Problem of Establishing Grain Structure Correspondence

## 4. Modelling Results and Their Analysis

## 5. Conclusions

- We put forward a method for establishing statistical consistency of the grain structure during its rearrangement in order to keep the results of numerical simulation of the inelastic deformation process, with an account of unchanged recrystallization.
- We determined the main variables of the model; namely, the stored energy difference distribution, mutual misorientation angle distribution, and grain sizes.
- We posed the problem of minimizing the difference between these variables in a statistical sense during the grain structure rearrangement. To solve it, a genetic algorithm was applied.
- In preliminary calculations, the parameters of the weight coefficients included in the definition of the objective function and the parameters of the genetic algorithm were determined.
- It is shown that the agreement between the variables under consideration provides the smallest (among the considered variables) deviation of the material response during the grain structure rearrangement. If one or more of these parameters are removed from the optimization problem statement, the material behavior (response) may deviate significantly after the grain structure rearrangement.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The scheme of the grain structure rearrangement in the statistical model with new recrystallized grains introduced into consideration.

**Figure 2.**The scheme of the method to establish correspondence between the initial grain structure and the rearranged structure, due to emergence of recrystallized grains.

**Figure 3.**(

**a**) The histogram for the distribution of mutual misorientation angles in the ${\mathsf{\phi}}_{m}$ in the copper polycrystal at temperature T = 773 and deformation ${\epsilon}_{u}=0.15$. (

**b**) The normalized histogram for the angles ${\mathsf{\phi}}_{m}^{norm}$. (

**c**) The evolution of the empirical (${\mathsf{\phi}}_{m}^{norm}$) distribution function during the optimization procedure.

**Figure 5.**The normalized histograms for the distributions of: (

**a**) ${d}_{gr0}^{norm}$ and ${\widehat{d}}_{gr}^{norm}$, (

**b**) ${e}_{dst0}^{norm}$ and ${\widehat{e}}_{dst}^{norm}$, and (

**c**) ${\mathsf{\phi}}_{m0}^{norm}$ and ${\hat{\mathsf{\phi}}}_{m}^{norm}$, obtained in the framework of the statistical model and by solving of the optimization problem for the grain structures being compared.

**Figure 6.**Dependence of the objective function of the best data comparison option on the step number (generation) k when solving the optimization problem: (

**a**) without mutation and crossing, (

**b**) without mutation, and (

**c**) without crossing.

**Figure 7.**Stress versus accumulated strain curves for the representative volume element of a polycrystal without grain structure rearrangement (black), with optimization (blue), and without optimization (red). The grain structure is reconstructed at the deformation instant of 15%.

**Table 1.**The deviations of distributions $\parallel \Delta {e}_{dst}^{norm}\parallel $ and $\parallel \Delta {\mathsf{\phi}}_{m}^{norm}\parallel $ at different values of the genetic algorithm parameters.

Parameters | p_{c} = 0.63, p_{m} = 0.37,α = 0.3, β = 1, γ = 0.0028, K = 100 | p_{c} = 0.63, p_{m} = 0.37,α = 0.5, β = 1, γ = 0.0028, K = 100 | p_{c} = 0.63, p_{m} = 0.37,α = 1, β = 1, γ = 0.0028, K = 100 |

Deviations $\parallel \Delta {e}_{dst}^{norm}\parallel $, $\parallel \Delta {\mathsf{\phi}}_{m}^{norm}\parallel $ | 17.59 4.59 | 16.3 4.68 | 15.25 4.94 |

Parameters | p_{c} = 0.63, p_{m} = 0.37,α = 0.8, β = 0.3 γ = 0.0028, K = 100 | p_{c} = 0.63, p_{m} = 0.37,α = 0.8, β = 0.5, γ = 0.0028, K = 100 | p_{c} = 0.63, p_{m} = 0.37,α = 0.8, β = 0.8, γ = 0.0028, K = 100 |

Deviations $\parallel \Delta {e}_{dst}^{norm}\parallel $, $\parallel \Delta {\mathsf{\phi}}_{m}^{norm}\parallel $ | 15.26 8.44 | 15.67 7.17 | 15.74 5.29 |

