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

Domain: physics
Published: 2026-03-16
Editorial outcome: major_revision

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 3,700 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.

Introduction
Despite being phenomena that have been studied for more than 150 years, fatigue fractures remain a predominant cause of metallic structural failures1–4, 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 material5,6, 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 microscopy7,8. 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 cycle3,4,9. 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 variables17,18. Numerical simulations, which integrate multiscale modelling 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 crystals19,20. 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 2,094 major fatigue-based accidents, causing approximately 13,000 fatalities; and the bridge engineering sector had at least 172 major fatigue-based accidents, resulting in 3,600 fatalities (Methods).

Beyond ships, aviation and bridges, objects such as elevators, railways, and various pieces of large-scale load-bearing equipment harbour 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 sensitivity 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 bond force can serve as indicators for characterizing the fatigue states of materials.

While quantifying interatomic binding forces under static conditions is feasible because stable atomic positions and electron cloud distributions enable techniques such as transmission electron microscopy (TEM)33, 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 (∆𝐹 ) and exchange interaction (∆𝐹 ) in the material layers can be simplified to approximately linear relationships with the changes in the average atomic spacing (∆𝑅̅): ∆𝐹 = 𝐾 ∆𝑅̅ (Fig. 1e and Methods) and ∆𝐹 = 𝐾 ∆𝑅̅ (Fig. 1d and Methods). This relationship reflects the synchronized degradation between interatomic binding forces and exchange interactions during fatigue processes: ∆𝐹 = 𝐾∆𝐹 (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 affectNs its interlamellar exchange interactions, with both phenomena being strongly correlated and synchronous.

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 neighbouring 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).

According to the experimental results, we discovered a significant linear correlation between the average MagDrift change rate (𝑘 ) 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 values: longer fatigue lives correspond to smaller 𝑘 values. These findings confirm that the average MagDrift change rate (𝑘 ) can be used to effectively characterize the macroscopic variation patterns of interlamellar binding forces during fatigue processes.

Analysis of the statistical effects of macroscopic magnetic flux and interlamellar binding force changes
To analyse 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 (Figs. 3b and 3c, 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 PSB3,4. 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. 4c). The correlation between the MagDrift slope and the number of cycles far exceeds that of the S-N curve.

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 and the MagDrift slope 𝑘 (Methods):
log𝑁 f = (log|𝑘 av |+𝐶) (1)

The model indicates that the fatigue life 𝑁 is linearly related to the average magnetic flux change rate 𝑘 on a logarithmic scale. Here, 𝑘 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 and 𝑁 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 bonds39,40.

Drawing from the existing experimental data and an analysis of the results, we found that the smaller the value of 𝑘 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 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 3,700 hours of testing (the criteria for judgement 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 by processing collected data and calculating 𝑘 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% and 7.54%, 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 (Figs. 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.

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 centre 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, 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 analysing 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 = 𝑒2 +∬ 𝜓∗ ∙𝑒2( 1 − 1 − 1 )∙𝜓 𝑑𝜏 𝑑𝜏 (2)
C ab ab 1 2
𝑅 𝑟12 𝑟b1 𝑟a2

where 𝑅 is the distance between atoms, 𝑟 is the distance between electrons, and 𝑟 and 𝑟 are the distances between atoms and adjacent electrons, respectively. 𝜓 and 𝜓∗ 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 Equation (2) as follows:

𝑈 = ∑( e2 +∬𝜓∗ ∙𝑒2( 1 − 1 − 1 )∙𝜓 𝑑𝜏 𝑑𝜏 ) (3)
C ab ab 1 2
𝑅𝑖𝑗 𝑟𝑖𝑗 𝑟b𝑖 𝑟a𝑗

where 𝑅 is the distance between atoms 𝑖 and 𝑗, 𝑟 is the distance between electrons 𝑖 and 𝑗, and 𝑟 and 𝑟 are the distances between atom a and adjacent electron 𝑖 and between atom 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 𝑟̅ , to simplify the equivalent overall wave function.

1 ≡ ∑( 1 +∬𝜓∗ ∙( 1 − 1 − 1 )∙𝜓 𝑑𝜏 𝑑𝜏 ) (4)
𝑟c ̅ 𝑅𝑖𝑗 ab 𝑟𝑖𝑗 𝑟b𝑖 𝑟a𝑗 ab 1 2

where 𝑟̅ represents the average Coulomb interaction distance between two layers of atoms.

Therefore,

𝑈 = 𝑒2 (5)
C 𝑟c ̅

Taking the derivative of 𝑈 with respect to the average interatomic distance 𝑅̅ between two layers of atoms yields the following:

𝐹 = 𝑑𝑈C = − 𝑒2 𝑑𝑟c ̅ (6)
C 𝑑𝑅̅ 𝑟c ̅ 2 𝑑𝑅̅

where 𝐹 represents the average binding force between two layers of atoms.

By expanding 𝑟̅ in terms of 𝑅̅ using Taylor series expansion, we can obtain 𝑟̅ = 𝐶 + 𝐶 𝑅̅ +𝐶 𝑅̅2 +⋯, where 𝐶 , 𝐶 and 𝐶 are constants.

Ignoring the higher-order terms and considering only the case where the atomic spacing increases slightly from its lowest energy position, where 𝑟̅ linearly changes with respect to 𝑅̅, we can obtain the following equation:

𝐹 = − 𝑒2𝐶C1 = − 𝑑C0 (7)
C (𝐶C0+𝐶C1𝑅̅)2 (𝑅̅+𝑑C1 )2

where 𝑑 = 𝐶C0 and 𝑑 = 𝑒2 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 = ∬𝜑∗ ∙𝑚( 1 − 1 − 1 )∙𝜑 𝑑𝜏 𝑑𝜏 (8)
m ab ab 1 2
𝑟12 𝑟b1 𝑟a2

The wave functions of the electrons that contribute to the magnetic exchange energy are represented by 𝜑 , and their conjugates are denoted as 𝜑∗ . 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 equation (8) as follows:

𝑈 = ∑∬𝜑∗ ∙𝑚( 1 − 1 − 1 )∙𝜑 𝑑𝜏 𝑑𝜏 (9)
m ab ab 1 2
𝑟𝑖𝑗 𝑟b𝑖 𝑟a𝑗

where 𝑟 is the distance between electrons 𝑖 and 𝑗 and 𝑟 and 𝑟 are the distances between atom a and adjacent electron 𝑖 and between atom b and adjacent electron 𝑗, respectively.

We define the following:

1 ≡ ∑∬𝜑∗ ∙( 1 − 1 − 1 )∙𝜑 𝑑𝜏 𝑑𝜏 (10)
𝑟m ̅ ab r𝑖𝑗 𝑟b𝑖 𝑟a𝑗 ab 1 2

Therefore,

𝑈 = 𝑚 (11)
m 𝑟m ̅

The overlined notation 𝑟̅ represents the equivalent exchange interaction distance between two layers of atoms. Similarly, 𝑟̅ is a function of the average distance 𝑅̅ between these two layers of atoms. Under other constant conditions, once the average distance 𝑅̅ between the layers is determined, 𝑟̅ takes a unique value.

By taking the derivative of 𝑈 with respect to 𝑅̅, we can obtain the following:

𝐹 = 𝑑𝑈m = − 𝑚 𝑑𝑟𝑚 ̅ (12)
m 𝑑𝑅̅ 𝑟m ̅ 2 𝑑𝑅̅

where 𝐹 represents the average magnetic force between two layers of atoms.

By expanding 𝑟̅ in terms of 𝑅̅ using Taylor series expansion, we can obtain 𝑟̅ = 𝐶 + 𝐶 𝑅̅ +𝐶 𝑅̅2 +⋯, where 𝐶 , 𝐶 and 𝐶 are constants.

Similarly, ignoring the higher-order terms and only considering the linear changes exhibited by 𝑟̅ with respect to 𝑅̅, we can obtain the following equation:

𝐹 = − 𝑚𝐶m1 = − 𝑑m0 (13)
m (𝐶m0+𝐶m1𝑅̅)2 (𝑅̅+𝑑m1 )2

where 𝑑 = 𝐶m0 and 𝑑 = 𝑚 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 𝑅̅ changes very slightly, an increase in 𝑅̅ leads to decreases in both the binding force and the magnetic force (we are only considering the effect of an increase in 𝑅̅).

When external cyclic loading is present, let the force exerted by the loading on the two layers of atoms be denoted as f(R𝜎), and let the interlamellar magnetic force caused by the loading be denoted as g(𝜎). The functional form of g(𝜎) 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 𝜎 −𝜎 , the combined effects of the binding force and loading action are as follows:

𝐹 −f(𝜎 −𝜎 ) (14)
C0 m L

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

𝐹 −g(𝜎 −𝜎 ) (15)
m0 m L

where 𝐹 = − 𝑑C0 , 𝐹 = − 𝑑m0 .
C0 (𝑅̅ 0+𝑑C1 )2 m0 (𝑅̅ 0+𝑑m1 )2

When the external loading is 𝜎 −𝜎 , the combined effects of the binding force and loading action are as follows:

𝐹 −f(𝜎 +𝜎 ) (16)
C1 m L

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

𝐹 −g(𝜎 +𝜎 ) (17)
m1 m L

where 𝐹 = − 𝑑C0 , 𝐹 = − 𝑑m0 , 𝜎 represents the average force during external cyclic loading, and 𝜎 is the fluctuation of the force during this cycle. 𝐹 is the average binding force inside the material when the external loading is 𝜎 −𝜎 , specifically at the onset of fatigue loading, and 𝐹 is the average binding force inside the material when the initial loading is set to 𝜎 +𝜎 , also at the onset of fatigue loading. Similarly, 𝐹 is the average magnetic force inside the material when the external loading is 𝜎 −𝜎 , specifically at the onset of fatigue loading, and 𝐹 is the average magnetic force inside the material when the external loading is 𝜎 +𝜎 , also at the onset of fatigue loading. Furthermore, 𝑅̅ denotes the average atomic spacing inside the material when the external loading is 𝜎 −𝜎 , specifically at the onset of fatigue loading, whereas 𝑅̅ represents the average atomic spacing inside the material when the initial loading is 𝜎 +𝜎 , also at the onset of fatigue loading.

Subtracting equation (14) from equation (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 = 𝐹 −f(𝜎 +𝜎 )−[𝐹 −f(𝜎 −𝜎 )] = f(𝜎 −𝜎 )−f(𝜎 +𝜎 )− 𝑑C0 + 𝑑C0 (18)
CPP C1 m L C0 m L m L m L (𝑅̅ 1+𝑑C1 )2 (𝑅̅ 0+𝑑C1 )2

Subtracting equation (15) from equation (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 = 𝐹 −g(𝜎 +𝜎 )−[𝐹 −g(𝜎 −𝜎 )] = g(𝜎 −𝜎 )−g(𝜎 +𝜎 )− 𝑑m0 + 𝑑m0 (19)
mPP m1 m L m0 m L m L m L (𝑅̅ 1+𝑑m1 )2 (𝑅̅ 0+𝑑m1 )2

When material fatigue is considered, the average atomic spacing 𝑅̅ increases by ∆𝑅̅, where ∆𝑅̅ represents the fatigue-induced dislocation change that occurs in the sample under external cyclic loading, and it is assumed that 𝑅̅ ≫ ∆𝑅̅.

For the external loading 𝜎 −𝜎 , the combined effects of the binding force and loading action can be written as follows:

𝐹′ −f(𝜎 −𝜎 ) (20)
C0 m L

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

𝐹′ −g(𝜎 −𝜎 ) (21)
m0 m L

where 𝐹′ = − 𝑑C0 and 𝐹′ = − 𝑑m0 .
C0 (𝑅̅ 0+∆𝑅̅+𝑑C1 )2 m0 (𝑅̅ 0+∆𝑅̅+𝑑m1 )2

When the external loading is 𝜎 −𝜎 , the increase in the distance between the two layers of atoms from their initial equilibrium position 𝑅̅ is ∆𝑅̅, and 𝑅̅ ≫ ∆𝑅̅.

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

𝐹′ −f(𝜎 +𝜎 ) (22)
C1 m L

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

𝐹′ −g(𝜎 +𝜎 ) (23)
m1 m L

where 𝐹′ = − 𝑑C0 and 𝐹′ = − 𝑑m0 .
C1 (𝑅̅ 1+∆𝑅̅+𝑑C1 )2 m1 (𝑅̅ 1+∆𝑅̅+𝑑m1 )2

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

𝐹′ = 𝐹′ −f(𝜎 +𝜎 )−[𝐹′ −f(𝜎 −𝜎 )] = f(𝜎 −𝜎 )−f(𝜎 +𝜎 )− 𝑑C0 + 𝑑C0 (24)
CPP C1 m L C0 m L m L m L (𝑅̅ 1+∆𝑅̅+𝑑C1 )2 (𝑅̅ 0+∆𝑅̅+𝑑C1 )2

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

𝐹′ = 𝐹′ −g(𝜎 +𝜎 )−[𝐹′ −g(𝜎 −𝜎 )] = g(𝜎 −𝜎 )−g(𝜎 +𝜎 )− 𝑑m0 + 𝑑m0 (25)
mPP m1 m L m0 m L m L m L (𝑅̅ 1+∆𝑅̅+𝑑m1 )2 (𝑅̅ 0+∆R̅+𝑑m1 )2

Subtracting equation (24) from equation (18) yields the relationship between the fatigue dislocation change ∆𝑅̅ and the average binding force change induced during an actual loading process:

∆𝐹 = 𝐹′ −𝐹 = [− 𝑑C0 + 𝑑C0 ]−[− 𝑑C0 + 𝑑C0 ] = [ 𝑑C0 − 𝑑C0 ]−[ 𝑑C0 − 𝑑C0 ] = 𝑑C0(∆𝑅̅2+2𝑅̅ 1∆𝑅̅+2∆𝑅̅𝑑C1) − 𝑑C0(∆𝑅̅2+2𝑅̅ 0∆𝑅̅+2∆𝑅̅𝑑C1) (26)
C CPP CPP (𝑅̅ 1+∆𝑅̅+𝑑C1 )2 (𝑅̅ 0+∆𝑅̅+𝑑C1 )2 (𝑅̅ 1+𝑑C1 )2 (𝑅̅ 0+𝑑C1 )2 (𝑅̅ 1+𝑑C1 )2(𝑅̅ 1+∆𝑅̅+𝑑C1 )2 (𝑅̅ 0+𝑑C1 )2(𝑅̅ 0+∆𝑅̅+𝑑C1 )2

Because 𝑅̅ ,𝑅̅ ≫ ∆𝑅̅, ∆𝑅̅2 in the above equation can be neglected. Therefore, equation (26) can be simplified as follows:

∆𝐹 ≈ 𝑑C0 (2𝑅̅ 1∆𝑅̅+2∆𝑅̅𝑑C1 ) − 𝑑C0 (2𝑅̅ 0∆𝑅̅+2∆𝑅̅𝑑C1 ) ＝2∆𝑅̅𝑑 [ 1 − 1 ] (27)
C (R̅ 1+dC1 )2(R̅ 1+∆R̅+dC1 )2 (𝑅̅ 0+𝑑C1 )2(𝑅̅ 0+∆𝑅̅+𝑑C1 )2 C0 (𝑅̅ 1+𝑑C1 )(𝑅̅ 1+∆𝑅̅+𝑑C1 )2 (𝑅̅ 0+𝑑C1 )(𝑅̅ 0+∆R̅+𝑑C1 )2

Equation (27) can be simplified to the following:

∆𝐹 = 𝐾 ∆𝑅̅ (28)
C C

where 𝐾 = 2𝑑 [ 1 − 1 ] and ∆𝐹 is the average variation in the interatomic binding force.
C C0 (𝑅̅ 1+𝑑C1 )(𝑅̅ 1+∆𝑅̅+𝑑C1 )2 (𝑅̅ 0+𝑑C1 )(𝑅̅ 0+∆𝑅̅+𝑑C1 )2 C

Owing to 𝑅̅ and 𝑅̅ 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 𝑅̅ and 𝑅̅ take on unique values. Given that 𝑅̅ −𝑅̅ ≫ ∆𝑅̅, ∆𝑅̅ has a negligible effect on the denominator term in 𝐾 , thus allowing us to approximate 𝐾 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 equation (25) from equation (19) yields the relationship between the change in fatigue dislocation ∆𝑅̅ and the change in the average magnetic force during actual loading:

∆𝐹 = 𝐹′ −𝐹 = [− 𝑑m0 + 𝑑m0 ]−[− 𝑑m0 + 𝑑m0 ] = [ 𝑑m0 − 𝑑m0 ]−[ 𝑑m0 − 𝑑m0 ] = 𝑑m0(∆𝑅̅2+2𝑅̅ 1∆𝑅̅+2∆𝑅̅𝑑m1) − 𝑑m0(∆𝑅̅2+2𝑅̅ 0∆𝑅̅+2∆𝑅̅𝑑m1) (29)
m mPP mPP (𝑅̅ 1+∆𝑅̅+𝑑m1 )2 (𝑅̅ 0+∆𝑅̅+𝑑m1 )2 (𝑅̅ 1+dm1 )2 (𝑅̅ 0+𝑑m1 )2 (𝑅̅ 1+dm1 )2(𝑅̅ 1+∆𝑅̅+𝑑m1 )2 (𝑅̅ 0+𝑑m1 )2(𝑅̅ 0+∆𝑅̅+𝑑m1 )2

Equation (29) can be simplified to the following:

∆𝐹 ≈ 𝑑m0 (2𝑅̅ 1∆𝑅̅+2∆𝑅̅𝑑m1 ) − 𝑑m0 (2𝑅̅ 0∆𝑅̅+2∆𝑅̅𝑑m1 ) ≈ 2∆𝑅̅𝑑 [ 1 − 1 ] (30)
m (𝑅̅ 1+𝑑m1 )2(𝑅̅ 1+∆𝑅̅+𝑑m1 )2 (𝑅̅ 0+𝑑m1 )2(𝑅̅ 0+∆𝑅̅+𝑑m1 )2 m0 (𝑅̅ 1+dm1 )(𝑅̅ 1+∆𝑅̅+𝑑m1 )2 (𝑅̅ 0+𝑑m1 )(𝑅̅ 0+∆𝑅̅+𝑑m1 )2

Equation (30) can be simplified to the following:

∆𝐹 = 𝐾 ∆𝑅̅ (31)
m m

where 𝐾 m = 2𝑑 m0 [ 1 − 1 ] . ∆𝐹 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 equation (28) and equation (31), the relationship between the binding force and magnetic force can be obtained as follows:

∆𝐹 = 𝐾∆𝐹 (32)
C C m

where 𝐾 = 𝐾C.
𝐾m

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. 2d):

𝐁 = 𝐁 ∙f(𝐹 ) (33)
1 0 m0

where 𝐁 represents the external magnetic field strength, 𝐁 represents the induced magnetic field strength of the first layer, 𝐹 denotes the interlamellar exchange interaction, and f(𝐹 ) 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:

𝐁 = 𝐁 ∙f(𝐹 ) (34)
2 1 m1

where 𝐁 is the induced magnetic strength value of the second layer and 𝐹 is the interlamellar exchange interaction.

In a similar manner, we can deduce the following:

𝐁 = 𝐁 ∙f(𝐹 ) (35)
𝑖 𝑖−1 m(𝑖−1)

where 𝐁 is the induced magnetic strength value of the 𝑖-th layer and 𝐹 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:

𝐁 = 𝐁 ∏𝑛−1f(𝐹 ) (36)
𝑛 0 𝑖=0 m𝑖

When the average interatomic distance between layers varies because of stress or fatigue, the interlamellar exchange interaction changes correspondingly by ∆𝐹 .

Then, the induced magnetic field strength of the n-th layer can be expressed as follows:

𝐁′ = 𝐁 ∏𝑛−1f(𝐹 +∆𝐹 ) (37)
𝑛 0 𝑖=0 m𝑖 m𝑖

In the case where ∆𝐹 is minimal, we have that

f(𝐹 +∆𝐹 ) ≈ f(𝐹 )+f′(𝐹 )⋅∆𝐹 (38)
m𝑖 m𝑖 m𝑖 m𝑖 m𝑖

Substituting this into 𝐁′ yields

𝐁′ = 𝐁 ∏𝑛−1[f(𝐹 )+f′(𝐹 )⋅∆𝐹 ] (39)
𝑛 0 𝑖=0 m𝑖 m𝑖 m𝑖

By taking the logarithms of 𝐁 and 𝐁 ′, we can obtain the following:

ln𝐁 = ln𝐁 +∑𝑛−1lnf(𝐹 ) (40)
𝑛 0 𝑖=0 m𝑖

ln𝐁′ = ln𝐁 +∑𝑛−1ln[f(𝐹 )+f′(𝐹 )⋅∆𝐹 ] (41)
𝑛 0 𝑖=0 m𝑖 m𝑖 m𝑖

Since ∆𝐹 is a small quantity, ln [f(𝐹 )+f′(𝐹 )⋅∆𝐹 ] ≈ lnf(𝐹 )+ f′(𝐹m𝑖 )⋅∆𝐹m𝑖 .

Therefore, equation (41) can be simplified as follows:

ln𝐁′ ≈ ln𝐁 +∑𝑛−1(lnf(𝐹 )+ f′(𝐹m𝑖 )∆𝐹m𝑖) (42)
𝑛 0 𝑖=0 m𝑖 f(𝐹m𝑖 )

Subtracting equation (40) from equation (42) yields

ln𝐁′ −ln𝐁 ≈ ∑𝑛−1f′(𝐹m𝑖 )⋅∆𝐹m𝑖 (43)
𝑛 𝑛 𝑖=0 f(𝐹m𝑖 )

Then,

ln( 𝐁𝑛 ′ ) ≈ ∑𝑛−1f′(𝐹m𝑖 )∆𝐹m𝑖 (44)
𝐁𝑛 𝑖=0 f(𝐹m𝑖 )

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

∆𝐁 = 𝐁′ −𝐁 = 𝐁 ( 𝐁𝑛 ′ −1) ≈ 𝐁 [exp(∑𝑛−1f′(𝐹m𝑖 )⋅∆𝐹m𝑖)−1] (45)
𝑛 𝑛 𝑛 𝑛 𝐁𝑛 0 𝑖=0 f(𝐹m𝑖 )

Expanding exp(𝑥) in a Taylor-wise manner and removing higher-order terms, exp(𝑥) ≈ 1+𝑥, where 𝑥 = ∑𝑛−1f′(𝐹m𝑖 )∆𝐹m𝑖, and equation (45) can be simplified as follows:

∆𝐁 = 𝐁′ −𝐁 ≈ 𝐁 [∑𝑛−1f′(𝐹m𝑖 )⋅∆𝐹m𝑖] (46)
𝑛 𝑛 𝑛 𝑛 𝑖=0 f(𝐹m𝑖 )

Substituting equation (36) into equation (46) yields:

∆𝐁 ≈ 𝐁 ∏𝑛−1f(𝐹 )∑𝑛−1f′(𝐹m𝑖 )∆𝐹m𝑖 (47)
𝑛 0 𝑘=0 m𝑘 𝑖=0 f(𝐹m𝑖 )

Owing to the correlation between ∆𝐹 and the change in the interlamellar atomic spacing ∆𝑅 (i.e., ∆𝐹 = 𝐾 ∆𝑅 ), we can conclude the following:

∆𝐁 ≈ 𝐁 ∏𝑛−1f(𝐹 )∑𝑛−1f′(𝐹m𝑖 )𝐾m∆𝑅𝑖 (48)
𝑛 0 𝑘=0 m𝑘 𝑖=0 f(𝐹m𝑖 )

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 (∆𝑅 ) between adjacent layers are equal, i.e., ∆𝑅 = 𝐾 ∆𝑅̅ . Furthermore, the exchange interaction functions 𝑓(𝐹 ) of different layers are also identical; that is, 𝑓(𝐹 ) = 𝐾 𝑓(𝐹 ).

Substituting ∆𝑅 = 𝐾 ∆𝑅̅ and f(𝐹 ) = 𝐾 f(𝐹 ), f′(𝐹 ) = 𝐾 f′(𝐹 ) into (48) yields:

∆𝐁 ≈ 𝐁 ∏𝑛−1𝐾 f(𝐹 )∑𝑛−1𝐾 f′(𝐹 )𝐾 ∆R̅ = 𝐁 ∏𝑛−1𝐾 f(𝐹 )∑𝑛−1f′(𝐹 )𝐾 ∆𝑅̅ = 𝐁 f(𝐹 )𝑛∏𝑛−1𝐾 ∑𝑛−1𝐾 𝑓′(𝐹 )𝐾 ∆𝑅̅ (49)
𝑛 0 𝑘=0 m𝑅𝑘 m 𝑖=0 m𝑅𝑖 m m 0 𝑘=0 m𝑅𝑘 𝑖=0 m𝑅𝑖 m 0 m 𝑘=0 m𝑅𝑘 𝑖=0 𝑅𝑖 m

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 (∆𝐹 = 𝐾 ∆𝑅̅) when a slight change in the atomic spacing ∆𝑅̅ occurs into equation (49), we obtain the following equation:

∆𝐁 ≈ 𝐁 f(𝐹 )𝑛−1f′(𝐹 )𝐾 ∆F𝐶∏𝑛−1𝐾 ∑𝑛−1𝐾 (50)
𝑛 0 m m m 𝑘=0 m𝑅𝑘 𝑖=0 𝑅𝑖

where 𝐴 = ∏𝑛−1𝐾 ∑𝑛−1𝐾 . Equation (50) can be rewritten as follows:
F 𝑘=0 m𝑅𝑘 𝑖=0 𝑅𝑖

∆𝐁 ≈ 𝐁 f(𝐹 )𝑛−1f′(𝐹 )𝐾 ∆𝐹C𝐴 (51)
𝑛 0 m m m F

In equation (51), except for the term related to the number of layers n contained in 𝐴 , all the other terms are independent of n. Let us derive 𝐴 separately below.
F

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 = 1+𝛿 ,|𝛿 | ≪ 1 (52)
m𝑅𝑘 𝑘 𝑘

𝐾 = 1+𝜖 ,|𝜖 | ≪ 1 (53)
𝑅𝑖 𝑖 𝑖

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

𝛿̅ = 1 ∑𝑛−1𝛿 (54)
𝑘=0 𝑘
𝑛

𝜖̅= 1 ∑𝑛−1𝜖 (55)
𝑖=0 𝑖
𝑛

Substituting 𝐾 = 1+𝛿 into the continuous multiplication term in 𝐴 yields
m𝑅𝑘 𝑘 F

∏𝑛−1𝐾 = ∏𝑛−1(1+𝛿 ) ≈ exp(∑𝑛−1(1+𝛿 )) ≈ exp(∑𝑛−1(𝛿 )) (56)
𝑘=0 m𝑅𝑘 𝑘=0 𝑘 𝑘=0 𝑘 𝑘=0 𝑘

Utilizing Taylor expansion ln(1+𝑥) ≈ 𝑥 (ignoring higher-order terms), then,

∏𝑛−1𝐾 ≈ exp (𝑛𝛿̅) (57)
𝑘=0 m𝑅𝑘

Substituting the continuous addition term ∑𝑛−1𝐾 into 𝐴 , where 𝐾 = 1+𝜖 , yields
𝑖=0 𝑅𝑖 F 𝑅𝑖 𝑖

∑𝑛−1𝐾 = ∑𝑛−1(1+𝜖 ) = 𝑛+∑𝑛−1(𝜖 ) = 𝑛(1+𝜖)̅ (58)
𝑖=0 𝑅𝑖 𝑖=0 𝑖 𝑖=0 𝑖

Substituting equations (57) and (58) into 𝐴 yields the following:
F

𝐴 = ∏𝑛−1𝐾 ∑𝑛−1𝐾 ≈ exp (𝑛𝛿̅)∙𝑛(1+𝜖)̅ (59)
F 𝑘=0 m𝑅𝑘 𝑖=0 𝑅𝑖

Equation (51) can be simplified as follows:

∆𝐁 = 𝐊 𝐴 ∆𝐹 (60)
𝑛 B F C

where 𝐊 = 𝐁0f(𝐹m )𝑛−1f′(𝐹m )𝐾m is related only to the material and excitation magnetic field and can be regarded as a constant under small variations. 𝐴 ≈ exp (𝑛𝛿̅)∙𝑛(1+𝜖)̅ 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 ∆𝐁 measured by an external Hall sensor is actually a mapping relationship of the internal magnetic intensity ∆𝐁 , we can conclude the following:

∆𝐁 = 𝐾 ∆𝐁 (61)
H H n

where 𝐾 is a constant, assuming that the measurement distance and environmental factors remain constant. Then, we can obtain

∆𝐁 = 𝐊 ∆𝐹 (62)
H H C

where 𝐊 = 𝐾 𝐊 𝐴 .
H E B 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 Δ𝐹 also increases in a power-law manner with the number of cycles 𝑁:

Δ𝐹 = 𝐴𝑁𝐵 (63)
C

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 equation (62), we can obtain the following:

Δ𝐁 = 𝐊 ⋅𝐴𝑁𝐵 (64)
H H

We define 𝐾 as the rate of change measured in the magnetic flux during the fatigue process with respect to the number of loading cycles:

𝐊 = d(Δ𝐁H) (65)
av d𝑁

Substituting equation (64) yields

𝐊 = 𝐊 𝐴⋅𝐵⋅𝑁𝐵−1 (66)
av H

where 𝐂 = 𝐊 𝐴𝐵; therefore, equation (66) can be simplified as follows:
k H

𝐊 = 𝐂 𝑁𝐵−1 (67)
av k

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

𝐊 = 𝐂 𝑁𝐵−1 (68)
av k f

Since 𝐾 is usually negative (indicating a gradual decrease in magnetic field changes), we take its absolute value as follows:

|𝐊 | = |𝐂 |𝑁𝐵−1 (69)
av k f

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

log|𝐊 | = log|𝐂 |+(𝐵−1)log𝑁 (70)
av k f

Therefore, 𝑁 can be obtained as follows:

log𝑁 = 1 log|𝐊 |− log|𝐂 | (71)
f 𝐵−1 av 𝐵−1

where 𝑘 = 𝐵−1, 𝐶 = − log|𝐂 | , can be obtained as follows:
c 𝐵−1

log𝑁 = 1 ⋅log|𝐊 |+𝐶 (72)
f 𝑘c av

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 (1+𝑎%)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 ∆𝐁 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 ∆𝐁 . During fatigue loading, when the external load is 𝜎 −𝜎 , the combined effects of the magnetic force, external load, surrounding environment, and changes in the surface temperature of the sample are as follows:

𝐹 −g(𝜎 −𝜎 ) −f(∆𝐁 )−f(∆𝐁 ) (73)
m0 m L E T

When the external load is 𝜎 −𝜎 , the combined effects of the magnetic force, external load, surrounding environment, and changes in the surface temperature of the sample are as follows:

𝐹 −g(𝜎 +𝜎 )−f(∆𝐁 )−f(∆𝐁 ) (74)
m1 m L E T

Among them, f(∆𝐁 ) and f(∆𝐁 ) 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 equation (74) from equation (73) as follows:

𝐹 = 𝐹 −g(𝜎 +𝜎 )−𝐹 +g(𝜎 −𝜎 ) (75)
mPP m1 m L m0 m L

f(∆𝐁 ) and f(∆𝐁 ) are completely eliminated, and the magnetic drift 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 analysing 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. 3).

