Time-Response-Histogram-Based Feature of Magnetic Barkhausen

sensors

Article Time-Response-Histogram-Based Feature of Magnetic Barkhausen Noise for Material Characterization Considering Inﬂuences of Grain and Grain Boundary under In Situ Tensile Test

Jia Liu 1,* , Guiyun Tian 1,2,* , Bin Gao 1 , Kun Zeng 1 , Yongbing Xu 3 and Qianhang Liu 1

1 School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China; [email protected] (B.G.); [email protected] (K.Z.); [email protected] (Q.L.) 2 School of Engineering, Newcastle University, Newcastle upon Tyne NE1 7RU, UK 3 Spintronics and Nanodevice Laboratory, Department of Electronics Engineering, University of York, York YO10 5DD, UK; [email protected] * Correspondence: [email protected] (J.L.); [email protected] (G.T.)

Abstract: Stress is the crucial factor of ferromagnetic material failure origin. However, the nondestructive test methods to analyze the ferromagnetic material properties’ inhomogeneity on the microscopic scale with stress have not been obtained so far. In this study, magnetic Barkhausen noise (MBN) signals on different silicon steel sheet locations under in situ tensile tests were detected by a high-spatial-resolution magnetic probe. The domain-wall (DW) motion, grain, and grain boundary were detected using a magneto-optical Kerr (MOKE) image. The time characteristic of DW motion Citation: Liu, J.; Tian, G.; Gao, B.; and MBN signals on different locations was varied during elastic deformation. Therefore, a time- Zeng, K.; Xu, Y.; Liu, Q. Time- response histogram is proposed in this work to show different DW motions inside the grain and Response-Histogram-Based Feature around the grain boundary under low tensile stress. In order to separate the variation of magnetic of Magnetic Barkhausen Noise for properties affected by the grain and grain boundary under low tensile stress corresponding to MBN Material Characterization Considering excitation, time-division was carried out to extract the root-mean-square (RMS), mean, and peak Inﬂuences of Grain and Grain Boundary under In Situ Tensile Test. in the optimized time interval. The time-response histogram of MBN evaluated the silicon steel Sensors 2021, 21, 2350. https:// sheet’s inhomogeneous material properties, and provided a theoretical and experimental reference doi.org/10.3390/s21072350 for ferromagnetic material properties under stress.

Academic Editor: Nerija Žurauskiene˙ Keywords: time-response histogram; magnetic Barkhausen noise; stress evaluation; grain/grain boundary; domain-wall motion Received: 5 March 2021 Accepted: 24 March 2021 Published: 28 March 2021 1. Introduction Publisher’s Note: MDPI stays neutral Stresses play a pivotal role in determining the durability and service life of compo- with regard to jurisdictional claims in nents. [1–3]. Jun et al. reported that components are vulnerable because of crack initiation published maps and institutional afﬁl- caused by the generation of high tensile stresses [4]. Lee et al. assessed the fatigue life of a iations. welded repaired rail based on the welding process and contact stresses [5]. In most cases, the material state evaluation is considered to be homogeneous in the macrolevel analysis; however, stress-measurement research is necessary to connect with the microstructural properties on the grain scale. Donegan et al. used a convolutional neural Copyright: © 2021 by the authors. network to predict which regions of a microstructure are susceptible to forming stress Licensee MDPI, Basel, Switzerland. concentration and serving areas of local damage accumulation [6]. Zarei et al. found that This article is an open access article by considering the various orientations and shapes of the grains, the ﬁnite element model distributed under the terms and of DP steel microstructure results in a better prediction of the stress–strain curve than the conditions of the Creative Commons homogeneous models [7]. Zhao et al. proposed an automatic 3D simulation to simulate Attribution (CC BY) license (https:// metallic materials’ microstructure and investigated each grain’s stress for polycrystalline creativecommons.org/licenses/by/ 4.0/). material [8]. Johnson et al. used high-resolution electron backscatter diffraction to show

Sensors 2021, 21, 2350. https://doi.org/10.3390/s21072350 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 2350 2 of 20

the intersection between material properties’ variation and grain boundaries for analysis of the crack-initiation process [9]. Analysis of the stress state affected by the grain and grain boundary is usually carried out in a simulation or a destructive manner [10]. For this reason, it would be beneﬁcial to propose a reliable nondestructive technique designed for the purpose above. The variation of the nondestructive test and evaluation method is used to characterize the electromagnetic properties for industrial materials’ state evaluation. Among them, magnetic Barkhausen noise (MBN) is a microstructure-sensitive and nondestructive evaluation technique for online inspection of structures to characterize both micro- and macrostresses of ferromagnetic material in industrial applications [11,12]. The MBN signal is found in the domain-wall (DW) motion, detected by a pick-up coil wound around the ferromagnetic sample [13,14]. The microstructure and the stress affect the alignment of the DWs, whereas lattice defects drive the free path of DW motion [15–17]. Nondestructive MBN determines both micro-and macroresidual stresses in silicon steel, steel weldments, and lightly deformed AISI 1070 steel [18–21]. MBN signals are combined with microstructure observation methods performed using magneto-optical Kerr (MOKE), atomic force microscopy (AFM), scanning electron microscopy (SEM), etc., to analyze the relationship between the ferromagnetic material’s microstructure and mechanical properties [22,23]. The peak value of MBN signals is calibrated as a function of the microstructure expressed in dislocation density [24]. MBN distinguishes the variable character of the microstructure and stress state for tool steel X210Cr12 [25]. The arrangement of magnetic domains along a grain boundary can be affected by the grain-boundary characteristic, and the MBN signal increased when increasing the angle of the grain-boundary misorientation [26]. The stress dependence on the MBN voltage and on the magnetic hysteresis loop is inﬂuenced by the grain size of commercial carbon steel [27]. Due to the MBN phenomenon observed in various directions for grain-oriented steels, MBN time-frequency distribution can analyze anisotropy distribution of the sample [28]. However, the effect of microstructure on material properties’ inhomogeneity is the key for the microscopic nondestructive stress characterization of the ferromagnetic material [29]. In particular, there is no reliable method for evaluating micromagnetic properties affected by the grain interior and grain boundary with different average stress via MBN. As mentioned above, the microstructure and MBN of silicon steel sheet under an in situ tensile test were detected by MOKE and MBN detection device in this study. By combining the time characteristic of DW motion and MBN signal on different locations, a time-response histogram of MBN was carried out to quantitatively analyze the effect of the grain and grain boundary on the magnetic properties’ inhomogeneity during elastic deformation. The optical time intervals were used to separate the variation of magnetic properties affected by the grain interior and grain boundary. The proposed work will analyze the inhomogeneous material properties affected by the stress, grain, and grain boundary for the ferromagnetic material.

2. Experimental Procedure and Methods MBN signals of a silicon steel sheet were detected by high-spatial-resolution sensing. Combining DW motion and MBN signals, a time-response histogram was carried out to evaluate the inhomogeneous magnetic properties with different stress inside the grain and around the grain boundary.

2.1. Sample Preparation The grain-oriented silicon steel sheet was investigated to analyze the correlation of DW motion and the time-response histogram inside the grain and around the grain boundary with stress state. The average grain size of the selected silicon steel sheet was over 10 mm, which is visible for DW motion and MBN signal detection on grains and grain boundaries. The dimension of the sample was 300 mm × 30 mm × 0.2 mm (length × width × thickness). The sample was subjected to a tensile load. Grain-oriented Sensors 2021, 21, 2350 3 of 20

silicon steel has the characteristics of high magnetic induction and low iron loss [30,31]. The chemical composition of the silicon steel sheet is found in Table1. In order to observe microstructure and DW motion by MOKE, the sample was mechanically polished, and annealing treatment was not performed before the tensile test. Therefore, the initial state of the sample surface’s thin layer contained small compressive stresses.

Table 1. Chemical composition of silicon steel sheets expressed as weight percentages.

Fe Si C Mn P S Al Balance 3.00~5.00 0.06 0.15 0.03 0.25 5.10~8.50

2.2. Experiment Setup The experimental setup for the MOKE and MBN observation for inhomogeneous material properties under the in situ tensile test is shown in Figure1. MBN and longitudinal MOKE microscopy captured the MBN signal and DW motion of the silicon steel sheet. This system included a high-spatial-resolution MBN probe, an MBN detection device (including signal amplifiers, filter, acquisition card, MBN signal processor), a digital CCD camera for DW observation, driving coils, and applied tensile stress [32]. The tensile stress was from 0 MPa to 123 MPa, which was in the elastic deformation. A C8484-03G02 digital CCD camera with a sampling rate of 16.3 Hz was used to observe the microstructure and DW motion directly. DW motion images and MBN signals were in the same excitation, which could bridge the correlation of the micro- and macromagnetic parameters and stress inside the grain and around the grain boundary. The applied excitation field was in the tensile stress direction. The driven coil with 355 turns wound around the sample could provide AC to magnetize the sample for periodicity. The excitation was a triangular wave, in which the magnetization frequency and amplitude were 0.5 Hz and 1 kA/m, respectively. This magnetic field strength was greater than the sample material’s maximum magnetic saturation level (about 0.9 kA/m). The frequency spectrum of MBN signals generated over different ranges of magnetization did not change significantly [33,34]. Since the MBN signal generated in the material was attenuated by the electromagnetic eddy current opposition to an extent that depended on the frequency of the signal, the measurement depth (skin-depth) was limited to a finite depth from the surface [33]. The electromagnetic skin-depth δ is given by the relation:

