# Applying Machine Learning to the Phenomenological Flow Stress Modeling of TNM-B1

^{1}

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## Abstract

**:**

## 1. Introduction

^{−1}. The same amount of experimental data was used to fit the PM, and to train the MLM and the HM. The models were analyzed and compared based on the accuracy of their predictions, efficiency in terms of development and computing time, and their ability to produce accurate predictions on interpolated and extrapolated inputs between and outside the experimental data.

## 2. Materials and Methods

#### 2.1. Material and Microscopy

#### 2.2. Compression Tests

_{3}N

_{4}-punches in the dilatometer. The tests were conducted in an argon atmosphere to protect from oxidation. The cylindrical samples were isothermally deformed using the following parameters; the constant strain rates 0.0013 s

^{−1}, 0.005 s

^{−1}, 0.01 s

^{−1}and 0.05 s

^{−1}, and the constant temperatures 1150 °C, 1175 °C and 1200 °C. The strain rate of 0.0013 s

^{−1}was used as this was the lowest strain rate allowed by the test equipment. The sample and molybdenum plates were first heated by induction using a heating rate of 10 K/s, and held at the testing temperature for 3 min to obtain a homogeneous microstructure and quasi-isothermal conditions prior to deformation. The samples were deformed from 0 to around 0.8 strain. Finally, the samples were cooled at a rate of 200 K/s using argon gas. The experiments were duplicated at least two times to ensure repeatability in the resulting flow curves.

#### 2.3. Phenomenological Model Fundamentals

#### 2.4. Machine Learning Model Fundamentals

#### 2.5. Hybrid Model Fundamentals

## 3. Results and Discussion

#### 3.1. Microstructure Overview and Deformation Behavior

_{3}Al (gray contrast) and ${\mathsf{\beta}}_{0}$-TiAl (bright contrast), all of which have different individual deformation behaviors [21,22]. The $\mathsf{\alpha}$ phase has a hexagonal close-packed (HCP) $\mathrm{D}{0}_{19}$ structure, with the highest strength and lowest formability in the TNM-B1 system. The $\mathsf{\gamma}$ phase has a face-centered cubic (FCC) $\mathrm{L}{1}_{0}$ crystal structure, with higher strength than the $\mathsf{\beta}$ phase, but higher deformability than the $\mathsf{\alpha}$ phase. The $\mathsf{\beta}$ phase has a body-centered cubic (BCC) A2 structure, and the ${\mathsf{\beta}}_{0}$ phase has a body-centered cubic (BCC) B2 structure. At high temperatures, it is the softest phase of the TNM-B1 alloy [23]. Therefore, the $\mathsf{\beta}$ phase carries a much higher strain during hot deformation than the nominal value, leading to highly misoriented subgrains and dynamic recrystallization (DRX). In general, the macroscopic deformation behavior of TNM-B1 is strongly dependant on DRX, dynamic recovery (DRV) and the orientation of the lamellar colonies.

^{−1}.

#### 3.2. Phenomenological Model Results

^{−1}. However, the PM was not able to capture the characteristic sharp peak for some conditions, which can be observed especially in the flow curves obtained for 0.01 s

^{−1}.

^{−1}. The computing time for generating the flow curves was 17.9 s. The error histogram for the PM is displayed in Figure 6, which shows the error distribution in MPa.

#### 3.3. Machine Learning Model Results

^{−1}, due to the MLM predicting a slightly steeper rise in stress than the experimental curve. The computing time for running the data through the finished ANN to predict the flow stress was around 0.06 s. The results reveal that the MLM was able to capture the sharp peak stress of the TNM-B1 flow curve, while avoiding overfitting to the noise in the data. In addition, the MLM is observed to adapt to the different flow curve shapes rather than outputting a generalized shape for every input condition. Figure 9 displays the error histogram for the MLM, with the errors given in MPa. By comparing this to the error histogram of the PM (Figure 6), it is apparent that the MLM predicts more values closer to zero error than the PM.

#### 3.4. Hybrid Model Results

^{−5}(equivalent to an average strain error of around 0.008). The training times were in the order of 0.1 to 0.2 seconds. The predicted positions of the characteristic points were then used as input for the phenomenological model, which produced the final flow curves.

^{−1}. The computing time for running the data through the finished ANNs to predict the characteristic stresses and strains was in the order of 0.05 s, and the computing time for generating the flow curves using the phenomenological model was 17.9 s. Figure 11 displays the error histogram for the HM, with the errors given in MPa.