Parameters | p_{c} = 0.63, p_{m} = 0.37,α = 0.8, β = 1, γ = 0.0028, K = 100 | p_{c} = 0.63, p_{m} = 0.37,α = 0.8, β = 1, γ = 0.01, K = 100 | p_{c} = 0.63, p_{m} = 0.37,α = 0.8, β = 1, γ = 0.1, K = 100 |

Deviations $\parallel \Delta {e}_{dst}^{norm}\parallel $, $\parallel \Delta {\mathsf{\phi}}_{m}^{norm}\parallel $ | 15.32 4.7 | 15.42 6.77 | 16.36 7.47 |

Parameter | Value |
---|---|

$\mathsf{\alpha}$ | 0.8 |

$\mathsf{\beta}$ | 1 |

$\mathsf{\gamma}$ | 0.0028 |

$K$ | 100 |

${p}_{c}$ | 0.63 |

${p}_{m}$ | 0.37 |

**Table 3.**The relative values of deviations of stress intensities $\parallel \Delta {\Sigma}_{u}\parallel $, $\parallel \Delta {e}_{dst}^{norm}\parallel $ and $\parallel \Delta {\mathsf{\phi}}_{m}^{norm}\parallel $ for different definitions of the objective function $\mathsf{\mu}\left(x\right)$.

Objective Function | Value $\parallel \mathbf{\Delta}{\mathbf{\Sigma}}_{\mathit{u}}\parallel $ | Value $\parallel \mathbf{\Delta}{\mathit{e}}_{\mathit{d}\mathit{s}\mathit{t}}^{\mathit{n}\mathit{o}\mathit{r}\mathit{m}}\parallel $ | Value $\parallel \mathbf{\Delta}{\mathsf{\phi}}_{\mathit{m}}^{\mathit{n}\mathit{o}\mathit{r}\mathit{m}}\parallel $ | Value $\mathsf{\mu}\left(\mathit{x}\right)$ |
---|---|---|---|---|

$\mathsf{\mu}$ | 0.03 | 15.32 | 4.7 | 0.31 |

${\mathsf{\mu}}_{{e}_{dst}^{norm}}$ | 0.07 | 13.93 | 11.37 | 0.1 |

${\mathsf{\mu}}_{{\mathsf{\phi}}_{m}^{norm}}$ | 0.04 | 17.19 | 6.59 | 0.15 |

${\mathsf{\mu}}_{{d}_{gr}^{nm}}$ | 0.07 | 16.79 | 8.26 | 0.11 |

${\mathsf{\mu}}_{{e}_{dst}^{norm},{\mathsf{\phi}}_{m}^{norm}}$ | 0.06 | 15.33 | 6.72 | 0.28 |

${\mathsf{\mu}}_{{e}_{dst}^{norm},}{}_{{d}_{gr}^{nm}}$ | 0.07 | 16.65 | 13.34 | 0.25 |

${\mathsf{\mu}}_{{\mathsf{\phi}}_{m}^{norm}}{}_{,{d}_{gr}^{nm}}$ | 0.06 | 24.64 | 7.15 | 0.27 |

${\mathsf{\mu}}_{M}$ | 0.07 | 16.76 | 8.07 | 0.23 |

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**MDPI and ACS Style**

Trusov, P.; Kondratev, N.; Podsedertsev, A.
Grain Structure Rearrangement by Means the Advanced Statistical Model Modified for Describing Dynamic Recrystallization. *Metals* **2023**, *13*, 113.
https://doi.org/10.3390/met13010113

**AMA Style**

Trusov P, Kondratev N, Podsedertsev A.
Grain Structure Rearrangement by Means the Advanced Statistical Model Modified for Describing Dynamic Recrystallization. *Metals*. 2023; 13(1):113.
https://doi.org/10.3390/met13010113

**Chicago/Turabian Style**

Trusov, Peter, Nikita Kondratev, and Andrej Podsedertsev.
2023. "Grain Structure Rearrangement by Means the Advanced Statistical Model Modified for Describing Dynamic Recrystallization" *Metals* 13, no. 1: 113.
https://doi.org/10.3390/met13010113