Issue 41

Carpinteri A. et alii, Frattura ed Integrità Strutturale, 41 (2017) 40-44; DOI: 10.3221/IGF-ESIS.41.06 42 where  is the pulsation. The symbols 11 S and 66 S represent the auto-spectral density function of the stress components 1 s and 6 s (i.e. x  and xy  ), respectively, whereas 16 S and 61 S represent the cross-spectral density functions of the above stress components. It is here assumed that the imaginary part of 16 S is equal to zero and, consequently, even the imaginary part of 61 S is equal to zero: that is, 16 61 S S  [8]. In such a case, the PSD matrix is symmetric. Now a rectangular spectrum is assumed for both 11 S and 66 S . Let us introduce the correlation coefficient given by: 0,16 16 0,11 0,66 r      (2) where 0,16  , 0,11  , and 0,66  represent the zero order moment of 16 S , 11 S and 66 S , respectively (more precisely, 0,16  is the co-variance of 16 S , whereas 0,11  , and 0,66  are the variances of 11 S and 66 S , respectively). Note that, for proportional stress components, 16 r tends to the unity, while 16 r tends to zero for highly uncorrelated loading. The central frequency, ,11 c  , and the height, 11 h , related to 11 S are assumed to be equal to 10Hz and 11MPa 2 /Hz, respectively. Different values of the ratio between the central frequencies, , ,66 ,11 / c r c c     are examined: , c r   0.1, 1.0, 1.1, 5.0, and 10.0. The maximum to minimum frequency ratio is equal to 1.1/0.9. The variance of the stress component x  turns out to be equal to 22 MPa 2 . Different values of the ratio between the zero order moments, 0, 0,66 0,11 / r     are examined: 0, r   0, 1, 100, and  . Three different types of spectral cross-correlation for 11 S and 66 S are examined, that is: (i) totally separated spectra (Fig.2(a)). In such a case: , c r   0.1, 5.0 and 10.0, and 16 r is equal to zero; (ii) completely overlapped spectra (Fig.2(b)). In such a case: , c r   1.0, and different values of 16 r are assumed, i.e. 16 r  0.00, 0.25, 0.50, 0.75, and 1.00; (iii) partially overlapped spectra (Fig.2(c)). In such a case: , c r   1.1, and different values of 16 r are assumed, i.e. 16 r  0.00, 0.25, 0.50, 0.75, and 1.00. By exploiting both Eq.(1) and the Schwartz inequality [8], which states that 16 S is non-negative only where 11 S and 66 S are overlapped and zero elsewhere, the height 16 h of the rectangular cross-spectrum is computed for each considered biaxial loading state.  c,11 h 11 h 66 ONE-SIDED PSD FUNCTION, 2 S or 2 S 11 66 PULSATION,  (a)  c,66 h 11 h 66 (b)  c,11 PULSATION,   c,66 h 11 h 66 (c)  c,11 PULSATION,   c,66 Figure 2 : One-sided PSD functions: (a) totally separated spectra (i), (b) completely overlapped spectra (ii), and (c) partially overlapped spectra (iii). The reference fatigue parameters used in the analysis are: , 1 af    79.37(10) -4 MPa, 0 N  2(10) 6 cycles, C  1.0 and 3.0 k  for fully reversed tension or bending, whereas , 1 af    , 1 / 3 af   for fully reversed torsion [8].