**1. Introduction**

Steel is susceptible to the harmful effects of certain external environmental factors. For this reason, it is necessary to subject steel products to examination at the stage of both production and operation. If the internal and external structure of the object must remain intact, nondestructive testing (NDT) is performed. Detection of small inhomogeneities in the material allows us to observe degradation at an early stage, reducing the possibility of a catastrophic failure and alternative repair costs. The good electrical conductivity and high permeability of steel create possibilities to detect discontinuities in its structure using electromagnetic methods of NDT. The electromagnetic methods have high sensitivity, so apart from detecting a defect, it is also possible to pinpoint its location and assess its dimensions. In the case of steel-made sheets rolled in the direction opposite to the grain orientation, magnetic anisotropy is a particular obstacle during an examination. In addition, anisotropy can also be induced by stress. This results in the need for the inspection to be carried out in two orthogonal directions of the material.

Ferromagnetic materials can be tested using several electromagnetic methods of NDT:

the magnetic flux leakage method is based on observation of the magnetic flux distribution over the material surface [1]. The primary magnetic field source causes a magnetic flux in the material. A barrier to the secondary flux is any inhomogeneity in the material structure that has a significant reluctance value [2]. The flux leakage method allows us to assess the tested object's surface and subsurface inhomogeneity [3]. The main advantages are high sensitivity, easiness of signal acquisition, and the possibility of automation [2,4]. However, this method also has some disadvantages, including sensitivity to material impurities and the need to magnetize the object [5]; the magnetic particle inspection method allows for the detection of both surface and subsurface heterogeneities [6]. First, the sample is exposed to an external magnetic field, whereby magnetic powder particles can be placed on the outer surface of the sample in two ways: during the magnetization or after switching the magnetic field source off. The magnetic flux dispersing on the inhomogeneities appears on the

**Citation:** Chady, T.; Łukaszuk, R. Examining Ferromagnetic Materials Subjected to a Static Stress Load Using the Magnetic Method. *Materials* **2021**, *14*, 3455. https:// doi.org/10.3390/ma14133455

Academic Editor: Giovanni Bruno

Received: 19 May 2021 Accepted: 15 June 2021 Published: 22 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

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■ material's surface and changes the distribution of the particles [7]. The resulting image contains foci of particles that indicate the material heterogeneities [8]. Instead of magnetic powder, a suspension liquid can also be used to enhance the inspection sensitivity. Nowadays, apart from traditional indicators such as magnetic powders and suspension liquids, GMR or Hall sensors are also applicable to the inspection. The magnetic particle method is a quick, inexpensive, and relatively uncomplicated inspection method that gives immediate indications of surface and near-surface defects.


This work presents an examination of steel-made samples subjected to a static tensile stress load. The raw data acquired during the observation of the local hysteresis loops in two perpendicular directions were normalized, visualized, and analyzed using an approach based on statistics.

### **2. Materials and Methods**

The subject of the examination consisted of specimens made of SS400 (JIS3101 standard) steel with an applied tensile stress. SS400 is one of the most commonly used hot-rolled general structural carbon and low-alloy steels designated for such structures as bridges, ships, and rolling stocks.

The mechanical properties of SS400 are as follows [28]: yield strength, 350 MPa; elastic modulus, 209 GPa; Poisson's ratio, 0.29; and chemical composition (wt), C—0.148%, Mn—0.458%, Si—0.213%, S—0.018%, P—0.012%, and Fe—bal.

The shape and dimensions of the SS400 specimens used in the experiments are shown in Figure 1. They are different from the standard tensile test specimens because a sufficient area for measurements has to be provided for the relatively large transducer. The samples were manufactured using a water jet cutting machine to avoid sharp edges, eliminate the heat-affected zone, and minimize introduced stress.

**Figure 1.** Shape and dimensions of the samples used in the experiments.

Before the experiment, each specimen (except the reference one, named S00) was subjected to a static tensile stress load at ambient temperature. The applied stress in the longitudinal direction (*x*-axis) was coincident with the rolling direction of the specimen.