1 δ = p (1) (π f σµ0µr)

where f is the frequency, σ is the conductivity of the material, µ0 is the permeability of vacuum, and µr is the relative permeability of the material. The frequency response strongly inﬂuences the maximum depth over which the changes in magnetization are detected as MBN signals. The magnetic probe with a dominant low-frequency response has greater skin-depth of detection, while the magnetic probe with more high-frequency response has shallow skin-depth. In this study, the thickness of the silicon steel sheet was 0.2 mm. Therefore, the magnetic probe could detect the MBN signal for the entire thickness of the sheet, when the frequency of the sample was 0.5 Hz. The time-response histogram was based on the correlation between the applied ﬁeld and the DW motion at different locations. The magnetic domain image was about 16.2 mm × 13.6 mm with 450 × 360 pixels, showing the microstructure and DW motion directly. The tape-recorder head was used for MBN detection. The probe was actually a hoof-shaped electromagnet. The probe was made by winding a coil around an iron core. When the excitation was applied to the sample, the probe read the data by sensing the magnetization on the sample. The probe had a high spatial resolution (about 3–4 mm), which could detect the different magnetization of the sample inside the grain and around the grain boundary. Linking the micromagnetic properties observed by MOKE, this system analyzed the difference of the time-response Sensors 2021, 21, x FOR PEER REVIEW 4 of 22

ness of the sheet, when the frequency of the sample was 0.5 Hz. The time-response histogram was based on the correlation between the applied field and the DW motion at different locations. The magnetic domain image was about 16.2 mm × 13.6 mm with 450 × 360 pixels, showing the microstructure and DW motion directly. The tape-recorder head was used for MBN detection. The probe was actually a hoof-shaped electromagnet. The probe was made by winding a coil around an iron core. When the excitation was applied to the sample, the probe read the data by sensing the magnetization on the sample. The probe had a Sensors 2021, 21, 2350 4 of 20 high spatial resolution (about 3–4 mm), which could detect the different magnetization of the sample inside the grain and around the grain boundary. Linking the micromagnetic properties observed by MOKE, this system analyzed the difference of the time-response histogramhistogram of of MBN MBN affected affected by by the the interior interior andand graingrain boundary,boundary, whichwhich interpretedinterpreted the effectef- offect the of grain the grain interior interior and grainand grain boundary boundary on the on inhomogeneousthe inhomogeneous material’s material’s properties properties with differentwith different average average tensile tensile stress. stress.

(a)

(b) (c)

FigureFigure 1. ( 1.a) ( Experimentala) Experimental set-up set-up for for DW DW and and MBN MBN observation; observation; ((b) schematic illustration illustration of of load, load, DW DW observation, observation, and and MBNMBN detection; detection; and and (c) ( schematicc) schematic for for high-spatial-resolution high-spatial-resolution MBNMBN probeprobe (about 3–4 3–4 mm) mm) and and MBN MBN detection detection device. device.

2.3. The Model of the Time-Response Histogram 2.3. The Model of the Time-Response Histogram The method for the time-response histogram on the grain and grain boundary with differentThe method stress state for theto analyze time-response the effect histogram of grain interior on the and grain grain and boundary grain boundary on the in- with differenthomogeneous stress material’s state to analyze properties the is effect outlined of grainin Figure interior 2. and grain boundary on the inhomogeneous material’s properties is outlined in Figure2. The microstructure and MBN signal of the silicon steel sheet were observed using a MOKE and MBN detection device. A tape-type probe was also designed as an MBN sensor to get the high spatial resolution of the MBN signal inside the grain and around the grain boundary. The combination of DW motion and MBN can analyze inhomogeneous material properties while considering the inﬂuence of the grain and grain boundary under tensile stress. Grain boundaries are important microstructural elements that affect DW motion, leading to a change in the differential susceptibility at all ﬁeld strengths, thereby changing the time characteristics of MBN [35,36]. The time-response histogram was proposed to divide the entire time range of MBN activities into a series of equal intervals, showing the time characteristics of DW motion and MBN at different locations with stress along time. As shown in Figure3, the time-response histogram was extracted from the MBN signal. Thus, the time-response histogram was modeled to analyze the variation of magnetic properties affected by the grain and grain boundary with different stress. Sensors 2021, 21, 2350 5 of 20 Sensors 2021, 21, x FOR PEER REVIEW 5 of 22

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Therefore, these methods analyzed the silicon steel’s inhomogeneous material properties affected by the stress, grain, and grain boundary. FigureFigure 2. The2. The approach approach of of the the time-response time-response histogram histogram toto evaluate the inhomogeneous inhomogeneous material’s material’s properties properties onon the the grain grain andand grain grain boundary. boundary.

The microstructure and MBN signal of the silicon steel sheet were observed using a MOKE and MBN detection device. A tape-type probe was also designed as an MBN sensor to get the high spatial resolution of the MBN signal inside the grain and around the grain boundary. The combination of DW motion and MBN can analyze inhomogeneous material properties while considering the influence of the grain and grain boundary under tensile stress. Grain boundaries are important microstructural elements that affect DW motion, leading to a change in the differential susceptibility at all field strengths, thereby changing the time characteristics of MBN [35,36]. The time-response histogram was proposed to divide the entire time range of MBN activities into a series of equal intervals, showing the Figure Figure3.time (a) Exemplary characteristics 3. (a) Exemplary distributions of distributionsDW of MBNmotion voltage ofand MBN and MBN excitation voltage at different andinside excitation thelocations grain acquired inside with thestress for grainexcitation along acquired frequencytime. for of 10 Hz andexcitationAs amplitude shown frequency in of Figure1 kA/m; of 103, (b Hzthe) the andtime-response envelope amplitude of MBN of histogram 1extracted kA/m; (from bwas) the the extracted envelope rectangular from of MBNregion the extracted ofMBN Figure signal. from3a; (c) the time- response histogram of MBN extracted from the rectangular region of Figure 3a. rectangularThus, the regiontime-response of Figure 3histograma; ( c) time-response was modele histogramd to analyze of MBN the extracted variation from of the magnetic rectangular properties affected by the grain and grain boundary with different stress. region of Figure3a. The mean, RMS,3. Experi andmental peak wereResults extracted from the time-response histogram to quan- To analyze the influence of grain interior and grain boundary on the inhomogeneous tify Thethe time mean, characteristics RMS, and peak of magnetic were extracted properties from at different the time-response positions. Based histogram on the to cor- quan- relation among themagnetic applied properties, field and the magnetic results of time the timecharacteristics-response histogram on the grain were and investigated grain to ana- tify the time characteristicslyze the variation of magnetic of MBN at properties different locations. at different positions. Based on the correlationboundary, amongtime-division the applied was carried ﬁeld out and to magnetic analyze the time time-response characteristics histogram on the features grain and in the optimized time intervals. The difference between the RMS, mean, and peak with grain boundary,3.1. time-division The Relationship was between carried the Time out- toResponse analyze Histogram the time-response and DW Motion histogram on the Grain and tensile stress and without tensile stress was determined to reduce the difference of the features in the optimizedGrain Boundary time intervals. The difference between the RMS, mean, and peak time-response histogram before the tensile test. The feature extraction and time-division with tensile stress andBased without on the tensile observation stress was of grain determined distribution to reduce and MBN the differenceof silicon steel, of the the time- quantitatively analyzed the inhomogeneous material’s properties affected by the grain, time-response histogramresponse histogram before the was tensile used to test. investigate The feature the influence extraction of the and grain time-division and grain boundary grain boundary, and stress. quantitatively analyzedon the micromagnetic the inhomogeneous properties material’sunder tensile properties stress. In Figure affected 4, when by the the grain, tensile stress grain boundary,was and 0 stress. MPa or 123 MPa, respectively, DW motion, hysteresis loops, and the time-response histogram were analyzed inside the grain and around the grain boundary. The distribution of the grain and grain boundary is shown in Figure 4a; S1-g1 and S1-g3 denote two adjacent grains, and S1-gb13 denotes the grain boundary between grains S1-g1 and S1-g3.

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Therefore, these methods analyzed the silicon steel’s inhomogeneous material properties affected by the stress, grain, and grain boundary.

3. Experimental Results To analyze the inﬂuence of grain interior and grain boundary on the inhomogeneous magnetic properties, the results of the time-response histogram were investigated to analyze the variation of MBN at different locations.

3.1. The Relationship between the Time-Response Histogram and DW Motion on the Grain and Grain Boundary Based on the observation of grain distribution and MBN of silicon steel, the time- response histogram was used to investigate the inﬂuence of the grain and grain boundary on the micromagnetic properties under tensile stress. In Figure4, when the tensile stress was 0 MPa or 123 MPa, respectively, DW motion, hysteresis loops, and the time-response histogram were analyzed inside the grain and around the grain boundary. The distribution of the grain and grain boundary is shown in Figure4a; S1-g1 and S1-g3 denote two adjacent grains, and S1-gb13 denotes the grain boundary between grains S1-g1 and S1-g3. The magnetic domain distribution is predicted through the minimization of general energy relations [37]: Etotal = Emag + Eex + Ea + Eσ (2)

where Emag, Eex, Ea, and Eσ respectively denote the magnetostatic energy, exchange energy, magnetocrystalline energy, and magnetoelastic and magnetostatic energies. The speciﬁc energy components are deﬁned by:

1 Emag = − ∑ ∑ ∑ mi·HI (3) 2 x y z

Eex = −2 ∑ ∑ ∑ JSi·Sj (4) x y z

h 2 2 2 2 2 2 2 2 2i Ea = −2 ∑ ∑ ∑ K1 α1 α2 + α2 α3 + α3 α1 + K2α1 α2 α3 (5) x y z

