A frequency-domain model assessing random loading damage by the strain energy density parameter

Highlights

• New frequency domain model to calculate damage degree with the use of the strain energy parameter.

• New probability distribution model for the strain energy density parameter.

• The strain energy parameter damage degree is calculated directly from the power spectral density.

• Theoretical proof of the equality of the energy and stress approach in frequency domain.

• The approach is verified with the use of experimental results for the narrowband and broadband loading case.

Abstract

The paper presents a new fatigue damage assessment model defined for the strain energy density parameter. The model is defined for calculations in the frequency domain for the case of narrowband loading. Analytical expressions are derived for the probability density function of peaks, the level crossing spectrum, and for the expected damage that is directly linked to the power spectral density of the strain energy parameter. The paper presents a theoretical proof which shows that the damage obtained for the energy model is equal to the damage obtained for the stress model. A numerical example is used to verify the correctness of the proposed frequency-domain damage estimation formula by comparison with experimental and time-domain fatigue life estimation results for two simulated random time-histories for the case of narrowband and broadband loading. The comparison between experimental and calculation results confirm the very good agreement between frequency and time domain approaches.

Introduction

Fatigue damage calculation algorithms can be divided into two major domains: time and frequency. The time domain uses cycle counting algorithms to assess the damage degree of the material. Beside this it has many advantages towards to the frequency domain which uses statistical information obtained from the power spectral density (PSD), but one huge disadvantage which is the computation time. During the last 70 years there has been an increasing growth in the use of advanced simulation and calculation approaches for random fatigue analysis that have been adapted to the use in the frequency domain. Since the early work of Bendat [1], which defined the basics for probability based random fatigue calculations, a great many spectral methods have been proposed. The first major paper by Dirlik [2] which incorporated the use of computers especially for calculations with the use of power spectral density (PSD) was a milestone for these techniques. Wirsching [3] was one of the first who defined probability based criteria for fatigue life estimation with the use of the frequency domain. Macha [4] was the first who defined the basics for multiaxial random calculations with the use of PSD. The frequency domain methods owe a lot of growth in renown to the papers by Pitoiset and Preumont [5], [6], due to the fact that they adapted the popular von Mises method to the frequency domain regime. The problem of wide band loading in spectral methods was one of the main problems in papers by Kihl and Sarkani [7]. Rychlik et al [8] have worked on the damage degree model for non-Gaussian random loads. Then at the beginning of the 21th century we deal with the beginning of the renowned series of publications dealing with the topic of frequency defined methods presented by Benasciutti and Tovo [9] which vastly influenced the spectral methods theory and applicability due to the effects of narrowband, broadband and non-Gaussian [10], [11], [12] random loading and presenting their popular probability density function based on the damage degree similar to the solution working in the rainflow algorithm. The evolution of spectral methods in terms of multiaxial fatigue was also the one of the main tasks of the group led by Carpinteri [13], [14]. Another problem which can be found in the literature on spectral methods is the non-stationarity effect which is widely discussed in the papers by Slavič et al [15], [16]. Problems related to mean stress effects in frequency domain have been described by Böhm et al. [17], [18]. A summation of the most important aspects in terms of variable amplitude loading and frequency domain calculations in terms of multiaxial loading can be found in the papers by Sonsino et al [19], [20].

Nevertheless it is rare to stumble upon a paper on fatigue damage estimation in frequency domain with the use of the strain energy density parameter such as the Banvilett et al [21]. The strain energy density parameter model often referred as the energy parameter model describes the signed strain energy of the combined stress–strain state of the material. This parameter allows taking into account either the elastic or plastic state of the material or a combination of these states [22], [23], [24]. This parameter is popular among the engineers responsible for composite structures especially related to rubber. It is very often discussed and used for tire fatigue design procedures [25], [26]. As it was noticed by many scientists like Garud or Kujawski the plastic strain amplitude alone has a major influence on the fatigue damage, but it is insufficient in many cases especially if we are analyzing the multiaxial state of material [27], [28]. For this reason, the information of both strain and stress should be taken into account. Another important fact that tips the scale on the side of this method is that for low cycle and high cycle fatigue life regions the strain energy density is a constant damage parameter, where it is not necessary to distinguish or choose the appropriate calculation method for low or high cycle fatigue. That is why the energy of the hysteresis loop seems to be a good factor to describe fatigue damage.

There are still unsolved issues like, inter alia, the proper use of the strain energy density models directly with the proper domain description. If one would look at a simple case of strain energy density described with the use of the stress descriptors then we can notice that, even though the stress loading time history is Gaussian, the energy time history will be non-Gaussian. For those occupied with frequency domain methods it is well known that non-Gaussianity is a major issue in damage assessment, as described by Benasciutti and Tovo [29].

Therefore, it is required to compensate the information about the non-Gaussianity in another form by either using transformations or, like in the case of this paper, to propose a direct damage intensity model that takes into account the kurtosis and skewness values of the base signal.

By extending the work presented in [30], this paper obtains the main statistical properties of the strain energy parameter for the case of random loading; it also derives the expression for the damage intensity for the narrowband case. The model proposed in the paper is valid, as all frequency defined methods for the linear elastic material state. For verification of the proposed model a comparison between time and frequency domain results has been performed. The calculations are performed for the time domain damage assessment with the use of the rainflow and Palmgren-Miner hypothesis for stress as well as strain energy density for a narrowband and broadband case on the basis of the experimental results of S355JR steel. The frequency domain damage calculations are performed with the Benasciutti-Tovo model and with the use of a new model, that is using spectral moment information of the narrowband power spectral density of stress, which is used in the strain energy density description process. The formulation of the model is explained stepwise. The important fact of the model is that it takes into account the non-Gaussian characteristic of the strain energy signal. The obtained results show good compatibility between the damage models.