Inhomogeneous strain associated with Lüders band propagation in an FeCo-2V alloy

Abstract

The inhomogeneous strains associated with Lüders bands in FeCo-2V have been studied. Mechanical testing revealed distinct elastic, Lüders band propagation, and work-hardening regimes. Digital Image Correlation (DIC) strain maps confirmed localised deformation during band propagation, transitioning to homogeneous plasticity post-saturation. Electron Backscatter Diffraction Kernel Average Misorientation (EBSD-KAM) analysis showed elevated misorientation angles in deformed regions, correlating with increased dislocation density. Synchrotron X-ray diffraction (XRD) showed peak broadening during Lüders band propagation, attributed to dislocation-induced lattice distortions that were quantified through Williamson-Hall analysis. The results demonstrate that Lüders bands in FeCo-2V arise from dislocation accumulation, with uniform peak width regions in XRD maps correlated with band propagation, while post-plateau work-hardening increased average peak widths. These findings provide insights into strain localisation in FeCo-2V.

Keywords

FeCo-2V alloy soft magnet Lüders band inhomogeneous strain

Introduction

In recent decades, the global movement towards low-carbon transportation has intensified, driven by environmental concerns and the need for sustainable energy solutions. This transformation is particularly evident in the development of electric vehicles and electric aircraft. One of the core technologies enabling this shift is high power density electric motors. These rely on soft magnetic materials. However, many conventional magnetic materials suffer from limitations such as high energy losses, insufficient thermal stability, and decreased magnetic performance under operational stress, leading to suboptimal motor efficiency. Thus, enhancing the performance of magnetic materials is central to increasing the overall efficiency and reliability of modern electric transport systems.

Soft magnetic materials are characterised by their low coercivity and high permeability, making them suitable for electromagnetic energy conversion. Common examples include pure iron, low-carbon steels, silicon-iron alloys, and nickel-iron alloys.¹ Among these, iron-cobalt (FeCo) alloys are of particular interest due to their outstanding magnetic properties, most notably the highest saturation magnetisation among soft magnetic materials.² This high saturation makes FeCo alloys ideal for use in high-performance devices such as aerospace actuators, transformers, and advanced electric motors. However, the high production costs of FeCo alloys limit their widespread use, restricting them to applications where performance benefits outweigh economic considerations.^1,3 The magnetic behaviour of FeCo alloys is influenced by external factors such as temperature, mechanical loading, and microstructural changes during service. In practical applications, these alloys often operate under mechanical stress, which can degrade magnetic properties and lead to functional failure. Therefore, understanding how mechanical deformation impacts their magnetic behaviour is vital for ensuring performance stability.

Recent studies have focused on the deformation behaviour of FeCo-2V alloys, a variant of FeCo materials with added V for enhanced thermal and structural stability. Notably, Li et al.,⁴ Keller et al.⁵ and Iordache and Hug⁶ investigated the tensile response of ordered FeCo-2V alloys and observed consistent features in their stress-strain behaviour. These features include an initial elastic region, followed by a stress plateau, and finally, a rapid work-hardening region. A distinctive characteristic observed in these studies is the emergence of Lüders bands, a form of plastic instability that appears during the plateau stage. Lüders bands are known to cause localised strain, resulting in non-uniform deformation across the material. This phenomenon is commonly attributed to interactions between dislocations and solute atoms, such as interstitial carbon, which form Cottrell atmospheres that temporarily pin dislocations^7,8 or short range order of substitutional solutes in the material.⁹ When such pinning is overcome, sudden dislocation motion occurs, leading to the formation and propagation of Lüders bands.

Lüders deformation poses several practical challenges in components manufactured from FeCo-2V alloys. The localised plastic flow associated with Lüders bands can cause surface markings, non-uniform elongation, and dimensional instability during manufacturing processes such as stamping, forming, and assembly of electrical machine laminations. In service, such inhomogeneous strain may also lead to magnetoelastic coupling effects, resulting in local degradation of magnetic permeability and increased power losses.¹⁰ These effects can reduce the overall efficiency and reliability of high-performance electric motors.¹¹ Given that localised plasticity can influence material magnetic properties at the microstructural level, a detailed understanding of Lüders band behaviour in FeCo-2V alloys is crucial.

