B Ducharne

S Zurek

L Daniel

G Sebald

An anisotropic vector hysteresis model of ferromagnetic behavior under alternating and rotational magnetic field

Keywords: multiscale model, dry friction model, hysteresis losses

Vector quantities and tensorial properties rule the magnetization mechanisms in ferromagnetic materials. Every ferromagnetic material is anisotropic to some degree in its magnetic response, so this anisotropy has to be considered for a general model. In this study, a multiscale approach based on a statistical description of the magnetic domain distribution and the knowledge of the crystallographic texture is used to predict the anhysteretic behavior along an arbitrary space direction. Combined with the vector Bergqvist dry-friction hysteresis model, qualitatively reliable simulation results are obtained under alternating and rotational magnetization. FeSi 3% grain-oriented (FeSi GO) electrical steel is chosen as study material: FeSi GO are widespread so that extensive data are available and strongly anisotropic, forcing the model toward the "worst-case" scenario from the viewpoint of anisotropy.

Moreover, under high excitation of rotational magnetization, losses drop due to the disappearance of the magnetic domains. This behavior is represented correctly by the proposed simulation method. In such conditions, the magnetization behavior is led mainly by the anhysteretic behavior, strengthening the predictive ability of the proposed model. In this manuscript, comparisons between simulations and measurements under many amplitudes of alternating and rotational magnetization for different levels of imposed excitation or magnetization are provided and used to validate the simulation method.

-Introduction

It is established that almost 3% of the electricity is lost in the magnetic cores [START_REF] Moses | Importance of rotational losses in rotating machines and transformers[END_REF]. In many applications (such as transformers), magnetization directions are well established and remain unchanged under operation. Alternating losses are then preponderant. For such applications, highly anisotropic ferromagnetic materials are industrially most suitable [START_REF] Moses | Problems in the design of power transformers[END_REF]. They favor magnetization in privileged directions and improve the conversion efficiency.

In many other applications (such as motors), it is more favorable to avoid highly anisotropic magnetic cores [START_REF] Binesti | Core losses and efficiency of electrical motors using new magnetic materials[END_REF]. In most of the geometry, the magnetization direction varies over time.

Alternating losses are still present, but the amount of rotational loss increases significantly (up to 50 % [START_REF] Moses | Importance of rotational losses in rotating machines and transformers[END_REF]).

The prediction of the magnetic core behavior is an essential task in the design of electromagnetic devices. Constitutive analytical equations [START_REF] Jiles | Introduction to magnetism and magnetic materials[END_REF] [START_REF] Bertotti | Hysteresis in magnetism[END_REF] and simulation methods [START_REF] Raulet | The magnetic field diffusion equation including dynamic hysteresis: a linear formulation of the problem[END_REF]- [START_REF] Zirka | Accounting for the influence of the tank walls in the zero-sequence topological model of a three-phase, three-limb transformers[END_REF] have been proposed, but progress is still to be made in this strongly nonlinear and vector environment.

Anisotropy in laminated electrical steel sheets is of genuine interest as witnessed by the vast list of article dealing with this topic [START_REF] Somkun | Magnetostriction Anisotropy and Rotational Magnetostriction of a Nonoriented Electrical Steel[END_REF]- [START_REF] Belahcen | Anisotropic and Strain-Dependent Model of Magnetostriction in Electrical Steel Sheets[END_REF].

According to [START_REF] Appino | Can rotational magnetization be theoretically assessed?[END_REF], ferromagnetic vector hysteresis models can be classified into two categories:

 "type I" is built from the superposition of scalar models continuously distributed along all possible directions.

 "type II" solves the hysteresis problem by integrating contributions of intrinsically vector elements.

The classical Preisach model has already been modified to become a "type I" [START_REF] Mayergoyz | Mathematical models of hysteresis[END_REF] [START_REF] Mayergoyz | Isotropic vector Preisach model of hysteresis[END_REF] or a "type II" [START_REF] Torre | Vector modeling Part I: Generalized hysteresis model[END_REF] vector model. In [START_REF] Aquino | A new vector model of magnetic hysteresis based on a novel class of play hysterons[END_REF], d'Aquino et al. proposed a "type I" Preisach vector model. This phenomenological approach is interesting but limited to isotropic materials. It is inadequate to simulate the strongly anisotropic FeSi GO laminations. This simulation method is descriptive (as opposed to predictive) as the experimental data used to validate and set the simulation parameters are the same. The model is then limited to pre-defined experimental conditions. Excellent fits are observed when those conditions are provided.

The also classical Jiles-Atherton model (JA) has as well already been converted to a "type II" to generate vector results [START_REF] Bergqvist | A simple vector generalization of the Jiles-Atherton model of hysteresis[END_REF]- [START_REF] Upadhaya | Alternating and rotational loss prediction accuracy of vector Jiles-Atherton model[END_REF].

All these examples follow the general trend observed in [START_REF] Appino | Can rotational magnetization be theoretically assessed?[END_REF]: "the extrapolation of solidly established scalar unidirectional models to the case where the field rotates in the lamination plane". All these simulation methods can be set precisely and provide accurate simulation results, but drawbacks remain, and the physical interpretations are limited.

In 1996, Bergqvist [START_REF] Bergqvist | Magnetic vector hysteresis model with dry friction-like pinning[END_REF] proposed a "type II" vector hysteresis model based on pseudo particles.

It introduces an ideal soft ferromagnetic material without hysteresis, which is ruled by a singlevalued magnetization curve called anhysteretic curve and expressed as M = Manh(Hsurf). M is the magnetization, Manh the anhysteretic magnetization, and Hsurf the tangential surface excitation field. Then, inside the real ferromagnetic material, he suggested the presence of potential wells generating sequences of small discontinuous jumps similar to dry frictions. If their number approaches infinity, the magnetic behavior results in a well-defined hysteresis shape. Bergqvist Dry Friction (B-DF) model is convenient as it relies on physical fundaments and behaves correctly under alternating and rotating magnetization. It furthermore reproduces some of the rotational losses peculiar behaviors, including the loss drop at high inductions.

In this study, we specifically focus on the anisotropy question. For this, the B-DF model will be combined with a multiscale model developed to predict anhysteretic behaviors [START_REF] Hubert | Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[END_REF]. Grainoriented electrical steel specimens (GO FeSi 3%) are chosen due to their highly anisotropic magnetic properties.

-Anisotropic vector hysteresis model

The dry friction vector hysteresis model (B-DF)

In physics, analogies are commonly used as indirect ways to understand and simulate phenomena impossible to observe in the first place [START_REF] Podolefsky | Use of analogy in learning physics: the role of representations[END_REF]. Hysteretic behavior of mechanical friction force has been observed and discussed for years [START_REF] Wojewoda | Hysteretic effects of dry friction: modelling and experimental studies[END_REF]- [START_REF] Lampaert | Online identification of hysteresis functions with non-local memory[END_REF]. The analogy with ferromagnetic hysteresis is evident, and it has already served to develop accurate simulation tools [START_REF] Henrotte | A dynamical vector hysteresis model based on an energy approach[END_REF] [START_REF] Baltensperger | Dry friction in micromagnetics[END_REF]. The B-DF model, as described by Bergqvist in [START_REF] Bergqvist | Magnetic vector hysteresis model with dry friction-like pinning[END_REF], is one of them. This model relies on energetical principles. More precisely, it is based on multiple discretized dry-friction elements.

Each element is characterized by its threshold κ and weight Spectrum(κ). A dry-friction element is ruled by sequence (1):

if |𝐇 𝐟 (t) -𝑓 (𝐌(t -dt))| > κ 𝐯 = 𝐇 ( ) (𝐌( )) |𝐇 ( ) (𝐌( ))| κ = |𝐇 (t) -𝑓 (𝐌(t -dt)) -A𝐯| 𝐌(t) = 𝑓(𝑓 (𝐌(t -dt)) + A𝐯) else 𝐌(t) = 𝐌(t -dt) (1)
Where A is a constant, and 𝐯 is a unit vector giving the direction of change [START_REF] Bergqvist | Magnetic vector hysteresis model with dry friction-like pinning[END_REF]. Their values are calculated at each simulation step time.

In [START_REF] Appino | Can rotational magnetization be theoretically assessed?[END_REF], η = 𝑓 -1 (M(t-dt))+Av is introduced as a rest field and is defined as the field that would produce the current magnetization in the absence of hysteresis. 𝑓 is a sigmoid function related to the anhysteretic behavior, Ms is the saturation magnetization and 𝑎 a constant:

𝐌 = M • tanh(𝑎 • 𝐇 ) (2)
Under unidirectional excitation (M and Hsurf supposed colinear), high amplitude and symmetrical Hsurf, the resolution of sequence (1) leads to a well-shaped major hysteresis cycle.

Unfortunately, sequence (1) is unable to simulate the first magnetization curve (Fig. 1 -a) correctly.

