# Parameter Optimization in High-Throughput Testing for Structural Materials

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. High-Throughput Method and Material

#### 2.1. Method ‘Farbige Zustände’

#### 2.2. Material

#### 2.3. Treatment and Testing of Samples

_{s}4 bar ≙ max. particle velocity of approx. 70 m/s for the investigated material) to impact on a hardened contact plate of 100Cr6 (60 HRC), which is located at a constant distance of 80 mm in front of the nozzle outlet. This avoids plastic deformation of the contact plate, while the particles are deformed by the impact according to their material properties (e.g., hardness or yield strength). Therefore the observed plastic deformation allows conclusions to be drawn about the mechanical properties of the samples. The particles, as well as the plastic deformation resulting from the impact, are analyzed using the Zeiss SteREO.V12 microscope in combination with the REOObjective.435200-0000-000 objective. The descriptor ‘linear plastic deformation’ is determined after the impact. It is defined as the difference between the initial particle radius r and the distance from the center to the flattened surface after the impact (see. Figure 2). The relationship between the determined plastic deformation and material properties has already been demonstrated in Kämmler et al [18]. For rather distinctive material properties (X210Cr12 vs. AlSi12), it was shown that different hardness values result in different descriptor values. As shown in [19], significantly differing DSC heat treatments can be characterized using the particle-oriented peening process. It is, therefore, assumed that a more fine-grained variation of the heat treatment parameters, as conceivable in the context of current investigations, will lead to distinguishable descriptor values.

#### 2.4. Design of Experiments and Routing of Processchains

## 3. Evaluation and Discussion of Descriptors

#### 3.1. Data

#### 3.2. Optimization Problem

#### 3.3. Method

#### 3.4. Results

_{s}= 1 bar, 2 bar, 4 bar), the statistical dispersion of the linear plastic deformation decreases with increasing jet pressure. This is due to the fact that comparatively small plastic deformations occur at a jet pressure of 1 bar. Although these can be measured, there is a higher uncertainty. Furthermore, a higher dispersion can be observed for 700 °C than for 1000 °C. At a heating temperature of 1000 °C, the austenitization temperature is already exceeded. However, this temperature, in combination with the holding time, is not sufficient for a complete phase transformation. Thus, a mixed microstructure with ferrite and a small amount of perlite still results. Lower heating temperatures such as 700 °C mainly result in tempering of the previously martensitic microstructure. This reduces the hardness compared to the initial state. Since a higher hardness results in a greater resistance to deformation, the determined linear plastic deformation has its lowest values for material states of high hardness. The higher plastic deformation determined at 1000 °C can be explained by the fact that when tempering at 700 °C there is still remaining martensite, which has a higher hardness than the microstructure resulting from a heating temperature of 1000 °C. Considering the results of the two iterations, an annealed martensite structure can also be assumed for the heating temperatures 460 °C, 830 °C and 880 °C. While the boxplots for 460 °C and 830 °C depict interquartile ranges comparable to 700 °C, the range for 880 °C is much larger. As observed in [19] this may be explained by a decarburization which has occurred near the surface of the particles. Since the temperature resulted in the material being in a two-phase region for the duration of the holding time, a mixed structure of ferrite, perlite, upper bainite and carbides was created. Since the particle-oriented peening mainly causes a deformation of the particle surface and subsurface area, the observed microstructural effects in the outer surface and subsurface layers (decarbonized) may affect the determined plastic deformation. Compared to [19] the jet pressure was lower during all the experiments in this paper. The lower pressure results in a smaller deformation and a higher relative error in the measurements. This explains the higher dispersion of the results. Based on the results of Toenjes et al. [19] it can further be assumed that a complete austenitization occurred at the temperature of 1100 °C. Therefore, a new microstructure was formed, resulting in a higher hardness and thus also a lower plastic deformation [19]. The results shown in Figure 8 can be attributed to the microstructure, which is caused by the DSC heat treatment. To ensure lower deviations of the descriptor values when considering the complete temperature range (from 400 °C to 1100 °C) an optimization of the problem should therefore be carried out on a sectoral basis. Materials knowledge could, for example, be used to redefine the limits of the search space.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${}^{\left(i\right)}$ | upper indices in parentheses contain the sample ID, here $i$ |

