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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its convenience for random loading, especially with large data sets such as those arising with long stress time histories. One disadvantage of the spectral approach, however, is its applicability to linear systems only. Indeed, almost all existing spectral methods deal with a stress-based approach valid for linear structures. In a non-linear structure, even if the input signal is Gaussian, the response might be non-Gaussian. One of the preliminary works on non-linear structural behavior was undertaken by Winterstein (1985, 1988). In his studies, Hermite moment models are proposed to transform a Gaussian process into a non-Gaussian response and used to include non-Gaussian effects in fatigue damage. This approach is independent of the source of non-linearity. On the other hand, one of the most common causes of non-linearity in engineering structures – not addressed in Winterstein ’s works − is material non-linearity. Although stress-based fatigue damage evaluation is well-established in spectral methods, in the presence of plasticity, a strain-based approach is required to consider plasticity accurately. Few methods exist in the literature that implement a strain-based fatigue approach in the frequency domain by spectral methods, e.g., Rognon et al. (2011, 2014) and Böhm et al. (2020). These approaches estimate the strain amplitude probability distribution by the use of Neuber’s rule and the Ramberg-Osgood constitutive material model. This paper aims to provide a review of the existing strain-based spectral methods dealing with material plasticity. It first introduces a reformulation of the approach of by Rognon et al. (2011, 2014). Then, it benchmarks the strain based spectral method against time-domain results, in which finite element simulations with material non-linearity (kinematic model) are used to model a notched cylindrical specimen subjected to a stationary zero-mean Gaussian random stress with narrow-band power spectral density. Results emphasize the role of multiaxial stress, which may presumably explain the degree of conservatism observed in the strain-based spectral approach based on Neuber ’s rule.

Nomenclature b

( ) auto-correlation function ( ) one-sided spectral density function i -th spectral moment 0+ zero up-crossing frequency , plastic strain amplitude ′ fatigue ductility coefficient ′ fatigue strength coefficient yield stress , kinematic hardening variables

fatigue strength exponent fatigue ductility exponent

c k

inverse slope fatigue constant fatigue damage elasticity modulus

C D E T

E[.]

mathematical expectation

number of cycles to failure at given stress amplitude

time length

2. Frequency domain approach and fatigue damage A zero-mean stationary random process ( ) (which represents the stress or strain in a structure) is characterized in time domain by an autocorrelation function, ( ) = [ ( ) ( + )] , where is the time lag, and in frequency domain by a two-sided power spectral density (PSD) function, ( ) , which is the Fourier transform of ( ) : ( ) = ∫ ( ) ∞ − ∞ − (1) Function ( ) provides the energy content of ( ) over frequencies. In a so-called narrow-band process, ( ) is centered around a narrow frequency range, whereas in a wide-band process, it extends over a broader frequency range. A one-sided spectrum, ( ) , defined over positive frequencies, can also be used. The shape of ( ) or ( ) is usually synthesized by means of spectral moments: =∫ | | ( ) ∞ − ∞ =∫ ( ) ∞ 0 , =1,2,… (2)