Cumulative damage model based on two-mode fatigue damage bounds

Based on earlier work by the author, a macro-model of damage accumulation is developed that circumvents crack tip micro-mechanics through introduction of two-mode damage curves. In such model, the actual microcrack configuration is replaced by an ‘equivalent’ crack while the actual heterogeneous medium is replaced by an effective homogeneous and isotropic medium in which the crack is embedded. By the simultaneous presence of slip and opening-mode damage during each cycle, the two-mode damage curves reflect conditions at a crack tip in the effective medium. Such curves are obtained from conditions imposed at four discrete microstructure barriers. With all barriers established in terms of the fatigue limit. The fatigue process is thus divided into three stages, the analogues of the three creep stages under constant stress. Two length scales are defined from order of magnitude considerations: a macroscale defining a representative volume element (RVE) and a microscale defining a much smaller volume of grain size order; both volumes surrounding the ‘equivalent’ crack tip. The presence of a sufficiently large number of grains of random size and orientation makes the RVE representative of the medium as a whole; in such a medium, the location and orientation of the ‘equivalent’ crack becomes immaterial. While the RVE is dominated by monotonic slip, the microvolume represents a region of cyclic slip; both volumes surround the crack tip, in line with fracture mechanics concepts. This model is then applied to cumulative damage and bounds on mean residual lifetime are obtained for two-stage high–low (H–L) and low–high (L–H) cycling. The presence of microstructure barriers proves essential to damage irreversibility in L–H cycling, leading to a division of the entire region behind the S-N curve into three regions dominated by part or by all three fatigue stages. A statistical analysis based on the two-mode bounds leads to a median residual lifetime and to bounds on the variance. It is argued that the variance can only be bracketed by the present model but not uniquely defined, unless ‘instantaneous’ cyclic hardening is introduced as an additional random variable. Bounds on crack growth rate under constant amplitude cycling are also derived that effectively coincide in the macrocrack regime, thus explaining the small crack rate scatter observed. An experimental method is proposed whereby long fatigue lifetimes, including the fatigue limit, can be indirectly determined from a crack growth rate drop that occurs at the transition from a predominantly slip to a predominantly opening-mode interval of crack growth.