Lüders band formation and propagation are governed by complex concepts of localised plastic deformation, dislocation dynamics, and microstructural features. To investigate these phenomena, a combination of advanced characterisation techniques, synchrotron X-ray diffraction (XRD), Digital Image Correlation (DIC), and Electron Backscatter Diffraction (EBSD) with Kernel Average Misorientation (KAM) analysis, has proven particularly effective. Synchrotron XRD enables in situ monitoring of internal strains and microstructural changes with high spatial resolution. For example, Zhang et al.¹² and Liu et al.¹³ employed synchrotron XRD to study medium-manganese steels, revealing increased dislocation densities and notable changes in diffraction patterns within Lüders band regions. Complementarily, DIC offers full-field, non-contact surface strain measurements, allowing real-time visualisation of Lüders band nucleation and propagation. Varanasi et al.¹⁴ combined DIC with electron channelling contrast imaging to show that Lüders bands in medium-manganese steels consist of strain-localised plastic zones. Han et al.¹⁵ used DIC to resolve strain gradients in high-strength pipe steel, while Qiu et al.¹⁶ applied DIC alongside extensometry to monitor Lüders band movement in ferrite-pearlite steels. In parallel, Electron Backscatter Diffraction Kernel Average Misorientation (EBSD-KAM) analysis provides post-deformation quantification of local misorientation, which reflects dislocation accumulation and plastic strain gradients. Han et al.¹⁵ reported elevated KAM values within Lüders bands, and Wang et al.¹⁷ used KAM mapping to identify dislocation-rich zones at Lüders band fronts in a Mg-Nd alloy.

Although there has been considerable research into the formation and behaviour of Lüders bands in various alloys, limited work has been conducted on FeCo-2V alloys, especially using in situ monitoring techniques. Most existing studies focus on post-deformation analysis or employ a single characterisation method, limiting the understanding of the real-time development of localised strain and its effects on material properties. Furthermore, the impact of Lüders bands on the functional performance of magnetic materials, such as the degradation of magnetic properties due to localised strain, remains underexplored in FeCo-based systems.

To address these gaps, this study aims to investigate the inhomogeneous strain and Lüders band propagation in FeCo-2V alloys using a multi-technique approach. This includes mechanical testing, DIC, EBSD-KAM, and synchrotron X-ray diffraction. The integrated use of these techniques provides a characterisation of deformation mechanisms in FeCo-2V. Such understanding is essential for optimising the effective design and application of FeCo-based soft magnets in advanced electric transportation and other high-performance technologies.

Experimental procedures

Sample preparation

This study utilised Vacoflux 50, a soft magnetic alloy produced by Vacuumschmelze. The alloy, composed of 49 wt.% iron, 49 wt.% cobalt, and 2 wt.% vanadium, was selected due to its high saturation magnetisation and its relevance to high-performance electromagnetic applications. The materials were provided in a cold-rolled condition with thicknesses of 0.15 mm and 0.2 mm. The gauge length of the DIC tensile specimen and synchrotron XRD experiment were 100 mm and 198 mm, respectively.

Two distinct batches of samples were prepared for different experimental procedures: one for in-situ mechanical testing with DIC, and the other for in situ synchrotron XRD measurements. Dog-bone shaped specimens for DIC experiment were fabricated using electro-discharge machining (EDM) to ensure precision and prevent unwanted mechanical deformation during cutting. Figure 1(a) and (b) illustrate the final geometries of the DIC and XRD samples, respectively. Dimensional accuracy was verified using a digital vernier calliper, ensuring consistency across samples. After machining, each specimen was thoroughly cleaned with acetone to remove surface contaminants and dried in a controlled cabinet before being stored in airtight containers to avoid oxidation or moisture exposure prior to heat treatment.

Figure 1.

Design specification of (a) dog-bone specimen for mechanical testing and DIC and (b) specimen for mechanical testing and in situ synchrotron XRD utilising IEC 60404¹⁸ SST / Epstein strip geometry (all dimensions are in mm).