This issue is solved once the distribution of dry elements is taken into account. In [START_REF] Ducharne | Low frequency modelling of hysteresis behaviour and dielectric permittivity in ferroelectric ceramics under electric field[END_REF], the spectrum function is introduced to condense all the dry-friction element weights:

∑ Spectrum(κ )𝐌 = 𝐌 (3)
This Gaussian-like distribution can be obtained numerically [START_REF] Guyomar | Fractional derivative operators for modeling the dynamic polarization behavior as a function of frequency and electric field amplitude[END_REF] or expressed analytically as follows:

Spectrum(κ ) = √ (4)
The number of dry-friction elements is not limited. Eq. ( 5) gives the unique condition to be fulfilled for coherent values on the induction axis.

∑ Spectrum(κ ) = 1 (5)
Fig . 1 -b shows a unidirectional Ba(Hsurf) first magnetization and major hysteresis loop simulated with the B-DF model for a typical GO FeSi. Where Ba is the average cross-section induction calculated from Eq. ( 6):

𝐁 = µ (𝐇 + 𝐌) (6)
The anhysteretic contribution is obtained with the analytical expression Eq. (2). Tab. 1 gives the simulation parameters, and the Spectrum function is depicted in Fig. 2 -c. The B-DF model parameters are set following two steps:

 An optimization window is established for both parameters (μd, θd). The product of μd and Δκ is close to the resulting unidirectional coercivity in the Ba(Hsurf) hysteresis loop. Similarly, θd is close to the ratio between the differential permeability read on the anhysteretic curve Banh(Hsurf) at Hsurf =0 and read at the coercivity of the unidirectional Ba(Hsurf) hysteresis curve.

 The minimization of the mean relative standard deviation error function:

Err (%) = ∑ 𝐁 (𝐇 ) 𝐁 (𝐇 ) 𝐁 (𝐇 ) (7)
Where n represents the discrete data points obtained experimentally. In the specific case of Fig. The B-DF model in its original form is frequency-independent. Thus its viability under rotational magnetization can only be observed by plotting the Ba dependence of the rotational losses Wrot under very-low-frequency circular magnetization, below the quasi-static threshold (Eq. ( 8))(Fig. 2).

W = ∫ ( ) H (t) + ( ) H (t) dt (8)
Where Bax, Bay, Hsurfx, and Hsurfy, are the x and y-axis projections of Ba and Hsurf, respectively, and T is the magnetization period. Fig. 2 shows a correct trajectory in the low induction range. However, it incorrectly keeps increasing even beyond the "saturation elbow" (≈ 1.5 T for this material). In practice, the quasistatic rotational losses decline in the high induction regime following the disappearance of the magnetic domains [START_REF] Zurek | Characterisation of Soft Magnetic Materials Under Rotational Magnetisation[END_REF]. The simulation method described above is therefore unable to reproduce this drop. This issue was solved in [START_REF] Bergqvist | Magnetic vector hysteresis model with dry friction-like pinning[END_REF] by conserving the Spectrum function but replacing the constant Δκi step between every element coercivities with a rest field-dependent distribution function:

Δκ = | | (9)
Here kd is an additional parameter set to optimize the simulation/measurement comparisons.

As the alternating losses are relatively independent of kd (Fig. 3 -b), the optimization process for a precise estimation of this parameter relies on Wrot.

Anhysteretic behavior: the MultiScale Model MSM

MSM [START_REF] Daniel | Reversible magneto-elastic behavior: a multiscale approach[END_REF][39] has been developed to predict the anhysteretic magnetic behavior of a ferromagnetic specimen. It relies on a statistical description of the distribution of the ferromagnetic domains. MSM provides vector information, and the anisotropy effects are naturally taken into account [START_REF] Hubert | Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[END_REF]. A polycrystalline ferromagnetic specimen is considered as an aggregate of single crystals (grains). Each grain is supposed to be divided into a finite number of

W rot (J m -3 ) W rot (J m -3 ) B ax (T)
a b c magnetic domain families. A little less than 3.5 10 4 families for each grain orientation have been considered for the FeSi GO tested in this study. Each domain family is characterized by its own magnetic orientation α and its own potential energy Wα (Eq. 10) where Wα k , Wα H , Wα σ , and Wα conf stand for the magneto-crystalline (Eq. ( 11)), magnetostatic (Eq. ( 12)), magneto-elastic (Eq. ( 13)), and initial configuration energy, respectively:

W = W + W + W + W (10)
W = 𝐾 (𝛾 𝛾 + 𝛾 𝛾 + 𝛾 𝛾 ) + 𝐾 𝛾 𝛾 𝛾 (11)
W = -μ 𝐇 . 𝐌 (12)
W = -𝛔 ∶ 𝛆 (13)
Hα, Mα, σα and 𝛆 are the magnetic field, the magnetization, the stress tensor, and the magnetostriction strain tensor defined at the magnetic domain scale, respectively. The inital configuration energy Wα conf is used to account for a potential misbalance in the initial domain configuration of the material in the absence of external magnetic field or mechanical stress. In GO steels, this initial misbalance can be interpreted as the result of a free-surface effect [START_REF] Hubert | Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[END_REF]. It was later shown [START_REF] Daniel | A multiscale model for magneto-elastic behaviour including hysteresis effects[END_REF] that initial configuration effects can be treated as the result of a fictitious initial stress σ0 in the material, this fictitious homogeneous mechanical stress being interpreted as the image of the initial configuration source (be it internal stress, plasticity, surface or geometrical effects, ...). The expression of Wα conf is then similar to Eq. ( 13), with σ0 instead of σα.

In the following σ0 will be taken as a uniaxial stress of amplitude σ0 along RD. Mα is defined by its norm (the material saturation magnetization Ms), and its direction is given by 𝛾1, 𝛾2, 𝛾3. 𝛆 is defined by the magnetostriction constants 𝜆100 and 𝜆111 [START_REF] Hubert | Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[END_REF]. 𝐾1 and 𝐾2 are the magnetocrystalline energy constants.

The volume fraction 𝑓 of a domain family is calculated from the knowledge of the potential energy of all domain families:

𝑓 = exp (- ) ∑ exp (- ) (14)
𝐴 is a material parameter that can be adjusted using the initial macroscopic susceptibility 𝜒 of the unstressed anhysteretic magnetization curve [START_REF] Daniel | Reversible magneto-elastic behavior: a multiscale approach[END_REF].

Once 𝑓 is calculated for all magnetic domains families, the magneto-elastic response at the grain scale is calculated by a volume average:

𝛆 = 〈𝛆 〉 = ∑ 𝑓 𝛆 (15)
𝐌 = 〈𝐌 〉 = ∑ 𝑓 𝐌 (16)
An orientation distribution function (crystallographic orientations) obtained from X-ray diffraction or Electron Back Scattering Diffraction (EBSD) measurements can be used to describe the crystallographic texture and return the behavior at the polycrystalline scale.

For simplification reasons, both magnetic H and mechanical σ external stimuli are supposed uniform within the material. The potential energy and the volume fraction of each magnetic domain family calculus come first. It is followed by each grain magnetization (Eq. ( 16)).

Eventually, an average over the whole volume is performed to obtain the entire specimen magnetization:

𝐌 = 〈𝐌 〉 (18)
This process allows constructing the stress-dependent anhysteretic magnetization curves based on a limited number of intrinsic material parameters. The magnetic induction is finally easily deduced from the magnetization using Eq. ( 6).

In this study, the simulation parameters and the crystallographic texture data for a typical GO FeSi come from [START_REF] Hubert | Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[END_REF] (Hi-B, 0.3 mm thick from Nippon Steel), except σ0 (not used in ref [START_REF] Hubert | Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[END_REF]). Fig. 4 -a shows simulated anhysteretic curves for the GO FeSi along different orientations in the lamination plane. θ = 0° is the rolling direction (RD -easy axis), θ = 90° is the transverse direction (TD). Fig. 4 -b gives the resulting induction levels for a given Hsurf as a function of θ.

The significant differences between RD and TD are noteworthy, and an especially unfavorable direction is observed at approximately 55° (usual observation for GO electrical steel [START_REF] Fiorillo | Comprehensive model of magnetization curve, hysteresis loops, and losses in any direction in grain-oriented Fe-Si[END_REF]- [START_REF] Zurek | Rotational power losses and vector loci under controlled high flux density and magnetic field in electrical steel sheets[END_REF]).

Integration of the multiscale model in the B-DF model

The B-DF model in its quasi-static and original form [START_REF] Bergqvist | Magnetic vector hysteresis model with dry friction-like pinning[END_REF], as described in section 2.1 relies on five parameters. The resolution of the accommodation and congruency issues by considering an effective field as input of the model is not discussed in this manuscript. Still, it is giving rise to a sixth parameter, α:

𝐇 = 𝐇 + α • 𝐁 (19)
Among these five parameters, Ms and 𝑎 are known as the anhysteretic ones. Together in Eq.

