# Magnetic Hysteresis and Barkausen Noise in Plastically Deformed Steel Sheets

^{*}

## Abstract

**:**

## 1. Introduction

^{−3}, is slightly anisotropic, showing a linear strain, around some 10

^{−5}, along the direction of the magnetization

**M**. The cell elongates along such a direction, until the sum of the decreasing magnetic energy and the correspondingly increasing elastic energy attains a minimum. Conversely, if the mechanical deformation is imposed, by residual or applied stresses, the magnetoelastic interaction will influence the local direction of

**M**and, globally, the magnetic state of the material. The magnetoelastic effect in a polycrystalline steel sample is conveniently expressed through the isotropic magnetostriction constant λ

_{s}, the relative elongation of the magnetically saturated sample with respect to a demagnetized state endowed with isotropically distributed domains. The energy of interaction between uniaxial stress

**σ**and magnetization

**M**forming an angle θ is then obtained, per unit volume, as E

_{σ}= (3/2)λ

_{s}σsin

^{2}θ. It is precisely this energy term that enters the magnetic phenomenology of the plastically deformed steels. Investigating such a phenomenology has obvious direct interest for steel sheets applied as magnetic cores of electrical devices, like transformers and motors, but it has also great appeal for the nondestructive assessment of the mechanical properties of the structural steels. It is remarkable that non-oriented and grain-oriented steel sheets can achieve their present-day excellent properties only through severe cold-rolling as intermediate preparatory steps [1,2], but their loss and permeability figures eventually suffer from the plastic strain engendered by sheet cutting and core forming [3,4]. On the other hand, both ferritic and austenitic steels undergoing structural changes upon machining and cold-working can nondestructively be tested by magnetic methods [5,6,7,8,9].

## 2. Dislocations, Residual Stress, Hysteresis Loop and Coercivity

#### 2.1. Plastic Strain and Magnetization Process

_{c}∝ (τ − τ

_{0})

^{1/2}, where τ is the resolved shear stress and τ

_{0}is the critical shear stress, along stage I and H

_{c}∝ (τ − τ

_{0}) along stages II and III [19]. This amounts to state that the coercivity is proportionally related to the dislocation density ρ as H

_{c}∝ ρ

^{1/2}. The distinction between the different deformation stages becomes blurred in polycrystalline materials, because the different orientations of the grains impose the activation of different slip planes since the inception of plastic straining, in order to preserve the mechanical coherence between neighbouring grains. Consequently, the work-hardening rate is the highest at low strains, as shown for interstitial-free (IF) steels [20] and ultra-low-carbon steels [21] of similarly small grain size (<s> = 17 μm and <s> = 16 μm, respectively), and for a very pure large-grained (<s> = 800 μm) Fe sample [22]. It is observed in fact (Figure 2a) that the work-hardening Δσ follows to good extent a square root dependence on plastic strain Δσ ∝ ε

_{p}

^{1/2}, so that the coercive field, according to the previously introduced linear dependence H

_{c}∝ (τ − τ

_{0}) occurring along stages II and III in single Fe crystals, is expected to approximate the law H

_{c}∝ ε

_{p}

^{1/2}. Figure 3 shows that such a law approximately holds in the tensile strained fine-grained IF and ultra-low carbon steels, where different deformation stages cannot be recognized. The H

_{c}dependence on ε

_{p}in the large-grained extra-pure Fe sample is instead reminiscent of the response of the single crystal, with stage II following stage I above about ε

_{p}= 2% [22].

_{p}= 1.72 T, up to plastic elongation ε

_{p}= 4.5%, in the tensile strained IF steel strip samples (length 300 mm, width 30 mm, thickness 1.25 mm). These strips were strained at a constant deformation rate dε/dt = 1.7 × 10

^{−4}s

^{−1}using an INSTRON machine. The dependence of the loop areas (hysteresis energy loss) on ε

_{p}is consistent with that of the coercive field reported in Figure 3. The dramatic drop of the remanent polarization value, already occurring with deformation values as low as ε

_{p}= 0.5%, is driven by the compressive residual stress, easily engendering, in the demagnetized state, the transition of the magnetization inside the domains towards the easy axes far-off the stress (i.e., field) direction. It is remarked in Figure 5b, in agreement with previous findings of the literature [24,25], that all the plastically strained loops tend to cross at the same point in the second and fourth quadrants. It has been suggested that the additional field H