Heat treatment

To relieve residual stress from cold rolling, all samples underwent a controlled heat treatment process in accordance with the manufacturer's recommendations for mechanically optimised magnetic properties.¹⁹ The cleaned Vacoflux 50 strips were sent to Kepston Ltd, for heat treatment in a dry, pure hydrogen atmosphere. The test specimens were heated at a rate of 15 °C/min to a target temperature of 750 °C, a condition chosen to promote the desired microstructural evolution while maintaining dimensional stability. The samples were held isothermally at this temperature for 3 h, after which the samples were cooled slowly at a rate of 100–300 °C per hour in the same hydrogen atmosphere to room temperature. The same procedure was applied to all samples, ensuring consistency between the mechanical and XRD test batches.

Mechanical testing and DIC

Before using DIC, a pattern must be created on the sample surface. VHT flameproof white paint was utilised to create a white background. The white paint was mixed with acetone in a 4:1 ratio (10 ml paint: 2 ml acetone). The mixture was then transferred to an Iwata Custom Micron B airbrush connected to a Sparmax TC-670H compressor. The spray distance was set to around 8–10 cm. Approximately 25 passes were required to spray over the sample surface to build up a continuous white background. VHT flameproof black paint was subsequently mixed with the same 4:1 ratio with acetone. Approximately 5 to 8 passes were performed over the white surface to build up a speckle pattern with spot density close to 50%. The speckle-sprayed sample is shown in Figure 2.

Figure 2.

Speckle pattern sprayed sample.

Key considerations for the optimal pattern generation technique include high-contrast features, dense coverage (approximately 50%), and appropriately sized features (3–8 pixels).²⁰ All samples were dried in the drying cabinet and stored with no contact to the sprayed surface.

Images of the sample during testing were acquired with a monochromatic LaVision Imager E-Lite 2M camera with 1636 pixels × 1236 pixels, fitted with a 100 mm fixed focal length lens and f/11 aperture. The camera was set at a distance of approximately 150 cm, corresponding to a field of view of around 15 cm × 4 cm and a pixel size of 86 µm. Raw images were recorded at 1 Hz during monotonic tensile tests. Two blue spotlight Light Emitting Diodes (LEDs) were used to illuminate the sample surface. Image correlation was performed using Davis version 8.4.0 software (LaVision).

An Instron 3367B testing frame with Instron 2712 pneumatic testing grips and a 30 kN load cell was used for mechanical testing. The sample was affixed to the crossheads with Instron 2712 pneumatic grips and equipped with a specimen alignment device to minimise bending. A strain rate of 0.02 mm s⁻¹ was used and controlled during testing through the Linear Variable Differential Transformer (LVDT) measured crosshead displacement. Data logging was performed in the Bluehill 3 software.

EBSD – KAM

To investigate dislocation density and local strain accumulation, EBSD was conducted on two samples: one undeformed and one subjected to 3.5% strain during the DIC experiment. These samples were selected to represent both the initial microstructure and the stage of plastic deformation where Lüders band activity is expected.

Sample preparation involved sequential grinding using silicon carbide (SiC) papers with grits of 600, 1200, and 2500. This was followed by final polishing with diamond paste, performed in four steps (6 µm, 3 µm, 1 µm, and 0.25 µm) for durations of 5, 5, 3, and 1 min respectively. An oil-based lubricant was applied during polishing to minimise oxidation. After polishing, the specimens were stored in a desiccator to prevent moisture absorption and surface degradation.

EBSD data collection was carried out using a Zeiss Gemini SEM 300 equipped with an Oxford Instruments Symmetry EBSD detector. The scanning conditions were as follows: 20 kV acceleration voltage, 60 µm aperture, 0.5 µm pixel size, 17.5 mm working distance, and 4 ms dwell time per pixel. Each scan covered an area of 500 µm × 500 µm, with the sample tilted at 70° to optimise signal quality. The resulting EBSD maps were analysed using the MTEX toolbox within MATLAB, focusing on KAM values to quantify local misorientation gradients related to dislocation structures and plastic deformation.