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 neighbouring 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 values. The fatigue failure lives and their corresponding 𝑘 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 | = 𝑘 log𝑁+𝐶 (76)
av c

Then, we obtain

𝑁 = 10(1/𝑘c)(log|𝑘av |+𝐶) (77)

where 𝑘 7-1.261 and 𝐶7-0.167.

Substituting the 𝑘 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 = ∑(𝑥−𝑥̄)(𝑦−𝑦̄) / √∑(𝑥−𝑥̄)2∑(𝑦−𝑦̄)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, 0.911, 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 5,000th loading cycle, the first window covers the 60 cycles immediately preceding the current time, and the second window covers the 1,000 cycles preceding the point 3,000 cycles before the current time plus an additional 60 cycles. Let 𝐵 represent the median MagDrift value within the first window and 𝐴 , 𝐴 , and 𝐴 represent the median, maximum, and minimum MagDrift values, respectively, within the second window. The alarm threshold is set as 𝐴𝑙𝑎𝑟𝑚 = 𝐵 −𝐴 + (𝐴max−𝐴min)/𝑆1 < 0 , where 𝑆 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 is 4.8, and the value for 𝐴 −𝐴 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 2,000 cycles preceding the point 40,000 cycles before the current time plus an additional 60 cycles. The alarm threshold is set as 𝐴𝑙𝑎𝑟𝑚 = 𝐵 −𝐴 − (𝐴max−𝐴min)/𝑆2 > 0, where 𝑆 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 is 1.2, and the value for 𝐴 −𝐴 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.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This study introduces a magneto-mechanical methodology that correlates fatigue-induced variations in magnetic flux with bond weakening in ferromagnetic metals, enabling accurate fatigue life prediction. The approach offers reliable early-warning signals prior to fracture, marking a promising step toward predictive maintenance for large-scale engineering systems.\n\nThe concept is both novel and compelling, particularly in its attempt to link dislocation density evolution to fatigue behavior through magnetic measurements. The manuscript is relatively well written, though it would benefit from thorough proofreading to improve scientific clarity and reduce the impression of AI-generated language.\n\nThe authors propose an innovative approach to estimate dislocation density and use it to assess fatigue strength. However, the manuscript currently includes several oversimplifications and lacks critical information necessary to fully validate the method for fatigue evaluation. Importantly, the proposed approach is based on averaged measurements, lacking the spatial resolution necessary to accurately capture the localized nature of fatigue damage, an issue the authors themselves acknowledge in the introduction. In its current form, the method may offer no more insight than conventional strain gauge measurements.\n\nMoreover, the interpretation of what is being measured magnetically requires further clarification. The dislocation structures that evolve during fatigue can vary substantially in nature, yet the proposed technique appears to treat them equivalently, potentially overlooking key distinctions in how different dislocation configurations influence fatigue behavior. For instance, dislocation annihilation, slip reversibility, an important mechanism that may still lead to micro-damage, is unlikely to be captured by the magnetic signal, despite its relevance to fatigue failure.\n\nThe influence of loading conditions is also insufficiently addressed. As an example, under ratcheting conditions, dislocation density may increase significantly, but the associated damage mechanisms can differ from those occurring under non-ratcheting fatigue. Such distinctions are critical and should be reflected in the analysis.\n\nA more appropriate first step would have been to investigate dislocation density evolution under monotonic tensile loading. This would have allowed the authors to better understand the limitations and resolution of their technique before applying it to more complex fatigue scenarios. Plenty of other comments are provided below. For all of these reasons, I recommend a major revision.\n\nThe citations associated with the data in Fig. 1 of the Extended Data should be mentioned.\n\nP1L36: The current body of fatigue research includes substantial work on the 3D characterization of microstructure effects. The authors have omitted this important body of literature and should include relevant references.\n\nP2L39: The term “progression of fatigue” is not commonly used in the fatigue community. The authors should clarify what they mean by this phrase.\n\nP2L43: The authors should cite key literature for each experimental technique mentioned to provide context and demonstrate the state of the art.\n\nWhile persistent slip bands (PSBs) are important features in fatigue damage, they are not the only structures involved. Other microstructural features such as veins, fatigue shear bands, and cell structures also play critical roles in controlling fatigue behavior. These are not addressed by the authors and should be included for a more comprehensive discussion.\n\n\"PSB formed during fatigue processes... occupy approximately 9.2%\": This value is highly material- and microstructure-dependent and should not be generalized. The authors should emphasize the conditional nature of this observation.\n\nThe claim that techniques such as HR-DIC cannot capture deformations below 100 nm is inaccurate. HR-DIC , as cited by the authors in the mentioned reference, can achieve displacement sensitivity in the 7–9 nm range and spatial resolutions around 20 nm.\n\nThe manuscript would benefit from a thorough proofreading to align with scientific standards in experimental mechanics and fatigue. Currently, the style resembles AI-generated text (e.g., ChatGPT), which may detract from its scientific credibility. The language should be revised for precision and field-appropriate terminology.\n\nThe authors have not discussed the use of advanced numerical simulations for fatigue life prediction and damage monitoring. Modern structural material design increasingly integrates such tools, particularly in industrial applications, and this should be acknowledged.\n\nThe phrase “tracking the sliding of grain boundaries at the atomic scale” requires clarification. If the authors are referring to grain boundary sliding, they should specify, as this mechanism differs fundamentally from PSB formation and is typically associated with high-temperature deformation.\n\n“Nanoscale atomic dislocation”: This expression is unconventional. Dislocations are already atomic-scale defects, so the phrasing is redundant or misleading. The authors should revise this for clarity and scientific accuracy.\n\nformatting: There is a typographical error in the phrase “and Methods)This relationship”, where a space is missing.\n\nThe authors should define the fatigue ratio and loading conditions used. More importantly, they should consider cyclic plasticity mechanisms such as ratcheting and slip irreversibility and also different loading conditions such negative and positive load ratios. In certain conditions, dislocation accumulation may occur without producing significant surface extrusion, which would influence the MagDrift signal identifically but produce different fatigue response.\n\nPSB morphology varies significantly between pure metals and high-strength alloys. The authors do not provide a rationale for how their average measurement technique is expected to capture these differences. It is difficult to imagine that such averaging can reliably identify damage processes, especially considering that fatigue failure is typically governed by the most extreme localized plastic events rather than average behavior.\n\nSimilarly, the evolution of the MagDrift signal must be interpreted in the context of strain localization, which the current approach may not capture. In this respect, the method is conceptually akin to applying a macro-scale extensometer or strain gauge, useful but lacking the spatial resolution needed for a predictive understanding of localized fatigue damage.\n\nIn fatigue, strain hardening or softening phenomena often occur, altering the dislocation structure over time. The manuscript does not discuss how these effects would influence MagDrift or whether the method can differentiate between them.\n\nDislocation structures evolve depending on the slip planarity, which has a direct impact on fatigue behavior. These structural changes should be considered if the authors aim to correlate dislocation behavior with fatigue properties.\n\nThe approach is reminiscent of nonlinear acoustic techniques used to monitor dislocation density evolution. However, as in those methods, without spatial correlation at the microstructural level, dislocation density alone cannot reliably predict fatigue life.\n\nFor example, in the Very High Cycle Fatigue (VHCF) regime, no measurable macroscopic strain evolution is typically observed, and this method would likely fail to detect damage evolution. Similarly, when fatigue is initiated at large defects, the deformation is highly localized and may not be captured by spatially averaged signals.", "reviewer_2": "This paper addresses the fatigue failure of metallic structures and proposes an innovative method to overcome the limitations of conventional fatigue detection techniques, which often suffer from insufficient accuracy and delayed damage recognition. The authors introduce a novel approach that predicts fatigue damage and provides fracture warnings by monitoring the external magnetic field of metallic specimens. By leveraging external magnetic field excitation to trigger the decoupling effects of exchange interactions, the method achieves cooperative amplification of spin correlation effects. A mapping relationship is established between fatigue-induced magnetic flux changes and the weakening of binding force.\n\nThe primary strength of the study lies in its non-destructive nature, overcoming the limitations of traditional fatigue testing methods that cannot capture microstructural changes during early fatigue damage. The paper successfully links microscopic structural parameters such as bond energy distribution and lattice defect evolution to macroscopic fatigue life prediction.\n\nOverall, the study is interesting and innovative. However, some methodological descriptions, including experimental design, require clearer elaboration. Certain technical details also need improvement. A major revision is recommended before the paper can be considered for acceptance. Specific comments are as follows:\n\n1. In the “Fatigue Life Prediction Model” section, the authors propose an M-N curve analogous to the classical S-N curve using MagDrift values. Line 335 mentions the use of an average slope across four metal materials. Why did the authors not apply individual slopes for each material?\n\n2. In the section “Macroscopic Magnetic Mapping of Microscopic Fatigue Signals: Spin Magnetic Moment Correlation Amplification Under External Magnetic Excitation,” the peak-to-peak value is chosen as the representative index (MagDrift). The rationale for selecting this index should be clearly explained.\n\n3. In Figure 3, the number of curves makes the plot cluttered and difficult to read. It is strongly recommended that the figure be enlarged or divided into separate plots for clarity.\n\n4. In the “Fatigue Failure Warning” section (Line 373), the manuscript states: “the Stage 2 system achieves 100% reliable early warning success rate.” What is the specific criterion for determining whether the early warning is successful?\n\n5. In the “Methods: Preparation of test specimens and experimental procedures” section, the reviewer notes that the loading grips and fixtures are likely made of ferromagnetic materials. Additionally, large instruments in the surrounding environment may generate ambient magnetic fields. Could these factors affect the test results?\n\n6. Also in this section, the authors mention using five N35 magnets. Please clarify whether these are permanent magnets, specify the initial magnetic field strength they generate, and explain the rationale for choosing this level of excitation.\n\n7. During loading, the authors use a frequency of 20 Hz. According to the reviewer’s knowledge, such a high frequency can cause surface temperature to rise in the specimen. Has this potential influence on the evolution of magnetization been considered or evaluated?\n\n8. When measuring the magnetic field, the Hall sensor is placed beneath the specimen. Why was this location selected? Although the manuscript states that a linear relationship exists between magnetic changes measured at the remote end and the variation in binding force at the weakest section, the specimen is not of uniform cylindrical geometry. Is this assumption still valid in the given configuration?\n\n9. Building upon the previous question: does the location of fatigue fracture affect the intensity of the measured magnetic signal?\n\n10. In the “Fatigue Fracture Early Warning Algorithm” section, the alarm mechanism needs to be more thoroughly explained. At minimum, the following points should be clarified:\n(1) What is the physical meaning of the two selected formulas?\n(2) What is the physical meaning of S₁ in the equations?\n(3) Describe the computational procedure in detail, including whether the method is applicable only after 90% of the fatigue life has elapsed, and whether a moving window is employed during prediction.", "reviewer_3": "the authors integrate the microscopic physical mechanism (synchronous weakening from bonding forces and exchange interactions) with macroscopic magnetic-signal features (the MagDrift metric) to establish the M-N model for fatigue-life prediction, and they propose a two-stage threshold algorithm for early-warning of fracture. My detailed queries are as follows:\n\n1. The M–N prediction model incorporates quantum‐mechanical concepts (bonding forces, exchange interactions, and COOP). However, it is unclear how the critical fitting parameters Kc and C are extracted from the fatigue tests. Although COOP is invoked to explain intrinsic differences in fatigue strength among materials, no actual COOP values or computations are presented. Are the quantum-mechanical quantities used purely for post hoc physical interpretation, or are they directly computed and input into the model? Do the parameters Kc and C rely entirely on fitting the macroscopic MagDrift data, with the quantum-mechanical concepts serving only an explanatory role?\n\n2. The core innovation, “magnetically induced spin-correlation amplification,” delivers high‐sensitivity detection. Yet the paper omits many details of the data-processing chain from the raw magnetic field to the final MagDrift metric: What form does the raw signal take (e.g., bandwidth, sampling rate)? Which filters (type and cutoff frequencies), baseline-correction or normalization routines, peak-detection algorithms, and SNR-enhancement measures are applied? Please provide either the key formulas or pseudocode that describe each processing step, and explain how you ensure that MagDrift variations reflect actual fatigue damage rather than measurement noise or environmental interference.\n\n3. The fatigue‐life prediction model is given by logN=(1/kc)(log|kac|+C), and the reported R2>0.9 refers to the fit of the Log M–N curve used to determine Kc and C. Figure 4d shows the correlation (r=0.907) between early-stage (e.g., 10 %) MagDrift data and final life. Were the parameters Kc and C obtained using the full fatigue‐life data from all specimens? Has the model’s predictive accuracy been tested on an independent validation set (i.e., specimens not used in parameter fitting)? Beyond the correlation coefficient, have you reported more intuitive metrics such as mean absolute error (MAE) or relative‐error distributions between predicted (Ne) and actual (Nf) fatigue lives?\n\n4. Conclusions are drawn from 193 specimens across five ferromagnetic materials, all tested under uniaxial tension–tension fatigue (R = 0), constant 20 Hz frequency, at room temperature. How might this approach perform under more complex service conditions (e.g., variable-amplitude loading, nonzero mean stress R≠0, different frequencies, corrosive environments, elevated temperatures)? Are there plans or preliminary data? For the same material subjected to different microstructural states (e.g., varied heat treatments or welded zones), how robust is the method? How can one be confident that the fitted parameters (Kc, C) and the warning thresholds (S1, S2)—obtained under limited material types and loading conditions—will generalize to other ferromagnets or operating regimes?\n\n5. Magnetic measurements are susceptible to environmental noise. The manuscript notes fixed sensor positions and distances, but does not detail countermeasures against:\n1. External field fluctuations (e.g., geomagnetic variations, nearby equipment),\n2. Temperature drift (both in the specimen and sensor),\n3. Electrical noise,\n4. Residual magnetization and magnetic‐history effects.\nWhat specific shielding, compensation, or signal-processing strategies were employed (e.g., magnetic shielding, temperature monitoring and correction algorithms, band-pass filtering, baseline drift correction)? Were control experiments conducted (e.g., static unloaded tests or tests on undamaged specimens) to quantify the noise floor and demonstrate that observed MagDrift changes predominantly arise from fatigue damage?\n\n6. The authors claim exceptionally high predictive accuracy (R2>0.9) and a 100 % early‐warning success rate. For the life‐prediction model, was performance assessed on an independent test set (entirely excluded from the fitting of Kc and C)? If so, what were the MAE or RMSE values? How were the two warning thresholds (S1, S2) determined in practice? Were they calibrated on a training subset and then validated on a separate test subset, or were both threshold setting and success‐rate assessment performed on the same 193 specimens? Can you provide false‐alarm rates for the first‐stage warning (≈90 % of life) and breakdowns by material type and stress level to check for any sample‐specific biases?\n\n7. The manuscript highlights advantages over traditional S-N curve methods in sensitivity, accuracy, and early warning and includes a quantitative comparison. To contextualize its novelty further, please provide (or cite) direct, quantitative comparisons against state-of-the-art fatigue nondestructive evaluation methods (e.g., acoustic emission (AE), nonlinear ultrasonic testing (UT), magnetic Barkhausen noise (MBN)):\n1. Number of cycles or damage size required to detect initial defects (e.g., first PSB or microcrack),\n2. Capability to quantify damage evolution and SNR,\n3. Remaining‐life prediction accuracy under comparable conditions,\n4. Reliability and lead time of fracture warnings.\nAdditionally, discuss potential advantages and challenges of your method in real‐world applications (e.g., on-line monitoring, field deployment, cost, system complexity, environmental robustness).\n\n8. In Methods, you derive formulas for spin-correlation amplification (△BH = KH△Fc) and for the relationship between bonding/exchange forces and interatomic spacing. Were these theoretical models used to make quantitative predictions of MagDrift values or trends in specific experiments? If so, please present a comparison (including error analysis) between the theory and measurements to validate model accuracy. If the models serve chiefly for qualitative explanation, what key assumptions (e.g., homogeneous deformation, neglect of pinning effects, chosen functional form f(Fm)) might limit their predictive power? How might these models be refined or further validated in future work?"}, "author_response": "Dear Reviewers:\nWe sincerely appreciate the valuable suggestions provided by yours. These\ncomments not only helped us identify areas for improvement, but also provided\nvaluable guidance for enhancing the scientific rigor of this study. We highly value the\ninsights shared by yours, and have now addressed them through targeted responses\nbased on theoretical analysis and experimental verification. The response section in the\nmain text details our revision rationale and provides point-by-point answers to the\nquestions raised. All supplementary experimental datasets, computational examples,\nand related analyses are included in the supplementary information for your review at\nany time.\nBelow, we provide a point-by-point response to the comments from yours (in\nblack), with our detailed replies in blue and the corresponding revisions in the\nmanuscript indicated in red.\nResponse to Reviewer 1\n<Reviewer 1>: 1. The authors propose an innovative approach to estimate dislocation\ndensity and use it to assess fatigue strength. However, the manuscript currently includes\nseveral oversimplifications and lacks critical information necessary to fully validate the\nmethod for fatigue evaluation. Importantly, the proposed approach is based on averaged\nmeasurements, lacking the spatial resolution necessary to accurately capture the\nlocalized nature of fatigue damage, an issue the authors themselves acknowledge in the\nintroduction. In its current form, the method may offer no more insight than\nconventional strain gauge measurements.\nResponse:\nWe realized that there was an issue of simplification in the presentation of\ntheoretical models in the original text, which may have given the impression that our\nmethod is \"average measurement\". Below, we will attempt to restate our measurement\nprinciples to demonstrate the mechanism, generation, and effects of localized fatigue\nphenomena, supplemented by experimental validation.\nDue to the correlation between interatomic binding forces and exchange\ninteractions with atomic spacing, when fatigue causes the atomic spacing between\natoms to gradually increase, the interatomic binding forces and exchange interactions\nchange synchronously.\nWe enhanced the macroscopic observability of this intrinsic correlation through\ndirectional excitation by an external magnetic field, achieving spin correlation\namplification. Unlike “averaging effect” measurement methods, this represents a\n“multiplicative effect” where the localized nature of fatigue damage exerts a dominant\ninfluence on the overall measurement value. The specific reasons are as follows:\nPhysical mechanism of spin correlation amplification method\nDuring the material fatigue process, the interlamellar atomic spacing gradually\nundergoes irreversible changes (\"interlamellar\" refers to the collective set of all atoms\nwithin a three-dimensional irregular thin-section perpendicular to the direction of\nstructural loading).\nThe variation of interlamellar atomic spacing refers to the collective effect of the\nvariation of atomic spacing between each atomic layer; the interlamellar bonding force\nis the collective effect of the changes in atomic bonding force between each atomic\nlayer; the interlamellar exchange interaction is the collective effect of exchange\ninteractions between the atomic layers.\nThe change affects not only the interlamellar bonding force but also synchronously\nalters the interlamellar exchange interaction, demonstrating a strong indirect correlation\nbetween these two properties.\nWhen a directional excitation is applied by an external magnetic field, the change\nin the interlamellar exchange interaction will cause a directional change in the\ninterlamellar spin magnetic moment at the microscale, and further alter the magnetic\ndomain pinning state at the mesoscopic scale.\nAs the external excitation magnetic field penetrates each atom layer disturbed by\nfatigue, change in spin orientation between atoms. This deflection accumulates\nprogressively as the magnetic field traverses additional damage concentration zone.\nUltimately, when the stimulating magnetic field penetrates all fatigue-disturbed layered\natom layers and outputs a measurable magnetic signal, its intensity reflects this\ncumulative deflection process.\nThis mechanism enhances the detectability of the weakening of interlamellar\nbonding forces and exchange interactions caused by changes in the interlamellar atomic\nspacing, allowing for more sensitive observation of material fatigue performance at the\nmacroscopic level (Fig. 1).\nFig. 1. Comparison of spin correlation multiplicative effect and strain additive effect. a, Under\nthe directional excitation of an external magnetic field, the slight weakening of interatomic binding\nforces can trigger a synergistic orientation change of spin magnetic moments at the microscale\nthrough exchange interactions; This micro spin reconstruction will significantly alter the pinning\nstate of magnetic domains at the mesoscopic scale, and then amplify into observable surface\nmagnetic flux changes at the macroscopic level through collective motion of magnetic domain walls.\nb, Under the action of fatigue, materials produce damage concentration zone, and strain gauges only\nmeasure local strain, which is independent of the microscopic mechanism and is an additive effect.\nMultiplication effect and its difference compared to strain gauges\nSpin correlation amplification is essentially a multiplication effect. During the\nfatigue process of materials, the initial atomic dislocation layer (such as the first layer)\ncan cause a deviation in electron spin (for example, with an amplitude of ). Through\ninterlamellar exchange interactions, this bias is gradually amplified i𝑎𝑎n% subsequent\ndamage concentration zone. Specifically, each layer amplifies the received deflection\nby a factor of one (the amplification factor may vary by layer; for simplicity, assume\nthat each layer has an amplification factor of ). So, after n-layer amplification,\nthe spin deflection amplitude of the final atom1ic+ d𝑎𝑎is%location layer will reach ( )\ntimes the initial deflection. Among them, the actual amplification factor of ea1ch+ l𝑎𝑎ay%er\nmay not be the same. Note that is only used to illustrate the operating principle of\nthe multiplication mechanism. 𝑎𝑎%\nFor the specific amplification model, please refer to the formula derivation from\nwave function to fatigue life and MagDrift slope relationship (Methods).\nThe strain gauge responds linearly to strain ac𝑘𝑘c𝑎𝑎u𝑎𝑎mulation: during material fatigue,\nminute displacements ( , , …, ) occur across each micro-section within the\nfatigue zone. The strain𝐿𝐿 ₁gau𝐿𝐿g₂e can 𝐿𝐿o𝑛𝑛nly linearly superimpose these displacements,\noutputting signal that reflects additive effect.\nSpin co𝛴𝛴r𝐿𝐿reᵢ/la𝐿𝐿tion amplification relies on the multiplicative coupling of interlamellar\nexchange interactions, and its signal exponentially amplifies with the number of\ndamage concentration zones; Strain gauges rely on linear accumulation (Fig. 1).\nThe dominant role of localized events\nDuring the fatigue process, material layers with defects or weak bonding forces\nbear the maximum dislocation (interlamellar atomic spacing) changes, and the\nmagnitude of changes in their interlamellar bonding forces and exchange interactions\nis significantly higher than in other regions. It is this local maximum effect that\ndominates the change in the final magnetic signal. Therefore, the spin correlation\namplification method has significant spatially varying response to fatigue damage, and\nits high sensitivity originates from the dominant role of interlamellar localized events\n(Fig. 2).\nFig. 2. Schematic diagram of localization effect under extreme conditions.\nTo verify the localization effect theory, we designed three types of progressive\nexperiments, which systematically analyzed the cross-scale conduction mechanism of\nlocal bonding weakening on global magnetic response from the perspectives of artificial\ncrack simulation, prefabricated defect evolution, and weld fusion zone evolution.\nAnalysis of steel bar cutting test\nThe experiment selected a soft magnetic iron rod with a diameter of 16 mm and a\nlength of 300 mm. Four measurement points (Measurement Points 1-4, MP1-MP4)\nwere set up every 50 mm along the axial direction, and a Cutting Point (CP) was marked\nbetween MP2 and MP3.\nIntroduce artificial cracks with a fixed width of 1 mm at this location: before the\ndepth reaches 10 mm, cut 2 mm each time, and after exceeding 10 mm, change to 1 mm\neach time until fracture. After each cutting, apply an 8-level increasing excitation\nmagnetic field of 50-430 mT to the fixed end of the specimen in sequence. After each\nlevel is loaded, use a handheld Gaussian meter to collect the magnetic induction\nintensity of each measuring point at a height of 3 mm. This artificial crack simulates\nthe process of atomic bond breaking. (Fig. 3)\nFig. 3. Schematic diagram of the comprehensive relationship between magnetic strength\nchanges at different measuring points and under different crack depths under different\nexternal magnetic excitations. a, Schematic diagram of steel bar cutting test and external magnetic\nfield configuration location. The preset crack position is located at the geometric center of the test\npiece, with measuring points 1 and 2 located in front of the crack, and measuring points 3 and 4\nlocated behind the crack. The external magnetic field is applied by neodymium-iron-boron\npermanent magnets, covering eight excitation levels from 50–430 mT. b, Schematic diagram of\nfinite element simulation of crack depth on magnetic induction line distribution. When there are\ncracks in the material, the magnetic flux in the damaged area at the crack location section decreases,\nwhile the magnetic flux in the undamaged area increases accordingly; and the increase in crack\ndepth leads to an increase in magnetic flux in the undamaged area. c, The evolution law of magnetic\nstrength at each measuring point under eight types of external magnetic excitation as a function of\ncrack depth. As the external magnetic excitation increases, the magnetic strength at each measuring\npoint increases synchronously. When the crack depth reaches a corresponding value, there is a trend\nof increasing magnetic strength at measuring points in front of the crack and decreasing magnetic\nstrength at measuring points behind the crack. d, Output the relationship diagram between the\nacceleration of magnetic field changes and the remaining depth, external magnetic excitation, and\nthe ratio of magnetic excitation to undamaged area.\nMagnetic field strength at the crack-front monitoring points (MP1, MP2) exhibits\na trend of initial stability followed by accelerated increase as the crack deepens, while\nthe crack-back monitoring points (MP3, MP4) show initial stability followed by\naccelerated decrease. Concurrently, a higher applied excitation magnetic field strength\nnot only enhances the stability of the measured magnetic field but also accelerates the\nonset of both the accelerated increase at MP1 and MP2 and the accelerated decrease at\nMP3 and MP4.