The safety and fatigue reliability of engineering structural components are critical challenges that demand urgent attention. For uniaxial fatigue problems, many researchers have developed comprehensive fatigue models based on empirical evidence [1]. However, structural failure resulting from multiaxial fatigue loading is a widespread phenomenon in practical engineering [2]. Key components in industries such as aviation, aerospace, and rail transportation are frequently subjected to complex multiaxial fatigue loading, which can lead to structural failure [3]. Data show that up to 80% of component failures are due to fatigue fractures [4]. The causes and mechanisms of such failures are highly complex. Even components primarily exposed to uniaxial loading may experience multiaxial fatigue loading at specific locations due to structural intricacies [5]. Metal fatigue failure behavior is commonly categorized based on the number of cycles to failure: low-cycle fatigue (<104), high-cycle fatigue (104−107), and ultra-high-cycle fatigue (>107) [6]. Non-proportional additional hardening caused by dislocation interactions through slip systems occurs in metallic materials [7–9]. Under identical equivalent strain amplitudes, this effect typically reduces low-cycle fatigue life compared to proportional loading [10]. Conversely, the fatigue behavior of metallic materials under high-cycle conditions differs significantly, as other studies have demonstrated [11–13]. Engineering structures like high-speed trains experience such complex multiaxial service loads combining tension, bending, shear, and torsion stresses. Consequently, fatigue life prediction methodologies demonstrating accurate and efficient performance across variable loading scenarios are imperative for operational safety.

To advance the study of material life prediction models under multiaxial fatigue loading, researchers globally have classified these models into two categories based on their principles and methods, derived through experiments and theoretical analysis [14–17]. The first category includes methodologies leveraging machine learning models trained on loading data [18–20]. Machine learning algorithms provide an efficient and rapid approach to establishing mapping relationships. With capabilities for continuous learning and adaptive parameter updating, they enhance computational efficiency and reduce conventional calculation errors. Their accurate data processing and precise, reliable modeling methods make them dependable tools for predicting material fatigue life and analyzing fatigue failure [21]. In recent years, fully connected neural networks have been widely applied to uniaxial fatigue life prediction for diverse materials. Based on their investigation into the failure mechanisms of 316L steel under complex service environments, Srinivasan et al. employed Artificial Neural Networks (ANNs) to establish complex functional mappings between variables (such as temperature) and fatigue life [22]. The work of Anastasios et al. demonstrated that with appropriate parameter settings and network architecture design, ANNs can predict fatigue life across the entire stress ratio range using only half the data required by conventional methods to construct multiple S-N curves. This enables more accurate life assessment and design work [23]. Addressing the fatigue life of metallic materials under mean stress, Barbosa et al. proposed an ANN-based probabilistic constant-life approach. This method achieves accurate predictions with minimal data, further validating the effectiveness of ANN methodologies [24]. Al-Assaf et al. experimentally and computationally demonstrated that ANNs for fatigue life prediction place high demands on model parameter settings, data quality, and purity; otherwise, they fail to ensure adequate accuracy and reliability of fitting results [25]. Furthermore, for fatigue analysis under random loading, ANN models—through rational input parameter design covering diverse materials and complex loading characteristics—maintain effectiveness and accuracy in predicting random loading fatigue life [26]. Jingye Yang et al. developed both semi-empirical and neural network models to study low-cycle fatigue behavior under non-proportional loading. Compared to the semi-empirical model, which has clear physical meaning but lower accuracy, the neural network more accurately captured the fatigue characteristics of materials under low-cycle fatigue loading, though it lacked corresponding physical interpretability [27]. However, such fully-connected neural network-based approaches for uniaxial fatigue life prediction are less suitable for multiaxial fatigue conditions. Building upon this research, scholars have proposed data-driven models applicable to multiaxial fatigue loading scenarios. Their aim is to develop a deep learning-based generic framework for predicting multiaxial fatigue life in metallic materials [28]. The application of machine learning to multiaxial fatigue life prediction enables the effective identification and optimization of damage coefficients, thereby facilitating the development of reliable fatigue life prediction models [29]. Yang et al. proposed a machine learning-based multiaxial fatigue life prediction model [30]. This model utilized a Long Short-Term Memory (LSTM) neural network to extract critical points from the load path, which were subsequently input into a Fully Connected Neural Network (FCNN). Similarly, Zheng et al. employed phase difference and normal and shear strain amplitude as inputs for a multiaxial fatigue prediction model [31,32]. They trained and tested their approach using fatigue data from pure titanium and titanium alloys. Their findings demonstrated that the proposed model outperformed conventional models based on physical mechanisms in predicting the fatigue life of titanium alloys. Zhan and Li integrated the theory of continuous medium damage mechanics (CDM) with machine learning to develop a predictive methodology for the lifespan of additively manufactured aerospace alloys [15]. This model was empirically validated through experimental testing. However, despite the success of machine learning in constructing multiaxial fatigue life prediction models and verifying their predictions with experimental data, certain limitations persist. Specifically, the selection of input feature parameters often neglects key loading parameters such as timing characteristics and amplitude. This omission prevents a comprehensive representation of the multiaxial fatigue loading process, resulting in discrepancies between predicted outcomes and experimental results.

