Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 355 Parameters D 1,i , D 2,i , D 3,i , Q i , R i are best-fitting coefficients; their expressions can easily be found in the literature (see, for example, [15-17]). Step 5 – Estimating the total fatigue damage Once the fatigue damage d ( Ω p,i ( t )) for every projection has been calculated, the total damage d ( Ω ) caused by stress vector Ω ( t ) (which coincides with the damage of the multiaxial stress x ( t )) can be estimated by the following expression:   2 3 1i 2 ) ( ref ref k k p,i d d                     (15) This expression employs a non-linear combination rule to handle damaging events in uncorrelated stress projections. This law is able to account for the phase shift between stress components. Its validity versus experimental data has been checked in previous publications [3]. Eqn. (15) is very general. Its mathematical expression depends on the particular spectral method used for estimating d ( Ω p,i ( t )). If the “TB method” is considered:       ref ref k k ,i TB,i ref ref TB k Γ C d                   3 1i 2 0 i0, 1 2 2 1    (16) The Dirlik’s method yields, instead, a slightly more complex equation:         2 3 1i 2 ,3 ,2 ,1 i0, 2 ip, 1 2 1 2 1 ref ref ref ref ref ref k k i k i i ref k ref k i i k ref DK D RD k k QD C d                                            (17) The fatigue life (time to failure) is T f = D cr / d ( Ω ), where D cr is a critical damage value (=1 in Palmgren-Miner rule). As a final remark, it is worth to observe that the PbP criterion provides damage estimations that are consistent with those for simple uniaxial random loadings (tension or torsion). For a pure axial loading (e.g. σ xx ( t )≠0 and σ yy ( t )=τ xy ( t )=0), the only non-null deviatoric component is C ' 11 = V xx /3, while the variance of hydrostatic stress is V H = V xx /9, which results in ρ ref =1 and 3/ ref A, A J   , k ref = k σ . The only non-null projection is 3/)( )( 1 t t xx p,    and the total damage d ( Ω )= d 1 ( Ω p,1 ( t )) equals the damage that would be obtained by applying Eqs. (13) or (14) to the uniaxial random stress σ xx ( t ). For a pure torsion loading (τ xy ( t )≠0 and σ xx ( t )=σ yy ( t )=0), the only non-null projection is Ω p,3 ( t )= τ xy ( t ), while the hydrostatic stress is zero ( V H =0) and ρ ref =0. The reference S-N line thus coincides with the torsion line ( J A,ref =τ A , k ref = k τ ), while the total damage d ( Ω )= d 3 ( Ω p,3 ( t )) matches the damage of the shear stress. PRACTICAL IMPLEMENTATION OF P B P CRITERION : NUMERICAL EXAMPLES Test cases with biaxial stress o further clarify the analysis steps sketched in Fig. 1, the PbP method is now applied to some simple case studies, in which 4 types of biaxial stress are combined with 3 types of material fatigue properties. The decision to use simple case studies is no coincidence. In fact, it permits a much clearer monitoring of results in each analysis step, which in turn makes the understanding of the PbP method much easier. On the other hand, the case studies here analyzed – though simple – do synthesize combinations of materials and stress states that are actually encountered in mechanical components subjected to multiaxial loadings. The examples, in particular, are also specifically conceived so to better emphasize some peculiarities of the PbP method. Each biaxial stress has components σ xx ( t ) , σ yy ( t ) , τ xy ( t ) that are zero-mean stationary and Gaussian, and with a band-limited rectangular PSD, see Fig. 4. Also the use of a rectangular PSD, in place of other more realistic (but irregular) spectra, contributes to make results much easier to be understood. The rectangular one-sided PSDs are centered on frequency T