# A Data-Driven Approach for Studying the Influence of Carbides on Work Hardening of Steel

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Synthetic Microstructures Generation

#### 2.2. Virtual Tensile Test

#### 2.3. Statistical Analysis

#### 2.4. Voce Model

#### 2.5. Data-Driven Approach

#### 2.6. Linear Mixed-Effects Model

**Fixed effects**, which concern parameters associated with the levels of the experimental factor or of the explanatory variable whose effect needs to be primarily investigated;**Random effects**, which concern parameters associated with the levels of the blocking factor or better associated with individuals or groups drawn at random from a population.

- ${y}_{i}|({X}_{i},1{\mathbf{\alpha}}_{0i})\sim {N}_{{n}_{i}}({X}_{i}\mathbf{\beta},1{\mathbf{\alpha}}_{0i},{\mathbf{\sigma}}^{2}{I}_{{n}_{i}})$
- ${\alpha}_{0i}\sim N(0,{\sigma}_{\alpha}^{2})$,

## 3. Results

#### 3.1. Goodness of Fit

## 4. Discussion

## 5. Conclusions

- The design of a randomized-block experiment allows to study the contribution of the ${\mathrm{M}}_{23}{\mathrm{C}}_{6}$ carbides on the stress–strain behavior of AISI 420 steel controlling for the possible confounding effect of the textures.
- Multi-level Voronoi diagrams prove to be a flexible model that allow to represent the microstructure under analysis.
- The approach is simulation-based, and hence it is fully reproducible and tuneable for other microstructure-features and mechanical-properties investigation.
- The FPCA model is a flexible approach that does not require any physical assumption and that can be applied also for modeling the mechanical behavior, highlighting the effect of the different sources of variations given by the microstructural features.
- Linear mixed-effects models are able to give a clear interpretation of the model parameters of both the Voce and the FPCA model in terms of the carbide volume fraction and the textures.
- The presented research methodology can be applied to other alloys with different precipitates such as graphite in cast iron or intermetallics in superalloys.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FPCA | Functional Principal Component Analysis |

DAMASK | Düsseldorf Advanced Material Simulation Toolkit |

ODF | Orientation Distribution Function |

RVE | Representative Volume Element |

FC | Full Constraints |

FFT | Fast Fourier Transform |

RMSE | Root Mean Square Error |

MAE | Mean Absolute Error |

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**Figure 1.**3D multi-level Voronoi diagrams with increasing level of ${\lambda}_{1}^{c}$: (

**a**) ${\lambda}_{1}^{c}=0$, (

**b**) ${\lambda}_{1}^{c}=0.01$, (

**c**) ${\lambda}_{1}^{c}=0.03$, (

**d**) ${\lambda}_{1}^{c}=0.05$, (

**e**) ${\lambda}_{1}^{c}=0.07$, (

**f**) ${\lambda}_{1}^{c}=0.09$, (

**g**) ${\lambda}_{1}^{c}=0.11$.

**Figure 2.**ODFs (${\phi}_{2}={45}^{\circ}$ ODF sections) corresponding to each of the ten different crystallographic textures in the ferrite phase under study. An orthorhombic sample symmetry was assumed. Above each texture, it is shown the maximum texture intensity in random units and the Taylor factor calculated using the full-constraints Taylor model (M).

**Figure 3.**Stress–strain curves. Different colors indicate different values of the volume fraction of the carbides in the range [0, 0.11], with different symbols indicating different textures in the ferrite phase.

**Figure 4.**Fitted stress–strain functions using Voce hardening law. Different colors indicate different values of the volume fraction of the carbides in the range [0, 0.11].

**Figure 5.**Mean stress–strain curve for microstructures without carbides (red line). Different colors indicate different textures in the ferrite phase with different Taylor factor M.

**Figure 6.**Stress–strain centered to the expected stress–strain for microstructures without carbides. Different colors indicate different values of the volume fraction of the carbides in the range [0, 0.11].

