Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 354 Parameter ρ ref quantifies the relative contribution of hydrostatic to deviatoric stress components in a multiaxial stress. Two limiting cases exist for uniaxial loading: a stress ratio ρ ref =1 for tension or bending (only normal stress), ρ ref =0 for torsion (only shear stress). A purely hydrostatic state of stress would have ρ ref  . On either two limiting cases (i.e. ρ ref =0 or 1), the reference S-N line coincides with the line of the corresponding uniaxial loading (tension or torsion). In any other case in which the loading is multiaxial (i.e. for any other value 0≤ ρ ref ≤1), the reference S-N curve would lie between those for tension and torsion, its position being established by ρ ref . Figure 3 : Relationship between S-N lines in Wöhler diagram (left) and Modified Wöhler Diagram (right). The reference S-N line for a general multiaxial stress ( ρ = ρ ref ) is also shown. Symbol J a stands for a2, J . In a log-log diagram, the reference S-N line is expressed by the equation ref CN J   ref k a , where C ref = N A ·(J A,ref ) kref is a strength constant; J A,ref is the amplitude strength (at N A cycles) and k ref the inverse slope. They are linearly interpolated as [7]:             k k k k J J J J ref ref A A ref A, A,ref         , , (12) from the amplitude strengths J A,  , J A,τ and inverse slopes k σ , k τ of the tension and torsion S-N curves. Step 4 – Fatigue damage calculated for each stress projection Each stress projection Ω p,i ( t ), obtained in Step 2, is a uniaxial random stress. Its damage in time unit (damage/s), say d ( Ω p,i ( t )), can be estimated in the frequency-domain by uniaxial spectral methods. No restriction is imposed on which method to use from those available in the literature [15,16], although “wide-band methods” are recommended if the PSD of each projection Ω p,i ( t ) is not narrow-band. For example, fairly accurate estimations are given by the “Tovo-Benasciutti (TB) method” [15,16]:              2 1 2 i0 1 0 ref k , ref ,i TB,i p,i TB k Γ C d ref    (13) where Γ ( - ) is the gamma function and η TB,i is a correction factor that accounts for the spectral bandwidth of S p,i ( f ) and it depends on a proper weighting coefficient b app (for their expressions, see [15,16]). In the limiting case of narrow-band PSD, η TB,i →1. Eqn. (13) has been proved to provide estimations close to those from Dirlik’s method, which the reader may be more familiar to [17]:                           i k i i ref k ref k i i k ,i ref p,i ip DK D RD k k QD C d ref ref ref ref ,3 ,2 ,1 0 , 2 1 2 1   (14) J a (log) N (log) N (log) σ,τ(log) σ A ρ =0 (torsion) ρ =1 (tension) ρ = ρ ref J A,τ J A,  J A,ref τ A torsion tension σ A /  3 k σ k τ k ref tension (scaled) k τ k σ