Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 352 Step 2 – From the original coordinate system to the (rotated) principal coordinate system In general, matrix C' is not diagonal, i.e. some correlation exists between the stress components of s ( t ). If this happens, it is necessary to compute a new covariance matrix C ' p in which all cross-correlations (out-of-diagonal elements) are zero. This outcome yields by solving the following eigenvalue/eigenvector problem for C ' :                 33 22 11 T 0 0 0 0 0 0 ' ' ' p p p p p C' C' C C UCU C (9) The eigenvalues are in the main diagonal of C ' p , the eigenvectors are the columns of U . In a time-domain perspective, the eigenvalue/eigenvector problem (9) can be viewed as the projection of the loading path  along the axes of a new (rotated) “principal coordinate system”. The new system is located by a rotation angle (whose direction cosines are the eigenvectors in the column of matrix U ) from the original system in which s ( t ) is initially defined. Projecting the loading path into the new coordinate system yields 3 new stress components Ω p,i ( t ) ( i =1, 2, 3), which are grouped in the vector Ω ( t ) = ( Ω p,1 ( t ), Ω p,2 ( t ) , Ω p,3 ( t )). Since matrix C ' p is diagonal, the stress components in Ω (t) are completely uncorrelated, that is Cov ( Ω p,h ( t ), Ω p,k ( t ))=0 for any h ≠ k . The diagonal elements in C ' p are the variances C pii = Var ( Ω p,i ( t )). Fig. 2 shows an example for a tension/torsion loading. The non-null deviatoric stress components s 1 ( t ), s 3 ( t ) give the two- dimensional loading path  . Once projected in the principal system this path gives rise to two stress projections Ω p,1 ( t ), Ω p,3 ( t ). Both vectors s ( t ) and Ω ( t ) then share the same loading path in the deviatoric space. Figure 2 : Example of random loading path  in the two-dimensional deviatoric space, resulting from a tension/torsion loading. The rotated “principal coordinate system” is located by axes ( s 1,0 , s 3,0 ) and it is used to obtain the stress projections Ω p,1 ( t ), Ω p,3 ( t ). In a frequency-domain approach, the direct projection  is yet not necessary, as it is replaced by a “rotation” of the PSD matrix from the “old” to the “new” coordinate system. Indeed, vector Ω ( t ) is characterized in the frequency-domain by the PSD matrix S ' p ( f ), which turns out by “projecting” the spectral matrix S ' ( f ) in the new rotated principal system:                         ) ( 0 0 0 ) ( 0 0 0 ) ( ' ' ' 33 22 11 p T p f S f S f S f f f p p p S U SU S (10) 1000 1500 2000 2500 s 3 ( t ) t t s 1 ( t ) s 1 s 3 Load path Ψ  xx ( t ) τ xy ( t ) t t