3 2 Eλ = − ∑ ∑ ∑ λσ cos θ (6) 2 x y z

where K1 and K2 are anisotropy constants, and α1, α2, and α3 are direction cosines of a moment at the center of the cube. Furthermore, λ, σ, and θ respectively denote the magnetostriction constant, magnetoelastic stress and angle between the magnetization and stress. Finally, HI denotes the interaction ﬁeld at mi due to all magnetic moment. Magnetic total energies change with the increasing low tensile stresses inside grains and around grains boundaries. These changes affect the distribution of the magnetic domain. The equilibrium size of the magnetic domains is determined by minimizing the magnetic total energies. One possible way to avoid or reduce this energy is to increase the volume of the main domains with magnetization parallel to one of the easy axes and reduce the volume of domains magnetized along the other axes [37]. When low tensile stress is applied to the material, the increased magnetoelastic energy can be reduced by the formation of adapted domain patterns in the grain interior and around the grain boundary. The grain interior has the same magnetic domain-wall distribution, on the assumption that the tensile stress and material magnetic properties are uniform with a ﬁnite size. Grain boundaries can affect the domain-wall distribution in polycrystalline materials, since grain boundaries are composed entirely of lattice defects [38]. Due to the differences in microstructure among grains and grain boundaries, when low tensile stresses are applied to the material, the changes of the magnetoelastic energy are different inside grains and around grains boundaries. Sensors 2021, 21, 2350 7 of 20 Sensors 2021, 21, x FOR PEER REVIEW 7 of 22

FigureFigure 4. 4. DifferentDifferent time-response time-response histograms histograms of ofthe the sample sample at diffe at differentrent locations locations when when the thetensile tensile stresses stresses were were 0 MPa 0 MPaand 123and MPa, 123 MPa, respectively: respectively: (a) denotes (a) denotes the locations the locations of S1-gb13, of S1-gb13, S1-g1, S1-g1, and S1-g3; and S1-g3; (b,c) show (b,c) showthe DW the motion DW motion at different at different loca- tionslocations under under 0 MPa, 0 MPa, (d,e ()d show,e) show different different DW DW motion motion on onthe the different different locations locations under under 123 123 MPa; MPa; (f (–f–hh) )show show the the related related similarsimilar time-response time-response histograms histograms of of MBN MBN under under 0 0 MPa; MPa; ( (ii––kk)) show show the the quite quite different different time-response time-response histograms histograms of of MBN MBN underunder 123 123 MPa. MPa.

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The microstructure and stress can affect the distribution of the magnetic domain and the behavior of DW motion around the grain boundary [39]. When the tensile stress is 0 MPa, DW motion is almost synchronous. Tensile stress aligns the magnetic domains parallel to the stress direction and makes the magnetization process easy when excitation is in the stress direction during the elastic deformation [40]. The formation of 180◦ DW increases inside the grain. The stable state of the magnetic domain structure is reached because the magnetic domain’s orientation makes the fewer free pole appear on the grain interface to reduce the magneto-static energy [41,42]. The 90◦ DWs appeared on the grain boundary S1-gb13 under 123 MPa to minimize the magnetic–static energy on the grain boundary affected by the stress. Thus, the distribution of DW was more complex on grain boundary S1-gb13 than on grains S1-g1 and S1-g3. Since stress made DW distribution more complex on the grain boundary, the DW motion became different at different locations with stress. The DW motion in magnetization occurring over a given time interval relates to MBN activity [43,44]. In particular, the change rate of magnetization occurring as Barkhausen emissions dMJS/dt is expected to be proportional to the change rate of magnetization occurring as magnetization changes dM/dt. The time-response histogram was modeled to divide the entire time range of MBN activities into a series of equal intervals. Each bin of the histogram is represented as the magnetization change with the dimensionless term. The time characteristic of Barkhausen noise is distributed as:

dMJS dM dH = γ (7) dt dH dt where H denotes the external magnetic field. In this study, the time-response histogram identified the amplitude of MBN activity along time. The grain boundary affects the movement of DWs and leads to different magnetization processes under stress in the previous research [45]. From Figure4d,e, there were more ◦ 90 domain walls around the grain boundary S1-gb13 during magnetization, and the time required to make material achieve saturation was longer around the grain boundary than inside the grain. The relationship between the difference of MBN activity ∆MJS, and the difference of magnetization intensity ∆M on different locations satisfies:

d∆MJS d∆M dH ∝ γ (8) dt dH dt When the tensile stress was 0 MPa, the difference in the time-response histograms for S1-g1, S1-g3, and S1-gb13 was small. When the tensile stress increased, MBN signals at these locations became much larger. The time-response histogram bins were quite different. When the tensile stress was 123 MPa, the histogram on S1-g3 was the smallest, and the histogram bins were the largest around the grain boundary S1-gb13 compared to grains S1-g1 and S1-g3. When the tensile stress was 123 MPa, the magnetic saturation was reached earliest on grain S1-g2 (from 0.3 s to 0.7 s), and magnetic saturation was reached latest on the grain boundary S1-gb12 under 123 MPa (from 0.1 s to 0.9 s). The different DW motion affected the time-response histogram at different locations. The magnetization of polycrystalline ferromagnetic materials is a complex phenomenon. However, there is a consensus about some of its main stages in magnetic materials: domain rotation, domain nucleation, ◦ ◦ ◦ 180 DW motion, and 90 DW motion [46]. From Figure4, it was found that the pure 180 ◦ domain-wall motion was inside the grain interior (from 0.3 s to 0.7 s), and the 90 domain walls were associated with the bins from 0.1 s to 0.3 s and from 0.7 s to 0.9 s around the grain boundary. Thus, the time-response histogram of MBN characterized the different DW motion inside the grain and around the grain boundary. Each bin of the time-response histogram Vn was proportional to the DW motion velocity vDW [35], where n denotes the number of Sensors 2021, 21, 2350 9 of 20

histogram bins. The relationship between the DW motion and time-response histogram’s bin at different locations satisﬁes:

∆Vn(σ) ∝ ∆v(σ) (9)

where ∆Vn(σ) and ∆v(σ) denote the difference of the time-response histogram’s bin and the difference of DW motion velocity under stress on the grain and grain boundary. Based on studying the DW motion at different locations, the time-response histogram quantiﬁes the micromagnetic, reﬂecting the variation of magnetic properties on the grain and grain boundary under stress.

3.2. Verification for Time-Response Histogram on Different Grains and Grain Boundaries To further analyze the influence of grain and grain boundary on material properties, the variation of the time-response histogram at different locations under the tensile test was analyzed. Figure5 shows the grain and the grain boundary distribution of the sample. The sam- Sensors 2021, 21, x FOR PEER REVIEW 10 of 22 ple was divided into 15 parts (5 × 3) by a dotted grid. The area of each grid area was 3.24 mm × 4.53 mm. The X-coordinate and Y-coordinate defined each part’s location, as shown in Figure5a. The size of the MBN probe was about 3–4 mm. The MBN signals were detectedshown aroundin Figure the 5a. grainThe size boundary of the MBN when probe the was MBN about probe 3–4 wasmm. onThe the MBN grain signals were boundaries S1-gb13detected andaround S1-gb23. the grain Due boundary to the high-spatial-resolutionwhen the MBN probe was magnetic on the grain probe, boundaries the time-responseS1-gb13 histogram and S1-gb23. of MBN Due was to detectedthe high-spatial-resolution on each part by scanning magnetic the probe, sample, the time-re- as shown in Figuresponse5 histogramb. The magnetic of MBN domainwas detected and MBNon each were part analyzedby scanning to investigatethe sample, as shown material propertiesin Figure at different5b. The magnetic locations. domain and MBN were analyzed to investigate material properties at different locations.

(a) (b)

Figure 5. TheFigure sample 5. wasThe divided sample into was 15 divided parts (5×3 into) by 15 a parts dotted (5 grid:× 3) ( bya) illustrates a dotted grid:the grain (a) illustratesdistribution the of grainthe sample and the coordinate of each part; (b) shows MBN detection in each part by scanning the sample. S1-g1, S1-g2, and S1-g3 distribution of the sample and the coordinate of each part; (b) shows MBN detection in each part denote three adjacent grains of the sample; S1-gb12 denotes the grain boundary between grain S1-g1 and grain S1-g2; S1- by scanning the sample. S1-g1, S1-g2, and S1-g3 denote three adjacent grains of the sample; S1- gb13 denotes the grain boundary between grain S1-g1 and grain S1-g3; and S1-gb23 denotes the grain boundary between grain S1-g2 andgb12 grain denotes S1-g3. the grain boundary between grain S1-g1 and grain S1-g2; S1-gb13 denotes the grain boundary between grain S1-g1 and grain S1-g3; and S1-gb23 denotes the grain boundary between grain S1-g2 and grainDW S1-g3. motion varied with the increased stress at different locations. Figure 6 illustrates the different DW motions inside the grain and around the grain boundary under 0 MPa DW motionand varied123 MPa. with The the difference increased of stressthe DW at motion different on locations.the grain and Figure the 6grain illustrates boundary under the different DW0 MPa motions was small, inside while the the grain DW andmotions around become the grainquite different boundary at underdifferent 0 MPalocations with and 123 MPa.the The increase difference of stress. of the When DW motion the average on the tensile grain stress and the was grain 123 boundaryMPa, the DW under distribution 0 MPa was small,was more while complicated the DW motions on the become grain boundary quite different than on at differentthe grain. locations From Figure with 6, the 90° the increase ofDWs stress. formed When around the average grain boundary tensile stress S1-gb13 was and 123 S1-gb12 MPa, the to DWreduce distribution the increasing total ◦ was more complicatedenergy caused on the by grainthe tensile boundary stress. than Especial on thely grain.on S1-gb12, From Figuremagnetic6 , thefield90 lines were DWs formedmostly around continuous grain boundary across the S1-gb13 grain boundary and S1-gb12 to reduce to reduce free thepoles increasing on the grain to- interface tal energy causedunder by 123 the MPa. tensile Due stress.to increasingly Especially complex on S1-gb12, domain magnetic reorganization ﬁeld linesaround were S1-gb12, the mostly continuousenergy across loss was the higher grain at boundary this location, toreduce so these free phenomena poles on satisfy the grain Equation interface (2) [47]. Thus, the MBN activities inside the grain and around the grain boundary became different with the increase of stress. The time-response histograms of MBN were different. Figures 7 and 8 show the time-response histograms on every part under the tensile test. Figure 7 a,b,d– g,i–j,n,o and Figure 8 a,b,d–g,j–l,n,o denoted the time-response histogram inside the grain under 123 MPa and 0 MPa, while Figure 7 c,h,i,m and Figure 8 c,h,i,m denoted the time- response on the grain boundary under 123 MPa and 0 MPa. From Figure 8, when tensile stress was 0 MPa, since the difference of DW motion was tiny inside the grain and around the grain boundary, the difference between the time-response histograms at these locations was tiny (see Figure 8 a,b,d–g,i–l,n,o). The tiny difference in the time-response histograms was caused by the sample surface’s mechanical polishing and material manufacturing. When the tensile stress increased, the time-response histograms became much larger, and the difference in the histograms at different locations became much larger. Since the strength of the external magnetic field required to make the material achieve saturation magnetization was much larger on the grain boundary (S1-gb12 and S1-gb13) than on the grains (S1-g1, S1-g2, and S1-g3), the amplitude of the time-response histograms became bigger on the grain boundary (S1-gb12 and S1-gb13) than on the grain (S1-

