# Evaluation of Ti–Mn Alloys for Additive Manufacturing Using High-Throughput Experimental Assays and Gaussian Process Regression

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Experiments

#### 2.2. Microstructure Analysis and Quantification

#### 2.3. Mechanical Characterization

^{2}(contact radius of roughly 7 μm) at indentation yield and hence reflected the effective response of the two-phase microstructures obtained in the sample library (micrographs presented later). The spacing between indentations was designed to be large enough to minimize the interference between neighboring indentations. However, it was also important to keep the spacing small enough so that the compositional variation between the indentation locations within each grid was very small.

#### 2.4. Gaussian Process Modeling

- (1)
- ${\sigma}_{f}$ is called the output scaling factor and determines the variance of the output values. A higher value of ${\sigma}_{f}$ indicates that the values of the output are widely spread. The ratio of ${\sigma}_{f}$ to the output noise ${\sigma}_{n}$ (discussed later) determines the uncertainty of the predictions made from the GP model.
- (2)
- ${l}_{T}$ and ${l}_{c}$ are the interpolation length scale parameters associated with the two input variables and capture the sensitivity of the output variable to the changes in the respective input values. Lower length scale values exhibit shorter memory, leading to sharper fluctuations and more complex nonlinear mapping between the inputs and the output. In other words, lower values of the interpolation length parameter indicate a higher sensitivity of the output to the input value (for the selected input variable). Conversely, larger values of the interpolation length parameters indicate low levels of correlation between the output and the corresponding input variable.
- (3)
- ${\sigma}_{n}$ is called the output noise hyperparameter and captures the variance in the training data. For the present study, where the training data are obtained from experiments, this variance can arise from variations in the execution of the experimental assays themselves or variations in the application of the analysis protocols (e.g., image segmentation). ${\sigma}_{n}$ is assumed to be the same for the entire input domain (also called homoscedasticity [104]).

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Illustration of the layered Ti–Mn cylindrical sample manufactured by the Laser Engineered Net Shaping (LENS) process in this study. (

**b**) Sample strip sectioned from (

**a**) with compositional gradient along the length of the sample. Five locations were chosen longitudinally in each sample strip for characterization. They were 8, 14, 20, 26 and 32 mm away from the pure titanium end of the strip (labeled as #1–#5, respectively). (

**c**) Three different sample strips were aged at three different temperatures (500, 600 and 700 °C, respectively) for four hours to produce the sample library used in this work. (

**d**) A grid of indentation and microscopy characterization was performed at each location illustrated in (

**b**). Each circle represents an indentation testing site, while the square represents the microscopy characterization site. Each measurement grid contained 5 by 5 indentation tests and the same number of microscopy characterizations. The test points in the grid were evenly spaced at 100 μm. Note all test sites shown in (

**b**) are intentionally kept away from the thin end of the sample strips, making sure the sample has at least 2 mm thickness at the indentation test sites.

**Figure 2.**(

**a**) Illustration of spherical indentation. (

**b**) Indentation stress–strain curve acquired from Location #4 (see Figure 1b) of the strip heat treated at 700 °C. The slope illustrated in the elastic portion of the indentation stress–strain curve is the effective modulus, ${E}_{eff}$. The red dot represents the indentation yield strength ${Y}_{ind}$ corresponding to a 0.002 offset indentation plastic strain, while the black segment (from 0.005 to 0.02 in offset indentation plastic strain) represents the data used to estimate the indentation work hardening rate ${H}_{ind}$.

**Figure 3.**(

**a**) Back-scattered electron (BSE)-SEM image for the sample strip aged at 500 ℃ for four hours and at the location where the Mn content was 5.8 wt.%. It depicts the dual-phase microstructure of the sample, where the darker phase is α-Ti and the brighter phase is β-Ti. (

**b**) Means and standard deviations from the energy dispersive spectroscopy (EDS) measurements of the Mn content at the five locations for all three high-throughput (HT) sample strips produced for this study. For clarity, all 500 °C and 700 °C values are intentionally shifted slightly in the negative and positive x directions, respectively. All points in each group correspond to the same nominal distance indicated by the axis ticks.

**Figure 4.**Means and standard deviations of the percentage volume fractions of the β phase obtained for the different Mn contents and post-build aging heat treatments.

**Figure 5.**Segmented SEM-BSE images for the sample library produced and studied in this work. The left, middle and right columns correspond to aging heat treatments of 500, 600 and 700 °C, respectively. The rows correspond to different locations exhibiting different manganese compositions (see Figure 1b and Figure 3b). The black phase in these micrographs represents α-Ti, while the white phase represents β-Ti.

**Figure 6.**Averaged chord lengths (CLs) of (

**a**) α phase and (

**b**) β phase at the selected five locations for all three high-throughput (HT) sample strips studied in this work.

**Figure 7.**Mechanical properties estimated from the spherical indentation stress–strain protocols: (

**a**) Young’s modulus, (

**b**) indentation yield strength and (

**c**) indentation initial hardening rate. The blue, green and red boxes correspond to the 500, 600 and 700 °C aged strips, respectively.

**Table 1.**Gaussian process regression (GPR) interpolation length hyperparameters and the mean absolute percentage error (MAPE) for each of the six outputs selected for these models. CL denotes the averaged chord length, VF is the volume fraction, Y is the indentation yield strength, and E is the Young’s modulus.

GPR Results | CL α | CL β | VF-β | Y | H | E |
---|---|---|---|---|---|---|

${l}_{T}$ | 667.17 | 141.85 | 263.23 | 199.12 | 143.38 | 211.78 |

${l}_{c}$ | 10.95 | 10.48 | 9.39 | 16.83 | 14.78 | 12.49 |

${\sigma}_{f}$ | 19.52 | 18.47 | 0.55 | 2.48 | 53.97 | 92.17 |

${\sigma}_{n}$ | 0.93 | 1.66 | 0.01 | 0.17 | 2.64 | 1.39 |

${\sigma}_{f}/{\sigma}_{n}$ | 20.88 | 11.13 | 45.59 | 14.50 | 20.41 | 66.10 |

MAPE | 6.89 | 9.87 | 3.54 | 6.26 | 3.32 | 2.02 |

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**MDPI and ACS Style**

Gong, X.; Yabansu, Y.C.; Collins, P.C.; Kalidindi, S.R.
Evaluation of Ti–Mn Alloys for Additive Manufacturing Using High-Throughput Experimental Assays and Gaussian Process Regression. *Materials* **2020**, *13*, 4641.
https://doi.org/10.3390/ma13204641

**AMA Style**

Gong X, Yabansu YC, Collins PC, Kalidindi SR.
Evaluation of Ti–Mn Alloys for Additive Manufacturing Using High-Throughput Experimental Assays and Gaussian Process Regression. *Materials*. 2020; 13(20):4641.
https://doi.org/10.3390/ma13204641

**Chicago/Turabian Style**

Gong, Xinyi, Yuksel C. Yabansu, Peter C. Collins, and Surya R. Kalidindi.
2020. "Evaluation of Ti–Mn Alloys for Additive Manufacturing Using High-Throughput Experimental Assays and Gaussian Process Regression" *Materials* 13, no. 20: 4641.
https://doi.org/10.3390/ma13204641