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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Fig. 4. Fitting of the kinematic model on the Ramberg-Osgood stabilized stress-strain curve.

As displayed in Fig. 4, the kinematic model fits perfectly the Ramberg-Osgood model in the range of interest of stress amplitudes between 845 MPa and 1220 MPa, with an average absolute relative error less than 1%. Also shown in the graph, the markers represent the stress-strain amplitudes of stabilized cycles obtained by a finite element analysis of a rectangular beam with the fitted kinematic hardening material model, which was adopted in a preliminary benchmark to confirm the correctness of the fitting procedure. 5. Comments on modeling aspects Before examining the results, it is worth commenting on some significant aspects of modeling: the sampling variability of the simulated stress-time histories and the use of turning points (peaks and valleys) instead of the whole actual stress-time history signal. As it is pointed out in Section 4.2, the simulated stress-time histories have a finite time length; hence, their statistical properties (e.g., variance) may differ from the exact theoretical values specified by their PSD. This leads to sampling variability: the shorter the time length, the higher the deviation of the statistical properties from their theoretical values. The sampling variability would not be present only if the stress-time history had an infinite time length, which nevertheless represents a mathematical abstraction not achievable in numerical simulations. In order to evaluate the amount of sampling variability, the empirical CDF computed from the simulated narrow band stress-time histories was compared to the theoretical Rayleigh CDF, which holds true for the stress amplitudes of a narrow-band process ( Slavič et al., 2021). Fig. 5 compares the empirical CDFs to the Rayleigh CDF for various numbers of simulated points. It is apparent that, as the number of points increases, the empirical CDF gets closer to the theoretical one, i.e., sampling variability decreases. Despite this, using a large number of points is not always feasible due to the significant increase in computational time of an elasto-plastic finite element analysis. For this reason, sample stress-time realizations with 500000 points were selected as a reasonable trade-off.

a.

b.

n=500.000

n=500.000

n=250.000

n=250.000

Rayleigh

Rayleigh

n=50.000

n=50.000

Fig. 5. Effect of time history length n on sampling variability for stress variance (a) 0 = 3000 2 and (b) 0 = 5000 2 .