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under 123 MPa. Due to increasingly complex domain reorganization around S1-gb12, the energy loss was higher at this location, so these phenomena satisfy Equation (2) [47]. Thus, the MBN activities inside the grain and around the grain boundary became different with the increase of stress. The time-response histograms of MBN were different. Figures7 and8 show the time-response histograms on every part under the tensile test. Figure7a,b,d–g,j–l,n,o and Figure8a,b,d–g,j–l,n,o denoted the time-response histogram inside the grain under 123 MPa and 0 MPa, while Figure7c,h,i,m and Figure8c,h,i,m denoted the time-response on the grain boundary under 123 MPa and 0 MPa. From Figure8 , when tensile stress was 0 MPa, since the difference of DW motion was tiny inside the grain and around the grain boundary, the difference between the time-response histograms at these locations was tiny (see Figure8a,b,d–g,i–l,n,o). The tiny difference in the time- response histograms was caused by the sample surface’s mechanical polishing and material manufacturing. When the tensile stress increased, the time-response histograms became much larger, and the difference in the histograms at different locations became much larger. Sensors 2021, 21, x FOR PEER REVIEWSince the strength of the external magnetic ﬁeld required to make the material achieve11 of 22

saturation magnetization was much larger on the grain boundary (S1-gb12 and S1-gb13) than on the grains (S1-g1, S1-g2, and S1-g3), the amplitude of the time-response histograms became bigger on the grain boundary (S1-gb12 and S1-gb13) than on the grain (S1-g1, g1, S1-g2, and S1-g3). The results shown in Figures 6–8 are consistent with the results S1-g2, and S1-g3). The results shown in Figures6–8 are consistent with the results shown shown in Figure 4. in Figure4.

Figure 6. The magnetic domain and DW motion inside the grain and around the grain boundary of the sample with the different tensile stresses. The stress was increased from 0 MPa to 123 MPa. When When the the tensile tensile st stressress increased, increased, the difference of DW motion increased on the grain and grain boundary.

The high-spatial-resolution MBN probe was sensitive enough for magnetic-property characterization, reﬂecting the grain and the grain boundary’s material properties. The time- response histogram for each part was consistent with the DW motion results, proving the reliable repeat of correlation between the DW motion and time-response histogram inside the grain and around the grain boundary. The material properties were inhomogeneous at

Figure 7. Time-response histograms at different locations when the tensile stress was 123 MPa. The time-response histograms increased with the increase of tensile stress, and the difference in MBN histograms on each part became much larger.

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g1, S1-g2, and S1-g3). The results shown in Figures 6–8 are consistent with the results shown in Figure 4.

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Figure 6. The magnetic differentdomain and locations, DW motion so the inside grain the boundary grain and andaround adjacent the grain grains boundary affected of the mechanical sample with behav- the different tensile stresses.ior The [8 stress,48]. Thewas time-responseincreased from 0 histogram MPa to 123 of MPa. MBN When had the enough tensile sensitivitystress increased, to characterize the difference the of DW motion increasedgrain on the and grain grain and boundarygrain boundary. effect on the inhomogeneous material’s properties.

FigureSensors 2021 7. Time-response, 21, x FOR PEER histograms REVIEW at different locations when the tensile stress was 123 MPa. The time-response histograms12 of 22

increasedFigure with 7. Time-response the increase of histogra tensile stress,ms at different and the difference locations inwhen MBN the histograms tensile stress on eachwas 123 part MPa. became The much time-response larger. histograms increased with the increase of tensile stress, and the difference in MBN histograms on each part became much larger.

Figure 8. Time-response histograms at different locations when the tensile stress was 0 MPa. The time-response histograms Figure 8. Time-response histograms at different locations when the tensile stress was 0 MPa. The time-response histograms exhibited a small difference before the tensile test. The difference was inﬂuenced by the material manufacturing. exhibited a small difference before the tensile test. The difference was influenced by the material manufacturing.

4. DiscussionThe high-spatial-resolution MBN probe was sensitive enough for magnetic-property characterization,In order to quantitatively reflecting the analyze grain and the differentthe grain inhomogeneousboundary’s material magnetic properties. properties The at differenttime-response locations histogram with stress,for each RMS, part was mean, consistent and peak with value the DW were motion extracted results, from prov- the time-responseing the reliable histograms. repeat ofBased correlation on the between time characteristics the DW motion of MBN and time-response inside the grain histo- and aroundgram theinside grain the boundarygrain and around under stress,the grain time-division boundary. The was material carried properties out to extract were inho- RMS, mean,mogeneous and peak at in different the optimized locations, time so interval,the grain which boundary highlighted and adjacent the variation grains affected of magnetic me- propertieschanical affected behavior by [8,48]. the stress, The time-response grain, and grain histogram boundary. of MBN had enough sensitivity to characterize the grain and grain boundary effect on the inhomogeneous material’s prop- 4.1.erties. Time-Response Histogram’s Feature Extraction with Different Stress The time-response histogram of MBN reﬂected the variation of the material state at 4. Discussion different locations. MBN characteristic parameters such as RMS, mean value, and peak In order to quantitatively analyze the different inhomogeneous magnetic properties at different locations with stress, RMS, mean, and peak value were extracted from the time-response histograms. Based on the time characteristics of MBN inside the grain and around the grain boundary under stress, time-division was carried out to extract RMS, mean, and peak in the optimized time interval, which highlighted the variation of magnetic properties affected by the stress, grain, and grain boundary.

4.1. Time-Response Histogram’s Feature Extraction with Different Stress The time-response histogram of MBN reflected the variation of the material state at different locations. MBN characteristic parameters such as RMS, mean value, and peak value had a high correlation with stress in previous studies [28]. RMS, mean, and peak value on the grains and grain boundaries were extracted from the histograms to investigate their statistical properties, which quantitatively analyzed the material properties. In this study, the magnetization frequency was 0.5 Hz. The interval of the time-response histogram bin was 0.1 s. The number of histogram bins was 10. Thus, the feature exaction for the quantitative method can be defined as:

∑ () () = (10)

() = (()) (11)

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value had a high correlation with stress in previous studies [28]. RMS, mean, and peak value on the grains and grain boundaries were extracted from the histograms to investigate their statistical properties, which quantitatively analyzed the material properties. In this study, the magnetization frequency was 0.5 Hz. The interval of the time-response histogram bin was 0.1 s. The number of histogram bins was 10. Thus, the feature exaction for the quantitative method can be deﬁned as: s N ∑ Vn(σ) RMS(σ) = i=1 (10) N

Peakvalue(σ) = max(Vn(σ)) (11)

mean(σ) = Vn(σ) (12)

where N denotes the number of histogram bins and Vn(σ) denotes the amplitude of each bin. However, due to the grain and grain boundary, the material properties at different locations were not uniform. Figure9 shows the curve of the three features of the time-response histograms on S1-g1, S1-g3, and S1-gb13. Figures 10–12 show the two-dimensional images for RSM, mean, and peak value of the time-response histograms at different locations. The mean, RMS, and peak values had the same increasing tendency with the increase of stress. However, these features were higher on the grain boundaries than on the grains. A clear positive correlation between the measured tensile stress state and the corresponding MBN was verified. The 180◦ DW motion inside the grain and 90◦ DW motion around the grain boundary [46] were directly proportional to the stress fluctuation, while the time- response histogram features reflected the variation of DW motion with stress. As shown in Figures9–12, the grain boundaries were more susceptible to external tensile stress, especially since the time-response histogram was much higher on the grain boundary S1-gb12. A minor failure was more likely to occur on the grain boundary S1-gb12 [9,10]. MBN emission increased along with tensile stresses and decreased with compressive stresses, especially for uniaxial loading and elastic regimes [13]. The total tensile stress acting normal to the grain boundary during elastic deformation was a sum of the internal stress due to the applied stress, resolved onto the grain boundary plane [9]. The domain- wall motion in this case was a function of microstresses (the function of grain and grain- boundary microstructure). The microstress was higher on the grain boundary than on the grain, making the grain boundary more prone to early minor failure formation [8]. The time-response histograms and their features characterized time characteristics of DW motion to quantitatively analyze the inhomogeneous material’s properties affected by the stress, grain, and grain boundary.