#### 3.5. Extrapolation and Interpolation

^{−1}. In (b), the strain rate was varied between 0.0005 and 0.1 s

^{−1}with 10 increments between each power, at a constant temperature of 1150 °C. The line plots on the surface illustrate the flow curves predicted at the experimental conditions used for training the models. The line plots at the edges are experimental flow curves obtained to evaluate the ability of the models to make extrapolated predictions, this data was not used for training the ANNs or as input for the PM. As 0.0013 s

^{−1}was the lowest strain rate allowed by the experimental setup (discussed in Section 2.2), flow stress data for 0.0005 s

^{−1}could not be obtained. Figure 12 displays the ability of the PM to give stress predictions on interpolated and extrapolated inputs. The PM appears to produce conservative predictions at both lower temperatures and higher strain rates. The PM produced a smooth surface of predictions for the interpolated and extrapolated inputs.

^{−1}is around 50 MPa higher than the experimental peak of 253.6 MPa. However, the surfaces of predicted stress values as a whole corresponds well with what can be expected from real experimental data.

^{−1}(b). While the networks produced very accurate predictions for the input data they were trained on, around half produced unrealistic predictions for inputs outside and between the training inputs. This suggests that although the MLM can give good predictions for the conditions used for training, its ability to extrapolate and interpolate varies significantly between training sessions and has to be verified. The probability that the MLM will learn a mapping function that produces realistic predictions for extrapolated and interpolated inputs is likely to increase with an increasing amount of training data.

## 4. Conclusions

^{−1}, to extract the experimental flow stress data. The three models were analyzed and compared based on the accuracy of their predictions, efficiency and their ability to produce accurate predictions on interpolated and extrapolated input data. The main conclusions drawn from this investigation are:

- Using the same amount of experimental data, the MLM made more accurate predictions than the PM. In addition, the MLM was more efficient in terms of both development and computing time. The MLM was also better able to capture distinct features of the TNM-B1 flow curves, such as the characteristic sharp peak stress. These findings indicate that the equations modeling the course of the flow curve in the PM can be replaced by a pure machine learning approach.
- The MLM was able to produce predictions for extrapolated and interpolated inputs that gave a smooth and realistic fit, and which was comparable to what can be expected from experimental data. However, many training iterations were required due to the random elements of generating neural networks.
- The HM was able to produce a better fit compared to the PM for extrapolated and interpolated inputs. This indicates that the traditional PM can be improved by replacing the equations used to model the dependence on the characteristic points with machine learning.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

HIP | Hot Isostatic Pressing |

TNM | TiAl with Nb and Mo as the main alloying elements |

ANN | Articifial Neural Network |

DRX | Dynamic Recrystallization |

DRV | Dynamic Recovery |

VAR | Vacuum Arc Remelting |

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**Figure 1.**Flow curve illustrating the characteristic points; peak stress/strain (${\mathsf{\sigma}}_{\mathrm{p}}/{\epsilon}_{\mathrm{p}}$), critical stress/strain (${\mathsf{\sigma}}_{\mathrm{cr}}/{\epsilon}_{\mathrm{cr}}$) and steady state stress/strain (${\mathsf{\sigma}}_{\mathrm{ss}}/{\epsilon}_{\mathrm{ss}}$). The thin line represents the modelled flow curve due to strain hardening (SH) and dynamic recovery (DRV). The thick line represents the modelled flow curve with dynamic recrystallization (DRX) kinetics included.

**Figure 3.**Schematic of the hybrid model (HM). ANNs was trained to predict the characteristic points (critical, peak, steady state stress and strain) as functions of temperature (T) and strain rate ($\dot{\epsilon}$). These predictions are used as input for the phenomenological model, which outputs the final flow curves.

**Figure 4.**SEM image of undeformed HIPed TNM-B1 and hot formed TNM-B1 at 1150 °C and 0.005 s

^{−1}, consisting of the ordered phases $\mathsf{\gamma}$-TiAl (dark contrast), ${\mathsf{\alpha}}_{2}$-Ti

_{3}Al (gray contrast) and $\mathsf{\beta}$/${\mathsf{\beta}}_{0}$-TiAl (bright contrast). The loading direction is along the y-axis.

**Figure 5.**Experimental flow curves and the PM predictions for TNM-B1 at the temperatures 1150, 1175 and 1200 °C and the strain rates 0.0013, 0.005, 0.01 and 0.05 s

^{−1}.

**Figure 6.**Error histogram for the PM with 30 bins and a bin width of around 2.2 MPa. The results displayed are the difference between the experimental and predicted stress values.

**Figure 8.**Experimental flow curves and the MLM predictions for TNM-B1 at the temperatures 1150, 1175 and 1200 °C and the strain rates 0.0013, 0.005, 0.01 and 0.05 s

^{−1}.