Scholars have also focused on analyzing the fatigue failure mechanisms of metallic materials, aiming to develop fatigue life prediction methods grounded in physical mechanisms. The most widely used models in this domain include the Equivalent Strain Energy Density Method (ESM) [16], the Energy Method [33,34], and the Critical Plane Method [19]. In applying these models, specimens are defined as being subjected to multiaxial loading conditions, as is typical in standardized fatigue testing. The aforementioned fatigue life prediction models are established for metallic materials under multiaxial fatigue loading conditions. Corresponding predictive mapping frameworks between damage parameters and fatigue life are developed based on specific failure modes. These models not only predict the fatigue life of metallic materials but also elucidate the underlying multiaxial fatigue failure mechanisms.

(1) Equivalence Transformation Method. The Manson-Coffin equation is a well-established approach for predicting the uniaxial tensile fatigue life of materials, with a proven track record of reliability. Many researchers have extended this method to multiaxial fatigue life prediction for metallic materials by incorporating strain coefficients into the Manson-Coffin framework. For instance, Yang et al. conducted tensile and shear fatigue tests on the critical parts of bolted joints based on the equivalent structural stress method, derived the equivalent structural stress calculation model and finite element model that accurately characterize the stress state of bolted joints, and finally obtained the equivalent structural stress at the crack initiation site of bolted joints, indicating that the method can provide a reference for the design calibration and fatigue life prediction of bolted joints [35]. Li et al. established and determined a fatigue assessment framework for the dominant failure modes of rib-panel welded joints based on the equivalent structural stress method for evaluating the fatigue performance of rib-panel welded structures, and validated it with experimental data, which showed that the framework can accurately predict the main fatigue failure modes of welded joints [36]. While the equivalent strain method is effective for predicting uniaxial tensile fatigue life, it has several limitations. Notably, it lacks a direct mapping relationship between strain and life, disregards the interaction between stress and strain during deformation, fails to account for the material’s response along the loading path, and exhibits limited adaptability [29]. Consequently, when applied to multiaxial non-proportional loading conditions, the method often produces significant prediction deviations.

