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Machine Learning-Assisted Sensitivity Analysis for Stochastic Fatigue Life Modeling of Metals
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Tran C. H. Nguyen1, N. Vu-Bac2, 3
International Journal of Mechanical System Dynamics | 2025, 5(3) : 481 - 494
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International Journal of Mechanical System Dynamics | 2025, 5(3): 481-494
• RESEARCH ARTICLE •
Machine Learning-Assisted Sensitivity Analysis for Stochastic Fatigue Life Modeling of Metals
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Tran C. H. Nguyen1, N. Vu-Bac2, 3
Affiliations
  • 1Reactor Center, Dalat Nuclear Research Institute, Dalat, Vietnam
  • 2Faculty of Civil Engineering, Ho Chi Minh University of Technology (HCMUT), Ho Chi Minh City, Vietnam
  • 3Vietnam National University Ho Chi Minh City, Linh Trung Ward, Ho Chi Minh City, Vietnam
doi: 10.1002/msd2.70024
Outline
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Predicting fatigue life with precision requires more than isolated evaluations of mechanical properties; it requires an integrated approach that captures the interdependencies between various parameters, including elastic modulus, tensile strength, yield strength, and strain-hardening exponent. Neglecting these correlations in sensitivity analyses can compromise prediction accuracy and physical interpretability. In this study, we introduce a dependency-aware sensitivity analysis framework, assisted by machine learning-based surrogate models, to evaluate the contributions of these mechanical properties to fatigue life variability. Tensile strength emerged as the most influential parameter, with significant second-order interactions, particularly between tensile and yield strength, highlighting the central role of coupled effects in fatigue mechanisms. By addressing these interdependencies, the proposed approach improves the reliability of fatigue life predictions and offers a solid foundation for the optimization of metallic components subjected to cyclic stresses.

