PSI - Issue 82

Sigbjørn Tveit et al. / Procedia Structural Integrity 82 (2026) 112–118 S. Tveit et al. / Structural Integrity Procedia 00 (2026) 000–000

113

2

cycle-counting techniques to convert multiaxial stress histories into a set of distinct stress cycles, the continuum mechanics fatigue model operates directly on the time-continuous stress tensor history. Using consistent loading unloading conditions and an endurance surface that moves in the stress space according to a back-stress tensor, the model allows a fatigue damage scalar to accumulate continuously in the time-domain until a critical value is reached, indicating that a macroscopic fatigue crack has formed. The format in Ottosen et al. (2008) captures the most important features of a successful multiaxial fatigue model; the fatigue limit depends on the magnitude of the stress amplitude, and the limit is sensitive to the mean value of normal stresses and invariant to the mean stress of shear stresses. The model was enhanced by Ottosen et al. (2018) to account for the stress gradient effect, and recently, Tveit et al. (2025) published a formulation that generalized the model to account for the difference in the shear stress fatigue limit ratio, i.e. !" !" ⁄ , where !" and !" are the infinite life fatigue limits of fully reversed shear and normal stress ( = #$% #&' ⁄ =−1) , respectively. In addition to this, a variety of specialized formats have been proposed in the literature, including the anisotropic formulation by Kouhia et al. (2025) and the series of endurance surfaces proposed in Lindström (2020) and Lindström et al. (2023) to provide a more accurate description of the mean stress effect. While these advances have greatly broadened the model’s applicability, its accuracy for certain nonproportional stress histories remains under scrutiny. Lindstrom et al. (2020) investigated the model’s ability to describe the finite fatigue lives of AA7050-T451 specimens subjected to nonproportional combinations of cyclic shear and normal stresses. For the specific stress history that was examined, a phase shift of 90º between the stress components resulted in an overestimation of the predicted fatigue life. Despite lack of experimental data that matched the specific scenario, the authors concluded that the prediction qualitatively disagreed with expectations. A similar investigation was carried out by Lilja et al. (2020), who used a slightly altered formulation and parameter set. This time, the 90º phase shift resulted in an underestimation of the fatigue life. Since then, various calibration methods have been employed and satisfying results have been obtained by controlling the value of the model’s back-stress parameter, see e.g. Rubio Ruiz et al. (2023) and Lindström et al. (2025). While the above discussion relates to finite fatigue lives, Tveit et al. (2024) tested the model’s ability to predict the multiaxial infinite fatigue limits of ductile materials. In this domain, the model behavior is unaffected by the above mentioned back-stress parameter. Despite this, the model was prone to nonconservative mispredictions for certain out-of-phase bending-torsion-combinations. In this work, we shall examine the continuum mechanics fatigue model’s ability to predict the strength of various metals under combined alternating bending and torsion with phase difference, by comparing the results to the infinite fatigue limit data presented by Nishihara and Kawamoto (1945). The following section provides a brief introduction to the continuum mechanics fatigue model, before the predictions for mild steel, hard steel, duraluminum, and 3%C cast iron are presented along with the experimental data in Section 3. Finally, we provide some concluding remarks. 2. Continuum mechanics fatigue model The endurance surface = 0 represents the boundary between safe and unsafe stress configurations in the stress space. We adopt the generalized format proposed by Tveit et al. (2025) where is taken as = 1 !" +,3 ̅ ( − 11 − 247 ̅ ) ( ̅ ( ) 5 + " − !" 9 (1) Here, σ !" is the normal stress fatigue limit for = −1 , is the mean stress sensitivity. The parameter λ = √3 !" − !" accounts for the effect of the shear stress fatigue limit ratio, !" !" ⁄ , and it is worth noting that Eq. (1) reduces to the original formulation by Ottosen et al. (2008) when λ=0 , which corresponds to !" !" ⁄ =1⁄√3 . The invariants " , ̅ ( , and ̅ ) are defined as " = ( ); ̅ ( = 12 D : D ; ̅ ) = ( D ) (2)