(2) Energy Method. To predict metal fatigue life, OSTERGREN applied the theory of calculating plastic work through stress-strain hysteresis loops, initially proposed by FELTNER [37,38]. Guard study further revealed an exponential relationship between crack life and the amount of plastic work performed, successfully extending this theory to multiaxial fatigue life prediction [39]. Considering the effects of mean stress and plasticity on the medium-high cycle uniaxial fatigue strength of common notched components, Benedetti refined and extended traditional elastoplastic methods, proposing a novel strain-energy-density-based fatigue criterion [40]. Gao et al. modified and generalized the conventional linear elastic strain energy density method by incorporating a novel exponential math model and material sensitivity parameters, significantly improving its applicability to medium-high cycle fatigue under wide stress ratios [41]. To address the unpredictability of failure locations and fatigue performance in aluminum alloy butt joints with fusion line defects under high-cycle fatigue, Chen established a defect competition behavior model based on linear elastic strain energy density. This model quantifies the effects of geometric parameters and residual stresses, with its accuracy validated through fatigue tests and finite element methods [42]. Sousa et al. developed a life prediction framework for bonded joints in composites and metals under medium-high cycle fatigue conditions using a strain energy density approach [43]. For thermo-mechanical fatigue life prediction, Sun et al. proposed a three-parameter power function equivalent energy method based on multiple conventional fatigue life prediction approaches. The validity of this method was verified through low-frequency fatigue test data of GH4133 material, demonstrating high computational accuracy and simultaneous fitting capability to fatigue test data at different strain ratios [44]. Although the energy method yields satisfactory predictions under uniaxial and multiaxial proportional loading conditions, its effectiveness varies depending on the specific energy form, applicable scenarios, and underlying physical mechanisms. For linear elastic strain energy methods: (1) These methods assume fully reversible energy dissipation. However, in medium/high-cycle fatigue, irreversible microscopic slip and dislocation motion occur [45], rendering elastic strain formulations inadequate for quantifying damage dissipation energy. (2) Formula modifications are required to characterize mean stress effects [46]. (3) Constrained by the macroscopic linear elasticity assumption, these methods fail in plasticity-dominated fatigue regimes [47]. For plastic strain energy methods: (1) As a scalar quantity, plastic work cannot fully capture fatigue damage mechanisms. (2) The approach neglects influences of mean stress and hydrostatic stress. (3) It becomes inappropriate when plastic strains are negligible [48].

(3) Critical Plane Method. The critical plane method, introduced by Brown and Miller integrates shear and normal strains within the plane of maximum shear strain [19]. Building on this foundation, Fatemi and Socie [20] proposed using stress instead of strain, enhancing the model’s utility. By incorporating a normal stress term, the method could account for the effects of additional reinforcement on fatigue life under low-circumferential fatigue conditions. To investigate proportional and non-proportional loading fatigue damage coefficients, Yakovchuk et al. compare the results of model calculations by Fatemi-Socie, Wang-Brown. with the experimental data obtained for 10 metal alloys and 6 multiaxial loading paths based on the theory of the critical plane method. The results show that the model correlates well with the experimental data for both proportional and non-proportional loading [49]. The critical plane method is widely used in the fatigue life calculation of metal materials. It reflects the microscopic mechanism of crack initiation and propagation to a certain extent, and the prediction results are good under non-proportional conditions of multi-axial loading. However, its applicability is also challenged under some special load conditions [50]. The study by Yang et al. integrates normal and shear stress failure criteria to implicitly capture phase-difference effects in non-proportional loading through multi-stress component combinations. This approach eliminates the dependency on explicit phase correction in conventional methods, providing a robust reference for life prediction of complex engineering components subjected to non-proportional loads [51]. Liu et al. proposed an enhanced critical plane approach for multiaxial fatigue life prediction by incorporating dynamic sensitivity coefficients, a shear stress intensification mechanism, and normal strain modification. This model accurately characterizes non-proportional additional hardening and crack propagation mechanisms, establishing a foundation for engineering applications in complex structural parts [52]. Qi et al. introduced the ratio of maximum shear stress amplitude to maximum normal stress amplitude, coupled with material fatigue parameters, to define distinct failure regimes on critical planes. Validated under multiaxial loading conditions including mean stresses, this model demonstrates superior prediction accuracy [53].

Although the critical plane approach demonstrates certain physical significance in damage parameter selection, its parameter determination has been controversial due to the lack of adherence to rigorous continuum mechanics theory. In recent years, some researchers have proposed the critical plane strain energy density method by integrating the advantages of energy-based methods and the critical plane approach [54–56]. Li et al. combined the critical plane method with strain energy density theory, introducing a damage proportionality model to simultaneously account for material failure modes and both normal and shear strain energy components. This provides a more comprehensive characterization of damage mechanisms under multiaxial loading [57]. Hao et al. improved the model by constructing a modified equivalent shear stress that incorporates a weighting function to characterize the influence of normal stress. The enhanced strain energy-based critical plane model effectively captures non-proportional additional hardening without requiring additional parameters [58].