(2), they control the anhysteretic behavior. Ms is a material constant. For isotropic materials, 𝑎 can also be set constant, independently from the magnetization azimuthal angle ϕ and polar angle θ. Oppositely, for anisotropic materials, accurate simulations results can only go through a ϕ and θ dependency of 𝑎(ϕ, θ). Also, to consider the amplitude variation (Fig. 4 -a), a new parameter is defined: M at Hsurf = 5 kA•m -1 also dependent on ϕ and θ: M5000(ϕ, θ).

MSM provides anhysteretic curves in every space direction and orientation. The simulated anhysteretic curves come as data files, and a preliminary stage consists of converting these data into analytical expressions (sigmoid function, Eq. ( 2)). In this study, all the experimental results

were measured in the lamination plane (i.e., ϕ = 0). Thus, the simulation results were limited to this condition. Due to their fabrication process, the studied materials are orthotropic, so that θ can also be restricted to [0 -90]° [START_REF] Naumoski | Investigation of the influence of different cutting procedures on the global and local magnetic properties of non-oriented electrical steel[END_REF].

As revealed clearly in [START_REF] Hubert | Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[END_REF][43], the magnetization curve of the GO FeSi laminations show a staircase shape in the TD. This staircase shape mostly affects the material response at low fields.

The use of the initial configuration energy Wα conf allows this effect to be modeled by the MSM.

In order to improve the quality of the analytical fits (including the staircase shape), the sigmoid function (Eq. ( 2)) has been slightly modified and enriched with two parameters, τ(θ), b(θ):

𝐌 = M (θ) • tanh 𝑎(θ) • 𝐇 • (1 -𝑒 𝐇 ( ) ) + b(θ) • 𝐇 (2.2)
Once the θ dependence of M5000, 𝑎, τ and b was established (Fig. 5), an analytical expression (harmonic type) of these functions was proposed to facilitate their incorporation in the B-DF model (becoming the MSM-DF model). Matlab® curve fitting toolbox was used to determine these expressions. Alternating excitation predictions are given in Fig. 6 and7. The staircase shape is visible when θ is getting closer to 90°.

-Comparison simulations/measurements under rotational magnetization

The importance of rotational magnetization behaviors in understanding the magnetization mechanisms was demonstrated a long time ago [START_REF] Moses | Importance of rotational losses in rotating machines and transformers[END_REF][2]. Still, characterization standards are in need, and getting reliable measurements is not an easy task. All the experimental results under rotational magnetization displayed in this manuscript have been measured with the same experimental setup (depicted in Fig. 9 below) [37][44]. A cylindrical magnetizing yoke was used. All the GO FeSi specimens tested were disks (80 mm diameter), their grade was M140-27. Two single turn local search coils, 20 mm wide each and positioned in quadrature in the center of the tested specimens, were used to measure Ba.

Similarly, two H-coils, also 20 mm wide and positioned tangent to the surface of the sample to be characterized, were used for Hsurf. All the tested specimens were placed in a way to align their RD with the system X-axis. A 2 mm gap between the specimen and the magnetizing yoke was set to ensure the magnetization homogeneity. Both Ba and Hsurf imposed amplitude tests were done.

Their consistencies were ensured through digital feedback. Characterizations were recorded in clockwise and anticlockwise directions, and the final results average both directions and multiple periods. More details on the characterization setup can be found elsewhere [37][44].  divergences between the crystallographic and magnetoelastic properties of the specimens tested and used to set the simulation parameters. In the same electrical steel grade, significant differences in the magnetic response have already been noticed and discussed in the scientific literature (material processing, cutting, punching, etc., are the source of residual stresses variations changing the magnetic response drastically [START_REF] Belahcen | Anisotropic and Strain-Dependent Model of Magnetostriction in Electrical Steel Sheets[END_REF][45] [START_REF] Weiss | Impact of punching parameter variations on magnetic properties of nongrain-oriented electrical steel[END_REF]). The saturation level is slightly lower in measurement, confirming divergences in some fundamental properties.

 Limitations of the simulation assumptions in the low induction range.

 Limitations of the experimental setup.

Even if RD exhibits softer magnetic properties in simulation and measurement, the supposedly unfavorable 55° direction is not evident in the experimental results (however, it is clearly distinguishable in the imposed Ba regime, as discussed below). It is also interesting to observe its progressive disappearance in both simulations and measurements for the high Hsurf levels. This observation is justified by the predominance of Wα H over the other energies.

In the following figures (Fig. 12, 13), the specimen is tested under circular Ba imposed conditions (orthogonal components Bax and Bay sine and cosine, respectively). The simulation results are obtained explicitly, using the Hsurf imposed model. For each simulation step time t, a window of Hsurf centered around its value at t = t -dt is tested (+/-ΔH, +/-Δθ). Where ΔH and Δθ are set depending on max(Ba) and dt. In Fig. 12, the magnetization level is limited to 1.9T. Similarly, Fig. 5 in Ito et al. [START_REF] Ito | Energy-Based Magnetization and Magnetostriction Modeling of Grain-Oriented Silicon Steel Under Vectorial Excitations[END_REF] is limited to 1.7 T. The experimental results in Fig. 12 show that Hsurf in the x-direction (RD) is getting higher when the magnetization levels increase. This effect is not visible in Fig. 12 simulation results nor in [START_REF] Ito | Energy-Based Magnetization and Magnetostriction Modeling of Grain-Oriented Silicon Steel Under Vectorial Excitations[END_REF]. It is now evident in Fig. 14, where simulations have been conducted for much higher magnetization levels, Hsurf starts increasing for magnetization levels higher than 2T. Ms the saturation magnetization is constant. For a given direction, once Ms is reached, no further magnetization variations are possible so that the Hsurf loci curve ends up in a quasi-perfect circle (Fig. 14).

It is essential to emphasize the predictive nature of the MSM-DF model, especially true under high magnetic field levels when the hysteresis loss is dropping. In such conditions, the MSM-DF relies mainly on the anhysteretic MSM contribution. It is worth mentioning that MSM is not a It is also remarkable that although the hardest direction (55°) is undoubtedly present (Fig. 11 confirms this, especially for the green curves, just below the magnetization elbow), it is not evident when measuring under imposed rotational Hsurf, and the model correctly reflects that behavior. It is true that the data in Fig. 10 does not match precisely between simulation/measurement but the relative changes of the imposed rotational Ba butterfly are much smaller in Fig. 10 than the imposed rotational Hsurf butterfly in Fig. 12.

A small hysteresis (difference between "ascending" and "descending" branches) can be observed in the experimental Hsurfx(Bax) and Hsurfy(Bay) cycles Fig. 12. The total rotational loss can be split into the Px and Py components which are proportional to the area of the corresponding B-H loops. However, especially at higher excitation, it is possible that the experimental B-H loop (as shown in Fig. 13) can have "negative" areas, due to the phase differences between the sensors resulting from their angular misalignment. This is one of the reasons why the value of power loss has to be averaged from clockwise and anticlockwise measurements. Investigation of this experimental phenomenon is beyond the scope of this paper, but extensive studies were already published elsewhere [START_REF] Zurek | Characterisation of Soft Magnetic Materials Under Rotational Magnetisation[END_REF][44].

-Conclusions and perspectives

1 ,Fig. 1 -

 11 Fig. 1 -a Major hysteresis cycle and first magnetization curve as obtained by solving sequence (1). Fig. 1 -b B-DF, first magnetization curve, and major hysteresis loop for a typical FeSi 3% GO laminated electrical steel sheet (rolling direction). Fig. 1 -c Spectrum distribution for q = 200.

Fig. 2

 2 Fig. 2 Magnetic losses under rotational magnetization as a function of the maximal magnetic field amplitude, as simulated with sequence (1).

Fig. 3 -(J m - 3 )

 33 a shows the Ba dependence of Wrot for the updated simulation method and increasing values of kd. Fig. 3 -b shows the Ba W rot dependence of Walt for the same simulation parameters. Eventually, Fig. 3 -c depicts the simulated unidirectional hysteresis cycles as obtained with and without the correction (kd = 80) and confirms the low influence of kd in a unidirectional situation.

Fig. 3 -

 3 Fig. 3 -a Wrot as a function of max(Ba) without the correction and for different values of kd. Fig. 3 -b Walt as a function of max(Ba) in the same conditions. Fig. 3 -c Comparison on the unidirectional hysteresis cycle with and without the correction (kd = 80).

Fig. 4 -

 4 Fig. 4 -a Simulated anhysteretic curves for different orientations in the lamination plane. Fig. 4 -b Same plot at a different x-axis scale. Fig. 4 -c Induction level for a given Hsurf as a function of θ.

Fig. 5 displays

 5 M5000, 𝑎, τ and b as a function of θ as simulated with the multiscale model and the comparison with the analytical expression obtained from the Matlab® curve fitting toolbox. Tab. 3 below gives these analytical expressions.

