High-accuracy fatigue life prediction and early fracture warning for ferromagnetic metals via spin correlation amplification

Abstract

Sudden fatigue failure of metallic structures triggers thousands of safety accidents annually, causing tens of thousands of casualties and economic losses approaching $100 billion. Traditional detection techniques face challenges in capturing early fatigue damage features like subnanometer atomic displacements and localized metal bond ruptures. Based on the synchronized weakening mechanism between binding forces and exchange interactions during fatigue processes in ferromagnetic metals, this study obtained high-sensitivity fatigue observations by exciting the quantum spin correlation amplification effect via an external magnetic field. We conducted fatigue tests on 193 ferromagnetic metal samples over 3700 cumulative hours, establishing a mapping relationship between fatigue-induced magnetic flux changes and the degree of binding force weakening. This work integrated macroscopic fatigue life prediction with ferromagnetic material microstructural parameters, achieving a prediction accuracy with R² > 0.9 and providing a reliable prefracture warning, potentially mitigating fatigue fractures in large-scale engineering structures and reducing the associated economic losses.

By employing a quantum spin correlation amplification method to magnify microscopic atomic changes into macroscopic magnetic signals, this study enables high-accuracy fatigue life prediction and early fracture warnings for large-scale structures.

Introduction

Despite being phenomena that have been studied for more than 150 years, fatigue fractures remain a predominant cause of metallic structural failures^1–4, with real-world structural failures often occurring suddenly and without prior warnings. Fatigue accidents involving vital public structures, such as airplanes, ships, trains, bridges, elevators, and buildings cause thousands of incidents and tens of thousands of casualties annually (Methods, Supplementary Fig. 1).

Extensive research efforts have been devoted to fatigue analysis, encompassing S-N curve characterizations, microscopic damage evolution mechanisms, multiphysical field monitoring methods, and numerical simulations, among other works.

The stress‒life (S-N) curve serves as a critical tool in structural fatigue analyses by quantitatively characterizing the relationship between the stress amplitude and fatigue life of a material^5,6, providing an essential basis for implementing fatigue-resistant designs. This curve enables rough structural life predictions and material performance evaluations. As the foundation of fatigue testing standards, its characteristic features include its slope and fatigue limit. Despite its low prediction accuracy (errors exceeding an order of magnitude), the S-N curve remains an indispensable core methodology for conducting fatigue life assessments because of its clear engineering significance and extensive experimental data support.

Microscopic studies of fatigue phenomena have established a multiscale and multidimensional observation framework, providing a solid foundation for revealing the underlying micromechanisms of material damage. This framework extensively employs advanced characterization techniques, such as scanning electron microscopy (SEM), transmission electron microscopy (TEM), high-resolution digital image correlation (HR-DIC), and three-dimensional orientation imaging-based microscopy^7,8. These methods enable the capture of nanoscale image details and provide direct visualizations of the initiation and evolution processes of fatigue damage, even enabling the quantitative prediction of macroscopic material lifetime through the localized strain field from a single cycle^3,4,9. These techniques have driven fatigue research from qualitative descriptions to quantitative characterizations.

Physical field detection techniques provide nondestructive, real-time, and efficient approaches for characterizing fatigue damage by monitoring the physical signals generated during the fatigue process, such as acoustic signals (detected via acoustic emission technology and ultrasonic testing)^10,11, magnetic signals (detected via magnetic crack-based pressure sensors)¹², and thermal signals (detected via infrared thermography)^13–15. These methods, which are based on the physical field variations induced by material damage evolution trends, can reflect the internal damage states of materials from different perspectives.

Numerical simulation enables quantitative fatigue prediction to be performed under complex working conditions through physical models. The crystal plasticity-based finite-element method (CPFEM) reveals the influences of grain orientation and dislocation mechanisms on the initiation and propagation of cracks at the microlevel¹⁶; continuum damage mechanics (CDM) predicts crack growth and fatigue life under multifield coupling effects using damage variables^17,18. Numerical simulations, which integrate multiscale modeling and advanced constitutive theories, enable quantitative prediction and provide theoretical support for material safety assessments.

However, these studies face certain inherent limitations in specific application scenarios: microdamage sensitivity at the macro scale, lifespan prediction accuracy issues, and fracture warning concerns.

The fundamental challenge encountered in fatigue detection tasks is the low sensitivity of macroresponses to structural microdamage, primarily because the various dislocation structures formed during fatigue account for only a small proportion of the affected area. For example, persistent slip bands (PSBs) constitute only approximately 9.2% of the fatigue-affected zone in some fatigue tests of single copper crystals^19,20. The resulting displacement increments have negligible effects on the overall strain. This atomic-scale damage evolution process cannot be captured by the conventional measurement technologies. Although they can offer displacement sensitivities as high as 7–9 nm and spatial resolutions of 20 nm at the microscale, advanced methods, such as digital image correlation (DIC)⁹ struggle to analyze the evolution of internal damage within materials. More critically, the stiffness of a material decreases by only marginally before fracture-critical states are reached²¹, causing almost no detectable changes in its macroscopic mechanical properties. This characteristic—significant microscopic damage but weak macroscopic responses—renders displacement-based fatigue detection methods inherently limited in engineering practice. The conventional methods can identify damage only when it accumulates into microcracks, which is one of the key factors why sudden fatigue fracture-based accidents are difficult to prevent²².

Furthermore, significant uncertainty exists when predicting the fatigue lives of metallic materials (with the induced errors often exceeding one order of magnitude), primarily stemming from the complex coupling of multiple factors. At the material level, slight fluctuations in smelting impurities (e.g., S and P), the random distributions of microdefects (gas pores and inclusions), and grain boundary characteristic variations lead to intrinsic property dispersion²³. Minor surface roughness variations, changes in the performance of welded joints, and fluctuations in the residual stress field can significantly affect fatigue lives during the manufacturing process²⁴. Overall, these factors pose considerable challenges for predicting the fatigue performance of structural components.

More critically, the field of fatigue monitoring lacks an effective failure warning mechanism. This technological gap directly leads to catastrophic and sudden brittle fracture accidents. The absence of such systems creates severe safety hazards.

Between 1900 and 2024, the shipping industry experienced at least 102 major fatigue-related accidents, resulting in 25,000 fatalities; the aviation industry had at least 2094 major fatigue-based accidents, causing approximately 13,000 fatalities; and the bridge engineering sector had at least 172 major fatigue-based accidents, resulting in 3600 fatalities (Methods).

Beyond ships, aviation and bridges, objects, such as elevators, railways, and various pieces of large-scale load-bearing equipment harbor substantial latent fatigue damage. Fatigue-related damage has become a significant contributor to thousands of safety incidents worldwide each year, resulting in annual economic losses approaching $100 billion (Methods).

Here, we established the correlation between dislocation motion and atomic spacing evolution under cyclic loading, with a fatigue evaluation principle bridging atomic-scale damage to macroscopic signal sensing proposed. Starting from the wave function, the synchronized weakening mechanism between binding forces and exchange interactions as atomic spacing increases in ferromagnetic materials is derived. A non-contact method based on magnetic field excitation is presented: via quantum spin correlation amplification, subtle changes in binding forces and exchange interactions are converted into detectable macroscopic magnetic signals. Leveraging this mechanism, we conducted fatigue tests on 193 ferromagnetic metal samples with a cumulative duration of 3700 hours, establishing a mapping relationship between fatigue-induced magnetic flux changes and the degree of binding force weakening. This work organically integrates macroscopic fatigue life prediction with the microscopic structural parameters and phase transition characteristics of ferromagnetic materials, establishing intrinsic correlations across the microscopic, mesoscopic and macroscopic scales to enable accurate macroscopic fatigue life prediction, with a predictive accuracy of R² > 0.9. Additionally, all samples exhibit clear and detectable pre-fracture warning signals before failure. This advancement may lay a solid foundation for the effective prevention and control of fatigue fractures in large-scale engineering structures and the reduction of unnecessary replacement of fatigue-prone components.

Results

The correlation mechanism between atomic spacing evolution and material fatigue damage

The fatigue behavior of materials is microscopically characterized by various dislocation evolutions, which ultimately lead to macroscopic fracture. The core of this process lies in the movement, interaction, and structural reconstruction of dislocations⁴, whose specific manifestations are closely related to the material’s crystal structure and loading conditions.

In the initial stage, cyclic loading activates dislocation sources within the material, prompting dislocations to slip back and forth along specific slip planes. Due to microstructural inhomogeneities, reverse slip cannot fully offset forward slip, resulting in partial slip irreversibility. Dislocations interact through mechanisms, such as coplanar stacking and heterogeneous annihilation, forming simple dislocation tangles or dipoles. As the number of cycles increases, dislocation motion enters a stable saturated stage, leading to the formation of dislocation substructures. In face-centered cubic (FCC) metals, dislocation tangles can develop into vein structures composed of edge dislocation dipoles; when the loading amplitude exceeds a critical threshold, local collapse of these vein structures occurs, generating persistent slip bands (PSBs) parallel to the slip planes²⁵. In contrast, in body-centered cubic (BCC) metals, screw dislocations exhibit poor mobility and asymmetric slip planes, resulting in earlier onset of irreversibility²⁶.

The evolution of these dislocation structures is a manifestation of cumulative changes in atomic spacing. Within the core region of lattice deformation, atoms deviate significantly from their ideal lattice positions, and atomic spacing is either compressed or stretched, forming a high-strain zone²⁷. Plastic strain is highly concentrated within shear bands, and this irreversible slip means that atoms cannot fully return to their initial positions after each cycle, leading to cumulative minor changes in atomic spacing.

In the stage of fatigue microcrack nucleation and initiation, the irreversibility of plastic strain becomes the key factor for crack initiation²⁸. During this stage, localized changes in atomic spacing gradually propagate from the interior of the material to the surface, eventually forming observable morphological features. Irreversible dislocation activity is locally concentrated in near-surface regions, resulting in the formation of extrusion and intrusion²⁹—one of the most intuitive manifestations of changes in atomic spacing: in extrusion regions, a large number of atoms migrate to the surface, significantly increasing the atomic spacing; in intrusion regions, atoms are displaced from their original positions, forming sharp microscale notch-like morphologies that act as free surfaces, thereby becoming the primary sites for fatigue crack initiation³⁰.

After microcrack initiation, it enters a stable propagation stage under cyclic loading. Cracks typically propagate in Stage I mode along the maximum shear stress planes (the direction of slip bands)³¹, and their propagation path is strongly influenced by local crystal orientation. When the crack size grows to a certain extent or encounters obstacles, such as grain boundaries, it gradually deflects and transitions to propagating perpendicular to the principal stress axis. At this point, the crack growth rate accelerates until it ultimately causes macroscopic fracture of the material. During crack propagation, drastic changes in atomic spacing are mainly concentrated in the crack tip high-strain zone. Under tensile loading, the atomic spacing at the crack tip increases abruptly; when it exceeds the maximum capacity of the bonding forces, atomic bonds break. This bond-breaking process constitutes the microscopic mechanism underlying crack propagation. As the crack advances, atomic bonds are sequentially broken, and atoms on both sides separate to form new free surfaces, which essentially represents the transition of atomic spacing from a finite value to an infinite state of separation.

Therefore, fatigue damage in materials is a multiscale evolutionary (Supplementary Fig. 3). From a microscopic perspective, this process is located within the damage concentration zone. This region consists of multiple atomic layers within the fatigue-affected zone and contains a series of microscopic defect clusters that influence the interlamellar atomic spacing (here, interlamellar refers to the collective set of all the atoms contained within a three-dimensional irregular thin section that is perpendicular to the direction of structural loading).

According to classical potential energy curve theory, the interatomic bonding force varies with atomic spacing. When material fatigue accumulation leads to an increase in interlamellar atomic spacing, the corresponding bonding force weakens. This attenuation is closely associated with dislocation activity¹⁹ (interatomic cohesion; Fig. 1a). A localized stress concentration reduces the interlamellar bonding force strength, facilitating dislocation nucleation and slip processes³². Throughout this process, damage to the interlamellar atomic bonding force continuously accumulates. When this force falls below the stress induced by the applied load, material fracture occurs. Hence, changes in the interatomic bonding force can serve as indicators for characterizing the fatigue states of materials.

Fig. 1. Taking PSB as an example, schematic of the microscopic development process of fatigue and the synchronous relationship between interatomic binding forces and exchange interactions.

a PSBs and interatomic dislocations of the fatigue section and weakening of the interatomic binding force caused by the PSBs in materials under cyclic loading. b The relationship between the interatomic binding forces and exchange interactions varies with the change in the atomic spacing. $R_0$, $R_{\mathrm{t}0}$; $R_i$, $R_{\mathrm{t}i}$; and $R_{\mathrm{f}}$ and $R_{\mathrm{tf}}$ represent the atomic spacings at time points $T_0$, $T_i$, and $T_{\mathrm{f}}$, respectively, under cyclic loading. The exchange interactions at points a and c represent the exchange interactions at the beginning of loading and at the maximum loading stress, respectively. Points b and d are the peak exchange interaction points that occur during the loading process. As the loading time increases, the number of atomic dislocations gradually increases, the atomic spacing changes, and the numbers of exchange interaction points a and c gradually shift. The peak change points b and d, however, remain stationary. The binding force $F_{\mathrm{C}}$ and the force resulting from the exchange interaction $F_{\mathrm{m}}$ increase but then decrease. The degree of breakage of the atomic bonds in the cross-section continues to increase, and fracture occurs when the overall binding force of the cross-section becomes less than the applied force. c. Electron clouds, interatomic bindings, and exchange interactions between adjacent iron atoms. d. Relationships between the magnetic potential energy, the exchange interactions of electrons, and the atomic spacings of iron atoms. e. Relationships between the binding energy, binding forces, and the atomic spacings of iron atoms.

