# Feasible piezomagnetic coefficients of Fe65Co35 alloys with Cr addition

Iris Braga Silva, Fernando Froes, Olivier Hubert, Cristina Bormio-Nunes

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Iris Braga Silva, Fernando Froes, Olivier Hubert, Cristina Bormio-Nunes. Feasible piezomagnetic coefficients of Fe65Co35 alloys with Cr addition. Smart Materials and Structures, 2026, 10.1088/1361-665X/ae485c hal-05530772

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Submitted on 27 Feb 2026

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# Feasible piezomagnetic coefficients of Fe₆₅Co₃₅ alloys with Cr addition.

Iris Braga Silva, Fernando Froes, Olivier Hubert, Cristina Bormio-Nunes.

[1] Escola de Engenharia de Lorena – USP, DEMAR, Estrada Municipal Chiquito de Aquino, N° 1000, Mondesir, 12612-550 Lorena/SP

[2] ENS - Paris-Saclay, LMPS, Avenue des Sciences 4, 91190 Gif-sur-Yvette/Île-de-France

## Abstract

The effect of Cr addition to Fe₆₅Co₃₅ alloy on magnetostriction, mechanical properties and magnetomechanical properties were accessed. Based on the high saturation induction and permeability of FeCo alloys, this study analyzed the potential of Fe₆₅Co₃₅ alloys with Cr as a cost-effective alternative for piezomagnetic applications. Chromium was added to this alloy at concentrations of 0.5 wt.% and 0.7 wt.% (Fe₆₅Co₃₅-0.5Cr and Fe₆₅Co₃₅-0.7Cr). The samples were arc-melted, hot rolled and annealed. A comprehensive characterization was performed of the microstructure, magnetic, mechanical, and magnetomechanical properties. Despite the small Cr content difference of only 0.2 wt.%, a significant microstructural variation was observed, with abnormal grain growth occurring (AGG) in the alloy containing 0.5 wt.% Cr. The Fe₆₅Co₃₅-0.7Cr alloy exhibited the best soft magnetic properties and the highest magnetostriction of 53 × 10⁻⁶. However, the highest piezomagnetic sensitivity reached was 6.2 kA·m⁻¹·MPa⁻¹ for the Fe₆₅Co₃₅-0.5Cr alloy. The source of differences of piezomagnetic behaviors was found to be associated to different internal stress levels

Keywords: Fe-Co, magnetostriction, magnetomechanical behavior, magnetic permeability.
# 1 Introduction

Devices that utilize magnetomechanical behavior for their operation require materials with strong magnetoelastic coupling which are related to high magnetostriction values. Therefore, these materials are expected to have a high rate of magnetization M and magnetic induction B) change with respect to stress σ at a constant magnetic applied field H, dB/dσ|<sub>H</sub>, or conversely a high magnetostriction λ change with respect to H, dλ/dH|<sub>σ</sub> at a constant stress σ. These properties are the piezomagnetic coefficients d<sub>33</sub>* = 1/μ<sub>0</sub> dB/dσ|<sub>H</sub> and d<sub>33</sub> = dλ/dH|<sub>σ</sub>. These parameters differ out of equilibrium (i.e. when hysteresis is considered), but in thermodynamic equilibrium d<sub>33</sub>* = d<sub>33</sub>. Their usual units are nm.A<sup>-1</sup> for d<sub>33</sub> and (kA.m<sup>-1</sup>).MPa<sup>-1</sup> for d<sub>33</sub>*. The sensitivity coefficient d<sub>33</sub>* is the adequate parameter in force sensors applications while the sensitivity d<sub>33</sub> is used to develop actuators. The minimum acceptable d<sub>33</sub> value is 1 nm.A<sup>-1</sup> as stated by [1] or equivalently d<sub>33</sub>* = 0.83 kA.m<sup>-1</sup>.MPa<sup>-1</sup> [2]. For instance, in the construction of piezomagnetic force sensors those materials should also support heavy loads, be adaptive with harsh environment, have low response time, energy consumption and heat generation [3].

In general, a well-known rare-earth based material, TERFENOL-D (Tb<sub>0.3</sub>Dy<sub>0.7</sub>Fe<sub>2</sub>) is used in such sensors and actuators due to: the significant magnetostrictive strain ranging from 800 ×10<sup>-6</sup> to 1600 ×10<sup>-6</sup>, a rapid reaction time, a high energy density, and a considerable output force reaching a sensibility of approximately 10 kA·m<sup>-1</sup>.MPa<sup>-1</sup> (~ 12 nA/m) [4][5][6]. However, the TERFENOL-D disadvantages are the high cost due to scarce reserves of rare earth elements, the brittleness and maximum tolerated compressions of only 55 MPa [7]. GALFENOL (Fe<sub>81</sub>Ga<sub>19</sub>) is another material exhibiting high magnetomechanical coupling, that can achieve a piezomagnetic constant of 28 mT.MPa<sup>-1</sup> (22 kA·m<sup>-1</sup>.MPa<sup>-1</sup>). Compared to TERFENOL-D, GALFENOL possesses a
higher experienced compression of 80 MPa. Both materials need to be texturized to take advantage of the favorable magnetic anisotropy direction and prestress to demonstrate their best properties. The texturing process adds high production costs to this material [8][9][10]. This material also suffers from low ductility.

The 2V-Permendur (HIPERCO 50 A), a Fe-Co alloy with composition of 48.5% of Co and 2% of V is the third most suitable commercial material with high magnetomechanical coupling [11] [12].

Table 1 presents a comparison between TERFENOL-D, GALFENOL, and the 2V-Permendur alloy that allows for a comparison of the main magneto-mechanical properties and price. It shows the maximum piezomagnetic sensitivity value and the magnetic field at which this maximum occurs, the manufacturing process, and the force application scenarios where they are used.

<table><caption>Table 1 – Commercial materials Terfenol-D, Galfenol, and the 2V-Permendur alloy showing price, the maximum sensitivity value and the magnetic field at which this maximum occurs, the manufacturing process, and the force application range. [12]</caption><thead><tr><th>Material</th><th>d<sub>33</sub><sup>*</sup>(max)<br>(kA.m<sup>-1</sup>.MPa<sup>-1</sup>)</th><th>H<sub>max</sub><br>(kA.m<sup>-1</sup>)</th><th>Material<br>processing</th><th>Usage in</th><th>Price<br>(US$/kg)</th></tr></thead><tbody><tr><td>2V-Permendur<br>(FeCo<sub>48.5</sub>-V<sub>2</sub>)</td><td>10</td><td>2</td><td>Rolled/Annealed</td><td>Compression<br>and traction</td><td>100 –<br>155 [13]</td></tr><tr><td>Terfenol-D<br>(Tb<sub>0.3</sub>Dy<sub>0.7</sub>Fe<sub>2</sub>)</td><td>10</td><td>20</td><td>DS*</td><td>Compression</td><td>10000 –<br>30000<br>[14]</td></tr><tr><td>Galfenol<br>(Fe<sub>81</sub>Ga<sub>19</sub>)</td><td>22.5</td><td>7</td><td>DS*/Pre-stress<br>annealing</td><td>Compression</td><td>9000 –<br>11000<br>[14]</td></tr></tbody></table>

<footnote>*DS = Directional Solidification</footnote>
Remarkably, the FeCo-2V alloy, besides being less expensive, can be used under both tension and compression, unlike the other materials, which are used only as compression sensors. Additionally, it is crucial to note that the magnetic bias field required to achieve optimal magnetomechanical performance in these materials is particularly low for 2V-Permendur.