Synchrotron XRD

To capture strain evolution and crystallographic changes during deformation, in situ synchrotron X-ray diffraction was conducted on heat-treated rectangular specimens. In situ diffraction data were collected on the I12 beamline at the Diamond Light Source using a transmission Debye-Scherrer configuration.²¹ The experiment utilised a high-energy monochromatic X-ray beam with an energy of 100.094 keV ( $λ$ = 0.1239 Å) and beam area of 0.5 × 0.5 mm².

A CeO₂ standard was used to calibrate the experimental geometry and determine instrumental parameters. Diffraction patterns were captured using a Pilatus 2M CdTe 2D area detector located downstream of the sample. Data processing followed a multi-step procedure: raw 2D patterns were first reduced using DAWN software,^22,23 and further quantitative analysis was conducted with Wavemetrics IgorPro.

Mechanical loading during the in situ XRD experiment was conducted using an Instron 100 kN uniaxial servo-hydraulic testing rig, equipped with an 8800MT controller and 100 kN hydraulic grips.²⁴ The specimen was strained at a constant rate of 0.02 mm s⁻¹. At eleven pre-defined strain intervals, the test was paused and unloaded to minimise motion blur, allowing diffraction data collection over an area of 15 × 15 mm². The area was rastered with the X-ray beam, which had an area of 0.5 × 0.5 mm² in 0.5 mm step. Each collection contained 961 diffraction patterns. After each collection, the sample was further strained by 1 mm, and the cycle was repeated until the final failure. This approach provided a time-resolved dataset capturing the structural evolution throughout plastic deformation.

XRD peak profiling and Williamson-Hall analysis

XRD peak profiling and Williamson-Hall analysis were performed to evaluate inhomogeneous strain development during deformation. One-dimensional diffraction patterns were processed using Wavemetrics IgorPro, and Gaussian functions were fitted to individual peaks to extract peak position, peak width, and peak area. To account for instrumental broadening, a CeO₂ reference sample was measured under identical experimental conditions. The peak widths from the reference were fitted using the Caglioti function (equation (1))²⁵ to obtain instrumental parameters U, V and W, allowing calculation of instrumental broadening ( $B_{i}$ ) as a function of diffraction angle ( $θ$ ).

B_{i}^{2} = U t a n^{2} θ + V \tan θ + W

(1)

The square of the total peak width ( $B_{t}$ ) was corrected by subtracting the square of $B_{i}$ , and the resulting data were analysed using the Williamson–Hall analysis. The corrected broadening was plotted as $(B_{t}^{2} - B_{i}^{2}) \cos^{2} θ$ versus $\sin^{2} θ$ as shown in equation (2):

(B_{t}^{2} - B_{i}^{2}) \cos^{2} θ = {(\frac{λ}{t})}^{2} + ε^{2} s i n^{2} θ

(2)

where

λ

is the X-ray wavelength, t is the crystallite size, and

ε

is the inhomogeneous strain. From the slope of the linear fit, the strain component (

ε

) was extracted.

Results

Mechanical testing and digital image correlation (DIC)

The measured stress-strain curve of FeCo-2V is presented in Figure 3. The diagram shows three stages. In the initial stages of deformation, the stress-strain curve exhibits classic linear elastic behaviour (1), where the material undergoes reversible deformation. At the end of this region, the stress-strain curve shows an upper yield point (2) (∼420 MPa) followed by a lower yield point (3) (∼370 MPa). After the lower yield point, there was a plateau region (4). During this stage, there is typically an increase in strain without a significant rise in stress. Beyond the plateau region, the stress-strain curve continues in the plastic deformation region (5), showing a more gradual increase in stress as the material undergoes further deformation. Eventually, the material reaches the ultimate tensile strength (6).

Figure 3.

Measured stress-strain curve of FeCo-2V alloy.

Figure 4 displays DIC screenshots acquired from different points on the stress-strain curve. These screenshots show the strain map of the entire gauge length of the sample. In the colour scale, blue represents the area with the lowest strain, while red indicates the region with the highest strain.

Figure 4.