Before the inspection, five samples (named S03, S04, S05, S06, and S07) were prepared and loaded using a standard Instron system. For each sample, the tensile stress test was stopped for different strain values (Figure 2). The residual strain ε was between 0.7% and 14%, while the maximum tensile stress was 389 MPa. Samples S04 and S05 were stress-loaded, respectively, up to the yield point (strain ε = 2%) and over it (strain ε = 2.4%). For illustrative purposes, all samples are marked on a single stress–strain curve, which is shown in Figure 2.

**Figure 2.** The stress–strain curve for the examined SS400 specimens.

Protection of the top specimen's surface from mechanical damage and the transducer from rupture of the measurement windings was provided by a polytetrafluoroethylene (PTFE) tape with a thickness of 0.2 mm. The samples were placed below the transducer one by one, attached to the scanning system's head, and the transducer was aligned with the *x*-axis direction (Figure 1). During the measurements, the scanning system's head was moved to the upper left corner of the area to be examined. The transducer moved along the *x*-axis at a 200 mm distance and stopped for 400 ms every 0.5 mm to read and record signal values. The procedure was repeated by analogy 25 times for other transducer positions (every 1 mm) varying along the *y*-axis. In the subsequent part of the inspection, the transducer was aligned with the *y*-axis direction (Figure 1) by rotating it 90 degrees clockwise. The measurement procedure with the rotated transducer was analogous to the measurements performed previously. The signal values were stored on a computer for further processing.

The main objective was to observe changes in the hysteresis loop caused by the stress applied to the samples before the inspection. The magnetization during the measurements was carried out twice on two perpendicular axes (one axis was parallel to the rolling direction).

Examination of the samples was carried out using a transducer. The transducer consists of a support plate, an auxiliary support, and three coils. The stillage carries a u-shaped ferrite core with two pick-up coils, while the excitation coil is attached to the auxiliary support (Figures 3 and 4). Selected transducer parameters are shown in Table 1.

**Figure 3.** Transducer model.

**Figure 4.** Photo of the transducer.


**Table 1.** Transducer parameters.

The transducer works in the system shown in Figure 5. The system consists of a scanning device, a signal generator, A/D signal converters, amplifiers, the transducer, and a control computer. Parameters of the requested excitation signal (amplitude and frequency *f* E = 4.4 kHz) are sent to the signal generator, which provides a relevant voltage signal to the power amplifier. The excitation frequency was selected considering the influence of noise, the level of induced voltages in the pick-up coils, and the results of preliminary experiments carried out with the SS400 sample. Subsequently, the boosted signal supplies an excitaon coil of the transducer. This coil induces a primary magnetic field, which flows through the transducer's ferromagnetic core and two pick-up coils (*B* and *H*) and penetrates the tested material. Then, the voltage signal from a field-sensing pick-up coil (*H*) reaches the instrumentation amplifier and, after being modified by a second-order Butterworth low-pass filter with a cutoff frequency of 50 kHz, enters an analog-to-digital converter. A signal from a flux-sensing pick-up coil (*B*) is passed directly to the second-order Butterworth low-pass filter with a cutoff frequency of 50 kHz and digitalized. Eventually, both voltage signal values are saved in the computer's memory for further analysis. Detailed information about the measurement system is provided in Table 1.

**Figure 5.** A block scheme of the measuring system.

**FRQYHUWHU**

The raw data collected from the two-dimensional scanning of the samples were processed using dedicated software. First, the data were normalized and corrected by removing distortions. Subsequently, the values of the hysteresis loop parameters were calculated for both axes, and graphs were plotted showing the relative change in a given parameter as a function of the coordinates of the point from the measurement area (Figure 1).