**Figure 9.**Error histogram for the MLM with 30 bins and a bin width of around 1.8 MPa. The results displayed are the difference between the experimental and predicted stress values.

**Figure 10.**Experimental flow curves and the HM predictions for TNM-B1 at the temperatures 1150, 1175 and 1200 °C and the strain rates 0.0013, 0.005, 0.01 and 0.05 s

^{−1}.

**Figure 11.**Error histogram for the HM with 30 bins and a bin width of around 2.2 MPa. The results displayed are the difference between the experimental and predicted stress values.

**Figure 12.**Surface plots of stress values predicted by the PM for interpolated and extrapolated inputs; (

**a**) temperatures between 1125 and 1225 °C with an increment of 5 °C; (

**b**) strain rates between 0.0005 and 0.1 s

^{−1}with 10 increments between each power.

**Figure 13.**Surface plots of stress values predicted by the MLM for interpolated and extrapolated inputs; (

**a**) temperatures between 1125 and 1225 °C with an increment of 5 °C, (

**b**) strain rates between 0.0005 and 0.1 s

^{−1}with 10 increments between each power.

**Figure 14.**Surface plots of stress values predicted by the HM for interpolated and extrapolated inputs; (

**a**) temperatures between 1125 and 1225 °C with an increment of 5 °C, (

**b**) strain rates between 0.0005 and 0.1 s

^{−1}with 10 increments between each power.

**Figure 15.**Cross section of interpolated and extrapolated surface plots from 20 tests at 0.5 strain.

Al | Nb | Mo | B | O | Fe | Ni | C | |
---|---|---|---|---|---|---|---|---|

at% | 43.7 | 4 | 1 | 0.1 | 0.161 | 0.027 | 0.008 | 0.038 |

wt% | 28.65 | 9.15 | 2.36 | 0.026 | 0.063 | 0.037 | 0.012 | 0.011 |

Zener-Hollomon parameter | $\mathrm{Z}=\dot{\epsilon}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}(\frac{{\mathrm{Q}}_{\mathrm{w}}}{\mathrm{R}\mathrm{T}})$ |

Strain hardening | $\mathsf{\sigma}(\epsilon )={\mathsf{\sigma}}_{\mathrm{p}}{\left(\right)}^{{\displaystyle \frac{\epsilon}{{\epsilon}_{\mathrm{p}}}}}$ |

Critical strain | ${\epsilon}_{\mathrm{cr}}=\mathsf{\alpha}{\epsilon}_{\mathrm{p}}$ |

Peak strain | ${\epsilon}_{\mathrm{p}}={\mathrm{a}}_{1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{d}}_{0}^{{\mathrm{a}}_{2}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{Z}}^{{\mathrm{a}}_{3}}$ |

steady state strain | ${\epsilon}_{\mathrm{ss}}={\mathrm{e}}_{1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\epsilon}_{\mathrm{m}}+{\mathrm{e}}_{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{d}}_{0}^{{\mathrm{e}}_{3}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{Z}}^{{\mathrm{e}}_{4}}$ |

Peak stress | $\mathrm{sinh}({\mathrm{f}}_{3}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathsf{\sigma}}_{\mathrm{p}})={\mathrm{f}}_{1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{Z}}^{{\mathrm{f}}_{2}}$ |

steady state stress | $\mathrm{sinh}({\mathrm{h}}_{3}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathsf{\sigma}}_{\mathrm{p}})={\mathrm{h}}_{1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{Z}}^{{\mathrm{h}}_{2}}$ |

DRX grain size | ${\mathrm{d}}_{\mathrm{DRX}}(\mathsf{\gamma},\mathsf{\beta})={\mathrm{b}}_{1}(\mathsf{\gamma},\mathsf{\beta})\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{Z}}^{{\mathrm{b}}_{2}(\mathsf{\gamma},\mathsf{\beta})}$ |

DRX kinetics | ${\mathrm{X}}_{\mathrm{DRX}}(\mathsf{\gamma},\mathsf{\beta})=1-\mathrm{exp}(\mathrm{k}(\mathsf{\gamma},\mathsf{\beta}){\left({\displaystyle \frac{\epsilon -{\epsilon}_{\mathrm{cr}}}{{\epsilon}_{\mathrm{ss}}-{\epsilon}_{\mathrm{cr}}}}\right)}^{\mathrm{q}(\mathsf{\gamma},\mathsf{\beta})}$ |