While quantifying interatomic binding forces under static conditions is feasible because stable atomic positions and electron cloud distributions enable techniques, such as TEM³³, atomic force microscopy (AFM)³⁴, and X-ray diffraction² to infer atomic arrangements and calculate binding forces via model theory, dynamic conditions pose significant challenges. These challenges are particularly pronounced when complex transient processes, such as dynamic dislocation annihilation, reorganization and reversible slip are involved. Under cyclic loading, the periodic changes exhibited by the atomic positions and electron cloud distributions complicate the task of accurately measuring the instantaneous binding force.

The magneto-mechanical coupling mechanism in fatigue processes: synergistic degradation of interatomic binding forces and exchange interactions

Ferromagnetic metals (iron, cobalt, nickel, and their alloys), as the most widely used metallic materials, fundamentally derive their magnetic properties from electron spin-based magnetic moments³⁵. Electron spin, which is an intrinsic angular momentum property coupled with orbital motion, induces degeneracy splitting of the electronic energy levels and magnetic level discretization through spin–orbit coupling effects, which fundamentally determine the magnetic characteristics of the target material³⁶.

Both interatomic binding forces and exchange interactions are intrinsically linked to electronic states and manifest within identical electron cloud configurations³⁶. When the interlamellar atomic spacing changes, the electronic states correspondingly change. Theoretical derivations based on iron atomic wave functions reveal that within minute ranges of variation in the atomic spacing, the average interatomic binding force (${\mathrm{\Delta }F}_{\mathrm{C}}$) and exchange interaction ($\mathrm{\Delta }{F}_{\mathrm{m}}$) in the material layers can be simplified to approximately linear relationships with the changes in the average atomic spacing ($\mathrm{\Delta }\overline{R}$): ${\mathrm{\Delta }F}_{\mathrm{C}}=K_{\mathrm{C}}\mathrm{\Delta }\overline{R}$ (Fig. 1e and Methods) and $\mathrm{\Delta }{F}_{\mathrm{m}}=K_{\mathrm{m}}\mathrm{\Delta }\overline{R}$ (Fig. 1d and Methods). This relationship reflects the synchronized degradation between interatomic binding forces and exchange interactions during fatigue processes: $\mathrm{\Delta }{F}_{\mathrm{C}}=K\mathrm{\Delta }{F}_{\mathrm{m}}$ (Fig. 1c and Methods). Crucially, the weakening of the interatomic binding force directly corresponds to diminished exchange interactions.

Building on this mechanism, a theoretical analysis suggests that the detection of exchange interaction variations could indirectly characterize binding force weakening. However, exchange interactions themselves cannot be directly measured and require indirect determination through observed magnetic moment alterations. The critical challenge lies in the extremely weak exchange interaction changes induced by subnanometer-scale binding force degradation; thus, this indirect detection method still faces significant challenges in practical observation cases.

Macroscopic magnetic mapping of microscopic fatigue signals: spin magnetic moment correlation amplification under external magnetic excitation

To effectively detect weak interatomic exchange interactions, we propose a spin magnetic moment correlation amplification method based on external magnetic field excitation (Fig. 2a). Its core mechanism is as follows. During the process of fatigue, the interlamellar atomic spacing undergoes gradual and irreversible changes. This change not only affects the bonding strength of the material layer but also simultaneously affects its interlamellar exchange interactions, with both phenomena being strongly correlated and synchronous.

Fig. 2. Schematic diagram of the trends of the magnetic field strength variations and cumulative amplification effects induced during the loading process based on the spin magnetic moment correlation amplification mechanism.

a Schematic diagram of the spin magnetic moment correlation amplification mechanism. b Comparison among the magnetic strength and strain, MagDrift, and displacement amplitude changes induced under cyclic loading. The median change rate of the magnetic strength was 69.22 times greater than the median change rate of the strain, whereas the median change from the beginning to the starting point of failure in MagDrift was 12.64 times greater than that in displacement. c Displacement amplitudes and MagDrift changes produced under typical fatigue stress loads. d Magnetic strength and strain change rates observed under typical fatigue stress loads. e Relationships between the change rates of the magnetic strength and strain for five ferromagnetic metals under static loading.

When an external magnetic field is applied, changes in the interlamellar exchange interactions induce oriented variations in the aggregate spin magnetic moment at the microscopic scale, which in turn alters the pinning states of magnetic domains at the mesoscopic scale.

As the external excitation-induced magnetic field penetrates each fatigue-affected damage (dislocation) layer, it undergoes irreversible deflection because of the spin‒spin interactions between atoms. This deflection accumulates progressively as the magnetic field traverses more fatigued dislocation layers. Essentially, this process functions as a spin correlation amplification method, manifesting as a multiplicative effect (Methods). Ultimately, the deflection characteristics that accumulate after the magnetic field passes through all the fatigue-induced dislocation layers are encoded within the magnetic signal, providing a basis for evaluating the fatigue state of the material (Fig. 1).

During the progressive development of fatigue, atoms in spin magnetic moment correlation states may experience microscopic changes in their electron cloud distributions and spin states because of the interference caused by various dislocation structures. These disturbances propagate to neighboring atoms through the synchronized weakening of interlamellar binding forces and interlamellar exchange interactions. Within the elastic limit of the microscopic crystal structure, once the stress is released, the initial positions of the atoms can be recovered, and the interlamellar binding forces and interlamellar exchanges interaction can also be restored accordingly. However, if the elastic limit is exceeded, some atoms will be unable to fully return due to plastic deformations, such as irreversible dislocation activities, leading to irreversible changes in the interatomic binding forces, exchange interactions, and magnetic field intensity at the surface. Under cyclic loading, some atoms cannot return to their initial states after each fatigue cycle because of dislocations or bond fractures, causing differences between the magnetic properties observed at the cycle endpoint and the initial state, which are macroscopically manifested primarily through magnetic field-based peak-to-peak variations. Consequently, we define the fatigue-induced changes in magnetic field-based peak-to-peak values as MagDrift. When ferromagnetic materials are subjected to external magnetic fields, these microscopic disturbances accumulate and increase along the direction of the magnetic field (Fig. 2a). This effect originates from the synergistic interaction between exchange interactions and the external magnetic field, thereby significantly amplifying the localized weakening of the interlamellar binding forces and interlamellar exchange interactions caused by fatigue. Such MagDrift variations directly reflect the evolution trends of interlamellar interatomic binding forces and interlamellar exchange interactions during fatigue. Therefore, MagDrift can characterize the cumulative fatigue damage suffered by materials.

We conducted fatigue tests on five types of ferromagnetic alloys to validate this hierarchical transmission process and its amplification effects. The magnetic signal changes exhibited by the spin magnetic moment system exhibited exponential amplification throughout the spin correlation conduction chain. The experimentally observed changes in the median magnetic field intensity were 69.22 times greater than those in the conventional mechanical parameters (Fig. 2b). For the first time, we obtained clear macroscopic observations of dynamic magnetic effects (interlamellar binding forces) throughout the entire long-term fatigue development process, revealing significantly accelerated changes preceding fracture (Fig. 2c, d).

According to the experimental results, we discovered a significant linear correlation between the average MagDrift change rate ($k_{\mathrm{av}}$) of the materials and the number of load cycles (Fig. 3 and Methods). The results demonstrate that the fatigue lives of materials are negatively correlated with their $k_{\mathrm{av}}$ values: longer fatigue lives correspond to smaller $k_{\mathrm{av}}$ values. These findings confirm that the average MagDrift change rate ($k_{\mathrm{av}}$) can be used to effectively characterize the macroscopic variation patterns of interlamellar binding forces during fatigue processes.

Fig. 3. Offset MagDrift and normalization curves of ferromagnetic materials.

a Offset MagDrift and normalized curves of a soft magnetic iron sample. The initial MagDrift value is offset by a certain amount to display the data. $N_{\mathrm{f}}$ represents the number of fatigue fracture cycles. Under different stress loads, the MagDrift levels of all soft magnetic iron samples tended to increase. The steeper $k_{\mathrm{av}}$ (the average change rate of MagDrift, where we adopted the slope of the linear fitting line as its representative value) was, the fewer cycles were observed. Notably, $k_{\mathrm{av}}$ represents the change rate of the cross-sectional interlamellar binding force (exchange interaction) under fatigue loading. b Offset MagDrift and normalized curves of 45 steel. c Offset MagDrift and normalized curves of Q235 steel. d Offset MagDrift and normalized curves of bridge stay cable steel. e Offset MagDrift and normalized curves of pure nickel.

Analysis of the statistical effects of macroscopic magnetic flux and interlamellar binding force changes

To analyze the macroscopic surface magnetic field variation responses and force–magnetism coupling effects of ferromagnetic materials under external magnetic fields after they underwent spin magnetic moment correlation amplification, we performed static tensile loading tests on five typical ferromagnetic materials (soft magnetic iron, Q235 steel, 45 steel, bridge stay cable steel, and pure nickel) under applied magnetic fields. These experiments were aimed at observing the magnetic induction intensity changes induced under external forces, with the specimen dimensions and experimental procedures detailed in the Methods section. The experimental results revealed that, except for pure nickel and bridge stay cable steel, all the other materials exhibited magnetic intensity variation rates that first increased but then decreased with strain (Fig. 2e). This phenomenon may be related to the coordinated magnetic domain arrangement state changes induced by stress through spin moment interactions. In the initial stage, external forces caused the magnetic moments to cooperatively alter the magnetic domain arrangements, promoting domain wall movement that aligned more domains along easy magnetization directions. This cooperative effect manifested macroscopically as a rapid magnetic induction intensity increase. At this stage, the external forces optimized the interlamellar exchange interactions by adjusting the interlamellar atomic spacing without damaging the integrity of the material, thereby achieving enhanced parallel domain alignment. When the dynamic equilibrium between the degree of exchange interaction-driven domain alignment and the magnetic anisotropy energy was reached, the magnetic induction intensity peaked. When the stress exceeded the critical value, plastic deformation-induced lattice defects (e.g., dislocations and vacancies) began to accumulate. These defects hindered domain wall motion through pinning effects and disrupted the original domain coordination. Simultaneously, these lattice defects significantly reduced the exchange integrals. Under the combined action of these two mechanisms, the dynamic equilibrium collapsed, resulting in a sustained decrease in the magnetic intensity variations.

Pure nickel, being a negative magnetostrictive material (the material elongates during magnetization), demonstrated decreasing magnetic intensity variation rates under the applied tensile stress because compressive stress impedes magnetization. Notably, the bridge stay cable steel showed similar trends to those of pure nickel, which may be attributed to its high carbon content inducing smaller or negative magnetostriction coefficients (analogous to pure nickel), where tensile stresses instead weakened the magnetic moment ordering scheme.

Statistical analysis of the macroscopic magnetic responses to fatigue: evolution of fatigue damage in ferromagnetic materials under fatigue loading (interlamellar binding force degradation)

To investigate the correlation between fatigue damage and magnetic responses in ferromagnetic metals, we conducted axial tensile fatigue tests on five typical ferromagnetic materials under an external magnetic field. The experimental results revealed that the evolution of magnetic signals during fatigue damage can be divided into three stages, which correspond to distinct characteristics of dislocation evolution. In the initial stage, the rapid change in magnetic signals reflects the formation of dislocation structures and the accumulation of initial damage. This is followed by a stage of steady magnetic signal growth, marking a prolonged accumulation phase dominated by the nucleation of micro-cracks. Finally, an accelerated change in the magnetic signal directly indicates the propagation of macroscopic cracks leading to fracture (Methods). Except for pure nickel and bridge stay cable steel, the overall MagDrift curves of all the materials tended to increase (Fig. 3). This phenomenon may be attributed to the synergistic effects of exchange interactions and magnetic domain rearrangement. Since cyclic loading stress generally operates within the elastic range, the atomic spacing gradually approaches the optimal values under stress adjustments. The exchange interaction energy drives the parallel alignment of magnetic moments, promoting the preferential orientation of magnetic domains along easy magnetization directions, which manifests macroscopically as an upwards MagDrift shift.

With respect to soft magnetic iron, the MagDrift value abruptly decreased immediately before fracturing occurred under high-stress conditions but displayed a sudden increase under low-stress conditions (Fig. 3a). This phenomenon is likely related to the stress-level-induced directional reversal of MagDrift. Under low-cycle fatigue, dislocations impede coordinated magnetic moment movements through domain wall pinning. Simultaneously, the increased lattice distortion within the target material causes a significant reduction in the exchange integrals, leading to disordered magnetic moment arrangements and a consequent sharp decline in MagDrift. However, under high-cycle fatigue, damage accumulates more slowly than it does under low-cycle fatigue. Until destruction occurs, the optimization effect of stress-induced domain rearrangement is always stronger than the weak negative impact caused by damage. In the final stage, the appearance and propagation of microcracks may have led to drastic changes in the stress field, which further promoted the optimized arrangement of the magnetic domains, resulting in an accelerated increase in MagDrift. Both 45 steel and Q235 steel exhibited sudden MagDrift increases immediately before fracturing (Fig. 3b, c, respectively). These two types of steel were similar to soft magnetic iron and had a positive magnetostriction coefficient. In the final failure stage, the intensification of microscopic damage (such as microcracks) may have triggered more severe local stress concentration and magnetic domain reconstruction effects, further optimizing the domain arrangement scheme and leading to an accelerated increase in MagDrift.