The following calculations were done to assess the condition of the examined samples. First, every single magnetic loop parameter in two perpendicular magnetization directions (1) was calculated.

$$
\Delta k\_{A\text{max}}(\mathbf{x}, \mathbf{y}) = k\_{A\text{max}}(\mathbf{x}, \mathbf{y}) - k\_{A\text{max}}(\mathbf{x}\_{0}, \mathbf{y}\_{0}), \tag{1}
$$

where: *k*—the chosen magnetic loop parameter, *A*—the selected magnetization direction, *x*, *y*—the measurement area point's coordinates, *kAmax*(*<sup>x</sup>*, *y*)—the maximum values of the chosen magnetic loop parameter, and *kAmax*(*<sup>x</sup>*0, *y*0)—a mean value calculated from measurements achieved for points located within a distance of 2 mm from a starting point having the coordinates (*<sup>x</sup>*0, *y*0). The starting point was located in the broader part of the sample, where the stress level was significantly lower. A similar point of construction will have to be selected as the reference point of the real tested structures.

Afterwards, normalized relative changes in the maximum (2) and minimum (3) magnetic induction, maximum (4) and minimum (5) magnetic field strength, as well as the area of the hysteresis loop (6) were calculated.

$$K\_{A1}(\mathbf{x}, \mathbf{y}) = \frac{\Delta B\_{Amax}(\mathbf{x}, \mathbf{y})}{\max(|\Delta B\_{Amax}(\mathbf{x}, \mathbf{y})|)} \tag{2}$$

$$K\_{A2}(\mathbf{x}, \mathbf{y}) = \frac{\Delta B\_{Amin}(\mathbf{x}, \mathbf{y})}{\max(|\Delta B\_{Amin}(\mathbf{x}, \mathbf{y})|)} \tag{3}$$

$$K\_{A3}(\mathbf{x}, \mathbf{y}) = \frac{\Delta H\_{A\text{max}}(\mathbf{x}, \mathbf{y})}{\max(|\Delta H\_{A\text{max}}(\mathbf{x}, \mathbf{y})|)} \tag{4}$$

$$K\_{A4}(\mathbf{x}, \mathbf{y}) = \frac{\Delta H\_{Amin}(\mathbf{x}, \mathbf{y})}{\max(|\Delta H\_{Amin}(\mathbf{x}, \mathbf{y})|)} \tag{5}$$

$$K\_{A5}(x,y) = \frac{\Delta P\_A(x,y)}{\max(|\Delta P\_A(x,y)|)}\tag{6}$$

where: *A*—the magnetization direction, *x*, *y*—the measurement area point's coordinates, <sup>Δ</sup>*BAmax*(*<sup>x</sup>*, *y*)—the relative change in the maximum magnetic induction, <sup>Δ</sup>*BAmin*(*<sup>x</sup>*, *y*)—the relative change in the minimum magnetic induction, Δ *HAmax*(*<sup>x</sup>*, *y*)—the relative change in the maximum magnetic field strength, Δ *HAmin*(*<sup>x</sup>*, *y*)—the relative change in the minimum magnetic field strength, and Δ *PA*(*<sup>x</sup>*, *y*)—the relative change in the local hysteresis loop's area.

The next step concerned performing statistical analysis to assess the conditions of the material (after the stress loading). Initially, frequency histograms were plotted for the hysteresis loop parameters measured for all samples. Then, the focus was on performing statistical analysis of the data.

### **3. Results and Discussion**

The measurements were done according to the procedure described in Section 2. As a result of these measurements, a set of signals was obtained for each sample, necessary to plot the hysteresis loop measured for magnetization in the *x-* and *y*-axis directions. The signals were achieved for all positions (*<sup>x</sup>*i, *y*i) over which the transducer was moved. Due to the design of the transducer and the shape of the samples, in the case of magnetization in the *y*-axis, the shift range in the *y*-axis was slightly smaller than the magnetization in the *x*-axis direction.

Next, the parameters defined by Equations (2)–(6) were calculated using the signals acquired for each of the measuring points. Examples of two-dimensional plots of these parameters are presented in Figures 6 and 7. The plots show changes in the parameter's value over the surface sample. It is possible to generally observe a good correlation between the quantities measured on both orthogonal axes, but also significant differences are visible. It allows for a hypothesis that these signals are complementary to each other. By analyzing the signals for samples S00, S03, and S04, it can be observed that the signal value increases significantly and then decreases in the case of S06 and S07. Similar trends occur both for the parameters *KA*4 (Figure 6) and *KA*5 (Figure 7).