Another interesting composition of Fe-Co alloys is Fe₃₀Co₇₀ because it has high magnetostriction value, of about 137 × 10⁻⁶ [15], and this material obtained by directional solidification resulted in (100) textured alloy with magnetostriction as large as 180 × 10⁻⁶ [16]. More recently, the additive manufactured Fe₃₀Co₇₀ alloy had its magnetostriction, crystal structure and piezomagnetic sensitivity investigated in [17]. However, the best piezomagnetic coefficient d₃₃ obtained was of only 400 pm/A = 0.4 nm/A, below the acceptable value for applications. In addition, the high Co alloy concentration is also a matter to be considered.

Fe-Co alloys magnetomechanical behavior studies with compositions of % Co < 50 were not found in the literature. The Fe₆₅Co₃₅ alloy, for example, is interesting since it has no negligible magnetostriction ~ 67 × 10⁻⁶ and has a high magnetization value of 245 Am²·kg⁻¹, the highest among Fe alloys [15]. For instance, this alloy has the prospect of an even lower cost compared to 2V-Permendur alloy, due to the lower content of Co, and consequently less expensive than TERFENOL-D and GALFENOL as well.

However, the BCC lattice ordering is known to occur in Fe-Co alloys from 29 to 70 at % Co while brittleness is reported between 25 and 65 at. % Co [18]. The temperature at which ordering occurs is dependent on cobalt content, with the ordering temperature for Fe₆₅Co₃₅ being approximately 690 °C [19]. To mitigate this brittleness Stanley and Yensen [20] conducted extensive studies on various alloying additions to binary Fe-Co alloys.
They concluded that only V and Cr were effective in improving workability. In their study [20], the addition of 0.4–0.5 wt.% Cr to Fe₆₅Co₃₅ was evaluated. However, this was not the only element introduced into the alloy; 0.1 wt.% carbon was also added [21]. Furthermore, during the melting process, slag was removed from the liquid metal, and approximately 0.1 wt.% titanium and 0.1 wt.% silicon was added as deoxidizers. Consequently, isolating the specific effect of Cr on the microstructure and mechanical properties becomes rather challenging.

In the present study, polycrystalline Fe₆₅Co₃₅ alloys (wt. %) with additions of 0.5% and 0.7% Cr were produced and characterized with respect to their microstructures and magnetic behavior, and also to access their magnetomechanical behavior.

## 2 Experimental methods

Three ingots of 40 g of each composition were arc-melted in argon atmosphere five times to ensure chemical homogeneity using Fe (99.8%), Co (99.9%) and Cr (99.99%). After melting, the alloys were encapsulated in quartz tubes with inert argon atmosphere and were carried out for homogenization annealing process at 1200 °C for 6 h and quenched in water. From now on the alloys will be denominated as Fe₆₅Co₃₅-0.5Cr and Fe₆₅Co₃₅-0.7Cr.

After the homogenization annealing process, the effect of Cr content on hardness was evaluated. One sample of each composition was subjected to cold rolling, and both samples reached a thickness reduction of approximately 63% without fracture. After each rolling pass, the plates were prepared metallographically for hardness testing. Hardness measurements were performed using a BUEHLER MICROMET 2004 microhardness tester. For each sample, 20 measurements were taken under a load of 100 mN applied for
15 s. The indentations were placed in the central region of the samples to minimize edge effects.

Subsequently, for the rolling process, the ingots were encapsulated in stainless steel tubes. Hot rolling was performed in a FENN 4, model 105 rolling mill, at a temperature of 1100 °C, the alloys reached a thickness reduction of approximately 80%. Afterwards the sheets were cut into rectangular shapes (plates) using the ISOMET 1000 precision cutter. Finally, the samples underwent an annealing process at 800 °C for 6h followed by quenching in water.

Sections of the plates parallel, transverse and normal to the rolling direction were cut from the plates to metallographic preparation for microscopy and chemical analysis for each composition.

<figure><img src="image_2.png" /><figcaption>Figure 1 – Schematic drawn of a rolled sheet showing the rolling (RD), normal (ND) and transversal (TD) directions.</figcaption></figure>

Optical microscopy was used to analyze the three sections of each composition. For SEM analysis, only the RD-TD plane (normal direction) was analyzed, using a HITACHI 4000 microscope coupled with an EDAX EBSD detector and an Oxford EDS detector for the composition analysis.
X-ray diffraction (XRD) analysis was performed on powder samples of the Fe₆₅Co₃₅-Cr alloys using an Empyrean diffractometer from PANalytical with Mo-Kα radiation, in an angular range of 10° to 65° and a step size of 0.01°. The powders were produced by filing and sieving to sizes less than 60 μm. The data were analyzed with PowderCell through the Rietveld method using the standard diffraction pattern for the disordered BCC phase to determine the lattice parameter $a_0$ [22].

The hysteresis and magnetostriction measurements without application of force were performed using two different SST-type devices [11,23], that warranted a closed magnetic circuit and consequently no demagnetization effect. For hysteresis, 10 hysteresis cycles were measured simultaneously for each sample using a 56-turn primary coil to produce the magnetic field between -11 kA/m and 11 kA/m in the RD direction of the sheet. A secondary coil with 30 turns was wound at the center and around the sheet to capture the magnetic induction.

To acquire the magnetostriction curves, three strain gauges were employed for each sample: two for measuring longitudinal magnetostriction ($\lambda_{long}$), to analyze if there's texture in the material, and one for transverse magnetostriction ($\lambda_{trans}$). The measurements were performed using an SST-type device in which the primary coil can deliver a magnetic field in the range of -45 kA/m to 45 kA/m [23]. The magnetostrictive deformation detected by the strain gauges was recorded using a highly sensitive National Instruments Wheatstone bridge, with a resolution of approximately $1 \times 10^{-6}$.

To determine the overall mechanical behavior of both alloys, two mechanical tests were conducted using a hydraulic MTS uniaxial machine: one leading the material to complete fracture and the other within the elastic region. For the elasticity test, a tensile force of 2000 N (~100 MPa) was applied to the Fe₆₅Co₃₅-0.5Cr alloy and 1500 N (~74 MPa) to
the Fe₆₅Co₃₅-0.7Cr alloy. Small extensometers were placed on both faces of the sample, and two loading and unloading cycles were performed. Using the data obtained from the elasticity test, the Young's modulus of each sample was determined.

To analyze the piezomagnetic behavior of the samples, magnetization measurements were conducted as a function of the applied magnetic field for constant values of tensile and compressive stresses applied to the material. The measurements were conducted to the samples, without applied stress (0 MPa) and under tensile stresses of 50 MPa and 100 MPa, as well as compressive stresses of -50 MPa and -100 MPa. To measure the magnetization, an external cylindrical coil was placed surrounding the specimen for applying the magnetic field and a secondary coil was wound around the material. The magnetization was measured in an open magnetic circuit, without magnetic cores to close the magnetic circuit due to the spatial limitations of the machine and reduced length of the samples, resulting then in the presence of a demagnetizing field. The samples dimensions were 2.48 × 8.10 ×79.9 mm for the Fe₆₅Co₃₅-0.5Cr alloy and 2.5 × 8.14 ×73.5 mm for the Fe₆₅Co₃₅-0.7Cr.