4.2. The Difference Elimination of Magnetic Properties before the Tensile Test Due to the residual stress and material manufacturing, the magnetic properties under 0 N exhibited a small difference, as shown in Figures9–12. It was better to reduce the difference before the tensile test to analyze the variation of the inhomogeneous material’s properties affected by the stress, grain, and grain boundary, as shown in Figure 13. The difference between these features with stress and without stress is deﬁned in the following, which was extracted from Equations (10)–(12):

∆RMS = RMS(σ) − RMS(0) (13)

∆mean = mean(σ) − mean(0) (14) ∆peak = peak(σ) − peak(0) (15) where RMS(σ), mean(σ), and peak(σ) denote RMS, mean, and peak value with stress, respectively; and RMS(0), mean(0), and peak(0) denote RMS, mean, and peak value without stress, respectively. From Figure 13, ∆RMS, ∆mean, and ∆peak were almost uniform at the different locations when the tensile stresses increased from 0 MPa to 53 MPa. Sensors 2021, 21, x FOR PEER REVIEW 13 of 22

() =() (12)

where denotes the number of histogram bins and () denotes the amplitude of each Sensors 2021, 21, 2350 bin. 13 of 20 However, due to the grain and grain boundary, the material properties at different locations were not uniform. Figure 9 shows the curve of the three features of the time- response histograms on S1-g1, S1-g3, and S1-gb13. Figures 10–12 show the two-dimen- This meanssional images that ∆ forRMS RSM,, ∆ mean,mean and, and peak∆ valuepeak ofremoved the time-response the difference histograms in at magnetic different properties beforelocations. the tensile The test.mean, When RMS, and the peak tensile values stresses had the increased same increasing from tendency 0 MPa to with 123 the MPa, ∆RMS, increase of stress. However, these features were higher on the grain boundaries than on ∆meanthe, and grains.∆peak A clearon positive the grain correlation boundaries between the were measured higher tensile than stress on thestate grains.and the The time- responsecorresponding histogram MBN on was the verified. grain The boundary 180° DW motion was more inside susceptiblethe grain and 90° to DW the mo- tensile stress; when thetion tensilearound stressthe grain increased boundary 123[46] MPa,were dire stressctly proportional made the degree to the stress of the fluctuation, material properties’ inhomogeneitywhile the time-response much more histogram significant features on reflected the grain the boundary variation of thanDW motion on the with grain. ∆RMS, ∆meanstress., and As∆ shownpeak reducedin Figure 9 theto Figure influence 12, the grain of the boundaries material were microstructure more susceptible onto the initial external tensile stress, especially since the time-response histogram was much higher on materialthe stategrain beforeboundary the S1-gb12. tensile A test,minor reflecting failure was themore inhomogeneity likely to occur on the of thegrain material bound- properties on theary grain S1-gb12 and [9,10]. grain boundary affected by stress.

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MBN emission increased along with tensile stresses and decreased with compressive stresses, especially for uniaxial loading and elastic regimes [13]. The total tensile stress acting normal to the grain boundary during elastic deformation was a sum of the internal stress due to the applied stress, resolved onto the grain boundary plane [9]. The domain- wall motion in this case was a function of microstresses (the function of grain and grain- boundary microstructure). The microstress was higher on the grain boundary than on the grain, making the grain boundary more prone to early minor failure formation [8]. The time-response histograms and their features characterized time characteristics of DW mo- Figure 9. TheFigure curves 9. The for curves thetion RMS, for to the quantitatively mean, RMS, mean, and and peak analyzepeak value value ofthe of time-responsetime-r inhomogeneousesponse histograms histograms material’s under understre propertiesss, reflecting stress, reﬂectingtheaffected inhomo- by the the inhomogeneity of magnetic properties at different locations. FD denotes the fitting data, and ID denotes the original data. geneity of magnetic propertiesstress, at grain, different and locations. grain boundary. FD denotes the ﬁtting data, and ID denotes the original data.

FigureFigure 10. 10.TheThe images images for the for mean the of mean the time-response of the time-response histograms histogramsat different locations at different to reflect locations to re- theﬂect inhomogeneous the inhomogeneous magnetic properties magnetic at properties different locations. at different The size locations. of each image The is size 16.2 ofmm each × image is 13.616.2 mm. mm × 13.6 mm.

Figure 12. The images for the peak value of the time-response histograms at different locations to reflect the inhomogeneous magnetic properties at different locations. The size of each image is 16.2 mm × 13.6 mm.

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MBN emission increased along with tensile stresses and decreased with compressive stresses, especially for uniaxial loading and elastic regimes [13]. The total tensile stress actingMBN normal emission to the grain increased boundary along during with tensile elastic stresses deformation and decreased was a sum with of thecompressive internal stressstresses, due especiallyto the applied for uniaxial stress, resolved loading onandto theelastic grain regimes boundary [13]. plane The total[9]. The tensile domain- stress wallacting motion normal in thisto the case grain was boundary a function during of microstresses elastic deformation (the functi wason aof sum grain of andthe internal grain- boundarystress due microstructure). to the applied stress, The microstress resolved on wasto the higher grain on boundary the grain planeboundary [9]. The than domain- on the grain,wall motionmaking in the this grain case boundary was a function more ofpron microstressese to early minor (the functi failureon formationof grain and [8]. grain- The time-responseboundary microstructure). histograms and The their microstress features wascharacterized higher on time the grain characteristics boundary of than DW on mo- the tiongrain, to quantitativelymaking the grain analyze boundary the inhomogeneous more prone to earlymaterial’s minor properties failure formation affected [8].by theThe stress,time-response grain, and histograms grain boundary. and their features characterized time characteristics of DW motion to quantitatively analyze the inhomogeneous material’s properties affected by the stress, grain, and grain boundary.

Figure 10. The images for the mean of the time-response histograms at different locations to reflect the inhomogeneous magnetic properties at different locations. The size of each image is 16.2 mm × 13.6Figure mm. 10. The images for the mean of the time-response histograms at different locations to reflect Sensors 2021the, 21 ,inhomogeneous 2350 magnetic properties at different locations. The size of each image is 16.2 mm × 14 of 20 13.6 mm.

Figure 11. The images for the RMS of the time-response histograms at different locations to reflect the inhomogeneous magnetic properties at different locations. The size of each image is 16.2 mm × Figure13.6Figure 11.mm.The 11. images The images for the for RMS the of RMS the time-response of the time-response histograms hi atstograms different locationsat different to reﬂect locations the inhomogeneous to reflect magnetic properties at different locations. The size of each image is 16.2 mm × 13.6 mm. the inhomogeneous magnetic properties at different locations. The size of each image is 16.2 mm × 13.6 mm.

Figure 12. Figure The12. imagesThe images for the for peak the value peak of thevalue time-response of the time-res histogramsponse at differenthistograms locations at different to reﬂect locations the inhomogeneous to magnetic properties at different locations. The size of each image is 16.2 mm × 13.6 mm. reflect the inhomogeneous magnetic properties at different locations. The size of each image is 16.2 mmFigure × 13.6 12. mm. The images for the peak value of the time-response histograms at different locations to 4.3. Time-Division to Separate the Magnetic Properties’ Variation Affected by Stress, Grain, reflect the inhomogeneous magnetic properties at different locations. The size of each image is 16.2 and Grain Boundary mm × 13.6 mm. Due to stress transfer on the grain and grain boundary, the magnetic properties

and material properties were inhomogeneous [11]. The time-division was carried out to optimize the time interval to separate the material properties’ variation affected by the grain and the grain boundary. The main factor of time-division is to the method of localizing the time bands [21]. The time-division had a high resolution and less redundant information for the material properties’ variation separation. From Figure 14, there are two methods for time-division for the RMS feature, which are the time-division with ﬁve time bands and the time-division with three time bands. In this study, the optimal time-division was in line with the excitation frequency. From Figures4 and 14, it was found that the bins from 0.1 s to 0.3 s and from 0.7 s to 0.9 s highlighted the different magnetic properties inside the grain and around the grain boundary. Based on the time-response histograms at different locations shown in Figure 14, the time-division with three time bands was enough to separate the material properties affected by stress, grain, and grain boundary. The time bands p1 (0–0.3 s) and p3 (0.7–1 s) highlighted the 90◦ DWs around the grain boundary, which identiﬁed the material properties’ variation affected by the tensile stress and microstructure. The time bands p2 (0.3–0.7 s) found 180◦ DW motion, which determined the material properties’ variation affected by the tensile stress. Sensors 2021, 21, 2350 15 of 20 Sensors 2021, 21, x FOR PEER REVIEW 16 of 22

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0.3 s) and p3 (0.7–1 s) highlighted the 90° DWs around the grain boundary, which iden- Figuretified 13. The the material different properties’ elimination variation of magnetic affected propertiesby the tensile before stress theand tensilemicrostructure. test. FD denotes the ° Figureﬁtting 13. The data,The timedifferent and bands ID elimination denotes p2 (0.3–0.7 the of originals)magnetic found data.properties180 DW motion,before the which tensile determined test. FD denotes the material the fitting data, and ID denotes theproperties’ original data. variation affected by the tensile stress. 4.3. Time-Division to Separate the Magnetic Properties’ Variation Affected by Stress, Grain, and Grain Boundary Due to stress transfer on the grain and grain boundary, the magnetic properties and material properties were inhomogeneous [11]. The time-division was carried out to optimize the time interval to separate the material properties’ variation affected by the grain and the grain boundary. The main factor of time-division is to the method of localizing the time bands [21]. The time-division had a high resolution and less redundant information for the material properties’ variation separation. From Figure 14, there are two methods for time-division for the RMS feature, which are the time-division with five time bands and the time-division with three time bands. In this study, the optimal time-division was in line with the excitation frequency. From Figures 4 and 14, it was found that the bins from 0.1 s to 0.3 s and from 0.7 s to 0.9 s highlighted the different magnetic properties inside the grain and around the grain boundary. Based on the time-response histograms at different locations shown in Figure 14, the time-division with three time bands was enough to separate the material properties affected by stress, grain, and grain boundary. The time bands p1 (0–

Figure 14. TwoFigure methods 14. Two methods for time-division. for time-division. (a) Grain(a) Grain distribution; distribution; (b (b) time-division) time-division with three with time three bands; time (c) bands; time-division (c) time-division with five time bands; (d) MBN features with three time bands; (e) MBN features with five time bands. with ﬁve time bands; (d) MBN features with three time bands; (e) MBN features with ﬁve time bands. As shown in Figure 15, the time-division was used for the feature separation on each part of the sample. For the time bands p1 and p3, the difference between the time-response histogram features (∆, ∆, and ∆) around the grain boundary increased much more than inside the grain under 123 MPa. However, the variation of these features for time band p2 was almost uniform.