**Figure 6.** Relative change in the *KA*4 parameter as a function of the coordinates of the point from the measurement area for the: (**a**) *x*-axis direction; and (**b**) *y*-axis direction.

Despite the visible trends, assessing the load condition of the sample directly based on the measured signals is complex and may lead to ambiguity, for example, when using only the maximum value of the parameter. For this reason, attempts were made to statistically analyze the determined parameters to become independent from random changes in the signal value and take into account only the general trends.

In the first phase of the analysis, a frequency histogram was prepared for each of the parameters determined from measurements carried out for a single sample and one direction of magnetization (Figures 8 and 9). In the beginning, frequency histograms were subjected to normalization, relying on reducing the influence of interference signals. Then, the focus was on performing statistical analysis of the data by calculating and visualizing the following values: the maximum and minimum value, expected value, median, mode, variance, standard deviation, kurtosis, and histogram intervals. The frequency histograms achieved for samples with various levels of introduced stress show significant differences. For example, in the case of the *KX*4 parameter (Figure 8a), interesting changes in skewness and distribution modality can be observed. The distribution is almost symmetrical and

unimodal for the undeformed sample, an evident skewness appears starting from sample S03, and from sample S06 the distribution becomes bimodal. The reason for the transition from a unimodal to a bimodal distribution is the increasing level of stress and dislocations in the internal structure of successive samples. In the second part of the analysis, statistical values for the *KA*4 and *KA*5 parameters on both axes were computed as already mentioned. However, some statistical values, such as variance, expected value, standard deviation, and kurtosis, could not eventually be taken into account due to non-monotonic changes and a problem with unequivocal identification of material conditions.

**Figure 7.** Relative change in the *KA*5 parameter as a function of the coordinates of the point from the measurement area for the: (**a**) *x*-axis direction; and (**b**) *y*-axis direction.

**Figure 8.** Frequency histograms of the *KA*4 parameter for the: (**a**) *x*-axis direction; and (**b**) *y*-axis direction.

**Figure 9.** Frequency histograms of the *KA*5 parameter for the: (**a**) *x*-axis direction; and (**b**) *y*-axis direction.

Figures 10 and 11 contain plots of selected statistical values, which create the best opportunity to evaluate the conditions of a given sample.

**Figure 10.** Chosen statistical values of the *KA*4 parameter for the: (**a**) *x*-axis direction; and (**b**) *y*-axis direction.

**Figure 11.** Chosen statistical values of the *KA*5 parameter for the: (**a**) *x*-axis direction; and (**b**) *y*-axis direction.

As shown in Figure 10, the minimum values of both the *KX*4 and *KY*4 parameters decrease from the undamaged sample S00 to sample S05. Around specimens S05 and S06, the stress level caused the yield point to be exceeded, which can also be observed in the graphs as an inflection of the curve. After passing the yield point and increasing the stress level, the values decrease (in the case of the *KX*4 parameter) or increase (in the case of the *KY*4 parameter). Mode and interval curves allow for the state of the samples to be determined straightforwardly as well. Graphs related to *KX*5 (Figure 11) indicate that the interval value increases up to the S05 sample and then the curves bend due to the growing number of inhomogeneities. Particularly beneficial are the interval and median values of the *KY*5 parameter, enabling the identification of the sample's condition unequivocally (both before and after yield).

The main goal of all the tests and analyses was to find the parameters of the measured signals that would allow for the assessment of the condition of the tested structure in terms of the stresses they were subjected to. An important guess was to choose such individual parameters or groups of parameters that guarantee assessment in both the elastic and plastic regions. The use of histograms and statistical features made the results independent of randomly changing signals caused by disturbances and other external factors, such as surface unevenness. Therefore, the parameters presented in Figures 10 and 11 create a good chance that the assumed goals have been achieved. One of the important conclusions is the need to carry out measurements in at least two directions (parallel and perpendicular to the expected direction of stresses). A good example is the *KA*5 parameter measured in the *y*-axis direction (Figure 11b), which allows for an unambiguous identification of the state. Such parameters are crucial for building an automatic identification system in the future, which will use several of the presented parameters and the rough sets algorithm.