# 3 RESULTS AND DISCUSSION

## 3.1 X-ray Diffraction

Figure 2 shows the X-ray diffraction (XRD) results alongside the standard BCC disordered lattice pattern. All peaks, except one at 2θ ≈ 18°, belong to the α – phase. The small impurity peak at 18° is likely due to Fe oxides formed during the grinding process of alloy powder production [24].
The calculated lattice parameters of the alloys with Cr are $a_0^{0.5Cr} = 2.8612$ Å and $a_0^{0.7Cr} = 2.8623$ Å and they increased compared to the lattice parameter of the alloy without Cr, a Fe<sub>65</sub>Co<sub>35</sub> alloy produced by the same process has $a_0 = 2.8595$ Å [24]. However, Han [15] published a value of $a_0 = 2.8658$ Å that is larger than the ones of the Cr added samples.

These differences are probably due to the degree of ordering of the alloys, since it is very sensitive to annealing temperature and cooling rate [20,25]. However, the ordering is difficult to detect by XRD because the superlattice reflections, which are unique to the ordered structure, can be very weak and easily masked by the more intense fundamental reflections of the disordered BCC structure. The very low intensity of superlattice reflections is due to small differences in scattering factors between Co and Fe, because their atomic numbers are close (26 and 27) and scattering factors are primarily determined by the number of electrons.

<figure><img src="image_3.png" /><figcaption>Figure 2 – XRD pattern of Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr and Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloys powders.</figcaption></figure>
In the XRD analysis, the peak broadening is influenced by the crystallite size (D) and the microstrain (ε<sub>XRD</sub>) of the lattice, which arises from point defects, grain boundaries, and stacking faults. The Williamson-Hall (W-H) method was employed to estimate both the crystallite size and the lattice strain [26] The calculated values of D and ε<sub>XRD</sub> are respectively 26 nm and 4.1 × 10<sup>-3</sup> for the Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr alloy, and 32 nm and 4.5 × 10<sup>-3</sup> for the Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy. Both parameters were higher for the alloy with the highest Cr content, meaning that the 0.7% Cr alloy is subjected to a higher residual stress level. These differences may be attributed to the BCC ordering [20][25][27], in which the main defects generated between BCC ordered and disordered phases are antiphase boundaries (planar defects) [28].

## 3.2 Optical microscopy

Figure 3 presents the optical micrographs of the Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr sheet surfaces, in which are shown the RD-TD, ND-TD and ND-RD surfaces. The images reveal a microstructure characterized by the presence of both small normal grains and very large grains which may signify abnormal grain growth (AGG). There are specific characteristics of AGG that are peninsular and island grain types as indicated in the micrographs by the numbers 1 and 2 respectively [29]. The AGG found in the Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr alloy contrasts with the findings in Fe-Co alloys (30 % Co) with Ti and Nb additions, as reported by Froes and Nunes (2023) [30], in which AGG was not observed. AGG occurrence is likely to have origin due to anisotropic grain boundary energy [31,32]. Therefore, somehow the AGG observed on the Fe<sub>65</sub>Co<sub>35</sub>-0.5 Cr alloy can be attributed to Cr causing some anisotropy on the grain boundaries.
<figure><img src="image_4.png" /><figcaption>Figure 3 – Optical micrographs of the Fe<sub>65</sub>Co<sub>35</sub> alloy with addition of 0.5% Cr of the surfaces: RD-TD (a), ND-TD (b) and (c) ND-RD.</figcaption></figure>

Similarly, Figure 4 displays micrographs of the Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy that do not show characteristics of AGG. A more equiaxed grain distribution is present compared to Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr alloy and the sizes of the grains are smaller than the giant grains noticed in Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr alloy. Based on the comparison of the grain structure between the two alloys, it is notable that an increase of only 0.2% of Cr has caused such a difference in the microstructure. This result points out to a more isotropic grain boundary energy in the Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy.
<figure><img src="image_5.png" /><figcaption>Figure 4 – Optical micrographs of the Fe<sub>65</sub>Co<sub>35</sub> alloy with addition of 0.7% Cr to the surfaces: RD-TD (a), ND-TD (b) and (c) ND-RD.</figcaption></figure>

## 3.3 Composition by EDS

The EDS elemental distribution maps (Figure S1, Supplementary File) show that Fe and Co, are mostly uniformly distributed in the Fe<sub>65</sub>Co<sub>35</sub>–0.5Cr and Fe<sub>65</sub>Co<sub>35</sub>–0.7Cr alloys, except for localized Cr enrichment highlighted by bright green points (Figures S2, Supplementary File). Similar behavior was reported in [33], where Cr-rich precipitates formed in Fe–49Co–2V–0.5Cr alloys at 760 °C, increasing in fraction with higher annealing temperatures, and contributing to grain size reduction. These results suggest that adding 0.5–0.7 at. % Cr brings the Fe–Co alloys close to the solubility limit. The
traces detected along grain boundaries could be nanoprecipitate precursors of a rich Cr
phase.

The EDS results of alloy's elements composition are very close to the nominal ones. The associated errors are approximately 6% for Cr and 0.2% for the remaining elements. Table S1 containing these results are in *Supplementary File*.

**3.4 Scanning electron microscopy - EBSD**

The complete texture EBSD results of the RD-TD surfaces are shown in *Supplementary File* in Figures S3, S4 and S5 for Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr and S5 for Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy.

For analyzing magnetic measurements, the relevant texture is the one parallel to the
rolling direction RD, the direction of the applied magnetic field. Therefore, in Figures 5
and 6 are depicted correspondingly the results of Inverse Pole Figure (IPF) of the
Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr and Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloys only in the RD direction. The intensity of the
texture is given in multiples of random distribution (mrd), in which a value of 1 m.r.d.
means the orientation is randomly distributed, values in the ranges of 2 – 3 mrd means
weak texture, 4-6 mrd means moderate texture and for > 7 mrd strong textures.
<figure><img src="image_6.png" /><figcaption>Figure 5 - Fe₆₅Co₃₅-0.5Cr alloy texture analysis results of an area of 1.8 × 1.8 mm in the ND direction. In (a) IPF of the total area; in (b) and (c) the analysis of the small and large grains, respectively, as marked in (a) by the rectangle and square.</figcaption></figure>

Figure 5 (a) shows Fe₆₅Co₃₅-0.5Cr alloy texture analysis of an area of 1.8 × 1.8 mm. While Figures 5 (b and c) are the separate analysis of the rectangular and square regions demarked in the IPF of Figure 5 (a), and those are the large grains and small grains regions, respectively. The analysis of the total area of Figure 5 (a) gives a more intense
pole in the region between <100> and <101> directions with an intensity of 2.442.

However, the region of small grain has barely texture with a maximum intensity of only 1.294 in three different directions. On the other hand, the region of big grains analyzed has obviously a stronger texture. The texture of the big grains dominates the overall texture of the total area analyzed as can be constated by the similarity of the IPF triangles of Figure 5 (a) and (c).

