Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 350 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 xy xy,xx xy yy yy xx yy xy xx yy xx xx C C C C C C C C C , , , , , C (2) The i -th diagonal term C ii = Var ( x i ( t )) is the variance of stress x i ( t ), the ij -th out-of-diagonal term is the covariance C ij = Cov ( x i ( t ), x j ( t )) between x i ( t ) and x j ( t ). Normalizing the covariance C ij allows a correlation coefficient to be defined as jj ii ij ij CC C r /  . The correlation coefficient represents, for multiaxial random loadings, the statistical equivalent of the 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]:   3 )( )( )( 1 0 0 0 2 1 0 0 32 1 3 1 where )( )( )( )( )( )( 2 1 )( 2 1 )( )( )( 2 32 1 )( 2 3 )( 3 2 1 t t t t t t t t s t t t s t t t t s yy xx H xy xy yy yy yy xx xx                                                    A xA s (3) 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:                    33 23 22 13 12 11 ' sym ' ' ' ' ' ' )( )( C C C C C C t t E T T C ACA s s C (5)