\nThe distribution map of magnetic induction lines shows that the magnetic flux at\nthe damaged part of the fracture area decreases, while the magnetic flux at the\nundamaged part of the fracture area increases accordingly, so the overall magnetic flux\nof the section remains stable at the initial stage. However, as the crack depth increases,\nthe magnetic flux in the damaged area continues to decrease, while the magnetic flux\nin the undamaged area steadily increases until reaching a critical saturation state. At\nthis time, the magnetic flux increment in the undamaged region can’t fully compensate\nfor the loss in the damaged region, resulting in the overall magnetic flux of the cross\nsection beginning to decrease. The decrease of the overall magnetic flux is the main\nreason for the decrease of the magnetic strength of the measuring points behind the\ncrack (MP3, MP4), while the increase of the magnetic strength of the measuring points\nin front (MP1, MP2) is related to the redistribution of local magnetic induction lines\n(concentrated to the undamaged area) caused by the decrease of the magnetic flux in\nthe damaged area (Fig. 3).\nTable. 1. Ratio of fracture area to undamaged area at critical filling under different excitation\nmagnetic strength.\nExternal Ratio of external\nmagnetic Crack depth Undamaged magnetic excitation to\nUndamaged rate\nexcitation (mm) depth (mm) undamaged area\n(mT) (mT/mm2)\n50 13 3 13% 0.28\n87 13 3 13% 0.5\n124 12.5 3.5 16.2% 0.74\n158 12 4 19.6% 0.977\n185 11 5 26.7% 1.3\n211 9 7 42.2% 1.82\n317 8 8 50% 3.2\n430 8 8 50% 4.3\nAccording to the data measured in Table. 1, we found that the ratio of external\nmagnetic excitation and undamaged area was linear, and the linear fitting slope was\n0.011, which could be regarded as the quantitative characterization of crack damage\ndegree and external magnetic excitation intensity in the critical filling state (Fig. 3d).\nWith the increase of the external magnetic excitation intensity, the crack depth\nrequired for the sharp change of the output magnetic field decreases significantly; When\nthe crack continues to deepen or the magnetic field continues to strengthen, the\nmagnetic flux at the damaged part of the crack area decreases rapidly, while the\nmagnetic flux at the intact part increases correspondingly until it reaches the critical\nfilling. The spin correlation amplification technique can capture the local magnetic field\ngradient changes induced by the magnetic flux redistribution, and the local defects\nsignificantly affect the final magnetic strength changes.\nFatigue test of defective specimen\nThe experiment utilized soft magnetic iron material. The geometric dimensions\nand loading parameters of the specimens were consistent with those described in the\noriginal text (Methods). Seven fatigue tests were conducted with a stress ratio of R=0\nand a maximum stress of 460 MPa. To construct the local damage characteristics, two\nartificial defects with a depth of 0.5 mm and a width of 1 mm were symmetrically pre-\nfabricated on both sides of the middle of the test piece (Fig. 4), and a defect-free control\ngroup was established for comparative analysis.\nFig. 4. Dimension diagram of defective test piece.\nThe experimental results show that the fatigue life of the defect group is\nconcentrated in 20,000-60,000 cycles, while that of the non-defect group is 200,000-\n300,000 cycles. Although the service lives of the two groups of specimens are quite\ndifferent, the change rate of magnetic strength (CRMS) of the two groups of specimens\npresents a consistent three-stage trajectory: a short acceleration decline in the initial\nstage, followed by a stable period, and finally a sharp decline again in the failure stage\nuntil fracture. The MagDrift curve also shows a similar evolution path, and the greater\nthe change rate of MagDrift, the shorter the service life, and the strength of local defects\ncan be shown through the change of MagDrift. In the normalized MagDrift diagram,\nthe MagDrift of the defect group and the defect free group have similar trends.\nThrough the comparative analysis of the two groups, the MagDrift slope of the\ndefect group is significantly higher than that of the non-defect group. According to this\ncharacteristic, whether the material contains defects can be distinguished. Therefore,\nspin correlation amplification technology can capture the changes of magnetic signal\ncaused by local defects. (Fig. 5).\nThis phenomenon reveals that the decay of bonding forces in localized stress\nconcentration zones is transformed into changes in the global MagDrift signal through\nmultiplicative effect, highlighting the localized detection nature of this method.\nFig. 5. Defect fatigue test. a, The diagram of Offset CRMS. b, The diagram of Normalization of\noffset CRMS. c, The diagram of MagDrift. d, The diagram of Normalization of MagDrift.\nFatigue test of weld specimen\nIn the experiment, soft magnetic iron plate with a width of 20 mm and a thickness\nof 10 mm was selected. A Y-shaped notch was prefabricated in the middle of the test\npiece and welded to form the weld fusion zone (Fig. 6). Carbon dioxide shielded\nwelding was used, and the welding wire was ER80S-G with a diameter of 1 mm. The\nloading parameters are the same as those in the original (Methods), the stress ratio R=0,\nand the maximum stress is 160 MPa. The fatigue life of the three groups were 57,000\ncycles, 9,100 cycles and 5,500 cycles, respectively.\n0\n1\n2\n8\n60°\n01\n0\ndifference, which is due to the different welding quality, so that the interlamellar atomic\nbond force in the weld fusion zone has a significant difference.\nThrough the comparative analysis of the evolution law of the change rate of\nmagnetic strength (Fig. 7), it is found that under the condition of fixed stress amplitude\n(160 MPa), the change rate of magnetic strength of different fatigue life specimens (57\nK, 9.1 K, 5.5 K) shows a slow and then rapid downward trend with the increase of the\nnumber of cycles, and the phenomenon of accelerated attenuation occurs near the\nfailure stage. Welding will also affect the normalization characteristics of magnetic\nstrength change rate: according to different welding quality, the final attenuation\ntrajectories of the three curves show significant differences. The specimen with fatigue\nlife of 5.5 K shows the steepest decline slope in the late normalization period, followed\nby 9.1 K, and 57 K decline is the most gradual, indicating that the worse the welding\nquality, the faster the magnetic signal attenuation. The strength difference of weld\nfusion zone can be directly reflected by the variation of CRMS. At the same time, the\nchange rate of the MagDrift curve increases with the decline of the weld quality,\nresulting in the shortening of the life of the specimen. This phenomenon reveals the\nlocalization effect: even if the welding quality is difficult to distinguish with the naked\neye, it can be sensitively characterized by the change of magnetic signal. (Fig. 7).\nThe evolution of MagDrift is determined by the bonding force state of the\ninterlamellar atoms, is highly sensitive to the decay of the bonding force, and can\ndirectly reflect the bonding force strength in the weld fusion zone.\nFig. 7. The weld fatigue test. a, CRMS of the weld specimen. b, Normalization CRMS of the weld\nspecimen. c, MagDrift of the weld specimen. d, Normalization MagDrift of the weld specimen.\nThese three groups of experiments confirmed that the local nature of fatigue\ndamage would have a dominant effect on the global change of MagDrift signal. The\nintrinsic reason is that the attenuation of the bonding force of the weakest cross section\nis reflected in the final measured magnetic signal through the multiplication effect of\nthe spin correlation amplification of the exchange interaction (magnetic moment).\nThree advantages of spin correlation amplification measurement over average\nmeasurement\nAlthough both the spin correlation amplification method and the strain gauge can\nbe employed for material damage detection, their underlying physical principles and\nresponse mechanisms are fundamentally distinct. The unique advantages of the spin\ncorrelation amplification method are manifested in the following three aspects:\n1. MagDrift features a self-compensation ability\nThe MagDrift metric is designed as a differential measure (peak-to-peak within a\nloading cycle), which inherently suppresses low-frequency environmental drift by\ncanceling common-mode magnetic perturbations. It is assumed that the influence of the\nsurrounding environment and geomagnetic field on the magnetic field strength of the\nspecimen is , and the influence of the surface temperature change of the specimen\non the magne∆ti𝐵𝐵c𝐸𝐸 field strength of the specimen is . Under cyclic loading:\nWhen the external load is , the com∆b𝐵𝐵in𝑇𝑇ed effects of magnetic and load\neffects, ambient environment eff𝜎𝜎e𝑚𝑚ct−s a𝜎𝜎n𝐿𝐿d specimen surface temperature changes are\nconsidered as follows:\n(R1-1)\nWhen the external l𝐹𝐹o𝑚𝑚ad0 i−s g(𝜎𝜎𝑚𝑚 −𝜎𝜎, 𝐿𝐿t)he− c𝑓𝑓o(m∆b𝐵𝐵i𝐸𝐸ne)d− e𝑓𝑓ff(e∆c𝐵𝐵ts𝑇𝑇 o)f magnetic force and\nload action, the influence of su𝜎𝜎rr𝑚𝑚ou+nd𝜎𝜎i𝐿𝐿ng environment and the change of specimen\nsurface temperature are as follows:\n(R1-2)\nAmong them, 𝐹𝐹𝑚𝑚 1a−ndg (𝜎𝜎𝑚𝑚 +𝜎𝜎𝐿𝐿 )re−sp𝑓𝑓e(c∆ti𝐵𝐵ve𝐸𝐸l)y −tak𝑓𝑓e( ∆in𝐵𝐵to𝑇𝑇 )account the influence\nof the surrounding e𝑓𝑓n(v∆i𝐵𝐵ro𝐸𝐸n)ment a𝑓𝑓n(d∆ t𝐵𝐵h𝑇𝑇e )surface temperature change of the specimen\nunder cyclic loading.\nSubtract R1-2 from R1-1 to obtain the MagDrift value during fatigue:\n(R1-3)\nIt can be seen 𝐹𝐹th𝑚𝑚a𝑚𝑚t𝑚𝑚 = 𝐹𝐹𝑚𝑚1 − agn(d𝜎𝜎 𝑚𝑚 +𝜎𝜎𝐿𝐿)− a𝐹𝐹re𝑚𝑚 0co+m𝑔𝑔p(le𝜎𝜎t𝑚𝑚ely− e𝜎𝜎l𝐿𝐿im) inated, and the\nMagDrift is only related to𝑓𝑓 t(h∆e𝐵𝐵 in𝐸𝐸t)ernal st𝑓𝑓at(e∆ o𝐵𝐵f𝑇𝑇 t)he material.\nWe added various fatigue loading tests under human interference to further verify\nthe rationality of choosing MagDrift as the representative index of the paper.\nThis experiment mainly simulates various possible influencing factors by exerting\nvarious human disturbances in the process of fatigue loading. These factors include\ntemperature noise, environmental magnetic strength change, laboratory power failure,\npoor channel contact of test data acquisition instrument, electrical noise interference of\nhigh-power electrical appliances and other factors. The corresponding magnetic\nstrength and MagDrift change data under these human interference factors are shown\nin Fig. 8.\nFig. 8. Comparison of magnetic strength and MagDrift data after interference. The shaded area\nin the figure indicates the time period when the connection of the data acquisition system is\ninterrupted. The sudden change or jump of magnetic strength data in this experiment is mainly due\nto temperature noise, environmental magnetic field fluctuation and electromagnetic noise\ninterference introduced by high-power equipment. In contrast, the MagDrift signal has an inherent\nability to suppress these interference factors by virtue of its inherent self-compensation mechanism,\nso it maintains a more stable change trend in the whole process.\nIt can be seen from the figure that the variation of the magnetic field strength value\nof the test piece has been greatly affected, including the loss of data in the magnetic\nfield strength value and MagDrift data caused by the poor contact of the data acquisition\ninstrument channel (this part is supplemented by adding shadows in the figure),\njumping, clutter and other phenomena. Due to the self-compensation feature of the\nMagDrift data, it can offset and slow down the impact of the above factors. Therefore,\nalthough there are some fluctuations in the MagDrift data, the overall trend is relatively\nflat.\nTherefore, compared with the magnetic field strength value, the MagDrift can\nbetter reflect the fatigue process and the change of the internal fatigue state of the\nmaterial. This is also the main reason why the MagDrift is adopted as the core indicator\nto evaluate the fatigue state of the material in this paper.\nUniversality of abnormal structure\nFrom the application perspective, strain gauges have significant limitations in the\nfatigue monitoring of non-standard structures (such as curved, wavy, or complex curved\ncomponents). Since strain measurement is difficult to directly apply to curved and non-\nsmooth surfaces, the following problems are prone to occur when bonding to curved\nsurfaces: (1) Uneven thickness of the bonding layer leads to distortion of stress transfer,\nand the measurement error increases significantly when the radius of curvature is small;\n(2) The multi-dimensional strain field of non-standard structures under dynamic\nloading can cause directional deviation in traditional uniaxial strain gauges; (3)\nMonitoring blind spots are likely to form at the locations of curvature changes on the\nstructure surface (Fig. 9).\nIn contrast, the magnetic measurement method shows unique advantages: through\nthe non-contact magnetic sensor array, the surface profile can be adaptive. In typical\nspecial-shaped structures such as arc beams, the magnetic measurement method can\nachieve full circumferential coverage monitoring, and the measurement accuracy of the\nmagnetic field gradient in the curvature change area can still be maintained. More\nimportantly, the magnetic sensor does not need to be fixedly connected with the\nstructure surface, which avoids the signal failure problem caused by the rubber layer\nfatigue of the strain gauge on the special-shaped structure. This adaptability is universal\nfor special-shaped structures and is not affected by the shape of the material structure\n(Fig. 9).\nFig. 9. Comparison of universality between magnetic measurement and strain measurement.\nMultiplication effect magnifies micro damage\nAs described in the section \" Physical mechanism of spin correlation amplification\nmethod\" earlier, the amplification effect of spin correlation transforms the quantum\ncharacteristics of nanoscale damage into observable flux jumps through multiplicative\ngain, with sensitivity increasing exponentially, far surpassing the linear superposition\nof traditional strains. When the external magnetic field penetrates the material, it\nproduces irreversible deflection due to spin synergy. Its cumulative multiplicative effect\nenables the system to give a clear warning signal before fracture.\nSpin correlation amplification possesses three major characteristics: noise\nresistance, universality, and signal amplification, which is significantly different from\naverage measurement.\nTo more clearly elaborate on the spin correlation method theory, the revised text\nin the manuscript is provided below (lines 197-211): “During the process of fatigue, the\ninterlamellar atomic spacing undergoes gradual and irreversible changes. This change\nnot only affects the bonding strength of the material layer but also simultaneously\naffects its interlamellar exchange interactions, with both phenomena being strongly\ncorrelated and synchronous.\nWhen an external magnetic field is applied, changes in the interlamellar exchange\ninteractions induce oriented variations in the aggregate spin magnetic moment at the\nmicroscopic scale, which in turn alters the pinning states of magnetic domains at the\nmesoscopic scale.\nAs the external excitation-induced magnetic field penetrates each fatigue-affected\ndislocation layer, it undergoes irreversible deflection because of the spin‒spin\ninteractions between atoms. This deflection accumulates progressively as the magnetic\nfield traverses more fatigued dislocation layers. Essentially, this process functions as a\nspin-correlation amplification mechanism, manifesting as a multiplicative effect\n(Methods). Ultimately, the deflection characteristics that accumulate after the magnetic\nfield passes through all the fatigue-induced dislocation layers are encoded within the\nmagnetic signal, providing a basis for evaluating the fatigue state of the material (Fig.\n1).”\nRevised text in the manuscript (lines 235-237): “The magnetic signal changes\nexhibited by the spin magnetic moment system exhibited exponential amplification\nthroughout the spin-dependent conduction chain.”\nTo illustrate the advantages of MagDrift, we have supplemented the Methods\nsection of the manuscript with the following explanation: (lines 944-963): “MagDrift\nfeatures a self-compensation ability.”\nIn our fatigue theory derivation model, we analyzed at the mesoscale, microscale,\nand macroscale. In the method section of the manuscript, we have supplemented the\nMethods section of the manuscript with the following explanation: (lines 880-923):\n“Macroscopic scale: Derivation of the magnetic flux change rate and fatigue life model.”\n<Reviewer 1>: 2. Moreover, the interpretation of what is being measured magnetically\nrequires further clarification. The dislocation structures that evolve during fatigue can\nvary substantially in nature, yet the proposed technique appears to treat them\nequivalently, potentially overlooking key distinctions in how different dislocation\nconfigurations influence fatigue behavior. For instance, dislocation annihilation, slip\nreversibility, an important mechanism that may still lead to micro-damage, is unlikely\nto be captured by the magnetic signal, despite its relevance to fatigue failure.\nResponse:\nIn the process of material fatigue, microscopic evolution mechanisms such as\ndislocation annihilation and slip reversibility indeed have significant effects on fatigue\nfailure. Dislocation annihilation typically occurs when two spatially adjacent\ndislocations with opposite Burgers vectors cause lattice distortions (such as\ncompressive and tensile regions) that cancel each other out, thereby restoring the lattice\nto a state of lower distortion. However, the changes in atomic spacing caused by\ndislocations near the annihilation site may be difficult to fully recover.\nSlip reversibility refers to whether dislocations generated by slip under forward\nloading can return along the original path during reverse loading in cyclic deformation,\nthereby restoring the local lattice to its initial state. The key lies in whether dislocations\ncan strictly retreat along their original path, which can be judged by whether the local\natomic spacing can fully recover: under cyclic loading, if the lattice distortions caused\nby forward slip can be eliminated through reverse slip, the atomic spacing may be\nrestored; otherwise, if dislocation retreat is impeded, the atomic spacing will change,\nleading to residual plastic strain that gradually accumulates into fatigue damage.\nRegarding the fatigue fracture mechanism in materials, we believe that: the\ninterlamellar atomic spacing to a certain extent determines the strength of interlamellar\nbonding and interlamellar exchange interactions; when the material system is fixed,\natomic spacing becomes the dominant variable regulating both; the essence of fatigue\nfracture is the continuous weakening of interlamellar bonding strength as the spacing\nincreases, and once the bonding strength falls below a critical threshold of the applied\nload, fracture occurs immediately; through the spin-correlation amplification effect,\nthe\nMagDrift signal amplifies this microscopic process of weakened bonding strength into\nan observable macroscopic response.\nThe above represents our current understanding of this issue. If there are any\noversights or shortcomings, we sincerely invite your feedback and suggestions.\nTo further clarify the interpretation of the magnetic measurement results. The\nrevised text in the manuscript is provided below (lines 108-141): “The fatigue\nbehaviours of materials manifest microscopically as the formation of various\ndislocation structures (e.g., persistent slip bands (PSBs), dislocation annihilation,\nreversible slip, and the ratcheting effect), which are characteristic features of localized\nplastic deformation under cyclic loading (Fig. 1a). At the microscopic level, the\nformation of these structures originates from the motion, interaction, and accumulation\nof dislocations at the nanoscale within the crystal lattice, gradually evolving into visible\nslip bands or other defect structures4. This evolutionary process is highly nonlinear and\nstrongly dependent on crystal structures, loading conditions, and environmental factors.\nFor instance, in face-centred cubic (FCC) metals, dislocations undergo rearrangement\nthrough reversible slip (where partial dislocations return along their original paths under\nalternating loads, leading to partial damage recovery) and dislocation annihilation\n(where dislocations with opposite signs meet and cancel each other out). This initially\nresults in the formation of vein-like structures, which further develop into cellular\nstructures. In contrast, for body-centred cubic (BCC) metals or objects under high-\nstress amplitude conditions, the ratcheting effect (the unidirectional accumulation of\nstrain under asymmetric cyclic loading) becomes more pronounced, promoting\nirreversible and rapid dislocation accumulation.\nThe nucleation and motion of dislocations are associated with the weakening of\ninteratomic bonding forces18 (interatomic cohesion; Fig. 1a). During cyclic loading, a\nlocalized stress concentration reduces the interatomic bonding strength, facilitating\ndislocation nucleation and slip processes25. This slip is not always uniform; its degree\nof localization can be characterized by slip planarity. Higher planarity indicates that the\nslip is concentrated on fewer planes, making it easier to form sharp persistent slip bands\n(PSBs). The interfaces between these bands and the matrix become high-risk regions\nfor crack initiation. In some alloys, microscopic plasticity may also appear in the form\nof fatigue shear bands, which are intensely localized shear bands that traverse grains.\nTheir formation is closely related to microstructural anisotropy.\nTherefore, fatigue damage in materials is essentially a multiscale evolutionary\nprocess spanning the atomic, microscopic, and macroscopic dimensions. From a\nmicroscopic perspective, this process is located within the damage concentration zone.\nThis region consists of multiple atomic layers within the fatigue-affected zone and\ncontains a series of microscopic defect clusters that influence the interlamellar atomic\nspacing (here, “interlamellar” refers to the collective set of all the atoms contained\nwithin a three-dimensional irregular thin section that is perpendicular to the direction\nof structural loading). Throughout this process, damage to the interlamellar atomic\nbonding force continuously accumulates. When this force falls below the stress induced\nby the applied load, material fracture occurs. Hence, changes in the interlamellar atomic\nbonding force can serve as indicators for characterizing the fatigue states of materials.”\nRevised text in the manuscript (lines 146-147): “These challenges are particularly\npronounced when complex transient processes such as dynamic dislocation annihilation,\nreorganization and reversible slip are involved.”\n<Reviewer 1>: 3. The influence of loading conditions is also insufficiently addressed.\nAs an example, under ratcheting conditions, dislocation density may increase\nsignificantly, but the associated damage mechanisms can differ from those occurring\nunder non-ratcheting fatigue. Such distinctions are critical and should be reflected in\nthe analysis.\nResponse:\nThe differences in the fatigue damage mechanisms between ratcheting and non-\nratcheting effects are important for accurately analyzing material fatigue behavior. Our\ndiscussion of this issue in the original manuscript was insufficient. The following is our\nunderstanding of the dislocation behavior under different loading conditions:\nThe essence of the ratcheting effect is the combined effect of gradual attenuation\nof the bonding force and incomplete dislocation annihilation caused by the irreversible\nnet increase of atomic spacing under cyclic loading. The core of non-ratcheting\nbehavior lies in the reversible elastic oscillation of atomic spacing to maintain bonding\nstability and realize full annihilation through dislocation reverse slip. Although the\nmicroscopic mechanisms of ratcheting and non-ratcheting effects are different, their\nessence is related to atomic spacing.\nTo elucidate the load-independent response of the MagDrift response, we have\nsupplemented the discussion with existing data, selecting 45 steel for both high-stress\nfatigue testing ( = 580 MPa, = 111 K) and low-stress fatigue testing ( =\nBy co𝑁𝑁m𝑓𝑓paring and analyzing the evolution behavior of MagDrift under different\nstress levels in Fig. 10, the following conclusions can be drawn: under the original cycle\nnumber scale, the MagDrift curve of the low stress specimen ( =510 MPa,\n=2,942 K) rises gently as a whole, and accelerates in the final failure\n𝜎𝜎\ns𝑚𝑚ta𝑎𝑎g𝑚𝑚e, indicating\n𝑁𝑁th𝑓𝑓at under the non-ratcheting dominated fatigue condition, the damage accumulation is\nslow, but the MagDrift has an upward evolution trend; The MagDrift of high stress\nspecimens ( =580 MPa, =111 K) changes rapidly as a whole, and also\naccelerates to𝜎𝜎 𝑚𝑚ri𝑎𝑎s𝑚𝑚e in the final fa𝑁𝑁i𝑓𝑓lure stage. This rapid change may be related to the\nsignificant increase of dislocation density under ratchet condition, indicating that under\nratcheting dominated fatigue condition, damage accumulation is faster, and the\nMagDrift also has a relatively obvious upward trend. The change rate of the\ninterlamellar atomic spacing is accelerated, which accelerates the attenuation of the\nbonding force of the interlamellar atoms. This shows that whether ratcheting effect or\nnon-ratcheting effect, its microscopic essence is attributed to the increase of the\ninterlamellar atomic spacing. The non-ratcheting effect shows a slow change of the\ninterlamellar atomic spacing, while the ratcheting effect shows a faster change of the\ninterlamellar atomic spacing, which directly represents the decline of the interlamellar\nbond force. The MagDrift signal accurately captures the decline of the bonding force\nthrough the quantum spin correlation chain, and establishes a cross-scale sensing path\nof atomic spacing bonding force magnetic response. This feature enables spin\ncorrelation amplification technology to span the difference of load conditions. The\nabove is our understanding of this issue, if there be any points that have not fully\ngrasped, we kindly request your further guidance.\nFig. 10. Comparison of MagDrift in 45 steel. a, Comparison of MagDrift in 45 steel at 510 MPa\nand 580 MPa stress; b, Comparison of Normalized MagDrift in 45 steel at 510 MPa and 580 MPa\nstress.\nThe revised text in the manuscript is the same as that described in <Reviewer 1>:\n2.\n<Reviewer 1>: 4. The citations associated with the data in Fig. 1 of the Extended Data\nshould be mentioned.\nResponse:\nAll the data in Extended Data Fig. 1 were independently collected and\nsystematically organized by our research team, without direct citation of published\nliterature. These data constitute an important part of the originality of this study. Like\nthe experimental data, these data are publicly stored in the DRYAD database\n(doi.org/10.5061/dryad.1rn8pk13t). We have updated the manuscript: Extended Data\nFig. 1 has been renamed as Supplementary Fig. 1. At the end of the title, we have\nexplicitly added a reference to this dataset.\n<Reviewer 1>: 5. P1L36: The current body of fatigue research includes substantial\nwork on the 3D characterization of microstructure effects. The authors have omitted\nthis important body of literature and should include relevant references.\nResponse：\nWe inadequately discussed the research achievements in this field in the initial\nmanuscript. To clearly elaborate on the contributions of 3D characterization in fatigue\nresearch. The revised text in the manuscript is provided below (lines 45-53):\n“Microscopic studies of fatigue phenomena have established a multiscale and\nmultidimensional observation framework, providing a solid foundation for revealing\nthe underlying micromechanisms of material damage. This framework extensively\nemploys advanced characterization techniques such as scanning electron microscopy\n(SEM), transmission electron microscopy (TEM), high-resolution digital image\ncorrelation (HR-DIC), and three-dimensional orientation imaging-based microscopy7,8.\nThese methods enable the capture of nanoscale image details and provide direct\nvisualizations of the initiation and evolution processes of fatigue damage. These\ntechniques have driven fatigue research from qualitative descriptions to quantitative\ncharacterizations.”\n<Reviewer 1>: 6. P2L39: The term “progression of fatigue” is not commonly used in\nthe fatigue community. The authors should clarify what they mean by this phrase.\nResponse:\nThe expression of this term in the first manuscript is not clear enough. To eliminate\nambiguity and improve clarity, the corresponding sections have been removed.\n<Reviewer 1>: 7. P2L43: The authors should cite key literature for each experimental\ntechnique mentioned to provide context and demonstrate the state of the art.\nResponse：\nWe recognize that there is a lack of literature support for some technical\nbackground in the first manuscript, and the latest technology of fatigue research is not\nfully displayed.\nThe latest technologies in fatigue research have been supplemented in the\nmanuscript with supporting references. The revised text in the manuscript is provided\nbelow (lines 54-60): “Physical field detection techniques provide nondestructive, real-\ntime, and efficient approaches for characterizing fatigue damage by monitoring the\nphysical signals generated during the fatigue process, such as acoustic signals (detected\nvia acoustic emission technology and ultrasonic testing)9,10, magnetic signals (detected\nvia magnetic crack-based pressure sensors)11, and thermal signals (detected via infrared\nthermography) 12-14. These methods, which are based on the physical field variations\ninduced by material damage evolution trends, can reflect the internal damage states of\nmaterials from different perspectives.”\n<Reviewer 1>: 8. While persistent slip bands (PSBs) are important features in fatigue\ndamage, they are not the only structures involved. Other microstructural features such\nas veins, fatigue shear bands, and cell structures also play critical roles in controlling\nfatigue behavior. These are not addressed by the authors and should be included for a\nmore comprehensive discussion.