Figure 6 depicts the Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy texture analysis of a RD-TD surface area of 1.0 × 1.0 mm. A stronger pole is observed in the <111> directions, however the intensity is low, with a maximum of only 2.948. A lower intensity pole is observed between<100> and <101> directions reaching 1.434.

<figure><img src="image_7.png" /></figure>

Figure 6 - Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy texture analysis results of an area of 1.0 × 1.0 mm in the ND direction.
The presence of another phase was not identified by EBSD as it was not by XRD. However, as already mentioned the particles may be nanometric and in very low concentration, then they were not detectable via XRD and EBSD. The presence of this second phase, as well as the different quantities in which it is present, could explain the distinctions observed in the microstructure of the two alloys analyzed. The Cr at grain boundaries identified by EDS would act as grain boundaries pinners. A lower Cr distribution at the grain boundaries in the 0.5% Cr alloy would permit abnormal grain growth (AGG), whereas the pinning particles were effective in preventing it.

The mean grain sizes are 560 µm and 216 µm for 0.5% Cr and 0.7% Cr alloys, respectively.

Another result from EBSD analysis that can help to understand magnetic properties, in particular, the coercive field $H_c$ result is the length of grain boundaries. For Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr alloy EBSD the length of grains boundaries is 11.23 cm and for the Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy 1.51 cm. However, the analyzed area of the 0.5% Cr alloy was larger than the 0.7% Cr alloy. A linear extrapolation of 0.7% Cr alloy grain boundaries length proportional to the area gives a length of 4.89 cm.

## 3.5 Magnetic Properties

Figure 7 (a) presents the magnetic induction hysteresis curves of the Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr and Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloys, while Figure 7 (b) shows an enlarged view of the origin of (a), highlighting the coercive field ($H_c$) and the remanent induction ($B_r$). Figure 7 (c) displays the magnetic permeability as a function of the applied magnetic field, obtained by differentiating the initial magnetization curve from the hysteresis loops. None of the
alloys reached magnetic saturation, since the maximum applied field in the experimental setup was limited to 11 kA/m.

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>
Figure 7 – Fe₆₅Co₃₅ alloys with 0.5% and 0.7% Cr: Magnetic induction hysteresis curves in (a); Magnification of the hysteresis curves near the origin in (b) and in (c) Relative magnetic permeability curves.

Figure 8 presents the longitudinal and transverse magnetostriction curves obtained for the Fe₆₅Co₃₅ alloys with addition of 0.5% and 0.7% Cr.

<figure><img src="image_11.png" /></figure>

Figure 8 - Fe₆₅Co₃₅ alloys with 0.5% and 0.7% Cr longitudinal and transverse magnetostriction curves.

The longitudinal and transverse values of the saturated magnetostrictions of each alloy taken from Figure 8 are $\lambda_{long} = 45.3 \times 10^{-6}$ and $\lambda_{trans} = -20.9 \times 10^{-6}$ for Fe₆₅Co₃₅-0.5Cr alloy and $\lambda_{long} = 48.0 \times 10^{-6}$ and $\lambda_{trans} = -31.2 \times 10^{-6}$ for Fe₆₅Co₃₅-0.7Cr alloy.

## 3.6 Mechanical behavior
Figure 9 shows the hardness results of Fe₆₅Co₃₅-0.5Cr and Fe₆₅Co₃₅-0.7Cr alloys as a function of true strain ε = ln(eᵢ/ef), in which eᵢ is the initial thickness of the ingot, without deformation, and ef is the final thickness for each cold rolling process step. The Fe₆₅Co₃₅-0.7Cr alloy initial hardness is 231 HV compared to 217 HV for Fe₆₅Co₃₅-0.5Cr alloy after ~63% of thickness reduction the hardness of both alloys reach 296 HV and curves show a saturation tendency. The initial hardness was taken in the as cast alloy ingots annealed at 1200°C and water quenched.

<figure><img src="image_12.png" /><figcaption>Figure 9 – Hardening curves as a function of the true strain ε for Fe₆₅Co₃₅-0.5Cr and Fe₆₅Co₃₅-0.7Cr alloys due to cold rolling.</figcaption></figure>

The stress-strain curves are shown in Figure 10. From this curve the Young's Modulus E values obtained are (222 ± 0.4) GPa and (214 ± 0.1) GPa respectively to the alloy with 0.5 % of Cr and 0.7% of Cr, and the errors are related to the linear fittings. The difference between E values of both alloys is only 3.6%, thus they have almost the same elastic behavior with a maximal deformation of 0.22% and 0.55%, both alloys can be considered as quasi-brittle.
The Fe₆₅Co₃₅-0.5% Cr alloy exhibits a slightly longer range of strain hardening Fe₆₅Co₃₅-0.7Cr. Brittleness indicates that ordering is present in both materials despite the quenching after the final annealing. In [19] a value of 211 GPa is presented for the Hiperco alloy Young's Modulus very close to the obtained here for the addition of purely Cr. However, they report an elongation of 6-8 %, which is more than 10 times higher than the obtained only with the addition of Cr. In this sense, the other elements as C, Ti and Si may improve significantly the ductility and Cr by itself is not effective to improve ductility. Its presence does not disturb the ordering process.

<figure><img src="image_13.png" /><figcaption>Figure 10 – Stress (MPa) vs deformation (%) tension curves of Fe₆₅Co₃₅ alloys with 0.5% and 0.7% Cr.</figcaption></figure>

Table 4 summarizes the magnetic and mechanical properties obtained from Figures 7-10 and the microstructure analysis. B<sub>max</sub>, B<sub>r</sub>, H<sub>c</sub>, μ<sub>i</sub>, μ<sub>max</sub>, and λ<sub>s</sub> are listed, with λ<sub>s</sub> calculated from λ<sub>total</sub> = 3/2 λ<sub>s</sub> = (λ<sub>long</sub> − λ<sub>trans</sub>). For comparison, data of the binary alloy from [24], as well as texture, hardness, and stiffness values, are also included.
<table><caption>Table 4 – Fe<sub>65</sub>Co<sub>35</sub> alloys with 0.5% and 0.7% Cr magnetic, mechanical and microstructure properties</caption><thead><tr><th>Property</th><th>Fe<sub>65</sub>Co<sub>35</sub></th><th>Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr</th><th>Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr</th></tr></thead><tbody><tr><td><i>B</i><sub>max</sub> (T)</td><td>--</td><td>2.27</td><td>2.35</td></tr><tr><td><i>B</i><sub>r</sub> (T)</td><td>--</td><td>0.583</td><td>0.670</td></tr><tr><td><i>H</i><sub>c</sub> (A/m)</td><td>970*</td><td>394</td><td>294</td></tr><tr><td>μ<sub>i</sub></td><td>767*</td><td>370</td><td>440</td></tr><tr><td>μ<sub>max</sub></td><td>767*</td><td>1220<br>(@290 A/m)</td><td>1650<br>(@ 220 A/m)</td></tr><tr><td>λ<sub>s</sub></td><td>15.4 × 10<sup>-6</sup>*</td><td>44.1 × 10<sup>-6</sup></td><td>52.8 × 10<sup>-6</sup></td></tr><tr><td>Mean grain size (μm)</td><td>329*</td><td>560</td><td>216</td></tr><tr><td>Texture intensity (RD)</td><td>1.606*</td><td>2.442</td><td>2.942</td></tr><tr><td>Texture pole</td><td>Between<br>&lt;001&gt; and &lt;111&gt;*<br>Between<br>&lt;001&gt; and &lt;101&gt;</td><td>&lt;111&gt;</td><td>&lt;111&gt;</td></tr><tr><td>Hardness<br>(HV)</td><td>--</td><td>217</td><td>231</td></tr><tr><td>Stiffness<br>(GPa)</td><td>--</td><td>222</td><td>214</td></tr></tbody></table>

Results presented in Table 4 show that the alloy with 0.7% of Cr presents better soft magnetic properties than the alloy with 0.5% of Cr, i.e., a higher value of initial and maximum relative magnetic permeability, as well as lower coercivity. On the other hand, the magnetostriction of the alloy with 0.5%Cr is lower than that of 0.7% Cr.