In bridge stay cable steel, the high carbon content may have resulted in a negative magnetostriction coefficient (λ < 0), meaning that the material shrunk along the magnetization direction when magnetized. The cementite phase strongly nailed the domain walls, making initial domain movement difficult. Throughout the entire fatigue process, the negative effects of domain rearrangement caused by cyclic tensile stress and the cumulative negative effects of fatigue damage combined, resulting in a slight downwards trend in MagDrift. In the final stage of destruction, the cumulative damage effect rapidly increased (with many lattice defects generated), which, together with the sustained negative magnetostriction effect, lead to an accelerated decrease in MagDrift (Fig. 3d).

Pure nickel, being a negative magnetostrictive material, demonstrated reduced magnetization intensity under tensile stress. The accumulation of cyclic stress gradually deviated the magnetic domains from their easy magnetization directions, resulting in continuous magnetic signal attenuation. However, during the final crack formation stage, rapid domain rearrangement occurred in the high-stress zone at the crack tips (e.g., magnetic charge concentration along the crack directions), causing a temporary magnetic signal enhancement. Therefore, pure nickel exhibited an overall decreasing MagDrift curve (Fig. 3e) with a rapid upwards surge immediately before fracture.

Fatigue life prediction model

Microscopic HR-DIC and TEM measurements reveal that early dislocation changes induced during fatigue affect the fracture location. Taking a PSB as an example, its slip marks are almost identical to the final fracture location. Significant correlations are present between the mechanical properties of materials, such as yield strength, ultimate tensile strength, and fatigue strength, and the early local slip of a PSB^3,4. Therefore, early changes in the dislocation structure can be used for fatigue life assessment purposes.

However, subject to uncertainties caused by material doping, structural design flaws, and load fluctuations, the traditional fatigue life prediction models exhibit significant scatter. An analysis of the MagDrift variations induced during fatigue tests of different ferromagnetic metals under cyclic stress loading revealed that larger MagDrift slopes correlate with shorter fatigue lives. Prior to fracture failure, the MagDrift slopes showed near-linear relationships with the cycle counts across all the samples under varying cyclic loading stresses, following similar fatigue evolution patterns. This suggests that MagDrift slopes inherently incorporate load fluctuation uncertainties, whereas material doping and structural defects are related solely to intrinsic material properties. Consequently, MagDrift slopes comprehensively reflect all the aforementioned uncertainties, with material fatigue damage progression rates being equivalent to MagDrift slope variations.

Referring to the S-N curve, we plotted the logarithmic MagDrift slope and cyclic loading stress values for each sample against the number of fracture loading cycles as scatter plots (Fig. 4a, c). The correlation between the MagDrift slope and the number of cycles far exceeds that of the S-N curve.

Fig. 4. Analysis chart of the experimental results.

a Comparison between the LogM-N and LogS-N curves produced for four ferromagnetic iron alloys. The average slope of the M-N fitting lines of the four ferromagnetic metals, -1.54, was used as the new slope ($k_{\mathrm{c}}$). The LogM-N fitting degrees were all above 0.9, far greater than the LogS-N values. b Relationship between the intercept $C$ of the linear fitting line of the LogM-N curve and the iron content. The higher the iron content is, the greater the intercept. c Comparisons between the M-N and S-N curves produced for the soft magnetic iron samples and comparisons between the LogM-N and LogS-N values of the soft magnetic iron samples. Compared with the S-N curve, the M-N curve has a much greater degree of fit. d The estimated lifespans $N_{\mathrm{e}}$ of soft magnetic iron under different early cyclic loading conditions (correlated with the actual lifespan $N_{\mathrm{f}}$). The correlation coefficient between the estimated lifespan and the actual lifespan under 9% loading already reached 0.907, which was significantly greater than the correlation coefficient between the maximum stress ${\sigma }_{\max }$ and the actual lifespan $N_{\mathrm{f}}$, which was −0.73. e Principle and effectiveness of the early warning scheme based on MagDrift. $S_1$ and $S_2$ are constants obtained through experimentation; for soft magnetic iron, the values of $S_1$ and $S_2$ were 4.8 and 1.2, respectively. $A_{\mathrm{med}}$ is the median of window 2, $A_{\max }$ is the maximum value of window 2, $A_{\min }$ is the minimum value of window 2, and $B_{\mathrm{med}}$ is the median of window 1. $N_{\mathrm{R}}$ is the remaining number of cycles until a fatigue fracture. f Prediction and warning diagram (taking a propeller as an example).

Starting from the wave function and by referring to various fatigue-related empirical formulas and our experimental data, we established the relationship between the fatigue life $N_{\mathrm{f}}$ and the MagDrift slope $k_{\mathrm{av}}$ (Methods):

$$\log N_{\mathrm{f}}=\frac{1}{k_{\mathrm{c}}}\left(\log ∣k_{\mathrm{av}}∣+C\right)$$ 1

The model indicates that the fatigue life $N_{\mathrm{f}}$ is linearly related to the average magnetic flux change rate $k_{\mathrm{av}}$ on a logarithmic scale. Here, $k_{\mathrm{c}}$ is the slope of the MagDrift rate-life (M-N) curve, reflecting the nonlinear accumulation characteristic of fatigue damage, and $C$ is the intercept of the M‒N curve, which is related to the intrinsic properties of the target material.

We calculated the linear fitting slopes (Methods) of the relationship between $k_{\mathrm{av}}$ and $N_{\mathrm{f}}$ for four types of ferromagnetic materials and used the average slope of these four materials (−1.54) as the correction slope (Fig. 4a). This value is closely aligned with the numerical relationship reported by Néel between the exchange integral and the interatomic distances of electrons in α-Fe (Fig. 4b)³⁷. This close agreement prompts speculation that there may be a physical correlation between the two metrics. For a metallic material, the strength of its metal bonds is closely related to the degree of overlap between the electron clouds of adjacent atoms, which can be quantified by the crystal orbital overlap population (COOP)³⁸. The higher the COOP value is, the stronger the bonding interactions between atoms, corresponding to stronger chemical bonds^39,40.

Drawing from the existing experimental data and an analysis of the results, we found that the smaller the value of $k_{\mathrm{c}}$ was, the longer the fatigue life of the corresponding material at the macroscopic level. This trend is similar to the physical meaning of the electron cloud overlap strength described by the COOP. In light of this correlation, we speculate that $k_{\mathrm{c}}$ may serve as a quantitative experimental indicator of COOP parameters. This indirectly reflects the impact of electron cloud overlap on material fatigue performance.

An electron probe analysis (Supplementary Figs. 4–8) of all the samples revealed a significant correlation between the intercept $C$ in the prediction formula and the iron content (Fig. 4b). A higher iron content corresponds to greater $C$ values, suggesting that $C$ may quantitatively reflect iron concentrations.

Utilizing the above formula, we predicted the final fatigue life using early MagDrift data (5% ~ 60% fatigue life) obtained from early cyclic loading. When 10% MagDrift data were used for prediction, the correlation between the estimated fatigue life and the final fatigue life reached 0.91 (Fig. 4d). With increasing data, the MagDrift was constantly revised, and the correlation between its estimated fatigue life and final life also increased and approached 1; the relative error also continuously decreased (Supplementary Fig. 9), further indicating that the fatigue life prediction model has excellent fatigue prediction capabilities.

Fatigue failure warning

Leveraging the high sensitivity of spin magnetic moment correlation amplification to the evolution of fatigue, we accurately captured the critical signals during the material fatigue accumulation process. All the samples (193 specimens of five ferromagnetic materials) exhibited clear and detectable prefracture warning signals before failing during 3700 h of testing (the criteria for judgment are described in the Methods section).

MagDrift exhibited significant long-term variations prior to material fractures. Leveraging this characteristic, we propose a two-stage early fatigue failure warning system. The first stage employs the fatigue life prediction formula, which estimates the fatigue life $N_{\mathrm{f}}$ by processing collected data and calculating $k_{\mathrm{av}}$ values in real time. When 90% of the predicted life is reached, a preliminary alarm is triggered, indicating that the material still has functionality but is approaching a dangerous state. The core evaluation indicator for this stage is prediction accuracy. The second stage triggers an extreme alert at the actual fatigue fracture point (Fig. 3), corresponding to a residual fatigue life between 0.04% ($N_{\mathrm{f}}$ = 7400 K; $N_{\mathrm{R}}$ = 2960) and 7.54% ($N_{\mathrm{f}}$ = 472 K; $N_{\mathrm{R}}$ = 38,540), where the Stage-2 system provides reliable early warning for all samples. Through this system, the prefracture warning signals of the samples were successfully captured (Fig. 4e, f), marking the first experimental realization of effective early warnings for material fatigue fractures (Supplementary Movie 1).

Stage-1, in our two-stage early fatigue failure warning system, serves as a preparatory early warning. Its core role is to provide a buffer period for maintenance preparation, addressing the practical challenge of rushed maintenance responses for critical engineering components (e.g., steel bridge welds, ship propellers). It should be clarified that Stage-1’s instability refers to initial prediction variability arising from limited early-stage data, not invalidity. With sufficient data, these fluctuations are effectively mitigated via real-time MagDrift updates, ensuring engineering applicability.

Stage-2, in our two-stage early fatigue failure warning system, serves as a critical fracture alarm, triggered by abrupt MagDrift changes preceding fracture. As the system’s core safety barrier, its primary function is to initiate emergency shutdowns in practical applications. After triggering, the material can safely withstand several thousand additional load cycles, providing ample time to terminate operations safely and avoid catastrophic fracture.

Discussion

A magnetically excited spin correlation amplification method is proposed in this study, pioneering a transformative approach for evaluating fatigue in ferromagnetic metals. The core breakthrough lies in leveraging spin correlation conduction mechanisms under external magnetic excitation to amplify localized microstructural changes into observable macroscopic magnetic responses (Fig. 2c), achieving macroscopic interlamellar binding force fatigue degradation detection.

The experimental investigation yielded three groundbreaking advancements. (1) The correlation of the spin magnetic moment is extremely sensitive to the weakening of the interlamellar binding force. Through multilayer conduction amplification, the amplification factor of the atomic bond displacement reached 86.72 (Fig. 2c, d), resulting in macroscopic observations of dynamic magnetic effects (interlamellar binding force evolution trends) throughout the fatigue process for the first time. (2) A multiparameter fatigue life prediction model was proposed; it achieved a notable improvement in prediction accuracy (R² > 0.9) and successfully solved the accuracy bottleneck faced by the traditional S-N curve-based methods. (3) The two-stage early fatigue failure warning system successfully warned of the impending fracture in every one of the 193 samples, addressing the lack of fatigue failure warning technology.

In different materials, a smaller $k_{\mathrm{c}}$ value typically corresponds to a longer fatigue life; correspondingly, a higher COOP value indicates stronger interatomic bonding, which also leads to an extended fatigue life. The trends exhibited by the two parameters are clearly similar, suggesting a potential correlation between $k_{\mathrm{c}}$ and the COOP. A further analysis revealed strong positive correlations between the parameter $C$ and the ferromagnetic element content (R² = 0.82; Fig. 3b), indicating that enhanced spin‒spin coupling derived from increased amounts of ferromagnetic elements may amplify the spin correlation effects induced under magnetic excitation. These findings provide new reference perspectives for relevant research on understanding the fatigue damage mechanisms of ferromagnetic materials.

Owing to the adoption of multiple simplified assumptions, the neglect of certain nonlinear factors during the theoretical derivations, and the simultaneous primary reliance on multiparameter models derived from experimental observations and data fitting, this study has the following limitations.

1. Inadequate mechanistic verification: the synergistic degradation mechanism between the interlamellar bond strength and interlamellar exchange interactions remains experimentally unverified because of the technical limitations exhibited by our current experimental methods.

2. Insufficient microscopic evidence: the spin magnetic moment correlation amplification effect induced under magnetic excitation lacks a microscopic-scale experimental validation.

3. Incomplete parameter correlations: the relationships between deterministic parameters and intrinsic material properties have not been fully established.

To address these challenges, further investigations may involve the following.

1. A quantitative correlation analysis between COOP parameters and fatigue strength could provide theoretical guidance for developing fatigue-resistant ferromagnetic alloys with high COOP values.

2. Precise fatigue life prediction models can be developed by incorporating nonlinear effects to enhance the robustness of predictions produced under complex service environments (e.g., alternating temperatures and corrosive media).

3. In situ observations of the magnetic domain evolution process induced during fatigue can be performed via Lorentz transmission electron microscopy (LTEM) to explore the microscopic regulatory effects of exchange interactions on magnetic moment deflection.

4. Multiscale simulations (DFT and micromagnetics) can be conducted to quantify the contributions of interface effects to magnetic domain rearrangement and analyze how magnetic moment reorientation influences macroscopic magnetic alignment.

Despite these limitations, experimental validations conducted across multiple material systems ensured the significant reliability of the core conclusions. This work successfully bridges microscopic phenomena with macroscopic engineering safety, marking a critical step toward a new predictive maintenance paradigm, with significant practical value for critical sectors, such as transportation and aerospace.

Methods

Preparation of the test samples and experimental procedures

The materials employed in the experiments included soft magnetic iron (DT4C), Q235 steel (PB240), 45 steel (C45E4), pure nickel (Nickel 201), and bridge stay cable steel (BSCS). The DT4C, PB240, C45E4, and nickel 201 were all 16-mm-diameter round rods, whereas the bridge stay cable steel had a diameter of 7 mm. Before loading, the central sections of the samples were machined into an arc form utilizing computer numerical control (CNC) techniques. The narrowest sections of the soft magnetic iron (DT4C), Q235 steel (PB240), 45 steel (C45E4), and pure nickel (Nickel 201) samples measured 8 mm, whereas the bridge stay cable steel sample had a minimum section diameter of 4 mm (Supplementary Table 1).