fatigue life prediction  /  machine learning  /  multiaxial loading  /  parameter dependency  /  variance-based sensitivity analysis
Tran C. H. Nguyen, N. Vu-Bac. Machine Learning-Assisted Sensitivity Analysis for Stochastic Fatigue Life Modeling of Metals[J]. International Journal of Mechanical System Dynamics, 2025 , 5 (3) : 481 -494 . DOI: 10.1002/msd2.70024
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1 | Introduction
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Fatigue remains one of the most significant failure mechanisms in metallic structures, particularly for components subjected to cyclic loading [1]. Unlike static failures, fatigue is characterized by the progressive accumulation of damage, often culminating in sudden and catastrophic fractures after prolonged exposure to repeated stress cycles. This phenomenon is especially concerning in metallic components, which are foundational in aerospace, automotive, and civil engineering applications due to their high strength, ductility, and versatility [2, 3]. Metals, while advantageous for their load-bearing capacity, exhibit microstructural behaviors under cyclic stresses, such as slip band formation and grain boundary degradation, that render them particularly susceptible to fatigue-related damage [4, 5]. These failures frequently occur without visible precursors, under nominally safe stress levels, and are exacerbated by multiaxial loading conditions.
In reality, loading is rarely uniaxial. Multiaxial fatigue involves simultaneous stresses along different axes, introducing complex interactions between normal and shear stress that significantly influence crack initiation and propagation. The challenge is further amplified under nonproportional loading, where phase differences between stress components induce additional cyclic hardening and material anisotropy [6, 7]. For instance, in high-strength steels, out-of-phase torsional and axial loading with a 90° phase difference induces dislocation pile-ups and shear-driven slip band formation, accelerating fatigue damage [4, 8].
Early approaches to fatigue life prediction were grounded in classical theories [9]. Methods such as the local stress–strain [10], stress field intensity approach [11], critical plane method [6, 12, 13], energy-based approaches [14], and damage tolerance frameworks [2] formed the foundation of early research that retains its influence to this day. These models were later enhanced by incorporating multiaxial, nonproportional loading effects, and advanced numerical simulations [15-17]. In particular, the critical plane approach was adapted to account for nonproportional loading by introducing cyclic hardening models and stress path-dependent parameters [8]. Finite element-based simulations further enabled detailed evaluations of localized stress–strain responses, thereby improving the accuracy of predictions for complex geometries [18]. More recently, physics-informed neural networks also offer promising computational approaches for solving differential equations in mechanics [19]. While these developments have improved the ability to model multiaxial fatigue to a great extent, their reliance on simplifying assumptions, such as isotropic material behavior and parameter independence, limits their applicability to realistic and varying conditions. This gives rise to data-driven approaches capable of extrapolating patterns beyond parametric or semiparametric models [20-23].
Fatigue life is strongly influenced by environmental and operational factors. Elevated temperatures can accelerate crack propagation, as demonstrated in nickel-based superalloys under jet engine conditions, while marine environments exacerbate corrosion fatigue in structural steels [5]. The effects of loading path sensitivity have also been extensively studied, with approaches such as the critical plane method evaluating stress orientation and amplitude [6, 24]. Recent advancements include deep learning frameworks incorporating self-attention mechanisms to model the effects of loading history and temperature variations on multiaxial fatigue life [21], and studies examining load path sensitivity under nonproportional loading conditions [25], showing the necessity of their inclusion in fatigue life predictions. Despite this progress, the role of material property interactions, particularly under multiaxial and nonproportional loading, remains an underexplored area, warranting further investigation.
Material properties play an instrumental role in the fatigue resistance of metals. Elastic modulus (), for instance, governs stiffness and stress redistribution, while yield strength () defines the stress threshold at which plastic deformation begins [2]. Tensile strength () is closely tied to fatigue strength, particularly in high-strength alloys, where fracture is the dominant failure mechanism [26]. Poisson's ratio () affects lateral strain and shear stress redistribution [27]. However, these properties do not act independently; instead, they are often strongly correlated due to shared metallurgical processes, including alloying and heat treatment [4]. For instance, yield strength and tensile strength in structural steels often correlate strongly (coefficients 0.85), reflecting shared microstructural attributes, like, grain size and dislocation density. In aluminum alloys, elastic modulus variations influence stress fields via interactions with Poisson's ratio, affecting crack propagation [27]. These interdependencies complicate efforts to isolate the individual effects of material properties on fatigue life, highlighting the need for integrated modeling approaches capable of addressing these correlations [28-30].