This paper proposes a deep learning-based model for predicting the fatigue life of metals under multiaxial loading conditions. First, the key characteristic parameters influencing multiaxial fatigue were identified through an extensive review of the literature [30,59,60]. Next, timing data were selected based on the loading waveform and integrated with the characteristic parameters to create the model’s input data, encompassing both timing and loading features [61–63]. Finally, the optimal model parameters were determined through training, enabling the establishment of a comprehensive mapping relationship between the input features and fatigue life. This approach facilitates the accurate and reliable prediction of the fatigue life of metallic materials [64].

A fully connected neural network consists of three main layers: the input layer, hidden layers, and the output layer. This structure is depicted in Fig. 1. Through iterative training, the network learns a complex non-linear mapping between the input feature set (x) and the output (y), enabling accurate modeling of intricate relationships [30]. (1)x→fFCNN⁡ywhere x denotes the input feature covariate and y denotes the output covariate obtained by model mapping, and both x and y can be multidimensional covariates.

Generally, traditional fatigue prediction models focus on establishing a functional mapping relationship between damage parameters and fatigue life [68]. However, the function mapping for multiaxial fatigue is inherently complex, primarily due to the intricate microfatigue mechanisms involved. Deriving this relationship through straightforward logical deduction is a challenging task. Nonetheless, this complex mapping relationship can be effectively derived using machine learning techniques.

Fig. 4 illustrates the STFAN model for predicting the fatigue life of metallic materials using deep learning techniques. In this example, the model’s prediction process is demonstrated with a sinusoidal loading waveform.

The following schematic illustrates the STFAN fatigue life prediction model. It is assumed that when the material is subjected to stress-controlled conditions, both the normal and shear stresses follow a sinusoidal waveform. The shape of the curves for these stresses may be influenced by the phase difference angle and the loading frequency ratio.

A series of data points are selected at equal intervals along the curve, based on a specific period, to form a feature matrix. This matrix serves as the input for both the Convolutional Neural Network (CNN) and the Long Short-Term Memory neural network (LSTM). The neural network then extracts the temporal characteristics of the multiaxial fatigue loading. The selected key parameters influencing the loading paths are used to create a new fatigue life prediction feature matrix, which serves as the input for the Fully Connected Neural Network (FCNN).

The newly developed feature matrix includes not only the time-series features of the fatigue loading condition but also key covariates that may influence the loading path, such as the phase difference angle and magnitude. Table 1 illustrates the parameters used during the model training process.

Currently, fatigue life prediction methods based on material physical mechanisms are widely used. These methods address the fatigue mechanisms under multiaxial loading conditions by establishing a mapping relationship between various damage coefficients and fatigue life. This approach is outlined in the referenced works [69]. Commonly used prediction models include the Equivalent Strain Method (ESM, Eqs. (9) and (10)), the Maximum Shear Strain Method (MSSM, Eq. (11)), and the Critical Plane Method, among others (FS, Eq. (12)) [17]. Their mathematical expressions are provided below: (9)Δεeq2=σf′E(2Nf)b+εf′(2Nf)c (10)εeq=(εa2+γa23)0.5 (11)γa,max=τf′G(2Nf)b0+γf′(2Nf)c0 (12)Δγmax2(1+kσn,maxσy)=τf′G(2Nf)b0+γf′(2Nf)c0

The fatigue life of TC4 titanium alloy under different working conditions is predicted using fatigue life prediction methods based on physical mechanisms [64], as indicated by the test data for various fatigue loading conditions of TC4 titanium alloy material. The Equivalent Effect Strain Method (ESM), the Maximum Shear Strain Method (MSSM), and the Critical Plane Method (FS) are employed, in addition to the STFAN-based fatigue life prediction model using deep learning proposed in this paper. The relevant parameters, as illustrated in Eqs. (9)–(12) and Table 2, are used in the calculation of fatigue life using the physical mechanism-based methods. In contrast, the fatigue life prediction for the material using the deep learning method is computed by the model shown in Fig. 4. The multiaxial loading condition test life is presented as an error band diagram for comparative analysis with the fatigue life predicted by the four methods, as shown in Fig. 5.