_{1}required to achieve the same polarization value attained in the unstrained material can be expressed as the product H

_{1}(σ, J) = h(σ) g(J) [24]. The coincidence points are assumed to be the ones where g(J) = 0, bringing the additional field to zero. While the physical mechanism by which the condition g(J) = 0 should arise is not clear, we observe in Figure 5b that the slope of the hysteresis loop in the second quadrant, across the region surrounding the coercive field H

_{c}(ε

_{p}), behaves in approximately inverse fashion with respect to the quantity ΔH

_{c}. This is identified with good approximation with the additional contribution to the coercivity deriving from plastic straining, the field value at the intersection point being practically coincident with the coercive field of the unstrained sample. By denoting with μ

_{a}the average differential permeability through this region, we thus have

_{c}(ε

_{p}) μ

_{a}~ const.

_{a}ensuing from plastic straining (e.g., from ~2 × 10

^{4}to ~7 × 10

^{3}) across the H

_{c}region on passing from ε

_{p}= 0 to ε

_{p}= 0.5%). If we then recognize a proportional relationship between ΔH

_{c}(ε

_{p}) and an equivalent internal demagnetizing factor N

_{d}~ 1/μ

_{a}, we arrive at justifying Equation (1).

#### 2.2. Tensile Straining versus Cold-Rolling

#### 2.3. Strain Hardening by Cutting

_{p}= 1.5 T are found to increase from 60 A/m to 97 A/m and from 51.9 mJ/kg to 62.9 mJ/kg, respectively. The somewhat inflated behavior of the loop around the coercivity region is reminding of the behavior of the loops after plastic straining shown in Figure 5 and Figure 6.

_{pc}to the hardened side bands of width L

_{c}and the value J

_{p0}> J

_{pc}to the inner region of width w

_{0}= w − 2L

_{c}(see inset of Figure 7b). The measured magnetization J

_{p}under defined H value is given by the weighted sum of J

_{p0}and J

_{pc}, according to

_{c}, and by J

_{p}= J

_{pc}for w ≤ 2L

_{c}. By measuring the magnetization curve at two strip widths, one can derive the quantities J

_{p0}and (J

_{p0}− J

_{pc}) L

_{c}as a function of H. We then obtain a family of curves for a convenient number of H values, which describe the dependence of J

_{p}on w, until the minimum width 2L

_{c}. The overall behavior of these curves permits one to estimate L

_{c}, which is typically of the order of 1–3 mm, depending on the cutting method [3,33]. Using then the same simplifying scheme leading to Equation (2), we write for the hysteresis loss W

_{h}at given J

_{p}value

_{h0}and W

_{hc}are the energy loss densities in the unscathed strip portion of width w

_{0}and the work-hardened edge bands, respectively, L

_{c}has been estimated via Equation (2), and w ≥ 2L

_{c}. The overall dependence of the normalized energy loss W

_{h,norm}= W

_{h}/W

_{h0}on w predicted by Equation (3) is shown for J

_{p}= 1.5 T in Figure 7b (solid lines). The curves refer to both punched and water-jet cut M440-65A (thickness d = 0.638 mm) and M440-50A (d = 0.470 mm) Fe–Si strips and agree with the evolution of the experimental W

_{h,norm}values (symbols). The lower magnetic deterioration brought about by water-jet cutting with respect to punching is apparent.

## 3. Barkhausen Noise in Plastically Deformed Steels

#### 3.1. Properties and Measurement of the Barkhausen Noise

_{m}is in the range 0.1 Hz–10 Hz and, in order to separate the noise from the continuous secondary signal, high-pass filtering is adopted; the higher f

_{m}the higher the cutoff frequency. By filtering, however, one may deprive the noise signal of significant information. In most cases the true rms value of the Barkhausen signal versus time (i.e., applied field) is measured along the cycle, and its envelope, resulting from an appropriately large number of averages, is empirically connected through its parameters (e.g., maximum value, integral over the half-period, pulse counting, amplitude distribution, etc.) to the specific state or property of the material. The effect of plastic deformation on these parameters is apparent, though the direction of the involved changes is not universally shared and contradictory results are often reported in the literature. We see in Figure 8, taken from [34], the case of Armco Fe sheets subjected to either 10% cold-rolling or 5% tensile deformation, whose BN envelopes versus applied field show quite different behaviors, with the tensile strained samples showing a little increase of their Barkhausen emission with respect to the unstrained samples. On the other hand, the results reported in Figure 9a, concerning ultra-low carbon steels, bring to light a drastic decrease of the BN in 10% tensile strained samples [21].