Sensitivity analysis (SA) aims to identify the influence of input parameters on model outputs. Variance-based global SA methods, such as those developed by Sobol [31] and refined by Saltelli et al. [32], have been widely applied in engineering for their ability to quantify both individual parameter effects and interactions. However, traditional SA methods assume parameter independence, which is often invalid in fatigue modeling where material properties exhibit strong correlations as described above. Neglecting parameter dependencies can lead to significant inaccuracies, as Vu-Bac et al. demonstrated in their modeling of Hi-Nicalon bundles, where accounting for correlations between stress and structural variations yielded more accurate fatigue predictions [28]. Similarly, their work on polymeric nanocomposites highlighted how capturing correlations among microstructural parameters strongly reinforces the reliability of mechanical property predictions [33]. Dependency-aware frameworks, such as those proposed by Kucherenko et al. [34] and Xu and Gertner [35], overcome these limitations by preserving the correlation structure of the inputs throughout the analysis. Alternative approaches like nonintrusive stochastic isogeometric analysis [36] have also addressed material uncertainty, though using continuity-preserving basis functions rather than our machine learning (ML) approach that may have an advantage in handling the irregular sampling and complex parameter dependencies present in multiaxial fatigue data. Additionally, Bayesian methodologies [37] provide probabilistic model assessment by updating prior knowledge, whereas our variance-based approach offers more direct quantification of input contributions, which is particularly advantageous when analyzing multiple correlated material properties under varied loading conditions.
The computational demands of fatigue life prediction further complicate sensitivity analyses. High-fidelity finite element analysis methods are often required to accurately simulate nonlinear behavior and multiaxial loading effects, but conducting the thousands of simulations needed for global SA is computationally prohibitive. Surrogate models, such as Random Forest (RF), Gaussian Processes, Artificial Neural Networks, and Extreme Gradient Boosting (XGBoost), provide an efficient alternative by approximating the behavior of high-fidelity models with high accuracy [38-42]. These models have been successfully applied in aerospace and mechanical systems to facilitate large-scale sensitivity studies and optimization [43-45]. In the context of fatigue modeling, surrogate models enable the integration of SA frameworks, particularly those addressing parameter correlations [28].
This study integrates ML-based surrogate models with dependency-aware SA to investigate the influence of material properties on multiaxial fatigue life. Using the data set collected by Chen et al. [26], which includes 1167 samples across 40 materials and 48 loading paths, we analyze the sensitivity of elastic modulus, yield strength, tensile strength, and Poisson's ratio to fatigue life. Surrogate models, including RF and XGBoost, are employed to approximate complex input–output relationships, enabling efficient sensitivity analyses. Dependency-aware SA frameworks, adapted from [34, 46], are used to capture the interdependencies among material properties, offering a deeper understanding of their combined effects on fatigue resistance.
The remainder of this paper is organized as follows. Section 2 describes the mechanical model and data set used in this study. Section 3 details the ML models and validation methods for surrogate modeling. Section 4 presents the SA framework, including dependency handling. Results and discussion are provided in Section 5, comparing model performances and sensitivity indices. Finally, Section 6 summarizes the findings and provides concluding remarks.
2 | Data Description
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2.1 | Material Properties and Their Physical Significance
The mechanical properties of materials play a central role in fatigue performance under cyclic loading [3, 4]. Here, we assess four particularly influential properties: elastic modulus (), tensile strength (), yield strength (), and Poisson's ratio (). Each governs specific mechanisms underlying fatigue crack initiation, propagation, and failure, as detailed below.
Elastic modulus quantifies stiffness, controlling resistance to elastic deformation [3]. During high-cycle fatigue, where deformation is predominantly elastic, regulates strain energy storage and dissipation, which influences the stress distribution around defects. This behavior is described by
where is the applied stress, is the elastic modulus, and is the elastic strain.
Figure 1A shows that materials in strain-controlled tests have higher values (clustered near 200 GPa), enabling them to endure significant cyclic strains. Conversely, stress-controlled tests focus on materials with GPa, reflecting their aptitude for elastic-dominated fatigue scenarios.
Tensile strength () measures the maximum stress a material withstands before failure and is critical for low-cycle fatigue involving substantial plastic deformation. The relationship between stress amplitude () and fatigue life () is expressed by Basquin's law:
where is the fatigue strength coefficient, and the fatigue strength exponent [47].
Figure 1B demonstrates that strain-controlled tests cover a wider range, indicating the use of diverse materials under severe loading. In contrast, stress-controlled tests cluster near MPa, indicative of materials optimized for elastic behavior.
Yield strength () defines the onset of plastic deformation. Cyclic loading near induces localized plasticity, forming persistent slip bands that often act as precursors to fatigue cracks [48]. The plastic stress–strain relationship can be represented as
where is the cyclic strength coefficient, is the plastic strain, and is the strain-hardening exponent.
As shown in Figure 2A, materials used in strain-controlled tests typically exhibit MPa, enabling them to sustain significant plastic deformation. Conversely, stress-controlled tests involve materials with MPa, reflecting elastic-dominated conditions.
Poisson's ratio () characterizes the coupling between lateral and axial deformations. This property is particularly significant in multiaxial fatigue, where it dictates shear stress effects and crack propagation under nonproportional loading [4, 6]. The shear modulus () is related to by
where is the elastic modulus, and is Poisson's ratio. Figure 2B shows that clusters near 0.3 for both test types, highlighting the prevalence of ductile materials in the data set.
The data set distinguishes between stress- and strain-controlled test regimes:
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Stress-controlled tests: Materials with lower , and dominate, reflecting conditions suited for high-cycle fatigue with predominantly elastic deformation (Figures 1 and 2).