Texture analysis of the alloys' samples indicates that both the 0.5% and 0.7% Cr-containing alloys exhibit low-intensity textures.

Although the texture intensity of 0.7%Cr alloy is more intense than of 0.5%Cr alloy, the texture of the 0.5% Cr alloy is more favorable, as its preferred crystallographic orientation along the rolling direction (RD) lies between easy and intermediate magnetization axes. In contrast, the 0.7% Cr alloy shows a less favorable texture pole, highlighting a hard

<footnote>*From [24]</footnote>
magnetization axis. The maximum relative permeability μr for the 0.7% Cr alloy occurs
at a lower magnetic field than for the 0.5% Cr alloy and this result cannot be also
attributed only to texture effects.

The decrease of 0.7% of Cr alloy Hc compared to 0.5% Cr one is counterintuitive, since
0.7% Cr alloy has smaller mean grains’ size and therefore supposedly more grain
boundaries, that are effective domain wall pinners. However, the uncommon
microstructure of the 0.5%Cr alloy showed that this alloy has a grain boundary length
about 2 times longer than that of 0.7% Cr alloy due to the presence of regions with very
small grains. This fact may explain the lower 0.7%Cr Hc value, as well as the higher
maximum μr and occurring at lower magnetic field compared to 0.5% Cr alloy.

The fact that $\lambda_s^{0.7 \text{ Cr}} > \lambda_s^{0.5 \text{ Cr}}$ cannot be attributed to texture effects either and these
differences are likely of another origin, that will be clarified with the piezomagnetic
results analysis.

In [20] the authors report μᵢ = 650 and μmax = 10.000 for the alloy with % Co from 34.5%
to 35.5%, and additions of 0.4-0.5% Cr, 0.1% of C, Ti and Si, affirming that these μ values
vary significantly with composition and heat treatment. These values are very much
higher than values obtained for Fe₆₅Co₃₅-0.5Cr. However, from the present work results,
the addition of only Cr seems not to be the origin of the very high permeabilities obtained
in [20]. Likewise, very important for the comparison is the annealing temperature that is
not stated in [20].

In addition, both alloys' compositions display significantly higher $\lambda_s$ than the Fe₆₅Co₃₅
alloy that is 15.4 × 10⁻⁶ [34], meaning 2.9 times higher for the Fe₆₅Co₃₅-0.5Cr alloy and
3.5 times higher for the Fe₆₅Co₃₅-0.7Cr alloy.
## 3.7 Magneto-mechanical behavior

Figure 11 (a-b) shows the B vs H curves for constant σ values, in which Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr data is in left column and Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr data is in right column. These magneto-mechanical behaviors were taken for applied tensile stresses σ = -100 MPa, - 50 MPa, 0, 50 MPa and 100MPa.

Figure 11 (c-d) shows the relative magnetic permeability (μ<sub>0</sub><sup>-1</sup>. dB/dH) calculated from Figure 11 (a-b).

<figure><img src="image_14.png" /></figure>

Figure 11 – Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr (left column) and Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr (right column) alloys initial magnetic induction B vs H for applied stresses σ = -100 MPa, - 50 MPa, 0, 50 MPa and
100MPa (a); in (b) Relative magnetic permeability (μ₀⁻¹. dB/dH) calculated from (a) curves.

Fe₆₅Co₃₅-0.7 alloy has a slightly higher value of B than Fe₆₅Co₃₅-0.5 alloy that is attributed to the highest magnetic permeability of the first, as can be seen in Figure 8 (b,c). These differences do not manifest clearly in Figure 11 (c,d) due to demagnetization effects.

Figure 12 (a-b) depicts the experimental curves for σ = 0 of the B vs H data with and without yoke for both alloys, in order to highlight the demagnetization effect present in the magneto-mechanical measurements, since, as already pointed out, it was not possible to use the yokes to close the magnetic circuit in magneto-mechanical measurements due to the small length of the sheet samples.

As previously mentioned, the sample sheet dimensions were 2.48 (a) × 8.10 (b) ×79.91 (c) mm - Fe₆₅Co₃₅-0.5Cr alloy and 2.5 (a) × 8.14 (b) ×73.56 (c) mm - Fe₆₅Co₃₅-0.7Cr.

The sheet aspect ratios were defined as c/√ab and a/b [35]. For the Fe₆₅Co₃₅-0.5Cr alloy, c/√ab = 17.73 and a/b = 3.266 and, while for Fe₆₅Co₃₅-0.7Cr alloy c/√ab = 16.31 and a/b = 3.256. Although these differences may appear negligible, Figures 12(a, b) show that the demagnetizing effect is significantly more pronounced in the Fe₆₅Co₃₅-0.7 alloy. This is attributed to its slightly smaller aspect ratios combined with its inherently higher magnetic susceptibility [35].
<figure><img src="image_15.png" /><figcaption>Figure 12- Experimental curves of B vs H for $\sigma = 0$, with and without yoke for Fe₆₅Co₃₅ alloys with 0.5% and 0.7% Cr.</figcaption></figure>

Demagnetizing corrections were not applied due to the complexity of calculating $H_d$, which depends on sample geometry, susceptibility, and the measurement method [29,35]. In plate samples, $H_d$ and M are nonuniform. Although permeability in piezomagnetic tests also varies with applied stress [Fig. 11(c,d)], the stress dependence of magnetization is evaluated at fixed fields and is therefore unaffected by the demagnetization factor. Only the absolute field values would require correction, since the effective internal field is smaller than the applied one. It must however be supposed that magnetization variation with stress has a small impact on demagnetizing field to consider the internal field as roughly constant whatever the stress level.

Figure 13 shows the piezomagnetic behavior of Fe₆₅Co₃₅-0.5Cr alloy (left column) and of Fe₆₅Co₃₅-0.7Cr alloy (right column) for magnetic fields in the range of 1 kA/m to 6 kA/m. The points plotted in Figure 13 were derived from Figure 11 (a, b), taking for each alloy at a constant value of the applied magnetic field, the variation of the magnetic
induction with applied stress. Piezomagnetic sensitivity was determined by fitting the B vs σ experimental data to a function based on physical principles.

