Transition of Fatigue Crack in Metals from Planar to Fractional Regime

This paper proposes a novel approach to modeling the dependence of the number of cycles to failure on the initial crack length and stress amplitude. This model is applicable in the context of linear fracture mechanics in polycrystal materials, particularly metal alloys. The principal benefit of the proposed functions is that they enable the potential for obtaining closed-form analytical solutions for the fractal crack growth law. This proposed fractal propagation function is based on the hypothesis that the load ratio stays the same throughout the load history, and it describes the small in-crease in crack length with each cycle. In contrast to the common propagation laws, the crack length growth depends on branching of the crack. This feature enables a straightforward representation of the transition from common, straight cracks to fractionally non-planar cracks in metals. We pro-pose two fractional spreading functions, each with a different number of fitting parameters. One advantage of the proposed closed-form analytical solutions is that they facilitate the universal fit-ting of the constants of the fatigue law across all stages of fatigue. Another advantage is that the closed-form solution simplifies the application of the fatigue law, as the solution of a nonlinear fractional differential equation is no longer necessary. In addition, the corresponding formulas for the length of the non-fractional crack over the number of cycles are derived in terms of Lerch functions.