\nResponse:\nThe discussion of microstructure in this paper is deficient. The discussion in this\npaper only involves persistent slip bands (PSBs). Other microstructure characteristics\n(vein structure, fatigue shear band and cell structure, etc.) have an important impact on\nfatigue evolution. The following is our understanding of the effect of microstructure\ncharacteristics on fatigue damage:\nIn the fatigue process of metal materials, the evolution of micro dislocation\nstructure is essentially a cumulative process of the shift of atomic spacing distortion\nfrom reversible elastic oscillation to irreversible plasticity under cyclic loading. As an\ninitial response, the venous structure forms a gradient change of atomic spacing in its\nhigh dislocation cluster region, which leads to the attenuation of local bonding force;\nPSB shows the phase balance effect. Reversible slip in the channel region weakens the\nperiodic oscillation of bonding force, while pinning dislocations in the dislocation wall\nregion increases the atomic spacing; In the cellular structure formed in the high cycle\nfatigue stage, the atomic hyperelastic distortion caused by dislocation entanglement in dislocation entanglement in\nthe cell wall continued to accumulate distortion energy, resulting in the gradual\nattenuation of the bonding force of interlamellar atoms.\nThese microstructure characteristics will lead to the attenuation of the bonding\nforce of interlamellar atoms, and the evolution of MagDrift is the macro\ncharacterization of the attenuation of the bonding force of interlamellar atoms, so the\nMagDrift signal contains these microstructure characteristics, which are shown by their\nchange rate. The above represents our current understanding of the three\nmicrostructures. If there are any oversights or shortcomings, we sincerely invite your\nfeedback and suggestions.\nIn order to more comprehensively elucidate the relationship between\nmicrostructural evolution and fatigue damage. The revised text in the manuscript is\nprovided below (lines 351-356): “Microscopic HR-DIC and TEM measurements reveal\nthat early dislocation changes induced during fatigue affect the fracture location. Taking\na PSB as an example, its slip marks are almost identical to the final fracture location.\nSignificant correlations are present between the mechanical properties of materials,\nsuch as yield strength, ultimate tensile strength, and fatigue strength, and the early local\nslip of a PSB3,4. Therefore, early changes in the dislocation structure can be used for\nfatigue life assessment purposes.”\nThe other revised text in the manuscript is the same as that described in <Reviewer\n1>: 2.\n<Reviewer 1>: 9. \"PSB formed during fatigue processes... occupy approximately 9.2%\":\nThis value is highly material- and microstructure-dependent and should not be\ngeneralized. The authors should emphasize the conditional nature of this observation.\nResponse:\nSince we did not fully emphasize the preconditions of the conclusion in the first\ndraft, the statement was not rigorous enough. The revised text in the manuscript is\nprovided below (lines 70-74): “The fundamental challenge encountered in fatigue\ndetection tasks is the low sensitivity of macroresponses to structural microdamage,\nprimarily because the various dislocation structures formed during fatigue account for\nonly a small proportion of the affected area. For example, persistent slip bands (PSBs)\nconstitute only approximately 9.2% of the fatigue-affected zone in some fatigue tests\nof single copper crystals18,19.”\n<Reviewer 1>: 10. The claim that techniques such as HR-DIC cannot capture\ndeformations below 100 nm is inaccurate. HR-DIC, as cited by the authors in the\nmentioned reference, can achieve displacement sensitivity in the 7–9 nm range and\nspatial resolutions around 20 nm.\nResponse:\nWe thank you for your valuable comment. We recognize that there was an error in\nthe original manuscript regarding the understanding of HR-DIC resolution, which\nfailed to adequately reflect the latest advancements in this technique. We have corrected\nthe relevant descriptions. The revised text in the manuscript is provided below (lines\n76-79): “Although they can offer displacement sensitivities as high as 7–9 nm and\nspatial resolutions of 20 nm at the microscale, advanced methods such as digital image\ncorrelation (DIC)20 still cannot correlate localized damage effects with macroscopic\nbehaviours.”\n<Reviewer 1>: 11. The manuscript would benefit from a thorough proofreading to align\nwith scientific standards in experimental mechanics and fatigue. Currently, the style\nresembles AI-generated text (e.g., ChatGPT), which may detract from its scientific\ncredibility. The language should be revised for precision and field-appropriate\nterminology.\nResponse:\nWe have carefully reviewed the entire manuscript according to your suggestions,\nwith a particular focus on improving the terminological accuracy and logical rigor of\nthe experimental descriptions and data analysis sections. To ensure the highest quality,\nwe subsequently sought professional editing assistance from Springer Nature Author\nServices to polish the language of the main text and methods section (Verification Code:\n7542-3227-89D9-9D2C-B66P).\n<Reviewer 1>: 12. The authors have not discussed the use of advanced numerical\nsimulations for fatigue life prediction and damage monitoring. Modern structural\nmaterial design increasingly integrates such tools, particularly in industrial applications,\nand this should be acknowledged.\nResponse:\nThere is lacking of discussion on advanced numerical simulations in the paper, we\nhave been supplemented in the introduction and the following references have been\ncited:\nTo more clearly elaborate on the application of numerical simulation in the field\nof fatigue. The revised text in the manuscript is provided below (lines 61-67):\n“Numerical simulation enables quantitative fatigue prediction to be performed under\ncomplex working conditions through physical models. The crystal plasticity-based\nfinite-element method (CPFEM) reveals the influences of grain orientation and\ndislocation mechanisms on the initiation and propagation of cracks at the microlevel15;\ncontinuum damage mechanics (CDM) predicts crack growth and fatigue life under\nmultifield coupling effects using damage variables16,17. Numerical simulations, which\nintegrate multiscale modelling and advanced constitutive theories, enable quantitative\nprediction and provide theoretical support for material safety assessments.”\n<Reviewer 1>: 13. The phrase “tracking the sliding of grain boundaries at the atomic\nscale” requires clarification. If the authors are referring to grain boundary sliding,\nthey should specify, as this mechanism differs fundamentally from PSB formation and is\ntypically associated with high-temperature deformation.\nResponse:\nThe expression “tracking the sliding of grain boundaries at the atomic scale” is\nindeed easy to be confused with the common mechanism of grain boundary sliding in\nhigh temperature creep, which is different from the fatigue mechanism we focus on. In\nthe manuscript, all terms related to “grain boundary sliding” have been removed to\nensure conceptual accuracy and avoid ambiguity.\n<Reviewer 1>: 14. “Nanoscale atomic dislocation”: This expression is unconventional.\nDislocations are already atomic-scale defects, so the phrasing is redundant or\nmisleading. The authors should revise this for clarity and scientific accuracy.\nResponse:\nThe description of “Nanoscale atomic dislocations” in the text is not rigorous in\nthe theory of crystal defects. To improve the accuracy of terminology, we have\nuniformly revised the expression to “dislocation” in the revised draft and eliminated\npotential ambiguity.\n<Reviewer 1>: 15. formatting: There is a typographical error in the phrase “and\nMethods)This relationship”, where a space is missing.\nResponse:\nThe phrase “and methods) this relationship” in the text does have typographical\nomissions. The missing space after the right parenthesis here is due to our mistake. The\noriginal text has been revised to the standard format. In addition, we check the full text\nand manually verify all similar punctuation marks to ensure that the format is\nstandardized.\n<Reviewer 1>: 16. The authors should define the fatigue ratio and loading conditions\nused. More importantly, they should consider cyclic plasticity mechanisms such as\nratcheting and slip irreversibility and also different loading conditions such negative\nand positive load ratios. In certain conditions, dislocation accumulation may occur\nwithout producing significant surface extrusion, which would influence the MagDrift\nsignal identifically but produce different fatigue response.\nResponse:\nThe difference of cyclic plasticity mechanism plays an important role in analyzing\nthe fatigue behavior of materials. Our discussion on this issue in the first manuscript is\ninsufficient. The following is our understanding of cyclic plasticity mechanism in\nfatigue, and the corresponding experiments are supplemented:\nAlthough the microscopic mechanism of ratcheting effect is different from that of\nnon-ratcheting effect, its essence is related to atomic spacing, and slip irreversibility is\nthe smallest unit of ratcheting effect, so the essence of slip irreversibility is also related\nto the change of atomic spacing. Therefore, these microstructure characteristics will\nlead to the attenuation of the interlamellar bonding force, and the evolution of MagDrift\nis the macro-characterization of the attenuation of the interlamellar bonding force, so\nthe MagDrift signal contains these microstructure characteristics.\nIn order to verify the influence of different load conditions on MagDrift signal,\nwe supplemented the existing data and selected 45 steel high stress fatigue test (580 MPa,\nanalysis: the high stress ratchet group caused significant strain growth due to the\ncontinuous accumulation of plastic deformation, and the MagDrift showed a significant\njump; Although the small stress non ratchet group has no macroscopic plastic\ndeformation, the MagDrift still evolves significantly, and the evolution rate of the large\nstress ratchet group is faster. For a detailed explanation, please refer to <Reviewer 1>:\n3.\nAt the same time, we also supplemented the fatigue experiments with different\nstress ratios. Soft magnetic iron materials were used in the experiments, which were\npositive stress ratio R=0.1, the maximum stress was 495 MPa, and negative stress ratio\nR=-0.5, the maximum stress was 380 MPa. The structural dimensions and other loading\nparameters of the specimens were the same as those in the paper (Methods), and fatigue\nfracture occurred after 90 K and 25 K, respectively.\nAccording to the experimental data, the change rate of magnetic strength with\ndifferent stress ratios has the same evolution trend, which initially goes smoothly and\nthen accelerates to decline. From the MagDrift diagram, it has the same change trend\nbefore and after the acceleration change, which first rises steadily and then accelerates\nto decline, indicating that the change of magnetic signal intensity with the equivalent\natomic spacing shows the same rule. It is confirmed that the signal change only depends\non the degree of bond force attenuation and is independent of load conditions.\nFig. 11. Fatigue test analysis of different stress ratios. a, 495 MPa, 90,000 cycles, 0.1 stress ratio,\nCRMS and MagDrift. b, 380 MPa, 25,000 cycles, -0.5 stress ratio, CRMS and MagDrift.\nAlthough different cyclic plasticity mechanisms are different, the increase of\natomic spacing caused by them will trigger the change of MagDrift. It can be seen from\nthe experimental diagram that the MagDrift has the same trend under different load\nconditions. This feature enables spin correlation amplification technology to span the\ndifference of load conditions.\nThe above represents our current understanding of this issue. If there are any\noversights or shortcomings, we sincerely invite your feedback and suggestions.\nThe revised text in the manuscript is the same as that described in <Reviewer 1>:\n2.\n<Reviewer 1>: 17. PSB morphology varies significantly between pure metals and high-\nstrength alloys. The authors do not provide a rationale for how their average\nmeasurement technique is expected to capture these differences. It is difficult to\nimagine that such averaging can reliably identify damage processes, especially\nconsidering that fatigue failure is typically governed by the most extreme localized\nplastic events rather than average behavior. Similarly, the evolution of the MagDrift\nsignal must be interpreted in the context of strain localization, which the current\napproach may not capture. In this respect, the method is conceptually akin to applying\na macro-scale extensometer or strain gauge, useful but lacking the spatial resolution\nneeded for a predictive understanding of localized fatigue damage.\nResponse:\nThere are some differences in plastic deformation mechanism between pure\nmetals and high strength alloys. This is also the limitation of traditional macro or\naverage measurement methods. We noticed that the previous description of the\ntheoretical model was oversimplified, and there was a lack of clarity in the\ninterpretation of the basic principles, which may lead to a misunderstanding of the\nnature of the measurement method, believing that it only captures the global average.\nThe essence of spin correlation amplification technology is not spatial average\nmeasurement. Its essence is to transform the spin correlation change triggered by the\nchange of interlamellar atomic spacing into a macro measurable MagDrift signal\nthrough the effect of multiple amplification. Its core mechanism is the multiple\namplification mechanism dominated by local damage.\nUniversal response mechanism to different material damage modes: Although\nregular and coarse persistent slip bands (PSBs) are usually formed in pure metals (such\nas nickel), plastic strain localization in high-strength alloys (such as high-strength steel)\ntends to be more dispersed or discontinuous in the form of microstrip, grain boundary\nslip or phase boundary cracking due to the existence of barriers such as grain boundary\nand second phase particle. However, whether the specific microscopic morphology is\npersistent slip band (PSB) or other types of strain localization regions, the physical\nessence behind it is due to lattice distortion, the increase of atomic spacing and the\nweakening of atomic bonding force. The change of spin correlation is a direct physical\nresponse highly sensitive to these microstructure changes. Therefore, the detection\ntarget of this technology can be directly traced to the attenuation of atomic bonding\nforce, which is unified in physical nature and does not differ from PSB or other damage\nmodes.\nWe have supplemented three kinds of experiments, including artificial crack\nsimulation, prefabricated defect evolution and weld fusion zone damage, to jointly\nconfirm that the attenuation of the bonding force of the weakest section in the fatigue\nprocess is reflected in the final measured MagDrift signal through the multiplication\neffect of the spin correlation amplification of the exchange interaction (magnetic\nmoment). The local nature of fatigue damage will have a dominant effect on the global\nchange of MagDrift signal. For a detailed explanation, please refer to <Reviewer 1>:\n1.\nThis technology has the adaptive ability of environmental interference, the\nuniversal adaptability to special-shaped structures and the exponential amplification of\nmicro damage, forming a unique advantage different from the traditional linear\nsuperposition of strain measurement.\n<Reviewer 1>: 18. In fatigue, strain hardening or softening phenomena often occur,\naltering the dislocation structure over time. The manuscript does not discuss how these\neffects would influence MagDrift or whether the method can differentiate between them.\nResponse:\nCyclic hardening and cyclic softening are two typical responses of metal materials\nunder fatigue loading. Cyclic hardening shows that the ability of the material to resist\ndeformation is enhanced, and its macroscopic characteristic is that the dynamic stiffness\nincreases with time, which usually results from the increase of dislocation density and\nits entanglement; On the contrary, cyclic softening shows that the material softens and\nthe dynamic stiffness decreases, and its microscopic mechanism is related to the\nformation of low-energy dislocation structure by dislocation recombination.\nWe have drawn dynamic stiffness and MagDrift curves for representative\nspecimens of soft magnet iron, pure nickel, 45 steel and Q235 steel respectively (Fig.\n12). It was found that all specimens exhibited initial cyclic hardening stage (although\nthe duration was short), then entered a stable cyclic softening stage, and finally showed\naccelerated softening phenomenon in the failure stage. In terms of magnetic response,\nthe dynamic stiffness of soft magnet iron, 45 steel and Q235 steel increased at first and\nthen increased steadily at the beginning of loading, showing cyclic hardening, while the\nMagDrift had no sudden change in this process, showing an accelerated upward trend,\nrising steadily in the stable softening stage, and changing rapidly with the softening\nacceleration at the failure stage. On the contrary, the MagDrift of Pure Nickel shows an\naccelerated downward trend during cyclic hardening, and it also accelerates with\nsoftening acceleration in the failure stage.\nFig. 12. Comparative analysis of dynamic stiffness and MagDrift of four materials. a, Dynamic\nstiffness and MagDrift diagram of soft magnetic iron specimen under the condition of maximum\nstress of 425 MPa and fatigue failure life of 97 K. b, Dynamic stiffness and MagDrift diagram of\npure nickel specimen under the condition of maximum stress of 430 MPa and fatigue failure life of\nmaximum stress of 560 MPa and fatigue failure life of 226 K. d, Dynamic stiffness and MagDrift\ndiagram of Q235 steel specimen under the condition of maximum stress of 520 MPa and fatigue\nfailure life of 230 K.\nThe above results show that different materials may affect the response sensitivity\nof magnetic signal to micro plastic deformation, but the synergistic behavior of\nMagDrift and stiffness change plays an auxiliary role in distinguishing the cyclic\nhardening and softening processes.\n<Reviewer 1>: 19. Dislocation structures evolve depending on the slip planarity, which\nhas a direct impact on fatigue behavior. These structural changes should be considered\nif the authors aim to correlate dislocation behavior with fatigue properties.\nResponse:\nIn the process of material fatigue research, we focus on the quantum of the\nsynchronous attenuation of bonding force driven by the change of interlamellar atomic\nspacing. The evolution of dislocation behavior (dislocation annihilation, slip,\nrecombination) can trigger the response of electron spin correlation amplification as\nlong as its essence is related to the change of atomic spacing. Although the slip planarity\nmodulates the spatial distribution of bit stagger, its core effect on fatigue still leads to\nthe attenuation of bonding force through the increase of atomic spacing. The MagDrift\nsignal is directly related to the interlamellar bonding force and the exchange interaction.\nThe change of the interlamellar atoms spacing becomes the dominant variable to control\nthe strength of the two, which transcends the microscopic mechanism limitation of the\nslip geometry difference.\nThe revised text in the manuscript is the same as that described in <Reviewer 1>:\n2.\n<Reviewer 1>: 20. The approach is reminiscent of nonlinear acoustic techniques used\nto monitor dislocation density evolution. However, as in those methods, without spatial\ncorrelation at the microstructural level, dislocation density alone cannot reliably predict\nfatigue life. For example, in the Very High Cycle Fatigue (VHCF) regime, no\nmeasurable macroscopic strain evolution is typically observed, and this method would\nlikely fail to detect damage evolution. Similarly, when fatigue is initiated at large\ndefects, the deformation is highly localized and may not be captured by spatially\naveraged signals.\nResponse:\nAlthough both spin correlation amplification and nonlinear acoustic techniques\ncan be used for material damage detection, their physical principles and response\nmechanisms are essentially different.\nNonlinear acoustic technology relies on nonlinear distortion of lattice strain field:\ndislocation evolution under cyclic loading or microcracks lead to high-order harmonics\nin acoustic wave propagation, and the damage degree is inversed by measuring acoustic\nnonlinear parameters (such as β coefficient). The response of this method is essentially\ndue to the enhancement of phonon defect scattering caused by periodic lattice damage.\nIts signal strength is modulated by macroscopic parameters such as material density\nand elastic modulus, and it has no direct ability to perceive atomic scale changes.\nIn contrast, spin correlation amplification technology achieves detection by\ncapturing the electron spin correlation effect caused by changes in atomic spacing in\nferromagnetic materials: when fatigue leads to an increase in interlamellar atomic\nspacing, the attenuation of interlamellar exchange interactions triggers directional\ndeflection of the spin magnetic moment. This local perturbation is exponentially\namplified through the spin correlation chain and ultimately manifests as measurable\nmagnetic signal changes at the macroscopic level. Its core advantage lies in its\nsensitivity to electronic changes and its nonlinear signal gain characteristics, which\namplify quantum effects as they pass through the fatigue dislocation layer.\nEffectiveness of high cycle fatigue\nIn order to verify the effectiveness of this technology in VHCF scenario, we added\nhigh cycle fatigue test: 45 steel, the loading parameters were consistent with the original\n(Methods), and the maximum stress was 480 MPa. After two weeks of continuous\nloading, the experiment was forced to stop after 22.17 million times due to time.\nIn Fig. 13, no macroscopic measurable evolution of displacement and\ndisplacement amplitude is observed, but the magnetic signal has a macroscopic\nmeasurable evolution, especially the MagDrift signal, which shows a clear downward\ntrend, indicating that the interlamellar bonding force is gradually weakening to some\nextent. This confirms that under high-cycle fatigue conditions, MagDrift can still\nachieve early damage warning by capturing the gradual attenuation of the atomic\nbonding force.\nFig. 13. The high cycle fatigue test data chart. a, The MagDrift and displacement amplitude. b,\nThe CRMS and strain.\nFor the localized deformation caused by large defects, we added three types of\nprogressive experiments, artificial crack simulation, prefabricated defect evolution and\nweld fusion zone evolution. These three groups of experiments confirmed that the local\nnature of fatigue damage would have a dominant effect on the global change of\nMagDrift signal. The lamination bond force attenuation through the weakest cross\nsection is reflected in the final measured magnetic signal through the multiplication\neffect of the spin correlation amplification of the exchange interaction (magnetic\nmoment). For a detailed explanation, please refer to <Reviewer 1>: 1.\nResponse to Reviewer 2\n<Reviewer 2>: 1. In the “Fatigue Life Prediction Model” section, the authors propose\nan M-N curve analogous to the classical S-N curve using MagDrift values. Line 335\nmentions the use of an average slope across four metal materials. Why did the authors\nnot apply individual slopes for each material?\nResponse:\nWe calculated the MagDrift slope values for the four iron-based metal materials\nand found that the difference in their slopes was not significant. The average slope\nabsolute value of their M-N curves is 1.54391, which aligns closely with the value of\n1.5 reported by Néel for the relative variation of the α-Fe exchange integral with respect\nto the proximity distance of neighboring electrons. Therefore, we speculate that there\nmay be a physical correlation between the two phenomena. A detailed explanation is\nprovided below:\nIndependent calculation of MagDrift slope for four types of metal materials\nRegarding the individual MagDrift slopes of the four metal materials, we did\nindeed calculate the individual MagDrift slope values for each metal material and\ncompared and analyzed the individual MagDrift slope values and their respective\ndegrees of fit, as shown in Fig. 14.\nFig. 14. MagDrift slope and fit degree of four metal materials separately.\nAs shown in Fig. 14, when the average slope across four metal materials is not\nconsidered, the slope of the M-N curve for soft magnetic iron material is -1.26075, with\na goodness-of-fit of 0.9723; the slope of the M-N curve for 45 steel material is -1.49818,\nwith a goodness-of-fit of 0.9626; The slope of the M-N curve for Q235 steel material\nis -1.80484, with a goodness-of-fit of 0.9697; the slope of the M-N curve for bridge\nstay cable steel material is -1.61185, with a goodness-of-fit of 0.9509. The average\nslope of the M-N curves for the four materials is -1.54391.\nMatching verification and adoption basis with an average slope of -1.54\nWe found that the absolute value of the average slope of the M-N curves for the\nfour metallic materials, 1.54391, is very close to the numerical relationship reported by\nNéel between the exchange integral and the interatomic distance of electrons in α-Fe,\nwhich is 1.5 (Fig. 15). This value is closely aligned with the numerical relationship\nreported by Néel between the exchange integral and the interatomic distance of\nelectrons in α-Fe. The close agreement prompts the speculation that there may be a\nphysical correlation between the two. To test this correlation, we calculated the\ngoodness of fit when using -1.54 to represent the average slope of the M-N curves for\nthe four ferromagnetic metals (Fig. 16), and found that the coefficient of determination\n(R²) remained greater than 0.9 for all materials-specifically, 0.9261 (soft magnetic iron),\n0.9649 (45 steel), 0.9525 (Q235 steel), and 0.9521 (bridge stay cable steel). Since all\nfour materials are ferromagnetic, we speculate that the average slope of their M-N\ncurves may represent a macroscopic manifestation of electron cloud overlap in\nferromagnetic metals. Therefore, in this paper, we adopted the average slope of the four\nmaterials, -1.54, as a unified slope parameter for the M-N curves of ferromagnetic\nmetals (Methods).\nFig. 15. Relative numerical values of exchange constants as a function of the proximity\ndistance of neighboring electrons.\nFig. 16. Fitting degree of four materials under average MagDrift of -1.54.\nTo justify the use of the average slope as the computational basis in this study. The\nrevised text in the manuscript is provided below (lines 379-384): “We calculated the\nlinear fitting slopes (Methods) of the relationship between and for four types\nof ferromagnetic materials and used the average slope of the𝑘𝑘s𝑎𝑎e𝑎𝑎 four m𝑁𝑁at𝑓𝑓erials (-1.54) as\nthe correction slope (Fig. 4a). This value is closely aligned with the numerical\nrelationship reported by Néel between the exchange integral and the interatomic\ndistances of electrons in α-Fe (Fig. 4b)30. This close agreement prompts speculation that\nthere may be a physical correlation between the two metrics.”\n<Reviewer 2>: 2. In the section “Macroscopic Magnetic Mapping of Microscopic\nFatigue Signals: Spin Magnetic Moment Correlation Amplification Under External\nMagnetic Excitation,” the peak-to-peak value is chosen as the representative index\n(MagDrift). The rationale for selecting this index should be clearly explained.\nResponse:\nThe main reason for selecting the magnetic peak to peak value (MagDrift) as a\nrepresentative indicator in the paper is that the MagDrift signal has a natural self-\ncompensating characteristic, which can offset and mitigate interference from various\nenvironmental factors during the test process. We have supplemented this\ncorresponding theoretical analysis and experimental verification to fully explain why\nMagDrift was selected as the representative indicator in the paper.\nSelf-compensation mechanism of MagDrift\nDuring the loading process, we assume that the effects of the surrounding\nenvironment and the Earth's magnetic field on the magnetic field strength of the\nspecimen are denoted as , and the effects of changes in surrounding temperature\nand specimen surface tem∆p𝐵𝐵er𝐸𝐸ature on the magnetic field strength of the specimen are\ndenoted as . During fatigue loading, when the external load is ,\nconsidering t∆h𝐵𝐵e 𝑇𝑇combined effects of magnetic force, external load, sur𝜎𝜎r𝑚𝑚ou−nd𝜎𝜎in𝐿𝐿g\nenvironment, and changes in the specimen's surface temperature are as follows:\n(R2-1)\nwhen the externa𝐹𝐹l𝑚𝑚 l0o−adg (is𝜎𝜎 𝑚𝑚 −𝜎𝜎𝐿𝐿)− , 𝑓𝑓th(e∆ 𝐵𝐵c𝐸𝐸om)−bin𝑓𝑓e(d∆ 𝐵𝐵e𝑇𝑇ff)ects of magnetic force, load,\nsurrounding envi𝜎𝜎ro𝑚𝑚nm+e𝜎𝜎n𝐿𝐿t, and changes in the specimen's surface\ntemperature are as follows:\n(R2-2)\nAmong them, 𝐹𝐹𝑚𝑚1 − agn(d𝜎𝜎 𝑚𝑚 +𝜎𝜎𝐿𝐿)− re𝑓𝑓s(p∆e𝐵𝐵c𝐸𝐸ti)ve−ly𝑓𝑓 c(o∆n𝐵𝐵si𝑇𝑇d)er the effects of changes\nin the surrounding e𝑓𝑓n(v∆i𝐵𝐵ro𝐸𝐸n)ment a𝑓𝑓n(d∆ t𝐵𝐵h𝑇𝑇e) surface temperature of the test piece during\ncyclic loading. The magnetic force difference can be obtained by subtracting equation\nR2-2 from equation R2-1:\n(R2-3)\nrepresent𝐹𝐹s 𝑚𝑚th𝑚𝑚e𝑚𝑚 p=ea𝐹𝐹k𝑚𝑚 v1a−lueg (o𝜎𝜎f𝑚𝑚 th+e m𝜎𝜎𝐿𝐿a)g−ne𝐹𝐹ti𝑚𝑚c 0fo+rc𝑔𝑔e (d𝜎𝜎u𝑚𝑚ri−ng𝜎𝜎 t𝐿𝐿h)e fatigue loading\nproce𝐹𝐹s𝑚𝑚s.𝑚𝑚 I𝑚𝑚t can be seen that the change in the magnetic force peak-to-peak value has\neliminated the effects of the surrounding environment, the test apparatus, and the\nloading frequency. According to the derivation in the original manuscript, it can be seen\nthat there is a direct correlation between the change in the magnetic force peak-to-peak\nvalue and the change in the magnetic strength peak-to-peak value.\nThe MagDrift metric is designed as a differential measure, which inherently\nsuppresses low-frequency environmental drift by canceling common-mode magnetic\nperturbations.\nFurthermore, these disturbances typically vary very little within a stress cycle, and\ntheir impact on the magnetic strength peak-to-peak value is far smaller than their impact\non the absolute magnetic field strength. Therefore, the magnetic strength peak-to-peak\nvalue exhibits good self-compensating characteristics, better reflecting changes in the\nmaterial's internal state compared to the absolute field strength value, and reducing\nerrors introduced by factors such as zero-point drift in the measurement system. This is\nalso a key reason why the paper adopts the MagDrift as the representative index for\nevaluating the material's fatigue state.