_{p}= ±1.17 T under both triangular H(t) (H

_{p}= ±130 A/m) and triangular J(t) (dJ/dt = 3.5 × 10

^{−2}T/s) at a frequency around 0.01 Hz. Each point of the profile is the result of noise averaging over a sufficiently high number of successive hysteresis cycles, which realize locally the stationary conditions of a statistical ensemble. However, by imposing a triangular H(t) waveform, the hysteresis loop is traversed with rapidly changing magnetization rate, peaking around the coercive field, where the differential permeability μ

_{diff}≅ dJ/dH is maximum. Consequently, the BN power is compressed inside a narrow field (time) interval and it is quite difficult to distinguish between the effect of the impressed magnetization rate and that of μ

_{diff}. It is apparent in Figure 10 how with constant dJ/dt one can highlight and meaningfully compare the specific features of the magnetization process at different points of the magnetization curve. It is noted, in particular, that P attains its maximum value close to the region where dμ

_{diff}/dJ is maximum, because of the rearrangement of the domain structure following the passage through the remanence. It was actually shown in [35] that the phenomenological relationship P = aμ

_{diff}+ b (dμ

_{diff}/dJ), with a and b constants, applies in grain-oriented and non-oriented sheets. Making the measurements under constant dJ/dt appears then a natural way to unambiguously assess the role of plastic straining on the noise.

#### 3.2. The Spectral Density of the Barkhausen Noise

_{p}= ±1.2 T. The 300 mm long strip samples are placed between the pole faces of a double-C soft yoke and the field is supplied by means of an enwrapping solenoid, supplied by an appropriately shaped and low-pass filtered current waveform, such as to impress the desired constant dJ/dt value. The noise is detected either by a couple of 200-turn windings connected in series opposition located at a distance of 90 mm or by a C-yoke of Mn-Zn ferrite endowed with a 100-turn pickup coil, whose pole faces are placed on the strip surface. To note that, by using the two counter-wound far-apart pickup coils, one can compensate the continuous signal without interfering with the noise signal. Details on the measuring method, including the analysis in the frequency domain, and setup are given in [37]. Figure 11 shows that plastic deformation affects in opposite fashion the low-frequency and high-frequency behavior of S(f), which is depressed below the kHz range and increased at high frequencies, where it attains a slope close to 1/f

^{2}.

_{g}is the average number of clusters in unit time, generating the macroscopic flux rate $\dot{\mathsf{\Phi}}$, and τ

_{0}is the average time interval between subsequent elementary events inside the cluster, the theory predicts, as fully discussed in [35], the following expression for the spectral density of the noise

_{0}is the average duration of a cluster. Equation (5) describes a function endowed with low and high cutoff frequencies f

_{1}= ν

_{g}/2π and f

_{2}= 1/2πρτ

_{0}, respectively, and a frequency dependence S(f) ∝ f

^{2}when f << f

_{1}and S(f) ∝ 1/f

^{2}when f >> f

_{2}. It is noted that the macroscopic flux rate $\dot{\mathsf{\Phi}}$ appears as a multiplying factor in Equation (5), further showing how good a simplification is to measure the noise under constant dJ/dt value. The gross features of the BN spectra and their dependence on plastic strain, shown in Figure 11 for the case of IF low-carbon steel sheet, are predicted by Equation (5), although the experimental low-frequency portion has somewhat limited extension, as imposed by the experimental requirement of a not too low magnetizing frequency (f

_{m}~ 0.02 Hz in the present case). In agreement with the experimental S(f) behavior and its theoretical formulation, it is convenient to separately analyze the low and high frequency ranges, exploiting the ubiquitously imposed condition of constant $\dot{\mathsf{\Phi}}$ value. For f << f

