Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 363 [26] Abramowitz, M., Stegun, I.A. (1965). Handbook of mathematical functions, with formulas, graphs, and mathematical tables, tenth ed., Dover. A PPENDIX A – S PECTRAL DESCRIPTION OF UNIAXIAL AND MULTIAXIAL RANDOM STRESS Let x ( t ) be a zero-mean uniaxial random stress. It is characterized, in time-domain, by the autocorrelation function R (  )= E [ x ( t )· x ( t +  )] (  is a time lag) and, in frequency-domain, by a two-sided Power Spectral Density (PSD) function S ( f ), ∞< f <∞. Both functions constitute a Fourier transform pair (Wiener–Khintchine relations) [20]:           f e fS R e R fS f i f i d ) ( ) ( d ) ( ) ( 2 2        (18) In practical applications, where negative frequencies have no direct physical meaning, the two-sided spectrum S ( f ) is replaced by a one-sided spectrum limited to positive frequencies only, which is defined as G ( f )=2 S ( f ), 0< f <∞ and zero elsewhere. It is customary to describe S ( f ) or G ( f ) by the set of spectral moments [21]: ... ,2,1,0 d) ( d) ( 0         n f fGf f fS f n n n  (19) Eqn. (19) shows that the variance of x ( t ) is the zero-order moment Var ( x ( t ))= R (0)=λ 0 , which corresponds to the area of G ( f ). If x ( t ) is Gaussian, the frequency of zero up-crossings, ν 0 , and the frequency of peaks, ν 0 , are [21]: 2 4 p 0 2 0 ;         (20) Spectral moments are also combined into bandwidth parameters, as for example [21]: 4 0 2 2 2 0 1 1 ,           (21) where 0≤α m ≤1 and α 1 ≥α 2 . Bandwidth parameters summarize the shape of a PSD. Two limiting cases exist: a narrow-band PSD with α 1 →1, α 2 →1, a wide-band PSD with α 1 <1, α 2 <1. The quantities in Eq. (19)-(21) enter the analytical expressions used by spectral methods for estimating the fatigue damage. The previous definitions can be generalized to a biaxial stress x ( t ) = (σ x ( t ), σ y ( t ), τ xy ( t )). By analogy with Eq. (18), x ( t ) is characterized in time-domain by a correlation matrix R (  )= E [ x ( t )· x ( t +  )] and in frequency-domain by a PSD matrix, which are a Fourier transform pair:           f e f e f f i f i d ) ( ) ( d ) ( ) ( 2 2        S R R S (22) In (22), R ii (  )= E [ x i ( t )· x i ( t +  )] ( i =1, 2, 3) is the auto-correlation function of x i ( t ) ( i =1, 2, 3) and R ij (  )= E [ x i ( t )· x j ( t +  )] ( i ≠ j ) is the cross-correlation functions between x i ( t ) and x j ( t ). In general R ji (  ) ≠ R ij (  ). The PSD matrix takes the form:            ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( * , * , , * , , , f S f S f S f S f S f S f S f S f S f xy xy yy xy xx xy yy yy yy xx xy xx yy xx xx S (23)