An Extension of the Vector-Play Model to the Case of Magneto-Elastic Loadings

Accurate modeling of the coupling between mechanical and magnetic behavior is a key challenge for designing many electromagnetic devices. The requirements for such modeling are notably the ability to consider multiaxial configurations, thermodynamic consistency to allow the calculation of losses, and the implementability into structural analysis tools. So far, the modeling approaches available in the literature do not usually combine these three features simultaneously. In this paper, for the first time, the influence of mechanical stress on the magnetic hysteretic behavior is modeled through the association of a reversible simplified multiscale approach and a macroscopic energy-based magnetic hysteresis model in a vector-play form. A phenomenological description of the dissipation parameters under mechanical stress is proposed. The non-monotonic effect of tensile stress on the magnetic permeability is modeled using a second-order development in the magneto-elastic energy. Material parameters for both reversible and irreversible behavior are identified from experimental characterization under mechanical stress performed on a DC04 electrical steel. The experimental tests include anhysteretic and hysteretic measurements. Modeling results of the anhysteretic magnetic permeability, the coercive field, and the remanent induction under several levels of peak magnetic field and uniaxial mechanical stress are satisfactorily compared with those obtained experimentally. The model is shown to reasonably predict the hysteresis losses under tensile and compressive stress, as well as the response of the material under a complex magnetic field waveform with harmonic content.


I. INTRODUCTION
Mechanical stresses strongly influence the losses of magnetic materials. Such effects have been notably illustrated in electrical steels under uniaxial [1], [2] or biaxial [3] mechanical loadings. Due to this coupled behavior, the overall efficiency of electromagnetic devices can be altered by different mechanical stress sources, such as centrifugal forces [4], or shrink-fitting [5]. Several methods to model the coupled magneto-mechanical hysteretic behavior are presented in the literature and can be defined from either a macroscopic approach, a multiscale approach, or a combination of both.
The associate editor coordinating the review of this manuscript and approving it for publication was Ladislau Matekovits .
In macroscopic models based on the classical scalar Jiles-Atherton (JA) approach -with anhysteretic behavior described by the Langevin function -the effects of stress on the magnetic behavior can be defined by an equivalent field incorporated in the effective field definition. Such an approach was used when uniaxial stress is applied along the magnetic field [6], [7], and when an angle between uniaxial stress and the magnetic field is considered [8]. The uniaxial mechanical limitation of these extensions of JA model can be overcome by considering the reversible behavior through the minimization of a Helmholtz free energy density, which is defined as a function of scalar invariants [9], [10]. In this approach, the scalar pinning parameter of the original JA model is replaced by a stress-dependent second-order tensor [11], [12]. One drawback of this approach is the need for several measurements under stress to identify the parameters of the Helmholtz free energy function. Whereas the magnetic JA model is easy to implement, and its parameters are relatively easy to identify, some limitations were observed in the representation of asymmetric minor loops [13] and in the definition of the anhysteretic behavior as a function of an effective field, which results in non-physical reversible behavior [14].
Another macroscopic approach consists of a magnetoelastic extension of the classical scalar Preisach magnetic hysteresis model. It can be defined by introducing stress in the effective field definition [15], or considering the influence of stress in each hysteron operator [16], or in a probability distribution function [17]. The mathematical approach yields parameters that are difficult to identify from measurements.
A macroscopic magnetic hysteresis model based on an analogy with plasticity is presented in [18]. The magnetic energy is represented by analytical expressions. A switching surface with a radius determined by the coercive field captures the hysteretic behavior. However, complementary parameters fitted on experimental data are required to capture the first magnetization curve and asymmetric minor loops. Following this approach, in [19] a magneto-elastic thermodynamic framework is presented, and the main focus is on the response of magnetorheological elastomers with permanent magnets particles.
The second class of modeling concerns multiscale approaches. Three scales are considered: the magnetic domain, the single-crystal (or grain), and the representative volume element (RVE or polycrystal) scale [20]. Magnetic and mechanical quantities are defined at different scales through localization and homogenization schemes. Originally proposed to model the anhysteretic magneto-mechanical behavior [20], the approach shows good prediction capabilities with a clear protocol for parameter identification. A multiscale model with hysteresis is proposed in [21] with the inclusion of a hysteresis energy term in the domain free energy definition that depends on the previous state of magnetization. As pointed out in [22], the localization process in multiscale approaches can result in time-consuming finite element simulation of electromagnetic devices. This can be overcome, in the reversible case, by using simplified versions of the full multiscale approach. A simplification consists in considering an equivalent simplified crystallographic texture with a reduced number of crystal orientations, leading to a simplified texture multiscale model (STMSM) [22]. A further simplification consists in representing the macroscopic behavior using a fictitious equivalent single-crystal, similar to the Armstrong approach [23], leading to the so-called simplified multiscale model (SMSM) [24], [25]. This latter option, when limiting the number of possible domain orientations to six, can provide analytical formulas for the magneto-elastic behavior [26], [27].
In an analogous multiscale approach called assembled domain structure model (ADSM), each ADSM is made of a simplified domain structure model (SDSM) formed by six domains. The magnetization direction of each domain is related to the easy axes of the material. A minimization of the energy balance (which comprises a magneto-elastic term) in a SDSM structure results in the magnetization state [28]. The hysteresis effects are reproduced by considering a pinning field defined by a Gaussian distribution [29].
The last class of magneto-mechanical hysteresis models consists of a combination approach: the reversible behavior is modeled with a magneto-mechanical multiscale approach, and the magnetic hysteresis is considered from a macroscopic description. In this case, examples are the combination of the full multiscale approach and Hauser model [30], the SMSM with the magnetic JA model [22], [31], [32], and the analytical multiscale model with the Kádár product model [33] or with the JA model [34]. Hysteresis effects are considered in the Armstrong model by defining a macroscopic energy dissipation term related to the defects of a material, which constrains the domain wall motion [35]. Improvements of this approach to represent asymmetric minor loops and an extension to consider variable stress under constant magnetic field are presented in [36].
These three classes here introduced contemplate only some examples of magneto-mechanical hysteresis models. Besides these classes, phase-field approaches can be highlighted. This modeling aims to describe the spatial and temporal evolution of magnetic domains in a microstructure. Numerical methods are used to minimize an energy function related to the domain wall motion of a discretized ferromagnetic material [37], [38]. This approach gives detailed information about the microstructure of a ferromagnetic material under magnetic and mechanical loadings. When the macroscopic response is sought, this modeling increases the computational cost.
Although many hysteretic magneto-elastic modeling approaches are available in the literature, as described above, none of them simultaneously combines three key features for the numerical analysis of electromagnetic devices. The first one is the ability to consider fully multiaxial loadings as encountered in practical applications, the second is thermodynamic consistency to compute losses accurately, and the last is the implementation into numerical analysis tools, which requires low computation time for behavior evaluation. In this paper, for the first time, a combination of a SMSM with an energy-based magnetic hysteresis model in a vector-play form [39], [40] is proposed. This rate-independent magnetic hysteresis model is based on an energetic description at the macroscopic scale and defined directly in a vector form. The association with the SMSM allows multiaxial stress configurations. The parameters are evaluated from a magneto-elastic characterization performed on DC04 steel under compression and tension, and the performance of the model is evaluated on a different set of experimental data. VOLUME 10, 2022