<figure><img src="image_16.png" /><figcaption>Figure 13 – Piezomagnetic behavior B vs. σ for at fixed values of magnetic field (1 ≤ H ≤ 6 kA/m), for Fe₆₅Co₃₅-0.5Cr alloy (left) and Fe₆₅Co₃₅-0.7Cr alloy (right).</figcaption></figure>

The derivation of a function to fit the experimental B vs. σ data is based on a multiscale approach of the magnetomechanical behavior [36-40]. The average magnetization <M> of the discrete medium is expressed along with the applied direction as a function of the magnetic field H and/or the applied stress σ. According to Boltzmann statistics, <M> can be determined by weighing the magnetization of each domain α with a probability function fα, as defined in equations (1) and (2). The overall average magnetization is then obtained through equation (3).

$$f_{\alpha} = \frac{\exp(-A_s g_{\alpha})}{\sum_{\alpha} \exp(-A_s g_{\alpha})} \quad (1)$$

$$A_s = \frac{3\chi_0}{\mu_0 M_s^2} \quad (2)$$
$$ \langle M \rangle = M = \sum_{\alpha} f_{\alpha} M_{\alpha} \qquad (3) $$

$\chi_0$ is the initial anhysteretic susceptibility, $\mu_0$ the vacuum magnetic permeability and $M_s$ the saturation magnetization of the material. The function $g_\alpha$ and the parameter $M_\alpha$ are the Gibbs free energy density and magnetization at the scale of a domain family $\alpha$, respectively.

The complete mathematical development to obtain $M = \langle M \rangle$ and consequently $B = \mu_0(H + M)$ is shown in Supplementary file 2 and the result is given by equation (4):

$$ B = \mu_0 H + \frac{\mu_0 M_m \exp(A_s \sigma \lambda_m) \sinh(A_s \mu_0 H M_m)}{\exp(A_s \sigma \lambda_m) \cosh(A_s \mu_0 H M_m) + 2 \exp\left(-\frac{A_s \sigma \lambda_m}{2}\right)} \qquad (4) $$

In equation (4) $M_m$ and $\lambda_m$ are the maxima magnetization and magnetostriction strain, respectively. $M_m$ can be reached by the isotropic material without rotation and can be lower than $M_s$. $\lambda_m$ is not the true saturation magnetostriction but can be thought as a parameter representative of the average magnetostriction [40].

Therefore, using equation (4) with $\chi_0$, $M_m$ and $\lambda_m$ as the free parameters, the curves shown in Figure 13 confirm the capability of the analytical function $M$ to accurately reproduce the experimental results. The corresponding fitting parameters are summarized in Tables S2 and S3 of Supplementary File.

The piezomagnetic sensitivity curves shown in Figure 14 are calculated as $d_{33}^* = 1/\mu_0$ dB/d$\sigma$ |$_H$ and the expression for the piezomagnetic sensitivity is given by equation (5).
$$d_{33}^* = \frac{1}{\mu_0} \left. \frac{dB}{d\sigma} \right|_H = \frac{3M_m A_s \lambda_m \exp\left(\frac{A_s \sigma \lambda_m}{2}\right) \sinh(A_s \mu_0 H M_m)}{\left[\exp(A_s \sigma \lambda_m) \cosh(A_s \mu_0 H M_m) + 2 \exp\left(-\frac{A_s \sigma \lambda_m}{2}\right)\right]^2} \quad (5)$$

Observe that the calculation of $d_{33}^*$ was extended to a stress range of -200 MPa to 200 MPa, but the experimental data cover only the range of -100 MPa to 100 MPa that are enclosed between dashed lines in Figure 14.

<figure><img src="image_17.png" /><figcaption>Figure 14 – Piezomagnetic sensitivity $d_{33}^* = 1/\mu_0 dB/d\sigma$ vs. $\sigma$ at fixed values of magnetic field ($1 \le H \le 6$ kA/m): Fe$_{65}$Co$_{35}$-0.5Cr alloy (left) and Fe$_{65}$Co$_{35}$-0.7Cr alloy (right).</figcaption></figure>

The sensitivities $d_{33}^*$ are higher for compressive forces $\sigma \le 0$, as expected, since the 0.5% Cr and 0.7% Cr alloys have positive magnetostriction. In the stress range -100 MPa to 100 MPa the highest $d_{33}^*$ of the Fe$_{65}$Co$_{35}$-0.5Cr and Fe$_{65}$Co$_{35}$-0.7Cr alloys varies from 6.2 to 2.2 kAm$^{-1}$.MPa$^{-1}$ (@6 kA/m) and 5.7 to 2.0 kAm$^{-1}$.MPa$^{-1}$ (@ 5 kA/m), respectively. The difference is not significant, but the origin can be explained as follows.
The stress $\sigma^*$ corresponding to maximum sensitivity is defined as the point where the second derivative of M equals zero. This value can be expressed as follows:

$$ \sigma^* = \frac{2}{(3A_s\lambda_m)} \ln\left(\frac{2}{\cosh(A_s\mu_0HM_m)}\right) \quad (6) $$

Figure 15 shows stress $\sigma^*$ leading to maximal sensitivities given by equation (6) as a function of applied magnetic field.

<figure><img src="image_18.png" /><figcaption>Figure 15 - Stress $\sigma^*$ corresponding to maximal sensitivities given by equation (6) as a function of magnetic field.</figcaption></figure>

Figure 15 shows that the stresses at maximum sensitivity are higher (in modulus) for the Fe<sub>65</sub>Co<sub>35</sub>-07Cr alloy than for the Fe<sub>65</sub>Co<sub>35</sub>-05Cr alloy, with a stress shift of $\Delta\sigma \approx 23$ MPa. This shift is attributed to a higher residual stress state in the 0.7% Cr alloy, likely arising from larger stresses between ordered and disordered phases during rapid quenching. The XRD results, showing larger crystallite size and higher lattice strain for the 0.7% Cr alloy,
support this interpretation. Similar differences in internal stress states due to ordering
have also been reported for Fe<sub>65</sub>Co<sub>35</sub> and Fe<sub>65</sub>Co<sub>35</sub> with 0.5%Cr added alloys [24].

The Fe<sub>65</sub>Co<sub>35</sub> alloys with Cr exhibit approximately half the sensitivity of 2V-Permendur.

## 4 Conclusion

The addition of small concentrations of Cr to Fe<sub>65</sub>Co<sub>35</sub> alloys resulted in materials with
very different microstructures due only to a difference of 0.2% Cr between the two studied
alloys. The alloy with 0.5% Cr presents AGG, while the alloy with 0.7% Cr had a mean
grain size reduction and a more isotropic distribution of grain sizes.

Fe<sub>65</sub>Co<sub>35</sub> alloy with 0.7% Cr is magnetically softer than the one with 0.5%, because 0.7% Cr alloy has the highest magnetic permeability and coherently the lowest coercive field H<sub>c</sub>

Both compositions demonstrated higher compressive strength range than rare-earth and gallium-based alloys and although their maxima sensitivities are lower, they exceed by 5 times the minimum required for practical applications (0.83 kA·m<sup>-1</sup>/MPa). Additionally, these alloys show promise for further optimization, particularly through texturing, stress and field annealing techniques to enhance sensitivity

The different piezomagnetic sensitivities behavior of the alloys is attributed to their different internal stress levels, associated with BCC lattice ordering due to the rapid solidification after annealing.

