PSI - Issue 61

100 Aliyye Kara et al. / Procedia Structural Integrity 61 (2024) 98–107 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 3 Spectral moments represent some important time domain properties of ( ) . For example, the zero-order spectral moment (i.e., the area under ( ) ) gives the variance of ( ) , 0 = ( ( )) . For a Gaussian process, the mean up-crossing frequency and the peak rate are, respectively: 0+ = 2 1 √ 2 0 , = 2 1 √ 4 2 (3) Combinations of spectral moments are also used to define the bandwidth parameters, = /√ 0 2 . Among them, 1 and 2 are the most used; they are close to unity for a narrow-band process, less than unity otherwise. The probabilistic definition of the Palmgren-Miner damage accumulation rule allows the expected damage for a stress time-history ( ) of time length T to be expressed as (Dirlik and Benasciutti, 2021): = ∫ 1 ( ) ∞ 0 ( ) (4) where ( ) designates the probability density function (PDF) of amplitudes, and ( ) is the number of cycles to failure at constant amplitude , which could represent either a stress or strain amplitude. Here, = is the number of rainflow cycles in time length T . For a narrow-band process, the expected number of cycles is = 0+ . 3. Strain-based spectral fatigue evaluation 3.1. Review of the approach based on Neuber’s rule Rognon et al. (2011, 2014) developed a strain-based spectral method for the first time. In their approach, Neuber’s rule was applied to the stress amplitude probability distribution in the elastic case to include material plasticity, while the Ramberg-Osgood model was used as the constitutive material model. In Rognon et al.’s approach, t he probability distribution of stress amplitudes is transformed into a corresponding probability distribution of strain amplitudes. This allowed the spectral method to be extended to the low-cycle fatigue regime as well. To expl ain the details of Rognon et al.’s approach, l et us consider a zero-mean stationary Gaussian random process, for which the probability density function (PDF) of rainflow stress amplitudes is ( ); see Fig. 1(a) (top). The function ( ) is a continuous over the values of stress amplitude, . According to Rognon e t al.’ s approach, the stress amplitude PDF is first discretized into bins of equal width, each characterized by its central value, , and the associated value, ( ) . Values, , are evenly spaced. The rectangular areas, = ( ) ∙ ( ) , i =1, 2,… , n are then calculated, their sum approximates the area under the curve ( ) . Area, , represents the probability to find cycles with stress amplitude, . All areas, , remain unchanged through Neuber’s transformation of amplitudes.

Fig. 1. (a) Flowchart of the approach by Rognon et al. (2011, 2014) and (b) its reformulation based on CDFs, presented in this article.