Fig. 5 -

 5 Fig. 5 -a M5000(θ), comparison multiscale model / analytical expression. Fig. 5 -b 𝑎(θ) comparison multiscale model / analytical expression. Fig. 5 -c τ(θ) comparison multiscale model / analytical expression. Fig. 5 -d b(θ) comparison multiscale model / analytical expression.

Tab. 3 -

 3 Analytical expression M5000(θ), 𝑎 (θ), τ(θ) and b(θ).

Fig. 6

 6 Fig. 6 Comparison simulation/measurement for quasi-static minor centered cycles under unidirectional alternating conditions and six different angles (same axis scale).

Fig. 7

 7 Fig. 7 Comparison simulation/measurement for quasi-static minor centered cycles under unidirectional alternating conditions and six different angles (optimized axis scale).

20 Fig. 8

 208 Fig. 8 gives an algorithm to summarize the MSM-DF model implementation, including step-by-

Fig. 8 -

 8 Fig. 8 -Algorithm for the MSM-DF model implementation.

Fig. 9

 9 Fig. 9 Experimental setup picture.

Fig. 10 below

 10 Fig. 11 shows the simulated and measured associated hysteresis cycles, projections of Hsurf and

Fig. 10

 10 Fig. 10 Ba and Hsurf loci curves, comparison simulations / measurements for different levels of imposed rotational Hsurf.

Fig. 11

 11 Fig. 11 Simulated and measured RD and TD hysteresis cycle for different levels of imposed rotational Hsurf. The colors of curves correspond to those from Fig. 8.

Fig. 12

 12 Fig. 12 Ba and Hsurf loci curves, comparison simulations / measurements for 4 levels of imposed rotational Ba.

Fig. 13

 13 Fig. 13 Simulated and measured RD and TD hysteresis cycle for 4 levels of imposed rotational Ba.

Fig. 14

 14 Fig. 14 Ba and Hsurf loci curves, for 7 levels of imposed rotational Ba.

  fitting procedure. The model parameters are identified from a limited number of independent experiments, and the simulation results are compared to very different types of experiments.

A

  vector anisotropic hysteresis model (MSM-DF) has been described in this manuscript. It is based on the combination of a multiscale model for the simulation of the anhysteretic magnetization and a hysteresis contribution relying on an analogy between ferromagnetic domain wall motions and mechanical dry-frictions. The model is validated by comparing simulation and experimental results obtained in the lamination plane of a GO FeSi electrical steel and circular magnetic field excitation (Ba or Hsurf imposed). The model succeeds in the restitution and prediction of multiple peculiar behaviors, including the loci curves' butterfly shape under rotational sinus Ba and sinus Hsurf imposed conditions (especially true at high magnetic field). The remarkably different influence of the unfavorable 55° direction under rotational circular Ba and circular Hsurf imposed conditions is also qualitatively well anticipated by the model. It reveals the fidelity with the magnetic structure of the model under completely different magnetization regimes. Under high-level, low-frequency circular magnetization where hysteresis effects are disappearing, magnetization processes rely mostly on the anhysteretic contribution. The MSM-DF model then reduces to the MSM and consequently becomes more predictive [27][39][48]. This behavior opens exciting perspectives in the domain of magnetic NDT [49][50]. Such experimental conditions could be used to isolate the anhysteretic contribution. The combination of experimental results and the MSM-DF model could also define an elegant way to anticipate residual mechanical stresses in ferromagnetic materials.

  It is worth mentioning that Ms has been slightly reduced to improve the comparisons with the experimental results. The texture of oriented grain electrical steel specimens being especially strong, accurate simulation results can be obtained with an orientation distribution function limited to 60 orientations. The simulation parameters are summarized in Tab. 2 below: Tab. 2 -MSM simulation parameters for the GO FeSi.

	Quantity	Ms	𝐾1 ; 𝐾2	𝜆100 ; 𝜆111	𝐴	σ
	Unit	A•m -1	kJ•m -3	-	m 3 •J -1	MPa
	For FeSi GO	1.37•10 6	38 ; 0	23•10 -6 ; -4.5•10 -6	2•10 -2	7

  𝑋 = 𝑼 𝟎 + 𝑼 𝟏 cos (𝜃 • 𝑼 𝒘 ) + 𝑽 𝟏 sin (𝜃 • 𝑼 𝒘 ) + 𝑼 𝟐 cos (2𝜃 • 𝑼 𝒘 ) + 𝑽 𝟐 sin (2𝜃 • 𝑼 𝒘 ) + 𝑼 𝟑 cos (3𝜃 • 𝑼 𝒘 ) + 𝑽 𝟑 sin (3𝜃 • 𝑼 𝒘 ) + 𝑼 𝟒 cos (4𝜃 • 𝑼 𝒘 ) + 𝑽 𝟒 sin (4𝜃 • 𝑼 𝒘 ) + 𝑼 𝟓 cos (5𝜃 • 𝑼 𝒘 ) + 𝑽 𝟓 sin (5𝜃 • 𝑼 𝒘 ) + 𝑼 𝟔 cos (6𝜃 • 𝑼 𝒘 ) + 𝑽 𝟔 sin (6𝜃 • 𝑼 𝒘 ) + 𝑼 𝟕 cos (7𝜃 • 𝑼 𝒘 ) + 𝑽 𝟕 sin (7𝜃 • 𝑼 𝒘 ) + 𝑼 𝟖 cos (8𝜃 • 𝑼 𝒘 ) + 𝑽 𝟖 sin (8𝜃 • 𝑼 𝒘 )

	X	M5000	𝑎	τ	b
	U0	1091400	0.08838	-3993000	9.969
	U1	218705	0.01667	1476000	-7.384
	V1	103530	-0.08604	7169000	-3.977
	U2	-16303	-0.04337	5158000	0.5014
	V2	35147.5	-0.009779	-2223000	-2.314
	U3	55913	0.000431	-2053000	-2.556
	V3	6845.05	0.002251	-2924000	0.3361
	U4	-3564.05	-0.01373	-1264000	0.2652
	V4	-15274.5	-0.00811	1352000	0.02128
	U5	10625	-0.006512	647400	-0.8497
	V5	3157.75	0.00944	390000	0.2104
	U6	-1923.55	0.000866	74430	0.2066
	V6	-9945	0.000472	-217800	0.3
	U7	-142.205	-0.003273	-46550	-0.2141
	V7	3842	0.002473	-5407	0.1101
	U8	2495.6	0.001683	463.1	0.1473
	V8	-2805	0.002382	4843	0.1862
	Uw	0.048416	0.06024	0.03117	0.05497


Patrick Fagan
email: patrick.fagan@insa-lyon.fr

Benjamin Ducharne
email: benjamin.ducharne@insa-lyon.fr

Laurent Daniel
email: laurent.daniel@centralesupelec.fr

Anastasios Skarlatos
email: anastasios.skarlatos@cea.fr

Mathieu Domenjoud
email: mathieu.domenjoud@centralesupelec.fr

Christophe Reboud
email: christophe.reboud@cea.fr

Effect of stress on the Magnetic Barkhausen Noise energy cycles: a route for stress evaluation in ferromagnetic materials

Keywords: Stress evaluation, Magnetic non-destructive testing, multiscale model, Hysteresis

The magnetic Barkhausen noise energy hysteresis cycles, MBNenergy(H), were evaluated as a nondestructive testing characterization method to identify intern mechanical stress in ferromagnetic parts. Oriented grains electrical steel, and iron-cobalt specimens were tested as model materials of opposite behaviors. Tensile stress tests were run and revealed MBNenergy(H) coercivity as the most correlated indicator. In parallel, a predictive multiscale model was developed to simulate the stress-dependent MBNenergy(H) anhysteretic curves. A hysteresis contribution was added, and the resulting hysteresis predictions were validated by comparison to the tensile-stress experimental tests. 2D simulation predictions reveal the identification of uniaxial tensile stress as more efficient when the magnetic field is applied within an angle between 30° and 75° from the stress direction. The simulation method allows the foresee of the most favorable sensor orientation configurations depending on the material tested and all available a priori knowledge of the stress configuration.

-Introduction

Mechanical internal stresses are a determinant factor to material performance, structural integrity, and lifetime of industrial systems. Many techniques have been developed for their evaluation [START_REF] Withers | Residual stress part 1-measurement techniques[END_REF] [START_REF] Guo | Recent Progress of Residual Stress Measurement Methods: A Review[END_REF]. The hole-drilling method [START_REF]American society for testing and materials; E837; Standard test method for determining residual stresses by the hole[END_REF], the contour method [START_REF] Pagliaro | Measuring Multiple Residual-Stress Components using the Contour Method and Multiple Cuts[END_REF], the crack compliance method [START_REF] Prime | Residual stress measurement by successive extension of a slot: the crack compliance method[END_REF], or the stripping method [START_REF] Guo | Recent progress of residual stress measurement methods: A review[END_REF] are the most common mechanical techniques [START_REF] Withers | Residual stress part 1-measurement techniques[END_REF].