II. MAGNETO-ELASTIC MODEL
The thermodynamics basis of the coupled magneto-elastic model is addressed in this section with a distinction between reversible and irreversible processes. The relationship between magnetic induction B, magnetic field H , magnetization M , magnetic polarization J and vacuum permeability µ 0 is expressed as: (1)

A. THERMODYNAMICS ASPECTS
In the framework of continuum thermodynamics, the first law of thermodynamics in a ferromagnetic material at the macroscopic scale can be written as [40]: • u is the time-derivative of the internal energy density, the dot product H · • B represents the magnetic power density, the double-contraction product σ : • between the second-order tensors of mechanical stress σ and strain rate • represents the mechanical power density, and q the heat flux. The second law of thermodynamics can be expressed as [41]: with s the entropy and T the temperature. Defining the Helmholtz free energy density as f ≡ u − Ts, combining (1), (2) and (3), and neglecting spatial and temporal thermal variations, the Clausius-Duhem inequality (CDI) for the magneto-mechanical case writes: with D the dissipation. The term µ 0 H · • H is omitted because, in general, it is negligible in comparison to the contribution of H · • J to the magnetic power density. A summary of the thermodynamics laws and approximations considered is presented in Table 1. Considering only the reversible behavior (D = 0), the Legendre transformation allows writing (4) as a function of the time-derivative of the Gibbs free energy • g:

B. REVERSIBLE BEHAVIOR
The reversible behavior is modeled using a simplified multiscale approach [25]. The scales considered are represented in Figure 1. In this work, we are interested in modeling the behavior of the RVE. The following assumptions are made: (a) the material behavior is isotropic, (b) demagnetizing effects are negligible, and (c) both applied magnetic field and mechanical stress are homogeneous at the domain scale (denoted by α).
In a domain family with direction α, the polarization J α is: with M s the saturation magnetization. The magnetostriction strain µ α for isotropic behavior is: where λ s denotes the maximum magnetostriction strain, ⊗ represents the tensor product, and I is the second-order identity tensor. The energy variation dg α for a time step dt at the domain scale is written [42]: with α the total strain at the domain scale. The magnetic part of the Gibbs free energy is defined by the integration over the magnetic field path: Considering small perturbations: α = e α + µ α , with e α the elastic strain. Supposing uniform strain in the single-crystal, the magneto-elastic part of the Gibbs free energy is written by integration over the stress path [42]: It was shown in earlier works that the effect of stress on magnetization is non-monotonic [30], [42] and this simplified approach does not capture such a tendency. This drawback can be solved by adding a stress-dependent demagnetizing term in the energy balance [30], or by adding a second order term -quadratic in stress -in the magneto-elastic energy definition [42]. We choose here this second option. The magneto-elastic energy is therefore defined as: In this definition, the second-order magnetostriction constant λ s is introduced and the equivalent stress σ eq [43] is written in terms of h = H / H and the deviatoric part of σ : The magnetostriction strain is composed of the sum of (7) with a second-order magnetostriction strain Considering a particular case under uniaxial stress σ eq = σ 11 applied along the magnetic field direction h = [1 0 0] t , the only non-null component of Therefore, introducing a second-order term in the magnetoelastic energy results in a magnetostriction strain that is stress-dependent. This allows capturing the non-monotonic effect of tensile stress on the magnetic behavior [42], [44]. λ s can be defined as (see Appendix A): where σ m is the value of the applied stress corresponding to the maximum magnetic permeability. Combining (11) and (15), the elastic part of the Gibbs free energy is: The volume fraction p α of a domain family with direction α is evaluated using a Boltzmann relation [45]: where the parameter A s is proportional to the initial susceptibility χ 0 of the stress-free anhysteretic curve [20]: In this simplified approach, the domain orientations are defined from a discretization of a unit sphere [46]. The macroscopic polarization and macroscopic strain are:

C. IRREVERSIBLE PROCESS AND VECTOR-PLAY MODEL
In a reversible framework at a macroscopic scale, a reversible magnetic field H rev and a reversible stress σ rev are defined: Combining (20) and (21) with the Clausius-Duhem inequality (4) yields: In the dissipative framework, an irreversible field H irr = ( H − H rev ) and an irreversible stress σ irr = (σ − σ rev ) are introduced. A simplification consists in considering mechanical stress only in a reversible way, and as a result σ irr = 0. The dissipation is modeled by analogy with a mechanical dry-friction system [39]. The defects that pin domain walls at specific positions are represented by a pinning field κ, a positive scalar in the isotropic case. The dissipation writes: As in the reversible case -where a relation between H rev and f was defined -H irr can be written as a function of the partial derivative of D. Since D is not differentiable at • J = 0, the subdifferential of a convex function is considered [39], [47]: This implies that for H irr < κ the polarization J will remain constant until H irr = κ [47]. From the previous definitions: The pinning parameter in real materials can be represented by a statistical distribution of pinning fields κ k with N − dryfriction systems (or cells) with normalized weights ω k that verify N k=1 ω k = 1 [40]. At each cell k, the polarization J k ( H k rev , σ ) and strain k ( H k rev , σ ) are related to the homogenized quantities: The following simplification can be made: the direction of H irr is written in terms of the reversible field at the previous time step H k rev(p) . This results in a vector-play model [39]. Using this approximation, the explicit update procedure of H k rev at each cell is: otherwise.
The field H k rev and σ are the inputs of the SMSM. J k ( H k rev , σ ) and k ( H k rev , σ ) are the outputs. J and are then defined by (26). B is given by (1). A simplified schematic of the algorithm is presented in Fig. 2.