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As shown in Figure 15, the time-division was used for the feature separation on each part of the sample. For the time bands p1 and p3, the difference between the time-response histogram features (∆RMS, ∆mean, and ∆peak) around the grain boundary increased much Sensors 2021, 21, x FOR PEER REVIEW 18 of 22 more than inside the grain under 123 MPa. However, the variation of these features for time band p2 was almost uniform.

FigureFigure 15.15. TheThe time-divisiontime-division toto separateseparate thethe magneticmagnetic properties properties affected affected by by stress, stress, grain, grain, and and grain grain boundary. boundary. (a ,Thed,g) histo- illus- trategram the features∆RMS in, ∆ p1mean (0–0.3, and s) ∆andpeak p3on (0. the7–1 S1-g1,s) identified s1-g3, andthe variation s1-gb13; (affectedb,e,h) illustrate by the tensile the ∆RMS stress, ∆ andmean microstructure,, and ∆peak on and the S1-g1,the histogram s1-gb13, features and s1-gb23; in p2 ( c(0.3–0.7,f,i) illustrate s) identified the ∆RMS the ,variatio∆mean,n and affected∆peak byon the the tensile S1-g1, stress. s1-g2, andFD denotes s1-gb12. the The fitting histogram data, featuresand ID denotes in p1 (0–0.3 the original s) and p3data. (0.7–1 s) identified the variation affected by the tensile stress and microstructure, and the histogram features in p2 (0.3–0.7 s) identified the variation affected by the tensile stress. FD denotes the fitting data, and ID denotes the original data. When the tensile stress was 53 MPa, the material properties on the grains and the grain boundaries were uniform. When the tensile stress increased to 123 MPa, due to the influence of stress and the material microstructure, material properties on grain boundaries varied significantly, reflected by the time bands p1 and p3. However, for time band p2, the variation in the time-response histogram features was almost uniform from 53 MPa to 123 MPa. The optimal time-division was based on the correlation between the applied field and the DW motion on different locations. When the stress increased, the 90° DW appeared

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When the tensile stress was 53 MPa, the material properties on the grains and the grain boundaries were uniform. When the tensile stress increased to 123 MPa, due to the influence of stress and the material microstructure, material properties on grain boundaries varied significantly, reflected by the time bands p1 and p3. However, for time band p2, the variation in the time-response histogram features was almost uniform from 53 MPa to 123 MPa. The optimal time-division was based on the correlation between the applied field and ◦ the DW motion on different locations. When the stress increased, the 90 DW appeared around the grain boundary. It took more time on the grain boundary to reach magnetic saturation during the magnetization process. As a result, the MBN activities lasted longer, and the time-response histogram features were larger on the grain boundary. After the difference elimination of material properties before the tensile test, the time-division separated the material properties affected by the grain and the grain boundary. The time bands p1 and p3 highlighted the different variations of micro- and macromagnetic properties on the grain and grain boundary under stress, while the time band p2 removed the variation of material properties affected by the microstructure under stress. In order to further analyze the sensitivity of the time-division for the material properties’ variation separation with different stress states directly, contrast values [49] were extracted from RMS, mean, and peak at the time intervals P1, P2, and P3. Contrast value is defined as: v u 2 u N−1 ( ) − t ∑x=0 Im x Im contrast = (16) N

where Im(x) denotes the curve of time-response histogram features Im denotes the mean of Im(x), N denotes the max of X-coordinate, and m and n denote the value of Y-coordinate and X-coordinate, respectively. From Table2, the contrast values of ∆RMS, ∆mean, and ∆peak in the time intervals P1 and P3 increased with increment of stress, while these values in the time interval P2 exhibited a minor change. These results for contrast values showed that the time-division separated the variation of magnetic properties affected by the stress, grain, and grain boundary. The time-response histograms characterized the different time characteristics of the MBN signal inside the grain and grain boundary. The optimal time-division highlighted variations of micro- and macromagnetic properties affected by the grain, grain boundary, and stress. Based on the approach proposed in this paper, the standard sample of the ferromagnetic material without external stress or the residual magnetic ﬁeld of the sample will be used as a reference to reduce the difference of material properties before tensile test to improve the results of the evaluation of inhomogeneous material properties affected by the grain, grain boundary, and stress [50]. After removal of background information, the different time responses at different locations can analyze inhomogeneity of material properties for ferromagnetic material on a microscopic scale.

4.4. Conclusions and Future Work In this paper, based on the observation of grain and grain-boundary distribution, the time-response histogram of MBN was established to nondestructively characterize silicon steel sheet’s inhomogeneous magnetic properties affected by the grain and grain boundary during elastic deformation. The main conclusions are summarized as follows: (1) The time-response histogram characterized the difference in DW motion at different locations. The time-response histogram became different on the grain and grain boundary with increased stress, characterizing the variation of the inhomogeneous magnetic properties under stress. (2) The varying degrees of magnetic properties are much higher around the grain boundary than the grain interior with the stress state. In particular, MBN signals around the Sensors 2021, 21, 2350 18 of 20

grain boundary S1-gb12 were much higher than at other locations, reﬂecting the fact that the grain boundaries were more susceptible to external tensile stress. (3) The RMS, mean, and peak were extracted from the time-response histograms to quantify the magnetic properties’ variation on the grain and grain boundary. These three features had the same tendency at different locations under stress. (4) After difference elimination of magnetic properties before the tensile test, the time- division was carried out to extract the RMS, mean, and peak in the optimized time interval, which separated the variations of magnetic properties affected by stress, Sensors 2021, 21, x FOR PEER REVIEW grain, and grain boundary. The optimal time intervals highlighted the different20 of varia-22 Sensors 2021, 21, x FOR PEER REVIEW 20 of 22 Sensors 2021, 21, x FOR PEER REVIEW tions in magnetic properties affected by the microstructure and stress, respectively.20 of 22

Table 2.Table The contrast 2. The contrast value value of ∆ of ∆,RMS ∆, ∆mean, and, and∆∆peak in inthe the time time intervals intervals P1, P2,P2, and and P3. P3. Table 2. The contrast value of ∆, ∆, and ∆ in the time intervals P1, P2, and P3. Table 2. The contrast value of ∆, ∆, and ∆ in the time intervals P1, P2, and P3. 5353 MPa 123 123 MPa MPa 53 MPa 123 MPa LocationLocation FeatureFeature P1P1 53 P2MPa P3 P3 P1 P1 123P2 MPa P2 P3 P3 Location Feature P1 P2 P3 P1 P2 P3 Location Feature∆ 0.0059P1 0.0079P2 0.0081P3 0.0202P1 0.0071P2 0.0191P3 ∆ 0.0059 0.0079 0.0081 0.0202 0.0071 0.0191 ∆∆ 0.00590.0068 0.00790.0102 0.00810.0125 0.02020.0241 0.00710.0085 0.01910.0201 ∆ 0.0068 0.0102 0.0125 0.0241 0.0085 0.0201 ∆mean∆ 0.00590.0068 0.0102 0.0079 0.0125 0.0081 0.0241 0.0202 0.0085 0.0071 0.0201 0.0191

∆ 0.0155 0.0157 0.0169 0.0361 0.0141 0.0279 ∆ 0.0155 0.0157 0.0169 0.0361 0.0141 0.0279 ∆RMS∆ 0.00680.0155 0.0157 0.0102 0.0169 0.0125 0.0361 0.0241 0.0141 0.0085 0.0279 0.0201 ∆peak 0.0155 0.0157 0.0169 0.0361 0.0141 0.0279

∆ 0.0071 0.0101 0.0066 0.0236 0.0174 0.0238 ∆ 0.0071 0.0101 0.0066 0.0236 0.0174 0.0238 ∆∆ 0.00710.0061 0.01010.0097 0.00660.0061 0.02360.0272 0.01740.0158 0.02380.0381 ∆ ∆ 0.0061 0.0097 0.0061 0.0272 0.0158 0.0381 ∆mean 0.00710.0061 0.0097 0.01010.0061 0.00660.0272 0.02360.0158 0.01740.0381 0.0238 ∆ 0.0112 0.0109 0.0071 0.0355 0.0164 0.0702 ∆ 0.0112 0.0109 0.0071 0.0355 0.0164 0.0702 ∆ 0.0112 0.0109 0.0071 0.0355 0.0164 0.0702 ∆RMS 0.0061 0.0097 0.0061 0.0272 0.0158 0.0381

∆peak 0.0112 0.0109 0.0071 0.0355 0.0164 0.0702 ∆ 0.0096 0.0121 0.0167 0.0293 0.0133 0.0329 ∆ 0.0096 0.0121 0.0167 0.0293 0.0133 0.0329 ∆∆ 0.00960.0124 0.01210.0114 0.01670.0177 0.02930.0364 0.01330.0184 0.03290.0551 ∆ 0.0124 0.0114 0.0177 0.0364 0.0184 0.0551 ∆ 0.0124 0.0114 0.0177 0.0364 0.0184 0.0551 ∆mean 0.0096 0.0121 0.0167 0.0293 0.0133 0.0329 ∆ 0.0179 0.0121 0.0248 0.0392 0.0181 0.0981 ∆ 0.0179 0.0121 0.0248 0.0392 0.0181 0.0981 ∆ 0.0179 0.0121 0.0248 0.0392 0.0181 0.0981