An MTS Landmark Model 370.10 electrohydraulic servo mechanical test system was used; five N35 magnets were employed, and the magnetic induction intensity measured at a height of 2 mm from the center of the upper surface was 230–250 mT. The Hall sensor adopted an EQ-730L, which was composed of an InAs Well Hall Element and a signal processing IC chip in a package. The supply voltage ranged from 3.0 to 5.5 V for a single power supply (Supplementary Table 1). The data were collected using the DH5922N dynamic signal analysis and acquisition system from Donghua Test.

We designed a material fatigue observation test device utilizing spin correlation amplification (Supplementary Fig. 2a) and conducted static load and fatigue tests through spin correlation amplification monitoring using soft magnetic iron (DT4C), Q235 steel (PB240), 45 steel (C45E4), pure nickel (Nickel 201), and bridge stay cable steel. All the fatigue tests had a loading frequency of 20 Hz and a stress ratio of 0. During this period, the magnet and Hall sensor were kept out of contact with the sample, and the distance between them remained constant.

Both ends of each sample were clamped on the high-frequency fatigue test machine of the MTS Landmark Model 370.10 electrohydraulic servo mechanical test system, and five N35 magnets were vertically superimposed and adsorbed to the upper fixture of the universal testing machine.

Detailed statistical method for addressing fatigue accidents and fatalities

In an effort to provide a comprehensive tally of the accidents and fatalities that stemmed from metal fatigue, we utilized targeted keywords: list of bridge failures, list of elevator accidents, list of industrial disasters, list of railway accidents, aviation safety network, list of marine disasters in the 21st century, list of marine disasters in the 20th century and list of building and structure collapses—to extract data from Wikipedia on incidents that occurred across various sectors, including the bridge, elevator, industry, railway, aviation, marine, and construction domains. We then compiled a statistical database documenting these accidents over the past century, which includes the time, location, number of fatalities, and detailed description of each event. To safeguard the integrity of our data, we meticulously reviewed the adjacent records to identify and remove any duplicates.

We subsequently utilized a series of keywords to compare and filter the causes of accidents across various industries. For the bridge industry, we used the keywords: metal fatigue, fatigue cracks, metal corrosion and fatigue, structural performance deterioration, structural collapse, overload, suspension cable fracture, and steel frame fracture. For the elevator industry, we screened for causes using the keywords: cable breakage, steel cable breakage, overload and mechanical failure. For the industrial sector, we applied the keywords: ultimate stress, corrosion cracking, mechanical failure, train derailment, cable wear, and cable fracture. For the railway industry, we searched for causes with the keywords: train derailment, axle failure, engine failure, rail breakage, boiler failure, and brake wear and overload. For the ship industry, we compared and screened the causes using the keywords: overload, outdated ship, engine failure, propeller shaft fracture, boiler failure, deck rupture, and ship vibration. For the construction industry, we used the keywords: bridges, masts, aircraft racks, steel structures, cranes and stadium roofs to screen the structural types. Finally, for the aviation industry, we used the keywords: wing flutter, engine failure, overload, mechanical failure, landing gear failure, fuselage bending, and engine vibration to compare and screen the causes of accidents. For the selected accidents, we manually searched each incident on Google and Wikipedia, carefully analyzing the authenticity of the events and the reliability of the corresponding data. Then, we used the keywords: metal fatigue, fatigue fracture, fatigue crack, fatigue cracking and fatigue failure to rescreen the accidents, ultimately classifying those that matched these keywords as metal fatigue-related accidents.

In addition, we utilized the keywords: metal fatigue, fatigue fracture, fatigue crack, fatigue cracking and fatigue failure to compare and screen the causes of aviation accidents, extracting the accident records that matched these keywords and directly classifying them as accidents caused by metal fatigue. Subsequently, we manually sorted and integrated the selected accidents in chronological order, ultimately obtaining a compilation of accidents caused by metal fatigue in the bridge, elevator, industrial, railway, aviation, ship and construction industries from 1900 to 2024.

Finally, we classified the accidents caused by metal fatigue, extracted the number of deaths from the raw data of each accident, and then calculated the cumulative number of deaths caused by metal fatigue in each industry every five years to obtain the total number of deaths caused by metal fatigue in each industry over the past century, as well as the number of deaths occurring every five years.

Method for calculating the maintenance costs of various industries in 2023

Infrastructure, such as bridges, elevators, railways, ships, and airplanes, is constantly affected by metal fatigue because of multiple factors, including long-term heavy loads, vibrations, and corrosion, leading to a gradual decline in performance. Therefore, metal fatigue, as the main driver of the maintenance costs incurred for these infrastructures, cannot be ignored in terms of its impact. To accurately and comprehensively reflect the latest trends in maintenance costs within the industry, especially the impact of metal fatigue on these costs, we adopted a systematic methodological approach. This involved conducting detailed multidatabase surveys and utilizing the predefined terminology: Global Bridge, Lift, Railway, Ship, and Aviation maintenance market size in 2023 to conduct a detailed statistical analysis of the maintenance cost data related to metal fatigue in the bridge, elevator, railway, ship, and aircraft industries for the year 2023.

After rigorous screening, we found that metal fatigue significantly impacted maintenance costs across various industries. The maintenance cost incurred for bridges due to metal fatigue was as high as $5.4 billion, while the elevator industry incurred costs of approximately $0.406 billion. The maintenance costs incurred for the railway and shipping industries reached $4.328 billion and $24.202 billion, respectively, and the aircraft industry had the highest maintenance costs, reaching $44.566 billion. These data not only reveal the direct impact of metal fatigue on maintenance costs but also reflect the economic pressures that different industries face in terms of addressing metal fatigue issues.

Atomic scale: theoretical model derivation of the relationships among the interlamellar binding force, exchange interaction, and atomic spacing

It is generally believed that binding energy manifests as Coulomb repulsion energy and attraction energy. When the distance between atoms is small, the repulsive forces between negatively charged electrons dominate, with the predominant energy manifested as repulsive energy. When the atomic distance is large, the main energy is manifested as attractive energy. In a free state, energy is at an equilibrium point where the sum of the repulsive and attractive energies is minimized.

In this study, only the tensile state of each material, which commenced from its free state, was considered; specifically, only the scenario where the distance between atoms underwent relatively minor changes was considered.

The measurements and simulation images of iron atom electron clouds reveal that electrons are distributed around atoms in the form of probability density functions of wave functions. In accordance with the representation method in solid-state physics (Schrödinger equation), we express the binding energy in the form of a wave function.

$$U_{\mathrm{C}}=\frac{e^2}{R}+\iint {\psi }_{\mathrm{ab}}^{*}\cdot e^2\left(\frac{1}{r_{12}}-\frac{1}{r_{\mathrm{b}1}}-\frac{1}{r_{\mathrm{a}2}}\right)\cdot {\psi }_{\mathrm{ab}}d{\tau }_{1}d{\tau }_2$$ 2

where $R$ is the distance between atoms, $r_{12}$ is the distance between electrons, and $r_{\mathrm{b}1}$ and $r_{\mathrm{a}2}$ are the distances between atoms and adjacent electrons, respectively. ${\psi }_{\mathrm{ab}}$ and ${\psi }_{\mathrm{ab}}^{*}$ are the wave function and conjugate of the electrons that form the binding energy, respectively, and $e$ represents the charge distribution between the electrons that form the binding energy and the atom.

At present, various models are available for determining the binding energy between iron atoms, such as the localized electron model, the itinerant electron model, and the band model. However, the basic forms of these models can all be represented by the above formula, with only minor differences.

If the binding energy between multiple pairs of atoms is the goal, we can write Eq. (2) as follows:

$$U_{\mathrm{C}}=\sum \left(\frac{{\mathrm{e}}^2}{R_{ij}}+\iint \underset{\mathrm{ab}}{\overset{*}{\psi }}\cdot e^2\left(\frac{1}{r_{ij}}-\frac{1}{r_{\mathrm{b}i}}-\frac{1}{r_{\mathrm{a}j}}\right)\cdot {\psi }_{\mathrm{ab}}d{\tau }_{1}d{\tau }_2\right)$$ 3

where $R_{ij}$ is the distance between atoms $i$ and $j$, $r_{ij}$ is the distance between electrons $i$ and $j$, and $r_{\mathrm{b}i}$ and $r_{\mathrm{a}j}$ are the distances between atom $\mathrm{a}$ and adjacent electron $i$ and between atom $\mathrm{b}$ and adjacent electron $j$, respectively.

Note that the binding energy of a material increases monotonically from the lowest energy state when it is stretched. We use a function of the average atomic spacing between the two layers of the tensile cross-section of the target material, denoted as ${\overline{r}}_{\mathrm{c}}$, to simplify the equivalent overall wave function.

$$\frac{1}{{\overline{r}}_{\mathrm{c}}}\equiv \sum \left(\frac{1}{R_{ij}}+\iint \underset{\mathrm{ab}}{\overset{*}{\psi }}\cdot \left(\frac{1}{r_{ij}}-\frac{1}{r_{\mathrm{b}i}}-\frac{1}{r_{\mathrm{a}j}}\right)\cdot {\psi }_{\mathrm{ab}}d{\tau }_{1}d{\tau }_2\right)$$ 4

where ${\overline{r}}_{\mathrm{c}}$ represents the average Coulomb interaction distance between two layers of atoms.

Therefore,

$$U_{\mathrm{C}}=\frac{e^2}{{\overline{r}}_{\mathrm{c}}}$$ 5

Taking the derivative of $U_{\mathrm{C}}$ with respect to the average interatomic distance $\overline{R}$ between two layers of atoms yields the following:

$$F_{\mathrm{C}}=\frac{d{U}_{\mathrm{C}}}{d\overline{R}}=-\frac{e^2}{{{\overline{r}}_{\mathrm{c}}}^2}\frac{d{\overline{r}}_{\mathrm{c}}}{d\overline{R}}$$ 6

where $F_{\mathrm{C}}$ represents the average binding force between two layers of atoms.

By expanding ${\overline{r}}_{\mathrm{c}}$ in terms of $\overline{R}$ using Taylor series expansion, we can obtain ${\overline{r}}_{\mathrm{c}}=C_{\mathrm{C}0}+C_{\mathrm{C}1}\overline{R}+C_{\mathrm{C}2}{\overline{R}}^2+\dots$, where $C_{\mathrm{C}0}$, $C_{\mathrm{C}1}$ and $C_{\mathrm{C}2}$ are constants.

Ignoring the higher-order terms and considering only the case where the atomic spacing increases slightly from its lowest energy position, where ${\overline{r}}_{\mathrm{c}}$ linearly changes with respect to $\overline{R}$, we can obtain the following equation:

$$F_{\mathrm{C}}=-\frac{e^2{C}_{\mathrm{C}1}}{{\left(C_{\mathrm{C}0}+C_{\mathrm{C}1}\overline{R}\right)}^2}=-\frac{d_{\mathrm{C}0}}{{\left(\overline{R}{+d}_{\mathrm{C}1}\right)}^2}$$ 7

where $d_{\mathrm{C}1}=\frac{C_{\mathrm{C}0}}{C_{\mathrm{C}1}}$ and $d_{\mathrm{C}0}=\frac{e^2}{C_{\mathrm{C}1}}$ are constants.

Therefore, we can use a function of the atomic spacing to express the change exhibited by the average interatomic binding force during tensile fatigue.

Exchange interactions are more complex, but here, they are simply calculated using the same formalism as that employed for the binding force. The magnetic exchange energy, which arises from the exchange interaction between the electrons of two atoms as well as between the electrons and the atomic nuclei, is expressed by a wave function.

$$U_{\mathrm{m}}=\iint {\phi }_{\mathrm{ab}}^{*}\cdot m\left(\frac{1}{r_{12}}-\frac{1}{r_{\mathrm{b}1}}-\frac{1}{r_{\mathrm{a}2}}\right)\cdot {\phi }_{\mathrm{ab}}d{\tau }_{1}d{\tau }_2$$ 8

The wave functions of the electrons that contribute to the magnetic exchange energy are represented by ${\phi }_{\mathrm{ab}}$, and their conjugates are denoted as ${\phi }_{\mathrm{ab}}^{*}$. $m$ represents the combined charge and magnetic moment of the electrons and atoms that form the magnetic exchange energy.

For the magnetic exchange energy between multiple pairs of atoms, we rewrite Eq. (8) as follows:

$$U_{\mathrm{m}}=\sum \iint \underset{\mathrm{ab}}{\overset{*}{\phi }}\cdot m\left(\frac{1}{r_{ij}}-\frac{1}{r_{\mathrm{b}i}}-\frac{1}{r_{\mathrm{a}j}}\right)\cdot {\phi }_{\mathrm{ab}}d{\tau }_{1}d{\tau }_2$$ 9

where $r_{ij}$ is the distance between electrons $i$ and $j$ and $r_{\mathrm{b}i}$ and $r_{\mathrm{a}j}$ are the distances between atom $\mathrm{a}$ and adjacent electron $i$ and between atom $\mathrm{b}$ and adjacent electron $j$, respectively.

We define the following:

$$\frac{1}{{\overline{r}}_{\mathrm{m}}}\equiv \sum \iint \underset{\mathrm{ab}}{\overset{*}{\phi }}\cdot \left(\frac{1}{{\mathrm{r}}_{ij}}-\frac{1}{r_{\mathrm{b}i}}-\frac{1}{r_{\mathrm{a}j}}\right)\cdot {\phi }_{\mathrm{ab}}d{\tau }_{1}d{\tau }_2$$ 10

Therefore,

$$U_{\mathrm{m}}=\frac{m}{{\overline{r}}_{\mathrm{m}}}$$ 11

The overlined notation ${\overline{r}}_{\mathrm{m}}$ represents the equivalent exchange interaction distance between two layers of atoms. Similarly, ${\overline{r}}_{\mathrm{m}}$ is a function of the average distance $\overline{R}$ between these two layers of atoms. Under other constant conditions, once the average distance $\overline{R}$ between the layers is determined, ${\overline{r}}_{\mathrm{m}}$ takes a unique value.