Chemical methods [START_REF] Guo | Recent progress of residual stress measurement methods: A review[END_REF] can also be found. These mechanical and chemical techniques, however, are destructive.

Non-destructive testing (NDT) methods for evaluating the internal mechanical stresses have also been developed [START_REF] Hellier | Handbook of non-destructive evaluation[END_REF]- [START_REF] Du | A review of miniaturized Non-destructive Testing technologies for in-situ inspections[END_REF]. They include diffraction methods (X-ray, Neutron), ultrasonic methods, or acoustic emission [START_REF] Höller | Nondestructive characterization of materials[END_REF]. These techniques can be applied successfully independently of the nature of the tested specimen. In the specific case of conductive and/or ferromagnetic materials, magnetic NDT techniques are available. Their applicability is more limited, but they offer many advantages such as fast response, low cost, small size, and easy maintenance.

Eddy currents testing [START_REF] Blodgett | Eddy current assessment of near-surface residual stress in shot peened nickel-base superalloys[END_REF], magneto-acoustic emission [START_REF] Su | Methodology for the Stress Measurement of Ferromagnetic Materials by Using Magneto Acoustic Emission[END_REF] [START_REF] Shibata | Magnetomechanical acoustic emission -a new method for non-destructive stress measurement[END_REF], electromagnetic acoustic transducer [START_REF] Guo | Recent Progress of Residual Stress Measurement Methods: A Review[END_REF], and Magnetic Barkhausen Noise (MBN) [START_REF] Gauthier | Measurement of residual stress in steel using the magnetic Barkhausen noise technique[END_REF]- [START_REF] Stewart | Magnetic Barkhausen noise analysis of stress in steel[END_REF] are some of the most used magnetic methods, alone or in combination with other NDT techniques [START_REF] Wilson | Magneto-Acoustic Emission and Magnetic Barkhausen Emission for Case Depth Measurement in En36 Gear Steel[END_REF]. This paper focuses on the use of MBN signals to detect stress in ferromagnetic materials.

Ferromagnetic materials exhibit a complex magnetic domain microstructure. The modification of this magnetic domain microstructure under the influence of an external stimulus (magnetic field, mechanical stress, or temperature variation) generates unpredictable and discontinuous domain wall motions, including successively pinning phases and large amplitude displacements [START_REF] Bozorth | Magnetism[END_REF] [START_REF] Kittel | Introduction to solid state physics[END_REF]. MBN signal is an indirect reflection of this microstructural reorganization.

It is linked to the magneto-mechanical state and the material properties [START_REF] Sakamoto | Theoretical analysis of Barkhausen noise in carbon steels[END_REF]. MBN is a stochastic process, and MBN raw measurements are not reproducible in practice. Their analysis is usually performed through time-independent indicators such as the Root Mean Square (RMS) [START_REF] Sakamoto | Theoretical analysis of Barkhausen noise in carbon steels[END_REF], the MBNenvelope [START_REF] Samimi | Multi-parameter Evaluation of Magnetic Barkhausen Noise in Carbon Steel[END_REF], or the MBNenergy [START_REF] Krause | Characterization of the magnetic easy axis in pipeline steel using magnetic Barkhausen noise[END_REF]- [START_REF] Ducharne | Phenomenological model of Barkhausen noise under mechanical and magnetic excitations[END_REF].

Many scientific works have described the use of MBN for the evaluation of residual mechanical stresses in ferromagnetic materials [START_REF] Gauthier | Measurement of residual stress in steel using the magnetic Barkhausen noise technique[END_REF]. Even if the first ones have been published at the beginning of the 20 th century [START_REF] Heaps | Magnetization of nickel under compressive stresses and the production of magnetic discontinuities[END_REF], it is still a timely subject, and the number of recent publications [START_REF] Ortega-Labra | A novel system for non-destructive evaluation of surface stress in pipelines using rotational continuous magnetic Barkhausen noise[END_REF]- [START_REF] Biesiekierski | 13 -Mechanical testing of metallic biomaterials[END_REF] is very high. Whatever the MBN indicator monitored, a preliminary calibration step is usually required. Non-destructive (X-ray diffraction [START_REF] Withers | Residual stress part 1-measurement techniques[END_REF][2], strain gauges [START_REF] Vourna | Residual stress analysis in nonoriented electrical steel sheets by barkhausen noise measurements[END_REF]) or destructive (Vickers impact [START_REF] Biesiekierski | 13 -Mechanical testing of metallic biomaterials[END_REF], hole drilling [START_REF] Guo | Recent Progress of Residual Stress Measurement Methods: A Review[END_REF] [START_REF]American society for testing and materials; E837; Standard test method for determining residual stresses by the hole[END_REF]) methods can be used for this calibration.

MBN measurements can provide information up to several hundreds of μm under lowfrequency excitation (< 1 Hz) [START_REF] Moorthy | Magnetic Barkhausen emission technique for evaluation of residual stress alteration by grinding in case-carburised En36 steel[END_REF]. Accurate surface inspections can be obtained with much higher frequencies (> 100 Hz) [START_REF] Moorthy | Magnetic Barkhausen emission technique for evaluation of residual stress alteration by grinding in case-carburised En36 steel[END_REF].

Several indicators have been used for the estimation of residual stress: MBNenvelop [START_REF] Samimi | Multi-parameter Evaluation of Magnetic Barkhausen Noise in Carbon Steel[END_REF], RMS value [15][16], MBNenergy [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF], or statistical markers, including the number of avalanches [START_REF] Šrámek | Influence of sample size and magnetizing voltage on Barkhausen noise during bending and uniaxial tensile tests[END_REF]. The relation between mechanical stress and MBN is nonlinear and is affected by plastic deformation and fatigue [START_REF] Roskosz | Magnetic methods of characterization of active stresses in steel elements[END_REF]. Most of the time, empirical relations are used [START_REF] Šrámek | Influence of sample size and magnetizing voltage on Barkhausen noise during bending and uniaxial tensile tests[END_REF] [START_REF] Roskosz | Magnetic methods of characterization of active stresses in steel elements[END_REF]. However, numerical simulations based on magnetoelasticity [START_REF] Roskosz | Magnetic methods of characterization of active stresses in steel elements[END_REF][37] or stochastic approaches [START_REF] Liu | Time-response-histogram-based feature of magnetic barkhausen noise for material characterization considering influences of grain and grain boundary under in situ tensile test[END_REF] have also been developed to interpret the mechanisms behind these empirical relations. For example, a linear relationship between the reciprocal of the MBNenvelope peak amplitude and the residual stress is established in [START_REF] Kypris | Measuring stress variation with depth using Barkhausen signals[END_REF]. However, the slope of this linear fit is identified from measurement since its theoretical determination would require delicate calculations and precise knowledge of the magnetostrictive behavior [START_REF] Kypris | Measuring stress variation with depth using Barkhausen signals[END_REF].

Although the stress state in industrial parts is usually multi-axial, the multiaxiality of stress is seldom considered in MBN studies [START_REF] Vengrinovich | Magnetic Barkhausen Effect in Steel Under Biaxial Strain/Stress: Influence on Stress Measurement[END_REF] [START_REF] Vengrinovich | Bi-axial stress quantitative evaluation with Barkhausen noise: the past and perspectives[END_REF]. Similarly, material anisotropy and its influence on MBN and stress evaluation are usually ignored. Material anisotropy, however, can be considered by controlling the magnetic excitation direction (rotating magnetic field [27][35] or inductor [START_REF] Le Manh | Model for the correlation between magnetocrystalline energy and Barkhausen noise in ferromagnetic materials[END_REF] [START_REF] Le Manh | Novel method for the accurate determination of magnetocrystalline energy from Barkhausen noise in ferromagnetic materials[END_REF]) or controlling MBN measurement direction (directional sensor [START_REF] Ducharne | Directional Barkhausen noise magnetic measurement using the magnetic needle probe method[END_REF]).

The objective of this work is to establish a modeling framework for the effect of stress on MBN and reach a quantitative understanding of stress evaluation. This work follows the footsteps of a recent modeling approach for the description of MBN [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF] [START_REF] Fagan | Magnetic Barkhausen noise: a simulation tool[END_REF]. MBN is studied through the MBNenergy(H) hysteresis cycles simulated by combining an anhysteretic behavior and a hysteresis contribution. Compared to the previous contribution [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF], the modeling approach is now capable of describing the effect of mechanical stress on the MBN response.

In the domain of mechanical stress evaluation, a unified theory relating magnetic signals to basic magnetic parameters is still lacking. This study undertakes the first steps toward an alternative method involving the combination of a predictive simulation tool and experimental observations. The classic MBN models' domain of validity is limited to the experimental conditions where the data used to parameter the model has been measured. Our simulation allows foreseeing stress distribution and Barkhausen noise-related behaviors impossible to observe experimentally.