∆RMS 0.0124 0.0114 0.0177 0.0364 0.0184 0551

∆peak 0.0179 0.0121 0.0248 0.0392 0.0181 0.0981 4.4. Conclusion and Future Work 4.4. Conclusion and Future Work 4.4. ConclusionIn this paper, and basedFuture on Work the observation of grain and grain-boundary distribution, the In thisThis paper, study based provides on the a theoretical observation and of experimentalgrain and grain-boundary basis for a better distribution, understanding the time-responseIn this paper, histogram based on of theMBN observation was established of grain to and nondestructively grain-boundary characterize distribution, silicon the time-responseof the effect histogram of grain and of MBN grain was boundary established on the to nondestructively evaluation of inhomogeneous characterize silicon material time-responsesteel sheet’s inhomogeneous histogram of MBN magnetic was established properties toaffected nondestructively by the grain characterize and grain bound- silicon steelproperties sheet’s inhomogeneous for other ferromagnetic magnetic material properti (especiales affected Q235, by steel. the grain etc.) byand using grain MBN bound- [51,52]. steelary during sheet’s elastic inhomogeneous deformation. magnetic The main properti conclusionses affected are summarizedby the grain asand follows: grain bound- aryThe during time-response elastic deformation. histogram The of main the MBN conclusions signal canare summarized analyze the variationas follows: of magnetic ary(1) propertiesduringThe time-response elastic on thedeformation. grain histogram and The grain characterized main boundary conclusions th duringe difference are plasticsummarized in deformation, DW motionas follows: at especially different the (1) The time-response histogram characterized the difference in DW motion at different (1) variationThelocations. time-response in inhomogeneityThe time-response histogram of magnetic characterizedhistogram properties be thcamee difference before different crack in onDW formation the motion grain on at aand different microscopic grain locations. The time-response histogram became different on the grain and grain scalelocations.boundary [2]. The Thewith high-sample-rate time-response increased stress, acquisition histogram characteri method bezingcame the improves different variation the on timeof the the to graininhomogeneous resolve and DW grain motion boundary with increased stress, characterizing the variation of the inhomogeneous detection,boundarymagnetic and propertieswith the increased high-spatial-resolution under stress, stress. characteri magneticzing the probe variation improves of the the inhomogeneous resolution for the magnetic properties under stress. (2) characterizationmagneticThe varying properties degrees of the under inhomogeneous of magnetic stress. properties material’s are properties much higher on the microscopicaround the scale.grain (2) The varying degrees of magnetic properties are much higher around the grain (2) Theboundary varying than degrees the grain of magneticinterior with properties the stress are state.much In higher particular, around MBN the signals grain boundary than the grain interior with the stress state. In particular, MBN signals Authorboundaryaround Contributions: the than grain the boundary grainConceptualization, interior S1-gb12 with were B.G. the much andstress higher Y.X.; state. data than In curation, atparticular, other K.Z.; locations, MBN investigation, signalsreflect- J.L. around the grain boundary S1-gb12 were much higher than at other locations, reflect- andarounding Q.L.; the factthe methodology, grainthat the boundary grain J.L. and boundaries S1-gb12 K.Z.; resources, were were much more G.T. higher andsusceptible B.G.; than supervision, at to other external locations, G.T., tensile B.G., reflect- stress. and Y.X.; writing—originaling the fact that draft, the grain J.L.; writing—review boundaries were and more editing, susceptible J.L. All authors to external haveread tensile and stress. agreed to (3) ingThe the RMS, fact mean,that the and grain peak boundaries were extracted were more from susceptible the time-response to external histograms tensile stress. to (3) theThe published RMS, mean, version and of the peak manuscript. were extracted from the time-response histograms to (3) Thequantify RMS, the mean, magnetic and properties’peak were variationextracted onfrom the thegrain time-response and grain boundary. histograms These to quantify the magnetic properties’ variation on the grain and grain boundary. These quantifythree features the magnetic had the sameproperties’ tendency variation at different on the locations grain and under grain stress. boundary. These three features had the same tendency at different locations under stress. (4) threeAfter featuresdifference had elimination the same tendency of magnetic at different properties locations before underthe tensile stress. test, the time- (4) After difference elimination of magnetic properties before the tensile test, the time- (4) Afterdivision difference was carried elimination out to extractof magnetic the RMS, properties mean, beforeand peak the intensile the optimized test, the time- time division was carried out to extract the RMS, mean, and peak in the optimized time divisioninterval, waswhich carried separated out to the extract variations the RMS, of magneticmean, and properties peak in the affected optimized by stress, time interval, which separated the variations of magnetic properties affected by stress, interval,grain, and which grain separated boundary. the The variations optimal time of magnetic intervals propertieshighlighted affected the different by stress, var- grain, and grain boundary. The optimal time intervals highlighted the different var- grain,iations and in magnetic grain boundary. properties The affected optimal by time the microstructureintervals highlighted and stress, the different respectively. variations in magnetic properties affected by the microstructure and stress, respectively. iationsThis study in magnetic provides properties a theoretical affected and expeby therimental microstructure basis for and a better stress, understanding respectively. This study provides a theoretical and experimental basis for a better understanding of theThis effect study of grainprovides and agrain theoretical boundary and onexpe therimental evaluation basis of for inhomogeneous a better understanding material of the effect of grain and grain boundary on the evaluation of inhomogeneous material ofproperties the effect for of othergrain andferromagnetic grain boundary material on the(especial evaluation Q235, of steel. inhomogeneous etc.) by using material MBN properties for other ferromagnetic material (especial Q235, steel. etc.) by using MBN properties[51,52]. The for time-response other ferromagnetic histogram material of the MBN (especial signal Q235, can analyze steel. etc.)the variation by using of MBNmag- [51,52]. The time-response histogram of the MBN signal can analyze the variation of mag- [51,52].netic properties The time-response on the grain hist andogram grain of thebounda MBNry signal during can plastic analyze deformation, the variation especially of magnetic properties on the grain and grain boundary during plastic deformation, especially neticthe variation properties in inhomogeneityon the grain and of grainmagnetic bounda propertiesry during before plastic crack deformation, formation on especially a micro- the variation in inhomogeneity of magnetic properties before crack formation on a micro- thescopic variation scale [2].in inhomogeneity The high-sample-rate of magnetic acquis propertiesition method before improves crack formation the time on to a resolvemicro- scopic scale [2]. The high-sample-rate acquisition method improves the time to resolve scopic scale [2]. The high-sample-rate acquisition method improves the time to resolve

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Funding: This work is funded by the National Natural Science Foundation of China (Grant Nos. 61527803, 61960206010 and 61971093). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available as the data forms part of an ongoing study. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References 1. Wang, H.; Dong, L.; Wang, H.; Ma, G.; Xu, B.; Zhao, Y. Effect of tensile stress on metal magnetic memory signals during on-line measurement in ferromagnetic steel. NDT E Int. 2021, 117, 102378. [CrossRef] 2. Habib, K.; Koyama, M.; Tsuchiyama, T.; Noguchi, H. Dislocation motion at a fatigue crack tip in a high-nitrogen steel clarified through in situ electron channeling contrast imaging. Mater. Charact. 2019, 158, 109930. [CrossRef] 3. Meng, C.; Hu, W.; Sandlöbes, S.; Korte-Kerzel, S.; Gottstein, G. The effect of large plastic deformation on elevated temperature mechanical behavior of dynamic strain aging Al-Mg alloys. Acta Mater. 2019, 181, 67–77. [CrossRef] 4. Jun, H.-K.; Seo, J.-W.; Jeon, I.-S.; Lee, S.-H.; Chang, Y.-S. Fracture and fatigue crack growth analyses on a weld-repaired railway rail. Eng. Fail. Anal. 2016, 59, 478–492. [CrossRef] 5. Lee, S.-H.; Kim, S.H.; Chang, Y.S. Fatigue life assessment of railway rail subjected to welding residual and contact stresses. J. Mech. Sci. Technol. 2014, 28, 4483–4491. [CrossRef] 6. Donegan, S.P.; Kumar, N.; Groeber, M.A. Associating local microstructure with predicted thermally-induced stress hotspots using convolutional neural networks. Mater. Charact. 2019, 158, 109960. [CrossRef] 7. Zarei, S.; Nedoushan, R.J.; Atapour, M. The sources of the micro stress and strain inhomogeneity in dual phase steels. Mater. Sci. Eng. A 2016, 674, 384–396. [CrossRef] 8. Zhao, Y.; Tryon, R. Automatic 3-D simulation and micro-stress distribution of polycrystalline metallic materials. Comput. Methods Appl. Mech. Eng. 2004, 193, 3919–3934. [CrossRef] 9. Johnson, D.C.; Kuhr, B.; Farkas, D.; Was, G.S. Quantitative linkage between the stress at dislocation channel–Grain boundary interaction sites and irradiation assisted stress corrosion crack initiation. Acta Mater. 2019, 170, 166–175. [CrossRef] 10. Kumar, S.; Kurtz, S.K.; Agarwala, V.K. Micro-stress distribution within polycrystalline aggregate. Acta Mech. 1996, 114, 203–216. [CrossRef] 11. Ryu, K.S.; Akinaga, H.; Shin, S.C. Tunable scaling behaviour observed in Barkhausen criticality of a ferromagnetic film. Nat. Phys. 2007, 3, 547. [CrossRef] 12. Burgess, J.A.J.; Fraser, A.E.; Sani, F.F.; Vick, D.; Hauer, B.D.; Davis, J.P.; Freeman, M.R. Quantitative magneto-mechanical detection and control of the Barkhausen effect. Science 2013, 339, 1051–1054. [CrossRef] 13. Jiles, D.C. Dynamics of domain magnetization and the Barkhausen effect. Czechoslov. J. Phys. 2000, 50, 893–924. [CrossRef] 14. Herranen, T.; Laurson, L. Barkhausen Noise from Precessional Domain Wall Motion. Phys. Rev. Lett. 2019, 122, 117205. [CrossRef] 15. Colaiori, F. Exactly solvable model of avalanches dynamics for Barkhausen crackling noise. Adv. Phys. 2008, 57, 287–359. [CrossRef] 16. Vazquez, A.; Sotolongo-Costa, O. Dynamics of a domain wall in soft-magnetic materials: Barkhausen effect and relation with sandpile models. Phys. Rev. Lett. 2000, 84, 1316. [CrossRef] 17. Zhukova, V.; Blanco, J.M.; Corte-Leon, P.; Ipatov, M.; Churyukanova, M.; Taskaev, S.; Shukov, A. Grading the magnetic anisotropy and engineering the domain wall dynamics in Fe-rich microwires by stress-annealing. Acta Mater. 2018, 155, 279–285. [CrossRef] 18. Sánchez, J.C.; De Campos, M.F.; Padovese, L.R. Magnetic Barkhausen emission in lightly deformed AISI 1070 steel. J. Magn. Magn. Mater. 2012, 324, 11–14. [CrossRef] 19. Yelbay, H.I.; Cam, I.; Gür, C.H. Non-destructive determination of residual stress state in steel weldments by Magnetic Barkhausen Noise technique. NDT E Int. 2010, 43, 29–33. [CrossRef] 20. Ding, S.; Tian, G.; Sutthaweekul, R. Non-destructive hardness prediction for 18CrNiMo7-6 steel based on feature selection and fusion of Magnetic Barkhausen Noise. NDT E Int. 2019, 107, 102138. [CrossRef] 21. Maciusowicz, M.; Psuj, G. Use of Time-Dependent Multispectral Representation of Magnetic Barkhausen Noise Signals for the Needs of Non-Destructive Evaluation of Steel Materials. Sensors 2019, 19, 1443. [CrossRef][PubMed] 22. Lan, L.; Jin, X.; Gao, S.; He, B.; Rong, Y. Microstructural evolution and stress state related to mechanical properties of electron beam melted Ti-6Al-4V alloy modified by laser shock peening. J. Mater. Sci. Technol. 2020, 50, 153–161. [CrossRef] 23. Wang, P.; Lao, C.S.; Chen, Z.W.; Liu, Y.K.; Wang, H.; Wendrock, H.; Eckert, J.; Scudino, S. Microstructure and mechanical properties of Al-12Si and Al-3.5 Cu-1.5 Mg-1Si bimetal fabricated by selective laser melting. J. Mater. Sci. Technol. 2020, 36, 18–26. [CrossRef] 24. Neslušan, M.; Minárik, P.; Cillikovˇ á, M.; Kolaˇrík, K.; Rubesova, K. Barkhausen noise emission in tool steel X210Cr12 after semi-solid processing. Mater. Charact. 2019, 157, 109891. [CrossRef] Sensors 2021, 21, 2350 20 of 20

