Estimation of remaining fatigue life under two-step loading based on kernel-extreme learning machine
Introduction
Fatigue due to variable amplitude loading (VAL) is one of the main causes of unexpected failures of structural components in practice [1], [2]. Compared to the constant amplitude loading, VAL makes the intensity and direction of each loading component randomly change throughout the entire loading process, in which case estimating the remaining fatigue life of materials is a challenging task. In general, all stress and corresponding strain information under VAL can be regularized and reordered to a sequence of loading steps by cycle counting techniques [3], [4]. However, as the underlying physical mechanism remains unclarified, which is sensitive to diverse factors, e.g., material properties, loading modes, and component geometries [5], the fatigue failure modeling under these loading steps can be a rather complicated process, severely limiting its application.
Currently, the remaining life estimation of engineering structures is generally based on damage models for quantifying the irreversible fatigue damage, in which the crack initiation is assumed to occur when the integrated damage reaches a critical value that is usually set as a scalar constant. In this respect, the Palmgren-Miner (PM) model [6], [7], assuming that fatigue damage is linearly accumulated within and between loading steps, could be the most prevalent one because of its conceptual simplicity and low computational cost. However, it should be noted that the PM model is actually not a physical law, and remains questionable as it overlooks the nonlinear nature of the fatigue damage accumulation as well as the effect of loading history, which would inevitably compromise the accuracy of remaining life estimation especially under two-step loading conditions [8], [9]. To compensate for such flaws, numerous models have been proposed based upon the nonlinear damage characteristics of uniaxial or multiaxial fatigue, most of which focus on the following two aspects:
- (1)
to seek a quantifiable fatigue damage indicator, which is usually based on the degradation of material property [10], [11], the dissipation of elastoplastic strain energy [12], [13], the evolution of fatigue crack system [14], [15] or the analysis of probability statistic [16], [17];
- (2)
to accurately describe the fatigue damage transition between different loading steps, with emphasis on modeling the loading history effect, including the effects of loading sequence and loading step interaction [18], [19].
It is important to note that even though considerable attempts have been made over the past decades, so far not a single nonlinear model exhibits the potential to replace the PM model in engineering practice [8], [19]. One significant point in modeling fatigue damage is how to capture the damage increments during the cyclic loading, when there is no universal fundamental physical law applicable to different materials or structures. In reality, many damage models are developed empirically by incorporating various material constants and/or fitting parameters [20], [21], [22], which significantly increases the computational cost, limiting their engineering application. Besides, most models, based on limited experimental research, cannot cover a wide application, and usually require extra empirical modifications or calibrations when addressing emerging materials or new structure types.
In fact, the main objective of any damage models is to reproduce the damage accumulation process as accurately as possible and to estimate the remaining life of materials and structures; from the mathematical point of view, that is to progressively characterize the mapping relationship between the given inputs (e.g., the known loading spectrum and loading history) and the expected output (i.e., the remaining life). However, considering the complexities and uncertainties involving material properties, component geometries, loading conditions, it is obviously a sophisticated and laborious task. In this regard, compared with ‘empiricism-based models’, many data-driven techniques have been proved more effective and self-adaptable in describing complex mapping relationships without requiring any apriori hypothesis. Among them, the neural network techniques are increasingly being used in various engineering fields from materials design, topology optimization to structural health monitoring [23], [24], [25].
In fatigue research, the back-propagation neural network (BPNN), as one of the most conventional neural network techniques, has been attracting considerable attentions in recent years [26], [27]. For example, Jimenez-Martinez et al. [28] applied BPNN to the damage analysis of chassis components, effectively capturing the effects of loading sequence and temperature on fatigue damage. However, since the learning process of BPNN accompanies with iterative updates of built-in parameters according to the chain rule, BPNN often suffers the curse of local minima issues and is very time-consuming for engineering problems where input data are updated quickly. These inherent deficiencies hinder its application in practice.
