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

115

4

3. Shear-normal stress combinations with phase differences: Infinite fatigue life As demonstrated by Tveit et al. (2025), the formulation accurately captures the behaviour of a wide range of materials subjected to in-phase multiaxial loading in the finite and infinite fatigue life domain. To further examine the model’s validity, the model is here tested against the fatigue data presented by Nishihara and Kawamoto (1945). Their experimental study investigates the effects of phase differences in fully reversed bending-torsion ( = −1) , where the history of the nonzero stress components can be expressed as '' ( ) = & ∙ ( ) '* ( ) = & ∙ ( − ) = 2 & ∙ ( − ) (6) Here, & is the normal stress amplitude, & is the shear stress amplitude, is the angular frequency, and is the phase shift. We have adopted the use of a stress angle to signify the ratio between the amplitude components, where tan = & 2 & ⁄ . Moreover, we have adopted the in-phase maximum shear stress %,#&' = 12 c & ( + 4 & ( (7) as a scalar to represent the multiaxial stress amplitude across various stress angles and phase shifts . Nishihara and Kawamoto (1945) present data from tests carried out on mild steel ( !" !" ⁄ =0.583 ), hard steel ( !" !" ⁄ =0.625 ), duraluminum ( !" !" ⁄ =0.641 ), and 3%C cast iron ( !" !" ⁄ =0.949 ). The dataset includes fatigue limits for stress angles = 0° , 22.5° , 45° , 67.5° and 90° ( = 22.5° and 67.5° is missing for duraluminum) that are rendered in Table 1 in this paper. When both & and & are nonzero, the experimental fatigue limits are given for various phase angles, e.g. = 0° , 30° , 60° and 90° . Table 2 summarizes the model parameters for the different materials. As the experimental dataset only contains experiments for = −1 loading, the mean stress sensitivity parameter was estimated using the recommendations in the FKM Guideline (Hänel et al., 2003), cf. Table 2. As the investigation is confined to infinite fatigue limits where the endurance surface eventually becomes stationary, the parameter values of , , and could be arbitrarily selected as they only influence the finite fatigue life behavior. In the analyses, = 10 , (high values shorten the initial transient phase) and = = 1 were used for all materials. Fig. 2(a) shows the experimental fatigue limits and model predictions for in-phase combinations of normal and shear stresses in the space of & and & . The solid line represents the generalized formulation presented in Section 2, while the dashed lines show the situation for = 0 (i.e. !" !" ⁄ =1⁄√3 ), corresponding to the original format from Ottosen et al. (2008). The two formulations are almost identical for mild steel where the ratio !" !" ⁄ is close to 1⁄√3 , and the difference is also small for hard steel and duraluminum. For 3%C cast iron, which has a higher !" !" ⁄ ratio, application of the generalized formulation drastically improves the prediction accuracy for shear stress dominated loading, although it somewhat overestimates the fatigue limit for = 22.5° . The fatigue limits for in-phase loading can be obtained analytically (Ottosen et al., 2008; Tveit et al., 2025). To determine the results for out-of-phase stress histories, the fatigue limits were found using a brute-force approach. The evolution equations in Section 2 were integrated numerically over several cycles of the experimental stress history until steady-state behavior was reached. This produced a binary output: below the fatigue limit (no steady-state damage evolution) or above the fatigue limit (nonzero steady-state damage evolution). Based on the outcome, the stress amplitude was scaled up or down, respectively, and the process was repeated until the predicted fatigue limit was determined to an accuracy of 0.1%. The results are presented in Fig. 2(b)-(d), which show how %,#&' changes as the phase angle varies for the different materials. Fig. 2 (b), (c) and (d), respectively display the results for the stress angles = 22.5° , 45° , and 67.5° . As also shown in Table 1, the experimental values of %,#&' exhibit a moderate increase as the phase angle → 90° . Studying the very limited experimental dataset, no apparent relations can be observed between the properties of the different materials, and the relative fatigue limit increase that is observed as the phase angle changes from 0° to 90° .