Issue 47

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

in which diagonal terms are auto-spectra and out-of-diagonal terms cross-spectra; f is the frequency (Hz). Matrix S ( f ) is Hermitian (see Appendix A). The PSD matrix allows the covariance matrix C (symmetric) to be computed:

    

    

xy yy C C C C C C C C C , , xy xx

xx

yy xx

,

(2)



C

xx yy

yy

,

xy,xx

yy xy

xy

,

The i -th diagonal term C ii

= Var ( x i

( t )) is the variance of stress x i ( t ). Normalizing the covariance C ij

( t ), the ij -th out-of-diagonal term is the covariance allows a correlation coefficient to be defined as

C ij

= Cov ( x i

( t ), x j

( t )) between x i CC C r

( t ) and x j

. The correlation coefficient represents, for multiaxial random loadings, the statistical equivalent of the



/

ij

ij

ii

jj

phase angle for sinusoidal loadings. It discriminates between two limiting cases: r ij

= 1 for perfectly correlated processes (i.e.

proportional stresses), r ij

= 0 for uncorrelated processes (i.e. not-proportional stresses).

Description of deviatoric and hydrostatic stress The PbP method is an invariant-based multiaxial criterion, so a frequency-domain description of the deviatoric and hydrostatic stresses is needed. The PbP criterion works with the amplitude a2, J of the square root of the second invariant J 2 of the deviatoric stress tensor σ' ( t ), where the decomposition σ ( t )= σ' ( t )+σ H ( t ) I into deviatoric and hydrostatic tensors is used. The definition )( )( )( 2 t t t J s s   relates the second invariant to the stress vector s ( t ) = ( s 1 ( t ), s 2 ( t ), s 3 ( t )) in the three-dimensional deviatoric space. The following transformation rules apply [6]:   1 1 )( )( 2 t t t t s       

        

2 3 )( xx

32 1 )( xx

       

       



0

yy

1

3

32

2 1 )( t yy 

2 1 )( t yy 

2 1

t s 

)(     t xy 



t   xA s



t

)(

)(

where )(

0 0

0

A

2

(3)

1 0



t s

t

)(

)(

xy

3



3  )(



t

t

)(

xx

yy



)( 

t

H

Expressions (3) allow both the hydrostatic and deviatoric stress components to be directly computed from the stress vector x ( t ) in the physical space. When the stress tensor σ ( t ) changes over time, the tip of vector s ( t ) describes a curve (loading path  ) in the deviatoric space. In a time-domain approach, conventional definitions (e.g. Longest Chord, Minimum Circumscribed Circle, etc. [6]) are used to identify the amplitude a2, J of the loading path. In a frequency-domain approach, this time-domain procedure is replaced by a spectral characterization of the stress quantities in Eq. (3). As a first step, the correlation matrix of s ( t ) is defined as:         T T T T t t E t tE A RA A x x A s s R'            ) ( ) ( )( ) ( )( ) (     (4) where R (  ) is the correlation matrix of x ( t ), with  a time lag (see Appendix A). The (symmetric) covariance matrix of s ( t ) results in the special case  =0:

     C C C C C C 33 23 13 ' ' '

'

'

           22 12 11 sym ' '





(5)

)( )( T

T



t t E

C

C ACA s s

350