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Strain-controlled tests: Higher values of , and characterize materials designed for low-cycle fatigue with significant plastic deformation (Figures 1 and 2).

Understanding the distributions of – (Figures 1 and 2) is conducive to generating representative samples and interpreting SA results (Section 5.2).
2.2 | Loading Path Characterization
Loading paths are fundamental to understanding fatigue behavior under cyclic loading as they directly influence stress distribution, crack initiation, and material response [6, 8, 25]. Here, each loading cycle in the data set is vectorized as a time series of 241 points, capturing the evolution of stress or strain components with high resolution. Stress-controlled tests are defined by axial stress and shear stress , while strain-controlled tests involve axial strain and shear strain . This representation encodes critical features, such as amplitude, phase, and path geometry.
Three primary loading path classifications, uniaxial, proportional multiaxial, and nonproportional multiaxial, are included in the data set (Table 1):
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Uniaxial paths: Cyclic deformation occurs along a single axis, serving as a benchmark for evaluating fundamental fatigue mechanisms.

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Proportional paths: Axial and shear components maintain a fixed phase relationship, resulting in linear trajectories in stress or strain space. These paths are key for assessing materials under stable phase conditions.

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Nonproportional paths: Characterized by variable phase relationships, these paths produce complex trajectories (e.g., circular or elliptical). Subdivided into six types (I-VI), nonproportional paths induce principal stress rotations and activate additional hardening mechanisms [8, 17].

