# Effect of pre strain on magnetic ageing in Fe-Si steels.

Ajay Kumar Jagannatha Rao, Lea Saleh, Sruthi Mohan, Dominique

Mangelinck, Claude Alfonso, Abdelkader Benabou, Oualid Messal, Stéphane

Clénet, Laurent Barrallier, Myriam Dumont

► To cite this version:

Ajay Kumar Jagannatha Rao, Lea Saleh, Sruthi Mohan, Dominique Mangelinck, Claude Alfonso, et al.. Effect of pre strain on magnetic ageing in Fe-Si steels.. Materialia, 2026, 45, pp.102700. 10.1016/j.mtla.2026.102700. hal-05533954

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Full Length Article

Effect of pre strain on magnetic ageing in Fe-Si steels

Ajay kumar Jagannatha Rao<sup>a ID</sup>, Lea Saleh<sup>c ID</sup>, Sruthi Mohan<sup>d</sup>, Dominique Mangelinck<sup>d ID</sup>,
Claude Alfonso<sup>d</sup>, Abdelkader Benabou<sup>c</sup>, Oualid Messal<sup>c</sup>, Stéphane Clénet<sup>c ID</sup>,
Laurent Barrallier<sup>b ID</sup>, Myriam Dumont<sup>a, *</sup>

ARTICLE INFO

Keywords:

Magnetic ageing

Pre-straining

Precipitation

Magnetic properties

Segregation

Modeling

A B S T R A C T

The durability and performance of electrical steels, essential for electric motors, are compromised by magnetic ageing, which mainly arises from microstructural changes such as carbide precipitation. This study examines how plastic pre-strain influences the magnetic ageing behaviour and microstructural evolution of an ultra-low carbon Fe-Si electrical steel during isothermal ageing at 200°C. Strain quantification was performed using controlled digital image correlation; samples were pre-strained to 2% and 5% plastic deformation prior to ageing. Magnetic properties were measured with single strip testers, and microstructural changes were quantitatively analysed through SEM imaging and image analysis at different ageing intervals. The results show that increasing pre-strain significantly slows down carbide nucleation and growth kinetics, contrary to the usual expectation that dislocations accelerate precipitation. This slowdown is explained by carbon atoms segregating to dislocations, forming Cottrell atmospheres that reduce the carbon available for carbide formation. Although magnetic losses increase with pre-strain due to enhanced domain wall pinning by dislocations, overall magnetic ageing is reduced because precipitate formation is suppressed. A modified precipitation model that includes dislocation density and carbon segregation effects successfully predicts the experimental data. These findings provide new insights into the complex relationship between plastic deformation, precipitation kinetics, and magnetic ageing in electrical steels, guiding strategies to improve motor efficiency and lifespan.

1. Introduction

The integration of electric motors into various applications is essential for reducing global warming emissions and advancing environmental sustainability. Electric motors rely heavily on electrical steels, whose efficiency and durability are intrinsically linked to their microstructure. Reduction of core losses in electrical steels is of prime importance for improving motor efficiency [1]. Increase in losses occurs at many steps of the motor life cycle: from manufacturing to operation. During operation of the electrical machine, heat will be dissipated in the iron core, resulting in the decrease of magnetic characteristics known as magnetic ageing. Magnetic ageing occurring during the operation of the machine is a significant challenge as it causes a gradual deterioration of magnetic characteristics over time, resulting in higher energy losses thus

lower motor efficiency. This ageing is primarily caused by microstruc-
tural changes due to temperature ageing in the range (150°C - 200°C)
[2] typical of the operating temperature of electrical machines. The
microstructural change responsible for the magnetic ageing is the for-
mation and growth of precipitates such as iron carbides, which impede
magnetic domain wall mobility [2,3] and thus, increase hysteresis losses
[4–6]. The interaction between domain walls and precipitates plays a
critical role in determining hysteresis losses, which are primarily influ-
enced by the precipitate volume fraction and size distribution. Very fine
precipitates (<0.1 µm) predominantly reduce domain wall energy,
whereas coarse precipitates (>1 µm) hinder domain wall motion mainly
through the formation of spike or closure domain structures [11]. Par-
ticles within the intermediate size range (approximately 0.1–1 µm) [9,
10] are particularly detrimental because they combine both interaction

<footnote><sup>a</sup> Arts & Métiers Institute of Technology. (MSMP-EA7350) Mechanics, Surfaces and Materials Processing, 8 bd Louis XIV, Lille Cedex, 59046, France</footnote>

<footnote><sup>b</sup> Arts & Métiers Institute of Technology. (MSMP-EA7350) Mechanics, Surfaces and Materials Processing, 2, cours des Arts et Métiers, 13617 Aix-en-Provence, France</footnote>

<footnote><sup>c</sup> Univ. Lille, Arts & Métiers Institute of Technology, Centrale Lille, HEI. (L2EP-EA2697) Laboratoire d'Électrotechnique et d'Électronique de Puissance, Bâtiment ESPRIT, Avenue Henri Poincaré, Villeneuve d'Ascq, 59655, France</footnote>

<footnote><sup>d</sup> Université Aix-Marseille, CNRS, IM2NP, UMR 7334 Marseille, France</footnote>

<footnote>* Corresponding author.
E-mail address: myriam.dumont@ensam.eu (M. Dumont).</footnote>
mechanisms and then exert the most pronounced effect on magnetic losses. The main parameters identified in the literature expected to influence magnetic ageing are: the thermal cycle (temperature and time in isothermal conditions) as precipitation is a thermally-activated phenomenon, the quantity of carbon available for precipitation and the amount of silicon that is known to inhibit the precipitation of cementite [14,15]. However, nothing is reported concerning the effect of deformation or stress on magnetic ageing.

Magnetic ageing in electrical steels is substantially temperature dependent, as evidenced by systematic experiments using isothermal heat treatments. Elevated ageing temperatures (160°C-200°C) accelerate both precipitate nucleation and growth [2], leading in increase of iron losses and coercivity over time, and enhanced magnetic ageing is observed at higher temperatures due to its precipitation kinetics. Experimental ageing methods [4,7,8] revealed that temperature exposure causes significant increases in hysteresis losses, indicating that motor cores operate primarily under aged circumstances. Microscopic analysis and kinetic modelling show that the nature and stability of precipitate phases change with temperature, influencing the rate and mechanism of domain wall pinning.