In the Fig. 5, panels (a), (b), (c), and (d) present a comparative analysis of the prediction outcomes against the experimental results obtained using the Equivalent Strain Method (ESM), the Maximum Shear Strain Method (MSSM), the Critical Plane Method (FS), and the Deep Learning Model (STFAN), respectively. The black dotted line represents the 1.5 times error band, the black solid line represents the 2 times error band, the horizontal axis indicates the predicted fatigue life, and the vertical axis represents the true fatigue life of the material obtained from testing. The model’s prediction accuracy is considered satisfactory if the result falls within the specified error band. A higher level of accuracy is indicated when the result lies within the 1.5 times error band, and an even higher level of accuracy is indicated when the result falls on the fitted straight line [30].

As illustrated in Fig. 5a, the fatigue life prediction of TC4 using the ESM method is largely satisfactory for low-cycle fatigue with the data predominantly falling within the 2-fold error band. However, as the number of fatigue cycles increases (104), the model’s predictive accuracy declines significantly. When the number of fatigue cycles exceeds, the prediction results fall outside the 2-fold error band. As shown in Fig. 5b, the MSSM method exhibits poor accuracy in predicting TC4 fatigue life, both for low-cycle fatigue (less than 104 cycles) and high-cycle fatigue (greater than 104 cycles), as indicated by the data falling outside the 2-fold error band. Similarly, once the number of fatigue cycles surpasses, most predictions also lie outside the 2-fold error band.

Fig. 5c shows that the predicted fatigue life of TC4 using the FS method remains largely within the acceptable range, with most data points falling within the 2-fold error band. However, as the number of test cycles exceeds, some data points deviate from the 2-fold error band, suggesting potential inaccuracies in the prediction at this stage. As depicted in Fig. 5d, the projected fatigue life of TC4 using the STFAN model is generally accurate, with only a few data points falling outside the 2-fold error band, especially when the number of fatigue test cycles exceeds. In most cases, the prediction results fall within the 1.5-fold error band, demonstrating the model’s high accuracy and satisfactory performance. In particular, at low fatigue cycles (less than 104 cycles), sporadic data fall outside the 2-fold error band. Therefore, considering whether the model may have a problem of reduced prediction accuracy under low fatigue cycle conditions. Next, aiming at the possible problems, different four materials and different loading paths are selected to verify the prediction accuracy of the model under low cycle times.

As shown in Fig. 5, the accuracy of the prediction results of the four material fatigue life prediction methods on TC4 titanium alloy is: STFAN > FS > ESM > MSSM. Notably, when the STFAN model is used to predict the fatigue life of TC4 material, some data points fall outside the 2-fold error band for low-cycle fatigue (less than cycles). However, the predictions tend to be conservative compared to the actual fatigue life test results. This raises the question of whether the model might have a tendency to provide conservative predictions with reduced accuracy under low-cycle fatigue conditions. The next phase of the study will address potential issues with the model and evaluate its generalization capacity, or its ability to make accurate predictions on datasets it has not encountered before. To this end, four different materials [59], each with distinct loading paths, have been selected for further analysis. The data have been divided into training, validation, and test sets to assess the accuracy of the STFAN model’s predictions for low-cycle fatigue. The loading paths of the test materials are illustrated in Fig. 6.

Fig. 6 illustrates the relationship between the multiaxial fatigue loading path and the stress-strain response for four distinct materials. For 45 steel [61–63] and the aluminum alloys LY12CZ and 1Cr-Mo-V [59,62], three factors must be considered when examining the effects of multiaxial fatigue loading: uniaxial stress, torsion, and phase difference. In the case of material E335 [62], the loading frequency ratio, which varies between 0.25 and 6, must also be accounted for. Fig. 6a to c demonstrates how the material responds to uniaxial and torsional forces when the loading path is aligned with the coordinate axes, with a 0∘ phase difference. In these cases, the loading path appears as a straight line passing through the first and fourth quadrants. As the phase difference increases, the path gradually transforms into an ellipse, and when the phase difference reaches 90∘, the loading path approximates a circle. As the phase angle continues to increase towards 180∘, the path shifts from a circle to a straight line again. During this phase, the diagram becomes symmetric around the y-axis, exhibiting characteristics similar to those observed when the phase difference angle ranges from 0∘ to 90∘. As the frequency ratio increases, the effect on the loading path becomes more pronounced, with the path in Fig. 6d showing a noticeable decline in regularity.