**Figure 7.**First two eigenfunctions obtained with the modified FPCA performed on the 70 stress–strain curves.

**Figure 8.**Plot of the two FPCA scores obtained for the 70 stress–strain curves. Different co lours indicate in (

**a**) different values of the carbides’ volume fraction in the range [0, 0.11] and in (

**b**) different textures in the ferrite phase.

**Figure 9.**Plot of the FPCA scores correspondent to the first functional principal component ${\varphi}_{1}$ and the observed values of carbide volume fraction ${\lambda}_{1}^{c}$ for the 70 stress–strain curves. Different colors indicate different textures in the ferrite phase.

**Figure 10.**Fitted stress–strain functions using FPCA model (Equation (18)). Different colors indicate different values of the the carbides in the range [0, 0.11].

**Figure 11.**Plot of the estimated Voce-model parameters (${\tau}_{0}$ (

**a**), ${\tau}_{1}$ (

**b**), ${\theta}_{0}$ (

**c**), ${\theta}_{1}$ (

**d**)) and the observed volume fraction of carbides for the 70 different microstructures (different symbols indicate different textures in the ferrite phase).

**Figure 12.**Plot of the estimated stress–strain curve using the Voce model with parameters modeled in terms of the carbide volume fraction and texture in the ferrite phase via linear mixed models.

**Figure 13.**Effect of texture in the ferrite phase (

**a**) and of the carbides’ volume fractions (

**b**) in the stress–strain curves.

**Figure 14.**Histograms of the RMSE and the MAE (values in MPa) computed for all curves using the model based on the Voce law (

**a**–

**c**) and the FPCA approach (

**b**–

**d**).

**Table 1.**Estimated moments of the geometrical features of 1000 grains obtained by EBSD measurements.

(a) Ferrite | (b) Carbides | ||
---|---|---|---|

Volume Fraction | $0.968$ | Volume Fraction | $0.032$ |

Mean Volume ($\mathsf{\mu}{\mathrm{m}}^{3}$) | $2.58\pm 0.05$ | Mean Volume ($\mathsf{\mu}{\mathrm{m}}^{3}$) | $0.45\pm 0.03$ |

Mean Area ($\mathsf{\mu}{\mathrm{m}}^{2}$) | $4.43\pm 0.07$ | Mean Area ($\mathsf{\mu}{\mathrm{m}}^{2}$) | $0.70\pm 0.03$ |

Parameter | Unit | Ferrite | ${\mathbf{M}}_{23}{\mathbf{C}}_{6}$ Carbides |
---|---|---|---|

${C}_{11}$ | GPa | 233 | $550.8$ |

${C}_{12}$ | GPa | 135 | $225.9$ |

${C}_{44}$ | GPa | 128 | 140 |

${\dot{\gamma}}_{0}$ | s${}^{-1}$ | $0.001$ | $0.001$ |

${n}_{slip}$ | - | 10 | 200 |

${\tau}_{C}^{\eta}$ | MPa | 80 | 1600 |

${\tau}_{sat}$ | MPa | 250 | 1800 |

${h}_{0}$ | MPa | $549.4$ | 20,000 |

a | - | $2.25$ | $1.1$ |

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**MDPI and ACS Style**

Vittorietti, M.; Hidalgo, J.; Galán López, J.; Sietsma, J.; Jongbloed, G.
A Data-Driven Approach for Studying the Influence of Carbides on Work Hardening of Steel. *Materials* **2022**, *15*, 892.
https://doi.org/10.3390/ma15030892

**AMA Style**

Vittorietti M, Hidalgo J, Galán López J, Sietsma J, Jongbloed G.
A Data-Driven Approach for Studying the Influence of Carbides on Work Hardening of Steel. *Materials*. 2022; 15(3):892.
https://doi.org/10.3390/ma15030892

**Chicago/Turabian Style**

Vittorietti, Martina, Javier Hidalgo, Jesús Galán López, Jilt Sietsma, and Geurt Jongbloed.
2022. "A Data-Driven Approach for Studying the Influence of Carbides on Work Hardening of Steel" *Materials* 15, no. 3: 892.
https://doi.org/10.3390/ma15030892