Carbon is fundamentally involved in the magnetic ageing of non-oriented electrical steels, acting as the principal solute species precipitating during service and temperature exposures. The experimental analysis on steels with different carbon concentrations revealed that higher carbon levels resulted in significantly faster attainment of elevated ageing indices and larger mean carbide sizes [10], supporting the notion that increased carbon drives faster nucleation and growth of iron carbides resulting in faster magnetic ageing.

Silicon is commonly added to electrical steels to reduce core losses, as it increases electrical resistivity [12]; however, silicon is generally added in the limit of 3 wt.% as over this limit, silicon has a deleterious impact on properties, increasing brittleness and decreasing saturation magnetization and Curie temperature [12]. Silicon also has the effect to inhibit magnetic ageing (for Si levels > 3wt%) [7,13]. In fact, the addition of silicon slows precipitation kinetics and alters precipitate morphology, producing finer and more distributed carbide particles [14]. Even if Si atoms are not directly forming carbides at typical ageing temperatures, silicon influences iron carbide kinetics via modifying the solubility or interfacial energy of these precipitates [15]. This means that for all Fe-Si steels with less than 3wt.% Si, in particular for the so-called low-grade Fe-Si steels (<1.8wt.%Si) which represents more than 75% of the market of non-oriented electrical steels, magnetic ageing is expected to occur and to decrease the magnetic properties over operation time of electrical machines.

Manufacturing processes including cutting or rolling introduce stresses and strains into electrical steels. The process is now known to have a great impact on the magnetic characteristics. Studies have shown that at initial stress levels, moderate tensile strain can help pre-strained specimens regain some of their magnetic performance by promoting domain wall motion and reducing hysteresis losses [16,17]. However, as strain levels increase, this effect becomes significantly worse. While elastic tensile strain generates reversible magnetic changes, plastic strain causes permanent degradation by creating dislocations and altering the microstructure [18]. Dislocations increase domain wall pinning and coercivity, increasing hysteresis losses [19,20,21].

Despite progress in understanding mechanical impacts on domain behaviour and magnetic degradation, the impact of deformation occurring during the manufacturing steps on subsequent precipitation kinetics during operation of the machine and thus on magnetic ageing is still unknown. The current article will look into an approach for determining the effect of plastic strain on magnetic ageing while accounting both magnetic and microstructural variables. Pre-strained samples (2% and 5% plastic strain) were aged at 200°C and magnetic measurements and microstructural evolution were rigorously compared over all ageing intervals and pre-straining circumstances. A precipitation model was created and modified to account for both silicon concentration and the

effects of plastic strain, with predictions validated by experimental results for magnetic losses and microstructural evolution.

## 2. Material and methods

The primary goal of the study was to investigate the magnetic characteristics and microstructural evolution of electrical steels subjected to various levels of plastic deformations and subsequent thermal ageing, which reflects the deformations that may be encountered during electric motor manufacturing and operation.

### 2.1. Material selection

The material chosen for this study was an electrical steel with 80 ppm C and 1.65 wt.% Si, manufactured in a cold-rolled and annealed condition with a thickness of 0.33 mm, similar to the composition and thickness of electrical steels used in electric motors. The material was lab-cast and processed by OCAS. Regarding the literature [8], this grade is expected to undergo magnetic ageing in the temperature range (150°C-200°C) within tens of hours of ageing.

### 2.2. Pre-straining of sample

The material was pre-strained along the rolling direction to 2% and 5% tensile plastic strain imitating mechanical deformations caused by manufacturing processes such as forging, rolling and punching. The dimensions of the samples were 220mm×40mm×0.33mm to allow subsequent cutting of samples for magnetic and microstructural characterizations. Preliminary study included a 1% plastic deformation condition, but this was dropped due to the appearance of Lüders bands, indeed these localized slip bands cause non-uniform strain distribution making this condition unsuitable for consistent analysis. Tensile tests were performed up to a plastic deformation after unloading of 2% and 5% as shown in Fig.1, with continuous image collection for Digital Image Correlation (DIC). The unloading process recovers the elastic strains leaving only the permanent plastic strain. The DIC technique is one of the most often used optical and contactless methods for measuring strains on a large scale [22]. DIC gives critical information for analysing strain distributions across samples. To determine strain levels, the DIC software enables quantitative, full-field strain measurements with sub-micron accuracy, allowing detailed investigation of deformation at various scales. The strain distribution is quantified and mapped with DIC, throughout the sample for 2% and 5% pre-strained samples as shown in Fig.2, assuring homogeneity across samples.

### 2.3. Sample sectioning and ageing

Following pre-straining, samples were cut into pieces for magnetic and microstructural characterizations using Wire Electrical Discharge Machining (WEDM). This cutting method is known to minimize the influence of cutting on magnetic characteristics when compared to other cutting processes, like punching or laser cutting [23]. Sample extraction distribution for Single Sheet Tester (SST) magnetic measurements and SEM microstructural investigation are displayed in Fig.3.

All prepared specimens, including non-strained controls (0%), are isothermally aged at 200°C for up to 80 hours (Table 1), to simulate maximum temperature exposures experienced during electric motor operation. After each step of ageing, SST and SEM samples were cooled to room temperature for magnetic measurements and microstructural analysis.

### 2.4. Magnetic characterization

Magnetization hysteresis curves and iron losses were measured with the SST (MPG200D system Brockhaus Measurements) at 50 Hz under sinusoidal magnetic flux density, with peak values ranging from 0.1 T to
<figure><img src="image_3.png" /><figcaption>Fig. 1. Measured Stress-strain curves of samples subjected to pre-strain during tensile test.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Strain gradient mapping with DIC for full-field surface strains: a) 2% pre strain b) 5% pre strain.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Sample extraction planning for microstructural samples (SEM analysis) and SST (magnetic characterization).</figcaption></figure>
<table><caption>**Table 1**
Ageing steps (at 200°C) applied to the samples for the experimental study.</caption><thead><tr><th>Samples</th><th>Ageing Times (200°C) in hours</th></tr></thead><tbody><tr><td>0%, 2% and 5% pre strained</td><td>0h 10h 30h 50h 80h</td></tr></tbody></table>

1.7 T. The ageing impact was evaluated by calculating total iron losses ($P_s$) and tracking the progression of losses compared to the unaged state, with ($P_s - P_0$), where $P_0$ represents the losses of unaged samples.

## 2.5. Microstructural characterization

The specimens were subjected to typical metallographic polishing including SiC and diamond polish to achieve a mirror polish surface before and after ageing at 200°C for various time steps. The samples were then etched with a 4% nitral solution (a mixture of 4ml nitric acid and 96ml ethanol) to reveal the microstructure. TESCAN MIRA II was used to perform SEM analysis on 0%, 2%, and 5% -strained samples to characterize the evolution of precipitation during ageing. Images were quantitatively assessed using ImageJ software for critical features such as area fraction (AF) and mean precipitate radius (Feret radius), which reflect microstructural changes over time.