Flow stress | ${\mathsf{\sigma}}_{\mathrm{y}}=\{\begin{array}{cc}{\mathsf{\sigma}}_{0}& \mathrm{if}\phantom{\rule{4pt}{0ex}}\epsilon <{\epsilon}_{\mathrm{cr}}\\ \left(\right)open="("\; close=")">1-({\mathrm{X}}_{\mathsf{\gamma}}+{\mathrm{X}}_{\mathsf{\beta}}){\mathsf{\sigma}}_{0}+({\mathrm{X}}_{\mathsf{\gamma}}+{\mathrm{X}}_{\mathsf{\beta}}){\mathsf{\sigma}}_{1}\\ \mathrm{if}\phantom{\rule{4pt}{0ex}}\epsilon {\epsilon}_{\mathrm{cr}}\end{array}$ |

**Table 3.**Dataset used for training the ANNs to predict the entire course of the flow curves for TNM-B1, with the first three rows and the last row shown. Training inputs; temperature, strain rate and strain. Target output; flow stress.

Row | Temp. [°C] | Strain rate [s^{−1}] | Strain [-] | Stress [MPa] |
---|---|---|---|---|

1 | 1150 | 0.0013 | 0 | 0 |

2 | 1150 | 0.0013 | 0.000357 | 0.954 |

3 | 1150 | 0.0013 | 0.000439 | 1.067 |

... | ... | ... | ... | ... |

103,296 | 1200 | 0.05 | 0.771 | 74.739 |

**Table 4.**Performance of a selection of tested ANN architectures showing the number of hidden layers (HL), number of neurons (N), average achieved MSE (mean squared error), equivalent error in MPa and training time. 10 tests were conducted for each architecture.

HL | N | MSE | STD | Eq. Stress [MPa] | Training Time [s] | STD |
---|---|---|---|---|---|---|

1 | 5 | 172.3 | 43.3 | 13.0 | 47.0 | 20.45 |

1 | 10 | 98.0 | 70.8 | 7.5 | 228.6 | 98.0 |

1 | 20 | 14.1 | 1.5 | 3.8 | 254.6 | 137.6 |

2 | 10, 3 | 10.4 | 1.9 | 3.2 | 210.0 | 141.9 |

2 | 15, 3 | 11.6 | 1.4 | 3.4 | 383.4 | 135.6 |

2 | 15, 5 | 10.6 | 1.5 | 3.3 | 309.8 | 139.0 |

3 | 15, 5, 3 | 14.1 | 3.3 | 3.7 | 260.4 | 72.7 |

4 | 5, 5, 5, 5 | 12.1 | 0.8 | 3.5 | 400.8 | 161.2 |

**Table 5.**Dataset used for training the ANNs to predict the position of the characteristic strains (peak, critical and steady state strain), with the first three rows and the last row shown. Training inputs; temperature, strain rate and parameter type. Target output; characteristic strain.

Row | Temp. [°C] | Strain Rate [s^{−1}] | Type | Ch. Strain [-] |
---|---|---|---|---|

1 | 1150 | 0.0013 | Peak strain | 0.0328 |

2 | 1150 | 0.0013 | Peak strain | 0.0301 |

3 | 1150 | 0.005 | Peak strain | 0.0344 |

... | ... | ... | ... | ... |

72 | 1200 | 0.05 | steady state strain | 0.7708 |

**Table 6.**Dataset used for training the ANNs to predict the position of the characteristic stresses (peak, critical and steady state stress), with the first three rows and the last row shown. Training inputs; temperature, strain rate and parameter type. Target output; characteristic stress.

Row | Temp. [°C] | Strain Rate [s^{−1}] | Type | Ch. Stress [MPa] |
---|---|---|---|---|

1 | 1150 | 0.0013 | Peak stress | 66.555 |

2 | 1150 | 0.0013 | Peak stress | 63.494 |

3 | 1150 | 0.005 | Peak stress | 114.557 |

... | ... | ... | ... | ... |

72 | 1200 | 0.05 | steady state stress | 74.612 |

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**MDPI and ACS Style**

Stendal, J.A.; Bambach, M.; Eisentraut, M.; Sizova, I.; Weiß, S.
Applying Machine Learning to the Phenomenological Flow Stress Modeling of TNM-B1. *Metals* **2019**, *9*, 220.
https://doi.org/10.3390/met9020220

**AMA Style**

Stendal JA, Bambach M, Eisentraut M, Sizova I, Weiß S.
Applying Machine Learning to the Phenomenological Flow Stress Modeling of TNM-B1. *Metals*. 2019; 9(2):220.
https://doi.org/10.3390/met9020220

**Chicago/Turabian Style**

Stendal, Johan A., Markus Bambach, Mark Eisentraut, Irina Sizova, and Sabine Weiß.
2019. "Applying Machine Learning to the Phenomenological Flow Stress Modeling of TNM-B1" *Metals* 9, no. 2: 220.
https://doi.org/10.3390/met9020220