_{2}Equation (5) can be written as

_{g}, S(f) is decreased at low frequencies. This is quantified in Figure 12a by the behavior of the low-frequency noise power $P={\displaystyle {\int}_{10\mathrm{Hz}}^{200\mathrm{Hz}}S(f)\text{\hspace{0.17em}}\mathrm{d}f}$ versus ε

_{p}. Plastic deformation has a detrimental effect on the propagation of the Barkhausen reversal, because number and strength of the pinning centers are increased and the internal demagnetizing fields become stronger, breaking up the avalanches. Such an effect is stronger after tensile straining than after cold-rolling and is paralleled by a correspondingly stronger drop of permeability and remanence (see Figure 6b). It can be associated with the previously remarked different patterns of residual stresses expected to arise from these different deformation modes.

_{2}) can be written as

^{2}behavior and the dependence on the amplitude of the elementary Barkhausen jumps $\Delta \overline{\mathsf{\Phi}}$. The high-frequency portion of the spectrum reflects the fine structure of the noise, that is, the elementary jumps and their sequence in the avalanche and the preponderant role of the events occurring at and close to the surface, where flux variations are little shielded by the eddy currents. The high-frequency noise is therefore mainly related to the local interactions between dislocations and DWs in the surface/subsurface region, independent of the specific macroscopic pattern of the residual stress. It is noted that the high-frequency noise power (Figure 12b), the coercive field, and the work-hardening all follow a similar approximate square root dependence on the plastic strain ε

_{p}.

_{r}~ 5 × 10

^{3}) and operates as a linear transducer for the BN. In addition, by working in the reversible regime, it is not the source of additional magnetization noise. Figure 13, showing the spectral densities measured by the ferrite sensor in the starting sheet and after either cold-rolling or tensile straining up to elongations around 4.5%, is in good qualitative agreement with the results reported in Figure 11.

## 4. Conclusions

_{c}of the coercive field due to plastic deformation, so that the loops all cross at a same point in these two quadrants. Such an effect is ascribed to the rise of an equivalent internal demagnetizing factor proportional to ΔH

_{c}.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Coercive field versus strain-hardening of a plastically deformed intermediately oriented pure Fe single crystal. τ is the resolved shear stress and τ

_{0}is the critical (yield) shear stress (adapted from [18]).

**Figure 3.**Work-hardening of the fine-grained IF and ultra-low carbon steels of Figure 2 brings about a dependence of the coercive field on plastic straining approximately following the law H

_{c}∝ ε

_{p}

^{1/2}. Deformation occurs in a rather continuous fashion and different stages cannot be discriminated. Stage I and stage II are instead recognized below and above about ε

_{p}~ 2% in the large-grained (<s> = 800 μm) extra-pure Fe sample. The data shown in this figure have been adapted from Refs. [20,21,22].

**Figure 4.**(

**a**) Hysteresis loops in non-oriented Fe–Si strip samples subjected to tensile stress. The coercive field passes through a shallow minimum around σ = 75 MPa (adapted from [27]). (

**b**) Effect of compressive and tensile stresses in the range −50 MPa ≤ σ ≤ 50 MPa on the quasi-static major hysteresis loop of a non-oriented Fe–(3 wt %)Si sheet. A monotonic increase of the coercive field, going hand-in-hand with a decrease of permeability, is observed under increasing (conventionally negative) compressive stress (partly adapted from [28]).

**Figure 5.**(

**a**) Quasi-static major hysteresis loops measured at peak polarization J

_{p}= 1.72 T in interstitial-free (IF) steel sheets up to ε

_{p}= 4.5%. (

**b**) Enlarged view of the coercive field region, showing that the loops of the plastically deformed sheets tend all to cross at the same point in the second and fourth quadrants.

**Figure 6.**Magnetic hardening in cold-rolled IF low-carbon steel strips. (

**a**) The coercive field shows, for a same plastic elongation, milder increase upon cold-rolling than with tensile straining (adapted from [20]). (

**b**) The quasi-static major hysteresis loops (J

_{p}= 1.72 T) accordingly show slower decrease of permeability and remanence with respect to the strips tensile strained to the same elongation level.