DIC screenshots of different regions on the stress-strain curve of FeCo-2V: (a) at the start of the test, (b) the beginning of the plateau region, (c) middle of plateau region, (d) end of plateau region, (e) at the end of the test and (f) a plot of Lüders front position against time (colour online).

In Figure 4(a), the DIC screenshot shows the gauge length of the specimen at the beginning of the test. The strain map exhibits uniform blue with less than 0.1% strain across the entire area. There is no change in the strain map in the elastic region. The strain associated with the initiation of a Lüders band can be seen near the bottom of the strain map in Figure 4(b). During the plateau region of the curve, the localised deformed region expands and propagate (Figure 4(c)) until it fully covers the whole length (Figure 4(d)). After this point, there is no further propagation, but there is an approximately unform increase in strain along the length of the sample. This is shown in Figure 4(e) as the colour of the map transitions from yellow toward red. To track the propagation of the localised deformed region, the position of the Lüders front is plotted against time in Figure 4(f).

EBSD – KAM

The KAM maps obtained from the undeformed sample and the sample deformed to 3.5% reveal a pronounced distinction in the average misorientation density or individual grain's local distortion. In the sample deformed to 3.5% strain, regions corresponding to the locations of Lüders bands exhibited significantly elevated KAM values, indicating a heightened density of local misorientations. This observation is visually corroborated by the color-coded KAM maps, where distinct hues highlight regions with intensified misorientation gradients (bright yellow colour), as shown in Figure 5. The quantitative analysis further reinforces these findings, revealing a significant difference in average KAM values between the undeformed sample and the 3.5% strained sample. After quantitative analysis, the average misorientation angle per pixel of the undeformed sample is 0.23° ± 0.01°, while in the 3.5% strained sample it is 0.92° ± 0.02°.

Figure 5.

KAM map of (a) undeformed sample and (b) 3.5% strain deformed sample (colour online).

Synchrotron XRD

In situ synchrotron X-ray diffraction was performed on the sample during tensile loading. The test was interrupted, and diffraction patterns were collected at twelve different values of strain, marked as blue numbers in Figure 6. Similar to the stress-strain curve in the mechanical testing - DIC result, this stress-strain curve showed identical features: an elastic region followed by a yield drop, a plateau region during Lüders band propagation, and a work-hardening region.

Figure 6.

Stress – strain curve of the in situ synchrotron XRD interrupted test. Blues numbers represent the points at which diffraction pattern collections were performed (colour online).

Figure 7 shows diffraction rings collected from undeformed and deformed regions, respectively. The rings in Figure 7(b), collected from a region where a Lüders band had already propagated, appear smoother and more continuous. In contrast, the rings in Figure 7(a), taken from the undeformed region, are more spotty, indicative of a larger crystal mosaic size.

Figure 7.

Debye-Scherrer rings from: (a) Undeformed area (b) Lüders band propagated area (colour online).

Each diffraction pattern was processed using DAWN software for data reduction. The raw diffraction images were azimuthally integrated along the tensile direction over an angular range of 90° ± 5° (highlighted in yellow). Figure 8 presents an example of a processed diffraction pattern, where the y-axis represents intensity and the x-axis represents reciprocal space ( $Q$ ). The resulting pattern displays three distinct diffraction peaks corresponding to the {110} reflection at $Q$ ≈ 3.1 Å⁻¹, {211} at $Q$ ≈ 5.4 Å⁻¹, and {220} at $Q$ ≈ 6.2 Å⁻¹.

Figure 8.

Example of 1 dimension diffraction pattern after integration.

Figure 9 presents peak width maps for all collection points. In image 3 of Figure 9, two distinct regions can be identified: the upper region, which displays a randomly distributed colour pattern, and the lower region, which shows a more uniform colour distribution. The colour gradient ranges from red to blue, where red represents areas with the lowest peak width values and blue indicates the highest. Notably, the region with uniform peak width expands progressively, eventually encompassing the entire area and is therefore associated with the propagating Lüders bands.

Figure 9.

Area maps of {110} peak width and their correlation to stress-strain curve (colour online).

