Two-Stage Entropy-Based Constitutive Modeling for Rapid Fatigue Life Prediction

Fatigue failure tends to begin invisibly within the metal and in bridges, aircraft, or welded joints, these hidden changes gradually coalesce into cracks that can end a structure’s service life without warning. The traditional staircase and group methods still dominate how endurance limits and S–N curves are determined, but they’re cumbersome and each dataset takes weeks or months to obtain. For modern engineering, where time and efficiency matter, such slow testing methods feel increasingly outdated. Thermodynamic approaches, especially those linking temperature evolution to fatigue damage, opened a new pathway. Infrared thermography allowed fatigue evaluation through self-heating behavior during cyclic loading. The underlying idea—that thermal energy dissipation mirrors the microstructural evolution—led to temperature-based estimation of fatigue limits. However, temperature alone is unreliable because it is easily influenced by environmental and surface factors, and is limited in providing information on the irreversible thermodynamic mechanisms underlying fatigue.

In conclusion, the new innovative work of Professor Wei Wei and colleagues deliver a physically interpretable, fast, and statistically reliable framework for evaluating fatigue behavior across diverse metallic systems. Indeed, the improved model refines our understanding of energy dissipation mechanisms and introduces a robust pathway to predict fatigue life from minimal experimental input. Its validation across steels and welded joints underscores its material generality and its compatibility with both classical and infrared-based measurement schemes. We believe this is important work because introduction of two characteristic stress amplitudes resolves a long-standing ambiguity in distinguishing harmless reversible motion from irreversible damage accumulation. This clarification permits direct thermodynamic interpretation of the fatigue limit, which now emerges naturally from entropy–stress relationships rather than from empirical S–N data. Second, the incorporation of nonlinear response terms allows the model to remain accurate across wide stress ranges, unlike bilinear predecessors that faltered near transition regimes. Consequently, the improved constitutive form mirrors the real mechanical behavior of metals under cyclic loading with far greater fidelity. Moreover, the proposed statistical framework marks another major advancement. The coupling of the entropy-based constitutive model with a confidence-adjusted maximum-likelihood formulation enables construction of P–S–N curves reflecting both survival probability and confidence level. Such integration of physics and statistics transforms fatigue assessment from a largely empirical practice into a predictive, quantifiable discipline. In industrial applications, the new approach can significantly accelerate qualification of new alloys and welded structures while maintaining scientific rigor. Ultimately, the work led by Professor Wei Wei demonstrates how an elegant theoretical refinement, grounded in thermodynamics and verified by careful experimentation, can yield a practical, rapid, and physically meaningful tool for modern fatigue design and provide a bridge between microscopic physics and macroscopic engineering reliability.

Entropy-based theories subsequently emerged as a more fundamental framework and if we describe fatigue in terms of energy dissipation and entropy production we can connect mechanical work to irreversible microstructural motion. Entropy generation effectively quantifies disorder produced by anelastic and inelastic processes: the former reversible and non-damaging, the latter responsible for irreversible plasticity and crack initiation. Previous studies demonstrated that fatigue life could be correlated with the accumulation of inelastic entropy, while reversible contributions from internal friction could be separated analytically. However, despite these advances, most available models assumed bilinear relations between entropy generation and stress amplitude, which poorly captured the gradual transition from elastic to plastic regimes. To this account, new research paper published in European Journal of Mechanics – A/Solids, and led by Professor Wei Wei, Jianhui Li, Dr. Dan Lin and Dr. Fufa Wu from the Liaoning University of Technology alongside Dr. Xinhua Yang from the Dalian Jiaotong University, the researchers developed an improved two-stage constitutive model that couples linear and nonlinear entropy generation with microstructural motion under cyclic loading. The model introduces two critical stress amplitudes (Σc0, Σc1) to distinguish recoverable anelasticity from damage-inducing inelasticity, enabling accurate fatigue limit identification. They also formulated a rapid fatigue-life prediction and P–S–N curve estimation method that integrates confidence-level statistics with entropy-based damage variables.

