# Complexity Modeling of Steel-Laser-Hardened Surface Microstructures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

_{3}) reports positive values when the horizontal component of the laser beam agrees with the forward velocity of the laser beam (Figure 1c) and negative values when the above-mentioned component is in opposition to the forward velocity (Figure 1d).

_{3}= −30° according to the other notation and X

_{3}= 90° coincides with X

_{3}= −90°. The second notation is useful to highlight whether the laser beam agrees with or is in opposition to the forward velocity and will be used below.

#### 2.2. Data Mining and Processing

_{kxk}matrix containing all the k × k pixels in the SEM image; n is the number of the entire pixels forming the matrix; b indicates the number of black pixels in the considered matrix; and D

_{k}is the parameter relative to the geometric complexity. Equations (2)–(4) illustrate the procedure described, which is also shown in Figure 4b.

## 3. Complexity Prediction Model

^{n}to P24

^{n}(with n = 2, 3, …, 6), were tested with variable impact angle when both the surface temperature and the laser beam velocity changed.

_{1}parameter indicates the temperature [K], the X

_{2}parameter indicates the robot laser beam velocity [mm/s], and the X

_{3}parameter indicates the angles [°] of the laser beam by means of which the hardening of the specimens was produced. These were the three process parameters which had the strongest influence on the surface microstructural complexity.

_{1}, X

_{2}, and X

_{3}) used in the experimental tests. The results were summarized in Table 3 which shows the values of minimum, maximum, and medium values obtained with the complexity GP prediction model.

- (1)
- All 3 × 3 pixel matrices in the SEM image were evaluated and divided into three groups: totally black matrices, where all nine pixels were black; almost black matrices, where eight pixels were black and one pixel was white; hemi-black matrices, where two or more pixels were white.
- (2)
- Each square surface of 10 × 10 pixels was extruded into a 3D voxel (0.18 × 0.18 × 0.18 µm cube), forming the layer 0 of the 3D reconstruction.
- (3)
- The areas with percentages of totally black matrices lower than 30% were associated with layer 0; the areas with percentages of totally black matrices plus almost black matrices ranging from 30% to 50% were associated with layer 1 (higher than layer 0 by one voxel); the areas with percentages of totally black matrices plus almost black matrices between 50% and 70% were associated with layer 2 (higher than layer 1 by one voxel); the areas with percentages of totally black matrices plus almost black matrices between 70% and 90% were associated with layer 3 (higher than layer 2 by one voxel); and finally, the areas with percentages higher than 90% in totally black matrices plus almost black matrices were associated with layer 4 (higher than layer 3 by one voxel).
- (4)
- Labeling the layer 0 areas with a blue color, the layer 1 areas with a cyan color, the layer 2 areas with a green color, the layer 3 with a yellow color, and the layer 4 with a red color, the 3D voxel maps were obtained (Figure 5a,c).

## 4. Statistical Analysis of the Data and Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) RV60-40 robot and laser-hardened specimen. (

**b**) Schematic view of the robot laser hardening process: (

**b**) origin on the side of the hardened surface (impact angle φ, ranging from 0° to 180°); (

**c**) horizontal component of the laser beam in agreement with the forward velocity of the laser beam (impact angles X

_{3}, ranging from 0° to 90°); (

**d**) horizontal component of the laser beam in opposition to the forward velocity of the laser beam (impact angles X

_{3}, ranging from 0° to −90°).

**Figure 3.**Types of self-affine fractal transformations: (

**a**) self-similar; (

**b**) quasi self-affine; (

**c**) statistically self-affine.

**Figure 4.**(

**a**) Statistical self-similarity microstructure of laser-hardened material; (

**b**) number of black and white pixels in the microstructure matrix of laser-hardened specimens.

**Figure 5.**Roughness evaluation from a black and white pixel SEM image: (

**a**) logic of voxel stacking; (

**b**) 750 × 500 black and white pixel image example; (

**c**) 3D voxel roughness modeling; (

**d**) 3D NURBS roughness modeling.

Parameter | Values |
---|---|

Size of organism population | 500 |

Maximum number of generations | 100 |

Reproduction probability | 0.5 |

Crossover probability | 0.6 |

Maximum permissible depth in the creation of the population | 7 |

Maximum acceptable depth after the operation of crossover of two organisms | 10 |

Smallest acceptable depth of organisms in generating new organisms | 3 |

Tournament size used for the selection of organisms | 6 |

Groups of Specimens | Specimen | Temperature | Velocity | Impact Angle | Complexity | |
---|---|---|---|---|---|---|