\nInterference resistance test verification based on MagDrift self-compensation\nmechanism\nWe supplemented the fatigue loading tests under various human-induced\ndisturbances to further verify the rationality of selecting the magnetic strength peak-to-\npeak value as the representative index in the paper.\nThis experiment mainly simulates various possible influencing factors by applying\nvarious artificial disturbances during the fatigue loading process. These include\ntemperature noise, changes in environmental magnetic field strength, poor contact in\nthe test data acquisition instrument channels, and electrical noise interference from\nhigh-power electrical appliances. The corresponding magnetic strength and MagDrift\nchange data under these artificial disturbance factors are shown in Fig. 17.\nFig. 17. Comparison of magnetic field strength and MagDrift data after fatigue test\ninterference. The shaded areas indicate periods of data acquisition system disconnection. Sudden\nchanges or jumps in magnetic field strength data during this experiment were primarily attributed\nto factors such as temperature noise, fluctuations in ambient magnetic fields, and electromagnetic\nnoise interference introduced by high-power equipment. In contrast, the MagDrift signal, leveraging\nits inherent self-compensating mechanism, exhibits an intrinsic ability to suppress these interference\nfactors, thereby maintaining a more stable trend throughout the process.\nAs shown in the figure, the absolute magnetic field strength values of this test\nspecimen exhibit significant variations. These effects include data gaps in magnetic\nfield strength values and MagDrift data caused by poor contact in the data acquisition\ninstrument channels (shaded areas in the figure indicate where data was supplemented),\nas well as fluctuations in absolute magnetic field values due to surrounding magnetic\nfield interference. However, owing to its inherent self-compensation capability, the\nMagDrift data effectively mitigates the effects of these factors. Consequently, despite\nsome localized fluctuations, the MagDrift data demonstrates overall stability.\nFinally, since MagDrift better reflects the fatigue process relative to absolute\nmagnetic field strength values, revealing changes in the material's internal fatigue state,\nthis paper employs MagDrift as the core metric for evaluating material fatigue\nconditions.\nIn order to clarify why the paper chose MagDrift as a representative indicator, we\nhave supplemented the Methods section of the manuscript with the following\nexplanation: (lines 944-963): “MagDrift features a self-compensation ability.”\n<Reviewer 2>: 3. In Figure 3, the number of curves makes the plot cluttered and\ndifficult to read. It is strongly recommended that the figure be enlarged or divided into\nseparate plots for clarity.\nResponse:\nWe have revised Figure 3 as suggested. The MagDrift and normalized MagDrift\nplots for each material have been separated into individual Multiple figures. The\nmanuscript has been revised accordingly (line 305).\n<Reviewer 2>: 4. In the “Fatigue Failure Warning” section (Line 373), the manuscript\nstates: “the Stage 2 system achieves 100% reliable early warning success rate.” What\nis the specific criterion for determining whether the early warning is successful?\nResponse:\nThere are two specific criteria for determining successful early warning. First,\neven when the material shows no visible cracks or macroscopic deformation, the\nMagDrift index has already detected fatigue damage at the atomic scale through the\nspin correlation amplification mechanism, and the MagDrift trend has shown a clear\nfatigue fracture inflection point. At this point, the MagDrift inflection point is strongly\ncorrelated with the final fracture.\nOn this basis, the warning mechanism used in the paper has captured the sudden\nchange characteristics of MagDrift and issued a fracture warning. Subsequently, even\nafter the material issued a warning, it could still safely bear thousands of external loads\non average, providing a critical buffer period for shutdown and maintenance. This also\nsolved the fatal flaw of traditional fatigue detection methods, which were “unable to\ngive fatigue failure warnings.”\nWe used this early warning mechanism to conduct fatigue fracture early warning\ntests on all fatigue test specimens and found that the MagDrift curves of all 193 sets of\nspecimens showed obvious inflection points before fracture. Furthermore, all materials\nwere still able to safely withstand external loads for a period of time after the fracture\nwarning was issued. Therefore, the early warning system achieved a 100% success rate\nin the experiment.\nTo clearly explain the specific criteria for successful early warning, we have\nsupplemented the Methods section of the manuscript with the following explanation:\n(lines 1061-1076): \"Criteria for determining successful warning\".\n<Reviewer 2>: 5. In the “Methods: Preparation of test specimens and experimental\nprocedures” section, the reviewer notes that the loading grips and fixtures are likely\nmade of ferromagnetic materials. Additionally, large instruments in the surrounding\nenvironment may generate ambient magnetic fields. Could these factors affect the test\nresults?\nResponse:\nDuring fatigue testing, the loading fixtures and clamping devices-composed of\nferromagnetic materials-may generate ambient magnetic fields. However, their\ninfluence on experimental results is negligible. Moreover, owing to its inherent self-\ncompensation capability, MagDrift can effectively suppress these ambient interferences.\nTo verify this, we conducted the following supplementary experiments:\nMagnetic strength wave test analysis under no load conditions\nTo further discuss the impact of the loading device on the test results, we\nsupplemented the magnetic strength fluctuation exploration tests under no-load\nconditions (without test specimens on the testing machine). This divided into two cases:\nwith and without magnets. The lower fixture was vibrated 20,000 times at a frequency\nof 20 Hz to simulate the actual fatigue loading process. The changes in magnetic\nstrength and displacement during the test are shown in Fig. 18.\nAs can be seen from the figure, since no test specimen was placed on the testing\nmachine, the magnetic strength and displacement exhibit a linear trend, and no\nsignificant fluctuations can be observed with the naked eye. By fitting the equations\ndescribing the changes in magnetic strength under the two conditions, it was found that\nthe slopes of the magnetic strength fitting equations were -5.17581E-9 (with magnet,\nwithout specimen) and 1.63773E-7 (without magnet, without specimen), respectively.\nThese values are far smaller than the changes in magnetic field strength observed during\nnormal fatigue cycle loading tests.\nFurthermore, due to the self-compensating properties of MagDrift during data\nprocessing, these minor fluctuations exert only a negligible impact on the experimental\nresults.\nFig. 18. Changes in magnetic strength and displacement under no-load conditions. a, with\nmagnet, without specimen. b, without magnet, without specimen.\nMagnetic strength wave test analysis without magnet but with specimen\nTo highlight the rigor of the test data, we also supplemented the magnetic strength\nfluctuation exploration tests under test without the applied magnets but with a specimen\ninstalled. We analyzed the magnetic strength fluctuations upper and lower the test piece\nunder empty magnetic conditions. We selected a standard soft magnetic iron piece and\nconducted a fatigue loading test at 485 MPa without adding magnets. The test results\nare shown in Fig. 19.\nAs can be seen from the figure, the displacement change is no different from that\nin a normal fatigue test, but the fluctuations in magnetic strength remain imperceptible\nto the naked eye throughout the entire process. We also fitted the trends of magnetic\nstrength changes at the two positions and found that the slopes of the equations\ndescribing the trends in magnetic strength changes at the upper and lower positions\nwere -2.30467E-8 (upper magnetic strength) and -1.15808E-9 (lower magnetic\nstrength), respectively. These values are still far smaller than the magnetic strength\ntrend changes observed during normal fatigue loading processes.\nBased on the two supplementary experiments, it can be concluded that the\nmagnetic field generated by the experimental setup has a negligible impact on the\nexperimental results. Moreover, the representative metric used in this study-MagDrift-\ndemonstrates inherent self-compensation characteristics, effectively canceling out or\nmitigating interference from ambient magnetic fields. As a result, minor magnetic\nfluctuations originating from the experimental apparatus have negligible influence on\nthe reliability of the experimental outcomes.\nFig. 19. Changes in magnetic strength without adding magnets to the test piece.\nTheoretical verification of the MagDrift self-compensation mechanism for\neliminating interference from test instruments and the environment\nDuring loading, we assume that the surrounding geomagnetic field and the\ninfluence of the testing equipment on the magnetic field strength of the test specimen\nare . Therefore, when performing fatigue loading, when the external load is\n, t∆he𝐵𝐵 c𝐸𝐸ombined effects of magnetic force, load, surrounding environment, and te𝜎𝜎s𝑚𝑚tin−g\n𝜎𝜎eq𝐿𝐿uipment are as follows:\n(R2-4)\nwhen the external load 𝐹𝐹is𝑚𝑚 0 −g(𝜎𝜎,𝑚𝑚 Th−e 𝜎𝜎c𝐿𝐿o)m−b𝑓𝑓in(e∆d𝐵𝐵 e𝐸𝐸f)fects of magnetic force, load,\nsurrounding environment, and t𝜎𝜎e𝑚𝑚sti+ng𝜎𝜎 e𝐿𝐿quipment are as follows:\n(R2-5)\nAmong them, 𝐹𝐹ta𝑚𝑚k1es− ingt(o𝜎𝜎 𝑚𝑚ac+co𝜎𝜎u𝐿𝐿n)t −the𝑓𝑓 (i∆nf𝐵𝐵lu𝑇𝑇e)nce of the surrounding\nenvironment and the t𝑓𝑓e(s∆t a𝐵𝐵p𝐸𝐸p)aratus during cyclic loading. Using equation R2-5 minus\nequation R2-4, the magnetic force difference can be obtained, which represents the\nmagnetic force peak-to-peak value during the fatigue loading process:\n(R2-6)\nTherefore, it can𝐹𝐹 b𝑚𝑚e𝑚𝑚 s𝑚𝑚e=en 𝐹𝐹th𝑚𝑚a𝑚𝑚t 𝑚𝑚du=e 𝐹𝐹𝑚𝑚1 − ga(n𝜎𝜎d𝑚𝑚 +𝜎𝜎𝐿𝐿)− 𝐹𝐹a𝑚𝑚re0 e+lim𝑔𝑔i(n𝜎𝜎a𝑚𝑚ted−, a𝜎𝜎n𝐿𝐿d) the MagDrift is only\nrelated to the internal sta𝑓𝑓t(e∆ o𝐵𝐵f 𝐸𝐸th)e m ate𝑓𝑓ri(a∆l.𝐵𝐵 𝑇𝑇)\nIn addition, these interferences change very little within a stress cycle, so these\ninfluencing factors are offset or mitigated and have a relatively minor impact on the test\nresults.\n<Reviewer 2>: 6. Also in this section, the authors mention using five N35 magnets.\nPlease clarify whether these are permanent magnets, specify the initial magnetic field\nstrength they generate, and explain the rationale for choosing this level of excitation.\nResponse:\nThe primary reason for selecting five N35 magnets as the external magnetic\nexcitation is that during the fatigue process of the material, the change in magnetic\nstrength under this initial magnetic strength can maximize the fatigue signal, ensuring\nthat the change in magnetic strength under this condition can reflect the fatigue\ncharacteristics of the material itself. To elaborate more clearly on the reasons for\nchoosing this external magnetic excitation, we have supplemented the following\nexperimental and theoretical analysis:\nAnalysis of notched bar magnetic saturation test\nThe experiment utilized N35 neodymium-iron-boron permanent magnets. The\ninitial surface magnetic field strength of a stack of five N35 magnets measured between\n240 and 250 mT. Multiple calibrations were performed using a gauss meter to ensure\ndata reproducibility. To justify the use of this external magnetic excitation, a\nsupplementary comparative experiment was conducted to evaluate variations in the\nsaturation magnetic field strength of test specimens under different external magnetic\nexcitations and crack depths. The specific experimental procedure is detailed below.\nThe experiment used an iron rod measuring 16 mm in diameter and 25 cm in length,\nand employed neodymium-iron-boron magnets to apply eight distinct levels of external\nmagnetic field excitation: 50 mT, 87 mT, 124 mT, 158 mT, 185 mT, 211 mT, 317 mT,\nand 430 mT.\nSelect the midpoint of the iron rod (at 12.5 cm) as the preset crack point. When\nsetting the preset crack, the crack width is 1 mm. When the crack depth is less than 10\nmm, the cutting depth is 2 mm each time. After exceeding 10 mm, the cutting depth is\n1 mm each time until the crack depth reaches 15 mm.\nSelect four measuring points at 5 cm, 10 cm, 15 cm, and 20 cm along the length\ndirection of the iron rod. Use a handheld gauss meter to collect magnetic strength at the\nmeasuring points at a lifting height of 3 mm. The experimental setup and data are shown\nin Fig. 20.\nFig. 20. Schematic diagram of the comprehensive relationship between magnetic strength\nchanges at different measuring points and under different crack depths under different\nexternal magnetic excitations. a, Schematic diagram of notched bar magnetic saturation test and\nexternal magnetic field configuration location. The preset crack position is located at the geometric\ncenter of the test piece, with measuring points 1 and 2 located in front of the crack, and measuring\npoints 3 and 4 located behind the crack. The external magnetic field is applied by neodymium-iron-\nboron permanent magnets, covering eight excitation levels from 50–430 mT. b, Schematic diagram\nof finite element simulation of crack depth on magnetic induction line distribution. When there are\ncracks in the material, the magnetic flux in the damaged area at the crack location section decreases,\nwhile the magnetic flux in the undamaged area increases accordingly; and the increase in crack\ndepth leads to an increase in magnetic flux in the undamaged area. c, The evolution law of magnetic\nstrength at each measuring point under eight types of external magnetic excitation as a function of\ncrack depth. As the external magnetic excitation increases, the magnetic strength at each measuring\npoint increases synchronously. When the crack depth reaches a corresponding value, there is a trend\nof increasing magnetic strength at measuring points in front of the crack and decreasing magnetic\nstrength at measuring points behind the crack. d, Output the relationship diagram between the\nacceleration of magnetic field changes and the remaining depth, external magnetic excitation, and\nthe ratio of magnetic excitation to undamaged area.\nExploration of saturation magnetic strength of iron rods with different crack\ndepths\nThe experimental results indicate that when the measurement point is ahead of the\ncrack, the magnetic field intensity initially increases slowly and then at an accelerating\nrate as the crack deepens; when the point is behind the crack, the intensity first decreases\nslowly and then at an accelerating rate. The underlying physical mechanism can be\ndescribed as follows: Cracks force magnetic field lines to divert around intact regions.\nWhen the magnetic permeability in these regions remains unsaturated, the magnetic\nflux continues to vary gradually. As a result, the magnetic field intensity initially\nexhibits a slow decreasing or increasing trend as crack depth increases. With further\ncrack propagation and increased external magnetic excitation, the local magnetic\npermeability-particularly at the most vulnerable cross-section-begins transitioning from\nan unsaturated to a saturated state. Once the magnetic circuit around the crack reaches\nsaturation, the permeability in the region behind the crack decreases significantly,\nleading to a pronounced decline in magnetic field intensity. Simultaneously, due to the\nconcentration of magnetic flux ahead of the crack, the magnetic strength in this zone\ndemonstrates a marked increasing trend.\nWe analyzed the remaining cross-sectional undamaged ratio when the magnetic\nstrength reached saturation under different external excitation magnetic intensities, as\nshown in Table. 2.\nTable. 2. Ratio of damaged area to undamaged area when the fracture zone reaches magnetic\nsaturation under different excitation magnetic intensities.\nExternal Ratio of external\nmagnetic Crack depth Undamaged magnetic excitation to\nUndamaged rate\nexcitation (mm) depth (mm) undamaged area\n(mT) (mT/mm2)\n50 13 3 13% 0.28\n87 13 3 13% 0.5\n124 12.5 3.5 16.2% 0.74\n158 12 4 19.6% 0.977\n185 11 5 26.7% 1.3\n211 9 7 42.2% 1.82\n317 8 8 50% 3.2\n430 8 8 50% 4.3\nAnalysis of the data presented in Table 2 revealed a linear relationship between\nthe intensity of the external magnetic excitation and the ratio of the undamaged area.\nThe slope of the linear fit was determined to be 0.011, which serves as a quantitative\nmeasure of both the crack damage extent and the external magnetic excitation intensity\nunder critical saturation conditions.\nAnalysis of the residual thickness in the crack region at magnetic saturation under\nvarying external magnetic excitation intensities reveals that when the residual section\ndepth is 8 mm, the external excitation magnetic field begins to enter the saturation-\ncritical transition zone within an approximate range of 211–317 mT. According to the\nspecification (GB/T 3075-2021) for fatigue loading tests, the diameter of the clamped\nend of a curved specimen shall be more than twice the diameter of the thinnest section\nin the middle. Given that the clamped end diameter of the specimen is 16 mm, the\nminimum thickness at the midsection is taken as 8 mm (analogous to a residual depth\nof 8 mm). At this residual depth, an external magnetic excitation intensity of 240–250\nmT corresponds accurately to the median value of the magnetic saturation-critical\ntransition zone.\nThe fatigue test therefore employs an external magnetic excitation of 240–250 mT.\nThis intensity not only sufficiently amplifies the fatigue signal but also effectively\nisolates geometric effects, thereby highlighting the intrinsic evolution of material\nfatigue and ensuring the dominance of the material's fatigue-related response. This\nconsideration is a key rationale for selecting a stack of five N35 magnets as the external\nmagnetic excitation source in this experiment.\n<Reviewer 2>: 7. During loading, the authors use a frequency of 20 Hz. According to\nthe reviewer’s knowledge, such a high frequency can cause surface temperature to rise\nin the specimen. Has this potential influence on the evolution of magnetization been\nconsidered or evaluated?\nResponse:\nThe fatigue experiment addressed the influence of temperature rise in the specimen\nunder high-frequency loading. Three additional experiments were conducted to\nexamine the effect of surface temperature variation on the data in 20 Hz fatigue tests.\nThe results demonstrated that slight temperature variations on the specimen surface had\nno significant impact, and the self-compensation feature of MagDrift effectively\ncanceled out or mitigated interference caused by specimen surface temperature on\nmagnetic field measurements. Fatigue tests were also performed at 5 Hz and 10 Hz,\nexperiments have shown that the trend of MagDrift is less affected by the loading\nfrequency.\nExploratory experiment on surface temperature change of test piece under 20Hz\nloading frequency\nIn single-temperature control mode, an external air conditioning system maintains\nthe room temperature at 27 °C. Under dual-temperature control, while the room\ntemperature is kept at 27 °C, a fan is used to dissipate heat from the testing machine,\nensuring that the temperature around the specimen remains at approximately 28 °C. In\nthe non-temperature-controlled scenario, no active temperature control measures are\nimplemented, at this time, the temperature around the test piece is maintained at around\n32 ℃.\nDuring the experiment, we measured the temperature and magnetic field changes\non the surface of the test specimens under three different temperature control conditions\nwith the same applied force, as shown in Fig. 21.\nFig. 21. Supplementary test on surface temperature change of the test piece under a loading\nfrequency of 20Hz. a, Single temperature control, 485 MPa, 98 K cycles, CRMS and strain. b,\nSingle temperature control, 485 MPa, 98 K cycles, MagDrift and peak-to-peak displacement. c,\nDual temperature control, 485 MPa, 95 K cycles, CRMS and strain. d, Dual temperature control,\nMPa, 52 K cycles, CRMS and strain. f, No temperature control, 485 MPa, 52 K cycles, MagDrift\nand peak-to-peak displacement.\nThe measurements indicated that the specimen surface temperature fluctuated\nwithin a range of 2.0 to 2.8 °C under uncontrolled ambient conditions. When only room\ntemperature was maintained, the variation ranged from 1.7 to 2.3 °C. Under\nsimultaneous room temperature control and active heat dissipation around the specimen,\nthe temperature change was further reduced to a range of 1.3 to 1.8 °C.\nAlthough slight fluctuations in the surface temperature of the specimens were\nobserved at a loading frequency of 20 Hz, no significant variations or distinct trends\nwere identified in the magnetic strength change rate or MagDrift across the three\ntemperature control conditions. This suggests that temperature variations induced by\nthe loading frequency had a relatively minor impact on the test outcomes. To further\nanalyze the changes in magnetic strength and MagDrift under these conditions, we also\ngenerated normalized plots for both MagDrift and change rate of magnetic strength\nacross the three specimens, as illustrated in Fig. 22.\nFig. 22. Normalized magnetic strength change rate and MagDrift of three specimens under\nthree temperature control conditions at a loading frequency of 20 Hz.\nNormalized MagDrift and magnetic field intensity change rate under three\ntemperature control conditions exhibit consistent trends across all scenarios. The only\nobserved variation lies in the critical points of MagDrift mutation, which we attribute\nto differences in the number of loading cycles. We speculate that a loading frequency\nof 20 Hz induces minor surface heating on the specimen; however, this temperature\nvariation remains relatively weak and does not significantly affect the final\nexperimental outcomes. Furthermore, supplementary theoretical analysis confirms that\nthe self-compensation characteristic of MagDrift effectively mitigates the influence of\ntemperature variations.\nTheoretical analysis of MagDrift self-compensation mechanism to eliminate the\ninfluence of temperature changes on specimens\nWe consider the effect of surface temperature changes on the magnetic field\nstrength of the specimen as . Therefore, when performing fatigue loading,\nconsidering the combined eff∆ec𝐵𝐵t𝑇𝑇s of magnetic force, load action, and surface\ntemperature changes of the specimen when the external load is , the following\nare taken into account: 𝜎𝜎𝑚𝑚 −𝜎𝜎𝐿𝐿\n(R2-7)\nwhen the external load is𝐹𝐹 𝑚𝑚0 −g(𝜎𝜎,𝑚𝑚 Th−e 𝜎𝜎c𝐿𝐿o)m−b𝑓𝑓in(e∆d𝐵𝐵 e𝐸𝐸f)fects of magnetic force, load,\nand surface temperature change𝜎𝜎s𝑚𝑚 on+ t𝜎𝜎he𝐿𝐿 specimen are as follows:\n(R2-8)\nAmong them, 𝐹𝐹ta𝑚𝑚k1es− ingt(o𝜎𝜎 𝑚𝑚ac+co𝜎𝜎u𝐿𝐿n)t −the𝑓𝑓 (i∆nf𝐵𝐵lu𝑇𝑇e)nce of surface temperature\nchanges of the speci𝑓𝑓m(e∆n𝐵𝐵 d𝑇𝑇u)ring cyclic loading. Using equation R2-8 minus\nequation R2-7, the magnetic force difference can be obtained, which represents the\nmagnetic force peak-to-peak value during the fatigue loading process:\n(R2-9)\nIt can be seen 𝐹𝐹th𝑚𝑚a𝑚𝑚t 𝑚𝑚du=e 𝐹𝐹𝑚𝑚1 − ga(n𝜎𝜎d𝑚𝑚 +𝜎𝜎𝐿𝐿)− 𝐹𝐹a𝑚𝑚re0 e+lim𝑔𝑔i(n𝜎𝜎a𝑚𝑚ted−, a𝜎𝜎n𝐿𝐿d) the MagDrift is only\nrelated to the internal sta𝑓𝑓t(e∆ o𝐵𝐵f 𝐸𝐸th)e m ate𝑓𝑓ri(a∆l.𝐵𝐵 𝑇𝑇)\nIt can be seen that due to the good self-compensation characteristics of MagDrift,\nit can effectively offset or weaken the interference of surface temperature on magnetic\nstrength of the specimen. Therefore, although the 20 Hz loading frequency will cause\ntemperature changes on the surface of the specimen, it has a relatively small impact on\nthe test results.\nAnalysis of magnetic strength waveform changes under different fatigue loading\nfrequencies\nTo investigate the effect of different loading frequencies on the magnetization\nprocess, we conducted additional analysis experiments on the waveform changes of\nmagnetic strength at loading frequencies of 5 Hz, 10 Hz, and 20 Hz (fatigue tests were\nconducted using sine wave loading). The waveform changes of magnetic strength at\neach loading frequency are shown in Fig. 23.\nFig. 23. Schematic diagram of displacement waveform and magnetic strength waveform\nchanges under different loading frequencies. a, Corresponding diagram of displacement\nwaveform and magnetic strength waveform at a loading frequency of 5Hz. b, Corresponding\ndiagram of displacement waveform and magnetic strength waveform at a loading frequency of 10Hz.\nc, Corresponding diagram of displacement waveform and magnetic strength waveform at a loading frequency of 20Hz.\nFrom the graph, it can be seen that as the loading frequency increases, the\ndisplacement and magnetic strength waveforms gradually exhibit a \"non-linear\" trend\nvisually. This difference is mainly due to the variation in the number of sampling points.\nUnder high-frequency loading, an increase in the number of cycles per unit time will\nresult in a decrease in data points in a single cycle, and the waveform appears more\ndiscretized (or less smooth) due to the lower number of data points per cycle.\nHowever, the essence of the above changes is only the data collection method and\ndoes not reflect the changes in the intrinsic evolution of material fatigue. In order to\nfurther analyze the changes in magnetic strength and MagDrift under different loading\nfrequencies, we also plotted the magnetic strength change maps, MagDrift change maps,\nand corresponding normalization maps of the three specimens at three different\nfrequencies, as shown in Fig. 24.\nFig. 24. Schematic diagram of changes in magnetic strength waveform under different\nloading frequencies. a, The CRMS of three different loading frequencies. b, The normalized\nCRMS of three different loading frequencies. c, The MagDrift of three different loading frequencies.\nd, The normalized MagDrift of three different loading frequencies.\nAs shown in the figure, although different loading frequencies were applied, the\nmagnetic field intensity change rate, MagDrift curves, and their corresponding\nnormalized curves all exhibit similar trends under identical load conditions. Fatigue\nfracture points consistently emerged prior to final failure. This indicates that while\nsampling rate variations due to different loading frequencies influence the volume of\ndata acquired, the overall trend of MagDrift remains effective in revealing the\nmaterial’s fatigue characteristics.\nFrom the above experiments, it can be concluded that the change in loading\nfrequency only affects the waveform visual and sampling point quantity of data\nacquisition. However, since the paper focuses on the relative changes in magnetic\nstrength rather than surface features, the impact of different loading frequencies on the\nexperimental results and material magnetization process is relatively small.\n<Reviewer 2>: 8. When measuring the magnetic field, the Hall sensor is placed beneath\nthe specimen. Why was this location selected? Although the manuscript states that a\nlinear relationship exists between magnetic changes measured at the remote end and\nthe variation in binding force at the weakest section, the specimen is not of uniform\ncylindrical geometry. Is this assumption still valid in the given configuration?\nResponse:\nThe paper places the Hall sensor beneath the specimen, mainly based strategic\nsensor placement designed to optimally capture changes in magnetic flux, to ensure that\nthe magnetic flux changes caused by cracks can be directly and efficiently captured;\nThe paper adopts MagDrift as the core indicator, effectively eliminating the nonlinear\nerrors caused by non-uniform geometry of the sample, sensor position differences, and\nenvironmental interference, thereby ensuring the reliability of the linear relationship\nbetween macroscopic magnetic flux and microscopic bonding force changes in non-\nuniform cylindrical samples. We have supplemented theoretical analysis and fatigue\ntests at different locations to fully illustrate this issue.\nAnalysis of sensor layout selection based on accurately capturing magnetic flux\nchanges\nDuring the process of fatigue crack propagation, magnetic field lines will undergo\nsignificant changes at the fracture surface. Placing the sensor beneath the specimen can\nmaximize the capture efficiency of magnetic signal changes caused by fatigue fracture.\nDuring the fatigue process of materials, the magnetic domain structure inside the\nmaterial will undergo distortion, resulting in the redistribution of local magnetic flux\nfields. If the fracture surface is placed on the magnet sensor connection, the magnetic\nflux change generated by the crack can directly and unobstructed penetrate the sensitive\narea of the sensor, thereby maximizing the efficiency of capturing magnetic flux change\nsignals. This avoids magnetic flux attenuation or scattering caused by positional\ndeviation, significantly improving the sensitivity and accuracy of detection.\nIn addition, the MagDrift data measured under this deployment strategy can\nrespond in real-time to weak fatigue characteristic changes, providing a solid physical\nfoundation for fatigue failure warning and ensuring the high reliability of experimental\ndata.\nFatigue loading test in different positions of sensors\nRegarding the influence of sensor position on the magnetic signal, it can be\nobserved from previous comparative experiments on magnetic strength changes at\ndifferent measurement points under varying crack depths that: when the sensor is\npositioned ahead of the crack, the magnetic field strength at the measurement point\ninitially remains stable and then increases; when the sensor is located behind the crack,\nthe magnetic field strength initially remains stable and then decreases.\nWe have supplemented the fatigue loading tests of the sensor at different positions,\nand the magnetic strength change rate and strain are shown in Fig. 25, while the\nMagDrift change is shown in Fig. 26.\nIt can be seen that when the fatigue fracture position is below the specimen and\nthe sensor is arranged above the fracture position, the magnetic strength shows an\nupward trend and the MagDrift shows a downward trend. When the fatigue fracture\nlocation is above the specimen and the sensor is arranged below the fracture location,\nthe magnetic strength shows a downward trend and the MagDrift shows an upward\ntrend. Both MagDrift and magnetic strength change rates show opposite trends. This is\nalso in line with the previous notched bar magnetic saturation test.