**1. Introduction**

Hardness is a property that cannot be defined unequivocally. In fact, many other properties are measured in hardness tests. The tests can be classified depending on the material resistance observed during the testing. Static methods of hardness measurement consist of pushing the indenter into the tested material beyond the yield point to cause plastic strains. In such methods, hardness can be defined as the measure of the material strength to plastic strains arising due to the indenter pressing force.

During a traditional hardness measurement, the hardness testing machine indenter causes plastic strain to the surface domain where the measurement is performed. For some elements, such damage to the surface is unacceptable. For others, plastic deformation of the surface is not an operating problem but a diagnostic one: it is impossible to repeat the hardness test in exactly the same location under the same conditions. These obstacles can be overcome using so-called inverse problems of non-destructive testing, where mechanical properties are determined based on other physical quantities.

In the testing presented herein, this physical quantity is the Barkhausen noise (BN), which is created due to a time-variable impact of a magnetic field on a ferromagnetic. The variable magnetic field causes the stochastic displacement of domain walls, which involves a sudden change in magnetization detectable by the measuring sensor. More details about the theory of the Barkhausen noise can be found in several models [1–3].

The Barkhausen noise is used as a diagnostic signal to determine the material stress-and-strain state [4–7], the state of the microstructure [8], defects in surface treatment [9], and hardness [5,9–15].

Ortega-Labra et al. [4] presented a novel system for measuring stress applied to a pipe section using the rotational continuous magnetic Barkhausen noise (MBN). The presented results show that the shape and the amplitude of the MBN angular dependence change with an increase in applied stress. The outcome reveals that the applied stress value could be estimated using the amplitude of the angular dependence of the measured MBN signal. Additionally, the shape of the angular dependence is related to the applied stress direction. This suggests that using a proper procedure, the direction of the applied stress could also be estimated based on the MBN.

Liu et al. [5] presented the application of a multifunctional magnetic testing technique in the quantitative prediction of tensile stress, surface hardness, and the case depth in medium-carbon steel rods. Simultaneous measurements were carried out on the MBN signal, the tangential magnetic field, and the magnetic hysteresis curve in tensioned steel rods with different depths of the surface-hardened layer. A total of 16 feature parameters were extracted from four types of signal patterns as candidate inputs of the back-propagation neural network. Very promising results were obtained, where the prediction model error was less than 5.3% for tensile stress, 5% for the case depth, and 0.22% for surface hardness.

In [6], skewness is utilized as a feature for applied stress detection and compared with the root-mean-square (RMS) peak value and peak position, which are the most common features of the MBN signal. Different levels of compressive and tensile stresses are applied to En36 steel specimens to evaluate the new feature effectiveness. Its non-linear behavior and sensitivity are compared with the RMS peak value and peak position. The domain wall energy and the eddy current damping are discussed as the cause of the phenomenon.

In [9], He et al. presented a review of non-destructive testing methods for the metallic grinding burn detection. One of them makes use of measurements of the magnetic Barkhausen noise (MBN) and the magnetoelastic acoustic emission (MAE). It is found that the BN method itself produces only relative results that need to be compared with calibration blocks to define certain limits for rejection. Several aspects for further investigation are defined, such as the relationship between the BN signal multiparameters and the measured material properties (stress state, hardness), the impact factors of BN measurements including microstructural domain walls, and macroscopic operations, such as the influence of the holding force, the contact angle between the probe and the workpiece, and the probe moving speed.

Wilson et al. [10] studied P9 and T22 steel specimens subjected to different kinds of heat treatment used to modify the microstructure. Magnetic hysteresis loops and the Barkhausen noise were measured and compared to the Vickers hardness number. It is demonstrated that a rise in hardness involves a rise in coercivity, a change in the location of the maximum value of the Barkhausen noise envelope, and a drop in magnetic permeability.

Moorthy et al. [11] examined the impact of a post-hardening change in the specimen structure on the Barkhausen noise envelope on the carburized layer of steel EN36. Several frequencies of the excitation field were applied, and a frequency analysis of the measured signal was conducted in different ranges. It is proved that the envelope maximum value correlates with the hardness measured in the layer to the depth of 100 μm.

