PSI - Issue 61

106 Aliyye Kara et al. / Procedia Structural Integrity 61 (2024) 98–107 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 9 Fig. 7 compares the CDFs of notch strain amplitudes. Function , ( , ) (blue line) is a Rayleigh distribution, and it refers to the hypothetical situation of purely elastic strains at the notch; in this case, the variance of strain amplitudes can simply be calculated by dividing the variance of notch stress amplitudes by the elastic modulus squared, 0, ℎ = 0, ℎ / 2 . Function ( ) (dashed brown line) represents the CDF estimated by Neuber ’s rule -based spectral approach, whereas function , ( , ) (dashed red line) is the empirical cumulative distribution referred to time-domain finite element results. It may be observed that the CDFs of both Neuber ’s rule -based spectral approach and time-domain approach differ from the Rayleigh CDF. This indicates that, in contrast to a fully elastic case, in the elasto-plastic case, the CDF is no longer Rayleigh due to the effect of plasticity. In addition, while the strain amplitude CDF obtained by Neuber’s rule almost overlaps the Rayleigh distribution at low strain amplitudes, it is instead shifted to the right at larger strain amplitudes. At higher variance levels of remote stress (i.e., at increased amount of plasticity), the shifted portion increases as well. By contrast, the CDF for time-domain results is located on the left of the CDF of Neuber ’s rule based spectral approach. The difference is more pronounced at high strain amplitudes, and it could be attributed to the plasticity effect.

a.

b.

Fig. 7. CDF of strain amplitudes for (a) 0 = 3000 2 (b) 0 = 5000 2 . The deviation at lower strain amplitudes, instead, could be ascribed to the presence of a multiaxial stress state at the notch. Indeed, even though the specimen undergoes a uniaxial input remote stress, the local stress at the notch is multiaxial (see Fig. 2(c)), and the amount of notch hoop strain (along the Z-direction) is not negligible. Despite this, Neuber’s rule , adopted by the spectral approach, assumes a uniaxial stress state at the notch. It may therefore be inferred that Neuber’s rule -based spectral approach is more suitable for a uniaxial stress state.

Table 2. Fatigue damage from strain-based spectral approach and time-domain finite element simulations Damage 0 = 3000 2 0 = 5000 2 / 3.1 2.7

The probability density functions derived from the CDFs shown in Fig. 7 are eventually used to compute the fatigue damage by utilizing Eqs. (4), (6), and (8). Neuber’s rule -based spectral fatigue approach resulted in a larger damage estimation in comparison to the estimation from time-domain simulations, with a relative ratio of about three regardless of the variance of the input remote stress (see Table 2). 7. Conclusions This paper investigated the use of the strain-based spectral method in frequency domain proposed by Rognon et al. (2011, 2014), in which the non-linearity caused by material plasticity is included by means of Neuber’s rule and the Ramberg-Osgood cyclic stress-strain curve. After a brief review, the spectral method is reformulated in terms of cumulative distribution functions. This reformulation allows the issues of discretization inherent in the original method to be overcome. Estimations by the strain-based spectral method are then compared with time-domain results from an