The authors first reformulated the thermodynamic entropy balance by expressing total entropy generation as a combination of two microstructural mechanisms: reversible anelasticity and irreversible inelasticity. They built upon Guo’s previous constitutive framework and introduced two critical stress amplitudes—Σc0 and Σc1—to distinguish recoverable motion from damage-inducing plastic deformation and found that below Σc0, the entropy response remains linear with stress amplitude, reflecting harmless anelastic oscillations near dislocation lines but once stress surpasses Σc1, entropy generation exhibits a pronounced nonlinear increase, marking the onset of fatigue damage. Between these thresholds, both mechanisms coexist, resulting in a smooth transition captured by the new constitutive expression. The team developed a systematic fitting procedure using the coefficient of determination (R²) to isolate the linear range, followed by logarithmic transformation and correlation optimization to determine nonlinear coefficients μin, m, and Σc1. Entropy generation data were obtained from infrared thermography under cyclic loading, complemented by datasets from prior literature. Three representative materials were examined: Q310NQL2–Q345NQR2 welded joints, 304L stainless steel, and Q235 structural steel. The researchers employed two computational routes for entropy generation—ρcRθ/Ts and ρcΔT/τeq—to verify model generality across measurement methodologies.

The authors performed experimental fatigue tests using a PLG100 servo-hydraulic machine at a stress ratio of 0.1 and a frequency of 120 Hz. They recorded surface temperatures by a high-precision infrared camera, while specimens were coated with matte black paint to ensure consistent emissivity. They reported that across sixteen stress amplitudes ranging from 80 to 150 MPa, self-heating curves, it revealed the characteristic three-phase thermal response typical of high-cycle fatigue. Entropy production rates were then extracted and fitted to the model using least-squares optimization. It is important to mention, the team’s findings demonstrated excellent agreement between the theoretical predictions and experimental data and for all tested materials, the model successfully captured the inflection point marking the fatigue limit and reproduced the full nonlinear entropy–stress behavior. The fitted exponents (m ≈ 2.3) remained nearly constant across materials and entropy-computation methods, implying a universal scaling law for metallic systems. Using the entropy-derived damage variable, the researchers further constructed S–N curves that closely matched experimental lifetimes. Finally, by incorporating confidence-level statistics via a maximum-likelihood approach, they proposed a rapid method to estimate P–S–N curves with defined survival probabilities which required only a small set of tests to achieve statistically reliable predictions.

In conclusion, the new innovative work of Professor Wei Wei and colleagues deliver a physically interpretable, fast, and statistically reliable framework for evaluating fatigue behavior across diverse metallic systems. Indeed, the improved model refines our understanding of energy dissipation mechanisms and introduces a robust pathway to predict fatigue life from minimal experimental input. Its validation across steels and welded joints underscores its material generality and its compatibility with both classical and infrared-based measurement schemes. We believe this is important work because introduction of two characteristic stress amplitudes resolves a long-standing ambiguity in distinguishing harmless reversible motion from irreversible damage accumulation. This clarification permits direct thermodynamic interpretation of the fatigue limit, which now emerges naturally from entropy–stress relationships rather than from empirical S–N data. Second, the incorporation of nonlinear response terms allows the model to remain accurate across wide stress ranges, unlike bilinear predecessors that faltered near transition regimes. Consequently, the improved constitutive form mirrors the real mechanical behavior of metals under cyclic loading with far greater fidelity. Moreover, the proposed statistical framework marks another major advancement. The coupling of the entropy-based constitutive model with a confidence-adjusted maximum-likelihood formulation enables construction of P–S–N curves reflecting both survival probability and confidence level. Such integration of physics and statistics transforms fatigue assessment from a largely empirical practice into a predictive, quantifiable discipline. In industrial applications, the new approach can significantly accelerate qualification of new alloys and welded structures while maintaining scientific rigor. Ultimately, the work led by Professor Wei Wei demonstrates how an elegant theoretical refinement, grounded in thermodynamics and verified by careful experimentation, can yield a practical, rapid, and physically meaningful tool for modern fatigue design and provide a bridge between microscopic physics and macroscopic engineering reliability.