The aim of the studies presented in [12] was to use the Barkhausen noise to characterize hardness in steel AISI 4140 subjected to the Jominy test. Various types of time-frequency representations were used: the spectrogram, the Wigner–Ville distribution, the Capongram, the ARgram, the wavelet transform scalogram, and the Mellingram. It is proved that due to the Barkhausen noise non-stationary character, time-frequency representations are a rich source of information. A method is proposed of extracting information from these representations by creating an algorithm reducing the number of variables from 3 (time, frequency, amplitude) to 2 (time or frequency and amplitude).

In [13], O'Sullivan et al. measured magnetoacoustic emission (MAE) and the Barkhausen noise (BN) to determine the properties of AISI 430 steel specimens subjected to heat treatment to change their microstructure. Parameters such as the RMS value of the Barkhausen noise and the magnetoacoustic emission absolute value proposed by the authors were compared e.g., to the Vickers hardness number. It is shown that a rise in plastic strain causes a drop in the BN and MAE and that the parameters are linearly dependent on the Vickers hardness number.

Franco et al. [14] present relations between the Barkhausen noise parameters and Rockwell hardness using SAE 4140 and SAE 6150 steel specimens subjected to the Jominy hardenability test. The Barkhausen noise is described using the maximum value, the location, and the RMS value of the envelope. A linear relation is found between all parameters of the Barkhausen noise and Rockwell hardness.

In [15], many properties of the Barkhausen noise are analyzed, and a method is proposed of their selection and mutual complementation for the purposes of hardness evaluation. An algorithm is presented for the analysis of the signal main properties. Combined with the method of the analysis of the correlations between them, the algorithm is used to select the optimal properties. Multidimensional linear regression with selected properties is applied, and a statistical linear model of merging BN properties is built to forecast the material hardness.

This paper presents the results of testing aiming to develop diagnostic correlations between hardness and the Barkhausen noise total number of events *NoETOT* BN. A good correlation between the two quantities would enable periodic and multiple measurements of hardness in the same location, which cannot be done using traditional hardness measurement methods. Two cases of differences in the material hardness are considered. One is a change in hardness due to cold plastic strain and the other is the change caused by thermochemical treatment.

*NoETOT* is selected as the diagnostic signal property to be analyzed because the authors already obtained promising results from their previous preliminary research on hardness evaluation [16,17] and from their comprehensive studies on the assessment of the state of active stresses in steel elements [7]. The following quantities are analyzed in [7,16,17]: the BN RMS value of voltage (*URMS*) [18–20], the BN energy (*EBN*) [20,21], characteristic parameters related to the BN signal envelope [22,23], and the distribution of the BN total number of events (*NoETOT* BN) [24]. The best determination coefficients *R*<sup>2</sup> and the smallest standard deviations are obtained for the *NoETOT* BN. Another issue, not taken up in this paper, is the analysis of the intensity of changes in the features describing the Barkhausen noise that occur due to changes in hardness or in the stress-and-strain state. The signal feature that changes the most intensely is the one best suited to enable the development of a diagnostic correlation. The issue is related to digital processing of the signal on the one hand, and to analysis of the variability of the signal individual features on the other. Such analyses are presented for example in [25], where a comparison is made between changes in the Barkhausen noise profile skewness and the noise signal RMS value caused by active compressive and tensile stresses. Within certain stress ranges, the sensitivity of skewness is higher than that of the RMS value, which makes the former a better diagnostic signal.

Among others, the Barkhausen noise depends on the state of the material microstructure and on the material stress state. The impacts can be separated from each other assuming that the two factors are not changed at the same time. Changes in hardness due to strain hardening are accompanied by changes in internal (residual) stresses. Similarly, changes in hardness due to thermochemical treatment involve, apart from changes in the specimen microstructure, changes in the level of residual stresses. In the presented testing, the impact of internal stresses was not taken into consideration in the analysis of results. In consequence, the developed diagnostic correlations are less universal because they are related to the structural properties of specimens.

## **2. Measuring Apparatus**

The Barkhausen noise was measured using the MEB4-C system made by the Mag-Lab s.c. company (Gda ´nsk, Poland). This is an independent system of measuring, analyzing, and recording the Barkhausen noise (BN) voltage signals. The system diagram is shown in Figure 1. The measurements were performed using a surface measuring head with systems of the electromagnet, Barkhausen noise detection, and magnetization control. The head diagram is presented in Figure 2. The system enables

configuring settings in the range of the magnetizing current amplitudes, the rate of changes in the magnetizing current (the magnetizing current frequency), and the measured signal amplification.

**Figure 1.** Schematic diagram of the MEB4-C system.

**Figure 2.** Surface measuring probe.