Nonproportional paths, in particular, represent the most demanding scenarios for fatigue modeling due to the intricacies of interaction between stress rotations and multiaxial damage mechanisms [15]. Principal stress rotations amplify cyclic hardening and significantly alter crack propagation trajectories. Their inclusion in the data set is necessary to maintain comprehensive coverage of multiaxial fatigue phenomena.
Surrogate models rely on high-quality input–output relationships to approximate computationally expensive simulations [38, 45]. By incorporating diverse and realistic loading conditions, the data set strengthens the ability of these models to capture the nuances of fatigue behavior across varied stress states.
Table 1 illustrates the representative patterns of each classification, providing insights into the underlying stress or strain trajectories. This diversity in loading paths demonstrates the robustness and expansiveness of the data set, crucial for accurate surrogate model development.
Note that loading paths are visualized in the stress/strain space, where the horizontal axis represents axial stress/strain, and the vertical axis represents shear stress/strain. Each path represents one complete loading cycle, encapsulating the trajectory of multiaxial deformation.
2.3 | Data Set Description and Statistical Information
The data set used in this study contains 1167 multiaxial fatigue test samples from 40 different metallic materials, including stainless steels, aluminum alloys, titanium alloys, and nickel-based superalloys [26]. These samples cover 48 distinct multiaxial loading paths and include both stress- and strain-controlled conditions. This data set provides a comprehensive foundation for studying the mechanical and fatigue properties of metals under various conditions.
The data set's construction involved an extensive literature review, with 36 studies selected from over 70 sources. Selection criteria emphasized high-quality experimental data and complete reporting of material properties and loading conditions. Most tests were conducted at room temperature, with the exception of GH4169 and Hayes alloys, which underwent high-temperature testing [26].
Figure 3 shows the correlations among the four mechanical properties in consideration: elastic modulus (), tensile strength (), yield strength (), and Poisson's ratio (). Tensile strength and yield strength ( and ) are strongly correlated (), reflecting their shared dependence on material composition and processing, such as heat treatment and alloying. These properties play a key role in fatigue resistance, particularly in controlling plastic deformation and delaying crack initiation [49].
Elastic modulus () and Poisson's ratio () exhibit weaker correlations with other properties and with each other. This suggests that they contribute to fatigue behavior independently. Elastic modulus governs stress distribution and elastic deformation, while Poisson's ratio affects how materials deform laterally under axial loading. These distinct roles will be further illuminated through SA results.
Figure 4 illustrates the distributions and pairwise relationships among , and , separated by stress- and strain-controlled test conditions. Materials used in strain-controlled tests generally have higher tensile strength () and yield strength (), allowing them to endure significant plastic deformation during low-cycle fatigue. In contrast, stress-controlled tests focus on materials with lower elastic modulus (), which are better suited for high-cycle fatigue where elastic deformation dominates.
3 | Surrogate Models for Fatigue Life Prediction
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In this study, we use RF and XGBoost as surrogate models to predict fatigue life () under multiaxial loading. Both models are capable of capturing complex, nonlinear interactions among mechanical properties while providing computational efficiency that makes them advantageous over traditional physics-based simulations [50-52]. Their formulations are detailed below.
3.1 | Random Forest
RF is an ensemble learning algorithm that constructs decision trees using randomly bootstrapped subsets of the data set [50]. Each tree is built recursively, splitting the feature space to minimize impurity. For regression tasks, impurity is minimized by selecting the feature and threshold at each node that solves
where and are the number of samples in the left () and right () child nodes, respectively, and represents the variance of the target values in a given node.
The prediction from a single tree is denoted as , where is the tree index. The final RF prediction aggregates the outputs from all trees:
RF's ability to model nonlinear interactions and rank features by their contribution to variance reduction makes it an appropriate tool for fatigue life prediction [42].
3.2 | Extreme Gradient Boosting
XGBoost builds an ensemble of trees sequentially, where each tree corrects the residual errors of the previous iteration [51]. The model at iteration is given by
where is the learning rate, and represents the tree added at iteration .
XGBoost optimizes a regularized objective function:
where is the loss function (e.g., squared error for regression tasks), and is the regularization term which controls model complexity, and is defined as
where is the number of leaves in the tree, represents the weight of leaf penalizes the number of leaves, and penalizes large weights to control overfitting.
At each iteration, XGBoost employs a second-order Taylor approximation of the loss function:
where and are the first and second derivatives of the loss function with respect to the previous prediction .