25. Neslušan, M.; Cˇ ížek, J.; Kolaˇrík, K.; Minárik, P.; Cillikovˇ á, M.; Melikhova, O. Monitoring of grinding burn via Barkhausen noise emission in case-hardened steel in large-bearing production. J. Mater. Sci. Technol. 2017, 240, 104–117. [CrossRef] 26. Yamaura, S.; Furuya, Y.; Watanabe, T. The effect of grain boundary microstructure on Barkhausen noise in ferromagnetic materials. Acta Mater. 2001, 49, 3019–3027. [CrossRef] 27. Anglada-Rivera, J.; Padovese, L.R.; Capo-Sanchez, J. Magnetic Barkhausen noise and hysteresis loop in commercial carbon steel: Influence of applied tensile stress and grain size. J. Magn. Magn. Mater. 2001, 231, 299–306. [CrossRef] 28. Maciusowicz, M.; Psuj, G. Time-Frequency Analysis of Barkhausen Noise for the Needs of Anisotropy Evaluation of Grain- Oriented Steels. Sensors 2020, 20, 768. [CrossRef][PubMed] 29. Ghanei, S.; Alam, A.S.; Kashefi, M.; Mazinani, M. Nondestructive characterization of microstructure and mechanical properties of intercritically annealed dual-phase steel by magnetic Barkhausen noise technique. Mater. Sci. Eng. A 2014, 607, 253–260. [CrossRef] 30. Imafuku, M.; Suzuki, H.; Akita, K.; Iwata, K.; Fujikura, M. Effects of laser irradiation on iron loss reduction for Fe–3% Si grain-oriented silicon steel. Acta Mater. 2005, 53, 939–945. [CrossRef] 31. Xiuhua, G.; Kemin, Q.; Chunlin, Q. Magnetic properties of grain oriented ultra-thin silicon steel sheets processed by conventional rolling and cross shear rolling. Mater. Sci. Eng. A 2006, 430, 138–141. [CrossRef] 32. Liu, J.; Tian, G.Y.; Gao, B.; Zeng, K.; Zheng, Y.; Chen, J. Micro-macro characteristics between domain wall motion and magnetic Barkhausen noise under tensile stress. J. Magn. Magn. Mater. 2020, 493, 165719. [CrossRef] 33. Vashista, M.; Moorthy, V. Influence of applied magnetic field strength and frequency response of pick-up coil on the magnetic Barkhausen noise profile. J. Magn. Magn. Mater 2013, 345, 208–214. [CrossRef] 34. Wilson, J.W.; Tian, G.Y.; Barrans, S. Residual magnetic field sensing for stress measurement. Sens. Actuators A Phys. 2007, 135, 381–387. [CrossRef] 35. Aranas, C., Jr.; He, Y.; Podlesny, M. Magnetic Barkhausen noise characterization of two pipeline steels with unknown history. Mater. Charact. 2018, 146, 243–257. [CrossRef] 36. Bertotti, G.; Durin, G.; Magni, A. Scaling aspects of domain wall dynamics and Barkhausen effect in ferromagnetic materials. J. Appl. Phys. 1994, 75, 5490–5492. [CrossRef] 37. Smith, R.C.; Dapino, M.J.; Braun, T.R.; Mortensen, A.P. A homogenized energy framework for ferromagnetic hysteresis. IEEE Trans. Magn. 2006, 42, 1747–1769. [CrossRef] 38. Winning, M. Influencing grain boundary properties by the application of mechanical stress fields. Phys. Status Solidi 2004, 201, 2867–2879. [CrossRef] 39. Winning, M.; Gottstein, G.; Shvindlerman, L.S. On the mechanisms of grain boundary migration. Acta Mater. 2002, 50, 353–363. [CrossRef] 40. Liu, J.; Tian, G.Y.; Gao, B.; Zeng, K.; Qiu, F. Domain wall characterization inside grain and around grain boundary under tensile stress. J. Magn. Magn. Mater. 2019, 471, 39–48. [CrossRef] 41. Shin, S.; Schafer, R.; Cooman, B.C.D. Grain Boundary Penetration by Lancet Domains in Fe-3%Si Grain-Oriented Steel. IEEE Trans. Magn. 2010, 46, 3574–3581. [CrossRef] 42. Calkins, F.T.; Smith, R.C.; Flatau, A.B. Energy-based hysteresis model for magnetostrictive transducers. IEEE Trans. Magn. 2000, 36, 429–439. [CrossRef] 43. Alessandro, B.; Beatrice, C.; Bertotti, G.; Montorsi, A. Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. II. Experiments. J. Appl. Phys. 1990, 68, 2908–2915. [CrossRef] 44. Zhang, Y.; Zuo, T.T.; Tang, Z.; Gao, M.C.; Dahmen, K.A.; Liaw, P.K.; Lu, P.Z. Microstructures and properties of high-entropy alloys. Prog. Mater. Sci. 2014, 61, 1–93. [CrossRef] 45. Bobylev, S.V.; Ovid’Ko, I.A. Stress-driven migration of deformation-distorted grain boundaries in nanomaterials. Acta Mater. 2015, 88, 260–270. [CrossRef] 46. Perez-Benitez, J.A.; Espina-Hernandez, J.H.; Le Manh, T.; Caleyo, F. Identification of different processes in magnetization dynamics of API steels using magnetic Barkhausen noise. J. Phys. D Appl. Phys. 2015, 48, 295002. [CrossRef] 47. Duan, H.L.; Wang, J.; Huang, Z.P.; Luo, Z.Y. Stress concentration tensors of inhomogeneities with interface effects. Mech. Mater. 2005, 37, 723–736. [CrossRef] 48. Brünig, M.; Gerke, S.; Hagenbrock, V. Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. Int. J. Plast. 2013, 50, 49–65. [CrossRef] 49. Peli, E. Contrast in complex images. JOSA A 1990, 7, 2032–2040. [CrossRef][PubMed] 50. Neslušan, M.; Cˇ ížek, J.; Zgútová, K. Microstructural transformation of a rail surface induced by severe thermoplastic deformation and its non-destructive monitoring via Barkhausen noise. Wear 2018, 402, 38–48. [CrossRef] 51. Camarillo-Garcia, J.P.; Hernández-Navarro, F.; Soto-Parra, D.E.; Rios-Jara, D.; Flores-Zuniga, H. High sensitivity on caloric effects induced by stress or magnetic field in a polycrystalline Ni-Mn-In-Co-Cu shape-memory alloy. Scr. Mater. 2019, 166, 92–95. [CrossRef] 52. Hier-Majumder, S.; Leo, P.H.; Kohlstedt, D.L. On grain boundary wetting during deformation. Acta Mater. 2004, 52, 3425–3433. [CrossRef]