Basic Mathematical Tools for Stochastic Fatigue Analysis

Abstract

In the chapter is given a short description of basic mathematical tools from stochastic process theory, which will be used to develop the stochastic approach of thermal fatigue crack growth in the high cycle domain. Fatigue analysis is often performed in the time domain, in which all input loading and output stress or strain response are time-based signals. In some situations, however, the response stress and input loading are preferable expressed as frequency-based signals, usually in the form of a power spectral density (PSD) plot. In this case, a system function (or a characteristic of the structural system) is required to relate an input PSD of loading to the output PSD of response. The Fast Fourier Transform (FFT) of a time signal can be used to obtain the PSD of the loading, whereas the Inverse Fourier Transform (IFT) can be used to transform the frequency-based signal to the time-based loading. The transform of loading history between the time domain and frequency domain is subject to certain requirements, as per which the signal must be stationary, random, and Gaussian (normal). Thermal striping is a random phenomenon in a temporal sense. In order to have a better understanding of further developments in the present book, some knowledge of stochastic processes is required.