The optimal weights for each leaf in the tree are computed analytically by minimizing :
where is the set of samples assigned to leaf . This gradient-based refinement allows XGBoost to iteratively improve predictions and spot intricate patterns in data.
3.3 | Model Training and Hyperparameter Optimization Process
Both RF and XGBoost models were trained using mechanical properties () as inputs, with fatigue life () as the target. Preprocessing included log transformation of the target variable for normalization, standardization of features for consistent scaling. Data set partitioning into training (70%), validation (15%), and testing (15%) subsets was done to achieve robust and generalizable models. Additionally, hyperparameter optimization was performed using a combination of heuristic methods and grid search.
For RF, key hyperparameters included the number of trees (), features per split (), and minimum samples per leaf. Grid search across (in increments of 100) showed optimal performance at . Features per split were set to , and was selected to balance complexity and variance reduction.
For XGBoost, the learning rate (), maximum depth (), regularization (), and subsampling ratio were optimized. Coarse grid search and fine-tuning identified , and as the best configuration. Early stopping was employed, terminating training when validation performance did not improve for 20 iterations.
3.4 | Performance Metrics
Model performance was evaluated on the test set using the following metrics:
Here, quantifies the proportion of variance explained by the model ( is set as the minimum threshold for acceptable accuracy) [46]. Root mean squared error (RMSE) and mean absolute error (MAE) capture the average prediction error, while relative deviation (RD) provides a scale-independent measure of deviation.
A potential point of contention that should be addressed is the use of more traditional ML models (RF and XGBoost) over demonstrably excellent deep learning models, such as long short-term memory and gated recurrent unit [26]. However, their deployment as surrogate models in this study poses several practical challenges. First, deep learning models typically require substantial computational resources for training and inference, making them less suitable for rapid SA, where model evaluations are performed thousands of times. In contrast, RF and XGBoost offer comparable performance with significantly reduced computational overhead during both training and prediction. Second, the relatively small size of the data set (1167 samples) limits the advantage deep learning might provide, as such models are inherently data-hungry and prone to overfitting when the training data is scarce. RF and XGBoost are robust to smaller data sets, effectively leveraging their ensemble architectures to prevent overfitting. Finally, the interpretability of RF and XGBoost, through feature importance and SHAP values, aligns well with the objectives of SA, which require clear insight into input–output relationships. These considerations collectively make RF and XGBoost more efficient and practical choices for this study.
These surrogate models make it possible to quickly assess fatigue life across diverse conditions, facilitating not global SA, but also optimization tasks, and uncertainty quantification [43, 44], thereby bridging the gap between computational and experimental fatigue studies.
4 | SA Framework for Dependent Variables
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This section presents the SA framework tailored for fatigue life prediction models with dependent input variables. Following methodologies from [30, 34, 53, 54], the approach adapts traditional Sobol's indices to account for input correlations, enabling a more accurate quantification of variance contributions.
4.1 | Incorporating Input Dependencies
Material properties in fatigue life modeling exhibit significant correlations, as demonstrated in Section 2.1. Ignoring these dependencies can yield misleading sensitivity indices. To address this, the following steps incorporate these dependencies directly into the SA framework.
First, the correlation matrix is derived from the empirical data set, encapsulating the relationships among the input variables:
where each element denotes the Pearson correlation coefficient between and :
This matrix reflects the empirical relationships between material properties, ensuring that the input space retains realistic correlations.
The correlation matrix must be symmetric and positive definite for mathematical operations, including variance decomposition and Sobol's sensitivity index computation. If numerical errors render nonpositive definite, it is corrected using the nearest positive definite matrix algorithm [34]:
where is the identity matrix, and is a small constant ensuring positive definiteness.
Once the matrix is validated as positive definite, Cholesky decomposition is applied:
where is a lower triangular matrix. The elements of are computed as
This decomposition facilitates stable numerical operations required for SA.
Unlike synthetic sampling approaches, such as Latin Hypercube Sampling, this method directly uses the empirical data set. Material properties and their inherent correlations are preserved without transformation into standard normal space. This ensures that the SA reflects the actual experimental data.
4.2 | Variance Decomposition Using ANOVA
The variance decomposition of the output serves as the foundation for SA. The total variance is partitioned into contributions from individual inputs and their interactions. For input variables , the function is expressed as
where
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is the mean value of ,