The experimental measurement of stress-dependent MBN signal is first performed on two materials (section 2), chosen for their high difference in crystalline anisotropy and crystallographic texture. Section 3 describes the modeling approach and its extension to consider the effect of stress. In the last section, measurements and simulations are compared, and the potentialities of MBNenergy measurements for stress detection are discussed.

-Experimental characterization of the MBNenergy(H) and B(H) hysteresis cycles 2.1 -Materials

Two materials have been chosen for this work based on their different material properties connected with the magnetization process. The magnetization process in ferromagnetic materials is the result of two distinct mechanisms. The first mechanism is domain wall motion.

The magnetic domains with a magnetization oriented favorably to the applied magnetic field grow, while the domains unfavorably oriented decline in proportion. The second mechanism is magnetization rotation. The magnetization of a magnetic domain, initially oriented along an easy axis, coherently rotates towards the direction of the applied magnetic field. The magnetization process is schematically illustrated in Fig. 1. The first material is a Grain Oriented Iron Silicon steel (GO FeSi), typically used in transformer cores [48][49]. It contains 3wt% silicon. It is characterized by a strong crystallographic texture and a high crystalline anisotropy, resulting in a substantial macroscopic anisotropy. Due to the Goss texture, most grains in the material have an easy direction (<100>) close to the rolling direction (RD). As a result, the magnetization rotation mechanism is almost absent in this material when it is magnetized along RD. The domain wall motions mainly drive the magnetization. The saturation state is made of domains with magnetization along the easy direction parallel (or very close) to RD.

The second material is a high yield strength (of the order of 1000 MPa) Iron-Cobalt alloy (FeCo), typically used for high-performance rotating machines [50][51]. The composition is 49%

Fe, 49% Co and 2% V. The material exhibits an isotropic behavior in the sheet plane. The crystallographic texture is not very pronounced, and the magnetocrystalline anisotropy is weak.

This results in a more prominent role of the magnetization rotation mechanism in the magnetization process.

The dimensions of the GO FeSi and FeCo specimens studied are given in Fig. 2 below:

-Characterization setup

The objective is to characterize both the MBNenergy(H) and the standard B(H) magnetic responses of a specimen subjected to uniaxial tensile stress. It requires performing a magnetic measurement while constant stress is applied. A 2D view of the setup is depicted in Fig. 3 below:

Fig. 3 -Overall 2D view of the experimental setup for measuring the stress dependent MBN and magnetic responses of ferromagnetic specimens.

Mechanical stress

A tension-compression machine (Zwick/Roell Z030) and the associated computer are used for the stress application [START_REF] Domenjoud | Characterization of giant magnetostrictive materials under static stress: influence of loading boundary conditions[END_REF]. A 10 kN load cell (strain gauge sensor TC-LC010kN) is used for the force measurement. The resolution and accuracy of the force and displacement sensors are 0.2 N ±0.06% and 1 μm ±0.1%, respectively. The first series of measurements was done by increasing incrementally (Δσ = 5 MPa) the mechanical force up to 100 MPa for the GO FeSi and 500 MPa for the FeCo (far below the yield strength for each material). At each increment, the stress was maintained constant for the duration of the magnetic measurement. For the second series, the mechanical stress was decreased with the same force step, down to complete stress cancellation.

Both the sequences were repeated to check the reproducibility.

-Magnetic excitation

The magnetic part is made of two U-shaped FeSi 3% yokes, ensuring the closure of the magnetic flux. The tested specimen is placed and kept inside the testing machine thanks to a diamagnetic fixture (see Fig. 3). A series circuit of two primary coils (16 AWG wire), one wound around each yoke (672 turns each) and supplied in current by a Kepco 72-14MG amplifier, ensure the magnetic field generation. This power amplifier can generate 14 A and 72 V, with 0.2% accuracy. It is driven by a National Instrument DAQ USB-6346 acquisition device. The sampling frequency is 500 kHz for all channels.

-Magnetic instrumentation

The tangent magnetic field H is measured locally on the surface of the tested specimens using a GM08 Gaussmeter and a transverse Hall probe. The accuracy of this sensor is 1%, and the noise level is close to ± 40 A•m -1 . All the tested specimens are wound with two 100-turns coils, in opposite directions, as described in [START_REF] Moses | AC Barkhausen noise in electrical steels: influence of sensing technique on interpretation of measurements[END_REF]. The voltage drop over a single coil due to the lowfrequency variations is monitored as well. Both these differential and common-mode measurements are done simultaneously to acquire the time-dependent MBN and magnetic flux variations. The Differential method for the MBN raw signal acquisition leads to a high reduction of the parasitic noises and interferences affecting the two pick-up coils quasi simultaneously.

The B(H) loop is obtained by numerical integration of the common mode. A numerical correction is done to cancel the undesired drift due to the integration process.

-MBN energy

As the domain wall motions are a stochastic process, the MBN raw signal is unpredictable and not reproducible. To overcome this issue and converge towards repeatable results, time average indicators are used for the MBN analysis. Here, the MBNenergy is chosen. It is defined as [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF][54]:

𝑀𝐵𝑁 (𝑡) = ∫ 𝑠𝑖𝑔𝑛 (𝑠) 𝑉 (𝑠) 𝑑𝑠 ( 1
)
where 𝑉 is the electromotive force measured from the coil sensor. The construction of the MBNenergy hysteresis loop from the raw MBN measurement is illustrated in Fig. 4. The MBNenergy is not stricto sensu an energy, but as described in [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF], it can be considered an image of the kinetic energy of domain walls. To compare the MBNenergy(H) and the standard B(H) hysteresis loops, it is necessary to renormalize the MBNenergy measurement. This renormalization can be achieved by equalizing the hysteresis area between the two measurements or equalizing the measured values at the beginning of the saturation elbow. However, in an NDT situation where B(H) hysteresis loops are most of the time unavailable, this renormalization process is impossible and is replaced by a normalization to one.

In practice, during the experimentation, the MBNenergy quantity is returned through an analogic procedure, including:

-an amplification of the raw MBN signal (G = 50 dB) -a band-pass filter (MAX274ACN analogic filter). The cut-off frequencies were set to 4 and 12 kHz, and the filter response attenuation slope to 70dB/dec.

-an AD633 analog multiplier.

-A low noise operational amplifier LT1001 in an integrator setup and an external switch push button to reset the integration process at the beginning of each new measurement.

For a reduction of the parasitic noises, all the data were averaged over five periods of excitation.

-Experimental results

The B(H) and the MBNenergy(H) hysteresis loops obtained for the GO FeSi specimen are shown in Fig. 5 for three different levels of tensile stress. The B(H) and the MBNenergy(H) hysteresis loops obtained for the FeCo specimen are shown in Fig. 6 for three different tensile stress levels. A first observation is that the magnetic behavior of the FeCo is much more sensitive to the application of tensile stress compared to the GO FeSi. A tensile stress increases the permeability at low field for the two materials tested. It is consistent with classical observations on positive magnetostriction materials [18][55]. The tensile stress also reduces the coercivity. This effect is also consistent with previous observations [START_REF] Rekik | Equivalent stress model for magnetic hysteresis losses under biaxial loading[END_REF][56], and it is much more significant for the FeCo alloy. For both materials, the tensile stress decreases the slope in the saturation/remanence branch.

- Unlike the FeCo, the MBN energy cycles for the GO FeSi exhibit a very similar shape compared to the B(H) loop. This observation is consistent with the fact that only the domain wall motions contribute to the MBNenergy(H) hysteresis cycles, as described in [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF]. The magnetic behavior of the GO FeSi being mostly driven by domain wall motion, the B(H) and MBNenergy(H) curves are the images of the same processes. On the contrary, magnetization rotation is more prominent for the FeCo. The B(H) curve incorporates physical mechanisms absent from the MBNenergy(H) curves.

It is then expected that they show more significant differences.

It is also worth noting that increasing tensile stress tends to make the MBNenergy(H) and B(H) cycles more similar, which could be interpreted as a lesser contribution of rotation mechanism in tension strained specimens.

A good indicator of the effect of stress on the magnetic behavior is the coercive field Hc. Fig. 7 shows the coercive fields extracted from the B(H) measurements and the MBNenergy(H) measurements, for both materials. A shift can be observed between these two extractions, even though the magnetic field is obtained from the same sensor. This difference is due to the filtering step performed to detect the MBN noise, which affects the value of the magnetic field. This effect is more visible for the FeSi GO, which exhibits a much lower coercivity. The frequency analysis (Fig. 8) of the analytical filter confirms this statement. The magnitude plot shows (as expected) 4 and 12 kHz as cut-off frequencies, while the phase plot reveals a phase answer varying from to -12 rd. The phase is negative for f > 6 kHz, which is a large majority of the bandwidth. This negative phase shift is significant enough to delay the Barkhausen measurement and increase the MBNenergy(H) coercivities. From Fig. 7, it is observed that the coercivity of the GO FeSi is almost insensitive to the application of tensile stress, while the FeCo shows more sensitivity. The evolution of the coercive field 𝐻 under uniaxial tensile stress can be approximated by a decreasing exponential function in the form of Eq. 2.