${}^{T}$ | transpose of a matrix or vector |

$\mathsf{\beta}$ | vector of coefficients from weighted least squares (WLS)-regression in Equation (7), $\mathsf{\beta}=\left({\beta}_{1},\dots ,{\beta}_{k}\right)$ |

${d}^{\left(i\right)}$ | $i$-th observation of descriptor, see Equation (2) |

$\Phi $ | real relationship between predictors and descriptors, see Equation (1) |

$\tilde{\Phi}$ | real relationship between pseudo predictors and pseudo descriptors, see Equation (4) |

$\Delta l$ | linear plastic deformation [µm] |

$\Delta {l}^{*}$ | target value for linear plastic deformation; equals $35\mu m$, see page 7 |

$\delta $ | half width of the target region for the linear plastic deformation |

$E[\cdot ]$ | expectation |

$E[\cdot |\cdot ]$ | conditional expectation |

${\epsilon}^{\left(i\right)}$ | model error term in Equation (2) |

$f$ | functional relationship between the first pseudo predictor and pseudo descriptor, see Equation (5) |

${\widehat{f}}_{1}$ | regression model for ${x}_{1}$ and ${\Delta}_{l}$(8) |

$k$ | polynomial order in the model estimation step, see page 9 |

${\widehat{\mu}}_{T},{\widehat{\mu}}_{ps}$ | empirical mean of $T$, ${p}_{s}$ |

$n$ | number of joint observations of predictors and descriptors |

$P$ | matrix of predictors, see Equation (10) |

${p}^{\left(i\right)}$ | vector of predictors of the $i$-th observation, see page 7 |

${p}_{ps}^{\left(i\right)}$ | jet pressure of $i$-th observation, see page 7 |

${p}_{s}$ | jet pressure [bar] |

${p}_{s}^{*}$ | optimal jet pressure |

${p}_{T}^{\left(i\right)}$ | temperature of $i$-th observation, see page 7 |

${\tilde{p}}_{}^{\left(i\right)}$ | normalized predictors ${\tilde{p}}_{}^{\left(i\right)}=\left({\tilde{p}}_{T}^{\left(i\right)},{\tilde{p}}_{ps}^{\left(i\right)}\right)$, see page 9 |

${\tilde{p}}_{ps}^{\left(i\right)}$ | second coordinate of ${\tilde{p}}_{}^{\left(i\right)}=\left({\tilde{p}}_{T}^{\left(i\right)},{\tilde{p}}_{ps}^{\left(i\right)}\right)$ |

${\tilde{p}}_{T}^{\left(i\right)}$ | First coordinate of ${\tilde{p}}_{}^{\left(i\right)}=\left({\tilde{p}}_{T}^{\left(i\right)},{\tilde{p}}_{ps}^{\left(i\right)}\right)$ |

${\widehat{p}}^{*}$ | transformation of ${\widehat{x}}_{1}^{*}$ to the standardized predictor space of temperature and jet pressure, see Equation (11) |

${\widehat{p}}_{ps}^{*}$ | jet pressure corresponding to ${\widehat{x}}_{1}^{*}$ in the standardized predictor space, see page 10 |

${\widehat{p}}_{T}^{*}$ | temperature corresponding to ${\widehat{x}}_{1}^{*}$ in the standardized predictor space, see page 10 |

$r$ | radius of the particle [mm] |

${\widehat{\sigma}}_{T}$, ${\widehat{\sigma}}_{ps}$ | empirical standard deviations of $T$, ${p}_{s}$ |

$T$ | temperature [°C, K] |

${T}^{*}$ | optimal temperature |

$V$ | matrix of modified weights, defined by Equation (10) |

${V}_{1,\cdot}$ | first row of matrix $V$ |

$X$ | matrix of scores, see Equation (10) |

${x}^{\left(i\right)}$ | pseudo predictor of $i$-th observation, see Equation (4) |