By taking the derivative of $U_{\mathrm{m}}$ with respect to $\overline{R}$, we can obtain the following:

$$F_{\mathrm{m}}=\frac{d{U}_{\mathrm{m}}}{d\overline{R}}=-\frac{m}{{{\overline{r}}_{\mathrm{m}}}^2}\frac{d{\overline{r}}_m}{d\overline{R}}$$ 12

where $F_{\mathrm{m}}$ represents the average magnetic force between two layers of atoms.

By expanding ${\overline{r}}_{\mathrm{m}}$ in terms of $\overline{R}$ using Taylor series expansion, we can obtain ${\overline{r}}_{\mathrm{m}}=C_{\mathrm{m}0}+C_{\mathrm{m}1}\overline{R}+C_{\mathrm{m}2}{\overline{R}}^2+\dots$, where $C_{\mathrm{m}0}$, $C_{\mathrm{m}1}$ and $C_{\mathrm{m}2}$ are constants.

Similarly, ignoring the higher-order terms and only considering the linear changes exhibited by ${\overline{r}}_{\mathrm{m}}$ with respect to $\overline{R}$, we can obtain the following equation:

$$F_{\mathrm{m}}=-\frac{m{C}_{\mathrm{m}1}}{{\left(C_{\mathrm{m}0}+C_{\mathrm{m}1}\overline{R}\right)}^2}=-\frac{d_{\mathrm{m}0}}{{\left(\overline{R}{+d}_{\mathrm{m}1}\right)}^2}$$ 13

where $d_{\mathrm{m}1}=\frac{C_{\mathrm{m}0}}{C_{\mathrm{m}1}}$ and $d_{\mathrm{m}0}=\frac{m}{C_{\mathrm{m}1}}$ are constants.

From this, we can consider the average binding force and magnetic exchange force between adjacent layers of atoms in ferromagnetic materials to be homologous effects, and both exhibit inverse square relationships with the average atomic spacing between the layers. Therefore, in cases where $\overline{R}$ changes very slightly, an increase in $\overline{R}$ leads to decreases in both the binding force and the magnetic force (we are only considering the effect of an increase in $\overline{R}$).

When external cyclic loading is present, let the force exerted by the loading on the two layers of atoms be denoted as $\mathrm{f}\left(\sigma \right)$, and let the interlamellar magnetic force caused by the loading be denoted as $\mathrm{g}\left(\sigma \right)$. The functional form of $\mathrm{g}\left(\sigma \right)$ is relatively complex, and since it can be omitted in the subsequent derivation, its specific form is not considered here.

During cyclic loading, the stress gradually increases to the maximum level and then decreases back to zero, after which the cycle is repeated around an intermediate stress value.

When the external loading is ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, the combined effects of the binding force and loading action are as follows:

$$F_{\mathrm{C}0}-\mathrm{f}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)$$ 14

The combined effects of the magnetic force and loading action are as follows:

$$F_{\mathrm{m}0}-\mathrm{g}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)$$ 15

where $F_{\mathrm{C}0}=-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_0{+d}_{\mathrm{C}1}\right)}^2}$, $F_{\mathrm{m}0}=-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_0{+d}_{\mathrm{m}1}\right)}^2}$.

When the external loading is ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, the combined effects of the binding force and loading action are as follows:

$$F_{\mathrm{C}1}-\mathrm{f}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)$$ 16

The combined effects of the magnetic force and loading action are as follows:

$$F_{\mathrm{m}1}-\mathrm{g}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)$$ 17

where $F_{\mathrm{C}1}=-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_1{+d}_{\mathrm{C}1}\right)}^2}$, $F_{\mathrm{m}1}=-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_1{+d}_{\mathrm{m}1}\right)}^2}$, ${\sigma }_{\mathrm{m}}$ represents the average force during external cyclic loading, and ${\sigma }_{\mathrm{L}}$ is the fluctuation of the force during this cycle. $F_{\mathrm{C}0}$ is the average binding force inside the material when the external loading is ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, specifically at the onset of fatigue loading, and $F_{\mathrm{C}1}$ is the average binding force inside the material when the initial loading is set to ${\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}$, also at the onset of fatigue loading. Similarly, $F_{\mathrm{m}0}$ is the average magnetic force inside the material when the external loading is ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, specifically at the onset of fatigue loading, and $F_{\mathrm{m}1}$ is the average magnetic force inside the material when the external loading is ${\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}$, also at the onset of fatigue loading. Furthermore, ${\overline{R}}_0$ denotes the average atomic spacing inside the material when the external loading is ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, specifically at the onset of fatigue loading, whereas ${\overline{R}}_1$ represents the average atomic spacing inside the material when the initial loading is ${\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}$, also at the onset of fatigue loading.

Subtracting Eq. (14) from Eq. (16) results in the difference in the binding force, which represents the peak-to-peak value of the binding force during loading, specifically at the onset of fatigue loading:

$$F_{\mathrm{CPP}}=F_{\mathrm{C}1}-\mathrm{f}\left({\sigma }_{\mathrm{m}}+{\sigma }_L\right)-\left[F_{\mathrm{C}0}-\mathrm{f}\left({\sigma }_{\mathrm{m}}-{\mathrm{\sigma }}_{\mathrm{L}}\right)\right]=\mathrm{f}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)-\mathrm{f}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_1{+d}_{\mathrm{C}1}\right)}^2}+\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_0{+d}_{\mathrm{C}1}\right)}^2}$$ 18

Subtracting Eq. (15) from Eq. (17) results in the difference in the magnetic force, which represents the peak-to-peak value of the magnetic force during loading, specifically at the onset of fatigue loading:

$$F_{\mathrm{mPP}}=F_{\mathrm{m}1}-\mathrm{g}\left({\sigma }_m+{\sigma }_L\right)-\left[F_{\mathrm{m}0}-\mathrm{g}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)\right]=\mathrm{g}\left({\sigma }_{\mathrm{m}}-{\mathrm{\sigma }}_{\mathrm{L}}\right)-\mathrm{g}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_1{+d}_{\mathrm{m}1}\right)}^2}+\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_0{+d}_{\mathrm{m}1}\right)}^2}$$ 19

When material fatigue is considered, the average atomic spacing ${\overline{R}}_0$ increases by $\mathrm{\Delta }\overline{R}$, where $\mathrm{\Delta }\overline{R}$ represents the fatigue-induced dislocation change that occurs in the sample under external cyclic loading, and it is assumed that ${\overline{R}}_0\gg \mathrm{\Delta }\overline{R}$.

For the external loading ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, the combined effects of the binding force and loading action can be written as follows:

$$F_{\mathrm{C}0}^{\prime }-\mathrm{f}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)$$ 20

The combined effects of the magnetic force and loading action are as follows:

$$F_{\mathrm{m}0}^{\prime }-\mathrm{g}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)$$ 21

where $F_{\mathrm{C}0}^{\prime }=-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{C}1}\right)}^2}$ and $F_{\mathrm{m}0}^{\prime }=-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}$.

When the external loading is ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, the increase in the distance between the two layers of atoms from their initial equilibrium position ${\overline{R}}_1$ is $\mathrm{\Delta }\overline{R}$, and ${\overline{R}}_1\gg \mathrm{\Delta }\overline{R}$.

Then, the combined effects of the binding force and loading action can be written as follows:

$$F_{\mathrm{C}1}^{\prime }-\mathrm{f}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)$$ 22

The combined effects of the magnetic force and loading action are as follows:

$$F_{\mathrm{m}1}^{\prime }-\mathrm{g}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)$$ 23

where $F_{\mathrm{C}1}^{\prime }=-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{C}1}\right)}^2}$ and $F_{\mathrm{m}1}^{\prime }=-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}$.

The difference obtained by subtracting Eq. (20) from Eq. (22) represents the actual binding force change that occurs because of fatigue, specifically when a fatigue dislocation takes place during loading:

$$F_{\mathrm{CPP}}^{\prime }=F_{\mathrm{C}1}^{\prime }-\mathrm{f}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)-\left[F_{\mathrm{C}0}^{\prime }-\mathrm{f}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)\right]=\mathrm{f}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)-\mathrm{f}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{C}1}\right)}^2}+\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{C}1}\right)}^2}$$ 24

The difference obtained by subtracting Eq. (21) from Eq. (23) represents the actual magnetic force change that occurs because of fatigue, specifically when a fatigue dislocation takes place during loading:

$$F_{\mathrm{mPP}}^{\prime }=F_{\mathrm{m}1}^{\prime }-\mathrm{g}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)-\left[F_{\mathrm{m}0}^{\prime }-\mathrm{g}\left({\sigma }_{\mathrm{m}}-{\sigma }_L\right)\right]=\mathrm{g}\left({\sigma }_{\mathrm{m}}-{\sigma }_L\right)-\mathrm{g}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}+\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{\mathrm{R}}+d}_{\mathrm{m}1}\right)}^2}$$ 25

Subtracting Eq. (24) from Eq. (18) yields the relationship between the fatigue dislocation change $\mathrm{\Delta }\overline{R}$ and the average binding force change induced during an actual loading process:

$${\mathrm{\Delta }F}_{\mathrm{C}}=F_{\mathrm{CPP}}^{\prime }-F_{\mathrm{CPP}}=\left[-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_1+\mathrm{\Delta }\overline{R}+d_{\mathrm{C}1}\right)}^2}+\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_0+\mathrm{\Delta }\overline{R}+d_{\mathrm{C}1}\right)}^2}\right]-\left[-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_1+d_{\mathrm{C}1}\right)}^2}+\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_0+d_{\mathrm{C}1}\right)}^2}\right]=\left[\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_1+d_{\mathrm{C}1}\right)}^2}-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_1+\mathrm{\Delta }\overline{R}+d_{\mathrm{C}1}\right)}^2}\right]-\left[\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_0+d_{\mathrm{C}1}\right)}^2}-\frac{d_{\mathrm{C}0}}{{\left({\overline{R}}_0+\mathrm{\Delta }\overline{R}+d_{\mathrm{C}1}\right)}^2}\right]=\frac{d_{\mathrm{C}0}\left({\mathrm{\Delta }\overline{R}}^2+2{\overline{R}}_1\mathrm{\Delta }\overline{R}+2\mathrm{\Delta }\overline{R}{d}_{\mathrm{C}1}\right)}{{\left({\overline{R}}_1+d_{\mathrm{C}1}\right)}^2{\left({\overline{R}}_1+\mathrm{\Delta }\overline{R}+d_{\mathrm{C}1}\right)}^2}-\frac{d_{\mathrm{C}0}\left({\mathrm{\Delta }\overline{R}}^2+2{\overline{R}}_0\mathrm{\Delta }\overline{R}+2\mathrm{\Delta }\overline{R}{d}_{\mathrm{C}1}\right)}{{\left({\overline{R}}_0+d_{\mathrm{C}1}\right)}^2{\left({\overline{R}}_0+\mathrm{\Delta }\overline{R}+d_{\mathrm{C}1}\right)}^2}$$ 26

Because ${\overline{R}}_0,{\overline{R}}_1\gg \mathrm{\Delta }\overline{R}$, ${\mathrm{\Delta }\overline{R}}^2$ in the above equation can be neglected. Therefore, Eq. (26) can be simplified as follows:

$$\mathrm{\Delta }{F}_C\approx \frac{d_{\mathrm{C}0}\left(2{\overline{R}}_1\mathrm{\Delta }\overline{R}+2\mathrm{\Delta }\overline{R}{d}_{\mathrm{C}1}\right)}{{\left({\overline{\mathrm{R}}}_1{+\mathrm{d}}_{\mathrm{C}1}\right)}^2{\left({\overline{\mathrm{R}}}_1{+\mathrm{\Delta }\overline{\mathrm{R}}+\mathrm{d}}_{\mathrm{C}1}\right)}^2}-\frac{d_{\mathrm{C}0}\left(2{\overline{R}}_0\mathrm{\Delta }\overline{R}+2\mathrm{\Delta }\overline{R}{d}_{\mathrm{C}1}\right)}{{\left({\overline{R}}_0{+d}_{\mathrm{C}1}\right)}^2{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{C}1}\right)}^2}=2\mathrm{\Delta }\overline{R}{d}_{\mathrm{C}0}\left[\frac{1}{\left({\overline{R}}_1{+d}_{\mathrm{C}1}\right){\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{C}1}\right)}^2}-\frac{1}{\left({\overline{R}}_0{+d}_{\mathrm{C}1}\right){\left({\overline{R}}_0{+\mathrm{\Delta }\overline{\mathrm{R}}+d}_{\mathrm{C}1}\right)}^2}\right]$$ 27

Equation (27) can be simplified to the following:

$${\mathrm{\Delta }F}_{\mathrm{C}}=K_{\mathrm{C}}\mathrm{\Delta }\overline{R}$$ 28

where $K_{\mathrm{C}}=2d_{\mathrm{C}0}\left[\frac{1}{\left({\overline{R}}_1{+d}_{\mathrm{C}1}\right){\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{C}1}\right)}^2}-\frac{1}{\left({\overline{R}}_0{+d}_{\mathrm{C}1}\right){\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{C}1}\right)}^2}\right]$ and ${\mathrm{\Delta }F}_{\mathrm{C}}$ is the average variation in the interatomic binding force.

