Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 356 f c =30 Hz and are characterized by a fixed maximum-to-minimum ratio f max /f min = 15, which guarantees that PSDs are wide- band with α 1 = 0.893, α 2 = 0.771 (these parameters remain unchanged for the power spectra of deviatoric and hydrostatic stress, as we as for stress projections). Being the frequency range fixed, each PSD is fully defined by the height h of each rectangle. Through the rectangle area, the height controls the variance V ii = h i,i ·( f max –f min ) of auto-PSDs and the covariance C ij = h i,j ·( f max –f min ) of cross-PSDs (subscript i and j stands for the stress component xx , yy and xy ). It is easy to derive the relationship j j i i j i j i h h r h , , , ,   between the cross-PSD height and the correlation coefficient r i,j . The height h i,j is then bounded as j i j i h h h    , 0 , see Fig. 4. The cross-PSD height varies according to the different correlation degrees in the range r i,j =0  1. Figure 4 : Example of one-sided auto- and cross-PSD for a tension-torsion loading with non-zero correlation r xx,xy (stress case 3). Combining the stress cases (1, 2, 3, 4) in Tab. 2 with the materials models (A, B, C) in Tab. 3 results in the whole set of load cases (1A, 2A, 3B, …) in Tab. 4, which are examined in the numerical examples. The stress cases in Tab. 2 are:  Case 1: proportional normal stresses σ xx ( t ) , σ yy ( t ), i.e. correlation degree equal to one (“in-phase loading”);  Case 2: not-proportional normal stresses σ xx ( t ) , σ yy ( t ), i.e. correlation degree equal to zero (“out-of-phase loading”);  Case 3: tension-torsion loading with proportional stresses σ xx ( t ) , τ xy ( t ), i.e. correlation degree equal to one;  Case 4: tension-torsion loading with not-proportional stresses σ xx ( t ) , τ xy ( t ), i.e. correlation degree equal to zero. All through the examples, the normal stresses have variance V xx = V yy =3 and the shearing stress V xy =1, so that all components of the stress deviator share the same unitary variance. No. of stress case Description V xx V yy C xy r xx,yy r xx,xy r yy,xy 1 Biaxial tension (correlated) 3 3 0 1 0 0 2 Biaxial tension (not correlated) 3 3 0 0 0 0 3 Bending-torsion (correlated) 3 0 1 0 1 0 4 Bending-torsion (not correlated) 3 0 1 0 0 0 Table 2 : Variance and correlation degree characterizing the biaxial stress states considered in numerical simulations. Three hypothetical material models, with different fatigue properties, are investigated:  Material A: this material has tension and torsion S-N lines with identical slope ( k  = k τ ) and with amplitude strengths σ A and τ A = σ A /√3, i.e. the torsion line is scaled by 3 . In the MWD, the tension and torsion S-N lines overlap each other, and they also overlap the reference line for any value of ρ ref . This material model has specifically been conceived for being not sensitive to the type of applied multiaxial stress (i.e. the material is not sensitive to ρ ref ). This situation reflects the hypothesis that the material obeys the Von Mises equation under fatigue loadings, and it is only sensitive to the deviatoric stress component while not sensitive to hydrostatic stress;  Material B: this material model has tension and torsion S-N lines with identical slope ( k  = k τ ), but with fatigue strengths not scaled as in von Mises criterion, that is τ A ≠ σ A /√3. Unlike Material A, this material has a fatigue strength that depends on the hydrostatic stress; f G xx ( f ) h xx f c f G xy ( f ) h xy f c f Re {G xx,yy ( f ) } h xx,yy f c xy xx xy xx yy xx h h r h   , ,