${\widehat{x}}_{1}^{*}$ | estimator for optimal pseudo predictor coordinate, see Equation (9) |

${w}_{i}$ | weight for $i$-th observation in WLS-regression in Equation (7) |

$wt\%$ | percentage by weight |

$z$ | transformation of ${\widehat{x}}_{1}^{*}$ to the standardized predictor space, see Equation (11) |

## Appendix A

Iteration | Temperature in °C | Pressure in bar | Iteration | Temperature in °C | Pressure in bar |
---|---|---|---|---|---|

0 | 1000 | 4 | 0 | 700 | 1 |

0 | 1000 | 4 | 0 | 700 | 1 |

0 | 1000 | 4 | 0 | 700 | 1 |

0 | 1000 | 4 | 0 | 700 | 1 |

0 | 1000 | 4 | 0 | 700 | 1 |

0 | 1000 | 4 | 0 | 700 | 1 |

0 | 1000 | 4 | 0 | 700 | 1 |

0 | 1000 | 4 | 0 | 700 | 1 |

0 | 1000 | 4 | 1 | 1100 | 1.71 |

0 | 1000 | 4 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 1100 | 1.71 |

0 | 1000 | 2 | 1 | 880 | 3.06 |

0 | 1000 | 2 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 880 | 3.06 |

0 | 1000 | 1 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 4 | 1 | 460 | 3.28 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 2 | 2 | 830 | 3.73 |

0 | 700 | 1 | 2 | 830 | 3.73 |

0 | 700 | 1 |

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**Figure 2.**Definition of the determined linear plastic deformation (Adapted from [18].)

**Figure 7.**Boxplots for the new measurements; second boxplot corresponds to the suggested predictor values in iteration 1.

**Table 1.**Chemical composition of the used alloy SAE52100 in wt.% in comparison with the required chemical composition of the DIN EN ISO 683-17:2000-04.

Material | Chemical Composition in wt.% | ||||||||
---|---|---|---|---|---|---|---|---|---|

Fe | C | Cr | Mn | Ni | P | S | Si | ||

Samples Test 1 ^{b} | - | bal. | 1.03 ^{c} | 1.20 | 0.38 | 0.40 | 0.015 ^{c} | 0.35 ^{a} | |

Samples Test 2 ^{b} | - | bal. | 1.07 ^{c} | 1.31 | 0.35 | 0.17 | 0.018 ^{c} | 0.35 ^{a} | |

DIN EN ISO 683-17:2000-04 [16] | min max | bal. | 0.93 1.05 | 1.35 1.60 | 0.25 0.45 | 0.00 0.40 | - 0.025 | - 0.015 | 0.15 0.35 |

^{a}by optical emission spectroscopy (OES),

^{b}by atomic absorption spectrometry (AAS),

^{c}combustion analysis.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bader, A.; Toenjes, A.; Wielki, N.; Mändle, A.; Onken, A.-K.; Hehl, A.v.; Meyer, D.; Brannath, W.; Tracht, K.
Parameter Optimization in High-Throughput Testing for Structural Materials. *Materials* **2019**, *12*, 3439.
https://doi.org/10.3390/ma12203439

**AMA Style**

Bader A, Toenjes A, Wielki N, Mändle A, Onken A-K, Hehl Av, Meyer D, Brannath W, Tracht K.
Parameter Optimization in High-Throughput Testing for Structural Materials. *Materials*. 2019; 12(20):3439.
https://doi.org/10.3390/ma12203439

**Chicago/Turabian Style**

Bader, Alexander, Anastasiya Toenjes, Nicole Wielki, Andreas Mändle, Ann-Kathrin Onken, Axel von Hehl, Daniel Meyer, Werner Brannath, and Kirsten Tracht.
2019. "Parameter Optimization in High-Throughput Testing for Structural Materials" *Materials* 12, no. 20: 3439.
https://doi.org/10.3390/ma12203439