III. EXPERIMENTAL SETUP
The apparatus used to carry out the magneto-mechanical characterization of a sample of DC04 steel (250 mm x 20 mm x 2 mm) under uniaxial stress is detailed in [48] and shown in Fig. 3. The mechanical setup is composed of a tension/compression machine Zwick/Roell Z030 with the possibility to control in force or displacement. A force control is used with a resolution and accuracy of 0.2 N ± 0.06%, and force measurements are performed using a 10 kN load cell (strain gauge sensor TC-LC010kN).
The magnetic setup is composed of two U-shaped Fe-Si yokes to ensure the closure of the magnetic flux. A Kepco 72-14MG amplifier, that can deliver 14 A and 72 V with 0.2% accuracy, supplies current to an excitation coil (28 turns) positioned around the sample. The current is measured with a LA 125-P transducer with 0.6% accuracy. A Teslameter FM302 and a transverse Hall probe 20 mT AS-VTP, which can operate from DC to 1 kHz, measure the magnetic field with an accuracy of 0.5 % and measured noise of 19 A/m in the range of 0 -15.9 kA/m. The time integration of the induced voltage of a B-coil (85 turns) wound around the sample (measurement area of Fig. 3) results in the measured induction with measured noise of about 0.1 mT with accuracy of 0.2% [48].
The magnetostriction strain is measured with a strain gauge rosette glued on the measurement area surface (Fig. 3) of the sample. The signal is amplified with a 4-channel strain gauge conditioner Vishay 2120 B with about 0.5% of accuracy and measured noise of about 10 -6 . A DS 1006 dSPACE processor board performs the acquisition and control of signals with a sampling frequency of 50 kHz. More information on the control and acquisition system can be found in [48]. A measurement reproducibility error is found to be about 0.5% in the magnetic field and 0.3% in the induction. The procedure to measure the anhysteretic magnetic behavior is detailed in [49]. The controlled current is set as an exponentially decaying sine wave, defined with a frequency of 1 Hz, superimposed to a bias level. This approach is repeated at several bias levels and under mechanical compressive (-) and tensile (+) uniaxial stresses that vary between -100 MPa and 100 MPa. The stress levels are defined to be below the yield stress of the sample (120 MPa) and the Euler buckling critical load estimated as -296 MPa.
The hysteresis measurements are performed at the same typical levels of stress and electric current, but the frequency of the electrical loading is reduced to 25 mHz. Indeed, it was observed that it is the frequency at which the quasi-static regime is attained (see Appendix B). The procedure to correct the drift in the measured induction is presented in Appendix C. Table 2 shows the quantities involved in the characterization process.
The magnetic quantities H c (coercive field) and B r (remanent induction) are evaluated by a linear regression around the point of interest -B = 0 for H c and H = 0 for B r . A post-processing filtering is applied to the measurements using a 50-point moving average on the cycle. The error bars take into account the noise and reproducibility presented before. Measurement results are discussed in section V when compared to modeled estimates.

IV. IDENTIFICATION OF MATERIAL PARAMETERS
The magneto-elastic model is entirely defined by the parameters of the anhysteretic behavior, here based on the SMSM, and the probability distribution of pinning field ω(κ). The parameters are evaluated from anhysteretic and hysteretic characterizations under uniaxial stress.

A. REVERSIBLE PARAMETERS
The reversible parameters M s , λ s , and χ 0 are identified from anhysteretic characterization without applied stress. M s is the maximum magnetization measured on the stress-free M -H curve. λ s is the maximum longitudinal magnetostriction strain obtained on the stress-free magnetostriction curve. χ 0 is the slope, at H = 0, of the anhysteretic stress-free M -H curve. A s is calculated from χ 0 by using (18). Fig. 4 presents the anhysteretic relative magnetic permeability µ anh r as a function of the applied stress for different values of the magnetic field. A non-monotonic effect of tensile stress on the reversible behavior is observed. Such non-monotonicity justifies the use of a second-order term in the magneto-elastic energy. This approach introduces an additional material parameter λ s (see (11)) which can be identified from (15).
The identified reversible parameters are presented in Table 3. The modeled anhysteretic behavior results in Fig. 5 (bottom) and shows a good agreement with the measurements in Fig. 5 (top). Fig. 4 shows that the SMSM with a second-order term can capture the reversible behavior and the non-monotonic effect. Differences become apparent, especially for tensile stress of 100 MPa, where the model underestimates the relative permeability at low field. Such a tendency is inevitable with the proposed description (second-order  elastic energy term), which imposes the permeability curve to be symmetric with respect to σ m , as shown in Appendix A. A possibility to improve this drawback would be using a stress-dependent demagnetizing term in the free energy, as proposed in [30], instead of or as a complement to the second-order approach. Another option would be introducing higher order terms in the elastic energy, to the price of additional material parameters.