\nFig. 25. CRMS and strain of the sensor Fig. 26. MagDrift and peak to peak of\nabove and below the specimen under fatigue displacement of the sensor above and below\nloading. the specimen under fatigue loading.\nMagDrift mitigate the effects of nonlinearities\nWe established a relationship between the macroscopic magnetic flux change and\nthe variation in bonding force at the weakest cross-section based on the assumption of\nsmall deformation. During actual fatigue loading tests, higher-order terms, sensor\nposition and distance, material inhomogeneity, and other factors may lead to\nnonlinearities in the experimental data. Furthermore, as demonstrated by the\nexperimental validation and theoretical analysis above, the geometric inhomogeneity\nof the material or changes in the fracture location indeed affect the magnetic field\nvariation, but their primary influence is on the absolute measured values of the magnetic\nfield strength. In this paper, the core indicator used to characterize the material's fatigue\nperformance is not the absolute magnetic field strength, but rather MagDrift, which is\nadopted as the representative parameter for characterizing the material's fatigue\nbehavior.\nThe key advantage of MagDrift is to offset or reduce the influence of factors such\nas non-uniformity of test materials, positional differences, environmental interference,\nor test equipment on magnetic strength values by calculating peak to peak values.\nAnd MagDrift has inherent self-compensation characteristics, which effectively reduces\nmeasurement system errors compared to the absolute value of magnetic strength.\nTherefore, even if the sample is not a uniform cylindrical structure. In this configuration,\nthe underlying assumption of the model remains valid.\n<Reviewer 2>: 9. Building upon the previous question: does the location of fatigue\nfracture affect the intensity of the measured magnetic signal?\nResponse:\nThe specific location of fatigue fracture only affects the absolute value of magnetic\nstrength, but the influence of this positional difference can be effectively eliminated by\nthe relative change rate of magnetic strength. This ensures that the fatigue performance\nevolution of the material can still be stably and reliably characterized at different\nfracture locations. We have added the following content to elaborate on this issue in\ndetail:\nAnalysis of magnetic signal variations due to different fracture positions in\nnotched bar magnetic saturation test\nAs demonstrated in the previous notched bar magnetic saturation test (Fig. 20),\nwhen the fatigue fracture occurs beyond the magnet-sensor connection zone, the\nmagnetic field strength at the measurement point initially increases slowly and then\naccelerates as the crack propagates. In contrast, when the fracture is located within the\nmagnet-sensor connection zone, the magnetic field strength exhibits an initial slow\ndecrease followed by an accelerated decline. These results confirm that the location of\nfatigue fracture significantly influences the variation trend of magnetic signals.\nFatigue loading tests at different fracture locations\nThe notched bar magnetic saturation test explored the influence of different\nfracture positions on the change trend of magnetic signal under static conditions. In\norder to analyze the influence of different fracture positions on the change of magnetic\nstrength during the fatigue loading test, we supplemented the fatigue loading test under\ndifferent sensor positions (the change rate of magnetic strength and strain are shown in\nFig. 25 above, and the MagDrift is shown in Fig. 26).\nFrom the experimental data, it can be seen that the change in magnetic strength\nwhen the fatigue fracture location is within the connection between the magnet and the\nsensor is opposite to the change in magnetic strength when it is outside the connection\nbetween the magnet and the sensor, and the MagDrift also shows opposite trends in\nboth cases.\nThe opposite trend of change corresponds to the magnetic strength variation\nobserved in the previous notched bar magnetic saturation test. When the fracture\nlocation is outside the connection between the sensor and the magnet, the magnetic\nstrength shows an upward trend, and when the fracture location is inside the connection\nbetween the sensor and the magnet, the magnetic strength shows a downward trend.\nRepresentative indicator - MagDrift\nFrom fatigue tests at different fracture locations, it can be seen that it mainly\naffects the absolute measurement value of magnetic strength. However, the core\nindicator used in this study to characterize the fatigue performance of materials is not\nthe absolute magnetic field strength value, but rather the use of MagDrift as a\nrepresentative indicator to characterize the fatigue performance of materials. Compared\nto the absolute value of magnetic strength, MagDrift can effectively counteract or\nweaken the effects caused by differences in fracture location. And MagDrift has\ninherent self-compensation characteristics, effectively reducing system measurement\nerrors, etc.\n<Reviewer 2>: 10. In the “Fatigue Fracture Early Warning Algorithm” section, the\nalarm mechanism needs to be more thoroughly explained. At minimum, the following\npoints should be clarified:\n(1) What is the physical meaning of the two selected formulas?\n(2) What is the physical meaning of S₁ in the equations?\n(3) Describe the computational procedure in detail, including whether the method\nis applicable only after 90% of the fatigue life has elapsed, and whether a moving\nwindow is employed during prediction.\nResponse:\nThe two warning formulas proposed in the paper have clear physical meanings,\nmainly targeting the accelerated damage mutation of low cycle fatigue and the gradual\ndamage accumulation mechanism of high cycle fatigue. By capturing the mutation\ncharacteristics of MagDrift, the degradation state of the weakest section bonding force\nand material fatigue performance can be determined. The warning calculation adopts a\ndynamic dual moving window mechanism to dynamically slide and update data in real\ntime, without relying on fatigue life percentage. Its essence is to capture the sudden\nchange characteristics of MagDrift and achieve the purpose of warning. The following\nwill elaborate in detail from three aspects: MagDrift mutation mechanism, parameter\ndefinition, and warning dynamic window algorithm:\nMechanism of MagDrift fracture mutation - saturation magnetic strength critical\neffect\nOur warning algorithm is predicated on a physical model inspired by the magnetic\nsaturation behavior observed in our supplementary bar-cutting tests.\nIn the notched bar magnetic saturation test, we systematically explored the\ninfluence of crack depth on magnetic strength evolution under different external\nmagnetic excitations of 50-430 mT. The magnetic saturation law of ferromagnetic\nmaterials during damage propagation was revealed by setting different crack depths of\n2-15 mm and measuring the magnetic strength at measuring points. The core conclusion\nof this experiment is that under different external magnetic excitations, when the critical\nvalue of magnetic saturation is reached at the crack, the local magnetic permeability of\nthe material will enter a saturated state, leading to accelerated increase and decrease in\nmagnetic strength in the pre and post crack regions, respectively. This is due to the fact\nthat magnetic field lines are forced to orbit undamaged areas, and the initial changes in\nmagnetic flux are gentle; But once the crack reaches a saturated magnetic strength state,\nthe magnetic flux distortion significantly intensifies. For a detailed explanation, please\nrefer to <Reviewer 2>: 6.\nIn fatigue testing, the accumulation of material fatigue damage is analogous to the\ndeepening process of cracks in cutting tests. For example, the sudden change point of\nMagDrift is directly related to the achievement of the \"saturation magnetic strength\ncritical state\".\nDuring fatigue testing, when the material is in the early to mid-stages of fatigue\ndamage, microcracks propagate gradually while the magnetic circuit remains\nunsaturated. Under these conditions, the variation in MagDrift exhibits a stable and\nmoderate trend, consistent with the gentle variation observed in magnetic field strength\nat the measurement point under non-saturated conditions during the notched bar\nmagnetic saturation test.\nWhen the material accumulates fatigue damage and is about to fracture, the\nMagDrift change begins to undergo a sudden change, and the magnetic strength at the\nweakest section reaches a state of magnetic saturation. This is analogous to the\naccelerated change in magnetic strength at the measuring point when the crack position\nin the steel cutting test is about to reach saturation magnetic strength.\nAt this point, if the loading continues, the material will experience fatigue fracture,\nand the MagDrift will show a sudden rise or fall trend, which is similar to the\nacceleration of the magnetic strength at the measuring point after reaching saturation in\nthe notched bar magnetic saturation test.\nTherefore, we hypothesize that the sudden change in MagDrift signifies a critical\nshift in the material's magnetic state, potentially analogous to the approach to magnetic\nsaturation at a microscopic flaw. This critical transition, informed by the static bar-\ncutting experiments, serves as the physical basis for our warning algorithm.\nAnalysis of the physical meaning of fatigue warning formula\nThe warning calculation formula we use for low cycle fatigue is:\n(R2-10)\n𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚−𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚\nThe core physic𝐴𝐴a𝐴𝐴l𝑎𝑎 s𝐴𝐴i𝐴𝐴gn 𝑡𝑡 i ℎ fi=can𝐵𝐵c 𝑚𝑚 e 𝑚𝑚 o 𝑚𝑚 f− th𝐴𝐴is 𝑚𝑚 f 𝑚𝑚 o 𝑚𝑚 rm+ula is𝑆𝑆 1to cap<tu0re accelerated damage\nmutations. Among them, (MagDrift value of the first 60 cycles at the current\ntime point) mainly reflects𝐵𝐵 t𝑚𝑚he𝑚𝑚 𝑚𝑚latest fatigue state of the material. This window covers\nthe late stage of fatigue, at which point damage accumulation enters the acceleration\nphase. (1,000 cycles before 3,000 cycles before the current time plus an\naddition𝐴𝐴al𝑚𝑚 6𝑚𝑚0𝑚𝑚 cycles) represents the initial fatigue baseline. It can avoid recent damage\ninterference and provide stable reference. The negative values of the two windows,\n, indicate that MagDrift has recently been significantly lower than\n𝐵𝐵hi𝑚𝑚st𝑚𝑚o𝑚𝑚ri−ca𝐴𝐴l l𝑚𝑚e𝑚𝑚v𝑚𝑚els, reflecting accelerated damage.\nWhen the negative deviation amplitude of the fatigue threshold exceeds the noise\ntolerance for four consecutive times, it indicates that the material has entered the\nirreversible damage acceleration stage, at this point, it indicates that the weakest cross-\nsectional bonding force of the material is about to be insufficient to resist external forces.\nThe warning calculation formula we use for high cycle fatigue is:\n(R2-11)\n𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚−𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚\nThe core phys𝐴𝐴ic𝐴𝐴a𝑎𝑎l𝐴𝐴 s𝐴𝐴ig 𝑡𝑡 n ℎ if=ica𝐵𝐵n 𝑚𝑚 c 𝑚𝑚 e 𝑚𝑚 o−f t𝐴𝐴hi 𝑚𝑚 s 𝑚𝑚 f 𝑚𝑚 o−rmula 𝑆𝑆is2 to id>en0tify the accumulation of\ngradual damage. Among them, (the MagDrift value of first 60 cycles at current\ntime point) is mainly used for r𝐵𝐵ea𝑚𝑚l𝑚𝑚-t𝑚𝑚ime monitoring of material fatigue status.\n(the MagDrift values of 2,000 cycles before 40,000 cycles before current time pl𝐴𝐴us𝑚𝑚 a𝑚𝑚𝑚𝑚n\nadditional 60 cycles) covers a longer damage evolution cycle, reflecting the early\nfatigue intrinsic trend. The positive values of (two windows) indicate\nthat MagDrift continues to exceed historical ben𝐵𝐵c𝑚𝑚hm𝑚𝑚𝑚𝑚a+rks𝐴𝐴, 𝑚𝑚re𝑚𝑚f𝑚𝑚lecting slow accumulation\nof damage.\nWhen the positive deviation amplitude of the fatigue threshold exceeds the noise\ntolerance for four consecutive times, it indicates that the material damage has entered\nthe critical accumulation stage, and at this time, at this point, it also indicates that the\nweakest cross-sectional bonding force of the material is about to be insufficient to resist\nexternal forces.\nLifetime applicability and dynamic window mechanism\nThis method is not limited to being applicable only after the material's fatigue life\nreaches 90%. This warning algorithm mainly utilizes the dual moving window\nalgorithm to dynamically slide and update data in real time, which can cover the entire\nfatigue life cycle. The essence of the warning algorithm is to capture the mutation\ncharacteristics of MagDrift and achieve the purpose of warning, regardless of the\npercentage of fatigue life. Four consecutive triggers ensure the elimination of\ninterference from instantaneous MagDrift changes.\nPhysical meaning of and\nand are em𝑺𝑺p𝟏𝟏irically𝑺𝑺 d𝟏𝟏etermined parameters that fundamentally govern the\nsensi𝑆𝑆ti1vity of 𝑆𝑆t2he detection window to mutations in the MagDrift signal. Their values\nare derived from experimental calibration and are physically tied to the rate of change\nof the remanent saturation magnetic flux density-a faster change corresponds to a larger\nvalue of .\nThis𝑆𝑆 relationship is substantiated by the cutting experiment on the soft magnetic\niron rod. As artificial cracks were introduced and deepened, the resulting redistribution\nof magnetic flux-decreasing in the damaged zone and increasing in the undamaged\nregion-led to an accelerated change in local magnetic field strength, particularly under\nhigher external excitation. The faster the crack progressed (simulating accelerated\ndamage accumulation), the more pronounced the change in magnetic flux density\nbecame, necessitating a larger value to effectively track and threshold the resulting\nsignal mutation. For specific an𝑆𝑆alysis, please refer to <Reviewer 2>: 6.\nIn low-cycle fatigue (LCF), damage accumulates rapidly, analogous to the deep-\ncrack phase in the cutting test, when magnetic flux changes abruptly. Here, a smaller\n(4.8 for soft magnetic iron) is used to suppress noise and avoid false alarms from\nt𝑆𝑆r1ansient interference, ensuring reliable capture of swift magnetic mutations.\nIn high-cycle fatigue (HCF), damage evolves gradually, similar to the early, stable\nphase of crack propagation. Under these conditions, magnetic changes occur slowly but\npersistently, requiring a larger (1.2 for soft magnetic iron) to enhance sensitivity to\nweak signals and enable early d𝑆𝑆e2tection of gradual damage progression.\nThus, and serve as damage-sensitivity parameters that modulate the\ndetection alg𝑆𝑆o1rithm’s 𝑆𝑆r2esponse to magnetic field mutations-whether abrupt or gradual-\nas validated through both fatigue-driven MagDrift evolution and controlled crack\nsimulation experiments.\nThe core of our research lies in capturing the point of accelerated change in the\nMagDrift signal, which corresponds to the initiation of unstable crack propagation\nduring material fatigue. The next step will be to establish a quantitative relationship\nbetween fatigue life stages, crack growth rates, and the magnetic signal based on this\nfinding, and further elucidate the underlying physical mechanisms by incorporating the\nintrinsic properties of the material.\nResponse to Reviewer 3\n<Reviewer 3>: 1. The M–N prediction model incorporates quantum‐mechanical\nconcepts (bonding forces, exchange interactions, and COOP). However, it is unclear\nhow the critical fitting parameters Kc and C are extracted from the fatigue tests.\nAlthough COOP is invoked to explain intrinsic differences in fatigue strength among\nmaterials, no actual COOP values or computations are presented. Are the quantum-\nmechanical quantities used purely for post hoc physical interpretation, or are they\ndirectly computed and input into the model? Do the parameters Kc and C rely entirely\non fitting the macroscopic MagDrift data, with the quantum-mechanical concepts\nserving only an explanatory role?\nResponse：\nThe M-N model is a semi-empirical framework informed by quantum-mechanical\nconcepts—such as bonding forces, exchange interactions, and COOP. These concepts\nprovide the physical basis for describing microscopic damage mechanisms (the\nweakening of bonding forces and exchange interactions) and by referring to various\nfatigue-related empirical formulas. The following is a brief derivation process of the\ntheoretical model. Please refer to the “Methods” for the detailed derivation process:\nAtomic scale: theoretical model derivation of the relationship between\ninterlamellar binding force, exchange interaction, and atomic spacing\nIn ferromagnetic materials, the interaction between atoms is dominated by two\nkinds of basic energy: bonding energy and exchange energy. The bonding strength and\nmagnetic properties of the materials are determined, respectively. Based on the wave\nfunction, we define the interatomic bonding energy and the exchange interaction\nenergy"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have made changes to the manuscript and provided additional justification for their method. However, the\nproposed explanation of plastic localization, fatigue damage, and their connection to interlamellar bonding and interlamellar\nexchange interactions lacks a strong scientific rationale grounded in established fatigue mechanisms such as intrusions and\nextrusions. During fatigue, dislocations glide, accumulate, and emerge at the surface, yet these processes do not\nnecessarily result in residual changes in interlamellar bonding strength. I recognize that the proposed method is sensitive to\nmicrostructural changes, particularly those associated with stored dislocations. Nonetheless, the link between the observed\nmagnetic response and the known fatigue dislocation structures remains insufficiently supported in the current manuscript.\nThe work has merit, but I strongly encourage the authors to reinforce the mechanistic relationship between their\nmeasurements and the physics of fatigue processes and small-scale damage evolution. I therefore recommend major\nrevisions. Below I provide detailed rationale based on the authors' responses. I also encourage the authors to revisit\nfoundational concepts related to fatigue crack initiation.\nRegarding specific statements:\n\"the essence of fatigue fracture is the continuous weakening of interlamellar bonding strength as the spacing increases, and\nonce the bonding strength falls below a critical threshold of the applied load, fracture occurs immediately\"\nThis is not accurate. The essence of fatigue crack initiation, particularly in most metallic systems, is typically driven by\nsurface intrusions and extrusions associated with irreversible dislocation motion. The authors do not establish how\ninterlamellar bonding chanes governs these early surface structural changes.\n\"These microstructure characteristics will lead to the attenuation of the bonding force of interlamellar atoms, and the\nevolution of MagDrift is the macro characterization of the attenuation of the bonding force of interlamellar atoms, so the\nMagDrift signal contains these microstructure characteristics, which are shown by their change rate.\"\nThere may indeed be a connection, but the manuscript lacks a clear mechanistic rationale explaining how observed\nmagnetic changes relate quantitatively to the formation of fatigue-related dislocation structures, their evolution, or the\ntransition to crack nucleation.\n“Although they can offer displacement sensitivities as high as 7–9 nm and spatial resolutions of 20 nm at the microscale,\nadvanced methods such as digital image correlation (DIC) 20 still cannot correlate localized damage effects with\nmacroscopic behaviours.”**\nThis statement is incorrect. These methods are specifically used to correlate localized strain and damage evolution with\nmacroscopic response. In the present work, the absence of spatial information prevents the identification of critical local\nevents that govern fatigue damage. The authors do not address how their method detects or distinguishes these localized\nevents. It is a non-spatial measurement that cannot differentiate the extreme controlling events that will lead to crack\ninitiation. The authors claim that their method is sensitive to extreme events, yet they do not provide convincing evidence or\nclear examples demonstrating such detection.\n\"ore dispersed or discontinuous in the form of microstrip, grain boundary slip or phase boundary cracking due to the\nexistence of barriers such as grain boundary and second phase particle. However, whether the specific microscopic\nmorphology is persistent slip band (PSB) or other types of strain localization regions, the physical essence behind it is due to\nlattice distortion, the increase of atomic spacing and the weakening of atomic bonding force\"\nThis description is not correct, and several terms are not standard in the fatigue community. Persistent slip bands arise due\nto irreversible dislocation activity during cyclic loading, not simply inducing lattice distortion or atomic spacing changes.\nFatigue damage frequently develops through dislocation glide and surface emergence without necessarily producing\nmeasurable lattice distortion. The literature documenting these distinctions is extensive, and the manuscript should align\nwith widely accepted mechanisms.", "reviewer_2": "The authors have supplemented essential experiments, thereby strengthening the persuasiveness of the manuscript.\nTheoretical explanations have been provided regarding the influence of ambient magnetic fields and the selection of the\nmagnetic indicators. Necessary clarifications have also been added to the section on the Fatigue Fracture Early Warning\nAlgorithm. In addition, issues related to formatting and figures have been appropriately addressed. Overall, the revisions and\nresponses have adequately resolved the reviewers' questions concerning experimental and theoretical aspects. The\nmanuscript has been significantly improved and is now suitable for acceptance.", "reviewer_3": "Review of “High-Accuracy Fatigue Life Prediction and Early Fracture Warning for Ferromagnetic Metals via Spin Correlation\nAmplification”\nReviewer Comment #1: Domain of Validity for the Binding Force–Exchange Interaction Proportionality. The manuscript\nposits a linear proportionality between variations in interlamellar binding force and exchange interaction (ΔF_C = K·ΔFm)\nunder limited atomic spacing perturbations. While this assumption is noted to apply within a \"minute range\" of lattice\ndistortion, the temporal scope of its validity across fatigue progression remains insufficiently defined. The specific fatigue\nregimes (e.g., early-stage cyclic hardening, mid-life damage accumulation, or pre-macrocrack stages) over which this\nproportionality is expected to hold; Whether the relationship remains applicable after the onset of significant microstructural\ndegradation (e.g., dislocation patterning, void coalescence) or near crack-tip plastic zones where nonlinear lattice distortions\ndominate.\nReviewer Comment #2: Quantification of Prediction Uncertainty from MagDrift Slope Variability\nThe fatigue lifetime prediction framework relies critically on the parameter k_a derived from the linear regression slope of\nMagDrift versus cycle count. Given that experimental noise and fitting uncertainties inherently affect k_a. The core life\nprediction uses log N_f = (1/k_c)·(log |k_a| + C), where k_a is obtained from a linear fit of MagDrift versus cycle count. To\nassess the robustness of the model, could the authors provide a simple sensitivity analysis showing how perturbations in\nk_a (e.g., ±5% or ±10% due to measurement noise or fitting uncertainty) translate into uncertainty in the predicted N_f?\nReviewer Comment #3: Stress Amplitude as an Input? The manuscript demonstrates that the MagDrift slope exhibits a\nstronger correlation with fatigue life compared to stress amplitude, thereby emphasizing the superiority of the M–N approach\nover the traditional S–N curve methodology. However, in many practical applications, both stress history and magnetic\nresponse data are typically available. The authors investigated whether incorporating stress amplitude as an additional input\ninto a simple two-parameter model—such as one that uses both log|k_a| and σ_max—could yield statistically significant\nimprovements over the MagDrift-only model?\nReviewer Comment #4: Quantitative Validation of Cross-Alloy Generalizability\nThe manuscript establishes a unified M–N relationship using a global parameter k_c=−1.54 derived from averaged slopes\nacross four Fe-based alloys, reporting high apparent goodness-of-fit (R2>0.9 ) for all materials under full-data calibration. To\nfurther demonstrate generality, if k_c and C are calibrated using three alloys and then used to predict N_f for the fourth, how\nmuch does the prediction error increase compared to full-data calibration?\nReviewer Comment #5: Clarification of Roles and Performance Metrics for the Two-Stage Warning Scheme. While the\nmanuscript proposes a two-stage failure warning system (Stage-1 triggered at 90% predicted life; Stage-2 activated by\nabrupt MagDrift changes near failure) and reports 100% Stage-2 success in tested cases, the rebuttal discussion noting\nStage-1’s instability and Stage-2’s potential operational sufficiency creates ambiguity regarding the practical implementation\nstrategy; the authors must explicitly delineate in the main text."}, "author_response": "Response Letter\nDear Reviewers:\nWe sincerely appreciate the valuable suggestions provided by yours. These\ncomments not only helped us identify areas for improvement, but also provided\nvaluable guidance for enhancing the scientific rigor of this study. We highly value the\ninsights shared by yours, and have now addressed them through targeted responses\nbased on theoretical analysis and experimental verification. The response section in the\nmain text details our revision rationale and provides point-by-point answers to the\nquestions raised. All supplementary experimental datasets, computational examples,\nand related analyses are included in the supplementary information for your review at\nany time.\nBelow, we provide a point-by-point response to the comments from yours (in\nblack), with our detailed replies in blue and the corresponding revisions in the\nmanuscript indicated in red.\nResponse to Reviewer 1\n<Reviewer 1>: 1. The authors propose an innovative approach to estimate dislocation\ndensity and use it to assess fatigue strength. However, the manuscript currently includes\nseveral oversimplifications and lacks critical information necessary to fully validate the\nmethod for fatigue evaluation. Importantly, the proposed approach is based on averaged\nmeasurements, lacking the spatial resolution necessary to accurately capture the\nlocalized nature of fatigue damage, an issue the authors themselves acknowledge in the\nintroduction. In its current form, the method may offer no more insight than\nconventional strain gauge measurements.\nResponse:\nWe realized that there was an issue of simplification in the presentation of\ntheoretical models in the original text, which may have given the impression that our\nmethod is \"average measurement\". Below, we will attempt to restate our measurement\nprinciples to demonstrate the mechanism, generation, and effects of localized fatigue\nphenomena, supplemented by experimental validation.\nDue to the correlation between interatomic binding forces and exchange\ninteractions with atomic spacing, when fatigue causes the atomic spacing between\natoms to gradually increase, the interatomic binding forces and exchange interactions\nchange synchronously.\nWe enhanced the macroscopic observability of this intrinsic correlation through\ndirectional excitation by an external magnetic field, achieving spin correlation\namplification. Unlike “averaging effect” measurement methods, this represents a\n“multiplicative effect” where the localized nature of fatigue damage exerts a dominant\ninfluence on the overall measurement value. The specific reasons are as follows:\nPhysical mechanism of spin correlation amplification method\nDuring the material fatigue process, the interlamellar atomic spacing gradually\nundergoes irreversible changes (\"interlamellar\" refers to the collective set of all atoms\nwithin a three-dimensional irregular thin-section perpendicular to the direction of\nstructural loading).\nThe variation of interlamellar atomic spacing refers to the collective effect of the\nvariation of atomic spacing between each atomic layer; the interlamellar bonding force\nis the collective effect of the changes in atomic bonding force between each atomic\nlayer; the interlamellar exchange interaction is the collective effect of exchange\ninteractions between the atomic layers.\nThe change affects not only the interlamellar bonding force but also synchronously\nalters the interlamellar exchange interaction, demonstrating a strong indirect correlation\nbetween these two properties.\nWhen a directional excitation is applied by an external magnetic field, the change\nin the interlamellar exchange interaction will cause a directional change in the\ninterlamellar spin magnetic moment at the microscale, and further alter the magnetic\ndomain pinning state at the mesoscopic scale.\nAs the external excitation magnetic field penetrates each atom layer disturbed by\nfatigue, change in spin orientation between atoms. This deflection accumulates\nprogressively as the magnetic field traverses additional damage concentration zone.\nUltimately, when the stimulating magnetic field penetrates all fatigue-disturbed layered\natom layers and outputs a measurable magnetic signal, its intensity reflects this\ncumulative deflection process.\nThis mechanism enhances the detectability of the weakening of interlamellar\nbonding forces and exchange interactions caused by changes in the interlamellar atomic\nspacing, allowing for more sensitive observation of material fatigue performance at the\nmacroscopic level (Fig. 1).\nFig. 1. Comparison of spin correlation multiplicative effect and strain additive effect. a, Under\nthe directional excitation of an external magnetic field, the slight weakening of interatomic binding\nforces can trigger a synergistic orientation change of spin magnetic moments at the microscale\nthrough exchange interactions; This micro spin reconstruction will significantly alter the pinning\nstate of magnetic domains at the mesoscopic scale, and then amplify into observable surface\nmagnetic flux changes at the macroscopic level through collective motion of magnetic domain walls.\nb, Under the action of fatigue, materials produce damage concentration zone, and strain gauges only\nmeasure local strain, which is independent of the microscopic mechanism and is an additive effect.