𝐻 (𝜎) = 𝑎. 𝑒 + 𝑐 (2)

The MATLAB® curve fitting toolbox was used to identify the parameters a, b and c from the results shown in Fig. 7. These parameters are given in Tab. 1.

Tab. 1 -Parameters a, b, and c for the description of 𝐻 (𝜎) as defined in Eq. 2. For some materials, the measurement of 𝐻 (𝜎) could be an excellent macroscopic indicator to evaluate the level of stress in a material [START_REF] Hu | Micromagnetic modeling studies on the effects of stress on magnetization reversal and dynamic hysteresis[END_REF]. However, to obtain a more comprehensive description of the effect of stress on the magnetic signature of ferromagnetic materials, we are developing in the next section a modeling approach for the B(H) and MBNenergy(H) loops.

-Modelling approach

The choice of modeling approach presented here is based on the classical separation between reversible (anhysteretic) and irreversible (hysteretic) contributions to magnetic behavior. The overall behavior will be defined as the superimposition of an anhysteretic term with a hysteresis contribution. The approach was presented in [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF] for a purely magnetic loading and is extended hereafter to the case where a mechanical stress is applied.

-Anhysteretic behavior: the multiscale model

The multiscale model [START_REF] Daniel | Reversible magneto-elastic behavior: a multiscale approach[END_REF][59] has initially been developed to describe the anhysteretic magnetic behavior of a ferromagnetic specimen based on the statistical evolution of its domain microstructure. The multiscale model provides directional information and naturally includes anisotropy effects [START_REF] Daniel | A multiscale model for magneto-elastic behaviour including hysteresis effects[END_REF]. A polycrystalline ferromagnetic specimen is described as an aggregate of single crystals (grains). Each grain is divided into a finite number of magnetic domain families.

Each family is characterized by a given magnetic orientation α. The volume fraction fα of each magnetic domain family is calculated using their potential energy Wα:

𝑊 = 𝑊 + 𝑊 + 𝑊 (3
where Wα k , Wα H , Wα σ , stands for the magneto-crystalline (Eq. 4), magnetostatic (Eq. 5), and magneto-elastic (Eq. 6) energy, respectively. 𝐇 , 𝐌 , 𝝈 and 𝜺 are the magnetic field, the magnetization, the stress tensor, and the magnetostriction strain tensor defined at the magnetic domain scale. 𝐌 is defined by its norm (the material saturation magnetization Ms), and its direction is given by 𝛾1, 𝛾2, 𝛾3. 𝜺 is defined by the magnetostriction constants 𝜆100 and 𝜆111 [START_REF] Daniel | Reversible magneto-elastic behavior: a multiscale approach[END_REF].

𝐾1 and 𝐾2 are the magneto-crystalline energy constants. 𝐴 is a material parameter that can be set using the initial macroscopic susceptibility 𝜒 of the unstressed anhysteretic magnetization curve [START_REF] Daniel | Reversible magneto-elastic behavior: a multiscale approach[END_REF].

Once fα calculated for all magnetic domains families, the magneto-elastic response at the grain scale is obtained by a volume average:

𝜺 = 〈𝜺 〉 = ∑ 𝑓 𝜺 (8) 𝐌 = 〈𝐌 〉 = ∑ 𝑓 𝐌 (9)
Finally, an orientation distribution function (distribution of the crystallographic orientations) obtained from X-ray diffraction or Electron Back Scattering Diffraction (EBSD) is used to simulate the behavior at the polycrystalline scale.

In the simplified calculation process used here, both the magnetic H and mechanical σ external stimuli are supposed to be uniform within the material. The potential energy and the volume fraction of each magnetic domain family are calculated first. Then, the magnetization is calculated for each grain (Eq. 9), and finally, an average over the volume is performed to obtain the whole specimen magnetization:

𝐌 = 〈𝐌 〉 ( 10
)
This process allows the construction of the stress-dependent anhysteretic magnetization curves based on a limited number of intrinsic material parameters. They have to be complemented with the crystallographic texture data to predict the macroscopic material behavior. The magnetic induction is finally easily deduced from the magnetization using Eq.11.

B = µ (𝐻 + 𝑀) (11)
In all the following, the simulation parameters and the crystallographic texture data from [START_REF] Hubert | Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[END_REF] and [START_REF] Rekik | Mesure et modélisation du comportement magnéto-mécanique dissipatif des matériaux ferromagnétiques à haute limite élastique sous chargement multiaxial: application aux génératrices à grandes vitesses pour l'aéronautique[END_REF] will be used for the GO FeSi and the FeCo, respectively. The orientation distribution function consists of 60 orientations for GO FeSi (strong texture) and 650 for FeCo (almost isotropic). The material parameters are given in Tab. 2 below:

Tab. 2 -Multiscale model simulation parameters for the GO FeSi and the FeCo.

-B(H) hysteresis loops

From the knowledge of the stress-dependent anhysteretic magnetization curve, the full hysteresis must now be constructed. A popular approach is the Jiles-Atherton model [START_REF] Jiles | Theory of ferromagnetic hysteresis[END_REF]- [START_REF] Sai Ram | Inclusion of dynamic losses in a scalar magneto-elastic hysteresis model derived using multiscale and Jiles-Atherton approaches[END_REF].

Given the shape observed for the hysteresis loops, and given the fact that we are only dealing with major loops, we have opted for a simple hysteresis shift from the anhysteretic curve to the hysteresis loop. This approach is analogous to a mechanical dry-friction [START_REF] Wojewoda | Hysteretic effects of dry friction: modelling and experimental studies[END_REF]- [START_REF] Ducharne | Low frequency modelling of hysteresis behaviour and dielectric permittivity in ferroelectric ceramics under electric field[END_REF]. The B(H) loop is then defined as:

𝐵 = 𝑓 (𝐻 + 𝛿𝐻 (𝜎)) (12)
where 𝐻 (𝜎) is the stress-dependent function for the coercivity (Eq. 2 is only valid under tensile stress (𝜎 > 0)) and 𝛿 = ±1 a directional parameter.

𝛿 = 1 𝑖𝑓 𝑑𝐻 𝑑𝑡 > 0 𝛿 = -1 𝑖𝑓 < 0 (13)
Fig. 9 below depicts some comparisons between simulations and experimental results for both materials tested with and without tensile stress. The excellent agreement is considered a validation of the simulation method.

B (T) B (T)

High stress

No stress

No stress High stress 20

-MBNenergy(H) hysteresis loops

The adoption of a similar approach for the MBNenergy loops requires some adjustment. Indeed, it is reminded that domain wall motion is the only magnetization contribution observed through the MBNenergy measurements. The simulation of the anhysteretic MBNenergy(H) behavior is possible with the multiscale model, but it requires restricting the magnetization process to the domain wall contribution. This is done by setting K1 and K2 (the magneto-crystalline anisotropy coefficients) with exaggerated values [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF]. A "no-rotation" anhysteretic magnetization curve is obtained and used as the skeleton for the MBNenergy curve. As explained in [START_REF] Fagan | Multiscale modelling of the magnetic Barkhausen noise energy cyclesm[END_REF], the hysteresis shift is considered the same for the MBNenergy(H) and the B(H) hysteresis cycles. Therefore, the parameters defined for the B(H) curve are equally set for the MBNenergy(H) curve :

𝑀𝐵𝑁 = 𝑓 (𝐻 + 𝛿𝐻 (𝜎)) (14)
The MBNenergy curve can finally be normalized to one as described in the first section for comparison with the experimental measurements.

These comparisons are presented in Fig. 10. Except in the saturation elbow, good agreement is observed here once again.

-Feasibility of a stress evaluation technique based on MBN measurement

The evaluation of mechanical stress in structural ferromagnetic steel is fundamental. It allows anticipating undesired degradation and failure. As illustrated in Fig. 10, MBN is highly stressdependent. It is well adapted to modern NDT as it can be implemented with limited costs and space restrictions. It is shown in this section that the proposed approach for the modeling of MBNenergy can be an efficient tool to optimize stress evaluation strategies using MBN measurements. It is worth noting that such evaluation would require specific inversion procedures, which are not discussed in this paper focused on the effect of stress on the MBN signals.