Owing to ${\overline{R}}_1$ and ${\overline{R}}_0$ being related to the external force magnitude during loading and the inherent properties of the material, once the material and loading force are specified, the averages of ${\overline{R}}_1$ and ${\overline{R}}_0$ take on unique values. Given that ${\overline{R}}_1-{\overline{R}}_0\gg \mathrm{\Delta }\overline{R}$, $\mathrm{\Delta }\overline{R}$ has a negligible effect on the denominator term in $K_{\mathrm{C}}$, thus allowing us to approximate $K_{\mathrm{C}}$ as a constant. When the material experiences fatigue dislocations, the change in the average binding force is solely dependent on the change in the average atomic spacing.

Similarly, subtracting Eq. (25) from Eq. (19) yields the relationship between the change in fatigue dislocation $\mathrm{\Delta }\overline{R}$ and the change in the average magnetic force during actual loading:

$$\mathrm{\Delta }{F}_{\mathrm{m}}=F_{\mathrm{mPP}}^{\prime }-F_{\mathrm{mPP}}=\left[-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}+\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}\right]-\left[-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_1{+\mathrm{d}}_{\mathrm{m}1}\right)}^2}+\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_0{+d}_{\mathrm{m}1}\right)}^2}\right]=\left[\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_1{+d}_{\mathrm{m}1}\right)}^2}-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}\right]-\left[\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_0{+d}_{\mathrm{m}1}\right)}^2}-\frac{d_{\mathrm{m}0}}{{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}\right]=\frac{d_{\mathrm{m}0}\left({\mathrm{\Delta }\overline{R}}^2+2{\overline{R}}_1\mathrm{\Delta }\overline{R}+2\mathrm{\Delta }\overline{R}{d}_{\mathrm{m}1}\right)}{{\left({\overline{R}}_1{+d}_{\mathrm{m}1}\right)}^2{\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}-\frac{d_{\mathrm{m}0}\left({\mathrm{\Delta }\overline{R}}^2+2{\overline{R}}_0\mathrm{\Delta }\overline{R}+2\mathrm{\Delta }\overline{R}{d}_{\mathrm{m}1}\right)}{{\left({\overline{R}}_0{+d}_{\mathrm{m}1}\right)}^2{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}$$ 29

Equation (29) can be simplified to the following:

$$\mathrm{\Delta }{F}_{\mathrm{m}}\approx \frac{d_{\mathrm{m}0}\left(2{\overline{R}}_1\mathrm{\Delta }\overline{R}+2\mathrm{\Delta }\overline{R}{d}_{\mathrm{m}1}\right)}{{\left({\overline{R}}_1{+d}_{\mathrm{m}1}\right)}^2{\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}-\frac{d_{\mathrm{m}0}\left(2{\overline{R}}_0\mathrm{\Delta }\overline{R}+2\mathrm{\Delta }\overline{R}{d}_{\mathrm{m}1}\right)}{{\left({\overline{R}}_0{+d}_{\mathrm{m}1}\right)}^2{\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}\approx 2\mathrm{\Delta }\overline{R}{d}_{\mathrm{m}0}\left[\frac{1}{\left({\overline{R}}_1{+\mathrm{d}}_{\mathrm{m}1}\right){\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}-\frac{1}{\left({\overline{R}}_0{+d}_{\mathrm{m}1}\right){\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}\mathit{+}d}_{\mathrm{m}1}\right)}^2}\right]$$ 30

Equation (30) can be simplified to the following:

$$\mathrm{\Delta }{F}_{\mathrm{m}}=K_{\mathrm{m}}\mathrm{\Delta }\overline{R}$$ 31

where $K_{\mathrm{m}}=2d_{\mathrm{m}0}\left[\frac{1}{\left({\overline{R}}_1{+d}_{\mathrm{m}1}\right){\left({\overline{R}}_1{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}-\frac{1}{\left({\overline{R}}_0{+d}_{\mathrm{m}1}\right){\left({\overline{R}}_0{+\mathrm{\Delta }\overline{R}+d}_{\mathrm{m}1}\right)}^2}\right]$. $\mathrm{\Delta }{F}_{\mathrm{m}}$ represents the average variation in the magnetic force. The change in the average magnetic force when a material experiences fatigue dislocations is related only to the change in the average atomic spacing.

Combining Eq. (28) and Eq. (31), the relationship between the binding force and magnetic force can be obtained as follows:

$$\mathrm{\Delta }{F}_{\mathrm{C}}=K\mathrm{\Delta }{F}_{\mathrm{m}}$$ 32

where $K=\frac{K_{\mathrm{C}}}{K_{\mathrm{m}}}$.

Mesoscale: derivation of the spin correlation amplification model

When an external magnetic field is applied to ferromagnetic metals, the magnetic induction intensity is transmitted in a layer-by-layer manner along a certain direction because of the (virtual) interlamellar exchange interaction. Assuming that the interlamellar exchange interaction is the sole factor determining the transfer of the magnetic field strength within the magnetic field and neglecting the magnetic field strength transmitted through air, we can present the relationships among the induced magnetic field strength, the applied magnetic field strength, and the interlamellar force resulting from the exchange interaction after the external magnetic field has passed through one layer (Fig. 2a):

$${\mathbf{B}}_1={\mathbf{B}}_0\cdot \mathrm{f}\left(F_{\mathrm{m}0}\right)$$ 33

where ${\mathbf{B}}_0$ represents the external magnetic field strength, ${\mathbf{B}}_1$ represents the induced magnetic field strength of the first layer, $F_{\mathrm{m}0}$ denotes the interlamellar exchange interaction, and $\mathrm{f}\left(F_{\mathrm{m}0}\right)$ is a continuous function of the interlamellar force resulting from the exchange interaction.

The magnetic field strength of the subsequent layer is induced by the interlamellar exchange interaction from the preceding layer; thus, we have the following equation:

$${\mathbf{B}}_2={\mathbf{B}}_1\cdot \mathrm{f}\left(F_{\mathrm{m}1}\right)$$ 34

where ${\mathbf{B}}_2$ is the induced magnetic strength value of the second layer and $F_{\mathrm{m}1}$ is the interlamellar exchange interaction.

In a similar manner, we can deduce the following:

$${\mathbf{B}}_i={\mathbf{B}}_{i-1}\cdot \mathrm{f}\left(F_{\mathrm{m}\left(i-1\right)}\right)$$ 35

where ${\mathbf{B}}_i$ is the induced magnetic strength value of the $i$-th layer and $F_{\mathrm{m}\left(i-1\right)}$ is the interlamellar exchange interaction of the $i$-th layer.

Afterward, we can obtain the induced magnetic strength of the $n$-th layer, where we measure the external magnetic strength as follows:

$${\mathbf{B}}_n={\mathbf{B}}_0{\prod }_{i=0}^{n-1}\mathrm{f}\left(F_{\mathrm{m}i}\right)$$ 36

When the average interatomic distance between layers varies because of stress or fatigue, the interlamellar exchange interaction changes correspondingly by $\mathrm{\Delta }{F}_{\mathrm{m}i}$.

Then, the induced magnetic field strength of the $\mathrm{n}$-th layer can be expressed as follows:

$${\mathbf{B}}_n^{\prime }={\mathbf{B}}_0{\prod }_{i=0}^{n-1}\mathrm{f}\left(F_{\mathrm{m}i}+\mathrm{\Delta }{F}_{\mathrm{m}i}\right)$$ 37

In the case where $\mathrm{\Delta }{F}_{\mathrm{m}i}$ is minimal, we have that:

$$\mathrm{f}\left(F_{\mathrm{m}i}+\mathrm{\Delta }{F}_{\mathrm{m}i}\right)\approx \mathrm{f}\left(F_{\mathrm{m}i}\right)+{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\cdot \mathrm{\Delta }{F}_{\mathrm{m}i}$$ 38

Substituting this into ${\mathbf{B}}_n^{\prime }$ yields:

$${\mathbf{B}}_n^{\prime }={\mathbf{B}}_0{\prod }_{i=0}^{n-1}\left[\mathrm{f}\left(F_{\mathrm{m}i}\right)+{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\cdot \mathrm{\Delta }{F}_{\mathrm{m}i}\right]$$ 39

By taking the logarithms of ${\mathbf{B}}_n$ and ${\mathbf{B}}_n^{\prime }$, we can obtain the following:

$$\ln {\mathbf{B}}_n={\ln \mathbf{B}}_0+{\sum }_{i=0}^{n-1}\ln \mathrm{f}\left(F_{\mathrm{m}i}\right)$$ 40

$$\ln {\mathbf{B}}_n^{\prime }={\ln \mathbf{B}}_0+{\sum }_{i=0}^{n-1}\ln \left[\mathrm{f}\left(F_{\mathrm{m}i}\right)+{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\cdot \mathrm{\Delta }{F}_{\mathrm{m}i}\right]$$ 41

Since $\mathrm{\Delta }{F}_{\mathrm{m}i}$ is a small quantity, $\ln \left[\mathrm{f}\left(F_{\mathrm{m}i}\right)+{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\cdot \mathrm{\Delta }{F}_{\mathrm{m}i}\right]\approx \mathrm{lnf}\left(F_{\mathrm{m}i}\right)+\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\cdot \mathrm{\Delta }{F}_{\mathrm{m}i}}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}$. Therefore, Eq. (41) can be simplified as follows:

$$\ln {\mathbf{B}}_n^{\prime }\approx {\ln \mathbf{B}}_0+{\sum }_{i=0}^{n-1}\left(\mathrm{lnf}\left(F_{\mathrm{m}i}\right)+\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\mathrm{\Delta }{F}_{\mathrm{m}i}}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}\right)$$ 42

Subtracting Eq. (40) from Eq. (42) yields:

$$\ln {\mathbf{B}}_n^{\prime }-\ln {\mathbf{B}}_n\approx {\sum }_{i=0}^{n-1}\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\cdot \mathrm{\Delta }{F}_{\mathrm{m}i}}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}$$ 43

Then,

$$\ln \left(\frac{{\mathbf{B}}_n^{\prime }}{{\mathbf{B}}_n}\right)\approx {\sum }_{i=0}^{n-1}\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\mathrm{\Delta }{F}_{\mathrm{m}i}}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}$$ 44

The magnetic induction intensity generated between adjacent layers is as follows:

$$\mathrm{\Delta }{\mathbf{B}}_n={\mathbf{B}}_n^{\prime }-{\mathbf{B}}_n={\mathbf{B}}_n\left(\frac{{\mathbf{B}}_n^{\prime }}{{\mathbf{B}}_n}-1\right)\approx {\mathbf{B}}_0\left[\exp \left({\sum }_{i=0}^{n-1}\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\cdot \mathrm{\Delta }{F}_{\mathrm{m}i}}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}\right)-1\right]$$ 45

Expanding $\exp \left(x\right)$ in a Taylor-wise manner and removing higher-order terms, $\exp \left(x\right)\approx 1+x$, where $x={\sum }_{i=0}^{n-1}\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\mathrm{\Delta }{F}_{\mathrm{m}i}}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}$, and Eq. (45) can be simplified as follows:

$$\mathrm{\Delta }{\mathbf{B}}_n={\mathbf{B}}_n^{\prime }-{\mathbf{B}}_n\approx {\mathbf{B}}_n\left[{\sum }_{i=0}^{n-1}\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\cdot \mathrm{\Delta }{F}_{\mathrm{m}i}}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}\right]$$ 46

Substituting Eq. (36) into Eq. (46) yields:

$$\mathrm{\Delta }{\mathbf{B}}_n\approx {\mathbf{B}}_0{\prod }_{k=0}^{n-1}\mathrm{f}\left(F_{\mathrm{m}k}\right){\sum }_{i=0}^{n-1}\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)\mathrm{\Delta }{F}_{\mathrm{m}i}}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}$$ 47

Owing to the correlation between $\mathrm{\Delta }{F}_{\mathrm{m}i}$ and the change in the interlamellar atomic spacing $\mathrm{\Delta }{R}_i$ (i.e., $\mathrm{\Delta }{F}_{\mathrm{m}i}=K_{\mathrm{m}}\mathrm{\Delta }{R}_i$), we can conclude the following:

$$\mathrm{\Delta }{\mathbf{B}}_n\approx {\mathbf{B}}_0{\prod }_{k=0}^{n-1}\mathrm{f}\left(F_{\mathrm{m}k}\right){\sum }_{i=0}^{n-1}\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)K_{\mathrm{m}}\mathrm{\Delta }{R}_i}{\mathrm{f}\left(F_{\mathrm{m}i}\right)}$$ 48

If the entire material is subjected to a uniform force or fatigue, the atomic spacing between each pair of layers tends to remain consistent, and the exchange interactions within each layer also exhibit similar effects. To simplify the derivation process, we assume that the variation in the atomic spacings ($\mathrm{\Delta }{R}_i$) between adjacent layers are equal, i.e., $\mathrm{\Delta }{R}_i=K_{Ri}\mathrm{\Delta }\overline{R}$. Furthermore, the exchange interaction functions $\mathrm{f}\left(F_{\mathrm{m}i}\right)$ of different layers are also identical; that is, $\mathrm{f}\left(F_{\mathrm{m}i}\right)=K_{\mathrm{m}Ri}\mathrm{f}\left(F_{\mathrm{m}}\right)$.

