A frequency-domain model assessing random loading damage by the strain energy density parameter
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.
Section snippets
Strain energy density parameter
The strain energy density can be described with the use of a stress–strain relation. It describes the relationship between stress and strain and is a product of their multiplication:where: σ(t)-stress course, ε(t)- strain course.
W(t) has generally a positive mean and is non-Gaussian, even if σ(t) has a zero mean value and is Gaussian. Therefore, it is not possible to separate the cycle parts of the course which are responsible for tension or compression in the loading history.
Statistical properties of the energy parameter for random loading
This section extends the concept of strain energy parameter to the case of a stationary random loading; it obtains the closed-form expressions of some statistical properties of the energy density parameter (e.g. probability distribution of values, probability distribution of local peaks, level crossing spectrum).
Let represent a stationary Gaussian narrowband random stress with zero mean value. In the frequency domain, the random stress is characterized by the one-sided power spectral
Expected fatigue damage for the strain energy density parameter
For a narrowband and Gaussian random stress of time duration , the expected fatigue damage is [32]:where is the expected number of up-crossings of the mean value counted in , and is the expected damage per cycle, which corresponds to the -th moment of the amplitude probability distribution:
Symbol denotes the strength coefficient of the S-N curve for stress.
Note that well approximates the number of cycles because, in a
Results and discussion
Based on the previous paragraph it was possible to demonstrate that . This result was expected, because it states that the damage remains unchanged whichever quantity (stress or energy) is used to compute it. This statement is true for the High Cycle Fatigue regime as the fatigue slope values will remain constant for this area. This theoretical result is a formal proof of this concept, but it seems it would be interesting to check how the model behaves under broadband loading. This
Conclusions and observations
The paper presented a new fatigue damage calculation model for the strain energy density parameter in which the fatigue damage can be estimated from the power spectral density in the frequency domain. The paper also derived the closed-form expressions of the main statistical properties of the random process (i.e. the probability distribution of values, of peaks, and the level crossing spectrum). Some general conclusions and observations can be formulated as follows:
- •
The strain energy
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (35)
Fatigue reliability in welded joints of offshore structures
Int J Fatigue
(1980)- et al.
Spectral methods for multiaxial random fatigue analysis of metallic structures
Int J Fatigue
(2000) - et al.
Stochastic fatigue damage accumulation under broadband loadings
Int J Fatigue
(1995) - et al.
Fatigue damage assessment for a spectral model of non-Gaussian random loads
Probabilist Eng Mech
(2009) - et al.
Cycle distribution and fatigue damage assessment in broad-band non-Gaussian random processes
Probabilist Eng Mech
(2005) - et al.
Fatigue life assessment in non-Gaussian random loadings
Int J Fatigue
(2006) - et al.
Analogies between spectral methods and multiaxial criteria in fatigue damage evaluation
Probabilist Eng Mech
(2013) - et al.
Variability of the fatigue damage due to the randomness of a stationary vibration load
Int J Fatigue
(2020) - et al.
Critical plane criterion for fatigue life calculation: time and frequency domain formulations
Procedia Eng
(2015) - et al.
Spectral fatigue life estimation for non-proportional multiaxial random loading
Theor Appl Fract Mec
(2016)
Non-stationarity index in vibration fatigue: Theoretical and experimental research
Int J Fatigue
Non-Gaussianity and non-stationarity in vibration fatigue
Int J Fatigue
Fatigue life estimation of explosive cladded transition joints with the use of the spectral method for the case of a random sea state
Mar struct
Fatigue testing under variable amplitude loading
Int J Fatigue
Fatigue life under non-Gaussian random loading from various models
Int J Fatigue
New energy model for fatigue life determination under multiaxial loading with different mean values
Int J Fatigue
Spectral methods for lifetime prediction under wide-band stationary random processes
Int J Fatigue
Cited by (3)
A probabilistic model for low-cycle fatigue crack initiation under variable load cycles
2022, International Journal of FatigueA revised Johnson–Cook model comprised of acoustic plasticity for non-oriented silicon steel under transverse ultrasonic vibration
2022, Journal of the Brazilian Society of Mechanical Sciences and Engineering