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is the first-order contribution of ,

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represents the interaction between and , and so on.

This decomposition is grounded in the principles of ANOVA, as detailed in [53, 54], and maintains orthogonality, such that:
The total variance is then decomposed as
where
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is the first-order contribution of ,

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is the interaction variance between and .

The total variance is the sum of these contributions:
4.3 | Sobol's Sensitivity Indices
Sobol's indices quantify the contributions of inputs and their interactions to the total variance [34]:
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The first-order Sobol's index :

measures the contribution of Xi alone.

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The total-effect Sobol's index :

represents the overall contribution of Xi, including higher-order interactions.

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The second-order Sobol's index :

quantifies the interaction effect between Xi and Xj.

4.4 | Monte Carlo (MC) Estimation of Sobol's Indices
MC integration is used to estimate Sobol's indices by systematically sampling the input space and evaluating variance contributions [46, 53]. For a given model , two sample sets are required:
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Baseline samples (): Independently generated input samples.

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Perturbed samples (): Constructed by replacing the th column of with the corresponding column from another set :

The first-order index , representing the contribution of to the total variance, is approximated as
where

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: Model output for the th baseline sample.

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: Model output for the th perturbed sample.

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: Total variance over the baseline samples.

The total-effect index , which captures the combined contribution of and all its interactions, is estimated as
The numerator quantifies variance induced by perturbations of , isolating its total effect.
4.5 | Integration With Surrogate Models
Given the computational cost of high-fidelity models, a surrogate model (e.g., XGBoost) is trained to approximate (see Section 3). The surrogate's performance is validated using metrics, such as , RMSE, and MAE. Once validated, the surrogate replaces the computationally expensive model for efficient computation of sensitivity indices [30].
This framework ensures that the correlations among material properties are accounted for, improving the physical interpretability of the sensitivity indices. The quantified indices in subsequent sections will illustrate how these correlations influence fatigue life predictions via both direct contributions and interaction effects.
4.6 | Implementation Procedure
Below is a streamlined summary of the SA workflow, consolidating the detailed steps previously outlined for clarity and reference, for more details see [34, 46, 53, 54].

Step 1: Data set preparation and analysis

Utilize the existing data set containing fatigue life and mechanical properties.

Extract relevant variables, ensuring input–output consistency.

Step 2: Transforming dependent inputs

Extract the correlation matrix Σ directly from the data set.

Validate the positive definiteness of Σ using Cholesky decomposition.

Ensure transformed samples align with the dependent input structure.

Step 3: Surrogate model construction and validation

Partition the data set into training and testing subsets.

Train a surrogate model, for example, XGBoost, to approximate f(X).

Validate the surrogate model using metrics, such as R2, RMSE, and MAE.

Step 4: Variance decomposition using ANOVA

Partition the surrogate model output variance into contributions.

Compute orthogonal components:

First-order variance . Interaction variance .

Step 5: MC estimation of sensitivity indices

Apply MC sampling to compute Sobol's indices: .

Compute total-effect indices: .

Use dependent input samples to estimate higher-order indices.

This workflow integrates the data set's inherent dependency structure, reduces computational costs through surrogate modeling, and ensures reliable computation of sensitivity indices. Each step builds sequentially, emphasizing accuracy and efficiency.
5 | Results and Discussion
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5.1 | Performance of Surrogate Models
Both surrogate models demonstrated strong predictive capabilities for fatigue life estimation, with XGBoost showing slightly better performance across all evaluation metrics (Table 2). The XGBoost model achieved an R2 of 0.9258 and RMSE of 0.2006, compared with RF's R2 of 0.8973 and RMSE of 0.2355. The relative difference of 4.11% for XGBoost indicates excellent prediction accuracy across the range of fatigue life values.
The parity plots (Figure 5) illustrate consistent performance between the training and test sets, with most predictions except one falling within the factor of 1.5 scatter bands. Both models maintain good accuracy across the fatigue life range, though slightly higher scatter is observed at extreme values. The XGBoost model's superior performance can be attributed to its gradient boosting architecture and regularization techniques making use of second-order derivatives to capture nonlinear relationship in fatigue life prediction.
The achieved performance metrics indicate that both surrogate models can serve as reliable replacements for computationally expensive finite element simulations in subsequent sensitivity analyses. However, the XGBoost model, with its higher accuracy and lower prediction errors, was selected as the primary surrogate model.
5.2 | Sensitivity Analysis
The SA quantifies the influence of material properties () on the variability of fatigue life predictions, while incorporating dependencies among inputs. This approach provides a more nuanced and physically grounded understanding of how material characteristics contribute to fatigue performance under multiaxial loading. Figures 6 and 7 present the first-order () and total-effect () sensitivity indices for dependent and independent inputs, respectively.
5.2.1 | Dominant Parameter Influence
The analysis reveals that (tensile strength) is the dominant parameter influencing fatigue life, with the highest first-order sensitivity index () in the dependent case. Tensile strength directly controls a material's ability to resist cyclic loading by influencing both crack initiation and propagation phases. High tensile strength materials can sustain larger stress amplitudes before failure, which explains 's significant impact on fatigue life variability. This result is consistent with prior findings that link tensile strength to fatigue resistance across various metallic materials [6, 8].
Interestingly, (elastic modulus), which governs stiffness and stress redistribution, exhibits negligible first-order sensitivity () in the dependent case. This observation highlights the interplay between stiffness and other properties, such as and , which overshadow the isolated effect of . While elastic modulus influences local stress fields and energy storage during loading, its contribution to fatigue life variability is minimal when considered independently.
5.2.2 | Interaction Effects
The results highlight the importance of parameter interactions, particularly second-order effects, in determining fatigue life. As shown in Figure 8, the interaction between (tensile strength) and (yield strength) is the most significant (). This interaction reflects the synergistic role of these properties in controlling plastic deformation mechanisms and crack growth. Tensile strength sets the material's resistance to cyclic stresses, while yield strength determines the stress threshold for plastic deformation. Together, they govern the material's response to multiaxial loading, particularly under conditions involving significant strain localization [1].
Another notable interaction involves (elastic modulus) and (strain-hardening exponent) (). This combination influences stress redistribution and energy dissipation during cyclic loading. Strain hardening enhances the material's ability to resist localized plasticity, while elastic modulus dictates the stiffness of the surrounding matrix. Their interaction directly affects the redistribution of stress near cracks or defects.
5.2.3 | Comparison With Independent Inputs
To evaluate the impact of accounting for input dependencies, sensitivity indices were computed for both dependent and independent cases. Figure 7 and Table 3 highlight the significant differences in the results. In the independent case, (elastic modulus) is incorrectly identified as the most influential parameter (). This misrepresentation arises from neglecting the correlations among material properties, which artificially inflates the isolated contribution of . Physically, stiffness primarily moderates stress distributions and does not directly drive fatigue life variability [2].
Conversely, the dependent case correctly identifies (tensile strength) as the dominant factor. The total-effect index for (yield strength) and (strain-hardening exponent) also decreases in the dependent case, reflecting their reduced individual contributions once correlations with are accounted for. This underscores the importance of accounting for input dependencies to achieve accurate and physically consistent SA [34, 46].
The findings here provide the following key insights:
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The dominant influence of aligns with its established role in fatigue resistance, emphasizing the need for precise tensile strength characterization in fatigue modeling.