**Figure 7.**(

**a**) Effect of sheet cutting on the quasi-static hysteresis loop of non-oriented Fe–Si alloys (J

_{p}= 1.5 T). The loops in this figure have been measured on guillotine-cut strips of thickness d = 0.638 mm and widths w = 30 mm and w = 5 mm. (

**b**) Increase of hysteresis energy loss W

_{h}versus decrease of strip width w in two different types of non-oriented Fe–Si sheets (d = 0.470 mm and d = 0.638 mm), subjected either to punching or water cutting. The loss figure is normalized to the value expected for a very large sheet, whose properties are negligibly affected by cutting. Symbols: experimental W

_{h}values. Continuous lines: prediction by Equation (3). The theoretical curves attain an upper limiting value at the strip widths where the material is assumed to be fully degraded by cutting (adapted from [33]). The inset shows the assumed schematic subdivision of the strip width between damaged (width 2L

_{c}) and pristine regions.

**Figure 8.**Envelope of the Barkhausen noise versus applied field in Armco Fe sheets before and after plastic deformation by cold-rolling (

**a**) and tensile straining (

**b**). Adapted from [34].

**Figure 9.**(

**a**) The Barkhausen jump sum rate (JSR) in 1 mm thick ultra-low carbon steel sheets tensile strained up to 10% (adapted from [21]). (

**b**) Maximum amplitude of the Barkhausen noise versus plastic deformation measured in Armco Fe and low-carbon steel bar samples. Straining is performed applying either compressive or tensile stresses (adapted from [7]).

**Figure 10.**Profile of the Barkhausen noise power measured along a half hysteresis loop with peak polarization values J

_{p}= ± 1.17 T in an Fe–(3 wt %)Si sheet under either triangular applied field waveform H(t) or triangular polarization J(t) at a frequency of about 0.01 Hz. With triangular field, the Barkhausen pulses are crowded inside a relatively restricted time (field) interval around the coercive field, as imposed by the non-linear J(H) behavior (partly adapted from [35]).

**Figure 11.**Spectral density of the BN in IF low-carbon steel sheets magnetized at the constant polarization rate dJ/dt = 0.1 T/s between J

_{p}= ±1.2 T. The measured value of S(f) is normalized with respect to the flux rate dΦ/dt and the number of turns squared N

^{2}of the pickup coils S(f)

_{norm}= S(f)/(dΦ/dt⋅N

^{2}). Plastic strain engenders, both in tensile strained (

**a**) and cold-rolled sheets (

**b**), a decrease of the low-frequency portion of the spectrum and an increase at high frequencies. Tensile strain affects to larger extent, for a same permanent elongation ε

_{p}, the low-frequency dependence of S(f).

**Figure 12.**The noise power P

_{norm}shows opposite trends versus plastic deformation when obtained by integrating the normalized spectral density of the noise S(f)

_{norm}= S(f)/(dΦ/dt⋅N

^{2}) in the low-frequency range 10 Hz–200 Hz (

**a**) or in the high-frequency range 10 kHz–100 kHz (

**b**). The low-frequency power is mainly determined by the ease of propagation of the Barkhausen avalanches, less hindered in cold-rolled sheets. The high-frequency power, quite independent of the deformation mode, is related to the fine structure of the noise, and is chiefly sensitive to the domain wall (DW) displacements occurring at and close to the sheet surface.

**Figure 13.**The spectral density of the Barkhausen noise S(f)

_{norm}= S(f)/dJ/dt, normalized to the polarization rate, detected by a soft ferrite C-yoke placed on the sheet surface, shows qualitative agreement with the results obtained using search coils, while enhancing the specific differences occurring at low frequencies between cold-rolled and tensile strained samples.

© 2017 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Fiorillo, F.; Küpferling, M.; Appino, C.
Magnetic Hysteresis and Barkausen Noise in Plastically Deformed Steel Sheets. *Metals* **2018**, *8*, 15.
https://doi.org/10.3390/met8010015

**AMA Style**

Fiorillo F, Küpferling M, Appino C.
Magnetic Hysteresis and Barkausen Noise in Plastically Deformed Steel Sheets. *Metals*. 2018; 8(1):15.
https://doi.org/10.3390/met8010015

**Chicago/Turabian Style**

Fiorillo, Fausto, Michaela Küpferling, and Carlo Appino.
2018. "Magnetic Hysteresis and Barkausen Noise in Plastically Deformed Steel Sheets" *Metals* 8, no. 1: 15.
https://doi.org/10.3390/met8010015