Issue 33

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 33 (2015) 376-381; DOI: 10.3221/IGF-ESIS.33.41 377 normal stress and shear stress, the latter projected along the direction that maximises the variance of such a stress; (iii) the PSD of the above equivalent stress is used to estimate damage, and hence to determine fatigue life of the structural component via the Tovo-Benasciutti method [11]. The frequency-based criterion being presented is applied to combined bending and torsion random fatigue test data [12], which are also compared with the results determined by employing the time-domain criterion proposed in Refs [4,5]. C RITICAL ( VERIFICATION ) PLANE ORIENTATION he verification plane orientation, named critical plane, is here assumed to be dependent on the PSD matrix of the stress vector [13], as is explained in the following. Let us assume that: (i) the random features of the stress tensor,   xyz 1 2 3 4 5 6 ( ) T t s , s , s , s , s , s  s , defined with respect to the fixed frame PXYZ (being P a point of the structural component), can be described by a six-dimensional ergodic stationary Gaussian stochastic process with zero mean values; (ii) the corresponding PSD functions are known. The coefficients of the PSD matrix, xyz ( )  S , are given by: , , 1 ( ) ( ) 2 i i j i j S R e d           , 1, 6 i j   (1) where , ( ) i j R  are the auto- and cross-correlation functions and  is the angular frequency. The PSD matrix with respect to a rotated coordinate system PX'Y'Z' can be defined by employing the xyz ( )  S matrix and a rotation matrix C depending on the three rotation Euler angles , ,    . In such a coordinate system, the corresponding stress tensor components and PSD coefficients are indicated as ' i s and ', ' ( ) i j S  , with ', ' 1, 6 i j   . The above axes are made to vary as follows: the direction Z' (defined by the angles ,   ) is such that 3' s experiences the maximum in a statistical sense, according to Davenport [14]: 3' 0 0 0 0 0.5772 max ( ) 2 ln( ) 2 ln( ) t T E s t T T                (2) being 0  the spectral moment of order 0 of the PSD function 3',3' S , 0   the expected rate of mean zero-upcrossings of 3' s and T the observation time interval. The Y' axis (defined by the angle  ) is made to vary in order to maximize the variance of 6' s : 2 6',6' 6',6' 0 2 0 2 ( , ) max max S d                           (3) The directions Z' and Y' are regarded as the averaged principal directions ˆ1 and ˆ3 , respectively. Then, the normal to the critical plane, w , is defined in the ˆ1 ˆ3 plane by the off-angle  (clockwise rotation), function of the ratio between fully reversed shear fatigue limit and normal stress fatigue limit: 2 , 1 , 1 3 1 8 af af                         (4) PSD EVALUATION OF THE EQUIVALENT NORMAL STRESS et us define the PSD matrix with respect to a coordinate system Puvw , where u and v belong to the critical plane and w is the normal to the critical plane. In such a coordinate system, the corresponding stress tensor components and PSD coefficients are indicated as i s  and , ( ) i j S    , with '', '' 1, 6 i j   . The axes u and v are made to vary in the critical plane in order to maximize the variance of 6 s  : T L