Substituting $\mathrm{\Delta }{R}_i=K_{Ri}\mathrm{\Delta }\overline{R}$ and $\mathrm{f}\left(F_{\mathrm{m}i}\right)=K_{\mathrm{m}Ri}\mathrm{f}\left(F_{\mathrm{m}}\right)$, ${\mathrm{f}}^{\prime }\left(F_{\mathrm{m}i}\right)=K_{\mathrm{m}Ri}{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}}\right)$ into (48) yields:

$$\mathrm{\Delta }{\mathbf{B}}_n\approx {\mathbf{B}}_0{\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}\mathrm{f}\left(F_{\mathrm{m}}\right){\sum }_{i=0}^{n-1}\frac{K_{\mathrm{m}Ri}{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}}\right)K_{\mathrm{m}}{K}_{Ri}\mathrm{\Delta }\overline{\mathrm{R}}}{K_{\mathrm{m}Ri}\mathrm{f}\left(F_{\mathrm{m}}\right)}={\mathbf{B}}_0{\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}\mathrm{f}\left(F_{\mathrm{m}}\right){\sum }_{i=0}^{n-1}\frac{{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}}\right)K_{\mathrm{m}}{K}_{Ri}\mathrm{\Delta }\overline{R}}{\mathrm{f}\left(F_{\mathrm{m}}\right)}={\mathbf{B}}_0{\mathrm{f}\left(F_{\mathrm{m}}\right)}^n{\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}\frac{f^{\prime }\left(F_{\mathrm{m}}\right)K_{\mathrm{m}}\mathrm{\Delta }\overline{R}}{\mathrm{f}\left(F_{\mathrm{m}}\right)}{\sum }_{i=0}^{n-1}{K}_{R\mathrm{i}}={\mathbf{B}}_0{\mathrm{f}\left(F_{\mathrm{m}}\right)}^{n-1}{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}}\right)K_{\mathrm{m}}\mathrm{\Delta }\overline{R}{\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}{\sum }_{i=0}^{n-1}{K}_{Ri}$$ 49

Given that the variation in the atomic spacing during material fatigue mainly occurs within the DCZ, it can be assumed that this amplification factor corresponds precisely to the number of layers contained in the DCZ. As fatigue develops, the magnetic intensity changes induced by DCZ variations within each layer continue to accumulate and manifest at the macroscopic level as significant amplifications of the magnetic intensity changes at the measurement points.

By substituting the relationship between the interlamellar binding force and interlamellar atomic spacing variation (${\mathrm{\Delta }F}_{\mathrm{C}}=K_{\mathrm{c}}\mathrm{\Delta }\overline{R}$) when a slight change in the atomic spacing $\mathrm{\Delta }\overline{R}$ occurs into Eq. (49), we obtain the following equation:

$$\mathrm{\Delta }{\mathbf{B}}_n\approx {\mathbf{B}}_0{\mathrm{f}\left(F_{\mathrm{m}}\right)}^{n-1}{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}}\right)K_{\mathrm{m}}\frac{{\mathrm{\Delta }\mathrm{F}}_C}{{\mathrm{K}}_c}{\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}{\sum }_{i=0}^{n-1}{K}_{Ri}$$ 50

where $A_{\mathrm{F}}={\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}{\sum }_{i=0}^{n-1}{K}_{Ri}$. Equation (50) can be rewritten as follows:

$$\mathrm{\Delta }{\mathbf{B}}_n\approx {\mathbf{B}}_0{\mathrm{f}\left(F_{\mathrm{m}}\right)}^{n-1}{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}}\right)K_{\mathrm{m}}\frac{{\mathrm{\Delta }F}_{\mathrm{C}}}{K_{\mathrm{c}}}{A}_{\mathrm{F}}$$ 51

In Eq. (51), except for the term related to the number of layers n contained in $A_{\mathrm{F}}$, all the other terms are independent of n. Let us derive $A_{\mathrm{F}}$ separately below.

Owing to the small interlamellar atomic spacing variation caused by each cycle, that is, the difference between layers is small, and the proportion coefficient of each layer being close to 1,

$$K_{\mathrm{m}Rk}=1+{\delta }_k,∣{\delta }_k∣\ll 1$$ 52

$$K_{Ri}=1+{ϵ}_i,∣{ϵ}_i∣\ll 1$$ 53

Therefore, the average deviation of the proportionality coefficient between the interlamellar atomic spacing and the exchange interaction is as follows:

$$\overline{\delta }=\frac{1}{n}{\sum }_{k=0}^{n-1}{\delta }_k$$ 54

$$\overline{ϵ}=\frac{1}{n}{\sum }_{i=0}^{n-1}{ϵ}_i$$ 55

Substituting $K_{\mathrm{m}Rk}=1+{\delta }_k$ into the continuous multiplication term in $A_{\mathrm{F}}$ yields:

$${\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}={\prod }_{k=0}^{n-1}\left(1+{\delta }_k\right)\approx \exp \left({\sum }_{k=0}^{n-1}\left(1+{\delta }_k\right)\right)\approx \exp \left({\sum }_{k=0}^{n-1}\left({\delta }_k\right)\right)$$ 56

Utilizing Taylor expansion $\ln \left(1+x\right)\approx x$ (ignoring higher-order terms), then,

$${\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}\approx \exp \left(n\overline{\delta }\right)$$ 57

Substituting the continuous addition term ${\sum }_{i=0}^{n-1}{K}_{Ri}$ into $A_{\mathrm{F}}$, where $K_{Ri}=1+{ϵ}_i$, yields:

$${\sum }_{i=0}^{n-1}{K}_{Ri}={\sum }_{i=0}^{n-1}\left(1+{ϵ}_i\right)=n+{\sum }_{i=0}^{n-1}\left({ϵ}_i\right)=n\left(1+\overline{ϵ}\right)$$ 58

Substituting Eqs. (57) and (58) into $A_{\mathrm{F}}$ yields the following:

$$A_{\mathrm{F}}={\prod }_{k=0}^{n-1}{K}_{\mathrm{m}Rk}{\sum }_{i=0}^{n-1}{K}_{Ri}\approx \exp \left(n\overline{\delta }\right)\cdot n\left(1+\overline{ϵ}\right)$$ 59

Equation (51) can be simplified as follows:

$$\mathrm{\Delta }{\mathbf{B}}_n={\mathbf{K}}_{\mathrm{B}}{A}_{\mathrm{F}}\mathrm{\Delta }{F}_{\mathrm{C}}$$ 60

where ${\mathbf{K}}_{\mathrm{B}}=\frac{{\mathbf{B}}_0{\mathrm{f}\left(F_{\mathrm{m}}\right)}^{n-1}{\mathrm{f}}^{\prime }\left(F_{\mathrm{m}}\right)K_{\mathrm{m}}}{K_{\mathrm{c}}}$ is related only to the material and excitation magnetic field and can be regarded as a constant under small variations. $A_{\mathrm{F}}\approx \exp \left(n\overline{\delta }\right)\cdot n\left(1+\overline{ϵ}\right)$ is an amplification factor of the interlamellar binding force, which is related to the number of layers n, atomic spacing changes, and exchange interactions and has a multiplication effect.

Macroscopic scale: derivation of the magnetic flux change rate and fatigue life model

Because the magnetic intensity $\mathrm{\Delta }{\mathbf{B}}_{\mathrm{H}}$ measured by an external Hall sensor is actually a mapping relationship of the internal magnetic intensity $\mathrm{\Delta }{\mathbf{B}}_{\mathrm{H}}$, we can conclude the following:

$$\mathrm{\Delta }{\mathbf{B}}_{\mathrm{H}}=K_{\mathrm{E}}\mathrm{\Delta }{\mathbf{B}}_{\mathrm{n}}$$ 61

where $K_{\mathrm{E}}$ is a constant, assuming that the measurement distance and environmental factors remain constant. Then, we can obtain:

$$\mathrm{\Delta }{\mathbf{B}}_{\mathrm{H}}={\mathbf{K}}_{\mathrm{H}}{\mathrm{\Delta }F}_{\mathrm{C}}$$ 62

where ${\mathbf{K}}_{\mathrm{H}}=K_{\mathrm{E}}{\mathbf{K}}_{\mathrm{B}}{A}_{\mathrm{F}}$.

Therefore, we can observe the fatigue process of ferromagnetic materials, which is caused by the gradual accumulation and weakening of binding forces. This process is reflected through the significant amplification effect of magnetic flux on subtle internal changes caused by the strength of the external magnetic field.

Considering that classic analysis models for fatigue processes, such as the Basquin criterion (the relationship between fatigue lives and stress amplitudes), the Paris formula (the relationship between the fatigue crack propagation rate and the stress intensity factor range), and the Coffin–Manson model (the relationship between the degree of strain accumulation and the cycle number) are all power-law models, it is assumed that the cumulative degradation of the binding force $\mathrm{\Delta }{F}_{\mathrm{C}}$ also increases in a power-law manner with the number of cycles $N$:

$$\mathrm{\Delta }{F}_{\mathrm{C}}=A{N}^B$$ 63

The index $B$ reflects the nonlinearity of material fatigue-based damage accumulation, whereas the coefficient $A$ is related to the intrinsic properties of the material. By substituting it into Eq. (62), we can obtain the following:

$$\mathrm{\Delta }{\mathbf{B}}_{\mathrm{H}}={\mathbf{K}}_{\mathrm{H}}\cdot A{N}^B$$ 64

We define $K_{\mathrm{av}}$ as the rate of change measured in the magnetic flux during the fatigue process with respect to the number of loading cycles:

$${\mathbf{K}}_{\mathrm{av}}=\frac{\mathrm{d}\left(\mathrm{\Delta }{\mathbf{B}}_{\mathrm{H}}\right)}{\mathrm{d}N}$$ 65

Substituting Eq. (64) yields:

$${\mathbf{K}}_{\mathrm{av}}={\mathbf{K}}_{\mathrm{H}}A\cdot B\cdot N^{B-1}$$ 66

where ${\mathbf{C}}_{\mathrm{k}}={\mathbf{K}}_{\mathrm{H}}AB$; therefore, Eq. (66) can be simplified as follows:

$${\mathbf{K}}_{\mathrm{av}}={\mathbf{C}}_{\mathrm{k}}{N}^{B-1}$$ 67

During the fatigue process, when the bonding force of the layered atoms in the material is lower than the external load after $N_{\mathrm{f}}$ cycles, the material experiences overall failure. The magnetic flux change rate at this time is as follows:

$${\mathbf{K}}_{\mathrm{av}}={\mathbf{C}}_{\mathrm{k}}{N_{\mathrm{f}}}^{B-1}$$ 68

Since $K_{\mathrm{av}}$ is usually negative (indicating a gradual decrease in magnetic field changes), we take its absolute value as follows:

$$\mid {\mathbf{K}}_{\mathrm{av}}\mid =\mid {\mathbf{C}}_{\mathrm{k}}\mid N_{\mathrm{f}}^{B-1}$$ 69

By taking the logarithm of both sides of Eq. (69), we can obtain the following:

$$\log \mid {\mathbf{K}}_{\mathrm{av}}\mid =\log \mid {\mathbf{C}}_{\mathrm{k}}\mid +\left(B-1\right)\log N_{\mathrm{f}}$$ 70

Therefore, $N_{\mathrm{f}}$ can be obtained as follows:

$$\log N_{\mathrm{f}}=\frac{1}{B-1}\log \mid {\mathbf{K}}_{\mathrm{av}}\mid -\frac{\log \mid {\mathbf{C}}_{\mathrm{k}}\mid }{B-1}$$ 71

where $k_{\mathrm{c}}=B-1$, $C=-\frac{\log \mid {\mathbf{C}}_{\mathrm{k}}\mid }{B-1}$, can be obtained as follows:

$$\log N_{\mathrm{f}}=\frac{1}{k_{\mathrm{c}}}\cdot \log \mid {\mathbf{K}}_{\mathrm{av}}\mid +C$$ 72

Multiplicative effect of spin correlation amplification

During the process of material fatigue, the initial damage concentration zone (such as the first layer) can cause electron spin to deflect (for example, with an amplitude of $a\%$). Through interlamellar exchange interactions, this bias is gradually amplified by the subsequent fatigue dislocation levels. Specifically, each layer amplifies the received deflection by a factor of one (the amplification factor may vary by layer; for simplicity, we assume that each layer has an amplification factor of $1+a\%$). Therefore, after n-layer amplification, the spin deflection amplitude of the final damage concentration zone reaches ${\left(1+a\%\right)}^n$ times the initial deflection. Among them, the actual amplification factor of each layer may not be the same. Here, $a\%$ is only used to illustrate the operating principle.

MagDrift data processing

Displacement and magnetic strength data were recorded simultaneously during the experimental process. When the sine wave loading method was used, the displacement data exhibited a standard sine waveform in each cycle. By identifying the peak and valley positions of the displacement data within a cycle, the corresponding magnetic intensity data points at that moment could be determined. To distinguish the magnetic intensity changes caused by fatigue from those caused by the load, we defined MagDrift as the amplitude change between the valley point c and the peak point d (Supplementary Fig. 2d). Setting the initial MagDrift at the start of loading as the fatigue-free state, the MagDrift changes induced from the onset of loading to fatigue fracturing reflected the magnetic flux intensity variations that occurred during fatigue. (Theoretically, zero-stress magnetic strength could also indicate fatigue-induced changes, but achieving true zero stress at a 20-Hz loading frequency is impractical because of machine vibrations.) On this basis, the MagDrift data could be obtained by subtracting the peak value from the corresponding valley value in the magnetic intensity data and extracting the minimum value of the difference every second.