Figure 10 presents the evolution of peak width for the three main diffraction peaks as a function of applied strain. First, the range of peak widths narrowed as strain increased. Second, the average peak width increased with increasing strain. At the initial stages of deformation (0–0.5% strain), peak widths exhibited a broad distribution. As the strain increased to 1.5%, this distribution became more uniform, accompanied by a slight increase in the average peak width. This trend continued at higher strain levels (2.0–3.0%), indicating progressive strain accumulation within the material. Similar patterns were observed in the analyses of peak position and peak area. Initially, both parameters displayed a wide range of values, which narrowed as macroscopic strain increased, mirroring the trend seen in the peak width analysis. These additional data are provided in the Supplementary Material.

Figure 10.

Violin plot of peak width distribution of {110}, {211} and {220} peaks against macrostrain.

Williamson - Hall analysis

To quantify the effect of peak broadening, Williamson – Hall analyses were performed. Figure 11 presents the square of the peak width plotted against $\tan θ$ for the CeO₂ reference data, with the corresponding values of coefficients U, V and W shown in the same figure.

Figure 11.

Square of the measured X-ray diffraction peak width against $\tan θ$ .

An example of a Williamson-Hall plot is presented in Figure 12(a). The figure contains three clusters of data labelled {110}, {211} and {220}, representing the peak width data collected from the synchrotron XRD experiment, with each cluster containing 961 peak width data points from area scan. The linear trend line was created based on the linear relationship of the average $(B_{t}^{2} - B_{i}^{2}) \cos^{2} θ$ values of each cluster. Coefficients of the straight-line fit are shown in the text box, where a is the y-intercept, and b is the slope of the curve. Figure 12(b) shows the relationship between inhomogeneous strain and macrostrain. Despite the scatter in the peak width data, the plot of inhomogeneous strain shows a clear rise with increasing macrostrain.

Figure 12.

(a) A Williamson-Hall plot of sample strained to ∼ 2% and b) A plot of inhomogeneous strain against macrostrain.

Discussion

The combined use of DIC, EBSD–KAM, and synchrotron XRD in this study provides a multi-scale perspective on the deformation processes governing Lüders band behaviour in annealed FeCo-2V. At the macroscopic scale, DIC captures the strain localisation associated with band nucleation and propagation. At the microscopic scale, EBSD–KAM reveals the corresponding gradients in local misorientation that reflect underlying dislocation structures. At the crystallographic scale, synchrotron XRD quantifies lattice strain evolution and dislocation-induced peak broadening averaged over the bulk. Integrating these techniques enables a direct correlation between surface-measured strain localisation and the underlying microstructural and crystallographic mechanisms that drive plastic deformation.

The plastic deformation in the annealed FeCo-2V alloy is dominated by the formation and propagation of Lüders bands. DIC strain maps clearly revealed the initiation and progression of Lüders bands. These bands first appeared just beyond the lower yield point and propagated steadily during the stress plateau, where macroscopic strain increased with minimal change in stress. Once the band traversed the entire gauge length, the material transitioned into uniform plastic deformation, mirroring observations in both FeCo-based systems and low-carbon steels.^16,26 This propagation-dominated regime highlights a clear separation between localised plastic deformation and the macroscopic mechanical response.

The thermomechanical history of the alloy provides insight into this behaviour. The cold-rolled FeCo-2V sample was annealed at 750 °C for 3 h, allowing recrystallisation and grain growth.⁴ Notably, this annealing heat treatment exceeded the order–disorder transition temperature (∼730 °C), and the subsequent slow cooling facilitated the formation of a highly ordered structure, with an expected order parameter of 0.84.⁴ This extensive long-range order (LRO) contrasts sharply with quenched samples, which are expected to exhibit a significantly lower order parameter of 0.36. These results suggest that minor deviations from ideal ordering could give rise to short-range order (SRO) domains, which persist even in highly ordered structures and play a role in inhibiting initial dislocation motion.