The error bars associated with the measured area fraction and Feret radius account for uncertainties introduced during the image thresholding procedure. These uncertainties reflect the range of threshold values within which precipitates remain identifiable and arise from manual threshold adjustments applied during image processing in ImageJ.

Magnetic results from SST (1T/50Hz) and microstructural data from SEM were rigorously compared over all ageing time steps and pre-straining circumstances. Precipitation model was created and modified to account for both silicon concentration and the effects of plastic strain, with predictions validated by experimental results for magnetic losses and microstructural evolution.

## 2.6. TEM and APT characterizations

Thin lamellae for transmission electron microscopy (TEM) were fabricated from 0% and 5% plastically pre-strained samples using a Helios Nanolab 600 dual-beam focused ion beam/scanning electron microscope (FIB/SEM). For TEM, lamellae were carefully extracted from within individual grains, away from grain boundaries, to target representative bulk regions for precise microstructural characterization. TEM imaging was performed on an FEI Titan 80-300 microscope equipped with a spherical aberration (Cs) corrector and operated at 200 kV, enabling detailed visualization necessary for dislocation density measurements.

APT tips were prepared from similar grain-interior regions of the 5% deformed material using standard lift-out and sharpening procedures in the Helios Nanolab 600. APT analyses were conducted on a CAMECA LEAP 6000XR instrument in laser pulsing mode at 50 K, with a detection rate of 0.005 ions per pulse, laser energy of 30 picojoules, and pulse repetition rate of 200 kHz, optimized for detecting carbon segregation at the atomic scale. Data reconstruction and compositional analysis were carried out using IVAS software version 3.6.14.

# 3. Results

## 3.1. Influence of pre strain on magnetic properties

For samples subjected to different levels of plastic pre-strain (0%, 2%, and 5%) prior to ageing, the magnetic properties were measured under an applied field of 1 T at a frequency of 50 Hz. Fig. 4 shows the corresponding B-H hysteresis loops. Increasing the pre-strain level from 0% to 2% and 5% leads to progressively distorted hysteresis loops with a pronounced sheared S-shape [20], indicating that higher magnetic field strengths are required to achieve the same magnetic flux density as the degree of pre-strain increases. This behaviour is attributed to the increased density of deformation-induced dislocations generated during

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** B-H loops recorded for 0%, 2% and 5% pre-strained samples at 0 h and with the corresponding aged state at 80 h measured at 1T/50Hz.</figcaption></figure>
plastic deformation. These microstructural defects act as effective pinning sites for magnetic domain walls, thereby hindering their motion and increasing the resistance to magnetization reversal [24]. Consequently, magnetic losses increase with increasing pre-strain prior ageing, as illustrated in Fig. 5.

During thermal ageing at 200°C, all samples exhibit clear magnetic ageing. A comparison of the B-H hysteresis loops of the pre-strained specimens in the unaged condition (0 h) and after 80 h of ageing (Fig. 4) reveals a significant increase in coercivity (see inset in Fig. 4) and an expansion of the hysteresis loop area, leading to further increase in magnetic losses. Despite ageing, the non-pre-strained (0 %) sample consistently exhibits lower coercivity and reduced magnetic losses than the pre-strained materials.

As shown in Fig. 5, the unstrained sample exhibits an initial iron loss of approximately 1.3 W/kg in the unaged condition, which increases rapidly during the first 20 h of ageing and subsequently stabilizes at around 2.6 W/kg. This trend suggests that most of the microstructural changes affecting iron losses occur within the first 30 h of ageing.

In contrast, the pre-strained samples (2 % and 5 %) consistently display higher iron losses across all ageing times, with losses continuing to increase without reaching a steady state when compared to the un-strained sample. However, the extent of magnetic ageing, quantified by the increase in losses (Ps – Po), decreases with increasing pre-strain. This behaviour highlights the significant influence of pre-straining on the magnetic ageing response of the material.

## 3.2. Microstructural evolution

The microstructural evolution of the alloy during isothermal ageing at 200 °C was systematically examined by SEM to evaluate the precipitation kinetics as a function of applied pre-strain.

In the 0% pre-strained condition, no precipitates were visible at 0 h. After 5 h of ageing, fine precipitates appeared and were homogeneously dispersed throughout the microstructure. Prolonged ageing to 30 h resulted in substantial precipitate growth and stabilization as seen in the Fig. 6. Further extension of the ageing time up to 80 h produced no significant morphological changes, suggesting that the precipitates had

attained a near-equilibrium state.

In the 2% pre-strained specimens, the precipitation response was comparatively delayed as seen in the Fig. 7. At both 0 h and 5 h, precipitates were not observed. First precipitates became apparent at 10 h, followed by progressive growth at 30 h and 80 h. Relative to the 0% condition, the precipitates exhibited slower kinetics, forming later, and by 80 h they remained relatively fine, sparsely distributed, and heterogeneously dispersed.

The 5% pre-strained samples demonstrated the strongest retardation in precipitation kinetics as shown in the Fig. 8. Precipitates were absent at short ageing times and only became detectable after 30 h, with further but limited growth at 80 h. Comparative analysis across 0%, 2%, and 5% pre-strained conditions indicates that increasing pre-strain systematically delays the formation of precipitates and reduces the magnitude of the precipitation. At 80 h, the highly strained microstructures displayed smaller, fewer, and more heterogeneously distributed precipitates than those observed in the lower pre strained conditions.

## 3.3. Quantification of precipitation evolution

The area fraction and mean radius (Feret radius) of precipitates were calculated from microstructural images using the Image J software. To assess the evolution of precipitation, the area fraction and mean radius of precipitates of 0%, 2%, and 5% pre-strained samples were compared with ageing time.

Area fraction grows rapidly for 0% strained samples, reaching a stability within 30 hours of ageing, suggesting fast nucleation and growth in the absence of pre strain as seen in the Fig. 9. The 2% and 5% strained samples exhibit smaller area fraction as compared to 0% strained samples, even after 80 hours. With higher pre strain, the area fraction of precipitates decreases, showing that plastic deformation limits the formation and growth of precipitates.

