Issue 37

M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 142 where MS f  ( )  is a scalar mean stress function of the current 6D stress   to account for mean/maximum-stress effects, which can be defined e.g. from Goodman’s or Gerber’s  a  m relations when applicable; and , NP f n    ( )     is a NP function to account for the additional effects introduced by the non-proportionality of the load path. For materials that fail due to distributed damage in all directions, the mean stress function MS f  ( )  could be based on the current hydrostatic stress  h from   . On the other hand, for materials that fail due to a single dominant crack, like most metallic alloys (whose multiaxial fatigue damage parameters tend to be better described by the critical-plane approach), then MS f  ( )  could be based on the normal stress   perpendicular to the considered candidate plane. Except for the failure surface (which never translates), during this damage process the fatigue limit and all damage surfaces suffer translations , if or , if i i i i i i i i d d v dD r d r                          | | 0 | |      (7) where d  i are coefficients calibrated for each surface, and i v    are the damage surface translation directions adapted e.g. from the general translation rule from [11]. The current generalized damage modulus D  is then obtained from the consistency condition, which guarantees that the current stress state is never outside the fatigue limit surface, taken from an analogy to the NLK hardening formulation for plasticity problems   M T i i i D d v n            1   (8) allowing the calculation of the evolution of the damage vector D   using Eq. (6). The (scalar) accumulated damage D is then obtained from Eq. (5). This formulation can deal with any multiaxial stress history, proportional or NP, and eliminates the need to count cycles and find equivalent ranges, or even to define them. Indeed, for instance, Fig. 2 shows continuous IFD damage predictions for a material whose elastic Coffin-Manson’s parameters are  c  772.5MPa and b   0.09 , under the uniaxial loading history  x = {0  300   300  300}MPa . Jiang-Sehitoglu’s translation rule was adopted with M  16 surfaces, calibrated between logarithmically spaced damage levels 10  8 and 0.01 . x 10 -5 accumulated stress (MPa) accumulated damage D theoretical (discrete) damage calculated continuous damage x 10 -5 0 200 400 600 800 1000 1200 1400 1600 0 1 2 3 4 5 6 7 -1.5 -1 -0.5 0 0.5 1 1.5 -300 -200 -100 0 100 200 300 signed damage D 1 normal stress (MPa) unloading loading Figure 2 : Hysteresis loops relating applied stress and a signed damage state (left) and resulting accumulated damage (right) for a uniaxial constant amplitude loading history. Strain-based Incremental Fatigue Damage Formulation All the formulations and the example presented above assumed nominally linear elastic loading histories, whose damage can calculated from SN models such as Wöhler-Basquin’s and Goodman, but this is not a limitation for this methodology.