Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 357  Material C: this material model has arbitrary tension and torsion S-N lines, i.e. fatigue strengths as per Material B, but with different slopes for tension and torsion ( k  ≠ k τ ). This is the most general situation. Like Material A, this material is sensitive to the hydrostatic stress, too. In Tab. 3, the tension S-N curve is kept fixed, while only the torsion line moves upward from case A to C. Material type Description N A  A MPa)  A (MPa) k  k  A Parallel S-N lines, scaled by 3 2  10 6 100 57.7 3 3 B Parallel S-N lines, not scaled by 3 2  10 6 100 70 3 3 C Arbitrary S-N lines 2  10 6 100 70 3 5 Table 3 : Fatigue properties of three different material models, considered in numerical simulations. Tab. 4 gathers the results of all the load cases examined. In particular, it collects some quantities calculated throughout the analysis: the non-zero elements of covariance matrix C’ in deviatoric space and the variance of hydrostatic stress (Step 1), the non-zero elements of covariance matrix C’ p in the principal coordinate system (Step 2), the parameters of the reference S-N curve in the MWD (Step 3), the damage of each stress projection (estimated by narrow-band formula and TB method) (Step 4), the total damage estimated by PbP method using the narrow-band formula or TB method (final analysis Output). ———————— Step 1 ———————— –– Step 2 –– ––––– Step 3 ––––– ––– Step 4 ––– Output Case No. C' 11 C' 22 C' 33 C' 12 C' 13 C' 23 V H a C' p22 C' p33 ρ ref J A,ref k ref d TB,2 ( d NB,2 ) d TB,3 ( d NB,3 ) d TB ( d NB ) 1A 0.25 0.75 0 0.433 0 0 1.333 0 1 2.0 57.7 3 0 ( 0 ) 2.756 ( 3.283 ) 2.756 ( 3.283 ) 2A 1.25 0.75 0 -0.433 0 0 0.667 0.50 1.50 1.0 57.7 3 0.974 ( 1.161 ) 5.063 ( 6.032 ) 7.796 (9.287) 3A 1 0 1 0 1 0 0.333 0 2 0.707 b 57.7 3 0 ( 0 ) 7.796 ( 9.287 ) 7.796 ( 9.287 ) 3B 1 0 1 0 1 0 0.333 0 2 0.707 b 61.3 3 0 ( 0 ) 6.504 ( 7.748 ) 6.504 ( 7.748 ) 3C 1 0 1 0 1 0 0.333 0 2 0.707 b 61.3 3.586 c 0 ( 0 ) 1.054 ( 1.308 ) 1.054 ( 1.308 ) 4A 1 0 1 0 0 0 0.333 1 1 0.707 b 57.7 3 2.756 ( 3.283 ) 2.756 ( 3.283 ) 7.796 ( 9.287 ) 4B 1 0 1 0 0 0 0.333 1 1 0.707 b 61.3 3 2.300 ( 2.740 ) 2.300 ( 2.740 ) 6.505 ( 7.749 ) 4C 1 0 1 0 0 0 0.333 1 1 0.707 b 61.3 3.586 c 0.304 ( 0.377 ) 0.304 ( 0.377 ) 1.054 ( 1.308 ) Table 4 : Summary of step-by-step results (only non-zero values) of numerical case studies. In Step 4 and Output, the damage values are multiplied by 10 10 (Notes: a 1.333=4/3, 0.667=2/3, 0.333=1/3; b 2/1 707 .0  ; c 2 5 586 .3  ). Load Case 1A and 2A refer to biaxial normal stresses applied to material A; Case 1A has “in-phase” stresses ( r xx,yy =1), Case 2A “out-of-phase” stresses ( r xx,yy =0). The comparison of damage values shows that Case 2A is more damaging than Case 1A. This result is due to the greater contribution of deviatoric stress components in Case 2A compared to Case 1A (see Step 2 in Tab. 4), whereas it does not depend on the hydrostatic stress component, which is higher in Case 1A. Though the hydrostatic stress affects the damage by changing the S-N reference curve via ρ ref value, Material A has specifically been conceived for no reaction to a change in the hydrostatic stress.