Recently, a new type of single-hidden layer neural networks, called extreme learning machine (ELM) [29], [30], is developed, which may be more appropriate for addressing the complexities of remaining life estimation under multistep loading. Different from BPNN, in training process, ELM randomly assigns the hidden parameters, and only solves the weights between the hidden and the output layers by the minimum norm solution rule [31], [32]. Therefore, it holds a higher training speed due to the avoidance of the multivariate optimization process, and is also free of the local minima curse as the output error in training process can be minimized globally through the minimum norm solution rule. Very recently, Hou et al. [33] presented an ELM-based model for the estimation of fatigue stress concentration factor, and demonstrated that the model is superior to other machine learning-based models by using extensive experimental data.
Despite the considerable improvement, it must be pointed out that the output performance of ELM is unstable because the initial neuron parameters are randomly generated in each independent implementation [34], [35]. Such instability is undesirable especially for the life management of sophisticated mechanical systems. To avoid this issue, the kernel mapping theory has been introduced in ELM and thereafter it is usually called kernel-ELM (KELM) technique [36]. KELM replaces the randomly-generated feature mapping of hidden neurons by the implicit kernel mapping, alleviating the problem of the random initializing parameters, and it has been demonstrated to have a better generalization capability than the standard ELM in many areas, such as human activity recognition, carbon pricing prediction, and medical diagnosis [37], [38], [39].
In this work, a data-driven model based upon the KELM technique is presented to tackle the intricacies of remaining life estimation under two-step loading, and a database including 169 experimental results is established for network training and model validation. The proposed model is demonstrated to be more accurate and robust than three conventional models, and also be more stable than the standard ELM-based model. The remainder of this work is structured as follows: Section 2 describes in detail the conventional damage models and the KELM technique; Section 3 elaborates the experimental database established; Section 4 optimizes the KELM-based model with five different input combinations; Section 5 performs experimental validations and model comparisons, and finally the conclusions are provided in Section 6.
Section snippets
Conventional damage models
In essence, the fatigue damage accumulation is an irreversible and complex process which accompanies with various material performance degradations in macro-scale, whose external driving force is the applied cyclic loading, and internal force is the initiation and propagation of fatigue crack. To characterize such a process, numerous damage models for the remaining fatigue life estimation have been developed. Several review articles have been presented, e.g., by Fatemi and Yang [40], and by Xia
Experimental database
In order to train the KELM network and compare different remaining life estimation models, a database is established by incorporating 169 experimental results generated under two-step loading conditions. The database covers nine materials, which are, 2024-T42 aluminum alloy [48], 300CVM steel [48], 45 steel(A) [49], 16Mn steel [49], 45 steel(B) [50], 316L stainless steel [51], C35 steel [44], P355NL1 steel [52], and 30NiCrMoV12 [22]. Two types of loading spectrums are involved, i.e., the
Implementation procedure
The implementation procedure of the proposed KELM-based model for remaining life estimation is illustrated in Fig. 6. As shown in this figure, each implementation can be categorized into training phase and testing/prediction phase. The former phase is to optimize the hyper-parameter set and then to train the network, while the latter phase is to check the generalization capability of the trained network or to obtain the desired predictions. In this work, to avoid the arbitrary and
Experimental validations and model comparisons
Once the optimal inputs are determined, the availability of the proposed KELM-based model in predicting the remaining life under two-step loading can be evaluated. In this section, the standard ELM-based model and the three other conventional models are employed for comparison purposes. To be precise, the three conventional models are the PM and YW models as well as the one proposed in Ref [19], called ASML model. Besides, in the standard ELM-based model, the number of hidden neurons is set to
Conclusions
This work is aimed to propose a data-driven model for the remaining life estimation under two-step loading. To this end, an emergent machine learning technique, i.e., Kernel extreme learning machine (KELM), is employed to capture the optimal mapping relationship between the given input and the desired output. And a database of 169 experimental results of nine materials subjected to two-step loading is constructed for network training and model validation. The main conclusions of this work can
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.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 11932005, 11972255 and 11772106).
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