Concerning the Fig. 10, mean radius follows a similar time evolution as the area fraction, with 0% pre strained samples reaching the highest mean radius of precipitates and doing so quickly within the first 30 hours. When compared to 0% pre-strained samples, the mean radius for both 2% and 5% strained samples develop significantly slower, even

<figure><img src="image_7.png" /><figcaption>Fig. 5. Measured iron loss with time at 1T/50Hz of 0%,2% and 5% pre strained samples aged at 200°C.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 6. Iron-carbides evolution in the 0% pre strained samples aged: a)0h, b)5h, c)10h, d)30h, e)80h at 200°C.</figcaption></figure>

after 80 hours. It is worth noting that the precipitate size stands within the (approx.0.1-1µm) range, which is known to be the most deleterious for magnetic properties [9,10].

Plastic straining dramatically slows both area fraction and mean radius increase during ageing, demonstrating that pre straining limits phase change kinetics. The 0% strained sample rapidly achieves equilibrium, whereas strained samples exhibit more delayed changes during ageing period.

# 4. Discussions

## 4.1. Correlations of precipitation evolution and Loss evolution

When comparing the iron loss evolution (Ps(t) - Po) and precipitate area fraction over ageing time, as illustrated in Fig. 11, the 0% pre-strained samples exhibit a more pronounced increase in iron losses corresponding to their larger area fraction. However, both the area fraction and evolution iron loss decrease with increasing pre-strain. This
<figure><img src="image_9.png" /><figcaption>Fig. 7. Iron-carbides evolution in the 2% pre strained samples aged at a)0h, b)5h, c)10h, d)30h, e)80h at 200°C.</figcaption></figure>

reduction in precipitate fraction and iron loss evolution due to plastic pre-straining highlights a substantial restriction of microstructural evolution and its consequent effect on iron losses in samples subjected to pre-strain

The evolution of iron losses with ageing time, commonly termed magnetic ageing, mirrors the change in precipitate area fraction. Both the precipitate area fraction and the corresponding iron loss evolution

exhibit similar trends, confirming their direct correlation. As the applied pre-strain increases, both the evolution of iron losses and precipitate area fraction decrease; in other words, magnetic ageing becomes less pronounced with increasing pre-strain.

The fundamental processes that underlie the observed inverse relationship between pre-strain and magnetic ageing are yet unknown. A thorough literature review was done to determine the underlying causes
<figure><img src="image_10.png" /><figcaption>Fig. 8. Iron-carbides evolution in the 5% pre strained samples aged: a)0h, b)5h, c)10h, d)30h, e)80h at 200°C.</figcaption></figure>

of this unusual behaviour.

## 4.2. Reason behind the effect of pre-strain on magnetic ageing

In most systems likely to be subject to precipitation, higher dislocation densities are expected to accelerate precipitation kinetics by increasing the number of nucleation sites (nucleation on dislocations) and enhancing atomic diffusion by the so-called pipe-diffusion mechanism [25-27]. However, this is the opposite of what is observed in the

current study.

Pre strain has two main effects, according to the results from iron losses and magnetic ageing (Fig.5). First, magnetic losses are directly increased even before ageing, because pre-strain increases dislocation density that leads to higher magnetic domain wall pinning [24]. Such an increase in iron losses by presence of dislocations can be estimated and linked to iron losses. Second, as already observed in special steels, called ultra-low carbon steels, carbon atoms preferentially segregate to dislocations, creating Cottrell atmospheres [28,29]. This segregation
<figure><img src="image_11.png" /><figcaption>Fig. 9. Time-dependent evolution of precipitate area fraction at 200°C based on microstructural images.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Time-dependent evolution of precipitate mean radius at 200°C based on microstructural images.</figcaption></figure>

phenomenon was already evidenced by atom probe tomography inves-
tigation [30,31]. Dislocations are then pinned by this segregation, which
reduces their mobility. As pre-strain increases, so does the dislocation
density, and interstitial carbon is depleted in the remaining matrix as a
result of its segregation on dislocations, which reduces the amount of
carbon available for traditional carbide precipitation in later ageing
[30]. During ageing, two distinct mechanisms may occur: carbon may
remain in a segregated state, or nanoscale precipitates may nucleate

along dislocations; both are expected to have a negligible effect on
magnetic properties. At this stage of the study, this point will require
further investigation. In contrast to the traditional precipitation kinetics
found in less-deformed steels, the precipitates formed after pre-straining
become finer and increasingly heterogeneous with progressive ageing
time steps [32].

To test this hypothesis, the contrasting microstructural states
observed between the undeformed and plastically deformed samples
<figure><img src="image_13.png" /><figcaption>Fig. 11. Correlation between the precipitate area fraction and the evolution of iron loss for 0%, 2%, and 5% pre-strained samples aged at 200°C.</figcaption></figure>

underscore the critical role of dislocation structures in material behaviour. The 0% pre-strained sample exhibits a pristine microstructure without any observable dislocations, representative of its strain-free condition. In contrast, the 5% pre-strained sample contains well-developed dislocation networks (Fig. 12), characteristic of plastic deformation-induced lattice defects. Accurate quantification using energy-filtered TEM (EFTEM) determined an average foil thickness of about 40 nm, based on an inelastic mean free path of 77 nm for 200 kV electrons in Fe [33]. Dislocation density was calculated by measuring the total dislocation line length in TEM micrographs and normalizing by specimen volume (image area × foil thickness), yielding an average density on the order of $1×10^{14}/m^2$. This high dislocation density quantitatively confirms the substantial strain accommodation via dislocation multiplication within the grain interiors resulting from 5%

plastic strain.

To investigate the interactions between solute atoms and these dislocations, Atom Probe Tomography (APT) conducted on the 5% pre-strained sample prior to ageing revealed a distinct linear defect identifiable as a dislocation, accompanied by clear and convincing evidence of carbon segregation along the dislocations. The reconstructed tip (Fig. 13a) and elemental maps for Fe, Al, Si, and C (Figs. 13b-13e) further demonstrate this segregation. A carbon isosurface at 0.4% level, visualized from lateral, front, and top views (Figs. 13f-13h), highlights a pronounced line of carbon enrichment consistent with the classical Cottrell atmosphere phenomenon around dislocations [37,39]. The dislocation length was measured to be approximately 29 nm.

