Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26

[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]:





 

 

 e R fS f i 2   

 f e fS f i 2  

d ) ( 



R

) (

) (

d ) (

(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]:





 

 

(19)

n

n

n 



d) ( f fS f

d) ( f fGf

n

... ,2,1,0



0

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

0   2

(20)

0 





p 

;

Spectral moments are also combined into bandwidth parameters, as for example [21]:

1 

2 

(21)

1 



2 



,









2 0

4 0

where 0≤α m PSD with α 1

≤1 and α 1 ≥α 2 . Bandwidth parameters summarize the shape of a PSD. Two limiting cases exist: a narrow-band

→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:





  S

 

 f i 2  R S   e

f i 



 e f 2

d ) ( 



f

f

) (

) (

) (

d

(22)

R





(  )= E [ x i

( t )· x i ( t +  )] ( i =1, 2, 3) is the auto-correlation function of x i

(  )= E [ x i

( t +  )] ( i ≠ j )

In (22), R ii

( t ) ( i =1, 2, 3) and R ij

( t )· x j

(  ) ≠ R ij

(  ). The PSD matrix takes the form:

is the cross-correlation functions between x i

( t ) and x j

( t ). In general R ji

    

    

S

, S f yy xx

, f S f f S f f S f xy xy yy xy xx ,

) (

) (

) ( ) (

xx

(23)

*

 ) (

f

S S

S f yy

) ( ) (

) (

S

yy xx

,

*

* S f xy yy ,

) (

) (

xy xx

,

363