(X_{1}) [K] | (X_{2}) [mm/s] | (φ) [°] | (X_{3}) [°] | (CC) [-] | ||

I | P1′ | 1373.0 | 2.0 | 30 | 30 | 1.8681 |

P2′ | 1473.0 | 3.0 | 60 | 60 | 1.8524 | |

P3′ | 1573.0 | 4.0 | 75 | 75 | 1.8275 | |

P4′ | 1673.0 | 5.0 | 90 | 90 | 1.8735 | |

II | P5′ | 1373.0 | −2.0 | 150 | −30 | 1.8916 |

P6′ | 1473.0 | −3.0 | 120 | −60 | 1.8365 | |

P7′ | 1573.0 | −4.0 | 105 | −75 | 1.8921 | |

P8′ | 1673.0 | −5.0 | 90 | −90 | 1.8863 | |

III | P9′ | 1373.0 | 2.0 | 30 | 30 | 1.8809 |

P10′ | 1473.0 | 3.0 | 60 | 60 | 1.8923 | |

P11′ | 1573.0 | 4.0 | 75 | 75 | 1.7253 | |

P12′ | 1673.0 | 5.0 | 90 | 90 | 1.8771 | |

IV | P13′ | 1373.0 | −2.0 | 150 | −30 | 1.8924 |

P14′ | 1473.0 | −3.0 | 120 | −60 | 1.7952 | |

P15′ | 1573.0 | −4.0 | 105 | −75 | 1.8893 | |

P16′ | 1673.0 | −5.0 | 90 | −90 | 1.8846 | |

V | P17′ | 1373.0 | 2.0 | 30 | 30 | 1.8793 |

P18′ | 1473.0 | 3.0 | 60 | 60 | 1.8552 | |

P19′ | 1573.0 | 4.0 | 75 | 75 | 1.8614 | |

P20′ | 1673.0 | 5.0 | 90 | 90 | 1.7721 | |

VI | P21′ | 1373.0 | −2.0 | 150 | −30 | 1.8832 |

P22′ | 1473.0 | −3.0 | 120 | −60 | 1.8382 | |

P23′ | 1573.0 | −4.0 | 105 | −75 | 1.8718 | |

P24′ | 1673.0 | −5.0 | 90 | −90 | 1.8857 |

Specimen | GPmin (Y) [-] | GPmax (Y) [-] | GPmed (Y) [-] | Specimen | GPmin (Y) [-] | GPmax (Y) [-] | GPmed (Y) [-] |
---|---|---|---|---|---|---|---|

P1 | 1.8620 | 1.9053 | 1.8808 | P13 | 1.8620 | 1.9053 | 1.8808 |

P2 | 1.8342 | 1.8768 | 1.8527 | P14 | 1.8342 | 1.8768 | 1.8527 |

P3 | 1.8433 | 1.8861 | 1.8619 | P15 | 1.8433 | 1.8861 | 1.8619 |

P4 | 1.8580 | 1.9012 | 1.8768 | P16 | 1.8580 | 1.9012 | 1.8768 |

P5 | 1.8620 | 1.9053 | 1.8808 | P17 | 1.8620 | 1.9053 | 1.8808 |

P6 | 1.8342 | 1.8768 | 1.8527 | P18 | 1.8342 | 1.8768 | 1.8527 |

P7 | 1.8433 | 1.8861 | 1.8619 | P19 | 1.8433 | 1.8861 | 1.8619 |

P8 | 1.8580 | 1.9012 | 1.8768 | P20 | 1.8580 | 1.9012 | 1.8768 |

P9 | 1.8620 | 1.9053 | 1.8808 | P21 | 1.8620 | 1.9053 | 1.8808 |

P10 | 1.8342 | 1.8768 | 1.8527 | P22 | 1.8342 | 1.8768 | 1.8527 |

P11 | 1.8433 | 1.8861 | 1.8619 | P23 | 1.8433 | 1.8861 | 1.8619 |

P12 | 1.8580 | 1.9012 | 1.8768 | P24 | 1.8580 | 1.9012 | 1.8768 |

Set of Specimens | CC Values | Mean Value | St. Dev. |
---|---|---|---|

Set 1 (1373 K; 2 mm/s; 30°) | |||

P1 | 1.8681 | 1.8826 | 0.0090 |

P5 | 1.8916 | ||

P9 | 1.8809 | ||

P13 | 1.8924 | ||

P17 | 1.8793 | ||

P21 | 1.8832 | ||

Set 2 (1473 K; 3 mm/s; 60°) | |||

P2 | 1.8524 | 1.8450 | 0.0316 |

P6 | 1.8365 | ||

P10 | 1.8923 | ||

P14 | 1.7952 | ||

P18 | 1.8552 | ||

P22 | 1.8382 | ||

Set 3 (1573 K; 4 mm/s; 75°) | |||

P3 | 1.8275 | 1.8446 | 0.0630 |

P7 | 1.8921 | ||

P11 | 1.7253 | ||

P15 | 1.8893 | ||

P19 | 1.8614 | ||

P23 | 1.8718 | ||

Set 4 (1673 K; 5 mm/s; 90°) | |||

P4 | 1.8735 | 1.8632 | 0.0449 |

P8 | 1.8863 | ||

P12 | 1.8771 | ||

P16 | 1.8846 | ||

P20 | 1.7721 | ||

P24 | 1.8857 |

Set of Specimens | D Mean Value | GP Complexity Mean Value | % Error |
---|---|---|---|

Set 1 (1373 K; 2 mm/s; 30°) | 1.8826 | 1.8808 | 0.095 |

Set 2 (1473 K; 3 mm/s; 60°) | 1.8450 | 1.8527 | 0.419 |

Set 3 (1573 K; 4 mm/s; 75°) | 1.8446 | 1.8619 | 0.940 |

Set 4 (1673 K; 5 mm/s; 90°) | 1.8632 | 1.8768 | 0.729 |

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**MDPI and ACS Style**

Babič, M.; Marinkovic, D.; Bonfanti, M.; Calì, M.
Complexity Modeling of Steel-Laser-Hardened Surface Microstructures. *Appl. Sci.* **2022**, *12*, 2458.
https://doi.org/10.3390/app12052458

**AMA Style**

Babič M, Marinkovic D, Bonfanti M, Calì M.
Complexity Modeling of Steel-Laser-Hardened Surface Microstructures. *Applied Sciences*. 2022; 12(5):2458.
https://doi.org/10.3390/app12052458

**Chicago/Turabian Style**

Babič, Matej, Dragan Marinkovic, Marco Bonfanti, and Michele Calì.
2022. "Complexity Modeling of Steel-Laser-Hardened Surface Microstructures" *Applied Sciences* 12, no. 5: 2458.
https://doi.org/10.3390/app12052458