Quantitative analysis using a cuboid encompassing the dislocation allowed plotting of the carbon concentration profile along the z-axis (Fig. 13i), revealing significant carbon enrichment confined to the dislocation core, with no detectable segregation of Si, Al, or N atoms. Two independent methods were used to quantify the excess carbon. The first method [30] compared carbon atom counts within a cylindrical volume around the dislocation to an identical region devoid of segregation. This yielded an excess of 980 carbon atoms, corresponding to 34 atoms per nanometer of dislocation length. Adjustment for the detector's 50% efficiency doubled this value to 68 atoms per nanometer. The second method employed the Gibbsian interfacial excess approach [40] (Fig. 13j), fitting cumulative concentration data on either side of the dislocation, producing a closely matching estimate of 984 excess atoms and a segregation density of 34 atom/nm. After correcting the detection's 50% efficiency, the same segregation density of 68 atoms per nanometer was confirmed.

The APT results complement Transmission Electron Microscopy (TEM) measurements performed on the 5% pre-strained and annealed sample, which found a dislocation density on the order of $10^{14}/m^2$. Integrating these findings, the carbon segregated to the dislocations corresponds to approximately 68 carbon atoms per nanometer of dislocation line, or $6.8×10^{10}$ atoms/meter. Multiplying by the measured dislocation density produces an estimated carbon content of $6.8×10^{24}$ atoms/m$^3$ segregated to the dislocation network. Considering the atomic number density of body-centered cubic iron as approximately $8.5×10^{28}$ atoms/m$^3$, the segregated carbon concentration corresponds to roughly $8.0×10^{-5}$ atomic fraction, or about 0.0017 wt% carbon residing specifically at dislocations.

<figure><img src="image_14.png" /><figcaption>Fig. 12. Bright-field TEM image of the 5% pre-strained Fe sample showing dislocation network.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** (a) SEM image of the APT needle prepared from the 5% pre-strained steel sample. (b–e) Reconstructed atom maps showing the spatial distributions of (b) Fe, (c) Al, (d) C, and (e) Si. (f–h) carbon isosurface rendered at 0.4 at.% shown from (f) lateral, (g) front, and (h) top viewing directions, revealing a carbon-enriched dislocation line. (i) One-dimensional carbon concentration profile extracted perpendicular the dislocation line. (j) Calculation of C segregation using Gibbsian excess method.</figcaption></figure>

### 4.3. Precipitation model

The development of precipitation models is essential for understanding precipitation kinetics and predicting the evolution of microstructure and material properties during heat treatment. These models simulate the simultaneous nucleation and growth of precipitate phases in low-carbon steels, with particular emphasis on carbide formation. By integrating thermodynamic and kinetic parameters, they enable accurate predictions of precipitate size and volume fraction [34,35]. After evaluating various precipitation models for their respective advantages and limitations, the mean radius approach was selected due to its balance of simplicity and predictive capability. This approach is based on classical nucleation and growth theory and employs the fundamental equations of nucleation (1) and growth (2), which allow quantification of precipitate volume fraction and mean radius evolution over ageing time at a given isothermal temperature and carbon remaining in solid solution. The initial parameters [34] used to determine the precipitation nucleation and growth are mentioned in the Table 2.

$$ \text{Nucleation equation: } \frac{dN}{dt} = N_0 Z \beta^* \exp\left(-\frac{\Delta G^*}{kT}\right) \exp\left(-\frac{\tau}{t}\right) \quad (1) $$

<table><caption>**Table 2**

List of initial parameters for precipitation calculations [34].</caption><thead><tr><th>Parameter</th><th>Meaning</th><th>Value</th></tr></thead><tbody><tr><td>a</td><td>Lattice parameter of bcc iron</td><td>2.86<sup>-10</sup>m</td></tr><tr><td>v<sup>at</sup></td><td>Atomic volume of bcc iron</td><td>a<sup>3</sup>/2</td></tr><tr><td>N<sub>0</sub></td><td>Number of sites per unit volume</td><td>1/v<sup>at</sup></td></tr><tr><td>Z</td><td>Zeldovich factor</td><td>1/20</td></tr><tr><td>γ</td><td>Interfacial energy of precipitate</td><td>0.147 Jm<sup>-2</sup></td></tr><tr><td>X</td><td>Initial value of carbon in mole fraction</td><td>0.000366</td></tr><tr><td>X<sub>eq(0)</sub></td><td>Pre-exp. factor for the solubility limit</td><td>0.01</td></tr><tr><td>Q<sub>p</sub></td><td>Act. energy for the solubility limit</td><td>28.4 kJmol<sup>-1</sup></td></tr><tr><td>X<sub>p</sub></td><td>Mole fraction of stable precipitate</td><td>0.25</td></tr><tr><td>D<sub>0</sub></td><td>Pre-exp. factor for the diffusion coefficient</td><td>6.2 &times; 10<sup>-7</sup> Jm<sup>2</sup>s<sup>-1</sup></td></tr><tr><td>Q<sub>D</sub></td><td>Act. energy for the diffusion coefficient</td><td>80 kJmol<sup>-1</sup></td></tr></tbody></table>

Where *T* is the isothermal temperature in *K*, *β*\* is the Condensation Rate, *Z* is the Zeldovich factor, *k* is the Boltzmann constant, Δ*G*\* being the energy required to form a critical nucleus of radius *R*\*, *τ* is an incubation time for nucleation, *N*\* is the number of nucleation site per unit volume and *t* is time in seconds. For a detailed derivation of the equations, refer [34,35].
$$ \text{Growth equation: } \frac{dR}{dt} = \frac{D}{R} \frac{X - X^i(R)}{\alpha X^P - X^i(R)} + \frac{1}{N} \frac{dN}{dt} (R_{k\beta t}^* - R) \quad (2) $$

With $X$, $X^i$, $X^P$ the molar fraction of carbon at origin, at equilibrium and precipitate respectively. $R_{k\beta t}^*$ is the radius at which stable precipitates nucleate, $D$ is the diffusion and $\alpha$ is the ratio of matrix to precipitates atomic volumes.

The precipitation model was refined to account for pre-straining effects, where variations in dislocation density modify precipitation kinetics by reducing the carbon available in solid solution. Dislocations serve as preferential segregation sites for solute atoms, with their density dictating the degree of segregation. The modified Guttman segregation equation (3) incorporates dislocation density, demonstrating that higher densities enhance solute segregation [36,37]. The residual carbon in solid solution is then determined by subtracting the segregated fraction from the initial carbon concentration.

$$ \text{Carbon atom segregation: } N_C^d = \frac{\rho * I * N_C^b * \exp\left(\frac{Ec}{kT}\right)}{N_b + N_C^b * \exp\left(\frac{Ec}{kT}\right)} \quad (3) $$

Where $\rho$ is the dislocation density in m⁻², $Ec$ is the segregation energy of Carbon (eV) typically around 0.1eV to 0.5eV [36] and $N_C^b$ is the interstitial atoms occupying bulk sites, per unit volume. $N_b$ is the total number of bulk sites per unit volume, $N_b = 3/\sqrt{Fe}$ where $Fe$ is the iron atomic volume. $I$ is the number of segregation sites per unit length on a single dislocation considered the value of 150 based on theoretical data [41]. The remaining carbon can be calculated by subtracting the segregated carbon atoms from initial carbon in solid solution.

