# Predictive Modeling of Induction-Hardened Depth Based on the Barkhausen Noise Signal

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}and f

_{2}are the frequency limits of the filtered frequency range and x as the distance inside of the material. Applying typical settings for a hardened and tempered steel, conductivity of 10

^{6}Ω

^{−1}·m

^{−1}, µ

_{r}= 200, and frequency range, f

_{1}− f

_{2}, 70–200 kHz, equals to a depth of analysis of 0.1 mm. For a non-hardened, mild steel, the depth of analysis will decrease to approximately 40 µm. Harder steel provides a higher analyzing depth and softer steel provides a lower analyzing depth at the same frequency range.

## 2. Methodology

#### 2.1. Material

#### 2.2. Heat Treatment

#### 2.3. Sample Characterization

- MicroScan software from Stresstech of the conventional BN parameters such as: root mean square value (RMS), peak position, and full width half maximum (FWHM);
- PCCaseDepth software from Stresstech of the magnetizing voltage sweep slope (MVSS), which measured the ratio from the maximum slope of the sweeps of 200 Hz and 20 Hz. In total, the average slope ratios of four sweep measurements were used.

^{2}ψ, in modified χ mode, was used with 5 tilt angles in the interval of ±40°.

## 3. Results

#### 3.1. Non-Destructive Testing

^{2}values of linear trendlines, if the 7 mm sample was considered as an outlier. The FWHM values ranged in the interval 3–7°, showing a decrease with a higher HD. The correlation was s-shaped for set A while set B showed a great variation.

#### 3.2. Destructive Evaluation

## 4. Analysis and Predictive Modeling

- (i)
- All NDC predictors in a model using
**principal components**(PC), assuming laboratory set-up independence; - (ii)
- All NDC predictor in a model using
**principal component and principal component interaction**with Tests sets, assuming laboratory set-up dependence; - (iii)
- Modeling based on minimal set of NDC using BN predictors only, to reduce the monitoring demand.

#### 4.1. Prinicple Component Analysis (PCA) of Predictor Correlation

#### 4.2. Ordinary Multi-Parameter Linear Regression of Hardening Depth (HD) Using Principal Components—Model i

#### 4.3. Modeling Including Test Sets and Principal Component Interaction—Model ii

#### 4.4. Residual Analysis for Model Comparison

#### 4.5. Modeling Including Test Set and Principal Componets Based on BN-Characteristics—Model iii

^{2}= 0.92). The models predicting hardening depth based in BN characteristics scaled for the different test sets are shown in Equations (10) and (11):

## 5. Discussion

## 6. Conclusions

- The different Barkhausen noise parameters, RMS/FWHM and MVSS(200 Hz/20 Hz), correlate well to hardening depths down to 4.5 mm;
- The surface hardness and hardening depth correlate, which explains the Barkhausen noise signal sensitivity to the several-millimetres-deep correlation to the hardening depth;
- It is possible to predict the hardening depth using principal components based on all or a reduced set of BN and RS characteristics;
- The test set dependence indicate that the BN and RS characteristics capture subtle differences of the hardening result hidden in the microstructure whether it depends on variations of the base material or slightly different heating and cooling efficiency in the elaborative set-up;
- The surface hardness and hardening depth are correlated, which explains the correlation between the hardening depth and the Barkhausen noise signal.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Dubois, M. Fiset, Evaluation of case depth on steel by Barkhausen noise measurement. Mater. Sci. Technol.
**1995**, 11, 264–267. [Google Scholar] [CrossRef] - Moorthy, V.; Shaw, B.A. Magnetic Barkhausen emission measurements for evaluation of material properties in gears. Nondestruct. Test. Eval.
**2008**, 23, 317–347. [Google Scholar] [CrossRef] - Saquet, O.; Chicois, J.; Vincet, A. Barkhausen noise from plain carbon steels: Analysis of the influence of microstructure. Mater. Sci. Eng. A
**1999**, 269, 73–82. [Google Scholar] [CrossRef] - Blaow, M.; Evans, J.; Shaw, B. Magnetic Barkhausen noise: The influence of microstructure and deformation in bending. Acta Mater.
**2005**, 532, 279–387. [Google Scholar] [CrossRef] - Franco, F.; González, M.; de Campos, M.; Padovese, L. Relation Between Magnetic Barkhausen Noise and Hardness for Jominy Quench Tests in SAE 4140 and 6150 Steels, 2012. J. Nondestruct. Eval.
**2013**, 32, 93–103. [Google Scholar] [CrossRef] [Green Version] - Jiles, D. Dynamics of domain magnetization and the Barkhausen effect. Czechoslov. J. Phys.
**2000**, 50, 893–924. [Google Scholar] [CrossRef] - Tomkowski, R.; Lundin, P.; Holmberg, J.; Jonsson, S.; Hammersberg, P.; Kristoffersen, H.; Olavisson, J.; Archenit, A. The Barkhausen Noise Measurements, Good Practice Guide; KTH Royal Institute of Technology: Stockholm, Sweden, 2018; ISBN 978-91-7729-978-3. [Google Scholar]
- Kittel, C.; Galt, J.; Seitz, F.; Turnbull, D. Ferromagnetic Domain Theory. Solid State Phys.
**1956**, 3, 437–564. [Google Scholar] - Tam, P.; Hammersberg, P.; Persson, G.; Olavison, J. Case depth evaluation of induction-hardened camshaft by using magnetic Barkhausen noise (MBN) method. Nondestruct. Test. Eval.
**2021**, 36, 494–514. [Google Scholar] [CrossRef] - Santa-aho, S.; Vippola, M.; Sorsa, A.; Leivisk, K.; Lindgren, M.; Lepistö, T. Utilization of Barkhausen noise magnetizating sweeps for case-depth detection from hardened steel. NDT E Int.
**2012**, 52, 95–102. [Google Scholar] [CrossRef] - Santa-aho, S.; Hakanen, M.; Sorsa, A.; Vippola, M.; Leiviskä, K.; Lepistö, T. Case Depth Verification of Hardened Samples with Barkhausen Noise Sweeps. AIP Conf. Proc.
**2014**, 1581, 1307–1314. [Google Scholar] [CrossRef] - Sorsa, A.; Santa-aho, S.; Aylott, C.; Shaw, B.A.; Vippola, M.; Leiviskä, K. Case Depth Prediction of Nitrided Samples with Barkhausen Noise Measurement. Metals
**2019**, 9, 325. [Google Scholar] [CrossRef] - Tiitto, S.; Säynäjäkangas, S. Spectral Damping in Barkhausen Noise. IEEE Trans. Magn.
**1975**, 11, 1666–1672. [Google Scholar] [CrossRef] - Jiles, D. Introduction to Magnetism and Magnetic Materials; Chapman and Hall: London, UK, 1972. [Google Scholar]
- Saquet, O.; Tapuleasa, D.; Chicoise, J. Use of Barkhausen noise for determination of surface hardness depth. Nondestruct. Test. Eval.
**1998**, 14, 277–292. [Google Scholar] [CrossRef] - Swallem, M.; Blaow, M.; Adarrat, A.M. Optimizing detection parameters of magnetic Barkhausen noise using heat affected zone in welded ship steel plate. Adv. Mater. Res.
**2015**, 1119, 849–856. [Google Scholar] [CrossRef] - Vaidyanathan, S.; Moorthy, V.; Jayakumar, T.; Raj, B. Evaluation of induction hardened case depth through microstructural characterisation using magnetic Barkhausen emission technique. Mater. Sci. Technol.
**2000**, 16, 202–208. [Google Scholar] [CrossRef] - Augustis, V.; Ramanauska, R.; Čiuplys, A.; Vilys, J.; Čiuplys, V. Determination of Metal Surface Hardened Layer Depth Using Magnetic Barkhausen Noise. Test. Anal. Mater.
**2006**, 12, 84–87. [Google Scholar] - Wheeler, D.J. Twenty Things You Need to Know; SPC Press: Knoxville, TN, USA, 2009; Chapter 19. [Google Scholar]
- SAE. Methods of Measuring Case Depth—J423; SAE International: Warrendale, PA, USA, 1998. [Google Scholar]
- Bisgaard, S.; Kulachi, M. Quality Quandaries: The Application of Principal Component Analysis for Process Monitoring. Qual. Eng.
**2006**, 18, 95–103. [Google Scholar] [CrossRef] - SAS Institute. JMP
^{®}12 Design of Experiments Guide; SAS Institute: Cary, NC, USA, 2015; Chapter 3; ISBN 978-1-62959-443-9. [Google Scholar]

