PSI - Issue 54

Arvid Trapp et al. / Procedia Structural Integrity 54 (2024) 521–535 Arvid Trapp / Structural Integrity Procedia 00 (2023) 000–000

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(Trapp et al. (2022)). Consequently, an intuitive extension for a statistical fatigue assessment would be, to complement the PSD with a spectral decomposition of the kurtosis resp. its underlying fourth-order moment. Addressing this need, the authors have proposed the non-stationarity matrix (NSM; Trapp and Wolfsteiner (2021b)), capturing the kurtosis for modulated Gaussian processes, which represent a critical class of processes in practical applications. This matrix can be processed for structural responses (stress states) through the use of linear systems theory, so that stress states are characterized by its response NSM or, after its summation, by response kurtosis. This provides the basis for the investigation herein presented. The central idea of this paper is to accompany the PSD with the NSM to extent the statistical characterization of the input loads, which by processing through linear systems theory consequently extends the available statistical characterization of the acting stress states. Then, the response NSM are related to fatigue damage, more specifically to predict the di ff erence in damage due to the non-stationarity of stress states. Therefore, this paper makes use of artificial neural networks (ANN) as a powerful framework to learn the relevant mechanisms by providing the necessary data base. Over the last decade, the interest in ANN for improving fatigue characterization has steadily increased (Kang et al. (2014); Kim et al. (2016); Durodola et al. (2017); Sun et al. (2022)). However, these applications aimed — as least initially — at generating models that improve the accuracy of fatigue life predictions for stationary Gaussian processes over existing frequency-domain methods (Dirlik (1985); Benasciutti (2004)). As such, they have been generating stationary Gaussian training data within a wide range of spectral shapes and frequency configurations and have been trying to use the capability of ANN’s to approximate the underlying complex relationships. In all referenced studies the input parameters consist of a combination of PSD-based spectral moments and the material parameters necessary for the fatigue damage assessment. Kang et al. (2014) used the discrete distribution of the rainflow amplitudes as the output, while Kim et al. (2016) used twenty weighting factors of Gaussian bases to approximate the rainflow amplitude probability as a continuous function. While the physics behind this model is questionable, this minimized the number of output values. Durodola et al. (2017) and Sun et al. (2022) incorporated a wide range of material parameters to directly predict the specific damage value as a single output. In contrast to the other solutions, applicable for stationary Gaussian stress states, Durodola (2019) aimed to improve his previous model with a range of non-Gaussian data to extend the input features to include kurtosis and skewness for characterizing the non-Gaussianity of the data. To enable reliable statistical prediction of fatigue damage for more general vibration loading, we herein propose a data-driven damage estimator based on response kurtoses and related fourth-order spectral moments. These are cal culated using the non-stationarity matrix (red in Fig. 1 (b) ). The aim of the subsequent damage estimator is to improve fatigue strength predictions under realistic non-stationary vibration loading, while preserving the computational ef ficiency of a statistical ’frequency-domain’ fatigue assessment. The remainder of this paper is organized as follows: The next Section 2 presents some essential background of statistical characterization of random processes, fatigue damage evaluation and spectral estimators; The following Sec. 3 details our study into a data-driven damage estimator and covers its design principles, the data generation, the neural network architecture, and its training; Subsequently, this approach is showcased by exemplary data which compares the proposal with the established fatigue evaluation approaches (Sec. 4). Finally, the paper concludes with a discussion of the results (Sec. 5). In random vibration fatigue, loading inherently exhibits variable amplitudes, which requires to process structural responses y ( t ) (e.g. stresses) by counting-algorithms to abstract load spectra, with rainflow-counting (RFC) being by far the most popular. A subsequent lifetime prediction consists in linear damage accumulation of the counted stress cycles, taking into account the corresponding material properties. These are provided in the form of stress-life-curves (S-N curves; Radaj and Vormwald (2013)). The most basic form of an S-N curve — as defined by the Palmgren-Miner rule — is fully characterized by constant C and S-N curve exponent k (Fig. 1). It relates the number of cycles n with amplitude s to failure by C = s k n . To compare load spectra of variable amplitudes s j and corresponding counts n j by a concise descriptor, we define the equivalent response amplitude: s eq = 1 n eq J j = 1 n j s k j 1 k (1) 2. Fundamentals