The dislocation density $\rho$ was calculated using the Taylor equation (4) [38], which takes stress-strain data from tensile testing. This method allows for the determination of dislocation density evolution as a function of strain using experimentally recorded mechanical behaviour at various deformation levels.

$$ \text{Taylor equation: } \sigma_y = \sigma_0 + \alpha' G b \sqrt{\rho} \quad (4) $$

Where $\sigma_0$ and $\sigma_y$ is the base yield stress and maximum plastic stress attained for 2% and 5% strains respectively in MPa, $\alpha'$ is Tylor's constant related to material typically around 0.1-0.5, G is the shear modulus of the material in MPa and b is the Burgers vector of BCC lattice. With the data from stress strain curves (Fig.1) the dislocation density measured for 2% and 5% pre strained samples as shown in the Table 3.

The dislocation density values acquired via TEM and calculated using the Taylor equation for the 5% pre-strained sample have similar values, indicating that the two results are comparable.

The determined dislocation density from Tylor's equation values used to estimate the amount of carbon segregated to dislocations and the remaining carbon retained in solid solution. According to Table 4, the samples pre-strained by 5% exhibit a carbon segregation of approximately 22 ppm (around 0.0022 wt%) at the dislocations. In comparison, atom probe tomography (APT) measurements indicate a slightly lower value of 17 ppm (0.0017 wt%). This discrepancy of about 5 ppm may stem from the choice of fitting parameters, such as activation energy (Ec) and Tylor's constant $\alpha'$.

<table><caption>**Table 3**
Measured dislocation density of 2% and 5% pre strained samples using the values of stress strain.</caption><thead><tr><th>Sample</th><th>σ<sub>0</sub> (MPa)</th><th>σ<sub>y</sub> (MPa)</th><th>α'</th><th>ρ (m<sup>-2</sup>)</th></tr></thead><tbody><tr><td>2% Pre strain</td><td>214.54</td><td>240.92</td><td>0.3</td><td>1.87 × 10<sup>13</sup></td></tr><tr><td>5% Pre strain</td><td>220.98</td><td>293.90</td><td>0.3</td><td>1.42 × 10<sup>14</sup></td></tr></tbody></table>

<table><caption>**Table 4**
Measured carbon segregations and remaining carbon of 2% and 5% pre strained samples using the values of dislocation density from table 3.</caption><thead><tr><th>Sample</th><th>ρ (m<sup>-2</sup>)</th><th>Ec (eV)</th><th>Segregated carbon</th><th>Remaining carbon</th></tr></thead><tbody><tr><td>2% Pre strain</td><td>1.87 × 10<sup>13</sup></td><td>0.34</td><td>8ppm<br>(0.0008wt%)</td><td>72ppm<br>(0.0072wt%)</td></tr><tr><td>5% Pre strain</td><td>1.42 × 10<sup>14</sup></td><td>0.34</td><td>22ppm<br>(0.0022wt%)</td><td>58ppm<br>(0.0058wt%)</td></tr></tbody></table>

### 4.4. Validating the precipitation model with experimental results

The precipitation model was used to estimate the evolution of precipitate volume fractions in samples subjected to 0%, 2%, and 5% pre-straining, aged at 200°C for 80 hours. Pre-straining changes the initial dislocation density, which has a major impact on precipitate nucleation and growth rates. By including these factors, the model can make quantitative predictions for precipitate volume fractions under various deformation conditions. To validate the prediction capabilities, the estimated results were carefully compared to experimental measurements derived from image analysis of microstructural data, as seen in Fig.14. The image analysis gives the area fraction, and the equation (5) converts the area fraction to the volume fraction. The comparison shows a good level of consistency between the simulated and experimental volume fractions for all pre-strain conditions. Such consistency suggests that the model accurately represents the role of segregation on precipitation kinetics, allowing for realistic predictions of precipitation behaviour at various pre-strain levels. This validation indicates that the model accurately represents the relationship between plastic deformation, solute redistribution, and precipitation kinetics in low-carbon steels.

$$ \text{Volume fraction (X\%)} = \frac{\text{Area fraction (X\%)}}{\text{Area fraction (0\%)}} \times F_v \quad (5) $$

With X% represents pre strain condition (0%, 2% and 5%) and Fv (eq) is volume fraction at equilibrium.

Pre-straining leads to a pronounced change in precipitation characteristics at the microstructural scale, manifested primarily through increasing grain-to-grain heterogeneity. In contrast to the non-pre-strained condition, the pre-strained samples exhibit non-uniform precipitate distributions, with the degree of heterogeneity increasing with applied pre-strain. For the 2% pre-strained sample, SEM observations reveal that certain grains contain a relatively high density of precipitates, while neighbouring grains remain sparsely precipitated, indicating significant variations in local precipitation kinetics. This microstructural non-uniformity results in substantial scatter in precipitate volume fraction measurements when image analysis is performed on regions containing approximately 12–15 grains, depending on the relative representation of highly and weakly precipitated grains within each field of view. Consequently, this grain-scale heterogeneity contributes to the apparent discrepancy between the experimentally measured and predicted volume fractions shown in Fig. 14.

### 5. Conclusions

This study shows that plastic pre-straining of ultra-low carbon Fe-Si electrical steels has a significant impact on magnetic ageing, changing both magnetic domain behaviour and precipitation kinetics.

* Pre-straining increases initial magnetic losses by increasing dislocation densities (approximately $1.4 \times 10^{14}$ m⁻² for 5% pre-strain), which enhances domain wall pinning, but paradoxically reduces carbide precipitation during future thermal ageing.

* Quantitative microstructural analyses and magnetic measurements demonstrate that higher pre-strain levels produce smaller, fewer, and more heterogeneously dispersed precipitates, resulting in low
<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Comparison of experimental (based on microstructural images) and modelled volume fraction evolution of precipitates during ageing at 200°C for samples with 0%, 2%, and 5% pre-strain.</figcaption></figure>

magnetic ageing despite increased plastic deformation. The strong correlation between precipitate area fraction and iron loss evolution across all pre-strain conditions confirms that carbide precipitation is the dominant mechanism driving magnetic ageing. Consequently, reduced precipitation in pre-strained samples directly translates to mitigated magnetic losses.