The absence of interstitial carbon in FeCo-2V rules out classical Cottrell atmosphere pinning as the origin of the observed Lüders bands. Instead, the findings of Rowlands et al.⁹ on Ni-based superalloys provide a compelling alternative explanation. Their study demonstrated that even in the absence of long-range solute clustering, localised SRO of substitutional solutes can present significant barriers to dislocation motion. By analogy, the FeCo-2V samples investigated likely contain localised SRO that impede dislocation motion. Once the applied stress reaches a critical threshold, dislocations collectively overcome these barriers, resulting in macroscopic Lüders band propagation. This hypothesis aligns with the interpretation of Rowlands et al.,⁹ who attributed Portevin–Le Chatelier (PLC) effects to SRO-mediated pinning in Ni-based alloys, and offers a plausible mechanism for Lüders behaviour in FeCo-2V despite the absence of interstitial solutes.

The microstructural impact of Lüders band propagation was further investigated through EBSD-KAM analysis. A clear contrast emerged when comparing the undeformed and 3.5% strained specimens: the deformed sample exhibited significantly elevated local misorientation in regions coinciding with band propagation. These high-KAM zones reflect increased lattice distortion and are indicative of dislocation accumulation within the strain-localised bands.

EBSD–KAM analysis further elucidates the microstructural consequences of Lüders band propagation. Comparisons between undeformed and 3.5% strained specimens reveal substantial increases in local misorientation in regions swept by the Lüders band. These elevated KAM values indicate the accumulation of dislocations and enhanced lattice curvature concentrated within the band path. The spatial pattern of misorientation correlates strongly with the strain localisation recorded by DIC, reinforcing the interpretation that Lüders bands are zones of intense microscopic plastic deformation. This correspondence between misorientation and localised plasticity supports the conclusion that Lüders bands function as dislocation-rich zones during the early stages of plastic deformation. The results align closely with previous studies, including those by Han et al.¹⁵ and Wang et al.,¹⁷ who demonstrated that KAM intensity reliably tracks the strain gradients and dislocation structures at the leading and trailing edges of Lüders bands.

The Synchrotron XRD analysis provided valuable insight into the Lüders band propagation. One notable observation was the wide scatter in peak width values in the upper region of the peak width map (e.g., image 3 of Figure 9). This variability is attributed to an artefact introduced by the CdTe detector used during the experiment. As a direct-detection device, the CdTe sensor is highly sensitive to intensity fluctuations around the Debye-Scherrer rings caused by scattering from individual crystallites. When azimuthal integration is performed, these fluctuations result in noise and irregular peak broadening values. Attempts at Gaussian blurring did not fully remove this noise. In contrast, earlier datasets obtained using a Pixium detector²⁷ did not exhibit such artefacts, highlighting detector sensitivity as an important consideration when analysing peak broadening phenomena.

As deformation progressed and Lüders bands propagated across the sample, the diffraction rings became noticeably smoother in the affected regions. This change in ring morphology indicates a decrease in mosaic size and crystallographic alignment among deformed grains, resulting in more uniform azimuthal intensity distributions. Consequently, the quality of peak fitting improved, and the peak width values in these regions became more consistent. This behaviour aligns with previous findings by Wang et al.,¹⁷ who linked the smoothing of diffraction rings to lattice reorientation under plastic strain. Humphreys²⁸ similarly noted that deformation-induced mosaicity can enhance diffraction uniformity in strained polycrystalline materials.

Building upon the synchrotron XRD observations, peak broadening trends across the gauge length were found to closely correlate with the propagation of Lüders bands. As shown in the peak width maps and violin plots (Figures 9 and 10), the average peak width increased during the early stages of deformation, particularly within the strain range associated with Lüders band activity. This broadening reflects lattice distortion induced by dislocation accumulation. The steady increase in peak width up to approximately 2% macrostrain corresponds with the continued propagation of the bands. Beyond this stage, a slight increase in peak width suggests a transition into the work-hardening regime. Quantitative assessment using Williamson–Hall analysis further confirmed these trends. The results showed that inhomogeneous strain increased systematically with applied macrostrain. This pattern strongly correlates with the nucleation and progression of Lüders bands and the associated increase in dislocation density.