**Figure 6.**Hardness profiles, Vickers (HV1), for (

**A**) test set A and (

**B**) test set B, induction hardened samples with different scanning speed (S), power (P), and quenchant concentration (Q).

**Figure 7.**Surface hardness, Knoop (HK0.2), versus the hardness depth of the induction hardened samples from (

**A**) test set A and (

**B**) test set B.

**Figure 8.**Micrographs of selected samples, induction heat-treated with different scanning speed (S) and power (P), with gradually increasing hardening depth (HD).

**Figure 9.**Micrographs at different depths (EHD: effective hardness depth, THD: total hardness depth) for left: sample with scanning speed (S) of: 7 mm/s and right: sample S: 3 mm/s.

**Figure 10.**Varying correlation between actual hardness depths at 400 HV versus different predictors, surface hardness, RS, and BN monitoring characteristics. Possible outliers marked with (*).

**Figure 11.**Heavy pairwise correlation between the response and predictors, both within and between residual stress (RS) and Barkhausen noise (BN) characteristics.

**Figure 13.**Score plot of all observations projected on the principal component plane: Test set A (•) and test set B (x). Outliers, identified using default T2-statistics in JMP, are marked with stars (*).

**Figure 14.**(

**A**) Actual vs. predicted plot for the model (i), based on principal components only and (

**B**) stratified on test sets.

**Figure 16.**The residuals, stratified on test sets (red for set (

**A**) and blue for set (

**B**)), from (

**A**) modeling hardening depths using principal components only (model i) and (

**B**) the improved model (ii).

Sample Set | Scanning Speed [mm/s] | Power [% of Full Power] | Full Power [kW] | Quenchant Concentration [%] |
---|---|---|---|---|

A | 2–8 | 90/100 | 50 | 5 |

B | 2.5–12 | 42–55 | 150 | 5, 11 |

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**MDPI and ACS Style**

Holmberg, J.; Hammersberg, P.; Lundin, P.; Olavison, J.
Predictive Modeling of Induction-Hardened Depth Based on the Barkhausen Noise Signal. *Micromachines* **2023**, *14*, 97.
https://doi.org/10.3390/mi14010097

**AMA Style**

Holmberg J, Hammersberg P, Lundin P, Olavison J.
Predictive Modeling of Induction-Hardened Depth Based on the Barkhausen Noise Signal. *Micromachines*. 2023; 14(1):97.
https://doi.org/10.3390/mi14010097

**Chicago/Turabian Style**

Holmberg, Jonas, Peter Hammersberg, Per Lundin, and Jari Olavison.
2023. "Predictive Modeling of Induction-Hardened Depth Based on the Barkhausen Noise Signal" *Micromachines* 14, no. 1: 97.
https://doi.org/10.3390/mi14010097