* The decrease in precipitate nucleation and growth is attributed to carbon segregation at dislocations, which reduces the carbon available for carbide formation. Atom probe tomography measurements reveal approximately 68 carbon atoms per nanometer of dislocation line, forming Cottrell atmospheres that deplete matrix carbon and limit precipitation.

* These experimental results are accurately predicted by a precipitation model that accounts for dislocation-assisted segregation and carbon depletion. A modified model combining dislocation-dependent segregation and classical precipitation theory successfully captures the observed kinetics, thereby validating the segregation mechanism and confirming that carbon redistribution to dislocations is the key physical driver of suppressed precipitation and reduced magnetic ageing.

This study increases our understanding of magnetic ageing mechanisms in electrical steels and lays the groundwork for optimizing mechanical processing and thermal treatment to improve performance and durability in electrical machine applications.

## CRediT authorship contribution statement

**Ajay kumar Jagannatha Rao:** Writing – original draft, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Lea Saleh:** Writing – review & editing, Validation, Formal analysis, Data curation.
**Sruthi Mohan:** Writing – original draft, Validation, Investigation, Data curation.
**Dominique Mangelinck:** Writing – review & editing, Validation, Supervision.
**Claude Alfonso:** Writing – review & editing, Validation, Supervision, Conceptualization.
**Abdelkader Benabou:** Writing – review & editing, Validation, Supervision, Investigation, Funding acquisition.
**Oualid Messal:** Validation, Methodology.
**Stéphane Clénet:** Writing – review & editing, Validation, Supervision,

Methodology, Conceptualization. **Laurent Barrallier:** Writing – review & editing, Supervision, Methodology. **Myriam Dumont:** Writing – review & editing, Validation, Supervision, Project administration, Funding acquisition, Conceptualization.

## Declaration of competing interest

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.

## Acknowledgment

This work was conducted as part of the ANR-funded Mastermind 2 Project (ANR-22-CE50-0012-01). Thanks to Fabrice Guittonneau, Fadlallah Abouhadid, Jean Patrick Goulmy from MSMP, Martaine Cabié, Andrea Campos, Christian Dominici, Thomas Neisius from CP2M, and Marion Descoins (IM2NP) for their invaluable help with experimental measurements.

## References

[1] A. Moses, Energy efficient electrical steels: Magnetic performance prediction and optimization, Scripta Materialia 67 (2012) 560–565, https://doi.org/10.1016/j scriptamat.2012.02.027.

[2] H. Helbling, M. Toto-Jamil, M. Dumont, A. Benabou, S. Clénet, Temperature-dependent modelling of magnetic ageing of FeSi electrical steels, J. Magn. Mag. Mater. (2022), https://doi.org/10.1016/j.jmmm.2022.170051.

[3] K. Jenkins, M. Lindenmo, Precipitates in electrical steels, J. Magn. Mag. Mater. 320 (2008) 2423–2429, https://doi.org/10.1016/j.jmmm.2008.03.062.

[4] G. Negri, N. Sadowski, N. Batistela, J. Leite, J. Bastos, Magnetic Ageing Effect Losses on Electrical Steels, IEEE Trans. Mag. 52 (2016) 1–4, https://doi.org/10.1109/tmag.2016.2517942.

[5] M.F de Campos, Marília Emura, F. Landgraf, Consequences of magnetic ageing for iron losses in electrical steels, J. Magn. Mag. Mater. 304 (2006), https://doi.org/10.1016/j.jmmm.2006.02.185.

[6] A. Krings, J. Soulard, Overview and comparison of iron loss models for electrical machines, J. Electric. Eng.-elektrotechnicky Casopis 10 (2010) 162–169.

[7] S. Ray, O.N. Mohanty, Magnetic ageing characteristics of low silicon electrical steels, J. Magn. Mag. Mater. 28 (1982) 44–50, https://doi.org/10.1016/0304-8853 (82)90027-0.
[8] S. Ray, O.N. Mohanty, On predicting the extent of magnetic ageing in electrical steels, J. Magn. Mag. Mater. 78 (1989) 255-262, https://doi.org/10.1016/0304-8853(89)90276-x.

[9] L. Dijkstra, C. Wert, Effect of inclusions on coercive force of iron, Phys. Rev. 79 (1950) 979-985, https://doi.org/10.1103/physrev.79.979.

[10] J. De Oliveira Júnior, R. Monteiro, S. Paolinelli, A. Cota, Kinetics of magnetic ageing of 2%Si non-oriented grain electrical steel, Mater. Res.-ibero-Am. J. Mater. 21 (2017), https://doi.org/10.1590/1980-5373-mr-2017-0575.

[11] B.D. Cullity, C.D. Graham, Introduction to Magnetic Materials (2nd ed.), Wiley-IEEE Press, 2009.

[12] G. Chin, J. Wernick, Chapter 2 Soft magnetic metallic materials, Handbook Ferromag. Mater. 2 (1980) 55-188, https://doi.org/10.1016/s1574-9304(05)80103-8.

[13] W.C. Leslie, G. Rauch, Precipitation of carbides in low-carbon Fe-AI-C alloys, Metallurg. Trans. A 9 (1978) 343-349, https://doi.org/10.1007/bf02646383.

[14] B. Kim, C. Celada, D. Martín, T. Sourmail, P. Rivera-Díaz-del-Castillo, The effect of silicon on the nanoprecipitation of cementite, Acta Mater. 61 (2013) 6983-6992, https://doi.org/10.1016/j.actamat.2013.08.012.

[15] E. Kozeschnik, H. Bhadeshia, Influence of silicon on cementite precipitation in steels, Mater. Sci. Technol. 24 (2008) 343-347, https://doi.org/10.1179/174328408x275973.

[16] V. Iordache, E. Hug, N. Buiron, Magnetic behaviour versus tensile deformation mechanisms in a non-oriented Fe-(3 wt.%Si) steel, Mater. Sci. Eng. A-Struct. Mater. Properties Microstruct. Proc. 359 (2003) 62-74, https://doi.org/10.1016/s0921-5093(03)00358-7.

[17] O. Perevertov, J. Thielsch, R. Schäfer, Effect of applied tensile stress on the hysteresis curve and magnetic domain structure of grain-oriented transverse Fe-3% Si steel, J. Magn. Mag. Mater. 385 (2015) 358-367, https://doi.org/10.1016/j.jmmm.2015.03.040.