Although the alloys with Cr addition have good soft magnetic properties, their magnetostriction is too high for application as magnetic cores in AC electric machines, since the high magnetostriction would be source of noise and heating. The application as
force sensors would be more adequate as the d₃₃* value remains sufficiently high for a range of applications. Moreover, the simplicity of the material production process represents a significant advantage.

5 CRediT authorship contribution statement

**Iris Braga Silva:** Writing – original draft, Methodology, Formal analysis, Writing – review & editing, Validation, Investigation. **Fernando Froes:** Investigation. **Olivier Hubert:** Supervision, Validation, Formal analysis. **Cristina Bormio-Nunes:** Writing – review & editing, Supervision, Project administration, Conceptualization, Validation, Data curation, Resources, Funding acquisition

6 Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

7 ORCIDs

**Iris Braga Silva:** https://orcid.org/0000-0003-1846-9707

**Fernando Froes:** https://orcid.org/0000-0003-3821-2095

**Olivier Hubert:** https://orcid.org/0000-0003-1573-0260

**Cristina Bormio-Nunes:** https://orcid.org/0000-0002-0109-039X
# 8 Conflict of Interest Statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

# 9 Funding Statement

This study was mostly financed by the São Paulo Research Foundation (FAPESP), Brazil. Processes Numbers 2021/06350-0, 2022/08992-1 and 2022/15322-2, also in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

# 10. Data Availability Statements

The data that supports the findings of this study are available on request from the corresponding author, CBN.

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Supplementary data

1. EDS Results

Figure S1 presents the EDS maps of composition of the elements Fe, Co and Cr of the Fe65Co35-0.5Cr and Fe65Co35-0.7Cr alloys

<figure><img src="image_19.png" /><figcaption>Figure S1 – EDS map of Fe, Co and Cr distribution in the Fe<sub>65</sub>Co<sub>35</sub>-0.5 Cr (at left) and Fe<sub>65</sub>Co<sub>35</sub>-0.7 Cr (at right) alloys.</figcaption></figure>
<figure><img src="image_20.png" /></figure>

Figure S2 – Image processed of the Cr composition maps, by varying brightness-contrast to highlight only some brighter points.

Table S1 reports the average EDS composition for Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr and Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloys.

<table><caption>Table S1 – EDS chemical composition of the Fe<sub>65</sub>Co<sub>35</sub> alloys with addition of 0.5% and 0.7% Cr (% in wt.)</caption><thead><tr><th rowspan="2">Alloys</th><th colspan="3">Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr</th><th colspan="3">Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr</th></tr><tr><th>Fe</th><th>Co</th><th>Cr</th><th>Fe</th><th>Co</th><th>Cr</th></tr></thead><tbody><tr><td>1</td><td>64.6 ± 0.5</td><td>34.8 ± 0.5</td><td>0.6 ± 0.1</td><td>64.6 ± 0.4</td><td>34.7 ± 0.4</td><td>0.7 ± 0.1</td></tr><tr><td>2</td><td>64.8 ± 0.4</td><td>34.7 ± 0.4</td><td>0.5 ± 0.1</td><td>64.7 ± 0.2</td><td>34.5 ± 0.2</td><td>0.8 ± 0.1</td></tr><tr><td>3</td><td>64.8 ± 0.4</td><td>34.7 ± 0.4</td><td>0.6 ± 0.1</td><td>64.7 ± 0.5</td><td>34.5 ± 0.4</td><td>0.7 ± 0.2</td></tr></tbody></table>
## 2. EBSD Results

The complete texture EBSD results of the RD-TD surfaces show Figures S3, S4, S5 for Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr and S5 for Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy.

<figure><img src="image_21.png" /><figcaption>Figure S3 - Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr alloy texture analysis results of an area of 1.8 × 1.8 mm in the ND direction.</figcaption></figure>
The Figure S4 shows IPFs maps and texture of the total area calculated for the ND, RD and TD directions; Rectangle and square areas are analyzed separately in Figures S3 and S4.

<figure><img src="image_22.png" /></figure>

Figure S4 - Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr alloy texture analysis results of the square area marked in Figure S2. The IPFs maps and texture of the square area calculated for the ND, RD and TD directions.
<figure><img src="image_23.png" /><figcaption>Figure S5 - Fe<sub>65</sub>Co<sub>35</sub>-0.5Cr alloy texture analysis results of the rectangular area marked in Figure S2. The IPFs maps and texture of the square area calculated for the ND, RD and TD directions.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure S5 - Fe<sub>65</sub>Co<sub>35</sub>-0.7Cr alloy texture analysis results of an area of 1.0 × 1.0 mm in the ND direction. The Figure shows IPFs maps and texture of the total area calculated for the ND, RD and TD directions.</figcaption></figure>
### 3. Magnetomechanical Analysis

To obtain the piezomagnetic sensitivity from M vs σ experimental data, it was looked for a fitting function M vs σ the closest as possible to physics. The idea is based on multiscale approach of the magnetomechanical behavior [36-40 of the main text]. $<M>$ = M can be determined according to Boltzmann statistics by weighing the magnetization of each domain α by a function $f_α$ given by equations (S1), (S2) and (S3).

$$f_{\alpha} = \frac{\exp(-A_s g_{\alpha})}{\sum_{\alpha} \exp(-A_s g_{\alpha})} \quad (S1)$$

In which,

$$A_s = \frac{3\chi_0}{\mu_0 M_s^2} \quad (S2)$$

being $\chi_0$ the initial anhysteretic susceptibility, $\mu_0$ the vacuum magnetic permeability and $M_s$ the saturation magnetization of the material. The function $g_α$ is the Gibbs free energy density at the scale of a domain family α.

Considering no rotation mechanism, the definition of the magnetization in the domain frame is given by:

$$M = M_m \alpha \quad (S3)$$

The Gibbs free energy density of a magnetic domain is now:

$$g_{\alpha} = -\mu_{0}H \cdot M - \sigma: \varepsilon_{\alpha}^{\mu} \quad (S4)$$

A simplification of the multiscale problem can be achieved considering a finite number of representative domain families. Considering a model with six domains with magnetization along the six < 100 > axes of an elementary cube (Figure S6) or with $\alpha \in \{x, -x, y, -y, z, -z\}$. The model considers a magnetic field H and a uniaxial stress σ applied both along the x direction, $H = H \cdot x$ e $\sigma = \sigma \cdot x$.
<figure><img src="image_25.png" /><figcaption>Figure S6 - Domain assembly representative for the material submitted to magnetic field and stress.</figcaption></figure>

In addition, the magnetostriction strain tensor in the domain frame $\alpha$ is given by:

$$ \epsilon_{\alpha}^{\mu} = \begin{pmatrix} \lambda_m & 0 & 0 \\ 0 & -\frac{1}{2}\lambda_m & 0 \\ 0 & 0 & -\frac{1}{2}\lambda_m \end{pmatrix}_{\alpha} \qquad (S5) $$

$\lambda_m$ denotes a maximum magnetostriction strain. It does not correspond to the true saturation magnetostriction of the material but can be more considered as a parameter representative of its average magnetostriction (the same approach has been used recently in ref [40] of the main text).

