Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 358 The comparison Case 1A vs. 2A shows that not-proportional stress lead to a greater damage. This effect of stress correlation cannot be generalized, yet. It closely depends on the material type. For example, Tab. 4 makes clear that a different material (B or C) would behave differently. For example, Case 3B and 4B highlight that damage in Material B is not correlated to the proportionality degree. In Tab. 4, it is worth noting that Cases 2A and 3A yield the same damage value. This outcome may be explained by considering the role of deviatoric and hydrostatic stress components. On one side, either Cases have in common (see Step 2 in Tab. 4) the sum ( C' p22 + C' p33 ) of variances of deviatoric stress components in the principal coordinate system (the third projection has C' p11 =0). This sum represents a sort of “total variance” of stress projections Ω p,2 ( t ), Ω p,3 ( t ) and it enters the damage expression (16) in the special case in which the bandwidth parameters η TB,i , ν 0,i take on identical values for all stress projections (a case that actually occurs in all stress states of Tab. 2). It is trivial to verify that damage d ( Ω ) is proportional to 2/ 33 22 ref ) (2 k p p C C    . On the other side, damage (16) is computed on the same S-N curve for both Case 2A and 3A, despite they have a different variance V H for the hydrostatic stress and, hence, a different ρ ref value (see Tab. 4). However, as already said, for Material A the hydrostatic stress has no effect on damage. Cases 3A, 3B and 3C emphasize the effect of different materials under the same tension-torsion loading (stress case 3). It is not surprising that a change of material type determines a change in the estimated damage, too. This result is governed by the reference S-N line, which takes on different positions for each material type, as confirmed by the dissimilar values of parameters J A,ref , k ref (see Step 3 in Tab. 4). For example, the damage increases from material A to C, in relation to the different positions assumed by the reference S-N line. Figure 5 : Effect of hydrostatic stress on fatigue damage for different materials undergoing a tension-torsion loading with C s1 + C s3 = 1. The graph in Fig. 5 allows the role of material properties to be further pointed out. The figure depicts the trend of the fatigue damage d TB (Ω) in the case of a tension-torsion loading, as a function of ρ ref in the range 0  1. Parameter ρ ref synthesizes different types of tension-torsion multiaxial loading, bounded by the limiting uniaxial case of pure torsion ( ρ ref =0) to pure tension ( ρ ref =1). The PSD of normal and shearing stress are suitably normalized so to keep constant the variance sum C s1 + C s3 =1, as well as the corresponding sum const C iip    , for stress projections. This makes the damage only a function of the hydrostatic stress via ρ ref . Predictably, the trends in Fig. 5 confirm how different materials respond differently to the same multiaxial loading. While materials B and C show a moderate-to-great sensitivity to the type of loading, material A shows no sensitivity at all. In particular, Fig. 5 makes crystalline clear that material A is unable to discriminate between different multiaxial states of stress; the estimated damage is constant and always equal to the value for torsion loading. Materials like A, however, may be very common in engineering applications. It may characterize those unfortunate situations in which only the tension S- N curve is available from experiments, whereas the torsion S-N curve is only approximated by taking a line parallel and moved downward by 3 ( Von Mises rule). Although not recommended, this A-type material represents the only solution feasible for applying the PbP criterion, if no fatigue data for torsion loading are available. It has to be emphasized that