Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 353 Matrix S ' p ( f ) is diagonal, by definition, and it collects the auto-spectra S' ii ( f ) of stress projection Ω p,i ( t ), i =1, 2, 3. Thanks to the fact that, in the principal system, the stress projections Ω p,i ( t ) are completely uncorrelated, their fatigue damage can be computed separately. Knowing the power spectra S' pii ( f ), the damage can be estimated through spectral methods for uniaxial loadings (see Step 4). To this end, each spectrum needs to be characterized by several parameters (see Appendix A): spectral moments (λ 0i , λ 1i , λ 2i , λ 4i ), bandwidth parameters (α 1i , α 2i ), frequency of up-crossing and peaks (ν 0,i , ν p,i ). Step 3 – Reference S-N line in the Modified Wöhler Diagram The fatigue damage for each projected stress Ω p,i ( t ) is calculated on a “reference S-N curve” in a Modified Wöhler Diagram (MWD) [7]. This diagram relates the number of cycles to failure, N , to the amplitude of the square root of second invariant of stress deviator, a J J  a2, (from now on, this simplified notation will be used to avoid the square root symbol). Fig. 3 depicts the relationship between the MWD and the Wöhler diagram. The figure shows, on the right, the reference S-N line along with the tension and torsion S-N lines in the MWD, and it also clarifies how the tension/torsion lines in MWD have to be sketched from the corresponding lines in the Wöhler diagram on the left. From the first and third expression in Eq. (3), in the MWD the reference strength amplitudes at N A cycles are 3/ , A A J    and A A J    , , where σ A and τ A are the amplitudes for fully-reversed axial and torsion loading, respectively. The inverse slopes for tension and torsion remain unchanged. Fig. 3 considers the case in which 3 A A    and k τ > k σ . Other arrangements of S-N lines are yet possible depending on the combination of fatigue properties characterizing a specific material. The list in Tab. 1 shows that materials are indeed characterized by S-N properties over a wide range [8]. Material Ref. Type of loading σ A τ A k σ k τ σ A /τ A k σ /k τ aluminium alloy AlCuMg1 [9] B, T 161 97 7.027 6.868 1.66 0.98 carbon steel C40 (SAE1040) [10] A, T 264.2 195.8 17.09 18.2 1.35 1.06 structural steel CSN 41 1523 (S355 type) [11] B, T 231.7 144.5 21.21 15.04 1.60 0.71 medium alloy steel 34CrMo4 [11] B, T 375 261.1 15.33 11.36 1.44 0.74 low-alloy steel S20C (AISI 1020 type) [9] B, T 227 97.8 6.17 6.06 2.32 0.98 aluminium alloy D-30 [9] B, T 180 120 10.753 9.174 1.5 0.85 structural steel 18G2A (S255 J0) [12] B, T 189.6 141.9 7.9 12.3 1.34 1.56 brass CuZn40Pb2 [9] B, T 216 187 5.857 17.172 1.16 2.93 carbon steel C40 (SAE1040) * [10] A, T 117.8 152.8 4.62 8.2 0.77 1.77 carbon steel Ck 45 (SAE1045) * [13], [14] B, T 357 226 7.7 13.4 1.58 1.74 Table 1 : Fatigue parameters for several plain materials or notched specimen). * V-notched ( R =0.5 mm) The reference S-N line in the MWD is located by the stress ratio [3,4]:       3 1 , 2 2 3 i iip H ,mH ref C V   (11) which is a function of the mean value σ H,m and variance V H of the hydrostatic stress σ H ( t ), and of the square root of the sum of variances C p,ii of stress projections Ω p,i ( t ). Parameter ρ ref only depends on the multiaxial stress, not on material. In Eq. (11), the numerator approximates the maximum of hydrostatic stress, whereas the denominator estimates the equivalent amplitude of cycles counted in each projection. If all stress components have a zero mean value, it is σ H,m =0. In case of constant amplitude multiaxial loading (sinusoidal), the expression (11) returns the definition of ρ ref given in [1,2] for the time-domain criterion.