Multiaxial fatigue life prediction remains challenging because many components are subjected to complex, often nonproportional, loading. Such conditions arise from the superposition of stress and strain components due to phase shifts or differing loading frequencies 1-3. Despite the development of numerous models to address these effects, nonproportional loading continues to motivate new approaches in the literature 4, 5. Amid the ongoing publication of new multiaxial fatigue life prediction models, validating them under diverse loading conditions and across various materials has become equally important as developing novel approaches. Therefore, the objective of this study is to further assess the TMW model using experimental data distinct from that employed in the original publication 6. To this end, experimental results from the author's own investigations on four materials, as reported in 2, were utilized. The experimental data were obtained for PA38-T6 aluminum alloy, E235+N and E355+N nonalloy steels, and X5CrNi18-10+A austenitic steel. Fatigue tests were conducted using an axial–torsional system under strain control. The applied loading conditions encompassed fully reversed axial and torsional loading, in-phase and 90° out-of-phase combinations, as well as asynchronous cases. All experimental details are provided in 2. The data are available as raw signals, recorded for each loading cycle at a high sampling rate. This enables precise postprocessing to extract all relevant physical quantities. The plastic strain threshold was set to 0. 001 mm/mm for the steels and 0. 002 mm/mm for the aluminum alloy. Below these thresholds, the stress-based formulation was employed. The cycle counting procedure occasionally generated stress reversals of very low amplitude. To mitigate this issue, a condition was implemented: If Δ σ e q ∗ < σ 0 ₄ₐ^ <₀, the corresponding reversal was excluded from the damage accumulation process. Otherwise, applying the subtraction Δ σ e q − 2 σ 0 ₄ₐ-2₀ as in Equations (2) and (3) would result in artificially inflated values for small Δ σ e q ₄ₐ. A custom-developed MATLAB code was utilized to automate all computational procedures and to visualize the results. Application of the TMW model requires six parameters: ν, G, R s, w s, b, 0. 3em G, 0. 3em Rₛ, 0. 3em wₛ, 0. 3em b, and σ 0 ₀. ν and G G were taken from previous work 2. σ 0 ₀ was obtained by extrapolating the uniaxial S-N curve and was required only for PA38-T6. Typical Burgers vector values were taken for BCC ferrite, FCC aluminum alloys, and for FCC austenitic steels. The surface free energy w s wₛ is difficult to determine directly and was therefore estimated for PA38-T6, E235+N, and E355+N using values for the primary alloying element and similar materials 11-13, while for X5CrNi18-10+A, it was adopted from 14. The surface factor R s Rₛ is 1 for a smooth surface and 1/3 for a machined or ground surface 6, 8 but was calibrated here so that the effective surface energy w s ′ = R s w s wₛ^ =Rₛwₛ provided accurate axial fatigue life predictions. The material parameters are summarized in Table 1. The multiaxial fatigue life predicted by the TMW model is compared with experimental results in Figure 2. For E235+N steel, fatigue life is accurately predicted under proportional loading, with axial (TC), torsional (TOR), and in-phase (IP) data forming a narrow, slightly conservative band (Figure 2a). Fatigue life is overestimated by about a factor of 2 for 90° out-of-phase loading and slightly more for asynchronous loadings. For E355+N steel (Figure 2b), TOR and IP loadings are underestimated by factors of 2–3, while OP and selected asynchronous loadings (ASN2b, ASN3b) are predicted accurately; other asynchronous cases show slightly larger underestimation. For the aluminum alloy, most predictions fall within a factor-of-two scatter band, with lower accuracy for ASN1 and ASN5 (Figure 2c). The largest scatter occurs for X5CrNi18-10+A (Figure 2d), where TOR fatigue life is substantially underestimated relative to TC. Overall, nonproportional fatigue life is generally overestimated, typically within a factor of 3. Two main factors affect the fatigue life predictions: loading nonproportionality and the relative contributions of normal and shear components. In Figure 2b and especially Figure 2d, a clear discrepancy between TC and TOR results is observed. As shown in 15, early-stage cracks in X5CrNi18-10+A steel deflected from maximum shear to maximum normal strain planes, indicating a stronger influence of normal strain than assumed by the Huber–Mises criterion. For all materials, out-of-phase and asynchronous loadings are predicted to have longer fatigue lives than in-phase loadings, indicating systematic overestimation. This suggests that the TMW model does not fully capture nonproportional loading effects. Since Equations (1) and 2 depend only on Δ ε e q, p ₄ₐ, and Δ σ e q ₄ₐ, with other terms acting as scaling factors, any inadequacy of these equivalent parameters directly affects fatigue life predictions. In 6, Wu emphasized the importance of loading history effects. Using experimental data from asynchronous loading, the TMW model was validated beyond constant-amplitude conditions. Two approaches were proposed: summation of plastic strain ranges above the plastic strain threshold and application of the Wang–Brown cycle counting method to the equivalent stress history below the threshold. Both approaches yield similar results for asynchronous and out-of-phase loadings, indicating that loading history effects are adequately captured. The author declares no conflicts of interest.
Łukasz Pejkowski (Fri,) studied this question.