Although the synchrotron XRD and EBSD results indicate progressive dislocation accumulation within the propagating Lüders bands, this process does not immediately produce macroscopic work hardening. During the stress plateau, plastic flow is confined to the advancing Lüders front, while the remainder of the specimen remains elastic. The local multiplication of dislocations in the band front is continuously balanced by plastic relaxation as the front propagates into undeformed regions, resulting in an overall constant flow stress. This behaviour represents a classical form of plastic instability, in which the stress required for continued deformation remains invariant despite local strain hardening. Similar interpretations have been reported in TRIP-aided medium-Mn steels, where local work-hardening occurs within the Lüders zone without an accompanying increase in macroscopic stress due to strain localisation during band propagation.²⁹ Once the Lüders band traverses the entire gauge length, deformation becomes homogeneous, and conventional work hardening resumes as dislocation interactions accumulate throughout the material. This transition marks the end of the instability regime and the onset of uniform strain hardening observed beyond the plateau region.

These combined observations establish a coherent mechanistic picture: the Lüders band phenomenon in annealed FeCo-2V originates from localised resistance to dislocation motion, likely due to SRO domains within an otherwise highly ordered structure. Once the applied stress surpasses a critical threshold, dislocations collectively overcome these SRO-mediated barriers, leading to the abrupt onset of plastic flow that progresses in well-defined, propagating Lüders bands. This process generates strongly localised deformation and dislocation accumulation, producing significant inhomogeneous strain concentrated along the band path. The phenomenon is captured macroscopically by DIC, resolved microscopically by EBSD-KAM, and crystallographically through synchrotron XRD and Williamson–Hall analysis.

Conclusion

This study explored the deformation behaviour of an annealed FeCo-2V alloy with a specific focus on the origins and propagation of Lüders bands. Through an integrated methodology combining in situ mechanical testing with DIC, EBSD-KAM, synchrotron XRD, and Williamson–Hall analysis.

Lüders bands were found to dominate the early stages of plastic deformation, initiating just after the lower yield point and propagating across the gauge length during the stress plateau. DIC strain maps captured this behaviour clearly, identifying a sharp transition from uniform elasticity to localised plastic flow. EBSD-KAM analysis revealed that the regions traversed by the bands exhibited elevated local misorientation, indicating dislocation accumulation and lattice distortion. Synchrotron XRD measurements provided crystallographic insight, showing increased peak broadening in areas affected by Lüders bands. Quantitative analysis via Williamson–Hall plots demonstrated a progressive increase in inhomogeneous strain with applied macrostrain, which was strongly correlated with the progression of Lüders bands.

The multi-scale approach employed in this study provides an understanding of Lüders band behaviour in ordered FeCo-2V alloy. These insights and methods presented offer the electrical machine designer and lifing engineer with data to underpin mechanical aspects of electrical machine design and operation.

Supplemental Material

sj-pdf-1-mst-10.1177_02670836261443789 - Supplemental material for Inhomogeneous strain associated with Lüders band propagation in an FeCo-2V alloy

Supplemental material, sj-pdf-1-mst-10.1177_02670836261443789 for Inhomogeneous strain associated with Lüders band propagation in an FeCo-2V alloy by Sirapob Toyting, Christopher W. Harrison, Stefan Michalik, Alexis Lambourne and Howard J. Stone in Materials Science and Technology

Footnotes

Acknowledgments

The authors acknowledge Diamond Light Source for beam time on beamline I12-JEEP.

ORCID iDs

Sirapob Toyting

Christopher W. Harrison

Stefan Michalik

Alexis Lambourne

Howard J. Stone

Author contribution(s)

Sirapob Toyting: Conceptualization; Formal analysis; Investigation; Project administration; Visualization; Writing – original draft.

Christopher W. Harrison: Investigation; Methodology; Supervision; Writing – review & editing.

Stefan Michalik: Investigation; Methodology.

Alexis Lambourne: Formal analysis; Resources; Writing – review & editing.

Howard J. Stone: Conceptualization; Formal analysis; Project administration; Supervision; Writing – review & editing.

Funding

This work was financially supported by Royal Thai government scholarship and Rolls-Royce plc.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data will be made available on request to the corresponding authors.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the authors used ChatGPT in order to improve grammar. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

Supplemental material

Supplemental material for this article is available online.

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