B. DISSIPATIVE PARAMETERS
Considering a magnetic case with applied field along h = [1 0 0] t , an identification method of ω(κ) is presented in [50], [51]. This procedure is based on the homogenization of reversible field, where it is defined an auxiliary function F(H ) (see Appendix D). The second derivative of F(H ) is the probability distribution ω(κ). The identification of F(H ) (as explained in Appendix D) can be performed from a set of measured H c under increasing peak magnetic fields H peak . These experimental measurements are presented in Fig. 6 for the stress-free case. This curve is extrapolated outside the VOLUME 10, 2022 FIGURE 6. Experimental measurements of coercive field under increasing magnetic field for the stress-free case.

FIGURE 7.
Identified auxiliary function for the stress-free case, first and second derivatives that represent the pinning field cumulative distribution and probability distribution, respectively. measured range using (28) [51]: where H min c is the lower measured coercive field on the corresponding peak magnetic field H min . The identified F(H ) and its derivatives ∂ H F(H ) and ∂ 2 H F(H ), are presented in Fig. 7. The derivatives are evaluated with a finite differences method. The non-zero component for κ(0) represents the bending of Bloch walls [39]. The continuous probability distribution is then discretized into 25 cells (see (45) in Appendix D).  An applied compression increases the coercive field, as observed in the measured hysteresis curves of Fig. 8 (top). The pinning parameter κ is directly related to the coercive field. We propose to model the stress dependence of dissipation parameters as follows: starting from the identified discrete pinning field distribution for 0 MPa, the weight ω is kept constant under stress. The pinning field κ(σ ) evolves as: with κ(0) the identified pinning field for 0 MPa, and a(σ ) a function that is fitted in order to match with the measured  H c (σ )/H c (0). This coercive field characteristic under stress is presented in Figure 9 in the case of uniaxial stress applied parallel to the magnetic field direction. It can be noted an exponential behavior of H c under compression and a close to constant behavior under tension. For other materials, such as Fe-Si [52], the exponential tendency of the coercive field under compression is also observed.
A phenomenological description of a(σ ) is then defined by: with σ eq the equivalent stress (12). The parameters a 1 , a 2 and a 3 are fitted from four measured coercive fields under 0 MPa, -20 MPa, -40 MPa and -100 MPa, respectively, by using the Curve Fitting Toolbox of Matlab. The identified parameters are presented in Table 4. Fig. 9 shows that (30) is appropriate to represent the measured coercive field characteristic under uniaxial mechanical loading.
The identification procedure of the dissipation parameters can be summarized as follows: from the stress-free curve of coercive field with increasing magnetic field, the method presented in [50], [51] allows identifying ω(κ(0)). By using standard measurements of coercive field under stress, the function a(σ ) is fitted, and so the dependence κ(σ ) of (29) is defined.

V. VALIDATION
The proposed magneto-elastic model results in the hysteresis curves presented in Fig. 8 (bottom), and the tendency of slant under compression -as observed in measurements of Fig. 8 (top) -is captured by the simulation. However, the model does not reproduce inflections in the hysteresis curve -more evident under -100 MPa. This measured behavior is attributed to the crystallographic texture, whereas in this proposed model, only an equivalent single crystal representing the macroscopic behavior is considered. A simplified texture multiscale model (STMSM) [22] may overcome this limitation, but it is not treated in this work. Fig. 10 (top) presents a comparison of the modeled coercive field with the measured symmetric minor loops under uniaxial stress. It is insisted here that the validation is performed by comparison to the experiments that have not been used for identification purposes. For the sake of clarity, the measured values used for identification are explicitly labeled in Fig. 10. Differences are observed in the major loop under tensile stress (25% for 20 MPa and 5050 A/m) but the general   behavior under stress is captured by the simulation. The second-order term in the magneto-elastic energy definition (11) allows the representation of the non-monotonic effect of tensile stress on the remanent induction B r as seen in Fig. 10 (bottom).
The measured hysteresis losses under uniaxial stress as a function of the maximum induction are shown in Fig. 11, where the usual behavior -an increase of losses with increasing compression [1] -is observed.
The prediction of hysteresis losses under stress is plotted in Fig. 12 by numerical integration of the surface of each hysteresis loop and is presented as a function of the maximum induction level. This calculation is a blind validation of the modeling approach since no loss measurement was used for material parameter identification. The modeling results show that the tendency to increase losses under compression is reproduced. Significant differences are seen mainly in the major loop under high compression. As already discussed, the SMSM does not consider the inflections in hysteresis curves under compression, which explains the difference of about 30% for the worst case (-100 MPa and 1.7 T). Fig. 13 shows a magnetic field waveform that allows producing a material response with asymmetric minor loops presented in Fig. 14. The comparison of measurements and model is presented in Fig. 14 for two levels of uniaxial stress. Under a tensile stress of 80 MPa, because the hysteresis curve is less slanted, only one asymmetric minor loop is clearly visible, with the others remaining in a region above 1000 A/m. Again, this comparison is independent of the identification process, so it can serve as a validation for the model. A good agreement between the model and experiment is observed, despite the harmonic content of the H waveform.