As shown in Fig. 7, the fatigue loading conditions for 45 steel are relatively simple, as they consider only the effect of phase difference variations on the material’s fatigue life. The results of using the STFAN model to predict the material’s fatigue life are highly satisfactory, with most of the data falling within the range of the fitted straight line. The fatigue loading of LY12CZ material was analyzed with respect to its effect on the material’s fatigue life, considering phase differences ranging from 0∘ to 90∘.

As shown in Fig. 8, the results of using the STFAN model to predict the material’s fatigue life are favorable. The data points align closely with the fitted straight line, with most falling within the 1.5 times error band, and all points within the 2 times error band. A comparative analysis of the fatigue loading conditions for 1Cr-Mo-V and LY12CZ materials reveals that the 1Cr-Mo-V material exhibits improved performance when the phase difference exceeds 90∘ [70]. This analysis also considers the effect of significant phase differences on the material’s fatigue life. In real-world production environments, loading conditions with phase differences greater than 90∘ are common. Therefore, the model’s ability to accurately predict fatigue life under such substantial phase difference angles is a key indicator of its overall predictive accuracy.

As shown in Fig. 9, the results of using the STFAN model to predict the fatigue life of the 1Cr-Mo-V material are promising. A subset of the data points falls on the fitted straight line, the majority are within the 1.5-fold error band, and all data points lie within the 2-fold error band.

Fig. 10 illustrates the multiaxial fatigue life error bands for material E335 under various loading ratios. The STFAN model demonstrates strong predictive capability for material E335 subjected to different fatigue loading conditions, including frequency ratios between 0.25 and 6 and phase difference angles of 90∘. The model’s predictions are highly accurate, with most data points falling within 1.5 times the error band and all points within 2 times the error band.

The multiaxial fatigue prediction results of the above four materials show that the model has strong prediction ability under low cycle fatigue conditions.

Traditional multiaxial fatigue life prediction methods often lead to “distorted” predictions due to their limited focus on only part of the physical mechanisms influencing fatigue life—primarily the loading process. A critical factor that is frequently overlooked is the effect of load timing on fatigue life. To overcome the limitations of conventional methods, this study introduces a deep learning-based multiaxial fatigue prediction model (STFAN model) that simultaneously incorporates the physical mechanisms throughout the loading process and load sequence effects (the sequential coupling effect under non-proportional loading). The aim of this study is to assess the predictive accuracy of the proposed model by integrating key loading conditions from existing multiaxial fatigue studies. Five sets of fatigue test data from various materials were utilized, and the results confirm that the deep learning-based STFAN model demonstrates excellent predictive performance. This approach offers potential solutions to alleviate the industrial challenge of expensive full-scale fatigue testing for high-speed train components (e.g., wheelsets and bogies).

(1) The multiaxial fatigue life prediction model proposed in this paper leverages deep learning techniques and is applicable to a variety of common multiaxial loading conditions and their combinations, such as uniaxial and torsional loading. Additionally, it offers improved accuracy in predicting the fatigue life of materials subjected to both proportional and non-proportional loading paths.

(2) The model incorporates input parameters that simultaneously account for the load-time sequence characteristics, stress amplitude, mean value, frequency ratio, phase difference angle, and other key factors influencing the fatigue life of the material under multiaxial loading conditions. This comprehensive approach enables a more accurate representation of the material’s physical response to complex loading scenarios.

(3) The proposed prediction model is versatile and not restricted to specific materials or loading conditions. To evaluate its predictive capability, five distinct metal materials and various loading conditions were tested. The majority of the predicted data fell within 1.5 times the error band, and all data points remained within 2 times the error band, demonstrating the model’s strong predictive performance.