Then, the energy term of equation (S4) can be written for each of the six domain families subjected to magnetic field and stress as:

$$ \left\{ \begin{array}{l} g_1 = -\mu_0 H M_m - \sigma \lambda_m \\ g_2 = \mu_0 H M_m - \sigma \lambda_m \\ g_3 = g_4 = g_5 = g_6 = \sigma \frac{\lambda_m}{2} \end{array} \right. \qquad (S6) $$

The volume fraction can be written for each of the six domain families subjected to the uniaxial stress and magnetic field using equations (S1) and (S6) as:
$$
\left\{
\begin{aligned}
f_1 &= \frac{\exp(A_s(\sigma\lambda_m + \mu_0 HM_m))}{\exp(A_s(\sigma\lambda_m + \mu_0 HM_m)) + \exp(A_s(\sigma\lambda_m - \mu_0 HM_m)) + 4\exp(-A_s\sigma\lambda_m/2)} \\
f_2 &= \frac{\exp(A_s(\sigma\lambda_m - \mu_0 HM_m))}{\exp(A_s(\sigma\lambda_m + \mu_0 HM_m)) + \exp(A_s(\sigma\lambda_m - \mu_0 HM_m)) + 4\exp(-A_s\sigma\lambda_m/2)} \\
f_3 &= f_4 = f_5 = f_6 = \frac{\exp(-A_s\sigma\lambda_m/2)}{\exp(A_s(\sigma\lambda_m + \mu_0 HM_m)) + \exp(A_s(\sigma\lambda_m - \mu_0 HM_m)) + 4\exp(-A_s\sigma\lambda_m/2)}
\end{aligned}
\right.
\qquad (S7)
$$

The average magnetization $\langle M \rangle$ of the discrete medium is:

$$
\langle M \rangle = M = \sum_{\alpha} f_{\alpha} M_{\alpha} \qquad (S8)
$$

The parameter $M_\alpha$ is the magnetization at the scale of a domain family $\alpha$.

Finally, using equations (S4) and (S8), the magnetization along the magnetic loading
axis is obtained by simple averaging:

$$
M = \frac{M_m [\exp(A_s(\sigma\lambda_m + \mu_0 HM_m)) - \exp(A_s(\sigma\lambda_m - \mu_0 HM_m))]}{\exp(A_s(\sigma\lambda_m + \mu_0 HM_m)) + \exp(A_s(\sigma\lambda_m - \mu_0 HM_m)) + 4 \exp(-A_s\sigma\lambda_m/2)} \quad (S9)
$$

And with $B = \mu_0 (H+M)$ an after some mathematical manipulation:

$$
B = \mu_0 H + \frac{\mu_0 M_m \exp(A_s \sigma \lambda_m) \sinh(A_s \mu_0 H M_m)}{\exp(A_s \sigma \lambda_m) \cosh(A_s \mu_0 H M_m) + 2 \exp\left(-\frac{A_s \sigma \lambda_m}{2}\right)} \quad (S10)
$$

There are two ways to use the result of equation (S10). For example, it is possible to
determine a unique set of constants (x0, Mm and lam) that can describe all the measured
behaviors. The function given by equation (S10) can then be considered as a constitutive
model, allowing for example to simulate any magneto-mechanical loading. It is also
possible to consider the expression (S10) as a basis function allowing the fit of each of
the B experimental curves independently, which assumes a distinct set of constants for
each magnetic field level.
The second strategy has been applied, and Tables S2 and S3 give the parameters for each magnetic field level and each material.

<table><caption>Table S2: Parameters for Fe<sub>65</sub>Co<sub>35</sub>-05Cr alloy.</caption><thead><tr><th></th><th>χ<sub>0</sub></th><th>M<sub>m</sub> (10<sup>6</sup> A/m)</th><th>λ<sub>m</sub>(ppm)</th></tr></thead><tbody><tr><td>1kA/m</td><td>347</td><td>0.47</td><td>2.2</td></tr><tr><td>2kA/m</td><td>370</td><td>0.95</td><td>9.1</td></tr><tr><td>3kA/m</td><td>331</td><td>1.30</td><td>18.0</td></tr><tr><td>4kA/m</td><td>296</td><td>1.55</td><td>26.9</td></tr><tr><td>5kA/m</td><td>271</td><td>1.70</td><td>34.7</td></tr><tr><td>6kA/m</td><td>249</td><td>1.82</td><td>41.6</td></tr></tbody></table>

<table><caption>Table S3: Parameters for Fe<sub>65</sub>Co<sub>35</sub>-07Cr alloy.</caption><thead><tr><th></th><th>χ<sub>0</sub></th><th>M<sub>m</sub> (10<sup>6</sup> A/m)</th><th>λ<sub>m</sub>(ppm)</th></tr></thead><tbody><tr><td>1kA/m</td><td>556</td><td>0.45</td><td>2.3</td></tr><tr><td>2kA/m</td><td>462</td><td>0.90</td><td>8.7</td></tr><tr><td>3kA/m</td><td>380</td><td>1.24</td><td>16.3</td></tr><tr><td>4kA/m</td><td>325</td><td>1.50</td><td>23.9</td></tr><tr><td>5kA/m</td><td>284</td><td>1.70</td><td>30.9</td></tr><tr><td>6kA/m</td><td>254</td><td>1.88</td><td>36.8</td></tr></tbody></table>

After few calculations, the derivation of magnetic induction with respect to stress leads to:

$$ \frac{dB}{d\sigma} = \frac{3\mu_0 M_m A_s \lambda_m \exp\left(\frac{A_s \sigma \lambda_m}{2}\right) \sinh(A_s \mu_0 H M_m)}{\left[\exp(A_s \sigma \lambda_m) \cosh(A_s \mu_0 H M_m) + 2 \exp\left(-\frac{A_s \sigma \lambda_m}{2}\right)\right]^2} \quad (S10) $$

In addition, the stress σ* corresponding to maximum sensitivity is defined as the point where the second derivative of B equals zero. This value can be expressed as follows:

$$ \sigma^* = \frac{2}{(3A_s \lambda_m)} \ln \left( \frac{2}{\cosh(A_s \mu_0 H M_m)} \right) \quad (S11) $$

Stresses σ* leading to maximal sensitivity are listed in Table S4 for both alloys.
<table><caption>Table S4: Stress σ* leading to maximal sensitivity of Fe<sub>65</sub>Co<sub>35</sub>-05Cr and Fe<sub>65</sub>Co<sub>35</sub>-07Cr alloy.</caption><thead><tr><th></th><th>Fe-35Co-05Cr (MPa)</th><th>Fe-35Co-07Cr (MPa)</th></tr></thead><tbody><tr><td>1kA/m</td><td>-76</td><td>-101</td></tr><tr><td>2kA/m</td><td>-72</td><td>-96</td></tr><tr><td>3kA/m</td><td>-73</td><td>-95</td></tr><tr><td>4kA/m</td><td>-77</td><td>-99</td></tr><tr><td>5kA/m</td><td>-87</td><td>-104</td></tr><tr><td>6kA/m</td><td>-97</td><td>-111</td></tr></tbody></table>