VI. CONCLUSION
In this paper, an extension of the energy-based vector-play magnetic hysteresis model has been proposed for the first time in order to incorporate the effect of stress on magnetization. This extension essentially consists of the association of the vector-play model with an anhysteretic sim-plified multiscale approach. This combination results in a magneto-elastic vector model applicable to multi-axial stress configurations. Stress-dependent dissipation parameters can be identified from a few measurements, mostly under compression. An accurate prediction of coercive field and remanent induction under stress were observed when compared to experimental measurements performed on electrical steel. The inclusion of a second-order development in the magneto-elastic energy enables capturing the non-monotonic evolution of the magnetic permeability under stress.
The magneto-elastic model can predict the general behavior of hysteresis losses under mechanical loadings from a small set of parameters and reasonably reproduce asymmetric minor loops. This strategy can be an interesting approach in the coupled magneto-elastic finite element simulation of electromagnetic devices subject to multi-axial stress and fed from a PWM (pulse width modulation) power converter. The proposed model is fully multi-axial. However, it has been validated so far only for uniaxial stress conditions. Validation under complex configurations, such as rotating fields and/or biaxial mechanical loadings, is required. Moreover, only static mechanical loadings have been considered. The extension to dynamic mechanical loading is necessary to cover the piezomagnetic behavior case. Such investigations will be considered in future works.

APPENDIX A IDENTIFICATION OF THE MAGNETOSTRICTION CONSTANT λ s
The identification of the second-order magnetostriction constant λ s is obtained from the analysis of the analytical expression of the anhysteretic relative magnetic permeability µ anh r (σ ). Considering µ anh r (σ ) for isotropic materials, and the magnetic field in the direction of the uniaxial stress. The starting point is the expression of the magnetization given bellow (see (67) from [42]).
where B(σ ) is: The quantity −λ s /2λ s (homogeneous to a stress) is denoted by σ m . It can be noticed that, for any stress σ , one has B(σ m + σ ) = B(σ m − σ ). This shows that independently of the magnetic field, the magnetization as a function of stress is always symmetric with respect to σ = σ m . Such symmetry is naturally inherited by the relative permeability. Furthermore to prove that µ r is maximal at σ m , we first carry out the where erfi is the imaginary error function given as: Upon taking the limit of ∂M /∂H at H → 0 one gets: .
This gives an analytical expression of the relative magnetic permeability, in the case of isotropic materials when the uniaxial loading is applied parallel to the magnetic field. By studying the function µ anh r (σ ), one can show that: (a) it is maximal at σ m = −λ s /2λ s , (b) it has σ = σ m as an axis of symmetry, (c) it has µ anh r = 1 as a horizontal asymptote and (d) equals 1 + χ 0 for σ = 0 (using a second-order Taylor series expansion). These characteristics explain the bell shaped curve observed in Fig. 4 and allow an easy identification of λ s from the values of λ s and σ m .

APPENDIX B DEFINITION OF THE QUASI-STATIC REGIME
A characterization without stress indicates that a frequency of 1 Hz does not allow a quasi-static assumption for this material sample, as seen in Fig. 15, where a significant change is observed in the coercive field when comparing measurements at 1 Hz and 25 mHz. The remanent induction (Fig. 16) is less sensitive to changes in frequency for 0 MPa. The hysteresis measurements under uniaxial stress are performed considering that the frequency of 25 mHz allows reaching the quasi-static regime. Such a value cannot be considered general since it is dependent on the prescribed waveform for the current, but it was empirically determined as relevant for the measurements shown in this paper.

APPENDIX C CORRECTION OF DRIFT IN MAGNETIC INDUCTION
The integration DC drift -or a cumulative offset -in voltage measurements can be related with thermal variation of electronic components [53]. This becomes more problematic with the choice of the frequency of 25 mHz for the input waveform. The drift in the measured induction B mes is linearly corrected with (36), considering the difference between two peaks: in Fig. 17 they are taken as B (1) max at t = 0 s and B (2) max at t = 40 s, with time difference denoted by t.