Cyclic Strength and Nonlinear Material Fracture Mechanics (by the   example of steels)
  It was shown that a material fatigue fracture diagram can be viewed as a locus of points with $\sigma $ and $\sqrt l$ coordinates' product equal to $K_{1c}/2$, and $\sigma $ and $l$ product -- to $G_{1c}/2$, where $K_{1c}$ and $G_{1c}$ are non-linear fracture mechanics force and energy criteria. It was established that the average number of interatomic bonds destroyed within one alternate stress $\nabla_{1cs}$ cycle is directly proportional to $\sigma $ that is twice as large as a peak value of $\sigma^a$. It was found that low-cycle fatigue is characterized by $\sigma >\sigma_{0.2}$ and $\sigma_{1cs}> 1$, high-cycle fatigue -- by $\sigma = \sigma_{0.2}$ and $\nabla_{1cs} = 1$, and giga-cycle fatigue -- by $\sigma < \sigma_{0.2}$ and $\nabla_{1cs} < 1$. An individual interatomic bond cannot be destroyed part by part but as a single unit. The latter means that in giga-cycle fatigue a single interatomic bond is destroyed within several cycles rather than within a single cycle. The factors $F$ (collapsibility) and $R$ (resistibility) were proposed and mentioned as essential material physical constants. The introduced notion $\nabla_{1cs}$ and the established linear nature of $\nabla_{1cs}$ relationship allow to: a) clarify the fatigue crack growth physical nature in low-, high- and giga-cycle fracture zones; b) determine the nature of a fatigue fracture diagram disruption; c) plot the fatigue fracture diagram using the results obtained in a single specimen cyclic strength test with a selected value of $\sigma \ge \sigma_{0.2}$. For giga-cycle fatigue it is important (with similar purpose in mind) to determine this dependence for $\sigma < \sigma_{0.2}$. It is recommended to use $G_{1c}$ criterion to find the $l_{cr}$ length value which in contrast to $K_{1c}$ has a clear physical nature. 
Leonid S. Kremnev
cond-mat.mtrl-sci