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Strong second-order interactions, particularly between and , highlight the importance of considering coupled effects rather than isolating individual parameters.

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Ignoring input dependencies leads to misleading conclusions, as demonstrated by the overestimated influence of in the independent case. This reinforces the need for dependency-aware methodologies in SA.

By accounting for correlated inputs, this analysis enhances the physical interpretability of sensitivity indices and provides a more reliable foundation for fatigue life predictions in multiaxial loading scenarios.
6 | Conclusions
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A framework for multiaxial fatigue life assessment for metallic components was developed by combining ML-based surrogate models with SA accounting for dependent inputs. The main findings include:
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The XGBoost model achieved the highest predictive accuracy (, RMSE = 0.2006), demonstrating its effectiveness in modeling the nonlinear relationships inherent in multiaxial fatigue data. Thus, it was considered sufficient as a surrogate model.

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Tensile strength () was identified as the most influential parameter, with the highest first-order sensitivity index. Its critical role in fatigue life prediction is consistent with its established effect on cyclic stress resistance.

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Second-order interactions, especially between tensile strength () and yield strength (), were significant, reflecting the interdependence of material properties in fatigue mechanisms.

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Comparing dependent and independent input scenarios revealed that ignoring parameter correlations distorts sensitivity indices, overestimating the influence of properties, like, elastic modulus ().

These insights highlight the integral role of incorporating dependency-aware SA to achieve reliable and physically interpretable results in fatigue modeling. Future work can extend this framework to include additional factors such as microstructural effects or environmental influences for broader applicability.
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Article Info
doi: 10.1002/msd2.70024
  • Receive Date：2025-01-15
  • Online Date：2026-03-24
Article Data
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History
  • Received：2025-01-15
  • Revised：2025-03-04
  • Accepted：2025-03-25
Affiliations
    1Reactor Center, Dalat Nuclear Research Institute, Dalat, Vietnam
    2Faculty of Civil Engineering, Ho Chi Minh University of Technology (HCMUT), Ho Chi Minh City, Vietnam
    3Vietnam National University Ho Chi Minh City, Linh Trung Ward, Ho Chi Minh City, Vietnam

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N. Vu-Bac ()
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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