PSI - Issue 61

Aliyye Kara et al. / Procedia Structural Integrity 61 (2024) 98–107 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Regarding the use of turning points, although it is true that plasticity is path-dependent, it is a matter of fact that fatigue damage is a function of rainflow stress/strain amplitudes, which in turn only depend on the sequence and relative positions of peaks and valleys in the loading. By taking only turning points, the peak/valley positions remain unchanged, and the actual stress-time path is replaced by a sort of linearized trend between adjacent peaks and valleys of the stress-time realization. This approximation is acceptable; in fact, when Fig. 2(b) is examined, it is seen that the actual stress-time curve deviates only slightly from the linear curve connecting two adjacent turning points. The use of turning points then seems like a reasonable approximation. 6. Results and discussion In the elasto-plastic finite element analysis, the outputs of interest were the stress and strain time histories (as turning points) at the notch tip, along the axial Y direction. Such histories were first rainflow-counted to extract the cycle amplitudes; their statistical distribution was evaluated as an empirical cumulative distribution function (ECDF). Time domain and frequency domain results were compared in terms of CDFs of stress and strain amplitudes. Fig. 6 compares four CDFs for two values of the variance of the remote random stress (3000 and 5000 MPa 2 ), which correspond to moderate and large plasticity. The function , ( , ) (solid blue line) represents the Rayleigh cumulative distribution that refers to the hypothetical case in which plasticity is not present. The function ( ) (dashed gray line) is the strain-based spectral estimate calculated by Neuber’s rule applied to the stress amplitude probability distribution of the elastic notch stress, which follows a Rayleigh distribution with variance 0, ℎ = 2 0 . In fact, since the random remote stress is narrow-band, in a purely elastic situation, the stress at the notch would also be narrow-band with Rayleigh distributed amplitudes. Finally, functions , ( , ) (dashed red line) and , ( ) (solid pink line) represent, respectively, the ECDFs of stress amplitudes for the axial stress along the Y-axis and for the von Mises stress at the notch, obtained from time-domain finite element simulations. A significant difference exists between time and frequency domain results: while the spectral estimate of CDF based on Neuber’s rule is continuous, the time domain CDF s is not and has a knee point. When the position of the knee point was examined with reference to the von Mises stress CDF, it was observed that it corresponds to the cyclic yield stress of the material, equal to 814 MPa as explained in Section 4.2. The knee point divides the CDF into two portions with different trends. The left portion corresponds to stress amplitudes below yield stress and, in fact, it overlaps the CDF, , ( , ) , valid for the elastic case. The right portion, instead, refers to amplitudes of von Mises stress above yielding and corresponds to the plastic region. The relative width of the two portions depends on the amount of plasticity: as the variance of remote stress ‒ and of notch stress, too ‒ increases, the amount of plasticity increases as well, and the right portion of CDF above the knee point widens. A similar trend is also observed for the Neuber ’s rule -based CDF, which is close to the Rayleigh CDF only at lower stress amplitudes in the elastic region. It is interesting to note that, while the Neuber ’s rule -based CDF is closer to the von Mises CDF at lower amplitudes, it approaches the CDF of longitudinal stress at larger amplitudes. This behavior may be attributed to the effect of multiaxiality, which is not present in the Neuber ’s rule -based CDF.

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Fig. 6. CDF of stress amplitudes for two variance values of remote stress: (a) 0 = 3000 2 (b) 0 = 5000 2 .