MagDrift features a self-compensation ability

Assume that the effects of the surrounding environment and the Earth’s magnetic field on the magnetic field strength of the sample are denoted as ${\mathit{△}\mathbf{B}}_{\mathrm{E}}$ and that the effects of changes in the surrounding temperature and specimen surface temperature on the magnetic field strength of the sample are denoted as ${\mathit{△}\mathbf{B}}_{\mathrm{T}}$. During fatigue loading, when the external load is ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, the combined effects of the magnetic force, external load, surrounding environment, and changes in the surface temperature of the sample are as follows:

$$F_{\mathrm{m}0}-\mathrm{g}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)-\mathrm{f}\left({\mathit{\Delta }\mathbf{B}}_{\mathrm{E}}\right)-\mathrm{f}\left({\mathit{\Delta }\mathbf{B}}_{\mathrm{T}}\right)$$ 73

When the external load is ${\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}$, the combined effects of the magnetic force, external load, surrounding environment, and changes in the surface temperature of the sample are as follows:

$$F_{\mathrm{m}1}-\mathrm{g}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)-\mathrm{f}\left({\mathit{\Delta }\mathbf{B}}_{\mathrm{E}}\right)-\mathrm{f}\left({\mathit{\Delta }\mathbf{B}}_{\mathrm{T}}\right)$$ 74

Among them, $\mathrm{f}\left({\mathit{△}\mathbf{B}}_{\mathrm{E}}\right)$ and $\mathrm{f}\left({\mathit{△}\mathbf{B}}_{\mathrm{T}}\right)$ consider the effects of changes in the surrounding environment and the surface temperature of the test sample, respectively, during cyclic loading. The magnetic force difference can be obtained by subtracting Eq. (74) from Eq. (73) as follows:

$$F_{\mathrm{mPP}}=F_{\mathrm{m}1}-\mathrm{g}\left({\sigma }_{\mathrm{m}}+{\sigma }_{\mathrm{L}}\right)-F_{\mathrm{m}0}+\mathrm{g}\left({\sigma }_{\mathrm{m}}-{\sigma }_{\mathrm{L}}\right)$$ 75

$\mathrm{f}\left({\mathrm{\Delta }\mathbf{B}}_{\mathrm{E}}\right)$ and $\mathrm{f}\left({\mathrm{\Delta }\mathbf{B}}_{\mathrm{T}}\right)$ are completely eliminated, and the MagDrift is related only to the internal state of the material.

The MagDrift metric is designed as a differential measure, which inherently suppresses low-frequency environmental drift by canceling common-mode magnetic perturbations.

Fatigue testing under typical cyclic loading conditions

We selected three typical cyclic stress states for analyzing the strain, displacement, and magnetic strength changes induced under fatigue loading conditions.

Initially, there was a slight mean strain increase, followed by minimal variation throughout most of the cycle, culminating in a rapid increase immediately prior to fracture. Regardless of the degree of cyclic stress, the strain waveform remained consistent, with the peak-to-peak strain variation remaining virtually constant until a notable deviation occurred just before a fatigue fracture. The strain amplitude changes observed under the three stress conditions were approximately 0.42%, 0.43%, and 0.43%, respectively. The strain change proportions from the beginning to near fracture under the three stress conditions were approximately 2.17%, 1.01%, and 0.42%, respectively.

In contrast with the subtle strain changes observed during fatigue, the magnetic strength and peak-to-peak values of spin magnetic moment correlation amplification under cyclic loading exhibited pronounced variations. The magnetic strength decreased rapidly at the onset of loading and decreased gradually as cyclic loading progressed. Notably, a rapid decrease in magnetic strength preceded fatigue fracture, significantly earlier than the onset of rapid strain changes. Compared with the initial rate, the rate of change in magnetic strength prior to fracturing increased significantly. The proportions of magnetic strength amplitude changes induced under the three stress conditions were 31.46%, 32.37%, and 27.28%, respectively.

MagDrift underwent an abrupt change as fracturing approached, which was significantly earlier and greater than the displacement amplitude variation. The variation ratios of MagDrift under the three cyclic loading forces were 22.45%, 12.86%, and 9.67%, respectively.

Comparing the magnetic strength amplitude changes due to loading to the strain amplitude changes and comparing MagDrift changes to material elongation changes due to fatigue across 19 sample tests, we found that the magnetic strength changes induced due to loading were 28.61 to 86.72 times greater than the strain changes. Similarly, the rate of change in MagDrift due to fatigue was 8.05 to 40.20 times greater than the rate of material elongation caused by fatigue (Fig. 2).

Evidently, compared with mechanical variables, magnetic strength is significantly more sensitive to fatigue changes.

The correlation between magnetic changes and the formation, evolution, or crack nucleation of dislocation structures associated with fatigue

During the initial fatigue loading period, the accelerated change in the magnetic signal is a macroscopic response to the drastic microstructural transformations within the material. This process is accompanied by intense and localized changes in atomic spacing, which synchronously modulate the interatomic bonding forces and exchange interactions. The accumulation of such microdamage is converted into an observable macroscopic magnetic signal through a spin correlation amplification mechanism. Therefore, the rapid change in the magnetic signal during the initial stage is essentially a direct reflection of the rapid establishment of dislocation structures and the accumulation of microdamage inside the material (Supplementary Fig. 10b).

With the increase in the number of cycles, the magnetic signal enters a stage of slow and stable change, which corresponds to the crack initiation stage that accounts for the longest proportion of the fatigue life. At this point, the evolution of dislocation structures tends to saturate, forming structures, such as persistent slip bands (PSBs). Plastic deformation becomes highly localized, accompanied by cumulative yet relatively slow irreversible changes in atomic spacing. Meanwhile, the irreversible slip within these shear bands continuously generates extrusions and intrusions on the material surface, leading to the progressive accumulation of surface damage. Due to the relatively constant rate of damage evolution during this stage, the macroscopic manifestation is a stable, linear, or near-linear slow increase in the magnetic signal. This stage constitutes the majority of the fatigue life and represents the accumulation process during which damage gradually accumulates until it reaches a qualitative change. (Supplementary Fig. 10c).

When cyclic loading enters the final phase, the renewed accelerated change in the magnetic signal marks the transition of the fatigue process into the crack propagation stage. Once the stress concentration at the surface intrusions reaches a critical threshold, microcracks begin to initiate and propagate. The most severe periodic changes in atomic spacing occur at the crack tip, where atomic bonds are repeatedly stretched until fracture, resulting in a sharp increase in the local damage rate. This highly localized and intense damage is again converted into an accelerated change in the macroscopic magnetic signal through a spin correlation amplification method. Thus, the second acceleration of the magnetic signal serves as a clear indicator of the transition of fatigue cracks from initiation to stable propagation, predicting the imminent final fatigue fracture of the material. When the magnetic signal exhibits a near-cliff-like change, the metallic material fractures immediately (Supplementary Fig. 10d).

Basis for adopting an average slope of -1.54

For the four types of metal materials, we calculated the slopes of their respective M-N curves and evaluated their fitting degrees: the slope of pure iron was -1.26075, and its fitting degree was 0.9723; the slope of 45 steel was -1.49818, and its goodness of fit was 0.9626; the slope of Q235 steel was -1.80484, and its goodness of fit was 0.9697; and the slope of the lasso steel was -1.61185, and its fitting degree was 0.9509. The average slope of the M-N curves for the four materials was -1.54391. The absolute value of the average slope (1.54391) was very close to the relative value of the α-Fe exchange integral as a function of the distance between neighboring electrons, as summarized by Néel. Therefore, we further evaluated the fitting effect of the M-N curves for each material using -1.54 as the uniform slope value. The results revealed that the fitting degrees of the four ferromagnetic metals were still greater than 0.9; specifically, pure iron had a fit of 0.9261, 45 steel had a fit of 0.9649, Q235 steel had a fit of 0.9525, and lasso steel had a fit of 0.9521. Since the selected materials were all ferromagnetic metals, we speculate that the average slope may macroscopically reflect the degree of electron cloud overlap in ferromagnetic metals. Therefore, this article ultimately adopted -1.54 as the unified slope value to define the M-N curve slopes of ferromagnetic materials.

Fatigue life evaluation method

To evaluate the final fatigue life using early cyclic loading data, we devised an analysis method, taking soft magnetic iron as an illustrative example.

First, the spin magnetic moment correlation detection method was employed to conduct fatigue experiments on 19 samples under varying stress levels, during which time the magnetic strength was determined throughout the fatigue process. The MagDrift curve, which represents the maximum peak-to-peak magnetic intensity per second during the loading cycle, was calculated. A linear fitting curve was then fitted to the average rate of change in MagDrift to determine the corresponding $k_{\mathrm{av}}$ values. The fatigue failure lives and their corresponding $k_{\mathrm{av}}$ values, obtained at different stress levels, were subsequently plotted as M-N curves. From these curves, the M-N linear fitting equation was derived as follows:

$$\log ∣k_{\mathrm{av}}∣=k_{\mathrm{c}}\log N+C$$ 76

Then, we obtain:

$$N=10^{\frac{1}{k_{\mathrm{c}}}\left(\log ∣k_{\mathrm{av}}∣+C\right)}$$ 77

where $k_{\mathrm{c}}$ = -1.261 and $C$ = -0.167.

Substituting the $k_{\mathrm{av}}$ obtained from the MagDrift values observed for the early loading cycles (5%, 6%, 7%, 8%, 9%, 10%, 20%, 30%, 40%, 50%, and 60% of the fatigue failure life) of the 19 samples into the aforementioned equation enabled their respective fatigue failure lives to be calculated. The calculated fatigue failure lives were subsequently correlated with the actual fatigue failure lives of the 19 samples using the following equation.

$$r=\frac{\sum \left(x-\overline{x}\right)\left(y-ȳ\right)}{\sqrt{\sum {\left(x-\overline{x}\right)}^2\sum {\left(y-ȳ\right)}^2}}$$ 78

where $x$ is the calculated fatigue failure life and $y$ is the actual fatigue failure life.

The correlations between the calculated fatigue life and actual life before 5%, 6%, 7%, 8%, 9%, 10%, 20%, 30%, 40%, 50%, and 60% of the loading cycles were 0.737, 0.762, 0.834, 0.886, 0.907, 0.912, 0.961, 0.977, 0.986, 0.993, and 0.994, respectively.

Early warning algorithm for fatigue fracture

We propose a two-window algorithm. When the number of cycles does not exceed 300,000, starting from the 5000th loading cycle, the first window covers the 60 cycles immediately preceding the current time, and the second window covers the 1000 cycles preceding the point 3000 cycles before the current time plus an additional 60 cycles. Let $B_{\mathrm{med}}$ represent the median MagDrift value within the first window and $A_{\mathrm{med}}$, $A_{\max }$, and $A_{\min }$ represent the median, maximum, and minimum MagDrift values, respectively, within the second window. The alarm threshold is set as ${Alarm}_{\mathrm{th}}=B_{\mathrm{med}}-A_{\mathrm{med}}+\frac{A_{\max }-A_{\min }}{S_1}<0$, where $S_1$ is a constant that is experimentally determined and is also related to the rate of change exhibited by the remanent saturation magnetic flux density—a faster change corresponds to a larger value of s. For soft magnetic iron, the value of $S_1$ is 4.8, and the value for $A_{\max }-A_{\min }$ is 0.0132. When the alarm threshold falls below 0 for more than four consecutive measurements, the alarm is triggered. Once the number of cycles exceeds 300,000, the first window covers the 60 cycles immediately preceding the current time, and the second window covers the 2000 cycles preceding the point 40,000 cycles before the current time plus an additional 60 cycles. The alarm threshold is set as ${Alarm}_{\mathrm{th}}=B_{\mathrm{med}}-A_{\mathrm{med}}-\frac{A_{\max }-A_{\min }}{S_2}>0$, where $S_2$ is also a constant that is obtained through experimentation and is related to the rate of change in the remanent saturation magnetic flux density—a slower change corresponds to a smaller value of s. For soft magnetic iron, the value of $S_2$ is 1.2, and the value for $A_{\max }-A_{\min }$ is 0.0132. When the alarm threshold exceeds 0 for more than four consecutive measurements, the alarm is triggered.

Criteria for determining successful warning

The success of early warning mainly relies on the following two specific criteria. First, when the material has not yet undergone any macroscopic changes, such as visible cracks or deformations, MagDrift detects atomic-scale fatigue damage through the spin correlation amplification mechanism, and its change trend shows a significant fracture characteristic mutation point, which must have a significant correlation with the final fracture.

Second, the warning system should promptly issue a fracture warning signal upon identifying the characteristics of MagDrift mutation. The key is that the material can still safely withstand an average of thousands of external loads after issuing warnings, providing a critical time buffer for performing downtime maintenance and making response decisions. This mechanism effectively overcomes the shortcomings of the traditional fatigue detection methods in terms of their early warning capabilities.

In accordance with the above standards, this study implemented fatigue fracture warning for all 193 sets of samples. The results revealed that the MagDrift curves of all the samples exhibited significant abrupt changes before fractures occurred, and all the materials could still withstand a certain number of loading cycles after the warnings were produced.

Author contributions

B.N.Z. conceived the study, derived the theoretical formula, designed the experiments, analyzed the experimental data, and wrote the manuscript. L.S.Z. performed the experiments, analyzed the experimental data, plotted the figures and wrote the manuscript. X.D.W. derived the theoretical formula, performed the experiments, analyzed the experimental data, plotted the figures and wrote the manuscript. J.G.Y. performed the electron probe microanalysis tests, plotted some of the figures and wrote the manuscript. X.C.C. designed the experiments and analyzed the experimental data. H.N.X. searched the literature and wrote the manuscript. Z.J.Z. participated in the experiments, searched the literature and plotted some figures. X.L. participated in the experiments and plotted some of the figures. F.P.Z. participated in the experiments, analyzed the experimental data and plotted some figures. J.L.P. participated in the experiments and searched the literature. H.F.J. wrote and revised the manuscript. G.Z. revised the manuscript. All the authors contributed to the preparation of the manuscript.

Peer review

Peer review information

Nature Communications thanks Xizhang Chen, Dawei Zhang and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The complete data used in this study are available in the Dryad database [10.5061/dryad.1rn8pk13t]⁴¹.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-70290-w.