[18] V. Iordache, F. Ossart, E. Hug, Magnetic characterisation of elastically and plastically tensile strained non-oriented Fe-3.2%Si steel, J. Magn. Mag. Mater. 254 (2003) 57-59, https://doi.org/10.1016/s0304-8853(02)00748-5.

[19] N. Leuning, S. Steentjes, K. Hameyer, Impact of grain size distribution on the magnetic deterioration due to cutting of electrical steel sheets, J. Magn. Mag. Mater. (2020), https://doi.org/10.1016/j.jmmm.2019.166080.

[20] N. Leuning, S. Steentjes, M. Schulte, W. Bleck, K. Hameyer, Effect of elastic and plastic tensile mechanical loading on the magnetic properties of NGO electrical steel, J. Magn. Mag. Mater. 417 (2016) 42-48, https://doi.org/10.1016/j.jmmm.2016.05.049.

[21] Kleiner Marques Marra, E.A. Alvarenga, V. Buono, Magnetic ageing anisotropy of a semi-processed non-oriented electrical steel, Mater. Sci. Eng. A-Struct. Mater. Properties Microstruct. Proc. 390 (2005) 423-426, https://doi.org/10.1016/j.msea.2004.08.066.

[22] A. Weidner, H. Biermann, Review on strain localization phenomena studied by high-resolution digital image correlation, Adv. Eng. Mater. (2021) 23, https://doi.org/10.1002/adem.202001409.

[23] M. Emura, F. Landgraf, W. Ross, J. Barreta, The influence of cutting technique on the magnetic properties of electrical steels, J. Magn. Mag. Mater. 254 (2003) 358-360, https://doi.org/10.1016/s0304-8853(02)00856-9.

[24] M. Sablik, S. Rios, F. Landgraf, T. Yonamine, M.F. Campos, Modeling of sharp change in magnetic hysteresis behavior of electrical steel at small plastic deformation, J. Appl. Phys. (2005) 97, https://doi.org/10.1063/1.1856191.

[25] Manon Bonvalet Rolland and A. Borgenstam, "Modeling precipitation kinetics in multicomponent alloys during deformation," 9 (2022), https://doi.org/10.3389/fmats.2022.958472.

[26] K. Gururaj, S. Pal, Influence of dislocation density and grain size on precipitation kinetics on P92 grade steel, in: Materials Today: Proceedings, 2019, https://doi.org/10.1016/j.matpr.2019.06.602.

[27] Yukiko Kobayashi, J. Takahashi, K. Kawakami, Effects of dislocations on the early stage of TiC precipitation kinetics in ferritic steel: A comparative study with and without a pre-deformation, Acta Mater. (2019), https://doi.org/10.1016/j.actamat.2019.06.055.

[28] A. De, K.D. Blauwe, S. Vandepitte, B.D. Cooman, Effect of dislocation density on the low temperature ageing behavior of an ultra low carbon bake hardening steel, J. Alloys Compounds 310 (2000) 405-410, https://doi.org/10.1016/s0925-8388(00)00973-7.

[29] D.V. Wilson, B. Russell, The contribution of atmosphere locking to the strain- ageing of low carbon steels, Acta Metallurgica 8 (Issue 1) (1960) 36-45, https:// doi.org/10.1016/0001-6160(60)90138-3.

[30] G.D. Rosa, P. Maugis, J. Drillet, V. Hébert, K. Hoummada, Co-segregation of boron and carbon atoms at dislocations in steel, J. Alloys Compounds 724 (2017) 1143-1148, https://doi.org/10.1016/j.jallcom.2017.07.096.

[31] Juan Macchi, J. Teixeira, F. Danoix, G. Geandier, Sabine Denis, Frédéric Bonnet, Sébastien Y.P. Allain, Impact of carbon segregation on transition carbides and cementite precipitation during tempering of low carbon steels: experiments and modeling, Acta Mater. (2024), https://doi.org/10.1016/j.actamat.2024.119919.

[32] A. De, S. Vandepitte, B.C. Cooman, Static strain ageing behavior of ultra low carbon bake hardening steel, Scripta Material. 41 (1999) 831-837, https://doi.org/10.1016/s1359-6462(99)00232-8.

[33] H. Shinotsuka, S. Tanuma, C. Powell, D.R. Penn, Calculations of electron inelastic mean free paths. X. Data for 41 elemental solids over the 50 eV to 200 keV range with the relativistic full Penn algorithm, Surf. Interface Anal. 47 (2015) 871-888, https://doi.org/10.1002/sia.5789.

[34] M. Perez, A. Deschamps, Microscopic modelling of simultaneous two-phase precipitation: application to carbide precipitation in low-carbon steels, Mater. Sci. Eng. A-Struct. Mater. Properties Microstruct. Proc. 360 (2003) 214-219, https:// doi.org/10.1016/s0921-5093(03)00431-3.

[35] M. Perez, M. Dumont, D. Acevedo-Reyes, Implementation of classical nucleation and growth theories for precipitation, Acta Mater. 56 (2008) 2119-2132, https:// doi.org/10.1016/j.actamat.2007.12.050.

[36] M. Guttmann, Thermochemical interactions versus site competition in grain boundary segregation and embrittlement in multicomponent systems, J. De Physique IV (1995), https://doi.org/10.1051/JP4:1995707, 05.

[37] W. Mottay, P. Maugis, M. Jouiad, F. Roch, C. Perrin-Pellegrino, K. Hoummada, Effect of dislocation density on competitive segregation of solute atoms to dislocations, SSRN Electron. J. (2023), https://doi.org/10.2139/ssrn.4186616.

[38] Kamyar M. Davoudi, J. Vlassak, Dislocation evolution during plastic deformation: equations vs. discrete dislocation dynamics study, J.Appl. Phys. 123 (2014) 085302, https://doi.org/10.1063/1.5013213.

[39] Michael K. Miller, Atom probe tomography characterization of solute segregation to dislocations, Microsc. Res. Tech. (2006) 69, https://doi.org/10.1002/jemt.20291.

[40] B. Krakauer, David N. Seidman, Absolute atomic-scale measurements of the Gibbsian interfacial excess of solute at internal interfaces, Phys. Rev.B, Cond. Matter 48 (9) (1993) 6724-6727, https://doi.org/10.1103/physrevb.48.6724.

[41] R.G.A. Veiga, M. Perez, C.S. Becquart, C. Domain, Atomistic modeling of carbon Cottrell atmospheres in bcc iron, J. Phys. Condens. Matter 25 (2013) 025401, https://doi.org/10.1088/0953-8984/25/2/025401.