\nMultiplication effect and its difference compared to strain gauges\nSpin correlation amplification is essentially a multiplication effect. During the\nfatigue process of materials, the initial atomic dislocation layer (such as the first layer)\ncan cause a deviation in electron spin (for example, with an amplitude of ). Through\ninterlamellar exchange interactions, this bias is gradually amplified i𝑎𝑎n% subsequent\ndamage concentration zone. Specifically, each layer amplifies the received deflection\nby a factor of one (the amplification factor may vary by layer; for simplicity, assume\nthat each layer has an amplification factor of ). So, after n-layer amplification,\nthe spin deflection amplitude of the final atom1ic+ d𝑎𝑎is%location layer will reach ( )\ntimes the initial deflection. Among them, the actual amplification factor of ea1ch+ l𝑎𝑎ay%er\nmay not be the same. Note that is only used to illustrate the operating principle of\nthe multiplication mechanism. 𝑎𝑎%\nFor the specific amplification model, please refer to the formula derivation from\nwave function to fatigue life and MagDrift slope relationship (Methods).\nThe strain gauge responds linearly to strain ac𝑘𝑘c𝑎𝑎u𝑎𝑎mulation: during material fatigue,\nminute displacements ( , , …, ) occur across each micro-section within the\nfatigue zone. The strain𝐿𝐿 ₁gau𝐿𝐿g₂e can 𝐿𝐿o𝑛𝑛nly linearly superimpose these displacements,\noutputting signal that reflects additive effect.\nSpin co𝛴𝛴r𝐿𝐿reᵢ/la𝐿𝐿tion amplification relies on the multiplicative coupling of interlamellar\nexchange interactions, and its signal exponentially amplifies with the number of\ndamage concentration zones; Strain gauges rely on linear accumulation (Fig. 1).\nThe dominant role of localized events\nDuring the fatigue process, material layers with defects or weak bonding forces\nbear the maximum dislocation (interlamellar atomic spacing) changes, and the\nmagnitude of changes in their interlamellar bonding forces and exchange interactions\nis significantly higher than in other regions. It is this local maximum effect that\ndominates the change in the final magnetic signal. Therefore, the spin correlation\namplification method has significant spatially varying response to fatigue damage, and\nits high sensitivity originates from the dominant role of interlamellar localized events\n(Fig. 2).\nFig. 2. Schematic diagram of localization effect under extreme conditions.\nTo verify the localization effect theory, we designed three types of progressive\nexperiments, which systematically analyzed the cross-scale conduction mechanism of\nlocal bonding weakening on global magnetic response from the perspectives of artificial\ncrack simulation, prefabricated defect evolution, and weld fusion zone evolution.\nAnalysis of steel bar cutting test\nThe experiment selected a soft magnetic iron rod with a diameter of 16 mm and a\nlength of 300 mm. Four measurement points (Measurement Points 1-4, MP1-MP4)\nwere set up every 50 mm along the axial direction, and a Cutting Point (CP) was marked\nbetween MP2 and MP3.\nIntroduce artificial cracks with a fixed width of 1 mm at this location: before the\ndepth reaches 10 mm, cut 2 mm each time, and after exceeding 10 mm, change to 1 mm\neach time until fracture. After each cutting, apply an 8-level increasing excitation\nmagnetic field of 50-430 mT to the fixed end of the specimen in sequence. After each\nlevel is loaded, use a handheld Gaussian meter to collect the magnetic induction\nintensity of each measuring point at a height of 3 mm. This artificial crack simulates\nthe process of atomic bond breaking. (Fig. 3)\nFig. 3. Schematic diagram of the comprehensive relationship between magnetic strength\nchanges at different measuring points and under different crack depths under different\nexternal magnetic excitations. a, Schematic diagram of steel bar cutting test and external magnetic\nfield configuration location. The preset crack position is located at the geometric center of the test\npiece, with measuring points 1 and 2 located in front of the crack, and measuring points 3 and 4\nlocated behind the crack. The external magnetic field is applied by neodymium-iron-boron\npermanent magnets, covering eight excitation levels from 50–430 mT. b, Schematic diagram of\nfinite element simulation of crack depth on magnetic induction line distribution. When there are\ncracks in the material, the magnetic flux in the damaged area at the crack location section decreases,\nwhile the magnetic flux in the undamaged area increases accordingly; and the increase in crack\ndepth leads to an increase in magnetic flux in the undamaged area. c, The evolution law of magnetic\nstrength at each measuring point under eight types of external magnetic excitation as a function of\ncrack depth. As the external magnetic excitation increases, the magnetic strength at each measuring\npoint increases synchronously. When the crack depth reaches a corresponding value, there is a trend\nof increasing magnetic strength at measuring points in front of the crack and decreasing magnetic\nstrength at measuring points behind the crack. d, Output the relationship diagram between the\nacceleration of magnetic field changes and the remaining depth, external magnetic excitation, and\nthe ratio of magnetic excitation to undamaged area.\nMagnetic field strength at the crack-front monitoring points (MP1, MP2) exhibits\na trend of initial stability followed by accelerated increase as the crack deepens, while\nthe crack-back monitoring points (MP3, MP4) show initial stability followed by\naccelerated decrease. Concurrently, a higher applied excitation magnetic field strength\nnot only enhances the stability of the measured magnetic field but also accelerates the\nonset of both the accelerated increase at MP1 and MP2 and the accelerated decrease at\nMP3 and MP4.\nThe distribution map of magnetic induction lines shows that the magnetic flux at\nthe damaged part of the fracture area decreases, while the magnetic flux at the\nundamaged part of the fracture area increases accordingly, so the overall magnetic flux\nof the section remains stable at the initial stage. However, as the crack depth increases,\nthe magnetic flux in the damaged area continues to decrease, while the magnetic flux\nin the undamaged area steadily increases until reaching a critical saturation state. At\nthis time, the magnetic flux increment in the undamaged region can’t fully compensate\nfor the loss in the damaged region, resulting in the overall magnetic flux of the cross\nsection beginning to decrease. The decrease of the overall magnetic flux is the main\nreason for the decrease of the magnetic strength of the measuring points behind the\ncrack (MP3, MP4), while the increase of the magnetic strength of the measuring points\nin front (MP1, MP2) is related to the redistribution of local magnetic induction lines\n(concentrated to the undamaged area) caused by the decrease of the magnetic flux in\nthe damaged area (Fig. 3).\nTable. 1. Ratio of fracture area to undamaged area at critical filling under different excitation\nmagnetic strength.\nExternal Ratio of external\nmagnetic Crack depth Undamaged magnetic excitation to\nUndamaged rate\nexcitation (mm) depth (mm) undamaged area\n(mT) (mT/mm2)\n50 13 3 13% 0.28\n87 13 3 13% 0.5\n124 12.5 3.5 16.2% 0.74\n158 12 4 19.6% 0.977\n185 11 5 26.7% 1.3\n211 9 7 42.2% 1.82\n317 8 8 50% 3.2\n430 8 8 50% 4.3\nAccording to the data measured in Table. 1, we found that the ratio of external\nmagnetic excitation and undamaged area was linear, and the linear fitting slope was\n0.011, which could be regarded as the quantitative characterization of crack damage\ndegree and external magnetic excitation intensity in the critical filling state (Fig. 3d).\nWith the increase of the external magnetic excitation intensity, the crack depth\nrequired for the sharp change of the output magnetic field decreases significantly; When\nthe crack continues to deepen or the magnetic field continues to strengthen, the\nmagnetic flux at the damaged part of the crack area decreases rapidly, while the\nmagnetic flux at the intact part increases correspondingly until it reaches the critical\nfilling. The spin correlation amplification technique can capture the local magnetic field\ngradient changes induced by the magnetic flux redistribution, and the local defects\nsignificantly affect the final magnetic strength changes.\nFatigue test of defective specimen\nThe experiment utilized soft magnetic iron material. The geometric dimensions\nand loading parameters of the specimens were consistent with those described in the\noriginal text (Methods). Seven fatigue tests were conducted with a stress ratio of R=0\nand a maximum stress of 460 MPa. To construct the local damage characteristics, two\nartificial defects with a depth of 0.5 mm and a width of 1 mm were symmetrically pre-\nfabricated on both sides of the middle of the test piece (Fig. 4), and a defect-free control\ngroup was established for comparative analysis.\nFig. 4. Dimension diagram of defective test piece.\nThe experimental results show that the fatigue life of the defect group is\nconcentrated in 20,000-60,000 cycles, while that of the non-defect group is 200,000-\n300,000 cycles. Although the service lives of the two groups of specimens are quite\ndifferent, the change rate of magnetic strength (CRMS) of the two groups of specimens\npresents a consistent three-stage trajectory: a short acceleration decline in the initial\nstage, followed by a stable period, and finally a sharp decline again in the failure stage\nuntil fracture. The MagDrift curve also shows a similar evolution path, and the greater\nthe change rate of MagDrift, the shorter the service life, and the strength of local defects\ncan be shown through the change of MagDrift. In the normalized MagDrift diagram,\nthe MagDrift of the defect group and the defect free group have similar trends.\nThrough the comparative analysis of the two groups, the MagDrift slope of the\ndefect group is significantly higher than that of the non-defect group. According to this\ncharacteristic, whether the material contains defects can be distinguished. Therefore,\nspin correlation amplification technology can capture the changes of magnetic signal\ncaused by local defects. (Fig. 5).\nThis phenomenon reveals that the decay of bonding forces in localized stress\nconcentration zones is transformed into changes in the global MagDrift signal through\nmultiplicative effect, highlighting the localized detection nature of this method.\nFig. 5. Defect fatigue test. a, The diagram of Offset CRMS. b, The diagram of Normalization of\noffset CRMS. c, The diagram of MagDrift. d, The diagram of Normalization of MagDrift.\nFatigue test of weld specimen\nIn the experiment, soft magnetic iron plate with a width of 20 mm and a thickness\nof 10 mm was selected. A Y-shaped notch was prefabricated in the middle of the test\npiece and welded to form the weld fusion zone (Fig. 6). Carbon dioxide shielded\nwelding was used, and the welding wire was ER80S-G with a diameter of 1 mm. The\nloading parameters are the same as those in the original (Methods), the stress ratio R=0,\nand the maximum stress is 160 MPa. The fatigue life of the three groups were 57,000\ncycles, 9,100 cycles and 5,500 cycles, respectively.\n0\n1\n2\n8\n60°\n01\n0\ndifference, which is due to the different welding quality, so that the interlamellar atomic\nbond force in the weld fusion zone has a significant difference.\nThrough the comparative analysis of the evolution law of the change rate of\nmagnetic strength (Fig. 7), it is found that under the condition of fixed stress amplitude\n(160 MPa), the change rate of magnetic strength of different fatigue life specimens (57\nK, 9.1 K, 5.5 K) shows a slow and then rapid downward trend with the increase of the\nnumber of cycles, and the phenomenon of accelerated attenuation occurs near the\nfailure stage. Welding will also affect the normalization characteristics of magnetic\nstrength change rate: according to different welding quality, the final attenuation\ntrajectories of the three curves show significant differences. The specimen with fatigue\nlife of 5.5 K shows the steepest decline slope in the late normalization period, followed\nby 9.1 K, and 57 K decline is the most gradual, indicating that the worse the welding\nquality, the faster the magnetic signal attenuation. The strength difference of weld\nfusion zone can be directly reflected by the variation of CRMS. At the same time, the\nchange rate of the MagDrift curve increases with the decline of the weld quality,\nresulting in the shortening of the life of the specimen. This phenomenon reveals the\nlocalization effect: even if the welding quality is difficult to distinguish with the naked\neye, it can be sensitively characterized by the change of magnetic signal. (Fig. 7).\nThe evolution of MagDrift is determined by the bonding force state of the\ninterlamellar atoms, is highly sensitive to the decay of the bonding force, and can\ndirectly reflect the bonding force strength in the weld fusion zone.\nFig. 7. The weld fatigue test. a, CRMS of the weld specimen. b, Normalization CRMS of the weld\nspecimen. c, MagDrift of the weld specimen. d, Normalization MagDrift of the weld specimen.\nThese three groups of experiments confirmed that the local nature of fatigue\ndamage would have a dominant effect on the global change of MagDrift signal. The\nintrinsic reason is that the attenuation of the bonding force of the weakest cross section\nis reflected in the final measured magnetic signal through the multiplication effect of\nthe spin correlation amplification of the exchange interaction (magnetic moment).\nThree advantages of spin correlation amplification measurement over average\nmeasurement\nAlthough both the spin correlation amplification method and the strain gauge can\nbe employed for material damage detection, their underlying physical principles and\nresponse mechanisms are fundamentally distinct. The unique advantages of the spin\ncorrelation amplification method are manifested in the following three aspects:\n1. MagDrift features a self-compensation ability\nThe MagDrift metric is designed as a differential measure (peak-to-peak within a\nloading cycle), which inherently suppresses low-frequency environmental drift by\ncanceling common-mode magnetic perturbations. It is assumed that the influence of the\nsurrounding environment and geomagnetic field on the magnetic field strength of the\nspecimen is , and the influence of the surface temperature change of the specimen\non the magne∆ti𝐵𝐵c𝐸𝐸 field strength of the specimen is . Under cyclic loading:\nWhen the external load is , the com∆b𝐵𝐵in𝑇𝑇ed effects of magnetic and load\neffects, ambient environment eff𝜎𝜎e𝑚𝑚ct−s a𝜎𝜎n𝐿𝐿d specimen surface temperature changes are\nconsidered as follows:\n(R1-1)\nWhen the external l𝐹𝐹o𝑚𝑚ad0 i−s g(𝜎𝜎𝑚𝑚 −𝜎𝜎, 𝐿𝐿t)he− c𝑓𝑓o(m∆b𝐵𝐵i𝐸𝐸ne)d− e𝑓𝑓ff(e∆c𝐵𝐵ts𝑇𝑇 o)f magnetic force and\nload action, the influence of su𝜎𝜎rr𝑚𝑚ou+nd𝜎𝜎i𝐿𝐿ng environment and the change of specimen\nsurface temperature are as follows:\n(R1-2)\nAmong them, 𝐹𝐹𝑚𝑚 1a−ndg (𝜎𝜎𝑚𝑚 +𝜎𝜎𝐿𝐿 )re−sp𝑓𝑓e(c∆ti𝐵𝐵ve𝐸𝐸l)y −tak𝑓𝑓e( ∆in𝐵𝐵to𝑇𝑇 )account the influence\nof the surrounding e𝑓𝑓n(v∆i𝐵𝐵ro𝐸𝐸n)ment a𝑓𝑓n(d∆ t𝐵𝐵h𝑇𝑇e )surface temperature change of the specimen\nunder cyclic loading.\nSubtract R1-2 from R1-1 to obtain the MagDrift value during fatigue:\n(R1-3)\nIt can be seen 𝐹𝐹th𝑚𝑚a𝑚𝑚t𝑚𝑚 = 𝐹𝐹𝑚𝑚1 − agn(d𝜎𝜎 𝑚𝑚 +𝜎𝜎𝐿𝐿)− a𝐹𝐹re𝑚𝑚 0co+m𝑔𝑔p(le𝜎𝜎t𝑚𝑚ely− e𝜎𝜎l𝐿𝐿im) inated, and the\nMagDrift is only related to𝑓𝑓 t(h∆e𝐵𝐵 in𝐸𝐸t)ernal st𝑓𝑓at(e∆ o𝐵𝐵f𝑇𝑇 t)he material.\nWe added various fatigue loading tests under human interference to further verify\nthe rationality of choosing MagDrift as the representative index of the paper.\nThis experiment mainly simulates various possible influencing factors by exerting\nvarious human disturbances in the process of fatigue loading. These factors include\ntemperature noise, environmental magnetic strength change, laboratory power failure,\npoor channel contact of test data acquisition instrument, electrical noise interference of\nhigh-power electrical appliances and other factors. The corresponding magnetic\nstrength and MagDrift change data under these human interference factors are shown\nin Fig. 8.\nFig. 8. Comparison of magnetic strength and MagDrift data after interference. The shaded area\nin the figure indicates the time period when the connection of the data acquisition system is\ninterrupted. The sudden change or jump of magnetic strength data in this experiment is mainly due\nto temperature noise, environmental magnetic field fluctuation and electromagnetic noise\ninterference introduced by high-power equipment. In contrast, the MagDrift signal has an inherent\nability to suppress these interference factors by virtue of its inherent self-compensation mechanism,\nso it maintains a more stable change trend in the whole process.\nIt can be seen from the figure that the variation of the magnetic field strength value\nof the test piece has been greatly affected, including the loss of data in the magnetic\nfield strength value and MagDrift data caused by the poor contact of the data acquisition\ninstrument channel (this part is supplemented by adding shadows in the figure),\njumping, clutter and other phenomena. Due to the self-compensation feature of the\nMagDrift data, it can offset and slow down the impact of the above factors. Therefore,\nalthough there are some fluctuations in the MagDrift data, the overall trend is relatively\nflat.\nTherefore, compared with the magnetic field strength value, the MagDrift can\nbetter reflect the fatigue process and the change of the internal fatigue state of the\nmaterial. This is also the main reason why the MagDrift is adopted as the core indicator\nto evaluate the fatigue state of the material in this paper.\nUniversality of abnormal structure\nFrom the application perspective, strain gauges have significant limitations in the\nfatigue monitoring of non-standard structures (such as curved, wavy, or complex curved\ncomponents). Since strain measurement is difficult to directly apply to curved and non-\nsmooth surfaces, the following problems are prone to occur when bonding to curved\nsurfaces: (1) Uneven thickness of the bonding layer leads to distortion of stress transfer,\nand the measurement error increases significantly when the radius of curvature is small;\n(2) The multi-dimensional strain field of non-standard structures under dynamic\nloading can cause directional deviation in traditional uniaxial strain gauges; (3)\nMonitoring blind spots are likely to form at the locations of curvature changes on the\nstructure surface (Fig. 9).\nIn contrast, the magnetic measurement method shows unique advantages: through\nthe non-contact magnetic sensor array, the surface profile can be adaptive. In typical\nspecial-shaped structures such as arc beams, the magnetic measurement method can\nachieve full circumferential coverage monitoring, and the measurement accuracy of the\nmagnetic field gradient in the curvature change area can still be maintained. More\nimportantly, the magnetic sensor does not need to be fixedly connected with the\nstructure surface, which avoids the signal failure problem caused by the rubber layer\nfatigue of the strain gauge on the special-shaped structure. This adaptability is universal\nfor special-shaped structures and is not affected by the shape of the material structure\n(Fig. 9).\nFig. 9. Comparison of universality between magnetic measurement and strain measurement.\nMultiplication effect magnifies micro damage\nAs described in the section \" Physical mechanism of spin correlation amplification\nmethod\" earlier, the amplification effect of spin correlation transforms the quantum\ncharacteristics of nanoscale damage into observable flux jumps through multiplicative\ngain, with sensitivity increasing exponentially, far surpassing the linear superposition\nof traditional strains. When the external magnetic field penetrates the material, it\nproduces irreversible deflection due to spin synergy. Its cumulative multiplicative effect\nenables the system to give a clear warning signal before fracture.\nSpin correlation amplification possesses three major characteristics: noise\nresistance, universality, and signal amplification, which is significantly different from\naverage measurement.\nTo more clearly elaborate on the spin correlation method theory, the revised text\nin the manuscript is provided below (lines 197-211): “During the process of fatigue, the\ninterlamellar atomic spacing undergoes gradual and irreversible changes. This change\nnot only affects the bonding strength of the material layer but also simultaneously\naffects its interlamellar exchange interactions, with both phenomena being strongly\ncorrelated and synchronous.\nWhen an external magnetic field is applied, changes in the interlamellar exchange\ninteractions induce oriented variations in the aggregate spin magnetic moment at the\nmicroscopic scale, which in turn alters the pinning states of magnetic domains at the\nmesoscopic scale.\nAs the external excitation-induced magnetic field penetrates each fatigue-affected\ndislocation layer, it undergoes irreversible deflection because of the spin‒spin\ninteractions between atoms. This deflection accumulates progressively as the magnetic\nfield traverses more fatigued dislocation layers. Essentially, this process functions as a\nspin-correlation amplification mechanism, manifesting as a multiplicative effect\n(Methods). Ultimately, the deflection characteristics that accumulate after the magnetic\nfield passes through all the fatigue-induced dislocation layers are encoded within the\nmagnetic signal, providing a basis for evaluating the fatigue state of the material (Fig.\n1).”\nRevised text in the manuscript (lines 235-237): “The magnetic signal changes\nexhibited by the spin magnetic moment system exhibited exponential amplification\nthroughout the spin-dependent conduction chain.”\nTo illustrate the advantages of MagDrift, we have supplemented the Methods\nsection of the manuscript with the following explanation: (lines 944-963): “MagDrift\nfeatures a self-compensation ability.”\nIn our fatigue theory derivation model, we analyzed at the mesoscale, microscale,\nand macroscale. In the method section of the manuscript, we have supplemented the\nMethods section of the manuscript with the following explanation: (lines 880-923):\n“Macroscopic scale: Derivation of the magnetic flux change rate and fatigue life model.”\n<Reviewer 1>: 2. Moreover, the interpretation of what is being measured magnetically\nrequires further clarification. The dislocation structures that evolve during fatigue can\nvary substantially in nature, yet the proposed technique appears to treat them\nequivalently, potentially overlooking key distinctions in how different dislocation\nconfigurations influence fatigue behavior. For instance, dislocation annihilation, slip\nreversibility, an important mechanism that may still lead to micro-damage, is unlikely\nto be captured by the magnetic signal, despite its relevance to fatigue failure.\nResponse:\nIn the process of material fatigue, microscopic evolution mechanisms such as\ndislocation annihilation and slip reversibility indeed have significant effects on fatigue\nfailure. Dislocation annihilation typically occurs when two spatially adjacent\ndislocations with opposite Burgers vectors cause lattice distortions (such as\ncompressive and tensile regions) that cancel each other out, thereby restoring the lattice\nto a state of lower distortion. However, the changes in atomic spacing caused by\ndislocations near the annihilation site may be difficult to fully recover.\nSlip reversibility refers to whether dislocations generated by slip under forward\nloading can return along the original path during reverse loading in cyclic deformation,\nthereby restoring the local lattice to its initial state. The key lies in whether dislocations\ncan strictly retreat along their original path, which can be judged by whether the local\natomic spacing can fully recover: under cyclic loading, if the lattice distortions caused\nby forward slip can be eliminated through reverse slip, the atomic spacing may be\nrestored; otherwise, if dislocation retreat is impeded, the atomic spacing will change,\nleading to residual plastic strain that gradually accumulates into fatigue damage.\nRegarding the fatigue fracture mechanism in materials, we believe that: the\ninterlamellar atomic spacing to a certain extent determines the strength of interlamellar\nbonding and interlamellar exchange interactions; when the material system is fixed,\natomic spacing becomes the dominant variable regulating both; the essence of fatigue\nfracture is the continuous weakening of interlamellar bonding strength as the spacing\nincreases, and once the bonding strength falls below a critical threshold of the applied\nload, fracture occurs immediately; through the spin-correlation amplification effect,\nthe\nMagDrift signal amplifies this microscopic process of weakened bonding strength into\nan observable macroscopic response.\nThe above represents our current understanding of this issue. If there are any\noversights or shortcomings, we sincerely invite your feedback and suggestions.\nTo further clarify the interpretation of the magnetic measurement results. The\nrevised text in the manuscript is provided below (lines 108-141): “The fatigue\nbehaviours of materials manifest microscopically as the formation of various\ndislocation structures (e.g., persistent slip bands (PSBs), dislocation annihilation,\nreversible slip, and the ratcheting effect), which are characteristic features of localized\nplastic deformation under cyclic loading (Fig. 1a). At the microscopic level, the\nformation of these structures originates from the motion, interaction, and accumulation\nof dislocations at the nanoscale within the crystal lattice, gradually evolving into visible\nslip bands or other defect structures4. This evolutionary process is highly nonlinear and\nstrongly dependent on crystal structures, loading conditions, and environmental factors.\nFor instance, in face-centred cubic (FCC) metals, dislocations undergo rearrangement\nthrough reversible slip (where partial dislocations return along their original paths under\nalternating loads, leading to partial damage recovery) and dislocation annihilation\n(where dislocations with opposite signs meet and cancel each other out). This initially\nresults in the formation of vein-like structures, which further develop into cellular\nstructures. In contrast, for body-centred cubic (BCC) metals or objects under high-\nstress amplitude conditions, the ratcheting effect (the unidirectional accumulation of\nstrain under asymmetric cyclic loading) becomes more pronounced, promoting\nirreversible and rapid dislocation accumulation.\nThe nucleation and motion of dislocations are associated with the weakening of\ninteratomic bonding forces18 (interatomic cohesion; Fig. 1a). During cyclic loading, a\nlocalized stress concentration reduces the interatomic bonding strength, facilitating\ndislocation nucleation and slip processes25. This slip is not always uniform; its degree\nof localization can be characterized by slip planarity. Higher planarity indicates that the\nslip is concentrated on fewer planes, making it easier to form sharp persistent slip bands\n(PSBs). The interfaces between these bands and the matrix become high-risk regions\nfor crack initiation. In some alloys, microscopic plasticity may also appear in the form\nof fatigue shear bands, which are intensely localized shear bands that traverse grains.\nTheir formation is closely related to microstructural anisotropy.\nTherefore, fatigue damage in materials is essentially a multiscale evolutionary\nprocess spanning the atomic, microscopic, and macroscopic dimensions. From a\nmicroscopic perspective, this process is located within the damage concentration zone.\nThis region consists of multiple atomic layers within the fatigue-affected zone and\ncontains a series of microscopic defect clusters that influence the interlamellar atomic\nspacing (here, “interlamellar” refers to the collective set of all the atoms contained\nwithin a three-dimensional irregular thin section that is perpendicular to the direction\nof structural loading). Throughout this process, damage to the interlamellar atomic\nbonding force continuously accumulates. When this force falls below the stress induced\nby the applied load, material fracture occurs. Hence, changes in the interlamellar atomic\nbonding force can serve as indicators for characterizing the fatigue states of materials.”\nRevised text in the manuscript (lines 146-147): “These challenges are particularly\npronounced when complex transient processes such as dynamic dislocation annihilation,\nreorganization and reversible slip are involved.”\n<Reviewer 1>: 3. The influence of loading conditions is also insufficiently addressed.\nAs an example, under ratcheting conditions, dislocation density may increase\nsignificantly, but the associated damage mechanisms can differ from those occurring\nunder non-ratcheting fatigue. Such distinctions are critical and should be reflected in\nthe analysis.\nResponse:\nThe differences in the fatigue damage mechanisms between ratcheting and non-\nratcheting effects are important for accurately analyzing material fatigue behavior. Our\ndiscussion of this issue in the original manuscript was insufficient. The following is our\nunderstanding of the dislocation behavior under different loading conditions:\nThe essence of the ratcheting effect is the combined effect of gradual attenuation\nof the bonding force and incomplete dislocation annihilation caused by the irreversible\nnet increase of atomic spacing under cyclic loading. The core of non-ratcheting\nbehavior lies in the reversible elastic oscillation of atomic spacing to maintain bonding\nstability and realize full annihilation through dislocation reverse slip. Although the\nmicroscopic mechanisms of ratcheting and non-ratcheting effects are different, their\nessence is related to atomic spacing.