-Evaluation of tensile stress configurations

The first case considered is a configuration of tension stress as considered in the experiments performed in this paper. The effect of stress on the coercive field, obtained either from the experimental MBNenergy(H) or from the B(H) hysteresis cycles, was shown in Fig. 7. Other indicators can be used, such as the MBNenergy(H) hysteresis area, or the magnetic field H95 measured at MBNenergy = 0.95 x max (MBNenergy) or H99 measured at MBNenergy = 0.99 x max (MBNenergy). These indicators are reported in Fig. 11 as a function of the applied stress. They have been normalized with their value at 𝜎 = 0. The dependence to tensile stress is clear for the FeCo but less significant for the GO FeSi. All indicators exhibit similar trends with increasing stress level. The effect of uniaxial stress on the coercive field is correctly described by Eq.2 (as shown in Fig. 7), but it requires previous calibration on experimental results. Moreover, no generalization to more complex mechanical loading can be expected since it is the result of a purely mathematical fitting. H95 and H99 on the other hand can be predicted independently from the measurement using the proposed multiscale approach. The results are shown in Fig. 12 for these two indicators, both for GO FeSi and FeCo. Fig. 12 shows a clear decrease of H95 and H99 with the application of stress and a saturation of this effect at very low stress. Although the trend is consistent with experimental results, the quantitative differences are significant. This disagreement can notably be explained by the difference between the anhysteretic function (used in the exploitation of the modelling results)

and the increasing and decreasing part of the hysteresis cycle envelope (used in the exploitation of the experimental results). Together with relatively low variations, it makes the direct use of the model uneasy in this configuration. However, the modelling approach can be used to explore the effect of different sensor orientations. Fig. 13 shows, in polar coordinates and for the FeCo alloy, the predicted normalized H95 and H99 indicators for a uniaxial stress applied along the rolling direction (RD), as a function of the orientation of the MBN sensor at various angles from the rolling direction (RD, 0°) to the transverse direction (TD, 90°). The results are not perfectly smooth because a discrete orientation distribution function (650 crystallographic orientations) has been used to describe the crystallographic texture, so that the angular response of the material is not exactly regular. The general trend is given by a spline approximation shown as a plain line. While it is evident that slight variation is observed when the MBN sensor is aligned with the applied stress (RD), much more significant variations are found when the sensor is positioned perpendicularly to TD. It is also found that the optimal orientation for maximum MBN sensitivity is obtained when the magnetic field is applied with an angle between 30° and 75° from RD. Our experimental setup and samples allow only a measure with the magnetic field aligned with RD, hence experimental validation of these results has not been done. The required setup would be similar to the one described in [START_REF] Neyra Astudillo | Magnetic Barkhausen noise and magneto acoustic emission in pressure vessel steel[END_REF] and the use of a biaxial stress device (as in [START_REF] Rekik | Influence of a multi-axial stress on the reversible and irreversible magnetic behaviour of a 3% Si-Fe alloy[END_REF]).

In a practical application, if the orientation of the uniaxial stress is known before its detection, the modeling can be used to choose the most sensitive orientation for the sensor. On the contrary, if the orientation of the uniaxial stress is unknown, several successive measurements can allow the identification of both intensity and orientation.

-Evaluation of multi-axial stress configurations

Most practical industrial parts are subjected to a multi-axial stress state and not only uniaxial tension. For an effective evaluation of stress, it is necessary to generalize the MBN 1D analysis to the real-life 3D multi-axial stress configurations. There are few experiments in the scientific literature that establish the link between magnetic response and multi-axial stress [START_REF] Rekik | Influence of a multi-axial stress on the reversible and irreversible magnetic behaviour of a 3% Si-Fe alloy[END_REF]- [START_REF] Aydin | Effect of multi-axial stress on iron losses of electrical steel sheets[END_REF].

Unfortunately, none of them report MBN observations. The multiscale modeling approach proposed in this study is another way to explore the influence of multi-axial stress on the MBNenergy(H) hysteresis cycles. Even though it has been parameterized and validated for a 1D tensile stress configuration, the model is intrinsically 3D.

As an example, Fig. 14 shows the modeling results obtained for H95 (very similar results are obtained for H99) in the case of the FeCo sample subjected to a biaxial stress. RD and TD are the principal axes for the stress, and the MBN sensor is aligned along RD. The stress tensor 𝝈 is given by Eq. 15 in the (RD, TD) coordinate system and the principal stresses 𝜎 and 𝜎 vary from -250 to 250 MPa. The results confirm again that a configuration with an MBN sensor positioned parallel to the direction of a uniaxial tension is the less sensitive configuration: the indicator H95 shows almost no variation on the horizontal line at 𝜎 = 0 for 𝜎 > 0. On the contrary, biaxial configurations with a compressive component parallel to the MBN sensor create the most significant variation of H95. It is again concluded that if the stress configuration is known a priori, the multiscale approach can be used to define the most appropriate orientation of the MBN sensor for maximum sensitivity in stress evaluation.

𝝈 = 𝜎 0 0 0 𝜎 0 0 0 0 , , (15)

-Conclusion

The stress dependence of B(H) and MBNenergy(H) hysteresis loops exhibit apparent similarities.

MBNenergy(H) hysteresis cycles can be modeled by combining an anhysteretic behavior and a hysteresis contribution. On the one hand, the anhysteretic contribution can be calculated using a multiscale approach. On the other hand, the hysteresis contribution can be kept identical to the one identified from the standard B(H) loops for a given specimen. In this paper, this extending the method described in [START_REF] Fagan | Magnetic Barkhausen noise: a simulation tool[END_REF] to the stress influence.

Fig. 1 -

 1 Fig. 1 -Schematic illustration of the magnetization process. (a) Demagnetised state (b) Domain wall motion (c) Magnetisation rotation. In practice, the two mechanisms can occur simultaneously.

Fig. 2 -

 2 Fig. 2 -Dimensions of the GO FeSi and FeCo specimens tested.

Fig. 4 -

 4 Fig. 4 -MBNenergy(H) hysteresis loop, construction process.

Fig. 5 -

 5 Fig. 5 -GO FeSi: experimental measurement for the stress-dependent B(H) -left -and MBNenergy(H) hysteresis cycles.

Fig. 6 -

 6 Fig. 6 -FeCo: experimental measurement for the stress-dependent B(H) -left -and MBNenergy(H) hysteresis cycles.

Fig. 7

 7 Fig. 7 Stress dependence of the measured B(H) and MBNenergy(H) coercivities.

Fig. 8 -

 8 Fig. 8 -MAX274ACN analogic filter Bode plots, comparison measurement/simulation.

) 4 )

 4 𝑊 = 𝐾 (𝛾 𝛾 + 𝛾 𝛾 + 𝛾 𝛾 ) + 𝐾 𝛾 𝛾 𝛾 (

Fig. 9

 9 Fig. 9 Comparisons simulations/measurements for the B(H) hysteresis cycles.

Fig. 10

 10 Fig. 10 Comparisons simulations/measurements for the MBNenergy(H) hysteresis cycles.

Fig. 11 -

 11 Fig. 11 -Tensile stress dependence of the MBNenergy(H) hysteresis area, coercive field, H95, and H99 normalized by the corresponding stress-free reference.

Fig. 12 -

 12 Fig. 12 -Multiscale model prediction H95 and H99 normalized by the corresponding stress-free reference, as a function of the applied tensile stress.

Fig. 13 -

 13 Fig. 13 -FeCo alloy : Evolution of the predicted normalized (a) H95 and (b) H99 indicators under 50, 100, 150 and 200 MPa tensile stress applied along RD as a function of the orientation of the MBN sensor from RD to TD. The dots show the result of the calculation while the lines are just guide for the eyes.

Fig. 14 -

 14 Fig. 14 -FeCo alloy: Evolution of the predicted normalized H95 under biaxial stress states. The MBN sensor is positioned along 𝜎 (RD).

  simulation approach has been investigated to analyze the potentialities of MBN techniques to detect mechanical stress in ferromagnetic materials. It is successfully applied on two exemplary ferromagnetic materials with significantly different behavior (an Iron-Silicon steel and an Iron-Cobalt alloy) subjected to uniaxial tensile stress. Several indicators have been tested for their sensitivity to stress. Based on MBNenergy(H) analysis, coercivity appears as a good indicator for both materials. Two other indicators, easily predicted by the proposed modeling approach, H95, and H99, have also shown interesting stress dependence, especially when the magnetic rotation mechanism is significant (FeCo).The simulation method has further been used to investigate the MBN variations of the Iron-Cobalt alloy under various sensor configurations and multi-axial stress states. It is concluded that the proposed approach can be a great help in the definition of optimal MBN sensor positioning depending on the stress configuration. It is also shown that uniaxial tension is one of the most unfavorable configurations for stress detection, although appropriate sensor positioning can increase measurement sensitivity. This work is a leap forward to the understanding and modeling of magnetization mechanisms for MBN interpretation. It belongs to a larger project with the final objective of using and interpreting MBN as a quantitative NDT tool for stress detection. The following steps include the development of more complex mechanical testing configurations such as compression or biaxial loadings for validation purpose, the introduction of temperature effects in the simulation of the MBNenergy(H) hysteresis cycles, or the simulation of stress-dependent MBN raw signals by

Data availability statement:

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.