\nTo elucidate the load-independent response of the MagDrift response, we have\nsupplemented the discussion with existing data, selecting 45 steel for both high-stress\nfatigue testing ( = 580 MPa, = 111 K) and low-stress fatigue testing ( =\nBy co𝑁𝑁m𝑓𝑓paring and analyzing the evolution behavior of MagDrift under different\nstress levels in Fig. 10, the following conclusions can be drawn: under the original cycle\nnumber scale, the MagDrift curve of the low stress specimen ( =510 MPa,\n=2,942 K) rises gently as a whole, and accelerates in the final failure\n𝜎𝜎\ns𝑚𝑚ta𝑎𝑎g𝑚𝑚e, indicating\n𝑁𝑁th𝑓𝑓at under the non-ratcheting dominated fatigue condition, the damage accumulation is\nslow, but the MagDrift has an upward evolution trend; The MagDrift of high stress\nspecimens ( =580 MPa, =111 K) changes rapidly as a whole, and also\naccelerates to𝜎𝜎 𝑚𝑚ri𝑎𝑎s𝑚𝑚e in the final fa𝑁𝑁i𝑓𝑓lure stage. This rapid change may be related to the\nsignificant increase of dislocation density under ratchet condition, indicating that under\nratcheting dominated fatigue condition, damage accumulation is faster, and the\nMagDrift also has a relatively obvious upward trend. The change rate of the\ninterlamellar atomic spacing is accelerated, which accelerates the attenuation of the\nbonding force of the interlamellar atoms. This shows that whether ratcheting effect or\nnon-ratcheting effect, its microscopic essence is attributed to the increase of the\ninterlamellar atomic spacing. The non-ratcheting effect shows a slow change of the\ninterlamellar atomic spacing, while the ratcheting effect shows a faster change of the\ninterlamellar atomic spacing, which directly represents the decline of the interlamellar\nbond force. The MagDrift signal accurately captures the decline of the bonding force\nthrough the quantum spin correlation chain, and establishes a cross-scale sensing path\nof atomic spacing bonding force magnetic response. This feature enables spin\ncorrelation amplification technology to span the difference of load conditions. The\nabove is our understanding of this issue, if there be any points that have not fully\ngrasped, we kindly request your further guidance.\nFig. 10. Comparison of MagDrift in 45 steel. a, Comparison of MagDrift in 45 steel at 510 MPa\nand 580 MPa stress; b, Comparison of Normalized MagDrift in 45 steel at 510 MPa and 580 MPa\nstress.\nThe revised text in the manuscript is the same as that described in <Reviewer 1>:\n2.\n<Reviewer 1>: 4. The citations associated with the data in Fig. 1 of the Extended Data\nshould be mentioned.\nResponse:\nAll the data in Extended Data Fig. 1 were independently collected and\nsystematically organized by our research team, without direct citation of published\nliterature. These data constitute an important part of the originality of this study. Like\nthe experimental data, these data are publicly stored in the DRYAD database\n(doi.org/10.5061/dryad.1rn8pk13t). We have updated the manuscript: Extended Data\nFig. 1 has been renamed as Supplementary Fig. 1. At the end of the title, we have\nexplicitly added a reference to this dataset.\n<Reviewer 1>: 5. P1L36: The current body of fatigue research includes substantial\nwork on the 3D characterization of microstructure effects. The authors have omitted\nthis important body of literature and should include relevant references.\nResponse：\nWe inadequately discussed the research achievements in this field in the initial\nmanuscript. To clearly elaborate on the contributions of 3D characterization in fatigue\nresearch. The revised text in the manuscript is provided below (lines 45-53):\n“Microscopic studies of fatigue phenomena have established a multiscale and\nmultidimensional observation framework, providing a solid foundation for revealing\nthe underlying micromechanisms of material damage. This framework extensively\nemploys advanced characterization techniques such as scanning electron microscopy\n(SEM), transmission electron microscopy (TEM), high-resolution digital image\ncorrelation (HR-DIC), and three-dimensional orientation imaging-based microscopy7,8.\nThese methods enable the capture of nanoscale image details and provide direct\nvisualizations of the initiation and evolution processes of fatigue damage. These\ntechniques have driven fatigue research from qualitative descriptions to quantitative\ncharacterizations.”\n<Reviewer 1>: 6. P2L39: The term “progression of fatigue” is not commonly used in\nthe fatigue community. The authors should clarify what they mean by this phrase.\nResponse:\nThe expression of this term in the first manuscript is not clear enough. To eliminate\nambiguity and improve clarity, the corresponding sections have been removed.\n<Reviewer 1>: 7. P2L43: The authors should cite key literature for each experimental\ntechnique mentioned to provide context and demonstrate the state of the art.\nResponse：\nWe recognize that there is a lack of literature support for some technical\nbackground in the first manuscript, and the latest technology of fatigue research is not\nfully displayed.\nThe latest technologies in fatigue research have been supplemented in the\nmanuscript with supporting references. The revised text in the manuscript is provided\nbelow (lines 54-60): “Physical field detection techniques provide nondestructive, real-\ntime, and efficient approaches for characterizing fatigue damage by monitoring the\nphysical signals generated during the fatigue process, such as acoustic signals (detected\nvia acoustic emission technology and ultrasonic testing)9,10, magnetic signals (detected\nvia magnetic crack-based pressure sensors)11, and thermal signals (detected via infrared\nthermography) 12-14. These methods, which are based on the physical field variations\ninduced by material damage evolution trends, can reflect the internal damage states of\nmaterials from different perspectives.”\n<Reviewer 1>: 8. While persistent slip bands (PSBs) are important features in fatigue\ndamage, they are not the only structures involved. Other microstructural features such\nas veins, fatigue shear bands, and cell structures also play critical roles in controlling\nfatigue behavior. These are not addressed by the authors and should be included for a\nmore comprehensive discussion.\nResponse:\nThe discussion of microstructure in this paper is deficient. The discussion in this\npaper only involves persistent slip bands (PSBs). Other microstructure characteristics\n(vein structure, fatigue shear band and cell structure, etc.) have an important impact on\nfatigue evolution. The following is our understanding of the effect of microstructure\ncharacteristics on fatigue damage:\nIn the fatigue process of metal materials, the evolution of micro dislocation\nstructure is essentially a cumulative process of the shift of atomic spacing distortion\nfrom reversible elastic oscillation to irreversible plasticity under cyclic loading. As an\ninitial response, the venous structure forms a gradient change of atomic spacing in its\nhigh dislocation cluster region, which leads to the attenuation of local bonding force;\nPSB shows the phase balance effect. Reversible slip in the channel region weakens the\nperiodic oscillation of bonding force, while pinning dislocations in the dislocation wall\nregion increases the atomic spacing; In the cellular structure formed in the high cycle\nfatigue stage, the atomic hyperelastic distortion caused by in dislocation entanglement in\nthe cell wall continued to accumulate distortion energy, resulting in the gradual\nattenuation of the bonding force of interlamellar atoms.\nThese microstructure characteristics will lead to the attenuation of the bonding\nforce of interlamellar atoms, and the evolution of MagDrift is the macro\ncharacterization of the attenuation of the bonding force of interlamellar atoms, so the\nMagDrift signal contains these microstructure characteristics, which are shown by their\nchange rate. The above represents our current understanding of the three\nmicrostructures. If there are any oversights or shortcomings, we sincerely invite your\nfeedback and suggestions.\nIn order to more comprehensively elucidate the relationship between\nmicrostructural evolution and fatigue damage. The revised text in the manuscript is\nprovided below (lines 351-356): “Microscopic HR-DIC and TEM measurements reveal\nthat early dislocation changes induced during fatigue affect the fracture location. Taking\na PSB as an example, its slip marks are almost identical to the final fracture location.\nSignificant correlations are present between the mechanical properties of materials,\nsuch as yield strength, ultimate tensile strength, and fatigue strength, and the early local\nslip of a PSB3,4. Therefore, early changes in the dislocation structure can be used for\nfatigue life assessment purposes.”\nThe other revised text in the manuscript is the same as that described in <Reviewer\n1>: 2.\n<Reviewer 1>: 9. \"PSB formed during fatigue processes... occupy approximately 9.2%\":\nThis value is highly material- and microstructure-dependent and should not be\ngeneralized. The authors should emphasize the conditional nature of this observation.\nResponse:\nSince we did not fully emphasize the preconditions of the conclusion in the first\ndraft, the statement was not rigorous enough. The revised text in the manuscript is\nprovided below (lines 70-74): “The fundamental challenge encountered in fatigue\ndetection tasks is the low sensitivity of macroresponses to structural microdamage,\nprimarily because the various dislocation structures formed during fatigue account for\nonly a small proportion of the affected area. For example, persistent slip bands (PSBs)\nconstitute only approximately 9.2% of the fatigue-affected zone in some fatigue tests\nof single copper crystals18,19.”\n<Reviewer 1>: 10. The claim that techniques such as HR-DIC cannot capture\ndeformations below 100 nm is inaccurate. HR-DIC, as cited by the authors in the\nmentioned reference, can achieve displacement sensitivity in the 7–9 nm range and\nspatial resolutions around 20 nm.\nResponse:\nWe thank you for your valuable comment. We recognize that there was an error in\nthe original manuscript regarding the understanding of HR-DIC resolution, which\nfailed to adequately reflect the latest advancements in this technique. We have corrected\nthe relevant descriptions. The revised text in the manuscript is provided below (lines\n76-79): “Although they can offer displacement sensitivities as high as 7–9 nm and\nspatial resolutions of 20 nm at the microscale, advanced methods such as digital image\ncorrelation (DIC)20 still cannot correlate localized damage effects with macroscopic\nbehaviours.”\n<Reviewer 1>: 11. The manuscript would benefit from a thorough proofreading to align\nwith scientific standards in experimental mechanics and fatigue. Currently, the style\nresembles AI-generated text (e.g., ChatGPT), which may detract from its scientific\ncredibility. The language should be revised for precision and field-appropriate\nterminology.\nResponse:\nWe have carefully reviewed the entire manuscript according to your suggestions,\nwith a particular focus on improving the terminological accuracy and logical rigor of\nthe experimental descriptions and data analysis sections. To ensure the highest quality,\nwe subsequently sought professional editing assistance from Springer Nature Author\nServices to polish the language of the main text and methods section (Verification Code:\n7542-3227-89D9-9D2C-B66P).\n<Reviewer 1>: 12. The authors have not discussed the use of advanced numerical\nsimulations for fatigue life prediction and damage monitoring. Modern structural\nmaterial design increasingly integrates such tools, particularly in industrial applications,\nand this should be acknowledged.\nResponse:\nThere is lacking of discussion on advanced numerical simulations in the paper, we\nhave been supplemented in the introduction and the following references have been\ncited:\nTo more clearly elaborate on the application of numerical simulation in the field\nof fatigue. The revised text in the manuscript is provided below (lines 61-67):\n“Numerical simulation enables quantitative fatigue prediction to be performed under\ncomplex working conditions through physical models. The crystal plasticity-based\nfinite-element method (CPFEM) reveals the influences of grain orientation and\ndislocation mechanisms on the initiation and propagation of cracks at the microlevel15;\ncontinuum damage mechanics (CDM) predicts crack growth and fatigue life under\nmultifield coupling effects using damage variables16,17. Numerical simulations, which\nintegrate multiscale modelling and advanced constitutive theories, enable quantitative\nprediction and provide theoretical support for material safety assessments.”\n<Reviewer 1>: 13. The phrase “tracking the sliding of grain boundaries at the atomic\nscale” requires clarification. If the authors are referring to grain boundary sliding,\nthey should specify, as this mechanism differs fundamentally from PSB formation and is\ntypically associated with high-temperature deformation.\nResponse:\nThe expression “tracking the sliding of grain boundaries at the atomic scale” is\nindeed easy to be confused with the common mechanism of grain boundary sliding in\nhigh temperature creep, which is different from the fatigue mechanism we focus on. In\nthe manuscript, all terms related to “grain boundary sliding” have been removed to\nensure conceptual accuracy and avoid ambiguity.\n<Reviewer 1>: 14. “Nanoscale atomic dislocation”: This expression is unconventional.\nDislocations are already atomic-scale defects, so the phrasing is redundant or\nmisleading. The authors should revise this for clarity and scientific accuracy.\nResponse:\nThe description of “Nanoscale atomic dislocations” in the text is not rigorous in\nthe theory of crystal defects. To improve the accuracy of terminology, we have\nuniformly revised the expression to “dislocation” in the revised draft and eliminated\npotential ambiguity.\n<Reviewer 1>: 15. formatting: There is a typographical error in the phrase “and\nMethods)This relationship”, where a space is missing.\nResponse:\nThe phrase “and methods) this relationship” in the text does have typographical\nomissions. The missing space after the right parenthesis here is due to our mistake. The\noriginal text has been revised to the standard format. In addition, we check the full text\nand manually verify all similar punctuation marks to ensure that the format is\nstandardized.\n<Reviewer 1>: 16. The authors should define the fatigue ratio and loading conditions\nused. More importantly, they should consider cyclic plasticity mechanisms such as\nratcheting and slip irreversibility and also different loading conditions such negative\nand positive load ratios. In certain conditions, dislocation accumulation may occur\nwithout producing significant surface extrusion, which would influence the MagDrift\nsignal identifically but produce different fatigue response.\nResponse:\nThe difference of cyclic plasticity mechanism plays an important role in analyzing\nthe fatigue behavior of materials. Our discussion on this issue in the first manuscript is\ninsufficient. The following is our understanding of cyclic plasticity mechanism in\nfatigue, and the corresponding experiments are supplemented:\nAlthough the microscopic mechanism of ratcheting effect is different from that of\nnon-ratcheting effect, its essence is related to atomic spacing, and slip irreversibility is\nthe smallest unit of ratcheting effect, so the essence of slip irreversibility is also related\nto the change of atomic spacing. Therefore, these microstructure characteristics will\nlead to the attenuation of the interlamellar bonding force, and the evolution of MagDrift\nis the macro-characterization of the attenuation of the interlamellar bonding force, so\nthe MagDrift signal contains these microstructure characteristics.\nIn order to verify the influence of different load conditions on MagDrift signal,\nwe supplemented the existing data and selected 45 steel high stress fatigue test (580 MPa,\nanalysis: the high stress ratchet group caused significant strain growth due to the\ncontinuous accumulation of plastic deformation, and the MagDrift showed a significant\njump; Although the small stress non ratchet group has no macroscopic plastic\ndeformation, the MagDrift still evolves significantly, and the evolution rate of the large\nstress ratchet group is faster. For a detailed explanation, please refer to <Reviewer 1>:\n3.\nAt the same time, we also supplemented the fatigue experiments with different\nstress ratios. Soft magnetic iron materials were used in the experiments, which were\npositive stress ratio R=0.1, the maximum stress was 495 MPa, and negative stress ratio\nR=-0.5, the maximum stress was 380 MPa. The structural dimensions and other loading\nparameters of the specimens were the same as those in the paper (Methods), and fatigue\nfracture occurred after 90 K and 25 K, respectively.\nAccording to the experimental data, the change rate of magnetic strength with\ndifferent stress ratios has the same evolution trend, which initially goes smoothly and\nthen accelerates to decline. From the MagDrift diagram, it has the same change trend\nbefore and after the acceleration change, which first rises steadily and then accelerates\nto decline, indicating that the change of magnetic signal intensity with the equivalent\natomic spacing shows the same rule. It is confirmed that the signal change only depends\non the degree of bond force attenuation and is independent of load conditions.\nFig. 11. Fatigue test analysis of different stress ratios. a, 495 MPa, 90,000 cycles, 0.1 stress ratio,\nCRMS and MagDrift. b, 380 MPa, 25,000 cycles, -0.5 stress ratio, CRMS and MagDrift.\nAlthough different cyclic plasticity mechanisms are different, the increase of\natomic spacing caused by them will trigger the change of MagDrift. It can be seen from\nthe experimental diagram that the MagDrift has the same trend under different load\nconditions. This feature enables spin correlation amplification technology to span the\ndifference of load conditions.\nThe above represents our current understanding of this issue. If there are any\noversights or shortcomings, we sincerely invite your feedback and suggestions.\nThe revised text in the manuscript is the same as that described in <Reviewer 1>:\n2.\n<Reviewer 1>: 17. PSB morphology varies significantly between pure metals and high-\nstrength alloys. The authors do not provide a rationale for how their average\nmeasurement technique is expected to capture these differences. It is difficult to\nimagine that such averaging can reliably identify damage processes, especially\nconsidering that fatigue failure is typically governed by the most extreme localized\nplastic events rather than average behavior. Similarly, the evolution of the MagDrift\nsignal must be interpreted in the context of strain localization, which the current\napproach may not capture. In this respect, the method is conceptually akin to applying\na macro-scale extensometer or strain gauge, useful but lacking the spatial resolution\nneeded for a predictive understanding of localized fatigue damage.\nResponse:\nThere are some differences in plastic deformation mechanism between pure\nmetals and high strength alloys. This is also the limitation of traditional macro or\naverage measurement methods. We noticed that the previous description of the\ntheoretical model was oversimplified, and there was a lack of clarity in the\ninterpretation of the basic principles, which may lead to a misunderstanding of the\nnature of the measurement method, believing that it only captures the global average.\nThe essence of spin correlation amplification technology is not spatial average\nmeasurement. Its essence is to transform the spin correlation change triggered by the\nchange of interlamellar atomic spacing into a macro measurable MagDrift signal\nthrough the effect of multiple amplification. Its core mechanism is the multiple\namplification mechanism dominated by local damage.\nUniversal response mechanism to different material damage modes: Although\nregular and coarse persistent slip bands (PSBs) are usually formed in pure metals (such\nas nickel), plastic strain localization in high-strength alloys (such as high-strength steel)\ntends to be more dispersed or discontinuous in the form of microstrip, grain boundary\nslip or phase boundary cracking due to the existence of barriers such as grain boundary\nand second phase particle. However, whether the specific microscopic morphology is\npersistent slip band (PSB) or other types of strain localization regions, the physical\nessence behind it is due to lattice distortion, the increase of atomic spacing and the\nweakening of atomic bonding force. The change of spin correlation is a direct physical\nresponse highly sensitive to these microstructure changes. Therefore, the detection\ntarget of this technology can be directly traced to the attenuation of atomic bonding\nforce, which is unified in physical nature and does not differ from PSB or other damage\nmodes.\nWe have supplemented three kinds of experiments, including artificial crack\nsimulation, prefabricated defect evolution and weld fusion zone damage, to jointly\nconfirm that the attenuation of the bonding force of the weakest section in the fatigue\nprocess is reflected in the final measured MagDrift signal through the multiplication\neffect of the spin correlation amplification of the exchange interaction (magnetic\nmoment). The local nature of fatigue damage will have a dominant effect on the global\nchange of MagDrift signal. For a detailed explanation, please refer to <Reviewer 1>:\n1.\nThis technology has the adaptive ability of environmental interference, the\nuniversal adaptability to special-shaped structures and the exponential amplification of\nmicro damage, forming a unique advantage different from the traditional linear\nsuperposition of strain measurement.\n<Reviewer 1>: 18. In fatigue, strain hardening or softening phenomena often occur,\naltering the dislocation structure over time. The manuscript does not discuss how these\neffects would influence MagDrift or whether the method can differentiate between them.\nResponse:\nCyclic hardening and cyclic softening are two typical responses of metal materials\nunder fatigue loading. Cyclic hardening shows that the ability of the material to resist\ndeformation is enhanced, and its macroscopic characteristic is that the dynamic stiffness\nincreases with time, which usually results from the increase of dislocation density and\nits entanglement; On the contrary, cyclic softening shows that the material softens and\nthe dynamic stiffness decreases, and its microscopic mechanism is related to the\nformation of low-energy dislocation structure by dislocation recombination.\nWe have drawn dynamic stiffness and MagDrift curves for representative\nspecimens of soft magnet iron, pure nickel, 45 steel and Q235 steel respectively (Fig.\n12). It was found that all specimens exhibited initial cyclic hardening stage (although\nthe duration was short), then entered a stable cyclic softening stage, and finally showed\naccelerated softening phenomenon in the failure stage. In terms of magnetic response,\nthe dynamic stiffness of soft magnet iron, 45 steel and Q235 steel increased at first and\nthen increased steadily at the beginning of loading, showing cyclic hardening, while the\nMagDrift had no sudden change in this process, showing an accelerated upward trend,\nrising steadily in the stable softening stage, and changing rapidly with the softening\nacceleration at the failure stage. On the contrary, the MagDrift of Pure Nickel shows an\naccelerated downward trend during cyclic hardening, and it also accelerates with\nsoftening acceleration in the failure stage.\nFig. 12. Comparative analysis of dynamic stiffness and MagDrift of four materials. a, Dynamic\nstiffness and MagDrift diagram of soft magnetic iron specimen under the condition of maximum\nstress of 425 MPa and fatigue failure life of 97 K. b, Dynamic stiffness and MagDrift diagram of\npure nickel specimen under the condition of maximum stress of 430 MPa and fatigue failure life of\nmaximum stress of 560 MPa and fatigue failure life of 226 K. d, Dynamic stiffness and MagDrift\ndiagram of Q235 steel specimen under the condition of maximum stress of 520 MPa and fatigue\nfailure life of 230 K.\nThe above results show that different materials may affect the response sensitivity\nof magnetic signal to micro plastic deformation, but the synergistic behavior of\nMagDrift and stiffness change plays an auxiliary role in distinguishing the cyclic\nhardening and softening processes.\n<Reviewer 1>: 19. Dislocation structures evolve depending on the slip planarity, which\nhas a direct impact on fatigue behavior. These structural changes should be considered\nif the authors aim to correlate dislocation behavior with fatigue properties.\nResponse:\nIn the process of material fatigue research, we focus on the quantum of the\nsynchronous attenuation of bonding force driven by the change of interlamellar atomic\nspacing. The evolution of dislocation behavior (dislocation annihilation, slip,\nrecombination) can trigger the response of electron spin correlation amplification as\nlong as its essence is related to the change of atomic spacing. Although the slip planarity\nmodulates the spatial distribution of bit stagger, its core effect on fatigue still leads to\nthe attenuation of bonding force through the increase of atomic spacing. The MagDrift\nsignal is directly related to the interlamellar bonding force and the exchange interaction.\nThe change of the interlamellar atoms spacing becomes the dominant variable to control\nthe strength of the two, which transcends the microscopic mechanism limitation of the\nslip geometry difference.\nThe revised text in the manuscript is the same as that described in <Reviewer 1>:\n2.\n<Reviewer 1>: 20. The approach is reminiscent of nonlinear acoustic techniques used\nto monitor dislocation density evolution. However, as in those methods, without spatial\ncorrelation at the microstructural level, dislocation density alone cannot reliably predict\nfatigue life. For example, in the Very High Cycle Fatigue (VHCF) regime, no\nmeasurable macroscopic strain evolution is typically observed, and this method would\nlikely fail to detect damage evolution. Similarly, when fatigue is initiated at large\ndefects, the deformation is highly localized and may not be captured by spatially\naveraged signals.\nResponse:\nAlthough both spin correlation amplification and nonlinear acoustic techniques\ncan be used for material damage detection, their physical principles and response\nmechanisms are essentially different.\nNonlinear acoustic technology relies on nonlinear distortion of lattice strain field:\ndislocation evolution under cyclic loading or microcracks lead to high-order harmonics\nin acoustic wave propagation, and the damage degree is inversed by measuring acoustic\nnonlinear parameters (such as β coefficient). The response of this method is essentially\ndue to the enhancement of phonon defect scattering caused by periodic lattice damage.\nIts signal strength is modulated by macroscopic parameters such as material density\nand elastic modulus, and it has no direct ability to perceive atomic scale changes.\nIn contrast, spin correlation amplification technology achieves detection by\ncapturing the electron spin correlation effect caused by changes in atomic spacing in\nferromagnetic materials: when fatigue leads to an increase in interlamellar atomic\nspacing, the attenuation of interlamellar exchange interactions triggers directional\ndeflection of the spin magnetic moment. This local perturbation is exponentially\namplified through the spin correlation chain and ultimately manifests as measurable\nmagnetic signal changes at the macroscopic level. Its core advantage lies in its\nsensitivity to electronic changes and its nonlinear signal gain characteristics, which\namplify quantum effects as they pass through the fatigue dislocation layer.\nEffectiveness of high cycle fatigue\nIn order to verify the effectiveness of this technology in VHCF scenario, we added\nhigh cycle fatigue test: 45 steel, the loading parameters were consistent with the original\n(Methods), and the maximum stress was 480 MPa. After two weeks of continuous\nloading, the experiment was forced to stop after 22.17 million times due to time.\nIn Fig. 13, no macroscopic measurable evolution of displacement and\ndisplacement amplitude is observed, but the magnetic signal has a macroscopic\nmeasurable evolution, especially the MagDrift signal, which shows a clear downward\ntrend, indicating that the interlamellar bonding force is gradually weakening to some\nextent. This confirms that under high-cycle fatigue conditions, MagDrift can still\nachieve early damage warning by capturing the gradual attenuation of the atomic\nbonding force.\nFig. 13. The high cycle fatigue test data chart. a, The MagDrift and displacement amplitude. b,\nThe CRMS and strain.\nFor the localized deformation caused by large defects, we added three types of\nprogressive experiments, artificial crack simulation, prefabricated defect evolution and\nweld fusion zone evolution. These three groups of experiments confirmed that the local\nnature of fatigue damage would have a dominant effect on the global change of\nMagDrift signal. The lamination bond force attenuation through the weakest cross\nsection is reflected in the final measured magnetic signal through the multiplication\neffect of the spin correlation amplification of the exchange interaction (magnetic\nmoment). For a detailed explanation, please refer to <Reviewer 1>: 1.\nResponse to Reviewer 2\n<Reviewer 2>: 1. In the “Fatigue Life Prediction Model” section, the authors propose\nan M-N curve analogous to the classical S-N curve using MagDrift values. Line 335\nmentions the use of an average slope across four metal materials. Why did the authors\nnot apply individual slopes for each material?\nResponse:\nWe calculated the MagDrift slope values for the four iron-based metal materials\nand found that the difference in their slopes was not significant. The average slope\nabsolute value of their M-N curves is 1.54391, which aligns closely with the value of\n1.5 reported by Néel for the relative variation of the α-Fe exchange integral with respect\nto the proximity distance of neighboring electrons. Therefore, we speculate that there\nmay be a physical correlation between the two phenomena. A detailed explanation is\nprovided below:\nIndependent calculation of MagDrift slope for four types of metal materials\nRegarding the individual MagDrift slopes of the four metal materials, we did\nindeed calculate the individual MagDrift slope values for each metal material and\ncompared and analyzed the individual MagDrift slope values and their respective\ndegrees of fit, as shown in Fig. 14.\nFig. 14. MagDrift slope and fit degree of four metal materials separately.\nAs shown in Fig. 14, when the average slope across four metal materials is not\nconsidered, the slope of the M-N curve for soft magnetic iron material is -1.26075, with\na goodness-of-fit of 0.9723; the slope of the M-N curve for 45 steel material is -1.49818,\nwith a goodness-of-fit of 0.9"}]

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